Heat Transfer Analysis and 

Modeling of a Parabolic Trough 

Solar Receiver Implemented in 

Engineering Equation Solver 

October 2003 • NREL/TP-550-34169 

R. Forristall 

National Renewable Energy Laboratory 

1617 Cole Boulevard 

Golden, Colorado 80401-3393 

NREL is a U.S. Department of Energy Laboratory 

Operated by Midwest Research Institute • Battelle • Bechtel 

Contract No. DE-AC36-99-GO10337 
National Renewable Energy Laboratory 

1617 Cole Boulevard 

Golden, Colorado 80401-3393 

NREL is a U.S. Department of Energy Laboratory 

Operated by Midwest Research Institute • Battelle • Bechtel 

Contract No. DE-AC36-99-GO10337 

October 2003 • NREL/TP-550-34169 

Heat Transfer Analysis and 

Modeling of a Parabolic Trough 

Solar Receiver Implemented in 

Engineering Equation Solver 

R. Forristall 

Prepared under Task No. CP032000 
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iii

HEAT TRANSFER ANALYSIS AND MODELING OF A PARABOLIC TROUGH 

SOLAR RECEIVER IMPLEMENTED IN ENGINEERING EQUATION SOLVER 

ABSTRACT 

This report describes the development, validation, and use of a heat transfer model implemented in 

Engineering Equation Solver (EES). The model determines the performance of a parabolic trough solar 

collector’s linear receiver, also called a heat collector element (HCE). All heat transfer and 

thermodynamic equations, optical properties, and parameters used in the model are discussed. The 

modeling assumptions and limitations are also discussed, along with recommendations for model 

improvement. 

The model was implemented in EES in four different versions. Two versions were developed for 

conducting HCE design and parameter studies, and two versions were developed for verifying the model 

and evaluating field test data. One- and two-dimensional energy balances were used in the codes, where 

appropriate. Each version of the codes is discussed briefly, which includes discussing the relevant EES 

diagram windows, parameter tables, and lookup tables. Detailed EES software instructions are not 

included; however, references are provided. 

Model verification and a design and parameter study to demonstrate the model versatility are also 

presented. The model was verified by comparing the field test versions of the EES codes with HCE 

experimental results. The design and parameter study includes numerous charts showing HCE 

performance trends based on different design and parameter inputs. Based on the design and parameter 

study, suggestions for HCE and trough improvements and further studies are given. 

The HCE performance software model compared well with experimental results and provided numerous 

HCE design insights from the design and parameter study. The two design versions of the EES codes of 

the HCE performance model are provided in the appendix. 

ACKNOWLEDGMENTS 

I would like to thank Associate Professor Sean Wright for his guidance and patience. I also would like to 

thank my supervisor at the National Renewable Energy Laboratory (NREL), Hank Price, for providing 

the opportunity to work on this project and his patience along the way. Further thanks go to Sandia 

National Laboratories (SNL) and Kramer Junction Operating Corporation for hosting visits to witness the 

solar trough in action and for providing test data to verify the software model. Finally, I would like to 

thank all those people who were kind enough to take time out of there busy schedules to answer questions 

I had during my work; specifically, Rod Mahoney, Tom Mancini, and Tim Reynolds of SNL; Mark 

Mehos, JoAnn Fitch, Mary Jane Hale, Tim Wendelin, and Vahab Hassani of NREL; and Professor Sam 

Welch of the University of Colorado at Denver (UCD). 
iv

CONTENTS 

Figures .............................................................................................................................viii 

Tables.................................................................................................................................xi 

Symbols ...........................................................................................................................xiii 

Chapter

1. Introduction.................................................................................................................... 1 

1.1 Background.................................................................................................................. 1 

1.2 Motivation.................................................................................................................... 4 

2. HCE Performance Model............................................................................................... 5 

2.1 One-Dimensional Energy Balance Model ................................................................... 5 

2.1.1 Convection Heat Transfer between the HTF and the Absorber................................ 8 

2.1.1.1 Turbulent and Transitional Flow Cases ................................................................. 9 

2.1.1.2 Laminar Flow Case................................................................................................ 9 

2.1.1.3 Annulus Flow Case................................................................................................ 9 

2.1.2 Conduction Heat Transfer through the Absorber Wall........................................... 10 

2.1.3 Heat Transfer from the Absorber to the Glass Envelope ........................................ 11 

2.1.3.1 Convection Heat Transfer .................................................................................... 11 

2.1.3.1.1 Vacuum in Annulus .......................................................................................... 12 

2.1.3.1.2 Pressure in Annulus .......................................................................................... 13 

2.1.3.2 Radiation Heat Transfer....................................................................................... 14 

2.1.4 Conduction Heat Transfer through the Glass Envelope.......................................... 14 

2.1.5 Heat Transfer from the Glass Envelope to the Atmosphere ................................... 14 

2.1.5.1 Convection Heat Transfer .................................................................................... 15 

2.1.5.1.1 No Wind Case................................................................................................... 15 
v

2.1.5.1.2 Wind Case......................................................................................................... 16

2.1.5.2 Radiation Heat Transfer....................................................................................... 16 

2.1.6 Solar Irradiation Absorption ................................................................................... 17 

2.1.6.1 Optical Properties................................................................................................. 17 

2.1.6.2 Solar Irradiation Absorption in the Glass Envelope ............................................ 19 

2.1.6.3 Solar Irradiation Absorption in the Absorber....................................................... 19 

2.1.7 Heat Loss through HCE Support Bracket............................................................... 20 

2.1.8 No Glass Envelope Case......................................................................................... 21 

2.2 Two-Dimensional Energy Balance Model................................................................. 22 

2.3 Assumptions and Simplifications............................................................................... 25 

2.4 Model Limitations and Suggested Improvements...................................................... 29 

3. EES Codes ................................................................................................................... 31 

3.1 One-Dimensional Heat Transfer EES Codes ............................................................. 32 

3.1.1 One-Dimensional Design Study Version................................................................ 32 

3.1.1.1 Lookup Tables ..................................................................................................... 33 

3.1.1.2 Diagram Window................................................................................................. 33 

3.1.1.3 Parametric Table .................................................................................................. 35 

3.1.2 AZTRAK Test Data Version .................................................................................. 36 

3.1.2.1 Diagram Window................................................................................................. 36 

3.1.2.2 Parametric Table .................................................................................................. 37 

3.2 Two-Dimensional Heat Transfer EES Codes ............................................................ 37 

3.2.1 Two-Dimensional Design Study Version ............................................................... 37 

3.2.1.1 Diagram Window................................................................................................. 38 

3.2.1.2 Parametric Table .................................................................................................. 39 

3.2.2 KJC Test-Loop Data Version.................................................................................. 39 
vi

3.2.2.1 Diagram Window................................................................................................. 39 

3.2.2.2 Parametric Table .................................................................................................. 40 

4. Comparison between the One- and Two-Dimensional Models ................................... 41 

5. Comparison with Experimental Data........................................................................... 46 

5.1 Comparison with SNL AZTRAK Data...................................................................... 47 

5.2 Comparison with KJC Test-Loop Data...................................................................... 53 

6. Design and Parameter Studies...................................................................................... 60 

6.1 Absorber Pipe Base Material ..................................................................................... 63 

6.2 Selective Coating ....................................................................................................... 64

6.3 Annulus Gas Type...................................................................................................... 66 

6.4 HCE Condition and Wind Speed ............................................................................... 68 

6.5 Annulus Pressure ....................................................................................................... 70 

6.6 Mirror Reflectance..................................................................................................... 73 

6.7 Solar Incident Angle .................................................................................................. 76 

6.8 Solar Insolation .......................................................................................................... 78 

6.9 HTF Flow Rate .......................................................................................................... 80 

6.10 HTF Type................................................................................................................. 82 

6.11 Glass Envelope Outer Diameter............................................................................... 84 

6.12 Parameter Variation along Receiver Length............................................................ 85 

6.13 Recommendations.................................................................................................... 87 

7. Conclusion ................................................................................................................... 90 

Appendix

A. Lookup Table References ........................................................................................... 94 

B. Optical and Material Property References .................................................................. 95 

C. Heat Transfer Fluid Change in Enthalpy..................................................................... 96 
vii

D. Radiation Heat Transfer Zonal Analysis for the Glass Envelope ............................. 100 

E. One-Dimensional Design Study Version of EES Code............................................. 104 

F. Two-Dimensional Design Study Version of EES Code ............................................ 126 

References....................................................................................................................... 147 
viii

FIGURES 

1.1 Aerial view of five, 30-MW SEGS solar plants (source: Price et al. 2002)................. 2 

1.2 Process flow schematic of large-scale parabolic trough solar power plant with 

thermal storage capability (source: Flabeg Solar International) .................................. 3 

1.3 Parabolic trough SCA located at a solar research test facility in Spain 

(source: Plataforma Solar de Almeria) ........................................................................ 3 

1.4 Schematic of an HCE (source: Solel Solar Systems Ltd.) ........................................... 4 

2.1 a) One-dimensional steady-state energy balance and b) thermal resistance model 

for a cross-section of an HCE ...................................................................................... 7 

2.2 Chart of heat loss per unit length of receiver as a function of annulus pressure........ 12 

2.3 Examples of solar trough receiver support brackets .................................................. 20 

2.4 Close-up of receiver support bracket attachment to absorber tube ............................ 20 

2.5 Schematic of the two-dimensional heat transfer model. ............................................ 24 

3.1 Diagram window from the Design Study version of the heat transfer code .............. 34 

3.2 Diagram window from the AZTRAK Test Data version of the heat transfer code ... 36 

3.3 Diagram window from the Two-Dimensional Design Study version of the heat 

transfer code............................................................................................................... 38 

3.4 Diagram window from the KJC Test-Loop Data version of the heat transfer code... 40 

4.1 Heat loss chart for different receiver length step sizes .............................................. 44 

4.2 a) Heat gain and b) efficiency charts comparing the one- and two-dimensional 

models for different HTF temperatures ..................................................................... 45 

4.3 Heat loss chart comparing the one- and two-dimensional models for different 

HTF temperatures. ..................................................................................................... 46 

5.1 AZTRAK rotating test platform at SNL. An LS-2 type collector module is shown 

installed on the platform. ........................................................................................... 47 

5.2 a) Efficiency and b) heat loss charts comparing the EES HCE code and SNL 

AZTRAK test data for Luz Cermet selective coating................................................ 49 
ix

5.3 Heat loss comparison chart between the EES HCE code and SNL AZTRAK test 

data for the off-sun case and Luz Cermet selective coating....................................... 50 

5.4 a) Efficiency and b) heat loss chart comparing the EES HCE code and SNL 

AZTRAK test data for Luz black chrome selective coating...................................... 51 

5.5 Heat loss comparison chart between the EES HCE code and SNL AZTRAK test 

data for the off-sun case and Luz black chrome selective coating............................. 52 

5.6 a) Efficiency trend and b) heat loss trend charts comparing the EES HCE code 

and SNL AZTRAK test data for varying wind speeds and no glass envelope .......... 53 

5.7 a) Heat gain and b) outlet temperature charts comparing the EES HCE code and 

KJC test-loop data for 01-02-01 ................................................................................ 55 

5.8 a) Heat gain and b) outlet temperature charts comparing the EES HCE code and 

KJC test-loop data for 03-22-01 ................................................................................ 56 

5.9 a) Heat gain and b) outlet temperature charts comparing the EES HCE code and 

KJC test-loop data for 06-21-01. ............................................................................... 57 

5.10 Power loss components and solar insolation chart from the KJC test-loop data 

 for 01-02-01 .............................................................................................................. 58 

5.11 Power loss components and solar insolation chart from the KJC test-loop data 

 for 03-22-01. ............................................................................................................. 58 

5.12 Power loss components and solar insolation chart from the KJC test-loop data 

for 06-21-01 ............................................................................................................... 59 

5.13 Average HTF temperature above ambient for each KJC test-loop data set.............. 60 

6.1 a) Efficiency and b) heat loss charts comparing different HCE absorber pipe base 

material types............................................................................................................. 64 

6.2 Energy rate component chart for each selective coating type.................................... 65 

6.3 a) Efficiency and b) heat loss charts comparing different HCE types ....................... 66 

6.4 a) Efficiency and b) heat loss charts comparing different annulus gas types............. 68 

6.5 a) Efficiency and b) heat loss charts comparing different HCE conditions as 

functions of wind speed ............................................................................................. 69 

6.6 Heat loss chart as a function of annulus pressure for air and hydrogen as the 

annulus gas................................................................................................................. 70 

6.7 a) Efficiency and b) heat loss charts for different annulus pressures and air as 

the annulus gas........................................................................................................... 71 
x

6.8 a) Efficiency and b) heat loss charts for different annulus pressures and hydrogen 

as the annulus gas ...................................................................................................... 72 

6.9 Energy rate component chart for different solar weighted reflectivities.................... 74 

6.10 a) Efficiency and b) heat loss charts comparing different solar weighted reflectivities. 

Both the one- and two-dimensional model results are included for comparison..... 75 

6.11 Energy rate component chart as a function of different solar incident angles ......... 76 

6.12 a) Efficiency and b) heat loss charts comparing different solar incident angles...... 77 

6.13 Chart showing solar incident angles for three different times of year from actual 

data for a north-south oriented SEGS plant in Kramer Junction, California ........... 78 

6.14 Energy rate component chart for different solar insolation values .......................... 79 

6.15 a) Efficiency and b) heat loss charts for different solar insolation values ............... 80 

6.16 a) Efficiency and b) heat loss charts comparing different flow rates and including 

both the one- and two-dimensional models ............................................................. 81 

6.17 a) Efficiency and b) heat loss charts comparing different HTF types and 

including both the one- and two-dimensional models. ............................................ 83 

6.18 a) Efficiency and b) heat loss charts comparing different glass envelope 

diameters.................................................................................................................. 85 

6.19 HCE cross-section temperature chart at position along receiver length .................. 86 

6.20 Energy rate components chart as a function of position along receiver 

(total receiver length = 779.5 m). ............................................................................ 87 

C.1 Specific heat at constant pressure for each HTF used in the HCE performance 

model. ....................................................................................................................... 97

C.2 Specific volume for each HTF used in the HCE performance model....................... 98 

D.1 Zone definitions for the generalized zone analysis of the radiation heat loss from 

the receiver. ............................................................................................................ 100 

D.2 Line segments used for the string method approximation for view factors ............ 102 

D.3 Radiation heat loss comparison chart showing difference between modeling the 

radiation loss with and without the collector effects.............................................. 105 
xi

TABLES 

2.1 Heat Flux Definitions................................................................................................... 8 

2.2 Nusselt Number for Laminar Annulus Flow with Uniform Heat Flux...................... 10 

2.3 Heat Transfer Coefficients and Constants for Each Annulus Gas ............................. 13 

2.4 Estimates of Effective Optical Efficiency Terms [Price 2001].................................. 17 

2.5 Optical Properties for Each Selective Coating........................................................... 18 

2.6 Temperature-Dependent Emittance Equations .......................................................... 19 

2.7 Assumptions and Simplifications Made in the HCE Performance Models ............... 27 

3.1 Part of the Lookup Table for Therminol VP1............................................................ 33 

3.2 Parametric Table for the Design Study Version of the Heat Transfer Code.............. 36 

3.3 Parametric Table for the AZTRAK Test Data Version of the Heat Transfer Code... 37 

3.4 Collector and Test Type Receiver Lengths................................................................ 39 

3.5 Parametric Table for the Two-Dimensional Design Study Version of the Heat 

Transfer Code ............................................................................................................ 39 

3.6 Parametric Table for the KJC Test-Loop Data Version of the Heat Transfer Code .. 41 

4.1 Iteration Time for Different Receiver Step Sizes....................................................... 43 

5.1 Mirror Reflectivity and Optical Efficiency for Each Test Type ................................ 48 

5.2 List of Figures and Comments for the AZTRAK Test Platform Data Comparison .. 48 

5.3 List of Figures and Comments for the KJC Test-Loop Data Comparison................. 54 

5.4 Daily Average Losses per Receiver Length............................................................... 58 

5.5 Estimated Heat Loss Percentages due to HCE Support Brackets .............................. 59 

6.1 HCE Design and Parameter Study Summary............................................................. 61 

6.2 Recommendations from HCE Design and Parameter Study...................................... 89 
xii

C.1 Orders of Magnitude of Change in Enthalpy Terms in Equation C.6 for Each 

HTF........................................................................................................................... 99 

D.1 Radiation Heat Transfer Zonal Analysis Parameters and Results........................... 104 
xiii

SYMBOLS 

a = accommodation coefficient of annulus gas 

Acs = absorber pipe cross-sectional flow area [m2

] 

Acs,b = minimum cross-section of receiver support bracket [m2

] 

Ai = circumferential area of receiver segment “i” [m2] 

b = interaction coefficient of annulus gas 

C, m, n = constants 

cave,i = HTF specific heat at average temperature of receiver segment “i” [o

C] 

D2 = inside diameter of absorber pipe [m] 

D3 = outside diameter of absorber pipe [m] 

D4 = inside diameter of glass envelope [m] 

D5 = outside diameter of glass enveope [m] 

Dh = hydraulic diameter of absorber pipe [m] 

Dp = outside diameter of absorber pipe flow restriction insert pipe 

(or cylinder) [m] 

f = Darcy friction factor 

f2 = friction factor for the inner surface of the absorber pipe 

g = gravitational constant, 9.81 m/s2

h1 = convection heat transfer coefficient for the HTF at T1 [W/m2

-K] 

h34 = convection heat transfer coefficient of annulus gas at T34

[W/m2

-K] 

h56 = convection heat transfer coefficient of air at T56 [W/m2

-K] 

hb,i

= convection heat transfer coefficient of air evaluated at the average receiver 

support bracket film temperature for receiver segment “i” [W/m2

-K] 

hi,in = enthalpy at inlet of receiver segment “i” [J/kg] 

hi,out = enthalpy at outlet of receiver segment “i” [J/kg] 

K = incident angle modifier 

k1 = thermal conductance of HTF at T1 [W/m-K] 

k23 = thermal conductance of absorber wall at T23 [W/m-K] 

k34 = thermal conductance of annulus gas at T34 [W/m-K] 

k56 = thermal conductance of air at T56 [W/m-K] 

kb = conduction heat transfer coefficient of air evaluated at the average receiver 

support bracket film temperature for receiver segment “i” [W/m-K] 

kstd = thermal conductance of annulus gas at standard temperature and pressure 

[W/m-K] 

m& = mass flow rate [kg/s] 

ni = number of receiver support brackets on receiver segment “i” 

NuD2 = Nusselt number of HTF based on D2

NuD5 = Nusselt number of air based on D5

Pa = annulus gas pressure [mmHg] 

Pb = exterior perimeter of receiver support bracket [m] 

Pinlet,i = HTF pressure at inlet of receiver segment “i” [o

C] 

Poutlet,i = HTF pressure at outlet of receiver segment “i” [o

C] 

Pr1 = Prandtl number of HTF evaluated at T1

Pr2 = Prandtl number of HTF evaluated at T2

Pr34 = Prandtl number of annulus gas at T34
xiv

Pr5 = Prandtl number of air at T5

Pr56 = Prandtl number for air at T56

Pr6 = Prandtl number of air at T6

Pw = absorber pipe inner wetted surface perimeter [m] 

conv q12 &′ = convection heat transfer rate between the heat transfer fluid and inside 

wall of the absorber pipe per unit receiver length [W/m] 

cond q23 &′ = conduction heat transfer rate through the absorber pipe wall per unit 

receiver length [W/m] 

SolAbs q3 &′ = solar irradiation absorption rate into the absorber pipe per unit receiver 

length [W/m] 

SolAbs i q3 , &′ = incident solar irradiation absorption rate into the absorber pipe of receiver 

segment “i” per unit segment length [W/m] 

conv q34 &′ = convection heat transfer rate between the outer surface of the absorber 

pipe to the inner surface of the glass envelope per unit receiver length 

[W/m] 

conv i q34 , &′ = convection heat transfer rate for receiver segment “i” between the outer 

surface of the absorber pipe to the inner surface of the glass envelope per 

receiver segment length [W/m] 

rad q34 &′ = radiation heat transfer rate between the outer surface of the absorber pipe 

to the inner surface of the glass envelope per unit receiver length [W/m] 

rad i q34 , &′ = radiation heat transfer rate for receiver segment “i” between the outer 

surface of the absorber pipe to the inner surface of the glass envelope per 

receiver segment length [W/m] 

conv q36 &′ = convection heat transfer rate between the outer surface of the absorber 

pipe to the atmosphere per unit receiver length [W/m] 

conv i q36 , &′ = convection heat transfer rate between the outer surface of the absorber 

pipe to the atmosphere per unit receiver segment length for receiver 

segment “i” [W/m] 

rad q57 &′ = radiation heat transfer rate between the outer surface of the glass envelope 

to the sky per unit receiver length [W/m] 

rad i q37 , &′ = radiation heat transfer rate between the outer surface of the absorber pipe 

to the sky per unit receiver segment length for receiver segment “i” [W/m] 

cond q45 &′ = conduction heat transfer rate through the glass envelope per unit receiver 

length [W/m] 

SolAbs q5 &′ = solar irradiation absorption rate into the absorber pipe per unit receiver 

length per unit receiver length [W/m] 

SolAbs i q5 , &′ = incident solar irradiation absorption rate into the glass envelope of receiver 

segment “i” per unit segment length [W/m] 

conv q56 &′ = convection heat transfer rate between the outer surface of the glass 

envelope to the atmosphere per unit receiver length [W/m] 

rad q37 &′ = radiation heat transfer rate between the outer surface of the absorber pipe 

to the sky per unit receiver length [W/m] 

cond bracket i q , , & = conductive heat loss through each receiver support bracket on receiver 

segment “i” [W] 

cond bracket total i q , , , & = total conductive heat losses through all receiver support brackets on 

receiver segment “i” [W] 

HeatLoss q&′ = total heat loss rate from the heat collecting element to the surroundings 

[W/m] 

HeatLoss i q , &′′ = heat loss rate per unit circumferential area to the atmosphere from receiver 

segment “i” (excluding bracket losses) [W/m2

] 
xv

HeatLoss total i q , , &′′ = total heat loss rate per unit circumferential area from receiver segment “i” 

[W/m2

] 

i q&′′ = net heat flux per unit circumferential area of receiver segment “i” [W/m2

] 

si q&′ = solar irradiance per receiver unit length [W/m] 

SolarAbs i q , &′′ = incident solar irradiation absorption rate per unit circumferential area of 

receiver segment “i” [W/m2

] 

RaD3 = Rayleigh number of annulus gas based on D3

RaD5 = Rayleigh number of air based on D5

ReD2 = Reynolds number of HTF based on D2

ReD2,ave,i = Reynolds number based on D2 and evaluated at the average HTF bulk 

temperature of receiver segment “i” 

ReD5 = Reynolds number of air based on D5

T1 = mean (bulk) temperature of the HTF [oC] 

T2 = absorber pipe inner surface temperature [oC] 

T23 = average absorber wall temperature, (T2 + T3)/2 [o

C] 

T3 = absorber pipe outer surface temperature [o

C] 

T34 = average temperature of annulus gas, (T3+T4)/2 [o

C] 

T4 = glass envelope inner surface temperature [o

C] 

T5 = glass envelope outer surface temperature [o

C] 

T56 = average temperature of air, (T5 + T6)/2 [o

C] 

T6 = atmosphere temperature [o

C] 

T7 = estimated effective sky temperature [o

C] 

Tbase,i = temperature at base of receiver support bracket for receiver segment “i” 

[

o

C] 

Tinlet,i = HTF temperature at inlet of receiver segment “i” [o

C] 

Toutlet,i = HTF temperature at outlet of receiver segment “i” [o

C] 

v56 = kinematic viscosity of air at T56 [m2

/s] 

vi,in = bulk fluid velocity at inlet of receiver segment “i” [J/kg] 

vi,out = bulk fluid velocity at outlet of receiver segment “i” [J/kg] 

∆hi = change in enthalpy through receiver segment “i” (hinlet,i – houtlet,i) [J/kg] 

∆Laperture = receiver segment length [m] 

∆Pi = change in HTF pressure through receiver segment “i” (Pinlet,i – Poutlet,i) [Pa] 

∆Ti = change in HTF temperature through receiver segment “i” 

(Tinlet,i – Toutlet,i) [K] 

ρave,i = HTF density at average temperature of receiver segment “i” [kg/m3

] 

Greek Letters

α = thermal diffusivity of air at T56 [m2

/s] 

αabs = absorptance of the absorber selective coating 

αenv = absorptance of the glass envelope 

β = volumetric thermal expansion coefficient of annulus gas or air [1/K] 

δ = molecular diameter of annulus gas [cm] 

ε = equivalent surface roughness [m] 

ε′# = optical efficiency terms 

ε3 = emissivity of absorber selective coating 

ε4 = emissivity of inner surface of glass envelope 

ε5 = emissivity of outer surface of glass envelope 
xvi

μave,i = dynamic fluid viscosity of HTF at the average temperature of receiver 

segment “i” [N-s/m2

] 

γ = ratio of specific heats of annulus gas 

ηabs = effective optical efficiency at the absorber 

ηenv = effective optical efficiency at the glass envelope 

λ = mean-free-path between collisions of a molecule of annulus gas [cm] 

π = ratio of the circumference to diameter of a circle, 3.14159 

σ = Stefan-Boltzmann constant [W/m2

-K4] 

τenv = transmittance of the glass envelope 

EES Lookup and Parametric Tables

Design Study Version: 

c = specific heat [J/kg-K] 

ηCol = collector efficiency [%] 

k = thermal conductance [W/m-K] 

μ = dynamic viscosity [kg/m-s] 

Pv = vapor pressure [kPa] 

qHeatGain = HTF heat gain per unit receiver length [W/m] 

qHeatLoss = heat loss per unit receiver length [W/m] 

ρ = density [kg/m3

] 

T = temperature [o

C] 

T1ave = average bulk HTF temperature between the receiver inlet and 

outlet [o

C] 

AZTRAK Test Data Version: 

ηCol = collector efficiency [%] 

Ib = direct normal incident solar irradiation per unit area [W/m2

] 

qHeatGain = heat gain per unit receiver length [W/m] 

qheat,loss,col,area = heat loss per projected collector area [W/m2

] 

T6 = ambient temperature [o

C] 

TdiffAir = temperature difference between the average HTF temperature and 

ambient temperature [o

C] 

Tin = HTF inlet bulk temperature [o

C] 

Tout = HTF outlet bulk temperature [o

C] 

v1 = HTF inlet volumetric flow rate [L/min] 

V6 = wind speed [m/s] 

KJC Test-loop Data Version: 

∆P = HTF pressure drop through receiver [Pa] 

ηCol = collector efficiency [%] 

Ib = direct normal incident solar irradiation per unit area [W/m2

] 

Θ = solar incident angle from the normal to the projected collector area 

[degrees] 

qcond,bracket = total conductive losses through receiver support brackets [W] 

qHeatGain = HTF heat gain per receiver length [W/m] 

qHeatLoss = heat loss per receiver length [W/m] 

qheat,loss,HCE,lengt

h

= heat loss per receiver length [W/m] 

qOptLoss = total optical loss per receiver length [W/m] 

qopt,loss = total optical loss per receiver length [W/m] 
xvii

qopt,loss,K = optical losses due to solar incident angle per receiver length 

[W/m] 

T1inlet = HTF bulk inlet temperature [o

C] 

T1outlet = HTF bulk outlet temperature [o

C] 

T6F = ambient temperature [o

F] 

Time = local time 

TinF = HTF bulk inlet temperature [o

F] 

ToutF = HTF bulk outlet temperature [o

F] 

ToutletF = HTF bulk outlet temperature [o

F] 

v1inlet = HTF bulk inlet velocity [m/s] 

v1outlet = HTF bulk outlet velocity [m/s] 

V1Volg = HTF volumetric inlet flow rate [gpm] 

V1volg,outlet = HTF volumetric outlet flow rate [gpm] 

V6mph = wind speed [mph] 
1

1. Introduction 

This report describes the development, validation, and use of a heat transfer model implemented in 

Engineering Equation Solver (EES). The model determines the performance of a parabolic trough solar 

collector’s linear receiver, also called a heat collector element (HCE). All heat transfer and 

thermodynamic equations, optical properties, and parameters used in the model are discussed, as are all 

model inputs and outputs. Inputs include collector and HCE geometry, optical properties, heat transfer 

fluid (HTF) properties, HTF inlet temperature and flow rate, solar insolation, wind speed, and ambient 

temperature. Outputs include collector efficiency, outlet HTF temperature, heat gain, and heat and optical 

losses. Modeling assumptions and limitations are also discussed, along with recommendations for model 

improvement. 

The model was implemented in EES in four versions: One-Dimensional Design Study, Two-Dimensional 

Design Study, AZTRAK Test Data, and KJC Test-Loop Data. The two design study versions were 

developed for conducting HCE design and parameter studies. The two test data versions were developed 

for verifying the model and evaluating field test data. One- and two-dimensional energy balances were 

used in the codes. When evaluating long receivers, the two-dimensional model becomes necessary. The 

one-dimensional model is valid for short receivers and for conducting design and parameter comparisons. 

The one-dimensional version offers faster convergence times. The discussion of each version of the codes 

includes the EES diagram windows, parameter tables, and lookup tables. Detailed EES software 

instructions are not included; however, references are provided. 

Model verification and a design and parameter study to demonstrate the model versatility are also 

presented. The model was verified by comparing the field test versions of the EES codes with 

experimental results. The field tests included data from a collector test platform at Sandia National 

Laboratories (SNL) in Albuquerque, New Mexico, and data from a test-loop of collectors in an operating 

solar plant in Kramer Junction, California. The design and parameter study section includes numerous 

charts showing HCE performance trends based on different design and parameter inputs. Based on the 

design and parameter study, suggestions for further studies and trough improvements are discussed. 

Additional model development analysis is provided in the appendices, along with a copy of the two 

design study versions of the EES codes, and numerous analysis and parameter references. The AZTRAK 

and KJC Test-Loop Data versions of the code are not included; however, they are very similar to the 

design study versions. They differ slightly because of the required inputs and output comparisons. 

The HCE performance software model compared well with the receiver field test results and provided 

numerous HCE design insights from the design and parameter study. 

1.1 Background 

Concentrating solar power (CSP) has been used for more than 100 years. Initially, CSP was used for 

small-scale solar thermal-mechanical applications, with outputs to 100 kW, mainly for water pumping 

[Duffie and Beckman 1991]. Only after the energy crises of 1973 did the idea of large-scale solar power 

plants take hold [Thomas and Guven 1993]. Starting in the late 1980s, nine solar electric generating 

systems (SEGSs) have been built and operated in the Mojave Desert of southern California. These SEGS 

plants range in size from 30 MW to 80 MW and total 354 MW of peak generating power for the Southern 

California Edison utility company. They supply enough electricity for more than 300,000 residents and 

displace more than 2 million barrels of oil per year [Solel 2002]. 

Five SEGSs located in Kramer Junction, California are shown in an aerial view (Figure 1.1). The SEGSs 

shown are operated by the Kramer Junction Corporation Operating Company (KJCOC). The plants are 
2

natural gas-supplemented steam Rankine turbine/generator systems. The natural gas supplements the 

required energy input during times of low solar insolation and allows the plants to operate 24 h/d at peak 

output. The buildings shown in the middle of the collector fields contain the heat exchangers, natural gas 

boilers, and pumping and generating equipment. Some SEGS plant designs are solar only; others have 

thermal storage capability. 

Figure 1.1 Aerial view of five, 30-MW SEGS solar plants (source: Price et al. 2002) 

A schematic of a SEGS power plant is shown in Figure 1.2. The plant consists of rows of solar troughs 

(also called solar collector assemblies [SCAs]), a storage system that stores the thermal energy (optional), 

heat exchangers that generate superheated steam, and a standard power cycle that converts the thermal 

energy to electrical. There may also be a supplemental boiler system (a gas boiler in this case). The solar 

energy is captured by an HTF – which could be synthetic oil or molten salt – and transferred to heat 

exchangers for thermal storage or steam generation. The thermal storage collects the solar energy during 

periods of high solar insolation, and supplements the heat input during periods of low insolation. The 

natural gas boiler would be used to further modulate the heat input and allow for 24-h plant operation. 

Typically, the trough field is designed to provide enough heat input to raise the temperature of an HTF to 

around 400o

C – high enough to generate superheated steam for a standard Rankine steam generating 

plant. The outlet temperature is maintained by the HTF flow rate, the thermal storage (if applicable), and 

the natural gas heater (if applicable). 

Figure 1.3 shows a row of SCAs from an experimental solar test facility in Spain. The figure illustrates 

the major components of a solar trough. It consists of the parabolic trough reflector, steel support 

structure, absorber pipe (HCE), and a single-axis drive mechanism. The trough reflector provides an 

aperture about 5 m wide that can extend for several thousand meters depending on the number of SCAs in 

a particular plant. The HCE is about 115 mm in diameter and 4 m long between the support braces that 

support it at the focal line. Each SCA system can move independently from the rest of the collector field. 

This allows for maintenance and a mechanism for controlling the HTF temperature, since one or a 

number of SCAs can be taken off sun during operation, which effectively controls the solar input. 
3

Figure 1.2 Process flow schematic of a large-scale parabolic trough solar power plant with 

thermal storage capability (source: Flabeg Solar International)

Figure 1.3 Parabolic trough SCA located at a solar research test facility in Spain 

(source: Plataforma Solar de Almeria) 

A diagram of an HCE is shown in Figure 1.4. The HCE consist of an absorber inside a glass envelope 

with bellows at either end. The absorber is typically a stainless steel tube about 70 mm in diameter with a 

special coating (selective coating) on the outside surface to provide the required optical properties. The 

selective coating has a high absorptance for radiation in the solar energy spectrum, and low emittance in 
4

the long wave energy spectrum to reduce thermal radiation losses [Duffie and Beckman 1991]. Section 

6.2 includes a discussion of the selective coating type, which has a strong influence on HCE performance. 

Figure 1.4 Schematic of an HCE (source: Solel Solar Systems Ltd.)

The glass envelope protects the absorber from degradation and reduces heat losses. It is typically made 

from Pyrex, which maintains good strength and transmittance under high temperatures. To reduce 

reflective losses, the glass envelope goes through an anti-reflective treatment – basically a slight chemical 

etching. The annulus space between the absorber and the glass envelope is under vacuum to reduce 

thermal losses and protect the selective coating [Price et al. 2002]. The getter bridge installed in the 

annulus consists of metallic compounds designed to absorb hydrogen – which naturally permeates from 

the HTF and if left in the annulus would decrease the HCE performance (see Sections 6.3 and 6.5). The 

getter is a vacuum loss indicator consisting of barium, which turns white when exposed to oxygen. 

The bellows provide a glass-to-metal seal and allow thermal expansion between the metal absorber and 

glass envelope. The bellows also allow the absorber to protrude beyond the glass envelope so that HCEs 

can be butt welded together to form a continuous receiver. Furthermore, the space between bellows 

provides a place to attach the HCE support brackets (see Figure 2.6 in Section 2.2). 

For years, a combination of operating problems, collector cost, and low oil prices has restricted the 

growth of CSP [Duffie and Beckman 1991]. With recent improvements in materials, optics, structures, 

and controls – along with a renewed awareness of the importance of cleaner energy sources – CSP has 

become a more viable option for large-scale electricity generation. Leading the technology has been 

parabolic trough solar technology, which is the most proven and lowest cost large-scale CSP technology 

available [Price, et al. 2002]. 

1.2 Motivation 

Early in the parabolic trough technology development, inefficient HCEs were a principal cause of poor 

SEGS performance [Price et al. 2002], so improving the HCE became a priority. A useful tool for 

improving the HCE was a heat transfer software model that could be used to evaluate new and current 

1. Bellow 4. Getter 

2. Glass Tube (Envelope) 5. Getter Bridge 

3. Absorber Tube (Pipe) 
5

HCE designs. The first software model was designed in the early 1990s, and upgraded several years later. 

The software played a useful part in HCE improvements, which have proven to be the primary reason for 

increases to the SEGS plants’ performance [Price et al. 2002]. However, the HCE performance software 

tended to be problematic and in need of another upgrade. 

The first version of the software model was developed by Virtus Energy Research Associates in 1992 and 

written in Microsoft QuickBasic [Cohen et al. 1999]. It (along with the later version and this current 

version) was based on a methodology proposed by SNL [1993]. Because QuickBasic had become 

obsolete, SNL developed a second version in 1999 written in Microsoft Excel Visual Basic. The second 

version improved algorithms and the user interface. However, it used a numerical solution technique that 

often could not converge on a solution and, because of simplifications in the heat transfer model, could 

not model many receiver configurations. 

In conjunction with an expanded R&D effort to develop higher performance parabolic trough receivers, 

the National Renewable Energy Laboratory (NREL) funded by the U.S. Department of Energy (DOE) 

decided an improved HCE performance model was needed to meet the analysis needs of the program. 

This work, sponsored by NREL, consists of developing and documenting this model. 

The new HCE performance model implemented in EES improves the heat transfer analysis, versatility, 

and speed of computation; provides additional model detail (two-dimensional energy balance model, 

HCE support bracket losses, pressure losses, etc.); improves the model robustness (valid for larger ranges 

of HTF flow properties, imbedded warning messages if correlations or HTF properties are used out of 

range of validity); and improves the modeling input and output capabilities. It was developed 

independently from the previous two models so it validates the heat transfer methodology originally 

proposed by SNL and all the changes to the original methodology (more detailed Nusselt number 

correlations, HCE support bracket losses, two-dimensional model etc.). Finally, it was developed in a 

more powerful programming environment that is more amenable to model changes and improvements. 

The model is described in Section 2. 

2. HCE Performance Model 

The HCE performance model is based on an energy balance about the collector and the HCE. The energy 

balance includes the direct normal solar irradiation incident on the collector, optical losses from both the 

collector and HCE, thermal losses from the HCE, and the heat gain into the HTF. For short receivers 

(< 100 m) a one-dimensional energy balance gives reasonable results; for longer receivers a twodimensional energy balance becomes necessary. All the equations and relationships used in both the oneand two-dimensional HCE performance models are described in the following sections. 

2.1 One-Dimensional Energy Balance Model 

The HCE performance model uses an energy balance between the HTF and the atmosphere, and includes 

all equations and correlations necessary to predict the terms in the energy balance, which depend on the 

collector type, HCE condition, optical properties, and ambient conditions. 

Figure 2.1a shows the one-dimensional steady-state energy balance for a cross-section of an HCE, with 

and without the glass envelope intact, and Figure 2.1b shows the thermal resistance model and subscript 

definitions. For clarity, the incoming solar energy and optical losses have been omitted from the 

resistance model. The optical losses are due to imperfections in the collector mirrors, tracking errors, 

shading, and mirror and HCE cleanliness (see Section 2.1.6.1). The effective incoming solar energy (solar 

energy minus optical losses) is absorbed by the glass envelope ( SolAbs q5 &′ ) and absorber selective coating 
6

( SolAbs q3 &′ ). Some energy that is absorbed into the selective coating is conducted through the absorber 

( cond q23 &′ ) and transferred to the HTF by convection ( conv q12 &′ ); remaining energy is transmitted back to the 

glass envelope by convection ( conv q34 &′ ) and radiation ( rad q34 &′ ) and lost through the HCE support bracket 

through conduction ( cond bracket q , &′ ). The energy from the radiation and convection then passes through the 

glass envelope by conduction ( cond q45 &′ ) and along with the energy absorbed by the glass envelope 

( SolAbs q5 &′ ) is lost to the environment by convection ( conv q56 &′ ) and radiation ( rad q57 &′ ). If the glass envelope is 

missing, the heat loss from the absorber is lost directly to the environment. The model assumes all 

temperatures, heat fluxes, and thermodynamic properties are uniform around the circumference of the 

HCE. Also, all flux directions shown in Figure 2.1a are positive. 

With the help of Figure 2.1, the energy balance equations are determined by conserving energy at each 

surface of the HCE cross-section, both with and without the glass envelope intact. 

with the glass envelope: 

conv cond q q 12 23 &′ = &′ (2.1a) 

SolAbs conv rad cond cond bracket q q q q q 3 34 34 23 , &′ = &′ + &′ + &′ + &′ (2.1b) 

conv rad cond q q q 34 34 45 &′ + &′ = &′ (2.1c) 

cond SolAbs conv rad q q q q 45 5 56 57 &′ + &′ = &′ + &′ (2.1d) 

qHeatLoss q56conv q57rad qcond,bracket &′ = &′ + &′ + &′ (2.1e) 

without the glass envelope: 

conv cond q q 12 23 &′ = &′ (2.2a) 

SolAbs conv rad cond cond bracket q q q q q 3 36 37 23 , &′ = &′ + &′ + &′ + &′ (2.2b) 

HeatLoss conv rad cond bracket q q q q 36 37 , &′ = &′ + &′ + &′ (2.2c) 

In the case without the glass envelope, Equations 2.1c and 2.1d drop out, and the 4 subscript for the 

convection and radiation from the absorber change to 6 and 7, respectively, since the heat loss from the 

absorber outer surface escapes directly to the environment instead of through the glass envelope. 

Note that the solar absorptance SolAbs q3 &′ and SolAbs q5 &′ are treated as heat flux terms. This simplifies the 

solar absorption terms and makes the heat conduction through the absorber pipe and glass envelope linear. 

In reality, the solar absorption in the absorber (opaque metal material) and glass envelope 

(semitransparent material) are volumetric phenomena. However, most of the absorption in the absorber 

occurs very close to the surface (about 6 angstroms) [Ozisik 1973], and although solar absorption occurs 

throughout the thickness of the glass envelope, the absorptance is relatively small (α = 0.02). Therefore, 

any error in treating solar absorption as a surface phenomenon should be relatively small. 

All the terms in Equations 2.1 and 2.2 are defined in Table 2.1. Dotted variables indicate rates and the 

prime indicates per unit length of receiver. A double prime will indicate per unit normal aperture area. 
7

cond q23 &′

conv q12 &′ q3SolAbs &′

conv q34 &′ rad q34 &′

q45cond &′

q5SolAbs &′

conv q56 &′

q57rad &′

q3SolAbs &′

q23cond &′

q12conv &′

conv q36 &′

q37rad &′

heat transfer fluid

absorber pipe

selective coating

glass envelope

With Glass Envelope Without Glass Envelope

qcond,bracket &′ cond bracket q , &′

(1) (2) (3) (4) (6) (5) (7)

convection conduction

radiation

convection conduction

radiation

convection

conduction

(HCE support bracket)

(1) heat transfer fluid (5) glass envelope outer surface

(2) absorber inner surface (6) surrounding air

(3) absorber outer surface (7) sky

(4) glass envelope inner surface

Figure 2.1 a) One-dimensional steady-state energy balance and b) thermal resistance model for a 

cross-section of an HCE

a) One-dimensional energy balance 

b) Thermal resistance model 
8

Table 2.1 Heat Flux Definitions 

Heat Flux Heat Transfer Path 

(W/m)* 

Heat Transfer 

Mode From To 

conv q12 &′ convection inner absorber pipe surface heat transfer fluid 

cond q23 &′ conduction outer absorber pipe surface inner absorber pipe surface 

SolAbs q3 &′ solar irradiation 

absorption incident solar irradiation outer absorber pipe surface 

conv q34 &′ convection outer absorber pipe surface inner glass envelope surface 

rad q34 &′ radiation outer absorber pipe surface inner glass envelope surface 

cond q45 &′ conduction inner glass envelope 

surface outer glass envelope surface 

SolAbs q5 &′ solar irradiation 

absorption incident solar irradiation outer glass envelope surface 

conv q56 &′ convection outer glass envelope 

surface ambient 

rad q57 &′ radiation outer glass envelope 

surface sky 

conv q36 &′ convection outer absorber pipe surface ambient 

rad q37 &′ radiation outer absorber pipe surface sky 

cond bracket q , &′ conduction outer absorber pipe surface HCE support bracket 

HeatLoss q&′ convection and 

radiation heat collecting element ambient and sky 

* Per unit aperture length. 

2.1.1 Convection Heat Transfer between the HTF and the Absorber 

From Newton’s law of cooling, the convection heat transfer from the inside surface of the absorber pipe 

to the HTF is 

( ) 12conv 1 2 2 1 q&′ = h D π T − T (2.3) 

with 

2

1

1 2 D

k h = NuD (2.4) 

where 

h1 = HTF convection heat transfer coefficient at T1 (W/m2

-K) 

D2 = inside diameter of the absorber pipe (m) 

T1 = mean (bulk) temperature of the HTF (o

C) 

T2 = inside surface temperature of absorber pipe (o

C) 

NuD2 = Nusselt number based on D2

k1 = thermal conductance of the HTF at T1 (W/m-K) 

In these equations, both T1 and T2 are independent of angular and longitudinal HCE directions, as will be 

all temperatures and properties in the one-dimensional energy balance model. 
9

The Nusselt number depends on the type of flow through the HCE. At typical operating conditions, the 

flow in an HCE is well within the turbulent flow region. However, during off-solar hours or when 

evaluating the HCE heat losses on a test platform, the flow in the HCE may become transitional or 

laminar because of the viscosity of the HTF at lower temperatures. Therefore, to model the heat losses 

under all conditions, the model includes conditional statements to determine type of flow. The Nusselt 

number used for each flow condition is outlined in Section 2.1.1.1. 

2.1.1.1 Turbulent and Transitional Flow Cases 

To model the convective heat transfer from the absorber to the HTF for turbulent and transitional cases 

(Reynolds number > 2300) the following Nusselt number correlation developed by Gnielinski [1976] is 

used. 

( )

( )

0.11

2

1

2 / 3

2 1

2 2 1

2 Pr

Pr

1 12.7 8 Pr 1

8 Re 1000 Pr

















+ −

− = ff Nu D

D (2.5) 

with 

( ) 2

2 1.82log10 (Re 2 ) 1.64 − f = D − (2.6) 

where 

f2 = friction factor for the inner surface of the absorber pipe 

Pr1 = Prandtl number evaluated at the HTF temperature, T1

Pr2 = Prandtl number evaluated at the absorber inner surface 

temperature, T2

Along with being valid for turbulent pipe flow, this Nusselt number correlation accounts for transitional 

flow states for Reynolds numbers between 2300 and 4000. Furthermore, the correlation adjusts for fluid 

property variations between the absorber wall temperature and bulk fluid temperature. The correlation is 

valid for 0.5 < Pr1 < 2000 and 2300 < ReD2 < 5E6. If used out of this range of validity, the code will 

display a warning message. Except for Pr2, all fluid properties are evaluated at the mean HTF 

temperature, T1. The correlation assumes uniform heat flux and temperature, and assumes the absorber 

has a smooth inner surface. 

2.1.1.2 Laminar Flow Case 

An option to model the flow as laminar is included in all the one-dimensional versions of the HCE heat 

transfer codes. When the laminar option is chosen and the Reynolds number is lower than 2300, the 

Nusselt number will be constant. For pipe flow, the value will be 4.36 [Incropera and DeWitt 1990]. For 

annulus flow, the value will depend on the ratio between the pipe insert diameter and absorber inside 

diameter, Dp/D2 (see Section 2.1.1.3). Pipe and annulus flow values were derived assuming constant heat 

flux. If the laminar option is not chosen and the Reynolds number is less than 2300, the code will use the 

turbulent flow model described in Section 2.1.1.1, but will display the following warning message: “The 

result may not be accurate, since 2300 < ReD2 < (5E6) does not hold. See Function fq_12conv. ReD2 = 

(value).” 

2.1.1.3 Annulus Flow Case 

One possible use of the HCE performance model will be to evaluate HCE testing conducted on an 

AZTRAK testing platform located at SNL (see Section 3.1.2). The test platform is essentially a scaleddown version of a trough plant. Unfortunately, the test platform does not have the same volumetric 

pumping capability and collector lengths as a trough plant. Therefore, to simulate field heat transfer 
10

characteristics, a pipe is inserted in the center of the HCE [Dudley et al. 1994]. This decreases the crosssectional flow area of the HTF, increasing the flow velocities, and thus increasing the Reynolds number. 

To apply to this type of HCE testing, the code is set up to model the HTF flow as annulus flow when 

prompted. 

The turbulent pipe flow correlations described in Section 2.1.1.2 can be used for annulus flow by 

substituting the inside pipe diameter, D2, with the following hydraulic diameter [Incropera and DeWitt 

1990]. 

p

w

cs

h D D

P

A

D = = 2 − 4

 (2.7) 

where 

Acs = flow cross-sectional area (m2

) 

Pw = wetted perimeter (m) 

Dp = outside diameter of pipe insert (m) 

D2 = inner diameter of absorber pipe (m) 

For laminar flow through an annulus, the Nusselt number depends on the ratio Dp/D2. Table 2.2 lists 

Nusselt numbers for various diameter ratios. The table is a modified version of a table for uniform heat 

flux occurring at both the inner and outer surfaces of the annulus [Incropera and DeWitt 1990]. 

Table 2.2 Nusselt Number for Laminar 

Annulus Flow with Uniform Heat Flux 

Dp/D2 NuDh

0 4.364 

0.05 4.792 

0.10 4.834 

0.20 4.833 

0.40 4.979 

0.60 5.099 

0.80 5.24 

1.00 5.385 

2.1.2 Conduction Heat Transfer through the Absorber Wall 

Fourier’s law of conduction through a hollow cylinder describes the conduction heat transfer through the 

absorber wall [Incropera and DeWitt 1990]. 

( ) ( ) 23 23 2 3 3 2 q&′ cond = 2πk T − T / ln D / D (2.8) 

where 

k23 = absorber thermal conductance at the average absorber 

temperature (T2+T3)/2 (W/m-K) 

T2 = absorber inside surface temperature (K) 

T3 = absorber outside surface temperature (K) 

D2 = absorber inside diameter (m) 

D3 = absorber outside diameter (m) 
11

In this equation the conduction heat transfer coefficient is constant, and is evaluated at the average 

temperature between the inner and outer surfaces. 

The conduction coefficient depends on the absorber material type. The HCE performance model includes 

three stainless steels: 304L, 316L, and 321H, and one copper: B42. If 304L or 316L is chosen, the 

conduction coefficient is calculated with the following equation. 

( ) 0.013 15.2 k23 = T23 + (2.9a) 

If 321H is chosen, 

( ) 0.0153 14.775 k23 = T23 + (2.9b) 

Both equations were determined by linearly fitting data from Davis [2000]. If copper is chosen, the 

conduction coefficient is a constant of 400 W/m-K [ASM Handbook Committee 1978]. Conductive 

resistance through the selective coating has been neglected. 

2.1.3 Heat Transfer from the Absorber to the Glass Envelope 

Convection and radiation heat transfer occur between the absorber and the glass envelope. The convection 

heat transfer mechanism depends on the annulus pressure [KJCOC 1993]. At low pressures (< ~1 torr), 

the heat transfer mechanism is molecular conduction. At higher pressures (> ~1 torr), the mechanism is 

free convection. The radiation heat transfer occurs because of the difference in temperatures between the 

outer absorber surface and the inner glass envelope surface. The radiation heat transfer calculation is 

simplified by assuming the glass envelope is opaque to infrared radiation and assuming gray (ρ = α) 

surfaces. 

2.1.3.1 Convection Heat Transfer 

Two heat transfer mechanisms are evaluated to determine the convection heat transfer between the 

absorber and glass envelope ( conv q34 &′ ), free-molecular and natural convection [KJCOC 1993]. In the HCE 

performance model, the larger of the two values is chosen. This results in a smooth transition between the 

two heat transfer modes, as shown in Figure 2.2. 
12

Heat Loss Vs. Annulus Pressure

375

425

475

525

575

625

675

725

0.0001 0.001 0.01 0.1 1 10 100 1000

Annulus Pressure (torr)

Heat Loss (W/m)

Figure 2.2 Chart of heat loss per unit length of receiver as a function of annulus pressure 

2.1.3.1.1 Vacuum in Annulus 

When the HCE annulus is under vacuum (pressure < ~1 torr), the convection heat transfer between the 

absorber and glass envelope occurs by free-molecular convection [Ratzel et al.1979]. 

( ) 34 3 34 T3 T4 q D h &′ conv = π − (2.10) 

with 

( ) 3 2ln( ) 4 3 ( ) 3 4 1 34+ + = D D D b D D

k h std

λ (2.11) 

( )( )

2 ( ) 1

2 9 5

+

− − = γ

γ

a

a b (2.12) 

( )( )

( ) 2

2.331 20 34 273.15

δ

λ

Pa

E − T + = (2.13) 

where 

D3 = outer absorber surface diameter (m) 

D4 = inner glass envelope surface diameter (m) 

h34 = convection heat transfer coefficient for the annulus gas at T34

(W/m2

-K) 

T3 = outer absorber surface temperature (o

C) 

T4 = inner glass envelope surface temperature (o

C) 

kstd = thermal conductance of the annulus gas at standard 

temperature and pressure (W/m-K) 

b = interaction coefficient 
13

λ = mean-free-path between collisions of a molecule (cm) 

a = accommodation coefficient 

γ = ratio of specific heats for the annulus gas 

T34 = average temperature (T3 + T4)/2 (o

C) 

Pa = annulus gas pressure (mmHg) 

δ = molecular diameter of annulus gas (cm) 

This correlation is valid for RaD4 < (D4 / (D4 –D3))4

, but slightly overestimates the heat transfer for very 

small pressures (~< 0.0001 torr). The molecular diameters of the gases, δ, were obtained from Marshal 

[1976] and are shown in Table 2.3. The table also compares the convection heat transfer coefficients (h34) 

and other parameters that are used in the calculation for each of the three gases included in the HCE 

performance model. 

Table 2.3 Heat Transfer Coefficients and Constants for Each Annulus Gas 

Annulus 

Gas 

kstd

[W/m-K] 

b λ

[cm] 

γ δ

[cm] 

h34

[W/m2

-K]

Air 0.02551 1.571 88.67 1.39 3.53E-8 0.0001115 

Hydrogen 0.1769 1.581 191.8 1.398 2.4E-8 0.0003551 

Argon 0.01777 1.886 76.51 1.677 3.8E-8 0.00007499

T1avg = 300 oC, Insolation = 940 W/m2

2.1.3.1.2 Pressure in Annulus 

When the HCE annulus loses vacuum (pressure > ~1 torr), the convection heat transfer mechanism 

between the absorber and glass envelope occurs by natural convection. Raithby and Holland’s correlation 

for natural convection in an annular space between horizontal cylinders is used for this case [Bejan 1995]. 

( )( ( ))

( ) ( ) 5 4 3 5

3 4

1 4

34 3 4 3 34

34

1

2.425 Pr 0.861 Pr

D D

k T T Ra

q D

conv

+

− + ′ = (2.14) 

( )

αν

β 3

3 4 3

3

g T T D RaD− = (2.15) 

For an ideal gas 

Tavg

1 β = (2.16) 

k34 = thermal conductance of annulus gas at T34 (W/m-K) 

T3 = outer absorber surface temperature (o

C) 

T4 = inner glass envelope surface temperature (o

C) 

D3 = outer absorber diameter (m) 

D4 = inner glass envelope diameter (m) 

Pr34 = Prandtl number 

RaD3 = Rayleigh number evaluated at D3

β = volumetric thermal expansion coefficient (1/K) 

T34 = average temperature, (T3 + T4)/2 (o

C) 
14

This correlation assumes long, horizontal, concentric cylinders at uniform temperatures, and is valid for 

RaD4 > (D4 / (D4 –D3))4

. All physical properties are evaluated at the average temperature (T3 + T4)/2. 

2.1.3.2 Radiation Heat Transfer 

The radiation heat transfer between the absorber and glass envelope ( rad q34 &′ ) is estimated with the 

following equation [Incropera and DeWitt 1990]. 

( )

( ) ( )( ) 3 4 3 4 4

4

4

4

3 3

34 1 1 D D

D T T

q rad

ε ε ε

σπ

+ −

− &′ = (2.17) 

where 

σ = Stefan-Boltzmann constant (W/m2

-K4) 

D3 = outer absorber diameter (m) 

D4 = inner glass envelope diameter (m) 

T3 = outer absorber surface temperature (K) 

T4 = inner glass envelope surface temperature (K) 

ε3 = Absorber selective coating emissivity 

ε4 = glass envelope emissivity 

Several assumptions were made in deriving this equation: nonparticipating gas in the annulus, gray 

surfaces, diffuse reflections and irradiation, and long concentric isothermal cylinders. Also, the glass 

envelope is assumed to be opaque to infrared radiation. Not all these assumptions are completely 

accurate. For instance, neither the glass envelope nor the selective coatings are gray, and the glass 

envelope is not completely opaque for the entire thermal radiation spectrum [Touloukian and DeWitt 

1972]. However, any errors associated with the assumptions should be relatively small. 

2.1.4 Conduction Heat Transfer through the Glass Envelope 

The conduction heat transfer through the glass envelope uses the same equation as the conduction through 

the absorber wall described in Section 2.1.2. The anti-reflective treatment on the inside and outside 

surfaces of the glass envelope is assumed to introduce no thermal resistance and to have no effect on the 

glass emissivity. This should be fairly accurate since the treatment is a chemical etching and does not add 

additional elements to the glass surface [Mahoney 2002]. As in the absorber case, the temperature 

distribution is assumed to be linear. Furthermore, the thermal conductance is assumed constant – as 

explained in Section 2.1 – with a value of 1.04 (Pyrex glass) [Touloukian and DeWitt 1972]. 

2.1.5 Heat Transfer from the Glass Envelope to the Atmosphere 

The heat will transfer from the glass envelope to the atmosphere by convection and radiation. The 

convection will either be forced or natural, depending on whether there is wind. Radiation heat loss 

occurs due to the temperature difference between the glass envelope and sky. 
15

2.1.5.1 Convection Heat Transfer 

The convection heat transfer from the glass envelope to the atmosphere ( conv q56 &′ ) is the largest source of 

heat loss, especially if there is a wind. From Newton’s law of cooling 

( ) 56 56 D5 T5 T6 q h &′ conv = π − (2.18) 

5

5

56

56 Nu D D

k h = (2.19) 

where 

T5 = glass envelope outer surface temperature (o

C) 

T6 = ambient temperature (oC) 

56 h = convection heat transfer coefficient for air at (T5 – T6)/2 

(W/m2

-K) 

k56 = thermal conductance of air at (T5 – T6)/2 (W/m-K) 

D5 = glass envelope outer diameter (m) 

NuD5 = average Nusselt number based on the glass envelope outer 

diameter 

The Nusselt number depends on whether the convection heat transfer is natural (no wind) or forced (with 

wind). 

2.1.5.1.1 No Wind Case 

If there is no wind, the convection heat transfer from the glass envelope to the environment will be by 

natural convection. For this case, the correlation developed by Churchill and Chu will be used to estimate 

the Nusselt number [Incropera and DeWitt 1990]. 

[ ] ( )

2

8 27 9 16

56

1 6

5

5

1 0.559 Pr

0.387 0.60

















+

= + D

D

Ra Nu (2.20) 

( )

( ) 56 56

3

5 6 5

5

α ν

gβ T T D RaD− = (2.21) 

56

1

T β = (2.22) 

56

56 Pr56

α

ν

= (2.23) 

where 

RaD5 = Rayleigh number for air based on the glass envelope outer 

diameter, D5

g = gravitational constant (9.81) (m/s2

) 
16

α56 = thermal diffusivity for air at T56 (m2

/s) 

β = volumetric thermal expansion coefficient (ideal gas) (1/K) 

Pr56 = Prandtl number for air at T56

ν56 = kinematic viscosity for air at T56 (m2

/s) 

T56 = film temperature (T5 + T6)/2 (K) 

This correlation is valid for 105

 < RaDo < 1012, and assumes a long isothermal horizontal cylinder. Also, 

all the fluid properties are determined at the film temperature, (T5 + T6)/2. 

2.1.5.1.2 Wind Case 

If there is wind, the convection heat transfer from the glass envelope to the environment will be forced 

convection. The Nusselt number in this case is estimated with Zhukauskas’ correlation for external forced 

convection flow normal to an isothermal cylinder [Incropera and DeWitt 1990]. 

1 4

5

6

5 5 6 Pr

Pr

Re Pr 













 = m n

D C D Nu (2.24) 

with 

ReD C m 

1-40 0.75 0.4

40-1000 0.51 0.5

1000-200000 0.26 0.6

200000-1000000 0.076 0.7

and 

n = 0.37, for Pr <=10 

n = 0.36, for Pr >10 

This correlation is valid for 0.7 < Pr6 < 500, and 1 < ReD5 < 106

. All fluid properties are evaluated at the 

atmospheric temperature, T6, except Pr5, which is evaluated at the glass envelope outer surface 

temperature. 

2.1.5.2 Radiation Heat Transfer 

As will be discussed in Section 2.1.6, the useful incoming solar irradiation is included in the solar 

absorption terms. Therefore, the radiation transfer between the glass envelope and sky, discussed here, is 

caused by the temperature difference between the glass envelope and sky. To approximate this, the 

envelope is assumed to be a small convex gray object in a large blackbody cavity (sky). The net radiation 

transfer between the glass envelope and sky becomes [Incropera and DeWitt 1990] 

( ) 4

7

4

q&57′ rad = σD5πε 5 T5 − T (2.25) 

where 

σ = Stefan-Boltzmann constant (5.670E-8) (W/m2

-K4) 

D5 = glass envelope outer diameter (m) 

ε5 = emissivity of the glass envelope outer surface 
17

T5 = glass envelope outer surface temperature (K) 

T7 = effective sky temperature (K) 

The sky, especially during less than clear conditions, does not act like a blackbody; however, it is 

common practice to model it as such and to use an effective to compensate for the difference as stated by 

Duffie and Beckman [1991]. As they stated, “the effective sky temperature accounts for the fact that the 

atmosphere is not at a uniform temperature and that the atmosphere radiates only in certain wavelength 

bands.” Furthermore, “the atmosphere is essentially transparent in the wavelength region from 8 to 14 

μm, but outside of this ‘window’ the atmosphere has absorbing bands covering much of the infrared 

spectrum.” Several relations have been proposed to relate the effective sky temperature for clear skies to 

measured meteorological data; however, to simplify the model, the effective sky temperature is 

approximated as being 8o

C below the ambient temperature – as determined in the original HCE heat loss 

model for an ambient temperature of 22o

C [KJC Operating Company 1993]. 

2.1.6 Solar Irradiation Absorption 

The optical losses and solar absorption – given the direct normal solar irradiation, solar angle, and optical 

properties of the trough mirrors and HCE components – is very difficult to model accurately with a set of 

equations that can be solved with a software program like EES. Because of this, optical efficiency terms 

are estimated and combined to form an effective optical efficiency, which in turn is used to determine the 

optical loss and solar absorption terms. 

2.1.6.1 Optical Properties 

The optical properties used in the HCE performance model were obtained from a combination of sources. 

Some were determined by SEGS plant performance modeling completed by NREL. Some were 

determined by tests conducted by SNL, and Solel Solar Systems Ltd. of Israel – the prime HCE 

manufacturer. Some were evaluated and revised with ray tracing software developed by NREL. 

Table 2.4 lists terms used to estimate the effective optical efficiencies. The table was generated from data 

published in a report by NREL [Price et al. 2002], which in turn was based on field tests [Dudley et al. 

1994], and software performance modeling . The first three terms, ε′1, ε′2, and ε′3, and the last term, ε′6, 

are strictly estimates. The clean mirror reflectance ρcl is a known value, and the two dirt effect 

approximations ε′4 and ε′5 are recommendations by Duffie and Beckman [1991]. The data in the table are 

valid only for solar incidence irradiation normal to the collector aperture. An incident angle modifier term 

is added to account for incident angle losses, which includes trough end shading, changes in reflection 

and refraction, and selective coating incident angle effects. 

Table 2.4 Estimates of Effective Optical Efficiency Terms [Price 2001] 

ε′1 = HCE Shadowing (bellows, shielding, supports) 0.974 

ε′2 = Tracking Error 0.994 

ε′3 = Geometry Error (mirror alignment) 0.98 

ρcl = Clean Mirror Reflectance 0.935 

ε′4 = Dirt on Mirrors* reflectivity/ρcl

ε′5 = Dirt on HCE (1 + ε′4)/2 

ε′6 = Unaccounted 0.96 

* reflectivity is a user input (typically between 0.88 and 0.93) 
18

In Table 2.4, there are terms for collector geometric effects (shadowing, tracking, alignment), mirror and 

glass envelope transmittance effects (mirror reflectance, and dirt), and a term for unexplained differences 

between field test data and modeled data. Testing continues to refine all these values and to better 

understand the optics of the HCEs. In the EES codes for the HCE performance model, the optical 

efficiency terms can also be manually entered (see Section 3). 

Another term, incident angle modifier, is needed for cases when the solar irradiation is not normal to the 

collector aperture. It is a function of the solar incidence angle to the normal of the collector aperture. The 

equation was determined from HCE testing conducted at SNL [Dudley et al. 1994]. 

2 K = cos(θ ) + 0.000884θ − 0.00005369θ (2.26) 

Other optical properties include the glass envelope absorptance, emittance, and transmittance; and the 

selective coating absorptance and emittance. The glass envelope absorptance and emissivity are constants 

(independent of temperature) and are constant for all selective coating types (α = 0.02, ε = 0.86). The 

transmittance and the selective coating properties (see Table 2.5) coincide with the selective coating 

types. Both the envelope transmittance and the coating absorptance are constants; the coating emittance is 

a function of temperature. All the selective coating types listed are included in the EES codes. The “Solel 

UVAC Cermet avg” selective coating is the reference value used in all the design study and comparison 

charts shown in this report, unless stated otherwise. 

Table 2.5 Optical Properties for Each Selective Coating 

Coating Emittance Selective Coating Envelope 

Transmittance 

Coating 

Absorptance @ 100o

C @ 400oC 

Luz Black Chrome 0.935 0.94 0.11 0.27 

Luz Cermet 0.935 0.92 0.06 0.15 

Solel UVAC Cermet a 0.965 0.96 0.07 0.13 

Solel UVAC Cermet b 0.965 0.95 0.08 0.15 

Solel UVAC Cermet avg 0.965 0.955 0.076 0.14 

Solel UVAC Cermet 

Proposed a 0.97 0.98 0.04 0.10 

Solel UVAC Cermet 

Proposed b 0.97 0.97 0.02 0.07 

The emittance equations used in the codes are shown in Table 2.6, and coincide with the emittance values 

in Table 2.5. For all the coating types, the emittance values between the two reference points, 400o

C and 

100o

C, are nearly linear. However, more test data were available for the UVAC (Universal Vacuum Air 

Collector) Cermets, and thus this emittance is described with a second-order polynomial fit; the other 

coating emittance values use linear fits. Also, the two proposed selective coatings (Solel UVAC Cermet 

Proposed a and Solel UVAC Cermet Proposed b) were specified by the manufacturer for a single 

temperature of 400o

C; therefore, the temperature dependency had to be approximated. This was done with 

a slope and intercept to resemble the trends of the results for the Cermet tested by SNL. 
19

Table 2.6 Temperature-Dependent Emittance Equations 

Coating Type Coating Emittance*

Luz Black Chrome 0.0005333 (T + 273.15) - 0.0856 

Luz Cermet 0.000327 (T + 273.15) - 0.065971 

Solel UVAC Cermet a (2.249E-7)T2 + (1.039E-4)T + 5.599E-2 

Solel UVAC Cermet b (1.565E-7)T2 + (1.376E-4)T + 6.966E-2 

Solel UVAC Cermet avg (1.907E-7)T2 + (1.208E-4)T + 6.282E-2 

Solel UVAC Cermet 

Proposed a (2.084E-4)T + 1.663E-2 

Solel UVAC Cermet 

Proposed b (1.666E-4)T + 3.375E-3 

* All temperatures are in degrees Celsius. 

2.1.6.2 Solar Irradiation Absorption in the Glass Envelope 

As stated in Section 2.1, the solar absorption into the glass envelope is treated as a heat flux to simplify 

the model. Physically this is not true. The solar absorption in the glass envelope is a heat generation 

phenomenon and is a function of the glass thickness. However, this assumption introduces minimal error 

since the solar absorptance coefficient is small for glass, 0.02 [Touloukian and DeWitt 1972], and the 

glass envelope is relatively thin, 6 mm. Also, an optical efficiency is estimated to calculate the solar 

absorption. With this stated, the equation for the solar absorption in the glass envelope becomes 

SolAbs si env env q&5

′ = q&′η α (2.27) 

with 

η env = ε 1

′ε 2′ε 3′ε 4′ε 5′ε 6′ρ clK (2.28) 

where 

si q&′ = solar irradiation per receiver length (W/m) 

ηenv = effective optical efficiency at the glass envelope 

αenv = absorptance of the glass envelope (Pyrex glass) 

Κ = incident angle modifier (as defined by Equation 2.26) 

All terms in Equation 2.28, except for K, are from Table 2.4. The solar irradiation term ( si q&′ ) in Equation 

2.27 is determined by multiplying the direct normal solar irradiation by the projected normal reflective 

surface area of the collector (aperture area) and dividing by the receiver length. All terms in both 

equations are assumed to be independent of temperature. 

2.1.6.3 Solar Irradiation Absorption in the Absorber 

The solar energy absorbed by the absorber occurs very close to the surface; therefore, it is treated as a 

heat flux (see Section 2.1). The equation for the solar absorption in the absorber becomes 

SolAbs si abs abs q&3

′ = q&′η α (2.29) 

with 

abs env env η =η τ (2.30) 

where 
20

ηabs = effective optical efficiency at absorber 

αabs = absorptance of absorber 

τenv = transmittance of the glass envelope 

In Equation 2.30, ηenv is the same value shown in Section 2.1.6.2. Again, all terms are assumed to be 

independent of temperature. 

2.1.7 Heat Loss through HCE Support Bracket 

The HCEs are supported at the collector focal line by support brackets that run from the collector 

structure to the absorber pipe (see Figures 2.3 and 2.4). There is a support bracket at each end of every 

HCE (about every 4 m of receiver length). The bracket losses are approximated by treating the support 

bracket as an infinite fin with base temperature 10 degrees less than the outer absorber surface 

temperature T3 at the point where the bracket is attached. This estimated base temperature accounts for 

heat losses along the short distance from the bracket attachment to the minimum cross-sectional area, 

which is assumed to be the base of the fin (~ 5 cm with ~ 4 cm insulated). 

Figure 2.3 Examples of solar trough receiver support brackets 

Figure 2.4 Close-up of receiver support bracket attachment to absorber tube 
21

The bracket heat loss is estimated with the following equation [Incropera and DeWitt 1990]. 

( ) cond bracket b b b Acs b Tbase T LHCE q h P k , = , − 6 &′ (2.31) 

where 

b h = average convection coefficient of bracket (W/m2

-K) 

Pb = perimeter of bracket (m) 

kb = conduction coefficient (W/m-K) 

Acs,b = minimum cross-sectional area of bracket (m2

) 

Tbase = temperature at base of bracket (o

C) 

T6 = ambient temperature (oC) 

LHCE = HCE length (m) 

The perimeter of the bracket Pb in Equation 2.31 is the perimeter around the two 1-in.x 1-in. square tubes 

(0.2032 m) that run from the absorber attachment bracket to the collector structure (see Figure 2.4). The 

cross-sectional area Acs,b is the cross-sectional area of the two connection tabs, 1 in. x 1/8 in. (1.613E-4 

m

2

), connecting the square tubes to the absorber attachment bracket. These dimensions were chosen 

because the square tubes are the parts of the support brackets that are exposed to convection heat transfer 

to the environment, and the two connection tabs form the smallest cross-section area near the base of the 

support bracket (fin) where conduction heat transfer occurs from the absorber pipe. The conduction coefficient (kb) for the HCE support bracket is a constant equal to 48.0 W/m-K (plain carbon steel at 600 K). 

The film coefficient b h in Equation 2.31 depends on the wind speed. If there is no wind (<= 0.1 m/s), the 

film coefficient is estimated with the same Churchill and Chu correlation described in Section 2.1.5.1.1 

for natural convection from a long isothermal horizontal cylinder. If there is wind (> 0.1 m/s), the film 

coefficient is estimated with Zhukauskas’ correlation described in Section 2.1.5.1.2. For both cases, 

“2 inches” is used as the effective diameter of the HCE support brackets and the average isothermal 

bracket temperature is estimated as (Tbase + T6) / 3 – the bracket base temperature plus the ambient 

temperature, divided by three. This estimate is based on intuition; however, the calculations gave 

reasonable results (see Section 5.2). Also, during the comparison with experimental data, the film 

coefficient was within expected values, 2-25 W/m2

-K for free convection and 25–250 W/m2-K for forced 

convection [Incropera and DeWitt 1990]. 

2.1.8 No Glass Envelope Case 

As mentioned in Section 2.1, the HCE is modeled with and without the glass envelope. Most of the 

equations discussed so far are for the glass envelope intact case. When the glass envelope is missing, the 

five energy balance equations, Equation 2.1, collapse down to three (as shown in Section 2.1). The 

equations for conv q12 &′ and cond q23 &′ remain unchanged for the two cases. Without the glass envelope, the 

convection and radiation heat transfer equations for the absorber are calculated with the same equations 

given in Section 2.1.5. Also, the solar absorption term for the absorber is adjusted to account for the solar 

flux that is no longer lost with the glass envelope. In this case ηabs equals ηenv as defined in Section 

2.1.6.2, without the ε5 term, which accounted for particulate matter on the HCE. 
22

2.2 Two-Dimensional Energy Balance Model 

This model was constructed by dividing the length of the receiver into “N” segments of equal length, with 

temperature continuity at the bounding surfaces (Figure 2.5). The radial heat fluxes are assumed uniform 

and normal to the surfaces for each segment and are evaluated at the average temperature between the left 

and right side of the segment ((Tright,i+Tleft,i)/2). The longitudinal temperature is assumed to be nearly 

linear (see Section 6.12) and the conduction coefficient constant. Therefore, the right and left side 

longitudinal conduction terms cancel out, and only the HTF will transfer energy in the longitudinal 

direction. With these assumptions, the radial heat transfer terms can be modeled with the one-dimensional 

energy balance (see Section 2.1). The steady-state energy balance for the receiver can be estimated with 

the following equation. 















 



 



  − +  



 ≈ ∑ ′′ + +

in N out

N

i i q A m h v h v

,

2

1,

2

1 2

1

2

1 0 & & (2.32) 

where 

i q&′′ = net heat flux per unit area (W/m2

) 

Ai = circumferential area of segment “i” (m2) 

m& = mass flow rate (kg/s) 

h = enthalpy (J/kg) 

v = bulk fluid velocity (m/s) 

In this equation, potential energy is neglected, and mass flow is constant. The kinetic energy is retained 

because the HTF expands thermally, especially for long receivers (>700 m). The velocity will be needed 

for the pressure drop and local Reynolds number calculations, which are used to determine the flow 

energy and radial HTF heat convection terms, respectively (see Section 2.1). 

For a receiver segment “i” of length ∆Laperture , the energy balance becomes 















 



 



  − +  



 = ′′ + +

i in i out

i i q A m h v h v

,

2

,

2

2

1

2

1 0 & & (2.33) 

The net heat flux ( )i q&′′ in Equations 2.32 and 2.33 includes solar absorption and heat loss. 

i i SolarAbs i i HeatLoss i Ai q A q A q , , &′′ = &′′ − &′′ (2.34) 

The solar absorption term includes both the absorber ( ) SolAbs i q3 , &′ and the glass envelope ( ) SolAbs i q5 , &′ . 

SolarAbs i i SolAbs i aperture SolAbs i Laperture q&′′ , A = q&3

′ , ∆L + q&5

′ , ∆ (2.35) 

These are the same solar absorption terms as described in Sections 2.1.6.2 and 2.1.6.3. 
23

The heat loss in Equation 2.34 includes the radiation and convection heat losses from the glass envelope 

and the conductive losses through the support brackets. 

HeatLoss i i rad i aperture conv i aperture cond bracket total i q A q L q L q , 57 , 56 , , , , &′′ = &′ ∆ + &′ ∆ + & (2.36) 

where 

cond bracket total i i cond bracket i q n q , , , , , & = & (2.37) 

Again, the radiation and convection terms are evaluated with the average longitudinal temperatures for 

each receiver segment (see Sections 2.1.5.1 and 2.1.5.2). The conductive heat loss through the bracket is 

described in Section 2.1.7. It is evaluated with the average longitudinal temperature for each segment. 

The parameter i n is the number of brackets attached to the HCE segment “i” and is a function of the 

segment length ∆Laperture . The code includes logic to determine the base temperature and number of 

brackets as determined by the segment length and location along the receiver. The code also normalizes 

the bracket conduction term when required. 

Assuming the HTF density is only a function of temperature (incompressible with pressure), the change 

in enthalpy in Equation 2.33 can be approximated with the following equation. 

i ave i Ti ∆h ≈ c , ∆ (2.38) 

where 

( ) ∆Ti = Tinlet,i − Toutlet,i

The specific heat in Equation 2.38 (Cave,i ) and the density (ρave,i ) are evaluated at the average HTF 

temperature along the length of the receiver segment. A complete derivation of Equation 2.38 is provided 

in Appendix C. 

Substituting all the above results into Equation 2.33 gives 

( )

( ) ( ) ave i in i out i in i out i cond bracket total i

solAbs i solAbs i rad i conv i aperture

m c T T v v q

q q q q L

, , ,

2

,

2

, , , ,

3 , 5 , 34 , 34 ,

2

1

0

& &

& & & &

−



 



 + − + −

= ′ + ′ − ′ − ′ ∆

 (2.39) 

And, solving for the outlet temperature gives 

(( ) )

( )

( )

in i

ave i

in i out i

ave i

solAbs i solAbs i rad i conv i aperture i cond bracket total i

out i

T

c

v v

mc

q q q q L q T

,

,

2

,

2

,

,

3 , 5 , 34 , 34 , , , , ,

,

2

1

+

−

+

′ + ′ − ′ − ′ ∆ − = &

& & & & &

(2.40) 
24

Figure 2.5 Schematic of the two-dimensional heat transfer model 

In this equation, the inlet velocity (ν in,i ) is determined from the absorber cross-sectional area and 

volumetric flow rate, both of which are inputs. The remaining velocities are calculated from conservation 

of mass and continuity at the segment boundaries. 

out i cs

out i A

m

v

,

, ρ

& = (2.41) 

in ( ) i out i v v , +1 = , (2.42) 

The change in pressure ( ) ∆Pi in Equation 2.40 is estimated with an equation to calculate pressure loss in 

a horizontal pipe with fully developed turbulent flow [Munson et al. 1990]. 

ave i

cs aperture

i D

A f L m

P

2 ,

2

2 ρ





 



 ∆

∆ =

&

 (2.43) 

∆Laperture

Laperture

i in

m h v

,

2

2

1 



 





& +

i out

m h v

,

2

2

1 



 





& +

SolarAbs i Ai q , &′′

HeatLoss i Ai q , &′′

niqcond,bracket,i &
25

Where f is the Darcy friction factor, and can be estimated for turbulent pipe flow with the following 

Colebrook equation [Munson et al. 1990]. 





















= − +

f

D Log f D2,ave,i

2

Re

2.51

3.7

2 1

ε

 (2.44) 

Here, ε is the equivalent roughness (~1.5E-6 m for drawn pipe), and ReD2,ave,i is the Reynolds number 

calculated at the average axial HTF bulk temperature for each receiver segment. 

ave i

ave i ave i

D ave i

v D

,

, , 2 Re 2, , μ

ρ

= (2.45) 

with 

μ ave,i = dynamic fluid viscosity (N-s/m2) 

The outlet temperature, outlet velocity, pressure drop, heat gain, and heat loss can be determined with 

these equations. If the calculations are done without the glass envelope, the following substitutions are 

made. 

conv i conv i

rad i rad i

solAbs i

q q

q q

q

56 , 36 ,

57 , 37 ,

5 , 0

′ → ′

′ → ′

′ →

& &

& &

&

2.3 Assumptions and Simplifications 

Numerous assumptions and simplifications made in the modeling of the HCE performance model are 

listed in Table 2.7. Many have already been discussed, but are included for completeness. Some items in 

Table 2.7 warrant further discussion. 

The heat transfer model assumes si q&′ is uniform both around the circumference (see Section 2.1) and 

along the length of the HCE. The actual solar flux pattern on the HCE will depend on the collector 

geometry, alignment errors, tracking errors, and any optical aberrations in the mirrors. The true solar flux 

profile around the circumference will be similar to an asymmetric normal distribution with the maximum 

at the point closest to the collector and the minimum located on the HCE side opposite the collector.1

 The 

profile along the length of the HCE will include hot points where the flux overlaps because of mirror 

misalignments and aberrations. The influence of the “true” solar flux profile still requires more study to 

better understand the loss in performance. 

The model assumes that the flow is uniform; however, because of the nonuniform solar flux around the 

circumference of the HCE, the flow will be heated asymmetrically and thus will be nonuniform. To 

 

1

 Thomas and Guven [1994] provide good examples of the flux distribution around the circumference of the 

absorber based on collector mirror optical errors. 
26

accurately predict the convective heat transfer rate for this type of flow, a computational fluid dynamics 

model would need to be used. The extent of the flow nonuniformity and how much that affects the heat 

transfer will require further study. For now, this is assumed to be a negligible effect, and besides, as stated 

earlier, the model assumes a uniform flux. 

Another effect of the nonuniform solar flux would be nonuniform temperature profiles around the 

circumference. The profiles would be nonlinear with the maximum temperatures closest to the collector 

and the minimum temperatures on the opposite side of the HCE, which would receive no concentrated 

solar. The error associated with assuming uniform circumferential temperatures requires further study; 

however, assuming uniformity in the flow and temperature profiles should cause the heat fluxes to be 

overestimated and not underestimated, if there is much difference. 

The model neglects the radiation heat transfer influence from the collector, ground, and surrounding 

troughs. A simple radiation zonal analysis was conducted to determine a first approximation to this error 

(see Appendix D). The results showed that neglecting these influences could add an error of 5%–10% to 

the radiation heat transfer from the glass envelope. However, since the error results in the radiation heat 

transfer being overpredicted, it was decided to leave the simplification in place. This was justified 

because the radiation loss is relatively small compared to the convective loss, and since the model 

typically underpredicts losses (see Section 5). 

Many correlations in the model are based on a uniform temperature in the longitudinal direction. 

However, because the heat gain per unit length decreases as the HTF temperature increases, and because 

the radiation heat loss is nonlinear, the true temperature profile will be nonlinear (see Section 6.12). 

Assuming a uniform temperature – or using a linear averaged temperature – may overpredict the heat gain 

and underpredict the heat loss; however, assuming uniform temperature profiles allows all the heat fluxes 

to be treated as one-dimensional and thus significantly simplifies the model. 

The film coefficients used to estimate forced convection from the HCE and HCE support brackets are 

based on a correlation that assumes the wind acts normal to the axis. In practice, the wind will be very 

turbulent and nonuniform in both magnitude and direction throughout a SEGS plant. Therefore, modeling 

the wind as normal to axis of the HCE and support bracket should result in the worst-case scenario for 

forced convection heat loss, and would still answer some questions a designer may have about the effect 

of heat loss caused by wind on an HCE. As shown in Section 6.4, when the annulus between the absorber 

and glass envelope is under vacuum, the heat losses are fairly insensitive to wind speed. 

As mentioned in Section 2.1.6.1, the optical properties are estimates based on experimental data and 

software models. Unfortunately, the testing has been limited, and the variation of the optical properties is 

not very well known. Tests have shown that there is some variation, even over a single HCE. Therefore, 

the assumption in the HCE performance model that the optical properties are uniform may not be valid. 

Furthermore, in the model, the absorptance terms and glass envelope transmittance and emittance are 

assumed to be independent of temperature. This is also known not to be true; however, dependence on 

temperature is assumed to be weak. 
27

Table 2.7 Assumptions and Simplifications Made in the HCE Performance Models 

Model Component Assumptions and Simplifications 

conv q12 &′ , convection heat transfer 

between the HTF and absorber 

(Section 2.1.1) 

• T1 is the bulk temperature. 

• Uniform flow. 

• For laminar flow in an annulus, uniform flux was assumed. 

cond q23 &′ and cond q45 &′ , conduction heat 

transfers through the absorber wall 

and glass envelope (Sections 2.1.2 

and 2.1.3.3) 

• Linear in radial direction. 

• Negligible conduction in circumferential and longitudinal HCE 

directions. 

• Negligible thermal resistance from absorber coating and glass 

envelope anti-reflection treatment. 

• Constant thermal conductance. 

conv q34 &′ , convection heat transfer 

between the absorber and glass 

envelope (Section 2.1.3.1) 

• Constant convection heat coefficient. 

• At low pressures (~< 0.0001), the heat transfer may be slightly 

overestimated. 

• Long, horizontal, concentric cylinders at uniform temperatures. 

rad q34 &′ , radiation heat transfer between 

the absorber and glass envelope 

(Section 2.1.3.2) 

• Annulus gas is nonparticipating. 

• Both surfaces are gray. 

• Diffuse irradiation and reflections. 

• Surfaces are formed from long concentric isothermal cylinders. 

• Glass envelope is opaque to radiation in the infrared range. 

conv q56 &′ , convection heat transfer 

between the glass envelope and 

atmosphere (Section 2.1.4.1) 

• Wind direction is normal to the axis of the HCE. 

• Long isothermal horizontal cylinders. 

rad q56 &′ , radiation heat transfer 

between the glass envelope and sky 

(Section 2.1.4.2) 

• Effective sky temperature is 8° below the ambient temperature. 

• Small convex gray object in a large blackbody cavity. 

• Collector has a 5% effect on radiation leaving glass envelope 

outer surface. 

optical properties (Section 2.1.6.1) 

• Uniform properties. 

• Negligible degradation with time. 

• Anti-reflection treatment has negligible affect on glass envelope 

emittance. 

• Incident angle modifier is the same for each HCE and each 

collector type. 

• Optical properties don’t vary from HCE to HCE. 

• Optical properties are independent of temperature, except for 

the selective coating emissivity. 

SolAbs q3 &′ and SolAbs q5 &′ (Sections 2.1.6.2 

and 2.1.6.3) 

• Can be treated as heat fluxes. 

si q&′ (Section 2.1.5.2) • Uniform along HCE circumference and length. 

• Neglected HCE and bracket shadowing. 
28

Table 2.7 (con’t) 

Model Component Assumptions and Simplifications 

cond bracket q , &′ (Section 2.1.5) 

• Estimated as an infinite fin. 

• Wind is normal to the bracket axis. 

• Convection and conduction coefficients are constant. 

• Conduction cross-sectional area is at the two 1-in. x 1/8-in. 

connection tabs from the square tubes to the absorber 

connection bracket. 

• Convection perimeter is the perimeter of the two 1-in. x 1=in. 

square tubes. 

• Convection coefficient is estimated by treating the bracket as 

a long horizontal isothermal cylinder with a 2-in. diameter. 

• Base temperature is estimated as (T3 - 10) and average 

temperature is estimated as (Tbase + T6)/3. 

thermal-physical property data 

• Minimal interpolation and extrapolation errors. 

• HTF data are based on average test data at saturation 

pressures. 

• The HTF thermal-physical properties could change over time. 

cond bracket q , &′ (Section 2.1.5) 

• Estimated as an infinite fin. 

• Wind is normal to the bracket axis. 

• Convection and conduction coefficients are constant. 

• Conduction cross-sectional area is at the two 1-in. x 1/8-in. 

connection tabs from the square tubes to the absorber 

connection bracket. 

• Convection perimeter is the perimeter of the two 1-in. x 1=in. 

square tubes. 

• Convection coefficient is estimated by treating the bracket as 

a long horizontal isothermal cylinder with a 2-in. diameter. 

• Base temperature is estimated as (T3 - 10) and average 

temperature is estimated as (Tbase + T6)/3. 

thermal-physical property data

• Minimal interpolation and extrapolation errors. 

• HTF data are based on average test data at saturation 

pressures. 

• The HTF thermal-physical properties could change over time. 

two-dimensional model (Section 2.2 and 

Appendix)

• Temperature continuity between receiver segment lengths. 

• Heat transfer analysis for each receiver segment assumes a 

one-dimensional model evaluated at the average 

temperatures along the length of the segment. 

• HTF density is only a function of temperature. 

• Constant specific heat for each segment. 

• Absorber pipe inner surface is smooth. 

• Conduction in the longitudinal direction is negligible. 

• All heat fluxes are normal to the surfaces. 

• Constant mass flow. 

General 

• Heat losses through cross piping are not included. 

• HCE end-losses from shielding and solar incident angle are 

not included. 

• Shadowing from adjacent troughs is neglected. 

• All heat fluxes, optical properties, thermodynamic properties, 

and temperatures are uniform around circumference and 

length of HCE segments. 
29 

Other optical property assumptions made in the model include: (1) the incident angle modifier term (K) is 

the same for each HCE and collector type, and (2) the selective coating emissivities follow the 

experimentally determined temperature functions. As mentioned earlier, the modifier is based on one set 

of test data conducted on a UVAC HCE on a LS-2 type collector, and most likely is different for different 

HCE and collector types. The selective coating emittance functions were determined from experimental 

measurements on HCEs at temperatures lower than normal HCE SEGS operating temperatures [Dudley et 

al. 1994]. Therefore, the emissivity functions may not be accurate at elevated temperatures. (The error, if 

any, associated with this is not completely known. Additional optical testing is planned to answer this 

question.) 

As one can see by the number of assumptions associated with cond bracket q , &′ , the calculation of the 

conduction losses through the HCE support brackets is a rough approximation. For instance, both the 

bracket base temperature and the bracket average temperature are based on calculated guesses. However, 

the estimated bracket film coefficient and resulting losses are within expected values – 2 to 25 W/m2

-K 

for free convection and 25 to 250 W/m2

-K for forced convection [Incropera and DeWitt 1990], and the 

comparisons with the KJC test-loop data revealed the bracket losses to contribute 1% to 4% of the total 

HCE thermal losses, depending on the ambient conditions and HTF temperature (see Section 5.2). 

Thermal-physical property data are based on tables in brochures provided by the manufacturer (see 

Lookup Table References in Appendix A). However, the manufacturer has warned that there could be 

variations in properties from batch to batch, and that properties change over time, especially after 

numerous thermal cycles and mechanical agitation. Furthermore, the thermal-physical property data are 

based on experiments conducted at saturation pressure. The actual pressures will be higher, depending on 

the pumping requirements of a particular SEGS plant on a particular day. Furthermore, the EES program 

interpolates and extrapolates the thermal-physical data from lookup tables. Nonetheless, errors associated 

with all these HTF property factors should have only a small influence on the data (see Section 6.10), and 

only when using the HCE performance model to compare with actual field data, whose HTF properties 

could differ slightly from the data in the code. 

A few general assumptions warrant further explanation. The model does not include losses through 

crossover piping. In an actual SEGS plant (see Figure 1.1), the solar field can consist of numerous rows of 

collectors, and can cover many acres of land. The actual crossover piping segments can become very 

long, and the quality of its insulation will vary; therefore, crossover piping can result in fairly significant 

heat loss from the HTF. The model accounts for none of these losses. In addition, in a SEGS plant one 

row of troughs can partially shade another during morning and late afternoon operations. Also, during the 

mornings and afternoons, the troughs at the ends of rows will experience some end shadowing. These 

factors are also not accounted for in the model. However, the crossover piping losses, trough to trough 

shading, and end losses become significant only when the model is used to evaluate actual SEGS data. 

2.4 Model Limitations and Suggested Improvements 

The HCE performance model has numerous limitations. For instance, because the model neglects the 

nonuniformity in the solar insolation around the circumference and length of the HCE, these effects 

cannot be evaluated. This also means that asymmetric design changes cannot be evaluated – such as 

adding a reflective coating to part of the inside surface of the glass envelope to reflect back to the 

absorber some of the concentrated solar flux that misses the absorber and to reduce some of the absorber 

thermal radiation loss.2

 

2

 See Duke Solar Energy [2001] for an example of such a study. 
30 

The HCE performance model simplifies other analysis, which limits the amount of information that can 

be gained. For instance, the model simplifies the heat transfer analysis between the absorber and HTF. 

This means the model cannot be used to evaluate HTF enhancement devices such as helical coil and 

twisted tape inserts, or machined longitudinal fins and helical ribs on the inside surface of the absorber 

pipe.3

 Also, as mentioned in Section 2.1.5.1, the wind effects are simplified. The wind is modeled as 

blowing normal to the receiver axis with no obstructions, so the model is not much help in evaluating 

actual wind losses that may occur as the wind blows from all directions and around adjacent SCAs. 

Furthermore, the model neglects the receiver radiation heat loss effects from the collector (see Appendix 

D), ground, and surrounding SCAs, and assumes an effective sky temperature for the radiation heat 

transfer loss (see Section 2.1.5.2), so studies of these effects also cannot be conducted with this model. 

The HCE performance model does not calculate location- or time-dependent parameters, or initial guesses 

or parameter bounds, so these values have to be manually changed when needed. One time- and locationdependent parameter is the solar incident angle. To evaluate the KJC test-loop data, the solar incident 

angle was calculated with a separate program and manually input into the parameter table (see Sections 

2.1.6.1, 3.2.2, and 5.2). Having to manually input the solar incident angle makes conducting studies on 

changing the tilt of the CSAs or changing the outer surface properties of the absorber difficult. Also, since 

the model does not calculate initial guesses or parameter bounds, these values have to be manually input 

into the EES variable information window. And for some design changes, this window may have to be 

updated for the model to converge. Changing the guesses and bounds can become tedious, especially 

when dealing with a long receiver that has a relatively small segment step size (resulting in large 

parameter arrays). 

The HCE performance model does not do optimization or exergy analysis, both of which would help 

identify HCE performance limitations. However, an incremental optimization parameter study, similar to 

Section 6.11 for the glass envelope outer diameter, can be conducted. As shown in Section 2, the HCE 

performance model is based on first law analysis, which provides values for collector efficiency, heat 

gain, and heat losses. However, it does not provide information on the limitations for improving these 

values or help identify specific components that have high irreversibility. Exergy analysis could not only 

provide second law efficiencies to identify improvement limitations, but could also help identify collector 

and HCE components that have the most room for improvement. As stated by Bejan, “With exergy 

analysis a designer can better focus effort so that losses are effectively pinpointed and reduced” [1988]. 

The HCE performance model can be improved, if necessary, to eliminate or reduce some of these 

limitations. For instance, the circumferential direction of the HCE cross-section could be modeled in 

discrete segments, similar to the two-dimensional model described in Section 2.2. The solar flux for each 

segment could be approximated with a weighted average of the expected profile4

 at the segment location. 

This would allow some analysis of the influence of the nonuniformity of the solar insolation around the 

HCE circumference. It would be best, however, to make this modification to the current one-dimensional 

version of the HCE performance model and not the two-dimensional, since a three-dimensional model 

may have convergence problems and require long iteration times. 

 

 

3

 See samples on pages 504 and 505 in Incropera and DeWitt [1990]. 

4

 The expected profile can be estimated from previous results listed in the literature (such as the results discussed by 

Thomas and Guven [1994] or from predictions with Ray Tracing techniques). 
31 

More meteorological data, date, time, and location inputs could be added to the HCE performance model 

to better represent actual field test data results. By adding wind direction, for example, an effective wind 

speed could be approximated by using the component of the wind vector that is normal to the receiver 

axis in the forced convection calculations. Also, knowing additional meteorological data will allow a 

calculation for a more accurate approximation of the effective sky temperature.5

 And with the date, time, 

and location inputs the solar incident angle could be calculated instead of input (see Section 5.2). 

Other improvements could include additional logic to smooth the transition between the turbulent and 

laminar models and adding a manual input for the receiver length. Currently, if the laminar selection is 

chosen, the model switches to the laminar Nusselt number when the Reynolds number drops below 2300. 

This will cause a discontinuity in the HTF convection calculation. Additional logic could smooth this 

transition, similar to the logic that was used for the convection heat transfer from the absorber to the glass 

envelope. The receiver length is currently dependent on the test type and collector type (see Table 3.4). 

To manually enter a receiver length additional logic would need to be added along with a “user-defined” 

entry similar to the “user-defined” option for the absorber selective coating option in the two-dimensional 

design version of the code (see Section 3.2.1). 

Additional model improvements could include temperature-dependent optical properties, changing the 

solar absorption terms to heat generation terms, and adding exergy analysis. Currently (see Section 

2.1.6.1), the absorptance terms are independent of temperature, and the emissivities were determined at 

temperatures lower than the HCE will see during normal operation. Once additional testing is complete, 

improved temperature-dependent functions may be determined. If so, these updated properties can easily 

be added to the HCE performance models. As mentioned in Section 2, the solar absorption is actually a 

heat generation phenomenon. This is especially true for the glass envelope, since the heat absorption 

occurs throughout its volume. The model could be changed to incorporate this effect; however, the 

thermal conductance would also need to reflect this change. Finally, an internal or external procedure 

(written in a software code such as Dynamic-C) could be used to add exergy analysis to the HCE 

performance model. This would allow component exergy generations terms, and an overall second law 

efficiency to be output, to better indicate the availability for HCE performance improvement and to help 

prioritize design changes to improve trough performance. 

3. EES Codes 

The HCE performance model has been coded into EES. The basic function of EES is to solve algebraic 

equations. EES is unique from other software because it automatically identifies all unknowns and groups 

the equations for the most efficient solution. Once a solution is found, the solution to all the unknowns 

can be viewed. This includes solutions to arrays. EES also supports user-defined procedures and 

functions, similar to FORTRAN, and provides built-in mathematical and thermal-physical property 

functions. Furthermore, EES supports user-generated property lookup tables and parametric tables. The 

lookup tables provide an easy means for adding user-defined thermal-physical properties, or any other 

data needed in the equations in the code. The parametric tables are similar to spreadsheets and are 

convenient for conducting parametric studies, such as heat losses as a function of HTF inlet temperature. 

EES also has a diagram window, which can be used for easy selection of input variables and can support 

simple drawings, such as a schematic of a thermodynamic flow cycle. EES can also solve differential and 

complex equations, and supports optimization, linear and nonlinear regression, uncertainty propagation, 

and generates user-defined plots. It can also be integrated with external functions and procedures written 

in a high-level language such as PASCAL, C, or FORTRAN. A complete EES user manual, provided by 

 

5

 Duffie and Beckman [1991] list several references for relations to estimate an effective sky temperature based on 

different meteorological data. 
32 

F-Chart Software, can be downloaded from the Web site http://fchart.com/ [Klein 2002] for more detailed 

explanations of the capabilities of EES and detailed instructions. 

Four versions of the EES code were written to solve the HCE performance model. Two include the onedimensional energy balance model and two include the two-dimensional model. Each can be used to 

evaluate and refine optical properties and HCE design parameters. Each is tailored for a different task. 

• The “Design Study” version is best suited for evaluating HCE design changes such as a proposed new 

selective coating or a new envelope material. It can also be used to evaluate the effects of damage to 

an HCE, like lost vacuum or missing glass envelope. 

• The “AZTRAK Test Data ” version was written specifically for evaluating the data from the 

AZTRAK test platform at SNL. This version is compatible with the AZTRAK and weather data 

output files and calculates the resulting heat loss and efficiency based on the input weather and HTF 

flow properties. In this version the energy absorbed in the glass envelope is included as part of the 

optical loss and not the heat loss. This enables comparisons with the heat loss as it is reported from 

the testing – the thermal energy leaving the absorber, not the glass envelope. The results can be used 

to evaluate the AZTRAK test data and to better explain the resulting heat losses and efficiencies. 

• The “KJC Test-Loop Data” version can be used to evaluate the data from two rows of collectors (testloop) at a SEGS plant located in Kramer Junction, California, and operated by KJCOC. This version, 

like the AZTRACK test data version, is compatible with the field test data and weather data outputs 

(type, units, etc.), and can be used to help understand the test data results. 

• The “Two-Dimensional Design Study” version was written to evaluate the effects of the pressure 

drop and HTF temperature changes along the length of the HCE. This version is valuable for 

evaluating effects associated with receiver lengths, and predicting the outlet temperature and velocity 

based on a given set of inlet data. 

These last three versions (AZTRAK Test Data version, KJC Test-Loop version, and Two-Dimensional 

Design Study version) can also be used to evaluate effects of design changes and HCE conditions with 

actual field criteria (wind speed, ambient temperatures, solar incident angles, solar insolation, flow rates, 

etc.). A brief description of each version of the codes is discussed below, and the two design study 

versions are provided in Appendices E and F. 

3.1 One-Dimensional Heat Transfer EES Codes 

The EES codes titled “One-Dimensional Design Study” and “AZTRAK Test Data” include only the onedimensional heat transfer analysis. These codes can be used to evaluate the HCE efficiencies for short 

receivers (~< 500 m, see Section 4), or when the effects of receiver length are not needed, for instance, 

when evaluating relative changes that result from different selective coating properties. In the case of the 

AZTRAK test data, receiver lengths are limited by the size of the rotating platform, which is only about 

10 m; therefore, HTF properties have little variation along the length of the receiver and a twodimensional model makes little difference in comparison to the one-dimensional model. 

3.1.1 One-Dimensional Design Study Version 

The one-dimensional design study version of the EES program is useful for evaluating relative design and 

parameter effects on HCE performance (see Section 6). It can also be used to evaluate current or proposed 
33 

selective coatings by selecting or manually inputting the optical properties for the case and to evaluate the 

effects of damage to the HCEs, such as lost vacuum or missing glass envelope. 

3.1.1.1 Lookup Tables 

As stated in Section 3, EES supports user-generated lookup tables. The design study version, as well as 

the other versions, has lookup tables for various HTFs and for argon gas. The HTF tables include 

• Therminol VP1 • Dowtherm RP 

• Therminol 59 • Syltherm 800 

• Therminol 66 • Xceltherm 600 

• Therminol XP • Water, salt (60% NaNO3, 40% KNO3) 

• Dowtherm Q • Hitec XL 

Part of the Therminol VP1 lookup table is shown in Table 3.1. The properties include temperature (o

C), 

density (kg/m3

), vapor pressure (Pa), specific heat (J/kg-K), dynamic viscosity (kg/m-s), and thermal 

conductance (W/m-K). All the data in the HTF tables are based on laboratory tests conducted by the 

manufacturers and were taken at the HTF saturation pressure. Therefore, small variances in the properties 

can be expected from sample to sample and because of pressure effects. However, temperature has the 

strongest influence on the properties. If the program evaluates properties outside the recommended HTF 

temperature range, a warning message will appear. HTFs can easily be added by inserting a lookup table 

with the appropriate properties and making a few changes to the code. 

Argon gas is included as a lookup table for evaluating its effects in the HCE annulus space between the 

absorber and the glass envelope (see Section 6.3). Pressurizing this space with argon has been proposed as 

an option to a vacuum or as a temporary fix if vacuum is lost. The properties in the argon table were 

compiled for a constant pressure of 100 kPa. The properties of the other two annulus gases (air and 

hydrogen) are already provided in the EES program as embedded thermal-physical property tables. For 

more information on the HTF fluids or argon data see the references in Appendix A. 

Table 3.1 Part of the Lookup Table for Therminol VP1

T 

[C] 

ρ 

[kg/m3] 

Pv

[kPa] 

c 

[J/kg-K] 

μ 

[kg/m-s] 

k 

[W/m-K] 

׃ ׃ ׃ ׃ ׃ ׃ ׃

Run 17 172 938 9.97 1970 0.000486 0.118

Run 18 182 929 13.8 2000 0.00045 0.1165

Run 19 192 920 18.9 2030 0.000418 0.115

Run 20 202 911 25.3 2050 0.000389 0.1135

Run 21 212 902 33.5 2080 0.000364 0.1119

Run 22 222 893 43.7 2110 0.000341 0.1103

׃ ׃ ׃ ׃ ׃ ׃ ׃

3.1.1.2 Diagram Window 

Figure 3.1 shows the diagram window used in the design study version of the EES HCE heat transfer 

code. The window enables the user to manually input data from pull-down menus and the keypad. The 

diagram window is divided into seven sections: Ambient Conditions, HCE and Collector Properties, 
34 

Optical Properties, Modeling Properties, Heat Transfer Fluid Properties, “User-Defined” Absorber 

Selective Coating Properties, and “User-Defined” Collector Type Properties. 

 Therminol VP1

Ambient Conditions

Heat Transfer Fluid Flow Rate =

 Solel UVAC Cerm et (SNL test avg)

Wind Speed =

Heat Transfer Fluid =

Absorber Selective Coating =

Gas in Annulus = Air

Annulus Absolute Pressure = 0.0001 [torr]

Glass Envelope Intact? Yes

HTF Flow Type = Pipe Flow

Absorber Material = 

Include laminar flow model? No

Collector Type = LS-2

 321H

Ambient Temperature =

HCE and Collector Properties Heat Transfer Fluid Properties

Modeling Properties

Direct Normal Incident Solar Irradiation =

Solar Incident Angle From Aperture Normal (0-75) = 

950 [W/m2]

0 [mph]

22 [C]

0 [degrees]

140 [gpm]

Optical Properties

Solar Weighted Mirror Reflectivity = 0.9

"User-Defined" Absorber Selective Coating Properties

Coating Absorptance = 0.98

Coating Emittance (@ 100 oC) = 

Glass Envelope Transmittance = 

0.04

Coating Emittance (@ 400 oC) = 0.1

0.97

(Only Valid if Absorber Selective Coating = "User-Defined")

HTF Annulus Inner Diameter =

(only valid for annulus HTF flow type)

0.0508 [m]

"User-Defined" Collector Type Properties

(Only Valid if Collector Type = "User-Defined")

0.066 [m]

0.07 [m]

0.109 [m]

5 [m]

0.115 [m]

0.974

0.994

0.98

0.935

0.963

0.981

0.96

Geometry

Absorber Inner Diameter =

Absorber Outer Diameter =

Glass Envelope Inner Diameter =

Glass Envelope Outer Diameter =

Projected Aperture Width =

Shadow ing =

Tracking Error =

Geometry =

Dirt on HCE =

Clean Mirror Reflectivity =

Dirt on Mirror =

General Error =

Optical Efficiency Terms

Total Optical Efficiency = 74.13 [%]

Total Optical Loss / Unit Length = 1229 [W/m]

Figure 3.1 Diagram window from the Design Study version of the heat transfer code 

• The Ambient Conditions section includes direct normal incident solar irradiation, wind speed, 

ambient temperature, and solar incident angle. All these inputs can be typed in by right clicking the 

curser on the input boxes. None of the inputs are bounded; typing in an erroneous value will either 

cause the program to fail or produce erroneous results. 

• The HCE and Collector Properties section has pull-down menus for selecting collector type, absorber 

material, absorber selective coating, gas in the annulus, and options for whether or not the glass 

envelope is intact. Clicking on the arrow next to the selection box and highlighting the desired choice 

selects the option. The collector type choices are LS-2, LS-3, and IST. The first two choices are 

standard collector types found in SEGS plants, and the IST collector type is a proposed smaller 

collector design. The LS-2 and LS-3 are second- and third-generation parabolic trough collectors built 

by Luz International Ltd. The IST is a parabolic trough collector built by Industrial Solar Technology. 

The optical properties and geometry for each collector type are written into the program and are 

discussed in Section 2.1.6.1. References for the optical properties are listed in Appendix B. 

The absorber material choices include three stainless steels: 304L, 316L, 321H, and one copper: B42 

(see Section 2.1.2). The absorber selective coating has eight choices: Luz black chrome and Luz 

Cermet coatings, five Solel UVAC Cermet coatings, and a user-defined choice. Two of the five Solel 
35 

UVAC coatings, 0.07 and 0.10 @ 400o

C, are not found in the SEGS plants, but rather are coating 

types the manufacturer Solel has proposed. All the other optical properties are based on tests 

conducted by SNL (see Section 2.1.6.1). The Solel UVAC Cermets labeled “SNL test a” and “SNL 

test b” are from two separate test results from either end of a single HCE that was tested. The Solel 

UVAC Cermet labeled “SNL test avg” is the average of these two results. The last choice, userdefined, is used to evaluate coating properties input manually. The remaining input in the HCE and 

Collector Properties window is the annulus absolute pressure, which is typed in. 

• The “Optical Properties” section has an input box for the trough solar weighted mirror reflectivity, 

and outputs for optical efficiency and optical loss. The reflectivity value is an estimated value of the 

mirror solar reflectance, and indicates mirror quality and cleanliness. The two outputs are provided 

for convenience when a user is trying to match a certain optical efficiency by adjusting the mirror 

reflectance. 

• The “Modeling Properties” section includes pull-down windows for HTF flow type and an option for 

including a laminar flow model in the analysis. The flow type choices are either pipe flow or annulus 

flow. If the annulus pipe flow is chosen, a valid annulus diameter needs to be typed in the box 

provided. The annulus flow type is included because HCEs are sometimes tested with an inner pipe or 

plug to help simulate Reynolds numbers that are typically achieved in a SEGS plant (see Section 

2.1.1.3). The option for laminar flow is included so losses associated with cooler HTF temperatures 

(~< 100o

C) can be evaluated. These temperatures could be seen during extended down times and 

during startups or testing. 

• The “Heat Transfer Fluid Properties” section has an input box for the HTF flow rate and a pull-down 

menu for the HTF type. Again, there are no restrictions on the value that can be input for the HTF 

flow rate. Values outside normal operating ranges could result in erroneous outputs. However, 

warnings are embedded into the code that will print out if the flow rate results in Reynolds numbers 

below the minimum in which the equations are valid. The choices for the HTF type are the same as 

described in Section 3.1.1.1 for the lookup tables. 

• The “‘User-Defined’ Absorber Selective Coating Properties” section contains input boxes for the 

coating optical properties and glass envelope transmittance. This section is valid only if the “UserDefined” selective coating option is chosen. If another selective coating option is chosen, optical 

properties written in the code are used (see Section 2.1.6.1). 

3.1.1.3 Parametric Table 

The parametric table for the one-dimensional design study version of EES is shown in Table 3.2. It acts as 

a spreadsheet. It contains user inputs and program outputs. For this case the input is the HTF temperature, 

T1ave (o

C), defined as the average bulk HTF temperature between the inlet and outlet of the receiver. The 

outputs include the HCE heat loss per receiver length, qHeatLoss (W/m); heat gain into the HTF fluid per 

receiver length, qHeatGain (W/m); and the collector efficiency, ηCol (%). The efficiency is defined as the 

total HTF heat gain per receiver length divided by the direct normal solar insolation at the collector 

aperture per receiver length. If any changes are required, the parametric table can easily be modified. 

Changing the input variables might also require one or more variable guesses or bounds to be changed. In 

order for the program to converge, the variable guesses need to be “reasonably” close. Instructions on 

modifying the table and variable information can be found in the general EES user manual. 
36 

Table 3.2 Parametric Table for the Design Study Version of the Heat Transfer Code 

> 

1..7 

T1ave

[C] 

qHeatLoss

[W/m] 

qHeatGain

[W/m] 

ηCol

[%] 

Run 1 100 11.03 3510 73.90 

Run 2 150 25.81 3495 73.59 

Run 3 200 49.83 3471 73.08 

Run 4 250 87.1 3434 72.30 

Run 5 300 142.8 3378 71.13 

Run 6 350 223.5 3298 69.43 

Run 7 400 337.3 3184 67.03 

3.1.2 AZTRAK Test Data Version 

This version of the EES program was created to help verify the model and to evaluate the data collected 

from testing HCEs and collectors on the AZTRAK rotating test platform at SNL (see Section 5.1). 

Information on the AZTRAK testing along with sample test results can be found in Dudley et al. [1994]. 

The AZTRAK Test Data version is very similar to the one-dimensional design study version. It is also 

based on the one-dimensional heat transfer model. However, the heat loss for this version of the code 

includes only the heat losses from the absorber and not from the glass envelope. This was done because of 

the way the heat loss was measured during the testing. The solar insolation absorption into the glass 

envelope is included as an optical loss. The lookup tables are all the same, and the diagram window is 

only slightly different. Furthermore, conversion statements were added to the code where appropriate, so 

output data from the testing could be input directly into the parametric table. 

3.1.2.1 Diagram Window 

The diagram window for the AZTRAK test data version of the EES code is shown in Figure 3.2. All the 

input options shown are the same as discussed in the design study version. For this version, some of the 

inputs have been moved to the parametric table, the option for pipe flow has been removed, and the 

“User-Defined” selective coating is not included. 

 Syltherm 800

Ambient Conditions

 Luz Cermet (SNL test)

Heat Transfer Fluid =

Absorber Selective Coating =

Gas in Annulus = Air

Annulus Absolute Pressure = 0.0001 [torr]

Glass Envelope Intact? Yes

Absorber Material = 

Include laminar flow model? Yes

Collector Type = LS-2

 321H

HCE and Collector Properties Heat Transfer Fluid Properties

Modeling Properties

Solar Incident Angle From Aperture Normal (0-75) = 0 [degrees]

Optical Properties

Solar Weighted Mirror Reflectivity = 0.9337

Total Optical Efficiency = 73.11 [%]

Total Optical Loss / Aperture Area = 242.9 [W/m2

]

Figure 3.2 Diagram window from the AZTRAK Test Data version of the heat transfer code 
37 

3.1.2.2 Parametric Table 

As shown as Table 3.3, the parametric table for the AZTRAK test data version of the EES program 

contains six inputs and four outputs. The input data are set up to match the data provided by the actual 

AZTRAK testing. The data include: 

• Direct normal insolation, Ib (W/m2) 

• Wind speed, v6 (m/s) 

• Ambient temperature, T6 (oC) 

• Inlet bulk HTF temperature, Tin (oC) 

• Outlet bulk HTF temperature, Tout (oC) 

• HTF volumetric flow rate, v1, (L/min) 

The output data include: 

• The difference between the average HTF and the ambient temperature, TdiffAir (oC) 

• HCE heat loss per collector aperture area, qheat, loss, col, area (W/m2) 

• HTF heat gain, qHeatGain (W/m) 

• Collector efficiency, ηCol (%) 

All necessary unit conversions are made in the code. 

Table 3.3 Parametric Table for the AZTRAK Test Data Version of the Heat Transfer Code 

> 

1..8 

Ib 

[W/m2] 

V6

[m/s] 

T6

[C] 

Tin

[C] 

Tout

[C] 

v1

[L/min] 

TdiffAir

[C] 

qAbsorbertHeatLoss 

[W/m2] 

qHeatGain

[W/m] 

ηCol

[%] 

Run 1 933.7 2.6 21.2 102 124 47.7 91.9 5.616 3402 72.5

Run 2 968.2 3.7 22.4 151 173 47.8 139.8 9.389 3510 72.1

Run 3 982.3 2.5 24.3 197 219 49.1 184.2 15.18 3533 71.6

Run 4 909.5 3.3 26.2 250 269 54.7 233.8 24.51 3218 70.4

Run 5 937.9 1.0 28.8 297 316 55.5 278.6 37.86 3256 69.1

Run 6 880.6 2.9 27.5 299 317 55.6 280.6 38.38 3042 68.7

Run 7 20.9 2.6 29.5 379 398 56.8 359.3 76.71 2998 64.8

Run 8 903.2 4.2 31.1 355 374 56.3 333.9 63.35 3000 66.1

3.2 Two-Dimensional Heat Transfer EES Codes 

The length of a receiver in an operating field can reach hundreds or thousands of meters. For instance, the 

receivers on the KJC test loops are almost 800 m long, yet consist of only two rows of collectors. For 

long receiver lengths the change in flow rate (caused by the density change) and pressure drop can have 

an appreciable effect on the heat transfer model (see Section 4). Because of this, and to study the effects 

of receiver lengths, two versions of the EES heat transfer code include a two-dimensional heat transfer 

model: “Two-Dimensional Design Study” and “KJC Test-Loop Data.” 

3.2.1 Two-Dimensional Design Study Version 

Like the One-Dimensional Design Study version, the two-dimensional version can be used to evaluate 

design and parameter changes. It can also be used to study property and temperature changes along the 

length of the receiver (see Section 6.12). 
38 

3.2.1.1 Diagram Window 

The diagram window for the Two-Dimensional Design Study version is shown in Figure 3.3. It is very 

similar to the One-Dimensional Design Study version shown in Figure 3.1. For the two-dimensional case, 

an input for the number of receiver length segments has been added to the Modeling Properties section. 

Also, the flow type and the “User-Defined” collector type properties have been removed; however, both 

can be added if needed. 

 Therminol VP1

Ambient Conditions

Heat Transfer Fluid Flow Rate =

 Solel UVAC Cermet (SNL test avg)

Wind Speed =

Heat Transfer Fluid =

Absorber Selective Coating =

Gas in Annulus = Air

Annulus Absolute Pressure = 0.0001 [torr]

Glass Envelope Intact? Yes

Test Type (only used with LS-2 collector) = KJC Test Loop

Absorber Material = 

Collector Type = LS-2

 321H

Ambient Temperature =

HCE and Collector Properties

Heat Transfer Fluid Properties

Modeling Properties

Direct Normal Incident Solar Irradiation =

Solar Incident Angle From Aperture Normal (0-75) = 

950 [W/m2]

0 [mph]

22 [C]

0 [degrees]

140 [gpm]

Optical Properties

Solar Weighted Mirror Reflectivity = 0.9

"User-Defined" Absorber Selective Coating Properties

Coating Absorptance = 0.98

Coating Emittance (@ 100 oC) = 

Glass Envelope Transmittance = 

0.04

Coating Emittance (@ 400 oC) = 0.1

0.97

(Only Valid if Absorber Selective Coating = "User-Defined")

Total Number of Discrete Segments (# of Nodes - 1) = 10

Total Optical Efficiency = 0 [%]

 Total Optical Loss / Receiver Length = 1185 [W/m]

Figure 3.3 Diagram window from the Two-Dimensional Design Study version of the 

heat transfer code

As shown in Section 4, the number of segments to use depends on the total length of the receiver. Too 

few segments result in larger errors in the calculations; too many results in excessive iteration times, with 

little gain in model accuracy. Also, adding more segments requires the initial guesses and bounds of the 

additional variables to be updated to the variable information window, which can become very time 

consuming. Reducing the number of segments from a previous run usually requires no modifications to 

the variable information window. 

The current version of the two dimensional model dynamic window does not have an explicit input for 

receiver length. The receiver length used in the model will depend on both the collector type and test type 

inputs. Table 3.4 summarizes the receiver lengths used in the code. A possible future modification to the 

program could be to add a manual input for the receiver length. 
39 

Table 3.4 Collector and Test Type Receiver Lengths

Collector Type Test Type Receiver Length (m) 

LS-2 SNL AZTRAK Platform 8.12 

LS-2 KJC Test-loop 779.52 

LS-3 N/A 97.44 

IST N/A 2.16 

3.2.1.2 Parametric Table 

The parametric table for the two-dimensional model is shown as Table 3.5. In addition to the heat gain, 

heat loss, and efficiency as shown with the one-dimensional design study version, the table includes the 

parameters resulting from the flow effects: HTF outlet temperature, T1outlet (o

C); inlet and outlet velocities, 

v1inlet and v1outlet (m/s); and the pressure drop, ∆P (Pa). The table also includes the total losses per receiver 

length caused by optical effects, qOptLoss (W/m). 

Table 3.5 Parametric Table for the Two-Dimensional Design Study Version 

of the Heat Transfer Code 

> 

1..7 

T1inlet

[C] 

T1outlet 

[C] 

v1inlet

[m/s] 

v1outlet

[m/s] 

∆P 

[Pa] 

qHeatLoss 

[W/m] 

qHeatGain

[W/m] 

ηCol

[%] 

Run 1 125 275.7 2.582 2.998 576151 56.50 3340 72.90

Run 2 150 298.6 2.582 3.018 551632 73.42 3324 72.53

Run 3 175 321.7 2.582 3.044 530224 94.12 3303 72.08

Run 4 200 344.8 2.582 3.073 510399 119.20 3278 71.53

Run 5 225 368.0 2.582 3.114 492940 149.40 3248 70.87

Run 6 250 391.1 2.582 3.165 477001 185.40 3212 70.09

Run 7 275 414.0 2.582 3.243 462750 227.80 3169 69.16

3.2.2 KJC Test-Loop Data Version 

The KJC Test-Loop Data version of the HCE heat analysis software was created to verify the software 

model and to evaluate the KJC test-loop data. The test data can be copied directly from the KJCOC 

performance spreadsheets and pasted into the parametric tables. The code has been written to convert all 

inputs into the correct units before running the iterations. 

3.2.2.1 Diagram Window 

An example of the diagram window for the KJC Test-Loop Data version is shown in Figure 3.4. The 

main difference between this diagram window and that for the Two-Dimensional Design Study version is 

that some inputs were moved to the parametric table, and like the AZTRAK version, the “user-defined” 

selective coating optioned is not included. The inputs that were moved to the parametric table include all 

the ambient conditions and the HTF flow rate. The option to select the test type was also removed. 
40 

 Therminol VP1

 Solel UVAC Cermet (SNL test avg)

Heat Transfer Fluid =

Absorber Selective Coating =

Gas in Annulus = Air

Annulus Absolute Pressure = 0.0001 [torr]

Glass Envelope Intact? Yes

Absorber Material = 

Collector Type = LS-2

 321H

HCE and Collector Properties

Heat Transfer Fluid Properties

Optical Properties

Solar Weighted Mirror Reflectivity = 0.904

Modeling Properties

Total Number of Discrete Segments (# of Nodes -1) = 10

Figure 3.4 Diagram window from the KJC Test-Loop Data version of the heat transfer code 

3.2.2.2 Parametric Table 

As shown in Table 3.6, the parametric table for the KJC Test-Loop Data version of the heat transfer code 

contains eight inputs and nine outputs. The inputs coincide with measured data from the KJC Test-Loop. 

The HTF temperatures, TinF and ToutF; HTF flow rate, V1volg; and ambient conditions, Ib, T6F, V6mph were 

all copied from the Excel spreadsheets created by KJCOC to evaluate the test-loop data. The time input is 

included to reference the data and is not used in any calculations. The solar incident angle, θ, has to be 

calculated independently. The incident angles used to generate the data in this report were copied from an 

Excel HCE spreadsheet used to evaluate the KJCOC performance data [Price 2000]. All parameter units 

in the parametric table are converted to metric within the EES code. 

The output data in the parametric table include: 

• Outlet temperature, T1outletF , (oF) 

• Outlet flow rate, v1volg,outlet, (gpm) 

• Pressure drop, ∆P (Pa) 

• Heat loss through the HCE support brackets, qcond,bracket (W) 

• Optical loss due to the solar incident angle, qopt,loss,K (W/m) 

• Total optical loss, qopt,loss, (W/m) 

• Heat loss, qHeatLoss, (W/m) 

• Heat gain, qHeatGain, (W/m) 

• Collector efficiency, ηCol 

All the unit length energy terms are based on the total receiver length. The optical loss caused by the solar 

incident angle is determined by reducing the solar insolation by the incident angle modifier given in 

Section 2.1.6.1. However, the total optical loss is determined by reducing the solar insolation by the 

effective optical efficiency, which includes the incident angle modifier term. 
41 

Table 3.5 Parametric Table for the KJC Test-Loop Data Version of the Heat Transfer Code

> 

1..288 Time TinF

[F] 

ToutF

[F] 

V1Volg

[gpm] 

Ib

[W/m2]

T6F

[F] 

V6mph

[mph] 

Θ

[deg] 

T1outlet 

[F] 

...

... ׃ ׃ ׃ ׃ ׃ ׃ ׃ ׃ ׃ ׃

Run190 1550 446 587 93.8 446.5 64.2 14.72 36.2 579.5 ...

Run191 1555 443 563 93.4 785.2 64.1 13.16 35.5 677.8 ...

Run192 1600 448 543 93.5 768.7 64.5 15.6 34.8 680.3 ...

Run193 1605 454 518 93.7 757 64.8 15.22 34.1 684.5 ...

Run194 1610 456 505 93.9 732.6 64.8 14.54 33.4 681 ...

Run195 1615 452 490 93.8 699.6 64.7 12.95 32.7 670.5 ...

Run196 1620 447 462 93.9 681.5 64.5 13.12 32.0 662 ...

... ׃ ׃ ׃ ׃ ׃ ׃ ׃ ׃ ׃ ׃

... v1outlet

[gpm] 

∆P 

[Pa] 

qcond,bracket 

[W] 

qopt,loss,K

[W/m] 

qopt,loss

[W/m] 

qHeatLoss

[W/m] 

qHeatGain

[W/m] 

ηCol

[%] 

׃ ׃ ׃ ׃ ׃ ׃ ׃ ׃ ...

... 102.2 229596 1078 25.27 109.3 500.6 966.5 50.0

... 110.5 231911 1963 28 150.2 844.9 1674 51.8

... 110.5 231283 1940 29.7 153.2 793.5 1615 52.3

... 110.8 231057 1928 29.81 157 748.5 1567 52.8

... 110.6 231403 1884 29.32 155.7 693.1 1494 53.3

... 109.6 231264 1820 27.82 149.3 632.7 1405 53.9

... 109.2 232594 1795 27.46 143.6 588 1349 54.6

׃ ׃ ׃ ׃ ׃ ׃ ׃ ׃ ...

4. Comparison between the One- and Two-Dimensional Models 

The one- and two-dimensional models both have advantages and disadvantages. The one-dimensional 

model is less complex; therefore, it is easier to modify, requires less iteration time, and has less trouble 

converging. The two-dimensional model consists of many arrays, requires longer iteration times, and can 

have problems converging, especially if all the array guess values are not within reasonable bounds and 

sequence. The code can also be more difficult to modify. However, it provides a higher degree of 

accuracy than the one-dimensional model when evaluating long receivers (> 700 m). 

The two-dimensional model includes HTF pressure losses, HTF velocity changes, and receiver support 

bracket conduction losses (see Section 2.2). The two-dimensional model also captures some of the 

nonlinearity of the HTF temperature changes along the length of the receiver. As the receiver length 

increases, each effect becomes more significant and the error associated with using the one-dimensional 

model increases. Two questions are explored here: 

• At what receiver length do the axial direction effects become significant? 

• When should the two-dimensional model be used instead of the one-dimensional model? 

The process used to compare the one- and two-dimensional models was to start with the KJCOC test-loop 

receiver length of 779.52 m (the KJC Test-Loop Data version of the EES Heat Transfer Code uses the 

two-dimensional model) and determine a step size that resulted in a “reasonable” level of accuracy for a 

“reasonable” convergence time. Then the two-dimensional model results (using the determined step size) 

were compared to results using the one-dimensional model. For an equal comparison, the average HTF 

temperature used in the one-dimensional model is the linear average between the inlet and outlet 

temperatures from the two-dimensional model. 
42 

Figures 4.1a and 4.1b show the inlet, outlet, and average temperatures used in the one- and twodimensional model comparison. The two-dimensional model captures the nonlinearity that results from 

the heat losses; therefore, the HTF temperature through the length of the receiver will be nonlinear and 

will be slightly higher on average as determined by the two-dimensional model then the linear average 

used in the one-dimensional model. This will result in the one-dimensional model underestimating the 

heat losses and overestimating the heat gain. Furthermore, the two-dimensional model is actually a 

summation of discrete one-dimensional segments and not a continuous two-dimensional model; as the 

discrete segment lengths approach zero the discrete model will approach a continuous model. 

Table 4.1 and Figure 4.2 show the results of the step size study. The “number of steps” in the first column 

in Table 4.1 is the number of discrete receiver segments modeled in the axial direction. The “step size” is 

the total receiver length (779.5 m) divided by the number of steps. In this case, the total receiver length of 

779.5 m was chosen simply because it was the length of the KJCOC test-loop. Any arbitrary length could 

have been chosen. Actual receiver lengths can be many times greater then the test-loop length. The testloop receiver length, however, is an appropriate length to help determine at what receiver length the axial 

direction effects become significant, ~ 80 m, according to Figure 4.2 where the heat loss first shows a 

drop as compared to smaller step sizes. The last column in Table 4.1 “time” is the time required for the 

EES program to reach convergence. Besides being dependent on the step size, the time is also dependent 

on how close the initial variable guesses are to the solution, and the speed of the microprocessor. The 

closer the initial guesses are to the solution, the shorter the iteration time. However, given the same guess 

values, the iteration time will always increase with decreasing step size (or increasing number of steps). 
43 

a) HTF Inlet and Outlet Temperature Comparison Between the 1D and 2D Models

245

265

285

305

325

345

365

385

405

0 100 200 300 400 500 600 700 800

Location Along Receiver Length (m)

HTF Temperature (oC)

2D Model 1D Model

b) HTF Average Temperature Comparison Between the 1D and 2D Models

245

265

285

305

325

345

365

385

405

0 100 200 300 400 500 600 700 800

Location Along Receiver Length (m)

HTF Temperature (oC)

1D Model 2D Model

Figure 4.1 a) HTF inlet and outlet and b) average HTF temperature comparisons between the oneand two-dimensional models.
44 

Table 4.1 Iteration Time for Different Receiver Step Sizes 

Number of 

Steps 

Step Size* 

(m) 

Time 

(s) 

100 7.795 91411.5 

50 15.59 13200 

20 38.98 11819 

10 77.95 311 

4 194.9 39 

1 779.5 6 

* Total length of solar receiver = 779.5 m 

Heat Loss for Different Step Sizes

0

50

100

150

200

250

125 150 175 200 225 250 275

HTF Inlet Temperature (o

C)

Heat Loss (W/m)

7.795 m 15.59 m 38.98 m 77.95 m 194.9 m 779.5 m

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

L = 779.5 m

Figure 4.2 Heat loss chart for different receiver length step sizes 

Figure 4.3 shows the heat gain and collector efficiency per receiver length and Figure 4.4 shows the heat 

loss per receiver length, for the one- and two-dimensional models. As expected, the one-dimensional 

model overpredicts heat gain and efficiency and underpredicts heat loss. Specifically, it gives heat gain 

and efficiency values that are approximately 0.5% different than the two-dimensional model, and heat 

loss values that are 5% to 10% different. The small differences between the one- and two-dimensional 

models for heat gain and efficiency values appear because the optical losses remain the same between the 

two models and the heat losses are relatively small compared to the optical losses (see Section 5.2). The 

charts also show that the higher the HTF temperature the larger the difference between the one- and twodimensional models. These differences will continue to increase with receiver length, and receivers can be 

many times longer than those used here. Additional comparisons between the one- and two-dimensional 

models are shown in Figures 6.10, 6-16, and 6.17 in Sections 6.6, 6.9, and 6.10, respectively. 
45 

a) Heat Gain Comparison Between the One and Two-Dimensional Models 

3150

3200

3250

3300

3350

3400

198.8 222.8 246.8 270.9 295 319 342.95

Average HTF Temperature (o

C)

Heat Gain (W/m)

Two-Dimensional Model One-Dimensional Model

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

L = 779.52 m

10 increments

b) Efficiency Comparison Between the One and Two-Dimensional Models

68

69

70

71

72

73

74

198.8 222.8 246.8 270.9 295 319 342.95

Average HTF Temperature (o

C)

Efficiency %

Two-Dimensional Model One-Dimensional Model

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

L = 779.52 m

10 increments

 

Figure 4.3 a) Heat gain and b) efficiency charts comparing the one- and two-dimensional models 

for different HTF temperatures
46 

Heat Loss Comparison Between the One and Two-Dimensional Models 

0

50

100

150

200

250

300

350

198.8 222.8 246.8 270.9 295 319 342.95

Average HTF Temperature (o

C)

Heat Loss (W/m)

Two-Dimensional Model One-Dimensional Model

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

L = 779.52 m

10 increments

Figure 4.4 Heat loss chart comparing the one- and two-dimensional models for different HTF 

temperatures 

5. Comparison with Experimental Data 

To verify the accuracy of the heat transfer models, the EES heat transfer codes were compared with two 

sets of data from HCE field tests. One set was from tests conducted from June 1992 through January 1993 

on the AZTRAK rotating test platform at SNL. The second set of data consists of three days of test data 

from a test-loop of collectors in a SEGS plant in Kramer Junction, California. A picture of the AZTRAK 

rotating test platform is shown in Figure 5.1. It consists of a rotating platform, a two-axis tracking control 

system, and all associated pumping equipment and instrumentation to test a segment of a trough collector 

and receiver. The trough shown consists of an LS-2 type collector segment with two Luz Cermet HCEs of 

4.06 m each. All the data used in the comparisons and a detailed explanation of the AZTRAK rotating test 

platform tests are reported in Dudley et al. [1994]. 

The second set of data, from a test-loop of collectors in a SEGS plant (Figure 1.1), includes data for 

January 2, March 22, and June 21, 2001. The test-loop consists of two rows of collectors containing 16 

LS-2 type SCAs in series each with 12 Luz Cermet selective coating HCEs at 4.06 m for a total receiver 

length of 779.52 m. The test-loop was instrumented to record HTF flow rates and HTF inlet and outlet 

temperatures. There was also instrumentation to record the direct normal solar irradiation, ambient 

temperature, and wind speed. 

When using the same HTF input property data and meteorological data as those from the field tests, the 

EES heat transfer codes gave results that agreed well with the field test data. 
47 

Figure 5.1 AZTRAK rotating test platform at SNL. An LS-2 type collector module is shown installed on 

the platform. 

5.1 Comparison with SNL AZTRAK Data 

The eight charts in Figures 5.2 to 5.6 show the comparison between the EES AZTRAK Test Data version 

of the heat transfer model and the 1992 data from the AZTRAK rotating platform tests [Dudley, et al. 

1994]. The AZTRAK test data included data for two different HCE selective coating types: Luz Cermet 

and Luz black chrome. For each coating, four test conditions were compared: 

• On-sun and vacuum in the annulus 

• On-sun and no vacuum in the annulus 

• Off-sun and vacuum in the annulus 

• Off-sun and no vacuum in the annulus 

The no vacuum condition was created simply by drilling small holes into the HCE bellows (see Figure 

1.4) allowing air to fill the annulus space. The AZTRAK off-sun tests were conducted after sunset with 

the collectors pointing toward the sky. Wind tests were conducted on the Luz black chrome HCE with the 

glass envelope removed; these data were compared with the EES program results. 

To simulate the test conditions with the EES code, the input data from the tests were copied into the 

parametric tables and the solar weighted reflectivity was adjusted until the optical efficiency calculated 

with EES matched the field tests results. The reflectivity values used are shown in Table 5.1. 

The one-dimensional heat transfer model compares well with the AZTRAK field test data. All trends are 

followed, and with a few exceptions, the data from the EES program are within the error bounds of the 

AZTRAK test data. However, the spread between data appears to increase with rising temperature for the 

Luz Cermet selective coating case. This could be due to either the changes in optical properties with 

temperature or to the possible sagging of the HCEs, causing a slight misalignment between HCE and 

collector. The optical properties used in the code are based on optical tests at lower temperatures. 

Therefore, the optical property equations given in Section 2.1.6.1 may be incorrect at elevated 
48 

temperatures for the Luz Cermet case – the Luz black chrome field test data show better agreement with 

the EES data. The HTF properties could also explain some of the differences. As the HTF manufacturers 

state in the information sheets, the thermophysical properties could vary slightly from batch to batch and 

could change over time, especially when exposed to thermal cycling (see Appendix A for references). 

Some of the extraneous points could also be due to typing or recording errors either in Dudley et al. 

[1994], actual test data recording, or in the EES parametric tables. 

Table 5.1 Mirror Reflectivity and Optical Efficiency for Each Test Type 

Test Type Solar Weighted Mirror 

Reflectivity 

Total Optical 

Efficiency 

(%) 

Luz Cermet, Vacuum 0.9337 73.1 

Luz Cermet, Loss Vacuum 0.9353 73.3 

Luz Black Chrome, Vacuum 0.9289 74.1 

Luz Black Chrome, Loss 

Vacuum 0.9221 73.3 

For convenience, all figures from the AZTRAK data comparisons are listed in Table 5.2, along with 

general comments about the comparisons. 

Table 5.2 List of Figures and Comments for the AZTRAK Test Platform Data Comparison

Figure Description Comments 

All All 

• Difference between the EES data and AZTRAK 

data for the Luz Cermet selective coating case 

tends to increase with increasing temperature. 

• The Luz black chrome selective coating resulted in 

better agreement between the EES data and 

AZTRAK data than the Luz Cermet selective 

coating. 

• EES data accurately depict all trends. 

• With a few exceptions, the EES data agree with 

the AZTRAK data within the instrument error 

bounds.

5.2 a) Luz Cermet, efficiency for vacuum and loss 

vacuum cases 

• Two EES data points lie outside the AZTRAK error 

bounds. 

5.2 b) Luz Cermet, heat loss for vacuum and loss 

vacuum cases with solar 

• Nine EES data points lie outside the AZTRAK error 

bounds. 

• EES data seem to underpredict the heat loss. 

• Shows the least agreement between the EES data 

and AZTRAK data. 

5.3 Luz Cermet, heat loss for vacuum and loss 

vacuum cases without solar 

• One EES data point lies outside the AZTRAK data 

error bounds. 

5.4 a) Luz black chrome, efficiency for vacuum and 

loss vacuum cases with solar 

• All EES data points are within the AZTRAK data 

error bounds. 

5.4 b) Luz black chrome, heat loss for vacuum and 

loss vacuum cases with solar 

• One EES data point lies outside the AZTRAK data 

error bounds. 

5.5 Luz black chrome, heat loss for vacuum and 

loss vacuum cases without solar 

• One EES data point lies outside the AZTRAK data 

error bounds. 

5.6 a) Luz Cermet, efficiency trend with wind 

5.6 b) Luz Cermet, heat loss trend with wind 

• Model tends to overpredict losses with increasing 

wind speed. 
49 

a) Efficiency Comparison

(Luz Cermet Selective Coating, Solar (NIP) ~ 800 to 970 W/m2)

50

55

60

65

70

75

80

50 100 150 200 250 300 350 400

Average Temperature Above Ambient (o

C)

Efficiency (%)

EES HCE Code SNL AZTRAK Tests

LS-2 Collector, Syltherm 800 

Vacuum

Loss Vacuum

b) Heat Loss Comparison

(Luz Cermet Selective Coating, Solar (NIP) ~ 800 to 970 W/m2)

0

20

40

60

80

100

120

140

160

50 100 150 200 250 300 350 400

Average Temperature Above Ambient (o

C)

[Heat Loss] / [Aperture Area] (W/m2

)

EES HCE Code SNL AZTRAK Tests

LS-2 Collector, Syltherm 800

Vacuum

Loss Vacuum

Figure 5.2 a) Efficiency and b) heat loss charts comparing the EES HCE code and SNL AZTRAK 

test data for Luz Cermet selective coating
50 

Heat Loss Comparison

(Luz Cermet Selective Coating, No Solar)

0

20

40

60

80

100

120

50 100 150 200 250 300 350 400

Average Temperature Above Ambient (o

C)

[Heat Loss] / [Aperture Area] (W/m2

)

EES HCE Code SNL AZTRAK Tests

LS-2 Collector, Syltherm 800

Vacuum

Loss Vacuum

Figure 5.3 Heat loss comparison chart between the EES HCE code and SNL AZTRAK test data for 

the off-sun case and Luz Cermet selective coating 

The heat transfer model overpredicts losses with wind (see Figure 5.6). This is expected, since in the 

program the wind direction is modeled as normal to the HCE axis, and the input wind speed is assumed to 

be the speed at the HCE height. In reality, many obstructions around the collectors and in general the 

wind won’t be normal to the HCE axis. In addition, the wind speed anemometer is located well above the 

collector and any obstructions; therefore the wind speed recorded will be higher than the speed at the 

collector height. Figure 5.6 shows that the error associated with the wind loss can be as high as 30% for 

the missing glass envelope case; however, the wind has a smaller effect on the heat loss when the glass 

envelope is in place, especially with a vacuum between the absorber and glass envelope (see Section 6.4). 

Since the HCE model overpredicts wind related losses, reducing the wind effects in the EES codes on a 

future revision may be worthwhile. 

There are many other explanations for the differences between the EES data and AZTRAK data. For 

instance, there could be other unaccounted optical effects during the testing, such as HCE and mirror 

alignment, aberration in mirrors, deflections in collector structure during tracking, and controller tracking 

errors. Furthermore, the solar flux along the receiver axial and circumferential directions is not uniform, 

because of some or all the effects just mentioned and collector geometry, yet the EES model assumes 

uniformity along both directions. Another difference between the test data and model data could be due to 

the vacuum level in the annulus and presence of hydrogen in the annulus. Both are explored more fully in 

Sections 6.3 and 6.5. Given all the possible sources of inaccuracies, the one-dimensional EES Heat 

Transfer Model does a good job modeling the HCE performance for a short segment of collector, such as 

the size that fits on the AZTRAK rotating test platform. 
51 

a) Efficiency Comparison

(Black Chrome Selective Coating, Solar (NIP) ~ 740 to 930 W/m2)

45

50

55

60

65

70

75

80

50 100 150 200 250 300 350 400 450

Average Temperature Above Ambient (o

C)

Efficiency (%)

EES HCE Code SNL AZTRAK Tests

LS-2 Collector, Syltherm 800

Vacuum

Loss Vacuum

b) Heat Loss Comparison

(Black Chrome Selective Coating, Solar (NIP) ~ 740 to 930 W/m2)

0

20

40

60

80

100

120

140

160

180

200

50 100 150 200 250 300 350 400

Average Temperature Above Ambient (o

C)

[Heat Loss] / [Aperture Area] (W/m2

)

EES HCE Code SNL AZTRAK Tests

LS-2 Collector, Syltherm 800

Loss Vacuum

Vacuum

Figure 5.4a) Efficiency and b) heat loss chart comparing the EES HCE code and SNL AZTRAK test 

data for Luz black chrome selective coating
52 

Heat Loss Comparison

(Black Chrome Selective Coating, No Solar)

0

20

40

60

80

100

120

140

50 100 150 200 250 300 350 400

Average Temperature Above Ambient (o

C)

[Heat Loss] / [Aperture Area] (W/m2

)

EES HCE Code SNL AZTRAK Tests

LS-2 Collector, Syltherm 800

Loss Vacuum

Vacuum

Figure 5.5 Heat loss comparison chart between the EES HCE code and SNL AZTRAK test data for 

the off-sun case and Luz black chrome selective coating 
53 

a) Comparison of Efficiency Trend with Increasing Wind Speed

(No Glass Envelope, Solar (NIP) ~ 800 to 960 W/m2)

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250 300 350 400

Average Temperature Above Ambient (o

C)

Efficiency (%)

EES HCE Code SNL AZTRAK Test

LS-2 Collector, Luz Cermet Selective Coating, Syltherm 800

0

2

4

6

8

10

0

2

4

6

8

wind speed (m/s) = 10

b) Comparison of Heat Loss Trend with Increasing Wind Speed

(No Glass Envelope, No Solar)

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350

Average Temperature Above Ambient (o

C)

[Heat Loss] / [Aperture Area] (W/m2

)

EES HCE Code SNL AZTRAK Tests

LS-2 Collector, Luz Cermet Selective Coating, Syltherm 800

= wind speed (m/sec)

0

2

2

4

4

6

6

8

10

8

10

0

Figure 5.6 a) Efficiency trend and b) heat loss trend charts comparing the EES HCE code and SNL 

AZTRAK test data for varying wind speeds and no glass envelope

5.2 Comparison with KJC Test-Loop Data 

Three days of data—a winter day, a spring day, and a summer day—were used to compare the KJC testloop with the EES model. The test-loop comprised LS-2 type collectors with Luz Cermet HCEs. In the 

EES heat transfer model, all HCEs were assumed to be in good condition and under vacuum. The testloop had a few bad HCEs, but the number was small compared to the total number in the test-loop. The 
54 

KJC Test-Loop Data version of the EES code consists of a two-dimensional model to account for the 

effects of the long receiver (pressure drop, thermal expansion of HTF, HCE support brackets). 

The EES data compared well with the KJC test-loop data, as shown in Figures 5.7 through 5.9, where the 

heat gain and outlet temperatures were compared for each of the three days of data. Like the AZTRAK 

test data comparison, there are plenty of reasons to help explain differences between the model and test 

data. In addition to those already mentioned in Section 5.1, startup and shutdown transients, cloud 

transients, and sections of collectors taken off line will cause differences between the model and test data. 

However, these differences are fairly noticeable in the data – sunrise and sunset periods in all the figures, 

and Figure 5.9 around 12:30 p.m. Also, end losses and shading will contribute to the differences, since the 

KJCOC test-loops track on one axis. All the figures along with general comments are listed in Table 5.3. 

Table 5.3 List of Figures and Comments for the KJC Test-Loop Data Comparison 

Figure Description Comments 

All All 

Transients explain the larger differences between the 

EES data and KJC data during startups and shutdowns. 

Except for the periods of transients, the EES data and 

KJC data showed good agreement. 

5.7 a) 01-02-03, Heat gain 

comparison 

5.7 b) 01-02-03, Outlet temperature 

comparison 

Transients caused by clouds explain the scattering and 

increase in differences during late afternoon. 

5.8 a) 03-22-01, Heat gain 

comparison 

5.8 b) 03-22-01, Outlet temperature 

comparison 

Increase in difference around 9:30 is most likely due to a 

collector segment being taken off line. 

Brief increase in outlet temperature around 11:30 a.m. is 

most likely due to a flow rate adjustment. 

5.9 a) 06-21-01, Heat gain 

comparison 

5.9 b) 06-21-01, Outlet temperature 

comparison 

Increase in difference after 12:30 p.m. is most likely due 

to a collector segment being taken off line. 

Transients caused by clouds explain the scattering and 

increase in differences during late afternoon. 

Because the SEGS troughs track in one axis only (east/west), the solar incident angle effect had to be 

included as an input. The incident angle, which is between the collector’s projected area normal and the 

direct normal incident solar, was determined with another spreadsheet. Its effects are estimated in the 

model with an equation that was developed from testing at SNL (see Section 2.1.6.1). 

In addition to the heat gain and outlet temperature comparisons, power loss components determined from 

the model were plotted and are shown in Figures 5.10 through 5.12. The losses include both heat and 

optical. Furthermore, the optical losses caused by the incident angle effect have been plotted separately. 

This demonstrates how the incident angle effect depends on the time of year – incident angle is smallest 

during the summer months. The plots also reveal that the optical losses are much larger than the heat 

losses. This means any error in the estimated optical losses could overshadow errors in the heat loss. 

Table 5.4 further illustrates this point by listing the actual calculated daily average values with 

percentages of the total losses. The plots also indicate that the heat losses are lower in the winter than the 

summer. This is because the plants typically operate at lower HTF temperatures during the winter (see 

Figure 5.13), when energy demand is lower. Also shown in the plots is the solar insolation to help 

indicate transient periods, such as sunrise, sunset, and cloudy conditions. 
55 

a) Heat Gain Comparison

(UVAC Data 01-02-01)

-500

0

500

1000

1500

2000

2500

3000

3500

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

Heat Gain (W/m)

KJC Test-Loop Data EES Model Data

b) Outlet Temperature Comparison

(UVAC Data 01-02-01)

0

100

200

300

400

500

600

700

800

900

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

Temperature (oF)

KJC Test-Loop Data EES Model Data

Figure 5.7 a) Heat gain and b) outlet temperature charts comparing the EES HCE code and KJC 

test-loop data for 01-02-01
56 

a) Heat Gain Comparison

(UVAC Data 03-22-01)

-500

0

500

1000

1500

2000

2500

3000

3500

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

Heat Gain (W/m)

KJC Test-Loop Data EES Model Data

b) Outlet Temperature Comparison

(UVAC Data 03-22-01)

0

100

200

300

400

500

600

700

800

900

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

Temperature (oF)

KJC Test-Loop Data EES Model Data

Figure 5.8 a) Heat gain and b) outlet temperature charts comparing the EES HCE code and KJC 

test-loop data for 03-22-01 
57 

a) Heat Gain Comparison

(UVAC Data 06-21-01)

-500

0

500

1000

1500

2000

2500

3000

3500

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

Heat Gain (W/m)

KJC Test-Loop Data EES Model Data

b) Outlet Temperature Comparison

(UVAC Data 06-21-01)

0

100

200

300

400

500

600

700

800

900

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

Temperature (oF)

KJC Test-Loop Data EES Model Data

Figure 5.9 a) Heat gain and b) outlet temperature charts comparing the EES HCE code and KJC 

test-loop data for 06-21-01
58 

Power Loss Components

(UVAC Data 01-02-01)

0

500

1000

1500

2000

2500

3000

3500

4000

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

Power Loss (W/m)

0

200

400

600

800

1000

1200

Insolation (W/m2

)

Heat Loss Incident Angle Optical Loss Other Optical Losses Total Optical Losses Solar Insolation

Figure 5.10 Power loss components and solar insolation chart from the KJC test-loop data 

for 01-02-01

Power Loss Components

(UVAC Data 03/22/01)

0

500

1000

1500

2000

2500

3000

3500

4000

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

Power Loss (W/m)

0

200

400

600

800

1000

1200

Insolation (W/m2

)

Heat Loss Incident Angle Optical Loss Other Optical Losses Total Optical Losses Solar Insolation

Figure 5.11 Power loss components and solar insolation chart from the KJC test-loop data 

for 03-22-01
59 

Power Loss Components

(UVAC Data 06/21/01)

0

500

1000

1500

2000

2500

3000

3500

4000

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

Power Loss (W/m)

0

200

400

600

800

1000

1200

Insolation (W/m2

)

Heat Loss Incident Angle Optical Loss Other Optical Losses Total Optical Losses Solar Insolation

Figure 5.12 Power loss components and solar insolation chart from the KJC test-loop data 

for 06-21-01

Table 5.4 Daily Average Losses per Receiver Length 

Optical Losses 

Date (W/m) Heat Loss 

(W/m) Incident Angle Other 

Total Losses 

(W/m)

01-02-01 74 

(7%) 

696 

(69%) 

238 

(24%) 1008 

03-22-01 130 

(17%) 

241 

(31%) 

409 

(52%) 

780 

06-21-01 113 

(18%) 

29 

(5%) 

484 

(77%) 

657 

The conductive losses through the HCE support brackets were evaluated with the KJC Test-Loop version 

of the code. Table 5.5 lists the results of this study. As shown, on average the brackets can contribute 1% 

to 1.5% of the total heat losses from the HCE. And the instantaneous heat loss through the brackets can be 

higher than 3%, depending on the operating and ambient conditions. This study assumes that all the 

brackets are similar in design to those pictured in Figures 2.3 and 2.4 and are not insulated from the HCE 

absorber. 

Table 5.5 Estimated Heat Loss Percentages Caused by HCE Support Brackets 

Percentage of Total Heat Loss Day maximum average 

01/02/01 3.2% 1.3% 

03/22/01 2.8% 0.9% 

06/21/01 2.5% 1.5% 
60 

Average HTF Temperature Above Ambient

0

100

200

300

400

500

600

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

∆Tave (oF)

1/2/2001 3/22/2001 6/21/2001

Figure 5.13 Average HTF temperatures above ambient for each KJC test-loop data set 

6. Design and Parameter Studies 

Numerous examples of design and parameter studies using the HCE performance models implemented in 

EES are shown below. The examples demonstrate the utility of the software program and identify and 

prioritize some methods for improving collector and HCE performance. The HCE performance models do 

not include an optimization routine; but rather, using the EES software, simultaneously solve a set of 

equations that describe the energy balance between the HTF and surrounding ambient conditions, given a 

set of inputs (see Section 2). Thus, the results described below were compiled by rerunning the EES 

program for each change in a design or parameter variable and then copying the outputs to an EXCEL 

spreadsheet for comparisons. 

Twelve design conditions and parameters were evaluated with the EES HCE performance codes. Table 

6.1 summarizes the results. Charts showing the various comparisons are discussed in Sections 6.1–6.12.
61 

Table 6.1 HCE Design and Parameter Study Summary 

Design Option or Parameter Purpose of Evaluation Results and Comm

Absorber Pipe Base Material (Section 6.1) 

To determine influence of less expensive materials (316L, B42 copper, and carbon steel) on the HCE performance 

• Negligible, yet material selection is a

strength, corrosion properties, instaapplication, and cost considerations

Selective Coating (Section 6.2) 

To compare the differences on HCE performance between all the absorber coatings that have been used or have been proposed 

• Chronologically, the improvements i

improved HCE performance. 

• Solel’s proposed UVAC Cermet with 

400oC gives the best HCE performa

• HCE performance would be sensitiv

selective coating optical properties. 

Annulus Gas Type (Section 6.3) 

To evaluate the differences on HCE performance between vacuum, air, argon, and hydrogen in the HCE annulus 

• Vacuum gives the best result. • Filling the annulus with an inert gas • Hydrogen permeation can degrade 

HCE Condition and Wind Speed (Section 6.4) 

To evaluate the loss in HCE performance because of loss vacuum in annulus, or a broken glass envelope. 

• A broken glass envelope on an HCE 

performance results, especially with 

• The wind has little influence on HCE 

annulus vacuum is intact, but does wlost. 

Annulus Gas Type (Section 6.3) 

To evaluate the differences on HCE performance between vacuum, air, argon, and hydrogen in the HCE annulus 

• Vacuum gives the best result. • Filling the annulus with an inert gas • Hydrogen permeation can degrade 

Annulus Pressure (Section 6.5) 

To determine sensitivity of HCE performance to the vacuum level inside the HCE annulus 

• Vacuum levels less than 0.1 torr sho

improvements from the 0.0001 torr 

• HCE performance declines apprecia

100 torr or greater in the annulus. 

• If hydrogen in present, the HCE perf

sensitive to annulus pressure. 

Mirror Reflectance (Section 6.6) 

To determine sensitivity of trough performance to the mirror reflectance. 

• The trough performance drops appr

weighted reflectivity less than 0.9. 

• Keeping mirrors clean is very import

performance. 

Solar Incident Angle (Section 6.7) 

Determine sensitivity of trough performance to solar incident angle. 

• Trough performance is very sensitiv

angle. 
62 

Table 6.1 (cont.) 

Design Option or Parameter Purpose of Evaluation Results and Comm

Solar Insolation (Section 6.8) Determine sensitivity of trough performance to solar insolation. • Trough performance very sensitive t

• Factors such as atmospheric polluta

should be considered when choosin

HTF Flow Rate (Section 6.9) Determine sensitivity of trough performance to HTF flow rate. • HCE performance has weak dependHTF Type (Section 6.10) To determine sensitivity of HCE performance to the type of heat transfer fluid 

• Trough performance has weak depe• Operation of the HCE at higher tem

the HCE performance yet increases efficiency. 

Glass Envelope Diameter (Section 6.11) 

Determine sensitivity of trough performance to glass envelope diameter. 

• Appears to be an optimal diameter t

losses. 

• Influence of diameter on heat loss is 

the annulus is not under vacuum. 

• Clearance for absorber pipe bowing 

Temperature and Heat Flux Variation along Receiver Length (Section 6.12) 

Determine temperature and heat flux profiles along length of receiver. 

• The temperatures along the length o

in a slightly nonlinear fashion. 

• The temperature differences betwee

absorber (T1 and T2), between the in

pipe surfaces (T2 and T3), and betw

glass-envelope surfaces (T4 and T5) 

• The temperature difference between 

glass-envelope (T3 and T4) changes 

fashion. 

• Radiation heat transfer fluxes increa• Heat gain per receiver length decrea

temperature increases. 

• Heat loss per receiver length increa

sectional temperatures increase. 

• Optical losses per unit receiver leng
63 

6.1 Absorber Pipe Base Material 

Four different absorber pipe base materials were evaluated; three stainless steels: 321H, 316L, and 304L; 

and one copper: B42. The materials were chosen to determine the influences of less expensive materials 

on the HCE performance. All the stainless steels evaluated have been used in HCEs; the copper has not. 

As shown in Figure 6.1, the differences in efficiency and heat loss between the materials are negligible. 

However, other factors such as material strength, corrosion properties, installation ease, coating 

application, and costs weigh in to determine the selection of the absorber pipe base material. 

The most recent HCE design uses 321H stainless steel as the absorber base material. It was selected 

because it is very strong and thought to reduce the problem of bending, and because it slows the hydrogen 

permeation rate. 

Both 304L and 316L stainless steels were used in earlier HCE fabrications; however, HCE bowing 

problems led to the belief that these materials were not strong enough. Later, after the decision to use the 

more expensive 321H material was made, the bowing problem was discovered to be due to the 

temperature cycling during plant startups rather than a material strength issue. Originally, the cost 

difference between 316L and 321H was significant. Now that vendors are set up for the 321H polishing 

process, the cost difference may be less important, especially when ordered in large quantities. However, 

316L remains a good candidate for HCEs, and further cost analysis may be warranted [Mahoney 2002; 

Cohen 2002]. 

Carbon steel is not included in the codes, but can be easily added; however, it has its own problems as an 

absorber pipe. For instance, carbon steel is very difficult to use with vacuum. If used in a vacuum, the 

steel would need to go through an expensive and timely process of removing out-gassing, which includes 

keeping the material in an oven at very high temperatures for days. The preparation process for applying 

selective coatings, including cleaning and polishing, is also lengthy and expensive. Furthermore, carbon 

steel may pose corrosion problems at the welded joints. In the past, carbon steel in SEGS plants has been 

used only for very specific applications; for instance, for the vacuum-insulated pipes in SEGS VIII 

headers [Mahoney 2002; Cohen 2002]. 

Copper pipe may cost less, but at current plant operating temperatures, it does not have proper strength 

and could lead to bowing and joint failure issues. However, if operating temperatures are reduced in 

future plants, copper piping may be worth re-evaluating as a means to reduce costs [Mahoney 2002; 

Cohen 2002]. 
64 

a) Efficiency Vs. Absorber Material Type

66

67

68

69

70

71

72

73

74

75

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

304L and 316L 321H B42 Copper Pipe

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

b) Heat Loss Vs. Absorber Material Type

0

50

100

150

200

250

300

350

400

450

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

304L and 316L 321H B42 Copper Pipe

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Figure 6.1 a) Efficiency and b) heat loss charts comparing different HCE absorber pipe base 

material types 

6.2 Selective Coating 

Six different selective coatings were evaluated to compare HCE performances for all the absorber 

coatings that have been used or proposed. The optical properties for the first four types, Luz black 

chrome, Luz Cermet, Solel UVAC Cermet (SNL test a), and Solel UVAC Cermet (SNL test b), were all 
65 

determined in testing at SNL. Two Solel UVAC Cermet coatings are included because the test results 

varied when testing either end of the same HCE [Mahoney 2002]. The last two coating types are 

theoretical, as proposed by Solel. 

The results of the selective coating type comparisons are shown in Figures 6.2 and 6.3. The bar chart in 

Figure 6.2 compares the magnitudes of the heat gain, heat loss, and optical loss for each. The selective 

coating type has a strong influence on each energy rate component, since each coating has different 

emittance and absorptance values. Figure 6.3 shows the efficiency and heat loss as a function of HTF 

temperature. Again, the shapes of the curves reflect the different optical property functions for each 

selective coating type (see Section 2.1.6.1). As shown in Figure 6.3, the 0.07 @ 400o

C proposed coating 

could increase the efficiency by as much as 8.5% (@ 400o

C HTF temperature) over current coatings. The 

figures also illustrate that improvements to the selective coatings have significantly increased collector 

efficiency. 

HCE performance is very sensitive to the optical properties of the selective coatings. Therefore, any 

manufacturing variances in coatings could result in significant energy losses. Further variance testing and 

manufacturing quality checks may thus be warranted. 

Energy Rate Components Vs. Selective Coating Type

0

500

1000

1500

2000

2500

3000

3500

4000

Luz Black Chrome Luz Cermet Solel UVAC Cermet

(SNL test a) 

Solel UVAC Cermet

(SNL test b)

Solel UVAC Cermet

Proposed (0.07

@400C)

Solel UVAC Cermet

Proposed (0.10

@400C)

Energy Rate (W/m)

Heat Loss Heat Gain Optical Loss

DNI 950 W/m2

LS-2 Collector

Therminol VP1

140 gpm

350 o

C

Figure 6.2 Energy rate component chart for each selective coating type 
66 

a) Efficiency Vs. Selective Coating

50

55

60

65

70

75

80

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

Luz Black Chrome Luz Cermet

Solel UVAC Cermet (SNL test a) Solel UVAC Cermet (SNL test b)

Solel UVAC Cermet Proposed (0.07 @400C) Solel UVAC Cermet Proposed (0.10 @400C)

DNI 950 W/m2

LS-2 Collector

Therminol VP1

140 gpm

b) Heat Loss Vs. Selective Coating

0

100

200

300

400

500

600

700

800

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

Luz Black Chrome Luz Cermet

Solel UVAC Cermet (SNL test a) Solel UVAC Cermet (SNL test b)

Solel UVAC Cermet Proposed (0.07 @400C) Solel UVAC Cermet Proposed (0.10 @400C)

DNI 950 W/m2

LS-2 Collector

Therminol VP1

140 gpm

Figure 6.3 a) Efficiency and b) heat loss charts comparing different HCE types 

6.3 Annulus Gas Type 

Figure 6.4 shows the efficiency and heat loss comparison of vacuum, air, argon, and hydrogen in the 

annulus space between the absorber and glass envelope in the HCE. A vacuum in the annulus results in 

the best HCE performance. An inert gas with a low thermal conduction coefficient, such as argon, is the 
67 

next best option shown. However, it provides only a slight improvement over air (argon is ~2.5% gain 

over air at 400o

C HTF temperature). Hydrogen decreases the HCE performance significantly. 

Currently, the HCEs are manufactured with the annulus space under a vacuum of 0.0001 torr (various 

other annulus pressures were evaluated and are discussed in Section 6.5). The problem with a vacuum in 

the annulus space is maintaining it. Historically, this has been a problem in SEGS plants. The main cause 

of vacuum loss has been failure of the glass-to-metal seal, which has been modified recently with 

promising results. Another cause of vacuum loss is hydrogen permeation. 

An inert gas, such as argon, has been considered as a possible alternative to a vacuum. Filling the annulus 

space with an inert gas instead of a vacuum could result in higher reliability and help mitigate the 

hydrogen permeation problem – further study is required to show whether this is true. Pumping an inert 

gas into the annulus space between the absorber and glass envelope could also be an alternative to 

replacing an HCE after the vacuum has been lost. 

Hydrogen is included in the study because it naturally permeates into the annulus space from the HTF, 

and historically has resulted in lower HCE efficiencies. This potential for a drop in HCE performance 

caused by hydrogen permeation was discovered early on in the HCE development. In fact, more than $1 

million has been spent on investigating the hydrogen permeation rate and on developing the getter bridge 

to absorb the gas [Cohen 2002]. At the permeation rates for the current HTF and absorber materials, the 

number of getters installed in an HCE have a lifetime of 30 years before they become saturated. However, 

some of the older HCEs installed in SEGS plants before the study was completed may have had too few 

getters installed to absorb the hydrogen properly [Mahoney 2002]. Furthermore, an attempt was made to 

implement a hydrogen removal (HR) device to provide a longer lifetime for the HCEs. Unfortunately, the 

HR device caused HCE glass envelope breakage, and therefore further development has been halted. As 

this study hints at, hydrogen may be one reason some SEGSs run at a lower than expected efficiency. 

To indicate whether a vacuum has been lost, the current HCE design has a barium marker (getter) in the 

annulus, which turns white when exposed to oxygen. Unfortunately, the barium marker is not sensitive to 

hydrogen. If the hydrogen getter bridge does not remove the hydrogen sufficiently, a vacuum could be 

replaced by hydrogen without an indication. To date, this is not known to be a problem; however, it does 

raise the following questions: 

• At what hydrogen concentration and annulus pressure does the HCE performance degrade? 

• At what pressure and concentration is the getter bridge in equilibrium with the permeation rate (see 

Section 6.5). 

• Is a different vacuum loss indicator, such as one based on temperature, needed? 
68 

a) Efficiency Vs. Annulus Gas Type

50

55

60

65

70

75

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

Vacuum Air Argon Hydrogen

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

b) Heat Loss Vs. Annulus Gas Type

0

200

400

600

800

1000

1200

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

Vacuum Air Argon Hydrogen

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Figure 6.4 a) Efficiency and b) heat loss charts comparing different annulus gas types 

6.4 HCE Condition and Wind Speed 

The charts in Figure 6.5 compare three HCE conditions – vacuum, loss vacuum, and broken glass 

envelope – all at different wind speeds. All three conditions have been experienced in SEGS plants. With 

a vacuum in the annulus, the heat losses from the HCE are affected little by wind. Wind has more 

influence when the vacuum is lost. The model shows that the decrease can be as large as 12% over the 
69 

vacuum case (@ 400o

C HTF temperature and 20 mph wind). If the glass envelope is broken, the HCE 

performance can drop by as much as 105% over the vacuum case (@ 400o

C and 20 mph wind). However, 

this decrease is probably smaller than modeled, since the wind will most likely be obstructed and the wind 

direction will be something other than normal to the HCE (see Section 5.1). 

a) Efficiency Vs. HCE Condition and Wind Speed

0

10

20

30

40

50

60

70

80

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

0 mph 5 mph 10 mph 20 mph

Vacuum

Loss Vacuum

Broken Envelope

DNI 950 W/m2

LS-2 Collector

Solel UVAC 

Cermet

Therminol VP1

140 gpm

b) Heat Loss Vs. HCE Condition and Wind Speed

0

500

1000

1500

2000

2500

3000

3500

4000

4500

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

0 mph 5 mph 10 mph 20 mph

Vacuum

Loss Vacuum

Broken Envelope

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Figure 6.5 a) Efficiency and b) heat loss charts comparing different HCE conditions as functions 

of wind speed
70 

6.5 Annulus Pressure 

Figures 6.6–6.8 show the changes in HCE performance as a function of the pressure in the annulus space 

between the absorber and glass envelope. Both air and hydrogen as the annulus gases are shown in the 

charts. The question being explored with this study is: “At what pressure deviation from the 0.0001 torr 

(manufactured pressure) does heat losses become a problem?” As the charts show, annulus pressures 

between 0.0001 torr and 0.1 torr have slight influence on the HCE performance. Pressures above 0.1 torr, 

however, can significantly affect the HCE performance, especially if hydrogen has permeated into the 

annulus space. In fact, partial vacuums from 1 torr to 760 torr decrease performance from 1% to 8.5% for 

air (@ 400o

C), and about triple those values for hydrogen. 

Heat Loss Vs. Annulus Pressure

200

300

400

500

600

700

800

900

0.0001 0.001 0.01 0.1 1 10 100 1000

Annulus Pressure (torr)

Heat Loss (W/m)

Air Hydrogen

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

350 o

C

Figure 6.6 Heat loss chart as a function of annulus pressure for air and hydrogen as the 

annulus gas

From the evidence in the charts, if some HCEs installed in the SEGS plants are at partial vacuum, the 

energy losses could become significant. Since there is no vacuum measurement on the HCEs, numerous 

questions go unanswered: 

• What are the actual pressures in the HCEs, both when they arrive from the manufacturer and after 

they’ve been installed over a period of time? 

• How much annulus pressure variance is there from HCE to HCE? 

• At what annulus pressure does the barium indicator trigger a lost vacuum? 

• What is the annulus pressure when the hydrogen concentration is in equilibrium with the getter 

bridge? 

Developing a means to measure the annulus pressures could help answer these questions. One such 

possible means may be to measure the outer glass envelope surface with an infrared camera and compare 

the results with the model predictions. Development of other means may be warranted. 
71 

a) Efficiency Vs. Annulus Pressure (Air)

55

57

59

61

63

65

67

69

71

73

75

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

0.0001 torr 0.1 torr 1 torr 10 torr 100 torr 400 torr 760 torr

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

b) Heat Loss Vs. Annulus Pressure (Air)

0

100

200

300

400

500

600

700

800

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

0.0001 torr 0.1 torr 1 torr 10 torr 100 torr 400 torr 760 torr

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Figure 6.7 a) Efficiency and b) heat loss charts for different annulus pressures and air as the 

annulus gas
72 

a) Efficiency Vs. Annulus Pressure (Hydrogen)

50

55

60

65

70

75

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

0.0001 torr 0.1 torr 1 torr 10 torr 100 torr 400 torr 760 torr

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

b) Heat Loss Vs. Annulus Pressure (Hydrogen)

0

200

400

600

800

1000

1200

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

0.0001 torr 0.1 torr 1 torr 10 torr 100 torr 400 torr 760 torr

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Figure 6.8 a) Efficiency and b) heat loss charts for different annulus pressures and hydrogen as 

the annulus gas

Being able to answer annulus pressure questions could lead to important HCE performance 

improvements. For instance, manufacturing quality control may need to be improved to tighten the 

annulus pressure variance from HCE to HCE. Also, perhaps the barium indicator does not turn white until 

the annulus pressure is close to atmospheric. If this is the case, a more precise vacuum indicator may be 

needed. Furthermore, actual HCE annulus pressure measurements could warrant a closer review of the 
73 

previous getter bridge studies, and thus an improvement on mitigating the hydrogen permeation into the 

annulus. 

6.6 Mirror Reflectance 

Mirror reflectance has a strong influence on collector efficiency (see Figures 6.9 and 6.10). For instance, 

at an HTF temperature of 400o

C, a decrease of 0.15 in reflectance results in efficiency decrease of 24.5%. 

This indicates that any error in mirror reflectance could easily explain the differences between the model 

predictions and field data (see Section 5). For example, a 5% error in reflectance could explain a 7% 

decrease in efficiency (@ 400o

C). This study also hints at the importance of keeping the trough mirrors 

and the glass envelopes on the HCEs clean. 

Reflectance is currently measured with an instrument that measures a very small point on the mirror with 

each reading; therefore, the technician taking the measurements is trusted to take a large enough sampling 

to represent the true average reflectance of a collector, or an entire SEGS plant. Does this current 

reflectance measuring technique give a dependable value of average mirror reflectance? Or, Should a new 

instrument be developed that takes a reading over a larger surface area to better estimate the overall 

mirror reflectance and reduce error associated with the measurement statistical dependency? 

Figure 6.9 shows that the HCE heat loss has only a slight dependence on mirror reflectance, whereas the 

heat gain and optical losses have a strong dependence. This is because the average HTF temperature and 

other parameters that influence the HCE heat loss directly were held constant during this study. This 

would not be completely true in an actual plant, where the HTF temperature would eventually drop with 

decreasing solar input. However, in actual operations small changes in the solar input would be 

compensated by changing the HTF flow rate, and thus maintaining the HTF temperature (and thus heat 

loss). 

Both the one- and two-dimensional model results were plotted in Figure 6.10 to further illustrate the 

differences between the two models. The two-dimensional model captures some of the nonlinearity in the 

temperature profiles along the longitudinal direction of the HCE, and includes the energy losses 

associated with the HTF pressure drop and thermal expansion of the HTF. As the total receiver length 

increases, the difference between the one- and two-dimensional models would become more prominent. 
74 

Energy Rate Components Vs. Solar Weighted Mirror Reflectivity

(Average HTF Temperature = 350 oC)

0

500

1000

1500

2000

2500

3000

3500

4000

0.8 0.85 0.9 0.935

Reflectivity

Energy Rate (W/m)

Heat Loss Heat Gain Optical Loss

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Figure 6.9 Energy rate component chart for different solar weighted reflectivities 
75 

a) Efficiency Vs. Solar Weighted Mirror Reflectivity

(Comparison between the One and Two Dimensional Models)

50

55

60

65

70

75

80

0 50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

0.8 0.85 0.9 0.935 0.8 2D 0.85 2D 0.9 2D 0.935 2D

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

L = 779.52 m

b) Heat Loss Vs. Solar Weighted Mirror Reflectivity

(Comparison between the One and Two-Dimensional Models)

0

50

100

150

200

250

300

350

400

450

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

0.8 0.85 0.9 0.935 0.8 2D 0.85 2D 0.9 2D 0.935 2D

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

L = 779.52 m

Figure 6.10 a) Efficiency and b) heat loss charts comparing different solar weighted reflectivities. 

Both the one- and two-dimensional model results are included for comparison.
76 

6.7 Solar Incident Angle 

The charts in Figures 6.11 and 6.12 show the sensitivity of HCE performance to solar incident angle. The 

HCE heat loss is affected only slightly in this study, since the average HTF temperature and other 

parameters directly affecting the heat loss are held constant. However, the solar incident angle has a large 

impact on the optical loss and heat gain. Specifically, with an incident angle of 30o

 the performance is 

reduced by approximately 15% and at 60o

 by approximately 65%. 

Energy Components Vs. Solar Incident Angle

(Average HTF Temperature = 350 oC)

0

500

1000

1500

2000

2500

3000

3500

4000

0 deg 10 deg 20 deg 30 deg 40 deg 50 deg 60 deg

Energy Rate (W/m)

Heat Loss Heat Gain Optical Loss

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Figure 6.11 Energy rate component chart as a function of different solar incident angles 

For a single-axis tracking system, like the SEGS plants in California, the solar incident angle depends on 

location and time of year. Figure 6.13 shows where the actual solar incident angles for a day in spring, 

summer, and winter. The largest incident angles occur during the winter months; the smallest angles 

occur during the summer. 

Although previous studies have shown that equipment costs associated with tilting the collectors offset 

the money gained in additional energy, new studies may be worth evaluating now that the actual incident 

angle losses are better understood. Furthermore, other options to reduce the solar incident angle effects 

such as roughening the outer surface of the absorber [Duffie and Beckman 1991] or developing coatings 

that are less sensitive to incident angle may be beneficial. 
77 

a) Efficiency Vs. Solar Incident Angle

0

10

20

30

40

50

60

70

80

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

0 deg 10 deg 20 deg 30 deg 40 deg 50 deg 60 deg

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

b) Heat Loss Vs. Solar Incident Angle

0

50

100

150

200

250

300

350

400

450

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

0 deg 10 deg 20 deg 30 deg 40 deg 50 deg 60 deg

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Figure 6.12 a) Efficiency and b) heat loss charts comparing different solar incident angles 
78 

Incident Angle

0

10

20

30

40

50

60

70

0:00 2:24 4:48 7:12 9:36 12:00 14:24 16:48 19:12 21:36 0:00

Time

degrees

1/2/2001 3/22/2001 6/21/2001

Figure 6.13 Chart showing solar incident angles for three different times of year from actual data 

for a north-south oriented SEGS plant in Kramer Junction, California 

6.8 Solar Insolation 

Figures 6.14 and 6.15 show the influence solar insolation has on HCE performance. Once again, the 

parameters with the strongest influence on HCE heat loss are held constant, so only the optical losses and 

heat gain increase significantly with increasing solar insolation. As shown, the HCE performance 

improves with increasing solar insolation. Therefore, besides clouds, sites with high pollutants and other 

particulates in the air – such as dust and sand – would have a negative impact on HCE performance. This 

study hints at the importance of choosing a SEGS plant site away from industrial areas with a lot of 

pollution, areas with high automotive traffic, or areas prone to frequent dust storms. 
79 

Energy Rate Components Vs. Solar Insolation

0

500

1000

1500

2000

2500

3000

3500

4000

300 (W/m2) 500 (W/m2) 700 (W/m2) 900 (W/m2) 1100 (W/m2)

Energy Rate (W/m)

Heat Loss Heat Gain Optical Loss

LS-2 Collector

Solel UVAC 

Cermet

Therminol VP1

140 gpm

Figure 6.14 Energy rate component chart for different solar insolation values 
80 

a) Efficiency Vs. Solar Insolation

50

55

60

65

70

75

80

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

300 (W/m2) 500 (W/m2) 700 (W/m2) 900 (W/m2) 1100 (W/m2)

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

b) Heat Loss Vs. Solar Insolation

0

50

100

150

200

250

300

350

400

450

500

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

300 (W/m2) 500 (W/m2) 700 (W/m2) 900 (W/m2) 1100 (W/m2)

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Figure 6.15 a) Efficiency and b) heat loss charts for different solar insolation values 

6.9 HTF Flow Rate 

Figure 6.16 shows that the HTF flow rate has little effect on HCE performance for the range of flow rates 

evaluated (100 gpm to 160 gpm). The figure includes the one- and two-dimensional model results. As 

expected, the two-dimensional model shows that the HCE performance depends more on the flow rate 

than the one-dimensional model. This is because the pressure loss and thermal expansion effects on the 
81 

HTF are included in the two-dimensional model. A more detailed study of the flow rate effects should 

include the power cycle side of the plant and losses through pumps and other equipment and piping. An 

exergy analysis would be more beneficial than an energy analysis for this type of study, since it would 

identify components that have the greatest influence on the HCE performance (and thus the most room for 

improvement). 

a) Efficiency Vs. HTF Flow Rate

(Comparison between the One and Two-Dimensional Models)

65

66

67

68

69

70

71

72

73

74

75

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

100 gpm 120 gpm 140 gpm 160 gpm 100 gpm 2D 120 gpm 2D 140 gpm 2D 160 gpm 2D

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

Reflectivity = 0.935

L = 779.52 m

b) Heat Loss Vs. HTF Flow Rate

(Comparison between the One and Two-Dimensional Models)

0

50

100

150

200

250

300

350

400

450

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

100 gpm 120 gpm 140 gpm 160 gpm 100 gpm 2D 120 gpm 2D 140 gpm 2D 160 gpm 2D

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

Reflectivity = 0.935

L = 779.52 m

Figure 6.16 a) Efficiency and b) heat loss charts comparing different flow rates and including both 

the one- and two-dimensional models
82 

6.10 HTF Type 

The charts in Figure 6.17 reveal that the HTF type has little effect on HCE performance. However, each 

fluid type has a range of recommended operating temperatures, and efficiency drops with increasing HTF 

temperature. Thus, fluids that can operate at lower temperatures would improve HCE performance. Each 

HTF type has other advantages. For instance, HTF types like salts can be used as thermal storage media, 

but also could require additional heating during off-sun hours to prevent solidification in piping and 

equipment. Also, cost and availability could dictate which HTF to use. Since power cycle efficiencies 

increase as the working fluid temperature increases, a complete study would need to include both the 

solar and power cycle sides. 
83 

a) Efficiency Vs. HTF Type

(Comparison between the One and Two-Dimensional Models)

50

55

60

65

70

75

80

50 150 250 350 450 550

Average HTF Temperature (o

C)

Efficiency (%)

Therminol VP1 Xceltherm 600 Syltherm 800 60-40 Salt Hitec XL Salt

Therminol VP1 2D Xceltherm 600 2D Syltherm 800 2D 60-40 Salt 2D Hitec XL Salt 2D

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

140 gpm

L = 779.52 m

b) Heat Loss Vs. HTF Type

(Comparison between the One and Two-Dimensional Models)

0

200

400

600

800

1000

1200

0 100 200 300 400 500 600

Average HTF Temperature (o

C)

Heat Loss (W/m)

Therminol VP1 Xceltherm 600 Syltherm 800 60-40 Salt Hitec XL Salt

Therminol VP1 2D Xceltherm 600 2D Syltherm 800 2D 60-40 Salt 2D Hitec XL Salt 2D

DNI 950 W/m2

LS-2 Collector

Solel UVAC 

Cermet

140 gpm

L = 779 52 m

Figure 6.17 a) Efficiency and b) heat loss charts comparing different HTF types and including 

both the one- and two-dimensional models 
84 

6.11 Glass Envelope Outer Diameter 

The charts in Figure 6.18 show that: 

• The glass envelope diameter has an optimal size to minimize heat losses from the HCE. 

• This effect is most sensitive when the annulus is not under vacuum. 

• The optimization reflects the fact that as the diameter is increased, the annulus gap between the 

absorber and glass envelope increases, which decreases the heat transfer between the absorber and 

glass envelope, but increases the surface area for heat transfer to the environment. 

• Decreasing the glass envelope diameter increases the heat transfer across the annulus gap, but 

decreases the surface area for heat transfer to the environment. 

Another problem with a too-small annulus gap is that it could cause tolerance problems with the absorber 

pipe, since the absorber pipe will tend to bow slightly when heated. A more detailed glass envelope 

optimization study would need to include the thermal deflections of the absorber. 
85 

a) Efficiency Vs. Glass Envelope Outer Diameter

55

57

59

61

63

65

67

69

71

73

75

0 50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Efficiency (%)

0.08 m 0.092 m 0.105 m 0.115 m 0.14 m 0.165 m

Vacuum

Loss Vacuum

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

b) Heat Loss Vs. Glass Envelope Outer Diameter

0

100

200

300

400

500

600

700

800

900

1000

50 100 150 200 250 300 350 400 450

Average HTF Temperature (o

C)

Heat Loss (W/m)

0.08 m 0.092 m 0.105 m 0.115 m 0.14 m 0.165 m

Vacuum

Loss Vacuum

DNI 950 W/m2

LS-2 Collector

Solel UVAC 

Cermet

Therminol VP1

Figure 6.18 a) Efficiency and b) heat loss charts comparing different glass envelope diameters 

6.12 Parameter Variation along Receiver Length 

The chart in Figure 6.19 shows the temperature variations along the length of the solar receiver (total 

length of 779.5 m) for each temperature in the cross-section of an HCE (also see Figure 2.1), as modeled 

with the two-dimensional HCE performance model. As expected, the temperatures along the length of the 

receiver increase in a slightly nonlinear fashion because: 
86 

• The radiation heat transfer functions are nonlinear. 

• Heat gain per receiver length decreases as the HTF temperature increases. 

• The heat loss per receiver length increases as the HCE cross-sectional temperatures increase. 

Also, the temperature differences between the HTF and absorber (T1 and T2), between the inner and outer 

absorber pipe surfaces (T2 and T3), and between the inner and outer glass-envelope surfaces (T4 and T5) 

all remain constant, since the heat transfer coefficients are constants (see Sections 2.1.1, 2.1.2, and 2.1.4). 

However, the temperature difference between the absorber and glass-envelope (T3 and T4) changes in a 

slightly nonlinear fashion because: 

• The selective coating emissivity is a function of temperature (see Section 2.1.6.1). 

• The radial heat transfer is nonlinear. 

Temperatures Along Length of Receiver

0

50

100

150

200

250

300

350

400

450

39.0 116.9 194.9 272.8 350.8 428.7 506.7 584.6 662.6 740.5

Position Along Receiver (m)

Temperature (oC)

T_1 T_2 T_3 T_4 T_5

DNI 950 W/m2

LS-2 Collector

Solel UVAC Cermet

Therminol VP1

140 gpm

L = 779.5 m

Figure 6.19 HCE cross-section temperature chart at position along receiver length 

The heat loss, heat gain, and optical loss along the length of the receiver are shown in Figure 6.20. Both 

the heat loss and heat gain reflect the trends shown with the temperature variations in Figure 6.19. The 

optical loss, however, is constant; since all the optical properties in the effective optical efficiency terms 

are assumed to be independent of temperature (see Section 2.1.6.1). 
87 

Energy Rate Components Along Receiver Length

0

500

1000

1500

2000

2500

3000

3500

39.0 116.9 194.9 272.8 350.8 428.7 506.7 584.6 662.6 740.5

Position Along Receiver (m)

Energy Rate (W/m)

heat loss heat gain optical loss

DNI 950 W/m2, LS-2 Collector, Solel UVAC Cermet, Therminol VP1, 140 gpm, L = 779.5 m

Figure 6.20 Energy rate components chart as a function of position along receiver (total receiver 

length = 779.5 m) 

6.13 Recommendations 

Table 6.2 lists recommendations based on the HCE design and parameter study. Many of the 

recommendations have already been mentioned throughout this report; however, where justified, further 

explanation is given. 

Understanding the manufacturing tolerances and degradation rates of the optical properties and HCE 

annulus vacuum levels is important (see Sections 6.2, 6.3, and 6.5). The selective coating optical 

properties may vary over the length of the HCE and from HCE to HCE. The testing done shows this to be 

the case. Since the optical properties have a strong influence on HCE performance, further testing of the 

selective coating variation and testing to determine the optical property degradation is recommended. This 

testing should include determining variations throughout a SEGS plant, especially one that has been 

operating for many years. 

Some HCEs in a SEGS plant are likely to have annulus pressures higher than the specified 0.0001 torr, 

because of manufacturing inconsistencies or hydrogen permeation (see Sections 6.3 and 6.5). If this is so, 

developing a means to measure the HCE annulus pressure on installed HCEs is important. One possible 

technique could be to measure the temperature of the outer glass envelope surface using an infrared 

camera and evaluating the data with the HCE performance model; however, this technique needs further 

evaluation. 

The model tends to overpredict wind-induced heat losses (see Section 6.4). Reducing forced convection 

effects by one-half in the model may lead to better results, especially for the missing glass envelope case. 

However, more comparisons with field test data need to be done. 
88 

Solar incident angle has a strong influence on HCE performance (see Section 6.7). Tilting the troughs 

slightly (~ 5o

) along the rotating axis, or possibly changing the absorber surface texture [Duffie 2000] or 

changing the selective coating properties to be less sensitive to solar incident angle could be cost 

effective. Conducting more incident angle tests on additional HCE types could also be beneficial. The 

HCE performance model could be used to give more insight to these investigations. 

Section 6.10 discusses how the HTF type has little effect on the HCE performance for any one 

temperature; however, the trends indicate that the HCE performance decreases with increasing HTF 

temperature. This indicates that methods for decreasing the HTF temperature could be beneficial, 

especially if it could be done in such a way as not to decrease the power cycle efficiency. A possible 

solution might be to run the power cycle side with a fluid with a lower vapor pressure, such as ammonia 

or a hydrocarbon. 

Finally, heat transfer enhancing mechanisms such as coiled spring inserts, twisted tape inserts, helical 

ribs, or roughening the inner absorber surface, may be worth evaluating. All these mechanisms would 

increase the convection surface area and enhance turbulent flow [Incropera and DeWitt 1990]. The 

current HCE performance model cannot be used to evaluate heat transfer enhancing devices without 

significant changes. However, an alternative to developing a more complex model may be to conduct 

field testing. For instance, measuring the radial temperature profile of the HTF and the temperature 

difference between the absorber inner surface and bulk HTF temperature, which would help determine 

what improvements, if any, a heat transfer enhancement device could make. If there is room for 

improvement, additional field testing of flow enhancement devices may continue to be less expensive 

than developing a more complex heat transfer analysis model. Of course any advantage to increasing the 

heat gain by an enhancement device would have to be balanced with the increase in pumping losses. 
89 

Table 6.2 Recommendations from HCE Design and Parameter Study 

Design Option or Parameter Recommendations 

Absorber Pipe Base Material (Section 6.1) 

• Re-evaluate 316-L. • Add carbon steel to study. 

Selective Coating (Section 6.2) 

• Re-evaluate optical properties at elevated temperatures. • Update optical properties. • Determine optical property variation. • Evaluate optical property degradation. • Determine the manufacturing variances. 

Annulus Gas Type (Section 6.3) 

• Add additional inert gasses to study (xenon, neon, etc.) • Determine effect of inert gases to hydrogen permeation. • Develop a better device for determining loss of vacuum (or partial vacuum). • Determine hydrogen concentration level when in equilibrium with getter bridge an

HCE Condition and Wind Speed (Section 6.4) 

• Reduce wind effects by approximately one-half. • Re-evaluate wind effects with different annulus pressures. • Re-evaluate wind effects with hydrogen as the annulus gas. 

Annulus Pressure (Section 6.5) 

• Determine annulus pressure variations throughout a SEGS plant (thermal imagin• Determine annulus pressure with hydrogen permeation. • Determine what the manufacturing variances are. 

Mirror Reflectance (Section 6.6) • Develop improved reflectance measuring techniques. • Measure true optical errors. Solar Incident Angle (Section 6.7) 

• Run a study simulating trough tilt angles on actual field test data. • Evaluate changes to absorber outer surface on reducing incident angle effects (ro

threads, different selective coating characteristics, etc.). 

HTF Type (Section 6.10) 

• Develop means to conduct HTF property testing in field. • Evaluate combine cycle operation to reduce HTF temperature (HTF to ammonia 

power cycle fluid). 

• Evaluate alternative working fluid to reduce HTF temperature (ammonia or hydro

Glass Envelope Diameter (Section 6.11) • Re-evaluate sizing now that HCE performance has been improved. Temperature and Heat Flux Variation along Receiver Length (Section 6.12) 

• Conduct an optimization study on length. 

General • Complete HCE testing with HTF heat transfer enhancing devices (coiled spring, t

insert, roughened surface) 
90 

7. Conclusion 

This report described the development, validation, and use of an HCE performance software model 

implemented in EES. All equations, correlations, and parameters used in the model were discussed in 

detail. The model was implemented in EES in four different versions. Two were developed for HCE 

design and parameter studies, and two for evaluating field test data. Both one- and two-dimensional 

analyses were used in the codes. Discussions of each version of the codes included relevant EES diagram 

windows, parameter tables, and lookup tables. The model was then shown to be accurate by comparing 

with actual HCE field test data. Following this, the model versatility was demonstrated by conducting 

various design and parameter studies. This also demonstrated how the HCE performance model 

implemented in EES could be used as a development tool for improving the HCE performance. Several 

recommendations were drawn from the design and parameter study (see Table 6.2). Several suggestions 

for improving the HCE performance model were also discussed. 
91 

APPENDIXES 
92 

A. Lookup Table References 

Argon: 

Ahlberg, K., ed. (1985) AGA Gas Handbook. Stockholm, Sweden: Almqvist & Wiksell International. 

Hitec XL:

Data from EXCEL spreadsheet provided by Mary Jane Hall of NREL and 

Coastal Chemical Company. “HITEC Heat Transfer Salt.” Sales Data Sheet. Pasadena, Texas. 

Nitrate Salt (60% NaNO3

, 40% KNO3): 

Data from EXCEL spreadsheet provided by Mary Jane Hall of NREL and 

Zavoico, A. B. (July 2001). Solar Power Tower, Design Basis Document, Revision 0. SAND2001-2100. 

Work performed by Nextant, San Francisco, California, and SANDIA National Laboratories, 

Albuquerque, New Mexico. 

Syltherm 800:

“Syltherm 800, Silicon Heat Transfer Fluid.” (November 2001). Form No. 176-01469-1101 AMS. The 

Dow Chemical Company, http:// www.dowtherm.com. 

Therminol VP1:

“Therminol, Heat Transfer Fluids, Computed Properties of Therminol VP-1 vs. Temperature (Liquid 

Phase).” (2001). Solutia Inc., www.therminol.com. 

Therminol 59:

“Therminol 59, Heat Transfer Fluid by Solutia.” (2001). Technical Bulletin 7239271. Solutia Inc., 

www.therminol.com. 

Therminol 66:

“Therminol 66, Heat Heat Transfer Fluid by Solutia.” (2001). Technical Bulletin 7239146C. Solutia Inc., 

www.therminol.com. 

Therminol XP:

“Therminol XP, Heat Heat Transfer Fluid by Solutia.” (2001). Technical Bulletin 7239262. Solutia Inc., 

www.therminol.com. 

Dowtherm Q:

“Dowtherm Q, Heat Transfer Fluid, Product Technical Data.” (June 1997). The Dow Chemical 

Company., www.dow.com/heattrans. 

Dowtherm RP:

“Dowtherm RP, Heat Transfer Fluid, Product Technical Data.” (November 1996). The Dow Chemical 

Company., www.dow.com/heattrans. 

Xceltherm 600: 

“Exceltherm 600, Engineering Properties.” (1998). Radco Industries, Inc., http://www.radcoind.com. 
93 

B. Optical and Material Property References 

Luz Black Chrome (SNL test): 

Visual Basic Heat Transfer Code and based on test data from Rod Mahoney of SNL, 8/93. 

Luz Cermet (SNL test): 

Visual Basic Heat Transfer Code and based on test data from Rod Mahoney of SNL, 11/06/98. 

Solel UVAC Cermet (SNL test a), Solel UVAC Cermet (SNL test b), Solel UVAC Cermet (SNL test 

avg): 

Tests conducted by Rod Mahoney of SNL and presented in a Power Point presentation dated April 23, 

2001, “HCE Issues; Cermet Preliminary Results Optical Properties & Construction Forum 2001.” 

Solel UVAC Cermet (0.07 @ 400C), Solel UVAC Cermet (0.15 @ 400C): 

Table in 

Price, H. (2002) Concentrated Solar Power Use in Africa. NREL/TP. Golden, Colorado: National 

Renewable Energy Laboratory. 

Glass Envelope (borosilicate glass): 

Estimated from Plots in 

Touloukian, Y. S., D. P. DeWitt, eds. (1972). Radiative Properties, Nonmetalic Solids. Thermophysical 

Properties of Matter, Vol. 8, New York: Plenum Publishing.

304L, 316L Stainless Steel: 

Linear best fit of data from tables in 

Touloukian, Y. S., R. W. Powell; et al., eds. (1970). Thermal Conductivity, Metallic Elements and Alloys. 

Thermophysical Properties of Matter, Vol. 1, New York: Plenum Publishing. 

321H Stainless Steel: 

Linear best fit of data from tables in 

Davis, J. R., ed. (2000). Alloy Digest, Sourcebook, Stainless Steels. Materials Park, Ohio: ASM 

International. 

and 

ASM Handbook Committee (1978). Metals Handbook. Properties and Selection – Stainless Steels, Tool 

Materials, and Special Purpose Metals, Vol. 3, Metals Park, Ohio: American Society for Metals. 

B42 Copper Pipe: 

ASM Handbook Committee (1978). Metals Handbook. Properties and Selection – nonferrous alloys and 

pure metals, Vol. 2, Metals Park, Ohio: American Society for Metals. 

Gas Molecular Diameters:

Marshal, N., transl. (1976). Gas Encylopedia. New York: Elsevier. 
94 

C. Heat Transfer Fluid Change in Enthalpy 

An estimate for the change in enthalpy of a heat transfer fluid (HTF) flowing through a heat collection 

element (HCE) for a solar trough collector can be derived by starting with the state postulate that any two 

independent, intensive, thermodynamic properties are sufficient to establish the stable thermodynamic 

state of a pure, simple, compressible substance [Howell and Buckius 1987]. This postulate allows the 

differential change in enthalpy and entropy to be determined as functions of temperature and pressure, 

dP

P

h dT

T

h dh T P

P T





 





∂

∂  + 

 





∂

∂ ( , ) = (C.1) 

dP

P

s dT

T

s ds T P

P T





 





∂

∂  + 

 





∂

∂ ( , ) = (C.2) 

Substituting equation C.2 into Gibb’s equation dh = Tds +νdP [Çengel and Boles 2002], gives 

dP

P

s dT T

T

s dh T

P T





 



 



 





∂

∂  + + 

 





∂

∂ = ν (C.3) 

Equating the coefficients of dT and dP in Equations C.1 and C.3 and using the definition of specific 

heat at constant pressure 

P

P T

h

c 



 





∂

∂ ≡ [Howell and Buckius 1987], gives 

T

c

T

s P

P

 = 

 





∂

∂ (C.4a) 

T P T

s T

P

h 



 





∂

∂  = + 

 





∂

∂

ν (C.4b) 

Using the fourth Maxwell relation P T T P

s 



 





∂

∂  = − 

 





∂

∂ ν [Çengel and Boles 2002], results in 

T T P

v T

P

h 



 





∂

∂  = − 

 





∂

∂

ν (C.5) 

Substituting all the above results into Equation C.1 and integrating determines the change in enthalpy 

∫ ∫ 



 



 



 





∂

∂ ∆ = + − dP

T

h T P c dT T

P

P

ν ( , ) ν (C.6) 

Equation C.6 can be simplified if the specific heat and specific volumes are linear in the regions of 

interest and if the second term on the right is negligible compared to the first term. The specific heats for 

all the HTFs used in the HCE performance models are plotted in Figure C.1 and the specific volumes are 
95 

plotted in Figure C.2. The change in HTF temperature along the length of the receiver is approximately 

15o

C for each 80 m of length (recommended increment for a two-dimensional heat transfer model). As 

shown, both the specific heat and specific volume are nearly linear for all the HTF’s over small 

temperature increments (< 25o

C). Therefore, both specific heat and specific volume can be treated as 

constants at there average values for relatively small HTF temperature changes. 

Specific Heat at Constant Pressure

1300

1500

1700

1900

2100

2300

2500

2700

2900

3100

0 100 200 300 400 500 600

T (oC)

Cp (J/kg-K)

Therminol VP1 Syltherm 800 Xceltherm 600 Salt (60-40) Hitec XL Therminol 59

Therminol 66 Therminol XP Dowtherm Q Dowtherm RP

Figure C.1 Specific heat at constant pressure for each HTF used in the HCE performance model 
96 

Specific Volume

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0 100 200 300 400 500 600

T (oC)

Specific Volume (m3/kg)

Therminol VP1 Syltherm 800 Xceltherm 600 Salt (60-40) Hitec XL Therminol 59

Therminol 66 Therminol XP Dowtherm Q Dowtherm RP

Figure C.2 Specific volume for each HTF used in the HCE performance model 

An order of magnitude study can be used to determine whether the second term in Equation C.6 is 

negligible in comparison to the first term. To accomplish this, all the HTFs used in the HCE performance 

model have been listed in Table C.1, along with the orders of magnitude of the relevant terms. The 

temperature ranges shown in the second column have been included to show where the orders of 

magnitude change relative to temperature for each term. As shown in the table, the second term in 

Equation C.6 (last column) is consistently significantly smaller then the first term (third column), 10° << 

104

; therefore the second term in Equation C.6 can be neglected without introducing any significant error. 

The change in enthalpy then becomes 

∆h ≈ c p,ave∆T (C.7) 

In most cases, for liquids the know specific heat will be cp. However, if cv is the known value, it can be 

substituted for cp with little error, since the specific heats will differ only slightly for liquids. 
97 

Table C.1 Orders of Magnitude of Change in Enthalpy Terms in Equation C.6 for Each HTF

HTF Temperature 

Range 

(Cp,ave)(∆T) 

[J/kg]

vave

[m3

/kg]

Tave(∆v/∆T) 

[kg/m-s]

(vave - Tave(∆v/∆T))∆P 

[J/kg]

Low 104 10-4 10-4 100

Mid 104 10-3 10-4 Therminol VP1 0 

High 104

 10-3 10-3 100

Low 104 10-3 10-4 100

Mid 104 10-3 10-4 100 Therminol 59 

High 104

 10-3 10-4 100

Low 104 10-4 10-4 100

Mid 104 10-3 10-4 100 Therminol 66 

High 104

 10-3 10-3 100

Low 104 10-3 10-4 100

Mid 104 10-3 10-4 100 Therminol XP 

High 104

 10-3 10-3 100

Low 104 10-3 10-4 100

Mid 104 10-3 10-4 100 Dowtherm Q 

High 104

 10-3 10-4 100

Low 104 10-4 10-4 100

Mid 104 10-3 10-4 100 Dowtherm RP 

High 104

 10-3 10-4 100

Low 104 10-3 10-4 100

Mid 104 10-3 10-4 Syltherm 800 0 

High 104

 10-3 10-3 100

Low 104 10-4 10-5 100

Mid 104 10-4 10-4 100 Salt 60-40 

High 104

 10-4 10-4 100

Low 104 10-4 10-4 100

Mid 104 10-4 10-4 100 Hitec XL 

High 104

 10-4 10-4 100

Low 104 10-3 10-4 100

Mid 104 10-3 10-4 100 Xceltherm 600 

High 104

 10-3 10-4 100
98 

D. Radiation Heat Transfer Zonal Analysis for the Glass Envelope 

A radiation heat transfer analysis for the glass envelope is conducted below to evaluate the effects of the 

collector and temperature distribution on the radiation heat losses from the receiver. A generalized zone 

analysis approach is used in the analysis, and as shown in Figure D.1, the collector and HCE cross-section 

is separated into four zones. Zone 1 is the receiver half closest to the collector. Zone 2 is the receiver half 

furthest from the collector. Zone 3 is the collector. Zone 4 is the sky opposite the collector. 

Figure D.1 Zone definitions for the generalized zone analysis of the radiation heat loss from 

the receiver 

With the four zones defined in Figure D.1, four separate cases can be evaluated: 

Case 1 is with one uniform receiver temperature and no collector (T1 = T2, T3 = T4 = Tsky). 

Case 2 is with two uniform receiver temperatures and no collector (T1 ≠ T2, T3 = T4 = Tsky). 

Case 3 is one uniform receiver temperature and collector present (T1 = T2, T3 = Tambient, T4 = Tsky). 

Case 4 is two uniform receiver temperatures and collector present (T1 ≠ T2, T3 = Tambient, T4 = Tsky). 

The collector temperature is set equal to ambient and the sky temperature is approximated as 8° less than 

ambient. 

Similar to other generalized zone analysis for radiative exchange in an enclosure, the following 

assumptions were made for the analysis that follows [Ozisk 1973]. 

• Radiative properties are uniform and independent of direction and frequency. 

• Surfaces are diffuse emitters and diffuse reflectors. 

• Radiative-heat flux leaving the surface is uniform over the surface of each zone. 

• Surfaces are opaque in the frequency of interest (α + ρ =1). 

• Temperatures are uniform over each surface. 

1

2

3

4

collector 

sky 

HCE
99 

The first step in the analysis is to define the radiative geometric configuration factors (view factors). For 

an enclosure with four surfaces there will be 16 view factors. Four can be set to zero, since the receiver 

surface is convex. 

F1−1 = 0 (D.1a) 

F1−2 = 0 (D.1b) 

F2−1 = 0 (D.1c) 

F2−2 = 0 (D.1d) 

From symmetry, five additional view factors can be eliminated. 

F1−3 = F2−4 (D.2a) 

F2−3 = F1−4 (D.2b) 

F3−1 = F4−2 (D.2c) 

F3−2 = F4−1 (D.2d) 

F3−3 = F4−4 (D.2e) 

View factor algebra can be used to define two additional view factors, which will be needed in the 

analysis. 

F3−( ) 1+2 = F3−1 + F3−2 (D.3) 

F4−( ) 1+2 = F4−1 + F4−2 (D.4) 

And now, taking advantage of reciprocity, another view factor relationship can be eliminated by 

symmetry. 

F() () 1+2 −3 = F1+2 −4 (D.5) 

Three additional view factors are defined by taking advantage of view factors catalogued in tables. From 

A Catalog of Radiation Configuration Factors, by Howell [1983] 

( ) 2

1 F1+2 −3 = (D.6) 

( ) ( ) ( )





































+ 





























 = − − + + − +

−

3

1 1 2

3

1 2

2

1

2

3

1 2

3 3 1 sin 2 1

r

r

r

r

r

r

F

π

 (D.7) 

( ) ( ) ( ) ( )

3

1 2

3

1 1 2

3

1 2

2

1

2

3

1 2

3 4 1 sin 2

r

r

r

r

r

r

r

r

F + + − + +

− −





































+ 





























 = − π

 (D.8) 
100 

Now, four more relationships are needed to define all the view factors. The first can be approximated with 

the help of Hottel’s crossed-string method [Siegel and Howell 2002]. Figure D.2 shows the required line 

segments for this method. 

( )

3

1 2

3 1 2

2 2

A

L L F + − = (D.9) 

with, 

L1 = Lab + Lbc (D.10a) 

()() 2 2 Lab = Lae′ + Le′b (D.10b) 

2

1r Lbc

π

= (D.10c) 

L2 = Lcd (D.10d) 

3 1 L r r cd = − (D.10e) 

And finally, from reciprocity the remaining three view factors can be found. 

( )

( )

( )

3

(1 2)

3

(1 2)

1 2 3

3

1 2

3 1 2 2

1

2 r

r

r

r

F

A

A

F + +

+ −

+

− + = = = (D.11) 

3 1

1

3

3 1

1

3

1−3 = − = F − r

r

F

A

A

F (D.12) 

3 2

2

3

3 2

2

3

2−3 = − = F − r

r

F

A

A

F (D.13) 
101 

a

b

d

e e' c

Figure D.2 Line segments used for the string method approximation for view factors 

The radiosities and heat fluxes can be determined from the following basic relations for radiative heat 

exchange in enclosures [Ozisik 1977]. 

∑=

= + −

4

1

4

j

Ri iσTi ρ i RjFi j ε (D.14) 

∑=

= − − ′′ 4

j 1

qi Ri RjFi j (D.15) 

i = 1,2,3,4

Using Equation D.1 and treating the sky as a blackbody give the following radiosity equations. 

( ) 1 3 1 3 4 1 4

4 R1 = 1σT1 + ρ R F − + R F − ε (D.16) 

( ) 2 3 2 3 4 2 4

4 R2 = 2σT2 + ρ R F − + R F − ε (D.17) 

( ) 3 1 3 1 2 3 2 3 3 3 4 3 4

4 R3 = 3σT3 + ρ R F − + R F − + R F − + R F − ε (D.18) 

4 R4 = σT4 (D.19) 

Solving for R3 gives 

{ ( ) [ ( ) ]}

[ ] ( ) 3 1 3 3 1 3 2 2 3 3 3

3 1 1 4 3 2 2 4 3 4

4

4

4

3 2 2

4

3 3 1 1

4

3 3

3 1 − − − − −

− − − − − − −

− + +

+ + + + + = F F F F F

T F T F T T F F F F F R

e

e e

ρ ρ

ε σ σρ ε ρ (D.20) 

collector 

HCE

sky 
102 

The heat flux leaving the glass envelope can be determined by eliminating the summations in Equations 

D.14 and D.15 and summing the heat fluxes from either side of the HCE. 

( ) (1 2) 1 1 2 q′ = 2 r q′′ + q′′ + π (D.21) 

( ) 1

4

1

1

1 q1

′′ = σT − R

ρ

ε

 (D.22) 

( ) 2

4

2

2

2 q′

2

′ = σT − R

ρ

ε

 (D.23) 

Assuming that the emissivity of the glass (pyrex) envelope and collector (low iron glass) is independent 

of temperature 

0.86 ε 1 = ε 2 = ε e =

0.9 ε 3 =

All the required information for solving the four cases is now available. Table D.1 shows an Excel 

spreadsheet used to solve the radiation heat transfer for each case. The results show that by assuming the 

collector has no effect and that the glass envelope temperature is uniform around the circumference, 

introduces an error in over predicting radiation heat transfer of 5%–10%. As shown in Figure D.3, this 

error is consistent regardless of the temperature difference between the front and back of the HCE – 

assuming the same average temperature around the circumference. Table D.1 also shows that neglecting 

the collector and modeling the glass envelope temperature as nonuniform would increase the radiation 

heat transfer by about 5%. 

Table D.1 Radiation Heat Transfer Zonal Analysis Parameters and Results 

R1 R2 R3 q1 q2 q1+2 % Diff

no collector, uniform envelope temperature 785.70 785.70 385.49 72.3 72.3 144.6 0

no collector, two uniform envelope temperatures 1018.26 597.99 385.49 114.3 38.4 152.7 5.6

collector, uniform envelope temperature 785.70 785.70 428.33 64.6 64.6 129.1 -10.7

collector, two uniform envelope temperatures 1018.26 597.99 428.72 106.5 30.6 137.1 -5.2

Collector Type = LS-2 F1-3 = 0.82212

Denvelope (m) = 0.115 F1-4 = 0.17788

Dcollector (m) = 4.803 F2-3 = 0.17788

εenvelope = 0.86 F2-4 = 0.82212

ρenvelope = 0.14 F3-1 = 0.019685

εcollector = 0.9 F3-2 = 0.004259

ρcollector = 0.1 F3-3 = 0.363198

∆T (K) = 50 F3-4 = 0.612859

Tave (K) = 350

Tave,e1 (K) = 375

Tave,e2 (K) = 325

Tamb (K) = 295.15

Tsky (K) = 287.15

σ (W/m2

-K4) = 5.67E-08

Inputs

Test Criterion Radiosities (W/m2

) Heat Fluxes (W/m)

View Factors
103 

Comparison of Radiation Heat Loss from the HCE

Modeled With and Without the Effect of the Collector

100.0

110.0

120.0

130.0

140.0

150.0

160.0

170.0

180.0

190.0

0 10 20 30 40 50 60 70 80 90 100

Temperature Difference Between Front and Back Sides of HCE

W/m

With Collector Without Collector

∆T = 0

Figure D.3 Radiation heat loss comparison chart showing difference between modeling the 

radiation loss with and without the collector effects
104 

E. One-Dimensional Design Study Version of EES Code 

(Tab settings have been shortened and numerous comment statements have been omitted to reduce 

length of following code.) 

"****************************************************************************************************************** 

 PROCEDURE Pq_12conv : Convective heat transfer rate from the HTF to the inside of the 

receiver tube 

******************************************************************************************************************"

PROCEDURE Pq_12conv(T_1ave, T_2, Fluid$: q_12conv) 

$Common D_2, D_p, D_h, L, v_1, Flow_Type$, IncludeLaminar$ 

" Warning Statements if HTF is evaluated out of recommended temperature ranges "

 If (Fluid$ = 'Syltherm 800') Then 

 If (T_1ave < -40) or (400 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Syltherm 800 fluid properties are out of recommended temperature range, -40 C < T < 400 C. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Therminol VP1') Then 

 If (T_1ave < 12) or (400 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Therminol VP1 fluid properties are out of recommended temperature range, 12 C < T < 400 C . 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Xceltherm 600') Then 

 If (T_1ave < -20) or (316 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Xceltherm 600 fluid properties are out of recommended temperature range, -20 C < T < 316 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Therminol 59') Then 

 If (T_1ave < -45) or (315 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Therminol 59 fluid properties are out of recommended temperature range, -45 C < T < 315 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Therminol 66') Then 

 If (T_1ave < 0) or (345 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Therminol 66 fluid properties are out of recommended temperature range, 0 C < T < 345 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Therminol XP') Then 

 If (T_1ave < -20) or (315 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Therminol XP fluid properties are out of recommended temperature range, -20 C < T < 315 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Dowtherm Q') Then 

 If (T_1ave < -35) or (330 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Dowtherm Q fluid properties are out of recommended temperature range, -35 C < T < 330 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Dowtherm RP') Then 

 If (T_1ave < -20) or (350 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Dowtherm RP fluid properties are out of recommended temperature range, -20 C < T < 350 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Salt (60% NaNO3, 40% KNO3)') Then 
105 

 If (T_1ave < 260) or (621 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Salt (60% NaNO3, 40% KNO3) fluid properties are out of recommended temperature range, 

260 C < T < 621 C. See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

If (Fluid$ = 'Hitec XL') Then 

 If (T_1ave < 266) or (480 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Hitec XL fluid properties are out of recommended temperature range, 266 C < T < 480 C. See 

Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

If (Fluid$ = 'Water') Then 

 If (T_1ave < 0) or (100 < T_1ave) Then CALL WARNING('The result may not be accurate, since 

Water fluid properties are out of recommended temperature range, 0 C < T < 100 C. See Procedure 

Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

" Thermophysical properties for HTF "

 MU_1 := INTERPOLATE(Fluid$,'MU','T',T=T_1ave) "[kg/m-s]"

 MU_2 := INTERPOLATE(Fluid$,'MU','T',T=T_2) "[kg/m-s]"

 Cp_1 := INTERPOLATE(Fluid$,'Cp','T',T=T_1ave) "[J/kg-K]"

 Cp_2 := INTERPOLATE(Fluid$,'Cp','T',T=T_2) "[J/kg-K]"

 k_1 := INTERPOLATE(Fluid$,'k','T',T=T_1ave) "[W/m-K]"

 k_2 := INTERPOLATE(Fluid$,'k','T',T=T_2) "[W/m-K]"

 RHO_1 := INTERPOLATE(Fluid$,'RHO','T',T=T_1ave) "[kg/m^3]"

 Re_D2 := (RHO_1 * D_h * v_1) / (MU_1) 

 Pr_2 := (Cp_2 * MU_2) / k_2 

 Pr_1 := (Cp_1 * MU_1) / k_1 

" Nusselt Number for laminar flow case if option to include laminar flow model is chosen "

 If (IncludeLaminar$ = 'Yes') and (Re_D2 <= 2300) Then 

 If (Flow_Type$ = 'Annulus Flow') Then 

 DRatio := D_p/D_2 

 Nu#_D2 := INTERPOLATE(Nu#, 'Nu#_D2','Dp/D2', Dp/D2=DRatio) "estimate for uniform 

heat flux case"

 Else 

 Nu#_D2 := 4.36 "uniform heat flux"

 EndIf 

 Else 

 " Warning statements if turbulent/transitional flow Nusselt Number correlation is used out of 

recommended range "

 If (Pr_1 <= 0.5) or (2000 <= Pr_1) Then CALL WARNING('The result may not be accurate, 

since 0.5 < Pr_1 < 2000 does not hold. See PROCEDURE Pq_12conv. Pr_1 = XXXA1', Pr_1) 

 If (Pr_2 <= 0.5) or (2000 <= Pr_2) Then CALL WARNING('The result may not be accurate, 

since 0.5 < Pr_2 < 2000 does not hold. See PROCEDURE Pq_12conv. Pr_2 = XXXA1', Pr_2) 

 If ( Re_D2 <= (2300) ) or (5*10^6 <= Re_D2 ) Then CALL WARNING('The result may not be 

accurate, since 2300 < Re_D2 < (5 * 10^6) does not hold. See PROCEDURE Pq_12conv. Re_D2 = 

XXXA1', Re_D2) 

 " Turbulent/transitional flow Nusselt Number correlation (modified Gnielinski correlation) " 

 f := (1.82 * LOG10(Re_D2) - 1.64)^(-2) 

 Nu#_D2 := (f / 8) * (Re_D2 - 1000) * Pr_1 / (1 + 12.7 * (f / 8)^(0.5) * (Pr_1^(0.6667) -1)) * (Pr_1 

/ Pr_2)^0.11 

 EndIf 
106 

 h_1 := Nu#_D2 * k_1 / D_h "[W/m^2-K]"

 q_12conv := h_1 * D_2 * PI * (T_2 - T_1ave) "[W/m]"

End 

"****************************************************************************************************************** 

 FUNCTION fq_34conv : Convective heat transfer rate between the absorber outer surface and 

the glazing inner surface 

******************************************************************************************************************"

FUNCTION fq_34conv(T_3, T_4) 

$Common D_3, D_4, L, P_a, P_6, g, v_6, T_0, T_6, T_std, AnnulusGas$, GlazingIntact$ 

 P_a1 := P_a * CONVERT(torr, kPa) "[kPa]"

 T_34 := (T_3 + T_4) / 2 "[C]"

 T_36 := (T_3 + T_6) / 2 "[C]"

 If (GlazingIntact$ = 'No') Then 

 

 " Thermophysical Properties for air "

 Rho_3 := Density(AIR, T=T_3, P=P_6) "[kg/m^3]"

 Rho_6 := Density(AIR, T=T_6, P=P_6) "[kg/m^3]"

 If (v_6 <= 0.1) Then 

 MU_36 := viscosity(AIR, T=T_36) "[N-s/m^2]"

 Rho_36 := Density(AIR, T=T_36, P=P_6) "[kg/m^3]"

 Cp_36 := CP(AIR, T=T_36) "[kJ/kg-K]"

 k_36 := conductivity(AIR, T=T_36) "[W/m-K]"

 NU_36 := MU_36 / Rho_36 "[m^2/s]"

 Alpha_36 := k_36 / (Cp_36 * Rho_36 * 1000) "[m^2/s]"

 Beta_36 := 1 / (T_36 + T_0) "[1/K]"

 Ra_D3 := g * Beta_36 * ABS(T_3 - T_6) * (D_3)^3 / (Alpha_36 * NU_36) 

 " Warning Statement if following Nusselt Number correlation is used out of recommended 

range "

 If (Ra_D3 <= 10^(-5)) or (Ra_D3 >= 10^12) Then CALL WARNING('The result may not be 

accurate, since 10^(-5) < Ra_D3 < 10^12 does not hold. See Function fq_34conv. Ra_D3 = XXXA1', 

Ra_D3) 

 " Churchill and Chu correlation for natural convection from a long isothermal horizontal 

cylinder "

 Pr_36 := NU_36 / Alpha_36 

 Nu#_bar := (0.60 + (0.387 * Ra_D3^(0.1667)) / (1 + (0.559 / Pr_36)^(0.5625))^(0.2963) )^2 

 h_36 := Nu#_bar * k_36 / D_3 "[W/m^2-K]"

 fq_34conv := h_36 * PI * D_3 * (T_3 - T_6) "[W/m]"

 Else 

 " Thermophysical Properties for air "

 MU_3 := viscosity(AIR, T=T_3) "[N-s/m^2]"

 MU_6 := viscosity(AIR, T=T_6) "[N-s/m^2]"

 k_3 := conductivity(AIR, T=T_3) "[W/m-K]"

 k_6 := conductivity(AIR, T=T_6) "[W/m-K]"

 Cp_3 := CP(AIR, T=T_3) "[kJ/kg-K]"

 Cp_6 := CP(AIR, T=T_6) "[kJ/kg-K]"
107 

 NU_6 := MU_6 / Rho_6 "[m^2/s]"

 NU_3 := MU_3 / Rho_3 "[m^2/s]"

 Alpha_3 := k_3 / (Cp_3 * Rho_3 * 1000) "[m^2/s]"

 Alpha_6 := k_6 / (Cp_6 * Rho_6 * 1000) "[m^2/s]"

 Re_D3 := v_6 * D_3 / NU_6 

 Pr_3 := NU_3 / Alpha_3 

 Pr_6 := NU_6 / Alpha_6 

 " Warning Statements if following Nusselt Number correlation if used out of range "

 If (Re_D3 <= 1) or (Re_D3 >= 10^6) Then CALL WARNING('The result may not be 

accurate, since 1 < Re_D3 < 10^6 does not hold. See Function fq_34conv. Re_D3 = XXXA1', 

Re_D3) 

 If (Pr_6 <= 0.7) or (Pr_6 >= 500) Then CALL WARNING('The result may not be accurate, 

since 0.7 < Pr_6 < 500 does not hold. See Function fq_34conv. Pr_6 = XXXA1', Pr_6) 

 " Coefficients for external forced convection Nusselt Number correlation (Zhukauskas's 

correlation) "

 If (Pr_6 <= 10) Then 

 n := 0.37 

 Else 

 n := 0.36 

 EndIf 

 If (Re_D3 < 40) Then 

 C := 0.75 

 m := 0.4 

 Else 

 If (40 <= Re_D3) and (Re_D3 < 10^3) Then 

 C := 0.51 

 m := 0.5 

 Else 

 If (10^3 <= Re_D3) and (Re_D3 < 2*10^5) Then 

 C := 0.26 

 m := 0.6 

 Else 

 If (2*10^5 <= Re_D3) and (Re_D3 < 10^6) Then 

 C := 0.076 

 m := 0.7 

 EndIf 

 EndIf 

 EndIf 

 EndIf 

 " Zhukauskas's correlation for external forced convection flow normal to an isothermal 

cylinder "

 Nu#_bar := C * (Re_D3)^m * (Pr_6)^n * (Pr_6 / Pr_3)^(0.25) 

 h_36 := Nu#_bar * k_6 / D_3 "[W/m^2-K]"

 fq_34conv := h_36 * D_3 * PI * (T_3 - T_6) "[W/m]" 

 EndIf 

 Else 

 " Thermophysical Properties for gas in annulus space "

 If (AnnulusGas$ = 'Argon') Then 

 MU_34 := INTERPOLATE('Argon', 'MU', 'T', T = (T_34 + T_0)) "[kg/m-s]"

 Cp_34 := INTERPOLATE('Argon', 'Cp', 'T', T = (T_34 + T_0)) "[kJ/kg-K]"
108 

 Cv_34 := INTERPOLATE('Argon', 'Cv', 'T', T = (T_34 + T_0)) "[kJ/kg-K]"

 Rho_34 := P_a1 / (0.20813 * (T_34 + T_0)) "[kg/m^3]"

 k_34 := INTERPOLATE('Argon', 'k', 'T', T = (T_34 + T_0)) "[W/m-K]"

 k_std := INTERPOLATE('Argon', 'k', 'T', T = (T_std + T_0)) "[W/m-K]"

 Else 

 MU_34 := VISCOSITY(AnnulusGas$, T=T_34) "[kg/m-s]"

 Cp_34 := CP(AnnulusGas$, T=T_34) "[kJ/kg-K]"

 Cv_34 := CV(AnnulusGas$, T=T_34) "[kJ/kg-K]"

 k_34 := CONDUCTIVITY(AnnulusGas$, T=T_34) "[W/m-K]"

 Rho_34 := DENSITY(AnnulusGas$, T=T_34, P=P_a1) "[kg/m^3]"

 k_std := CONDUCTIVITY(AnnulusGas$, T=T_std) "[W/m-K]"

 EndIf 

 " Modified Raithby and Hollands correlation for natural convection in an annular space 

between horizontal cylinders "

 Alpha_34 := k_34 /(Cp_34 * Rho_34 * 1000) "[m^2/s]"

 NU_34 := MU_34 / Rho_34 "[m^2/s]"

 Beta_34 := 1 / (T_34 + T_0) "[1/K]"

 Ra_D3 := g * Beta_34 * ABS(T_3 - T_4) * (D_3)^3 / (Alpha_34 * NU_34) 

 Ra_D4 := g * Beta_34 * ABS(T_3 - T_4) * (D_4)^3 / (Alpha_34 * NU_34) 

 Pr_34 := NU_34 / Alpha_34 

 Natq_34conv := 2.425 * k_34 * (T_3 - T_4) / (1 + (D_3/ D_4)^(0.6))^(1.25) * (Pr_34 * Ra_D3 / 

(0.861 + Pr_34))^(0.25) "[W/m]" 

 P := P_a * CONVERT(torr, mmHg) "[mmHg]"

 C1 := 2.331*10^(-20) "[mmHg-cm^3/K]"

 " Free-molecular heat transfer for an annular space between horizontal cylinders "

 If (AnnulusGas$ = 'Air') Then 

 Delta := 3.53 * 10^(-8) "[cm]"

 EndIf 

 If (AnnulusGas$ = 'H2') Then 

 Delta := 2.4 * 10^(-8) "[cm]"

 EndIf 

 If (AnnulusGas$ = 'Argon') Then 

 Delta := 3.8*10^(-8) "[cm]"

 EndIf 

 Lambda := C1 * (T_34 + T_0) / (P * Delta^2) "[cm]"

 Gamma := Cp_34 / Cv_34 

 a := 1 

 b := (2 - a) / a * (9 * Gamma - 5) / (2 * (Gamma + 1)) 

 h_34 := k_std / (D_3 / 2 * ln(D_4 / D_3) + b * Lambda * (D_3 / D_4 + 1)) "[W/m^2-K]"

 Kineticq_34conv := D_3 * PI * h_34 * (T_3 - T_4) "[W/m]" 

 " Following compares free-molecular heat transfer with natural convection heat transfer and 

uses the largest value for heat transfer in annulus "

 If (Kineticq_34conv > Natq_34conv) Then 

 fq_34conv := Kineticq_34conv "[W/m]"

 " Warning Statement if free-molecular heat transfer correlation is used out of range "

 If (Ra_D4 < 10^7) Then CALL WARNING('The result may not be accurate, since (D_4 / 

(D_4 - D_3))^4 < Ra_D4 < 10^7 does not hold. See Function fq_34conv. Ra_D4 = XXXA1', Ra_D4) 

 Else 
109 

 fq_34conv := Natq_34conv "[W/m]"

 EndIf 

 EndIf 

End 

"****************************************************************************************************************** 

 FUNCTION fq_34rad : Radiation heat transfer rate between the absorber surface and glazing 

inner surface 

******************************************************************************************************************"

FUNCTION fq_34rad(T_3, T_4) 

$COMMON D_3, D_4, L, T_7, T_0, sigma, EPSILON_3, EPSILON_4, GlazingIntact$ 

 If (GlazingIntact$ = 'No') Then 

 fq_34rad :=EPSILON_3 * PI * D_3 * sigma * ((T_3 + T_0)^4 - (T_7 + T_0)^4) "[W/m]"

 Else 

 fq_34rad := PI * D_3 * sigma * ((T_3 + T_0)^4 - (T_4 + T_0)^4) / (1 / EPSILON_3 + D_3 / D_4 

* ( 1 / EPSILON_4 - 1)) "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fq_56conv : Convective heat transfer rate between the glazing outer surface and the 

ambient air 

******************************************************************************************************************"

FUNCTION fq_56conv(T_5, T_6) 

$Common D_5, L, P_6, v_6, g, T_0, GlazingIntact$ 

 T_56 := (T_5 + T_6)/2 "[C]"

" Thermophysical Properties for air "

 MU_5 := VISCOSITY(Air,T=T_5) "[kg/m-s]"

 MU_6 := VISCOSITY(Air,T=T_6) "[kg/m-s]"

 MU_56 := VISCOSITY(Air, T=T_56) "[kg/m-s]"

 k_5 := CONDUCTIVITY(Air,T=T_5) "[W/m-K]"

 k_6 := CONDUCTIVITY(Air,T=T_6) "[W/m-K]"

 k_56 := CONDUCTIVITY(Air, T=T_56) "[W/m-K]"

 Cp_5 := SPECHEAT(Air,T=T_5) "[kJ/kg-K]"

 Cp_6 := SPECHEAT(Air,T=T_6) "[kJ/kg-K]"

 Cp_56 := CP(AIR, T=T_56) "[kJ/kg-K]"

 Rho_5 := DENSITY(Air,T=T_5, P=P_6) "[kg/m^3]"

 Rho_6 := DENSITY(Air,T=T_6, P=P_6) "[kg/m^3]"

 Rho_56 := DENSITY(Air, T=T_56, P=P_6) "[kg/m^3]"

" If the glass envelope is missing then the convection heat transfer from the glass envelope is 

forced to zero by T_5 = T_6 "

 If (GlazingIntact$ = 'No') Then 

 fq_56conv := (T_5 - T_6) "[W/m]"

 Else 

 If (v_6 <= 0.1) Then 

 " Coefficients for Churchill and Chu natural convection correlation "

 NU_56 := MU_56 / Rho_56 "[m^2/s]"
110 

 Alpha_56 := k_56 / (Cp_56 * Rho_56 * 1000) "[m^2/s]"

 Beta_56 := 1 / (T_56 + T_0) "[1/K]"

 Ra_D5 := g *Beta_56 * ABS(T_5 - T_6) * (D_5)^3 / (Alpha_56 * NU_56) 

 " Warning Statement if following Nusselt Number correlation is used out of range "

 If (Ra_D5 <= 10^(-5)) or (Ra_D5 >= 10^12) Then CALL WARNING('The result may not be 

accurate, since 10^(-5) < Ra_D5 < 10^12 does not hold. See Function fq_56conv. Ra_D5 = XXXA1', 

Ra_D5) 

 " Churchill and Chu correlation for natural convection for a horizontal cylinder "

 Pr_56 := NU_56 / Alpha_56 

 Nu#_bar := (0.60 + (0.387 * Ra_D5^(0.1667)) / (1 + (0.559 / Pr_56)^(0.5625))^(0.2963) )^2 

 h_6 := Nu#_bar * k_56 / D_5 "[W/m^2-K]"

 fq_56conv := h_6 * PI * D_5 * (T_5 - T_6) "[W/m]"

 Else 

 " Coefficients for Zhukauskas's correlation "

 Alpha_5 := k_5 / (Cp_5 * Rho_5 * 1000) "[m^2/s]"

 Alpha_6 := k_6 / (Cp_6 * Rho_6 * 1000) "[m^2/s]"

 NU_5 := MU_5 / Rho_5 "[m^2/s]"

 NU_6 := MU_6 / Rho_6 "[m^2/s]"

 Pr_5 := NU_5 / Alpha_5 

 Pr_6 := NU_6 / Alpha_6 

 Re_D5 := v_6 * D_5 * Rho_6 / MU_6 

 " Warning Statement if following Nusselt Number correlation is used out of range "

 If (Pr_6 <= 0.7) or (Pr_6 >= 500) Then CALL WARNING('The result may not be accurate, 

since 0.7 < Pr_6 < 500 does not hold. See Function fq_56conv. Pr_6 = XXXA1', Pr_6) 

 If (Re_D5 <= 1) or (Re_D5 >= 10^6) Then CALL WARNING('The result may not be 

accurate, since 1 < Re_D5 < 10^6 does not hold. See Function fq_56conv. Re_D5 = XXXA1 ', 

Re_D5) 

 " Zhukauskas's correlation for forced convection over a long horizontal cylinder "

 If (Pr_6 <= 10) Then 

 n := 0.37 

 Else 

 n := 0.36 

 EndIf 

 If (Re_D5 < 40) Then 

 C := 0.75 

 m := 0.4 

 Else 

 If (40 <= Re_D5) and (Re_D5 < 10^3) Then 

 C := 0.51 

 m := 0.5 

 Else 

 If (10^3 <= Re_D5) and (Re_D5 < 2*10^5) Then 

 C := 0.26 

 m := 0.6 

 Else 

 If (2*10^5 <= Re_D5) and (Re_D5 < 10^6) Then 

 C := 0.076 

 m := 0.7 

 EndIf 
111 

 EndIf 

 EndIf 

 EndIf 

 Nu#_6 := C * Re_D5^m * Pr_6^n *(Pr_6/Pr_5)^0.25 

 h_6 := Nu#_6 * k_6 / D_5 "[W/m^2-K]"

 fq_56conv := h_6 * PI * D_5 * (T_5 - T_6) "[W/m]"

 EndIf 

 EndIf 

End 

"****************************************************************************************************************** 

 FUNCTION fq_57rad : Radiation heat transfer rate between the glazing outer surface and the sky 

******************************************************************************************************************"

FUNCTION fq_57rad(T_5, T_7) 

$COMMON EPSILON_5, D_5, L, sigma, T_0, GlazingIntact$ 

" If glass envelope is missing then radiation heat transfer from glass envelope is set to zero; 

otherwise, radiative heat transfer for a small convex object in a large cavity "

 If (GlazingIntact$ = 'No') Then 

 fq_57rad := 0 "[W/m]"

 Else 

 fq_57rad := EPSILON_5 * PI * D_5 * sigma * ((T_5 + T_0)^4 - (T_7 + T_0)^4) "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fq_5SolAbs : Solar flux on glazing 

******************************************************************************************************************"

FUNCTION fq_5SolAbs(q_i) 

$COMMON GlazingIntact$, Alpha_env, OptEff_env 

" If glass envelope is missing then solar absorption in glass envelope is set to zero; otherwise, 

solar absorption is estimated with an optical efficiency term "

 If (GlazingIntact$ = 'No') Then 

 fq_5SolAbs :=0 "[W/m]"

 Else 

 fq_5SolAbs := q_i * OptEff_env * Alpha_env "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 PROCEDURE Pq_45cond : One dimensional energy equation about inside surface of glazing 

******************************************************************************************************************"

PROCEDURE Pq_45cond(q_34conv, q_34rad: q_45cond) 

$COMMON GlazingIntact$ 

" If glass envelope is missing then radial conduction through glass envelope is set to zero; 

otherwise, the energy balance is used "

 If (GlazingIntact$ = 'No') Then 

 q_45cond :=0 "[W/m]"
112 

 Else 

 q_45cond := q_34conv + q_34rad "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 PROCEDURE Pq_56conv : One dimensional energy equation about outside surface of glazing 

******************************************************************************************************************"

PROCEDURE Pq_56conv(q_45cond, q_5SolAbs, q_57rad: q_56conv) 

$COMMON GlazingIntact$ 

" If the glass envelope is missing then the convective heat transfer from the envelope is set to 

zero; otherwise, the energy balance is used "

 If (GlazingIntact$ = 'Yes') Then 

 q_56conv := q_45cond + q_5SolAbs - q_57rad "[W/m]"

 Else 

 q_56conv := 0 "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fA_cs : Inside cross sectional flow area of absorber 

******************************************************************************************************************"

FUNCTION fA_cs(D_2, D_p) 

$COMMON Flow_Type$ 

" If 'SNL AZTRAK Platform' then the HTF flow area accounts for the inserted plug "

 If (Flow_Type$ = 'Annulus Flow') Then 

 fA_cs := PI * (D_2 ^2 - D_p ^2) / 4 "[m^2]"

 Else 

 fA_cs := PI * (D_2 ^2 / 4) "[m^2]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fETA_Col : Collector efficiency 

******************************************************************************************************************"

FUNCTION fETA_Col(q_12conv, q_i) 

" If the solar insolation is zero or the efficiency is negative, then the collector efficiency is set to 

zero "

 If (q_i =0) Then 

 fETA_Col := 0 

 Else 

 If ((q_12conv/q_i)<=0.001) Then 

 fETA_Col := 0 

 Else 

 fETA_Col :=q_12conv/q_i 

 EndIf 

 EndIf 
113 

END 

"****************************************************************************************************************** 

 FUNCTION fk_23: Absorber conductance 

******************************************************************************************************************"

{ Based on linear fit of data from "Alloy Digest, Sourcebook, Stainless Steels"; ASM International, 

2000.}

FUNCTION fk_23(T_2, T_3) 

$COMMON AbsorberMaterial$ 

 T_23 := (T_2 + T_3) / 2 "[C]"

 If (AbsorberMaterial$ = '304L') or (AbsorberMaterial$ = '316L') Then 

 fk_23 := 0.013 * T_23 + 15.2 "[W/m-K]"

 EndIf 

 If (AbsorberMaterial$ = '321H') Then 

 fk_23 := 0.0153 * T_23 + 14.775 "[W/m-K]"

 EndIf 

 If (AbsorberMaterial$ = 'B42 Copper Pipe') Then 

 fk_23 := 400 "[W/m-K]"

 EndIf 

END 

"****************************************************************************************************************** 

 PROCEDURE pSelectiveCoatingProperties: Selective Coating Emissivity and Absorptance 

******************************************************************************************************************"

PROCEDURE pSelectiveCoatingProperties(T_3: EPSILON_3, Alpha_abs, TAU_envelope) 

$COMMON SelectiveCoating$, T_0, TAU_envelope_UD, Alpha_abs_UD, EPSILON_3_100_UD, 

EPSILON_3_400_UD 

" Calculates emissivity and determines optical properties for chosen selective coating type "

 If (SelectiveCoating$ = 'User-Defined') Then 

 TAU_envelope := TAU_envelope_UD 

 Alpha_abs := Alpha_abs_UD 

 EPSILON_3 := EPSILON_3_100_UD - (EPSILON_3_400_UD - EPSILON_3_100_UD)/3 + 

T_3 * (EPSILON_3_400_UD - EPSILON_3_100_UD)/300 

 Endif 

 If (SelectiveCoating$ = 'Black Chrome (SNL test)') Then 

 TAU_envelope := 0.935 

 Alpha_abs := 0.94 

 EPSILON_3 := 0.0005333 * (T_3+T_0) - 0.0856 

 If (EPSILON_3 < 0.11) Then 

 EPSILON_3 := 0.11 

 EndIf 

 Endif 

 If (SelectiveCoating$ = 'Luz Cermet (SNL test)') Then 

 TAU_envelope := 0.935 
114 

 Alpha_abs := 0.92 

 EPSILON_3 := 0.000327 * (T_3+T_0) - 0.065971 

 If (EPSILON_3 < 0.05) Then 

 EPSILON_3 := 0.05 

 EndIf 

 Endif 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (SNL test a)') Then 

 TAU_envelope := 0.965 

 Alpha_abs := 0.96 

 EPSILON_3 := 2.249*10^(-7)*(T_3)^2 + 1.039*10^(-4)*T_3 + 5.599*10^(-2) 

 EndIf 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (SNL test b)') Then 

 TAU_envelope := 0.965 

 Alpha_abs := 0.95 

 EPSILON_3 := 1.565*10^(-7)*(T_3)^2 + 1.376*10^(-4)*T_3 + 6.966*10^(-2) 

 EndIf 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (SNL test avg)') Then 

 TAU_envelope := 0.965 

 Alpha_abs := 0.955 

 EPSILON_3 := 1.907*10^(-7)*(T_3)^2 + 1.208*10^(-4)*T_3 + 6.282*10^(-2) 

 EndIf 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (0.10 @ 400C)') Then 

 TAU_envelope := 0.97 

 Alpha_abs := 0.98 

 EPSILON_3 := 2.084*10^(-4)*T_3 + 1.663*10^(-2) 

 EndIf 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (0.07 @ 400C)') Then 

 TAU_envelope := 0.97 

 Alpha_abs := 0.97 

 EPSILON_3 := 1.666*10^(-4)*T_3 + 3.375*10^(-3) 

 EndIf 

END 

"****************************************************************************************************************** 

 PROCEDURE pHCEdimensions: HCE dimensions based on HCE type 

******************************************************************************************************************"

PROCEDURE pHCEdimensions(CollectorType$: D_2, D_3, D_4, D_5, W_aperture) 

$COMMON D_2_UD, D_3_UD, D_4_UD, D_5_UD, W_aperture_UD 

" The following determines the HCE dimensions depending on the HCE type chosen in the 

Diagram Window "

 If (CollectorType$ = 'User-Defined') Then 

 D_2 := D_2_UD "[m]"

 D_3 := D_3_UD "[m]"

 D_4 := D_4_UD "[m]"

 D_5 := D_5_UD "[m]"

 W_aperture := W_aperture_UD "[m]"

 EndIf 
115 

 If (CollectorType$ = 'LS-2') Then 

 D_2 := 0.066 "[m]"

 D_3 := 0.070 "[m]"

 D_4 := 0.109 "[m]" 

 D_5 := 0.115 "[m]"

 W_aperture := 4.8235 "[m]"

 EndIf 

 If (CollectorType$ = 'LS-3') Then 

 D_2 := 0.066 "[m]"

 D_3 := 0.070 "[m]"

 D_4 := 0.115 "[m]"

 D_5 := 0.121 "[m]"

 W_aperture := 5.59 "[m]"

 EndIf 

 If (CollectorType$ = 'IST') Then 

 D_2 := 0.066 "[m]"

 D_3 := 0.070 "[m]"

 D_4 := 0.075 "[m]"

 D_5 := 0.0702 "[m]"

 W_aperture := 3.053 "[m]"

 EndIf 

END 

"****************************************************************************************************************** 

 PROCEDURE pOpticalEfficiency: Optical Efficiencies based on HCE type 

******************************************************************************************************************"

PROCEDURE pOpticalEfficiency(CollectorType$: OptEff_env, OptEff_abs) 

$COMMON GlazingIntact$, K, Reflectivity, TAU_envelope, Shadowing_UD, TrackingError_UD, 

GeomEffects_UD, Rho_mirror_clean_UD, Dirt_mirror_UD, Dirt_HCE_UD, Error_UD 

" The following determines the optical properties depending on the HCE type chosen in the 

Diagram Window 

 The properties should be modified as better data becomes available "

 If (CollectorType$ = 'User-Defined') Then 

 Shadowing := Shadowing_UD 

 TrackingError := TrackingError_UD 

 GeomEffects := GeomEffects_UD 

 Rho_mirror_clean := Rho_mirror_clean_UD 

 Dirt_mirror := Dirt_mirror_UD 

 Dirt_HCE := Dirt_HCE_UD 

 Error := Error_UD 

 EndIf 

 If (CollectorType$ = 'LS-2') Then 

 Shadowing := 0.974 

 TrackingError := 0.994 

 GeomEffects := 0.98 

 Rho_mirror_clean := 0.935 

 Dirt_mirror := Reflectivity/Rho_mirror_clean 

 Dirt_HCE := (1+ Dirt_mirror)/2 

 Error := 0.96 

 EndIf 
116 

 If (CollectorType$ = 'LS-3') or (CollectorType$ = 'IST') Then 

 Shadowing := 0.974 

 TrackingError := 0.994 

 GeomEffects := 0.98 

 Rho_mirror_clean := 0.935 

 Dirt_mirror := Reflectivity/Rho_mirror_clean 

 Dirt_HCE := (1+ Dirt_mirror)/2 

 Error := 0.96 

 EndIf 

 

" following if statement prevents Dirt_mirror and Dirt_HCE from being larger then 1 if the input for 

Reflectivity is larger then Rho_mirror_clean "

 If (Dirt_mirror > 1) Then 

 Dirt_mirror := 1 

 Dirt_HCE := 1 

 EndIf 

 

 If (GlazingIntact$ = 'No') Then 

 OptEff_env := 0 

 OptEff_abs := Shadowing * TrackingError * GeomEffects * Rho_mirror_clean * Dirt_mirror * 

Error * K 

 Else 

 OptEff_env := Shadowing * TrackingError * GeomEffects * Rho_mirror_clean * Dirt_mirror * 

Dirt_HCE * Error * K 

 OptEff_abs := OptEff_env * TAU_Envelope 

 Endif 

END 

"****************************************************************************************************************** 

 FUNCTION fD_h: Hydraulic diameter 

******************************************************************************************************************"

FUNCTION fD_h(Flow_Type$) 

$COMMON D_2, D_p 

" If 'SNL AZTRAK Platform' is chosen, then the inside absorber diameter accounts for the inserted 

plug "

 If (Flow_Type$ = 'Annulus Flow') Then 

 fD_h := D_2 - D_p "[m]"

 Else 

 fD_h := D_2 "[m]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fq_cond_bracket: Heat loss estimate through HCE support bracket 

******************************************************************************************************************"

FUNCTION fq_cond_bracket(T_3) 

$COMMON T_6, T_0, P_6, v_6, g 

" effective bracket perimeter for convection heat transfer"

 P_brac := 0.2032 "[m]"
117 

" effective bracket diameter (2 x 1in) "

 D_brac := 0.0508 "[m]"

" minimum bracket cross-sectional area for conduction heat transfer"

 A_cs_brac := 0.00016129 "[m^2]"

" conduction coefficient for carbon steel at 600 K"

 k_brac := 48 "[W/m-K]"

" effective bracket base temperature"

 T_base := T_3 - 10 "[C]"

" estimate average bracket temperature "

 T_brac := (T_base + T_6) / 3 "[C]"

" estimate film temperature for support bracket "

 T_brac6 := (T_brac + T_6) /2 "[C]"

" convection coefficient with and without wind"

 If (v_6 <= 0.1) Then 

 

 MU_brac6 := viscosity(AIR, T=T_brac6) "[N-s/m^2]"

 Rho_brac6 := Density(AIR, T=T_brac6, P=P_6) "[kg/m^3]"

 Cp_brac6 := CP(AIR, T=T_brac6) "[kJ/kg-K]"

 k_brac6 := conductivity(AIR, T=T_brac6) "[W/m-K]"

 NU_brac6 := MU_brac6 / Rho_brac6 "[m^2/s]"

 Alpha_brac6 := k_brac6 / (Cp_brac6 * Rho_brac6 * 1000) "[m^2/s]"

 Beta_brac6 := 1 / (T_brac6 + T_0) "[1/K]"

 Ra_Dbrac := g * Beta_brac6 * ABS(T_brac - T_6) * (D_brac)^3 / (Alpha_brac6 * NU_brac6) 

 " Warning Statement if following Nusselt Number correlation is used out of recommended 

range "

 If (Ra_Dbrac <= 10^(-5)) or (Ra_Dbrac >= 10^12) Then CALL WARNING('The result may not 

be accurate, since 10^(-5) < Ra_Dbrac < 10^12 does not hold. See Function fq_cond_bracket. 

Ra_Dbrac = XXXA1', Ra_Dbrac) 

 " Churchill and Chu correlation for natural convection from a long isothermal horizontal cylinder 

"

 Pr_brac6 := NU_brac6 / Alpha_brac6 

 Nu#_bar := (0.60 + (0.387 * Ra_Dbrac^(0.1667)) / (1 + (0.559 / Pr_brac6)^(0.5625))^(0.2963) 

)^2 

 h_brac6 := Nu#_bar * k_brac6 / D_brac "[W/m^2-K]"

 

 Else 

 

 " Thermophysical Properties for air "

 MU_brac := viscosity(AIR, T=T_brac) "[N-s/m^2]"

 MU_6 := viscosity(AIR, T=T_6) "[N-s/m^2]"

 Rho_6 := Density(AIR, T=T_6, P=P_6) "[kg/m^3]"

 Rho_brac := Density(AIR, T=T_brac, P=P_6) "[kg/m^3]"

 k_brac := conductivity(AIR, T=T_brac) "[W/m-K]"

 k_6 := conductivity(AIR, T=T_6) "[W/m-K]"

 k_brac6 := conductivity(AIR, T=T_brac6) "[W/m-K]"

 Cp_brac := CP(AIR, T=T_brac) "[kJ/kg-K]"

 Cp_6 := CP(AIR, T=T_6) "[kJ/kg-K]"
118 

 NU_6 := MU_6 / Rho_6 "[m^2/s]"

 NU_brac := MU_brac / Rho_brac "[m^2/s]"

 Alpha_brac := k_brac / (Cp_brac * Rho_brac * 1000) "[m^2/s]"

 Alpha_6 := k_6 / (Cp_6 * Rho_6 * 1000) "[m^2/s]"

 Re_Dbrac := v_6 * D_brac / NU_6 

 Pr_brac := NU_brac / Alpha_brac 

 Pr_6 := NU_6 / Alpha_6 

 " Warning Statements if following Nusselt Correlation is used out of range "

 If (Re_Dbrac <= 1) or (Re_Dbrac >= 10^6) Then CALL WARNING('The result may not be 

accurate, since 1 < Re_Dbrac < 10^6 does not hold. See Function fq_cond_bracket. Re_Dbrac = 

XXXA1', Re_Dbrac) 

 If (Pr_6 <= 0.7) or (Pr_6 >= 500) Then CALL WARNING('The result may not be accurate, 

since 0.7 < Pr_6 < 500 does not hold. See Function fq_cond_bracket. Pr_6 = XXXA1', Pr_6) 

 " Coefficients for external forced convection Nusselt Number correlation (Zhukauskas's 

correlation) "

 If (Pr_6 <= 10) Then 

 n := 0.37 

 Else 

 n := 0.36 

 EndIf 

 If (Re_Dbrac < 40) Then 

 C := 0.75 

 m := 0.4 

 Else 

 If (40 <= Re_Dbrac) and (Re_Dbrac< 10^3) Then 

 C := 0.51 

 m := 0.5 

 Else 

 If (10^3 <= Re_Dbrac) and (Re_Dbrac < 2*10^5) Then 

 C := 0.26 

 m := 0.6 

 Else 

 If (2*10^5 <= Re_Dbrac) and (Re_Dbrac < 10^6) Then 

 C := 0.076 

 m := 0.7 

 EndIf 

 EndIf 

 EndIf 

 EndIf 

 " Zhukauskas's correlation for external forced convection flow normal to an isothermal cylinder 

"

 Nu#_bar := C * (Re_Dbrac)^m * (Pr_6)^n * (Pr_6 / Pr_brac)^(0.25) 

 h_brac6 := Nu#_bar * k_brac6 / D_brac "[W/m^2-K]"

 

 EndIf 

" estimated conduction heat loss through HCE support brackets / HCE length "

 fq_cond_bracket := SQRT(h_brac6 * P_brac * k_brac * A_cs_brac) * (T_base - T_6)/4.06 "[W/m]"

END 
119 

"****************************************************************************************************************** 

 FUNCTION fq_HeatLoss: Heat loss definition 

******************************************************************************************************************"

FUNCTION fq_HeatLoss(q_34conv, q_34rad, q_56conv, q_57rad, q_cond_bracket) 

$COMMON GlazingIntact$ 

 If (GlazingIntact$ = 'Yes') Then 

 fq_HeatLoss := q_56conv + q_57rad + q_cond_bracket "[W/m]"

 Else 

 fq_HeatLoss := q_34conv + q_34rad + q_cond_bracket "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 Constants and conversions 

******************************************************************************************************************"

" Stefan-Boltzmann constant "

sigma = 5.67E-8 "[W/m^2-K^4]"

" Used to convert temperature from C to K "

T_0 = 273.15 "[C]"

" Gravitational constant "

g = 9.81 "[m/s^2]"

" Wind speed from MPH to m/s conversion "

v_6 = v_6mph * CONVERT(mph, m/s) "[m/s]"

"****************************************************************************************************************** 

 Optical properties 

******************************************************************************************************************"

Alpha_env = .02 

" Calls procedure that determines optical properties "

CALL pSelectiveCoatingProperties(T_3: EPSILON_3, Alpha_abs, TAU_envelope) 

" Calls procedure that determines effective optical efficiencies at the glass envelope and absorber "

CALL pOpticalEfficiency(CollectorType$: OptEff_env, OptEff_abs) 

" Inner and outer glass envelope emissivities (Pyrex)"

EPSILON_4 = 0.86 

EPSILON_5 = 0.86 

" Incident angle modifier from test data for SEGS LS-2 receiver "

K = COS(THETA) + 0.000884 * THETA - 0.00005369 * (THETA)^2 

"****************************************************************************************************************** 

 Heat collector element size 

******************************************************************************************************************"

" Calls procedure for determining HCE dimensions based on 'CollectorType' "

CALL pHCEdimensions(CollectorType$: D_2, D_3, D_4, D_5, W_aperture) 
120 

"****************************************************************************************************************** 

 Ambient conditions 

******************************************************************************************************************"

" Effective sky temperature estimated as 8 C below ambient "

T_7 = T_6 - 8 "[C]"

" Converts ambient pressure from 0.83 atm to kPa, ambient pressure is treated as a constant "

P_6 = 0.83 * CONVERT(atm, kPa) "[kPa]"

" Standard ambient air temperature "

T_std = 25 "[C]"

" Incoming solar radiation per aperture length "

q_i = I_b * W_aperture "[W/m]"

"****************************************************************************************************************** 

 Temporary outputs and inputs 

******************************************************************************************************************"

" Space used for any temporary outputs or inputs "

"****************************************************************************************************************** 

 Hydraulic diameter 

******************************************************************************************************************"

"Calls function to calculate HTF hydraulic diameter "

D_h = fD_h(Flow_Type$) 

"****************************************************************************************************************** 

 Heat transfer fluid flow rates 

******************************************************************************************************************"

" Calls function to calculate HTF cross-section flow area "

A_cs = fA_cs(D_2, D_p) "[m^2]"

" Converts HTF flow rate from gpm to m^3/s "

v_1volm = v_1volg* CONVERT(gpm, m^3/s) "[m^3/s]"

" HTF velocity "

v_1 = v_1volm / (A_cs) "[m/s]"

"***************************************************************************************************************** 

 q_12conv Convective heat transfer rate between the heat transfer fluid and absorber 

******************************************************************************************************************"

CALL Pq_12conv(T_1ave, T_2, Fluid$: q_12conv) 

"****************************************************************************************************************** 

 q_23cond Conduction heat transfer rate through the absorber 

******************************************************************************************************************"

" Absorber conductance, temperature and material type dependent "

k_23 = fk_23(T_2, T_3) "[W/m-K]"

q_23cond = 2 * PI * k_23 * (T_3 - T_2) / LN(D_3 / D_2) "[W/m]"
121 

"****************************************************************************************************************** 

 q_34conv Convective heat transfer rate between the absorber pipe and glazing 

******************************************************************************************************************"

q_34conv = fq_34conv(T_3, T_4) "[W/m]"

"****************************************************************************************************************** 

 q_34rad Radiation heat transfer rate between the absorber surface and glazing inner surface 

******************************************************************************************************************"

q_34rad = fq_34rad(T_3, T_4) "[W/m]"

"****************************************************************************************************************** 

 q_45cond Conduction heat transfer rate through the glazing 

******************************************************************************************************************"

" glass envelope conductivity "

K_45 = 1.04 "[W/m-K]" 

q_45cond = 2 * PI * K_45 * (T_4 - T_5) / LN(D_5 / D_4) "[W/m]"

"****************************************************************************************************************** 

 q_56conv Convective heat transfer rate from the glazing to the atmosphere 

******************************************************************************************************************"

q_56conv = fq_56conv(T_5, T_6) "[W/m]"

"***************************************************************************************************************** 

 q_57rad Radiation heat transfer rate between the glazing outer surface and the sky 

******************************************************************************************************************"

q_57rad = fq_57rad(T_5, T_7) "[W/m]"

"****************************************************************************************************************** 

 q_3SolAbs Solar flux on absorber pipe 

******************************************************************************************************************"

q_3SolAbs = q_i * OptEff_abs * Alpha_abs "[W/m]"

"****************************************************************************************************************** 

 q_5SolAbs Solar Flux on glazing Envelope 

******************************************************************************************************************"

q_5SolAbs = fq_5SolAbs(q_i) "[W/m]"

"****************************************************************************************************************** 

 q_cond_bracket HCE support bracket conductive losses 

******************************************************************************************************************"

q_cond_bracket = fq_cond_bracket(T_3) "[W/m]"

"****************************************************************************************************************** 

 One dimensional (Radial) model 

******************************************************************************************************************"

CALL Pq_45cond(q_34conv, q_34rad: q_45cond) "[W/m]"
122 

CALL Pq_56conv(q_45cond, q_5SolAbs, q_57rad: q_56conv) "[W/m]"

q_12conv = q_23cond "[W/m]"

q_3SolAbs - q_23cond - q_34conv - q_34rad - q_cond_bracket = 0 "[W/m]"

q_HeatLoss = fq_HeatLoss(q_34conv, q_34rad, q_56conv, q_57rad, q_cond_bracket) "[W/m]"

q_OptLoss = q_i * (1-ETA_EffectiveOptEff/100) "[W/m]"

q_HeatGain = q_12conv "[W/m]"

"****************************************************************************************************************** 

 ETA_Col Collector efficiency 

******************************************************************************************************************"

ETA_Col = fETA_Col(q_12conv, q_i)*100 "[%]"

ETA_EffectiveOptEff = (OptEff_abs * Alpha_abs + OptEff_env * Alpha_env)*100 "[%]"
123 

F. Two-Dimensional Design Study Version of EES Code 

(Tab settings have been shortened and numerous comment statements have been omitted to reduce 

length of following code.)

"****************************************************************************************************************** 

 PROCEDURE Pq_12conv : Convective heat transfer rate from the HTF to the inside of the 

receiver tube 

******************************************************************************************************************"

PROCEDURE Pq_12conv(T_1ave, v_1ave, T_2, Fluid$: q_12conv, Cp_1) 

$Common D_h, D_2, L, TestType$ 

" Warning Statements if HTF is evaluated out of recommended temperature ranges "

 If (Fluid$ = 'Syltherm 800') Then 

 If (T_1ave < -40) or (400 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Syltherm 800 fluid properties are out of recommended temperature range, -40 C < T < 400 C. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Therminol VP1') Then 

 If (T_1ave < 12) or (400 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Therminol VP1 fluid properties are out of recommended temperature range, 12 C < T < 400 C . 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Xceltherm 600') Then 

 If (T_1ave < -20) or (316 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Xceltherm 600 fluid properties are out of recommended temperature range, -20 C < T < 316 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Therminol 59') Then 

 If (T_1ave < -45) or (315 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Therminol 59 fluid properties are out of recommended temperature range, -45 C < T < 315 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Therminol 66') Then 

 If (T_1ave < 0) or (345 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Therminol 66 fluid properties are out of recommended temperature range, 0 C < T < 345 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Therminol XP') Then 

 If (T_1ave < -20) or (315 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Therminol XP fluid properties are out of recommended temperature range, -20 C < T < 315 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Dowtherm Q') Then 

 If (T_1ave < -35) or (330 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Dowtherm Q fluid properties are out of recommended temperature range, -35 C < T < 330 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Dowtherm RP') Then 

 If (T_1ave < -20) or (350 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Dowtherm RP fluid properties are out of recommended temperature range, -20 C < T < 350 C.. 

See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 If (Fluid$ = 'Salt (60% NaNO3, 40% KNO3)') Then 
124 

 If (T_1ave < 260) or (621 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Salt (60% NaNO3, 40% KNO3) fluid properties are out of recommended temperature range, 

260 C < T < 621 C. See Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

If (Fluid$ = 'Hitec XL') Then 

 If (T_1ave < 266) or (480 < T_1ave) Then CALL WARNING('The result may not be accurate, 

since Hitec XL fluid properties are out of recommended temperature range, 266 C < T < 480 C. See 

Procedure Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

If (Fluid$ = 'Water') Then 

 If (T_1ave < 0) or (100 < T_1ave) Then CALL WARNING('The result may not be accurate, since 

Water fluid properties are out of recommended temperature range, 0 C < T < 100 C. See Procedure 

Pq_12conv.T_1ave = XXXA1', T_1ave) 

 EndIf 

 

" Thermophysical properties for HTF "

 MU_1 := INTERPOLATE(Fluid$,'MU','T',T=T_1ave) "[kg/m-s]"

 MU_2 := INTERPOLATE(Fluid$,'MU','T',T=T_2) "[kg/m-s]"

 Cp_1 := INTERPOLATE(Fluid$,'Cp','T',T=T_1ave) "[J/kg-K]"

 Cp_2 := INTERPOLATE(Fluid$,'Cp','T',T=T_2) "[J/kg-K]"

 k_1 := INTERPOLATE(Fluid$,'k','T',T=T_1ave) "[W/m-K]"

 k_2 := INTERPOLATE(Fluid$,'k','T',T=T_2) "[W/m-K]"

 RHO_1 := INTERPOLATE(Fluid$,'RHO','T',T=T_1ave) "[kg/m^3]"

 Re_D2 := ABS((RHO_1 * D_h * v_1ave) / (MU_1)) 

 Pr_2 := ABS(Cp_2 * MU_2 / k_2) 

 Pr_1 := ABS(Cp_1 * MU_1 / k_1) 

" Warning statements if Nusselt Number correlation is used out of recommended range "

 If (Pr_1 <= 0.5) or (2000 <= Pr_1) Then CALL WARNING('The result may not be accurate, since 

0.5 < Pr_1 < 2000 does not hold. See PROCEDURE Pq_12conv. Pr_1 = XXXA1', Pr_1) 

 If (Pr_2 <= 0.5) or (2000 <= Pr_2) Then CALL WARNING('The result may not be accurate, 

since 0.5 < Pr_2 < 2000 does not hold. See PROCEDURE Pq_12conv. Pr_2 = XXXA1', Pr_2) 

 If ( Re_D2 <= (2300) ) or (5*10^6 <= Re_D2 ) Then CALL WARNING('The result may not be 

accurate, since 2300 < Re_D2 < (5 * 10^6) does not hold. See PROCEDURE Pq_12conv. Re_D2 = 

XXXA1', Re_D2) 

" Turbulent/transitional flow Nusselt Number correlation (modified Gnielinski correlation) "

 f_2 := (1.82 * LOG10(Re_D2) - 1.64)^(-2) 

 Nu#_D2 := (f_2 / 8) * (Re_D2 - 1000) * Pr_1 / (1 + 12.7 * (f_2/ 8)^(0.5) * (Pr_1^(0.6667) -1)) * 

(Pr_1 / Pr_2)^0.11 

 

 h_1 := Nu#_D2 * k_1 / D_h "[W/m^2-K]"

 q_12conv := h_1 * D_2 * PI * (T_2 - T_1ave) "[W/m]"

End 

"****************************************************************************************************************** 

 FUNCTION fq_34conv : Convective heat transfer rate between the absorber outer surface and 

the glazing inner surface 

******************************************************************************************************************"

FUNCTION fq_34conv(T_3, T_4, AnnulusGas$) 

$Common D_3, D_4, L, P_a, P_6, g, v_6, T_0, T_6, T_std, GlazingIntact$ 
125 

 P_a1 := P_a * CONVERT(torr, kPa) "[kPa]"

 T_36 := (T_3 + T_6) / 2 "[C]"

 If (GlazingIntact$ = 'No') Then 

 

 " Thermophysical Properties for air "

 Rho_3 := Density(AIR, T=T_3, P=P_6) "[kg/m^3]"

 Rho_6 := Density(AIR, T=T_6, P=P_6) "[kg/m^3]"

 If (v_6 <= 0.1) Then 

 MU_36 := viscosity(AIR, T=T_36) "[N-s/m^2]"

 Rho_36 := Density(AIR, T=T_36, P=P_6) "[kg/m^3]"

 Cp_36 := CP(AIR, T=T_36) "[kJ/kg-K]"

 k_36 := conductivity(AIR, T=T_36) "[W/m-K]"

 NU_36 := MU_36 / Rho_36 "[m^2/s]"

 Alpha_36 := k_36 / (Cp_36 * Rho_36 * 1000) "[m^2/s]"

 Beta_36 := 1 / (T_36 + T_0) "[1/K]"

 Ra_D3 := g * Beta_36 * ABS(T_3 - T_6) * (D_3)^3 / (Alpha_36 * NU_36) 

 " Warning Statement if following Nusselt Number correlation is used out of recommended 

range "

 If (Ra_D3 <= 10^(-5)) or (Ra_D3 >= 10^12) Then CALL WARNING('The result may not be 

accurate, since 10^(-5) < Ra_D3 < 10^12 does not hold. See Function fq_34conv. Ra_D3 = XXXA1', 

Ra_D3) 

 " Churchill and Chu correlation for natural convection from a long isothermal horizontal 

cylinder "

 Pr_36 := NU_36 / Alpha_36 

 Nu#_bar := (0.60 + (0.387 * Ra_D3^(0.1667)) / (1 + (0.559 / Pr_36)^(0.5625))^(0.2963) )^2 

 h_36 := Nu#_bar * k_36 / D_3 "[W/m^2-K]"

 fq_34conv := h_36 * PI * D_3 * (T_3 - T_6) "[W/m]"

 Else 

 

 " Thermophysical Properties for air "

 MU_3 := viscosity(AIR, T=T_3) "[N-s/m^2]"

 MU_6 := viscosity(AIR, T=T_6) "[N-s/m^2]"

 k_3 := conductivity(AIR, T=T_3) "[W/m-K]"

 k_6 := conductivity(AIR, T=T_6) "[W/m-K]"

 k_36 := conductivity(AIR, T=T_36) "[W/m-K]"

 Cp_3 := CP(AIR, T=T_3) "[kJ/kg-K]"

 Cp_6 := CP(AIR, T=T_6) "[kJ/kg-K]"

 NU_6 := MU_6 / Rho_6 "[m^2/s]"

 NU_3 := MU_3 / Rho_3 "[m^2/s]"

 Alpha_3 := k_3 / (Cp_3 * Rho_3 * 1000) "[m^2/s]"

 Alpha_6 := k_6 / (Cp_6 * Rho_6 * 1000) "[m^2/s]"

 Re_D3 := v_6 * D_3 / NU_6 

 Pr_3 := NU_3 / Alpha_3 

 Pr_6 := NU_6 / Alpha_6 

 " Warning Statements if following Nusselt Correlation is used out of range "

 If (Re_D3 <= 1) or (Re_D3 >= 10^6) Then CALL WARNING('The result may not be 

accurate, since 1 < Re_D3 < 10^6 does not hold. See Function fq_34conv. Re_D3 = XXXA1', 

Re_D3) 
126 

 If (Pr_6 <= 0.7) or (Pr_6 >= 500) Then CALL WARNING('The result may not be accurate, 

since 0.7 < Pr_6 < 500 does not hold. See Function fq_34conv. Pr_6 = XXXA1', Pr_6) 

 " Coefficients for external forced convection Nusselt Number correlation (Zhukauskas's 

correlation) "

 If (Pr_6 <= 10) Then 

 n := 0.37 

 Else 

 n := 0.36 

 EndIf 

 If (Re_D3 < 40) Then 

 C := 0.75 

 m := 0.4 

 Else 

 If (40 <= Re_D3) and (Re_D3 < 10^3) Then 

 C := 0.51 

 m := 0.5 

 Else 

 If (10^3 <= Re_D3) and (Re_D3 < 2*10^5) Then 

 C := 0.26 

 m := 0.6 

 Else 

 If (2*10^5 <= Re_D3) and (Re_D3 < 10^6) Then 

 C := 0.076 

 m := 0.7 

 EndIf 

 EndIf 

 EndIf 

 EndIf 

 " Zhukauskas's correlation for external forced convection flow normal to an isothermal 

cylinder "

 Nu#_bar := C * (Re_D3)^m * (Pr_6)^n * (Pr_6 / Pr_3)^(0.25) 

 h_36 := Nu#_bar * k_36 / D_3 "[W/m^2-K]"

 fq_34conv := h_36 * D_3 * PI * (T_3 - T_6) "[W/m]" 

 EndIf 

 Else 

 T_34 := (T_3 + T_4) / 2 "[C]"

 " Thermophysical Properties for gas in annulus space "

 If (AnnulusGas$ = 'Argon') Then 

 

 MU_34 := INTERPOLATE('Argon', 'MU', 'T', T = (T_34 + T_0)) "[kg/m-s]"

 Cp_34 := INTERPOLATE('Argon', 'Cp', 'T', T = (T_34 + T_0)) "[kJ/kg-K]"

 Cv_34 := INTERPOLATE('Argon', 'Cv', 'T', T = (T_34 + T_0)) "[kJ/kg-K]"

 Rho_34 := P_a1 / (0.20813 * (T_34 + T_0)) "[kg/m^3]"

 k_34 := INTERPOLATE('Argon', 'k', 'T', T = (T_34 + T_0)) "[W/m-K]"

 k_std := INTERPOLATE('Argon', 'k', 'T', T = (T_std + T_0)) "[W/m-K]"

 Else 

 MU_34 := VISCOSITY(AnnulusGas$, T=T_34) "[kg/m-s]"

 Cp_34 := CP(AnnulusGas$, T=T_34) "[kJ/kg-K]"

 Cv_34 := CV(AnnulusGas$, T=T_34) "[kJ/kg-K]"

 k_34 := CONDUCTIVITY(AnnulusGas$, T=T_34) "[W/m-K]"
127 

 Rho_34 := DENSITY(AnnulusGas$, T=T_34, P=P_a1) "[kg/m^3]"

 k_std := CONDUCTIVITY(AnnulusGas$, T=T_std) "[W/m-K]"

 EndIf 

 " Modified Raithby and Hollands correlation for natural convection in an annular space 

between horizontal cylinders "

 Alpha_34 := k_34 /(Cp_34 * Rho_34 * 1000) "[m^2/s]"

 NU_34 := MU_34 / Rho_34 "[m^2/s]"

 Beta_34 := 1 / (T_34 + T_0) "[1/K]"

 Ra_D3 := g * Beta_34 * ABS(T_3 - T_4) * (D_3)^3 / (Alpha_34 * NU_34) 

 Ra_D4 := g * Beta_34 * ABS(T_3 - T_4) * (D_4)^3 / (Alpha_34 * NU_34) 

 Pr_34 := NU_34 / Alpha_34 

 Natq_34conv := 2.425 * k_34 * (T_3 - T_4) * (Pr_34 * Ra_D3 / (0.861 + Pr_34))^(0.25) / (1 + 

(D_3/ D_4)^(0.6))^(1.25) "[W/m]" 

 " Free-molecular heat transfer for an annular space between horizontal cylinders "

 P := P_a * CONVERT(torr, mmHg) "[mmHg]"

 C1 := 2.331*10^(-20) "[mmHg-cm^3/K]"

 If (AnnulusGas$ = 'Air') Then 

 Delta := 3.53 * 10^(-8) "[cm]"

 EndIf 

 If (AnnulusGas$ = 'H2') Then 

 Delta := 2.4 * 10^(-8) "[cm]"

 EndIf 

 

 If (AnnulusGas$ = 'Argon') Then 

 Delta := 3.8*10^(-8) "[cm]"

 EndIf 

 Lambda := C1 * (T_34 + T_0) / (P * Delta^2) "[cm]"

 Gamma := Cp_34 / Cv_34 

 a := 1 

 b := (2 - a) * (9 * Gamma - 5) / (2 * a * (Gamma + 1)) 

 h_34 := k_std / (D_3 / 2 * ln(D_4 / D_3) + b * Lambda * (D_3 / D_4 + 1)) "[W/m^2-K]"

 Kineticq_34conv := D_3 * PI * h_34 * (T_3 - T_4) "[W/m]"

 " Following compares free molecular heat transfer with natural convection heat transfer and 

uses the largest value for heat transfer in annulus "

 If (Kineticq_34conv > Natq_34conv) Then 

 fq_34conv := Kineticq_34conv "[W/m]"

 " Warning statement if free molecular correlation is used our of range"

 If (Ra_D4 < 10^7) Then CALL WARNING('The result may not be accurate, since (D_4 / 

(D_4 - D_3))^4 < Ra_D4 < 10^7 does not hold. See Function fq_34conv. Ra_D4 = XXXA1', Ra_D4) 

 Else 

 fq_34conv := Natq_34conv "[W/m]"

 EndIf 

 EndIf 

End 

"****************************************************************************************************************** 
128 

 FUNCTION fq_34rad : Radiation heat transfer rate between the absorber surface and glazing 

inner surface 

******************************************************************************************************************"

FUNCTION fq_34rad(T_3, T_4, i) 

$COMMON D_3, D_4, L, T_7, T_0, sigma, EPSILON_3[1..imax], EPSILON_4, GlazingIntact$ 

 If (GlazingIntact$ = 'No') Then 

 fq_34rad :=EPSILON_3[i] * PI * D_3 * sigma * ((T_3 + T_0)^4 - (T_7 + T_0)^4) "[W/m]"

 Else 

 fq_34rad := PI * D_3 * sigma * ((T_3 + T_0)^4 - (T_4 + T_0)^4) / (1 / EPSILON_3[i] + D_3 / 

D_4 * ( 1 / EPSILON_4 - 1)) "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fq_56conv : Convective heat transfer rate between the glazing outer surface and the 

ambient air 

******************************************************************************************************************"

FUNCTION fq_56conv(T_5, T_6) 

$Common D_5, L, P_6, v_6, g, T_0, GlazingIntact$ 

 T_56 := (T_5 + T_6) / 2 "[C]"

" Thermophysical Properties for air "

 MU_5 := VISCOSITY(Air,T=T_5) "[kg/m-s]"

 MU_6 := VISCOSITY(Air,T=T_6) "[kg/m-s]"

 MU_56 := VISCOSITY(Air, T=T_56) "[kg/m-s]"

 k_5 := CONDUCTIVITY(Air,T=T_5) "[W/m-K]"

 k_6 := CONDUCTIVITY(Air,T=T_6) "[W/m-K]"

 k_56 := CONDUCTIVITY(Air, T=T_56) "[W/m-K]"

 Cp_5 := SPECHEAT(Air,T=T_5) "[kJ/kg-K]"

 Cp_6 := SPECHEAT(Air,T=T_6) "[kJ/kg-K]"

 Cp_56 := CP(AIR, T=T_56) "[kJ/kg-K]"

 Rho_5 := DENSITY(Air,T=T_5, P=P_6) "[kg/m^3]"

 Rho_6 := DENSITY(Air,T=T_6, P=P_6) "[kg/m^3]"

 Rho_56 := DENSITY(Air, T=T_56, P=P_6) "[kg/m^3]"

" If the glass envelope is missing then the convection heat transfer from the glass envelope is 

forced to zero by T_5 = T_6 "

 If (GlazingIntact$ = 'No') Then 

 fq_56conv := (T_5 - T_6) "[W/m]"

 Else 

 If (v_6 <= 0.1) Then 

 " Coefficients for Churchill and Chu natural convection correlation "

 NU_56 := MU_56 / Rho_56 "[m^2/s]"

 Alpha_56 := k_56 / (Cp_56 * Rho_56 * 1000) "[m^2/s]"

 Beta_56 := 1 / (T_56 + T_0) "[1/K]"

 Ra_D5 := g *Beta_56 * ABS(T_5 - T_6) * (D_5)^3 / (Alpha_56 * NU_56) 

 " Warning Statement if following Nusselt Number correlation is used out of range "

 If (Ra_D5 <= 10^(-5)) or (Ra_D5 >= 10^12) Then CALL WARNING('The result may not be 

accurate, since 10^(-5) < Ra_D5 < 10^12 does not hold. See Function fq_56conv. Ra_D5 = XXXA1', 

Ra_D5) 
129 

 " Churchill and Chu correlation for natural convection for a horizontal cylinder "

 Pr_56 := NU_56 / Alpha_56 

 Nu#_bar := (0.60 + (0.387 * Ra_D5^(0.1667)) / (1 + (0.559 / Pr_56)^(0.5625))^(0.2963) )^2 

 h_6 := Nu#_bar * k_56 / D_5 "[W/m^2-K]"

 fq_56conv := h_6 * PI * D_5 * (T_5 - T_6) "[W/m]"

 Else 

 " Coefficients for Zhukauskas's correlation "

 Alpha_5 := k_5 / (Cp_5 * Rho_5 * 1000) "[m^2/s]"

 Alpha_6 := k_6 / (Cp_6 * Rho_6 * 1000) "[m^2/s]"

 NU_5 := MU_5 / Rho_5 "[m^2/s]"

 NU_6 := MU_6 / Rho_6 "[m^2/s]"

 Pr_5 := NU_5 / Alpha_5 

 Pr_6 := NU_6 / Alpha_6 

 Re_D5 := v_6 * D_5 * Rho_6 / MU_6 

 " Warning Statement if following Nusselt Number coorelation is used out of range "

 If (Pr_6 <= 0.7) or (Pr_6 >= 500) Then CALL WARNING('The result may not be accurate, 

since 0.7 < Pr_6 < 500 does not hold. See Function fq_56conv. Pr_6 = XXXA1', Pr_6) 

 If (Re_D5 <= 1) or (Re_D5 >= 10^6) Then CALL WARNING('The result may not be 

accurate, since 1 < Re_D5 < 10^6 does not hold. See Function fq_56conv. Re_D5 = XXXA1 ', 

Re_D5) 

 " Zhukauskas's correlation for forced convection over a long horizontal cylinder "

 If (Pr_6 <= 10) Then 

 n := 0.37 

 Else 

 n := 0.36 

 EndIf 

 If (Re_D5 < 40) Then 

 C := 0.75 

 m := 0.4 

 Else 

 If (40 <= Re_D5) and (Re_D5 < 10^3) Then 

 C := 0.51 

 m := 0.5 

 Else 

 If (10^3 <= Re_D5) and (Re_D5 < 2*10^5) Then 

 C := 0.26 

 m := 0.6 

 Else 

 If (2*10^5 <= Re_D5) and (Re_D5 < 10^6) Then 

 C := 0.076 

 m := 0.7 

 EndIf 

 EndIf 

 EndIf 

 EndIf 

 Nu#_6 := C * Re_D5^m * Pr_6^n *(Pr_6/Pr_5)^0.25 

 h_6 := Nu#_6 * k_6 / D_5 "[W/m^2-K]"

 fq_56conv := h_6 * PI * D_5 * (T_5 - T_6) "[W/m]"

 EndIf 
130 

 EndIf 

End 

"****************************************************************************************************************** 

 FUNCTION fq_57rad : Radiation heat transfer rate between the glazing outer surface and the sky 

******************************************************************************************************************"

FUNCTION fq_57rad(T_5, T_7) 

$COMMON EPSILON_5, D_5, L, sigma, T_0, GlazingIntact$ 

" If glass envelope is missing then radiation heat transfer from glass envelope is set to zero; 

otherwise, radiative heat transfer for a small convex object in a large cavity "

 If (GlazingIntact$ = 'No') Then 

 fq_57rad := 0 "[W/m]"

 Else 

 fq_57rad := EPSILON_5 * PI * D_5 * sigma * ((T_5 + T_0)^4 - (T_7 + T_0)^4) "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fq_5SolAbs : Solar flux on glazing 

******************************************************************************************************************"

FUNCTION fq_5SolAbs(q_i) 

$COMMON Alpha_env, OptEff_env, GlazingIntact$ 

" If glass envelope is missing then solar absorption in glass envelope is set to zero; otherwise, 

solar absorption is estimated with an optical efficiency term "

 If (GlazingIntact$ = 'No') Then 

 fq_5SolAbs :=0 "[W/m]"

 Else 

 fq_5SolAbs := q_i * OptEff_env * Alpha_env "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 PROCEDURE Pq_45cond : One dimensional energy equation about inside surface of glazing 

******************************************************************************************************************"

PROCEDURE Pq_45cond(q_34conv, q_34rad: q_45cond) 

$COMMON GlazingIntact$ 

" If glass envelope is missing then radial conduction through glass envelope is set to zero; 

otherwise, the energy balance is used "

 If (GlazingIntact$ = 'No') Then 

 q_45cond :=0 "[W/m]"

 Else 

 q_45cond := q_34conv+ q_34rad "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 
131 

 PROCEDURE Pq_56conv : One dimensional energy equation about outside surface of glazing 

******************************************************************************************************************"

PROCEDURE Pq_56conv(q_45cond, q_5SolAbs, q_57rad: q_56conv) 

$COMMON GlazingIntact$ 

" If the glass envelope is missing then the convective heat transfer from the envelope is set to 

zero; otherwise, the energy balance is used "

 If (GlazingIntact$ = 'No') Then 

 q_56conv := 0 "[W/m]"

 Else 

 q_56conv := q_45cond+ q_5SolAbs - q_57rad "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fA_cs : Inside cross sectional flow area of absorber 

******************************************************************************************************************"

FUNCTION fA_cs(D_2, D_p) 

$COMMON TestType$ 

" If 'SNL AZTRAK Platform' then the HTF flow area accounts for the inserted plug "

 If (TestType$ = 'SNL AZTRAK Platform') Then 

 fA_cs := PI * (D_2 ^2 - D_p ^2) / 4 "[m^2]"

 Else 

 fA_cs := PI * (D_2 ^2 / 4) "[m^2]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fETA_Col : Collector efficiency 

******************************************************************************************************************"

FUNCTION fETA_Col(q_12conv, q_i) 

" If heat is leaving the HTF, then the collector efficiency is set to zero "

 If ((q_12conv/q_i) <= 0.001) Then 

 fETA_Col := 0 

 Else 

 fETA_Col := q_12conv/q_i 

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fk_23: Absorber conductance 

******************************************************************************************************************"

{ Based on linear fit of data from "Alloy Digest, Sourcebook, Stainless Steels"; ASM International, 

2000.}

FUNCTION fk_23(T_2, T_3) 

$COMMON AbsorberMaterial$ 
132 

 T_23 := (T_2 + T_3)/2 "[C]"

 If (AbsorberMaterial$ = '304L') or (AbsorberMaterial$ = '316L') Then 

 fk_23 := 0.013 * T_23 + 15.2 "[W/m-K]"

 EndIf 

 If (AbsorberMaterial$ = '321H') Then 

 fk_23 := 0.0153 * T_23 + 14.775 "[W/m-K]"

 EndIf 

 If (AbsorberMaterial$ = 'B42 Copper Pipe') Then 

 fk_23 := 400 "[W/m-K]"

 EndIf 

END 

"****************************************************************************************************************** 

 PROCEDURE pSelectiveCoatingProperties: Selective Coating Emissivity and Absorptance 

******************************************************************************************************************"

PROCEDURE pSelectiveCoatingProperties(imax, T_3[1..imax]: EPSILON_3[1..imax], Alpha_abs, 

TAU_envelope) 

$COMMON T_0, TAU_envelope_UD, Alpha_abs_UD, EPSILON_3_100_UD, EPSILON_3_400_UD, 

SelectiveCoating$ 

" Do-Loop to calculate emissivity for all the HCE increments, and to return optical properties for 

chosen selective coating type "

 j := 0 

 repeat 

 j := j+1 

 If (SelectiveCoating$ = 'User-Defined') Then 

 TAU_envelope := TAU_envelope_UD 

 Alpha_abs := Alpha_abs_UD 

 EPSILON_3[j] := EPSILON_3_100_UD - (EPSILON_3_400_UD - EPSILON_3_100_UD)/3 + 

T_3[j] * (EPSILON_3_400_UD - EPSILON_3_100_UD)/300 

 Endif 

 If (SelectiveCoating$ = 'Black Chrome (SNL test)') Then 

 TAU_envelope := 0.935 

 Alpha_abs := 0.94 

 EPSILON_3[j] := 0.0005333 * (T_3[j]+T_0) - 0.0856 

 If (EPSILON_3[j] < 0.11) Then 

 EPSILON_3[j] := 0.11 

 EndIf 

 Endif 

 If (SelectiveCoating$ = 'Luz Cermet (SNL test)') Then 

 TAU_envelope := 0.935 

 Alpha_abs := 0.92 

 EPSILON_3[j] := 0.000327 * (T_3[j]+T_0) - 0.065971 

 If (EPSILON_3[j] < 0.05) Then 

 EPSILON_3[j] := 0.05 

 EndIf 

 Endif 
133 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (SNL test a)') Then 

 TAU_envelope := 0.965 

 Alpha_abs := 0.96 

 EPSILON_3[j] := 2.249*10^(-7)*(T_3[j])^2 + 1.039*10^(-4)*T_3[j] + 5.599*10^(-2) 

 EndIf 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (SNL test b)') Then 

 TAU_envelope := 0.965 

 Alpha_abs := 0.95 

 EPSILON_3[j] := 1.565*10^(-7)*(T_3[j])^2 + 1.376*10^(-4)*T_3[j] + 6.966*10^(-2) 

 EndIf 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (SNL test avg)') Then 

 TAU_envelope := 0.965 

 Alpha_abs := 0.955 

 EPSILON_3[j] := 1.907*10^(-7)*(T_3[j])^2 + 1.208*10^(-4)*T_3[j] + 6.282*10^(-2) 

 EndIf 

 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (0.10 @ 400C)') Then 

 TAU_envelope := 0.97 

 Alpha_abs := 0.98 

 EPSILON_3[j] := 2.084*10^(-4)*T_3[j] + 1.663*10^(-2) 

 EndIf 

 If (SelectiveCoating$ = 'Solel UVAC Cermet (0.07 @ 400C)') Then 

 TAU_envelope := 0.97 

 Alpha_abs := 0.97 

 EPSILON_3[j] := 1.666*10^(-4)*T_3[j] + 3.375*10^(-3) 

 EndIf 

 until (j>=(imax)) 

END 

"****************************************************************************************************************** 

 PROCEDURE pHCEdimensions: HCE dimensions based on HCE type 

******************************************************************************************************************"

PROCEDURE pHCEdimensions(HCEtype$: D_4, D_5, A_aperture, L_aperture, L_HCE, 

Number_HCE) 

$COMMON TestType$ 

 If (HCEtype$ = 'LS-2') Then 

 D_5 := 0.115 "[m]"

 D_4 := 0.109 "[m]"

 L_HCE := 4.06 "[m]"

 If (TestType$ = 'SNL AZTRAK Platform') Then 

 A_aperture := 39 "[m^2]"

 Number_HCE := 2 

 L_aperture := Number_HCE * L_HCE "[m]"

 Else 

 If (TestType$ = 'KJC Test Loop') Then 

 A_aperture := 235 * 16 "[m^2]"

 Number_HCE := 12 * 16 

 L_aperture := L_HCE * Number_HCE "[m]"
134 

 Endif 

 Endif 

 EndIf 

 If (HCEtype$ = 'LS-3') Then 

 D_4 := 0.115 "[m]"

 D_5 := 0.121 "[m]"

 L_HCE := 4.06 "[m]"

 A_aperture := 545 "[m^2]"

 Number_HCE := 24 

 L_aperture := L_HCE * Number_HCE "[m]"

 EndIf 

 If (HCEtype$ = 'IST') Then 

 D_4 := 0.075 "[m]"

 D_5 := 0.0702 "[m]"

 A_aperture := 13.19 "[m^2]"

 L_HCE := 2.16 "[m]"

 Number_HCE := 1 

 L_aperture := L_HCE * Number_HCE "[m]"

 EndIf 

END 

"****************************************************************************************************************** 

 PROCEDURE pOpticalEfficiency: Optical Efficiencies based on HCE type 

******************************************************************************************************************"

PROCEDURE pOpticalEfficiency(HCEtype$: OptEff_env, OptEff_abs) 

$COMMON Alpha_abs, Alpha_env, K, Reflectivity, TAU_envelope, GlazingIntact$ 

" All the following optical properties should be modified as updated values are determined "

 If (HCEtype$ = 'LS-2') Then 

 Shadowing := 0.974 

 TrackingError := 0.994 

 GeomEffects := 0.98 

 Rho_mirror_clean := 0.935 

 Dirt_mirror := Reflectivity/Rho_mirror_clean 

 Dirt_HCE := (1+ Dirt_mirror)/2 

 Error := 0.96 

 EndIf 

 If (HCEtype$ = 'LS-3') or (HCEtype$ = 'IST') Then 

 Shadowing := 0.974 

 TrackingError := 0.994 

 GeomEffects := 0.98 

 Rho_mirror_clean := 0.935 

 Dirt_mirror := Reflectivity/Rho_mirror_clean 

 Dirt_HCE := (1+ Dirt_mirror)/2 

 Error := 0.96 

 EndIf 

" following if statement prevents Dirt_mirror and Dirt_HCE from being larger then 1 if the input for 

Reflectivity is larger then Rho_mirror_clean "

 If (Dirt_mirror > 1) Then 

 Dirt_mirror := 1 
135 

 Dirt_HCE := 1 

 EndIf 

 

 If (GlazingIntact$ = 'No') Then 

 OptEff_env := 0 

 OptEff_abs := Shadowing * TrackingError * GeomEffects * Rho_mirror_clean * Dirt_mirror * 

Error * K 

 Else 

 OptEff_env := Shadowing * TrackingError * GeomEffects * Rho_mirror_clean * Dirt_mirror * 

Dirt_HCE * Error * K 

 OptEff_abs := OptEff_env * TAU_Envelope 

 Endif 

END 

"****************************************************************************************************************** 

 FUNCTION fD_h: Hydraulic diameter 

******************************************************************************************************************"

FUNCTION fD_h(TestType$) 

$COMMON D_2, D_p 

" If 'SNL AZTRAK Platform' is chosen, then the inside absorber diameter accounts for the inserted 

plug "

 If (TestType$ = 'SNL AZTRAK Platform') Then 

 fD_h := D_2 - D_p "[m]"

 Else 

 fD_h := D_2 "[m]"

 EndIf 

END 

"****************************************************************************************************************** 

 FUNCTION fq_cond_bracket: Heat loss estimate through HCE support bracket 

******************************************************************************************************************"

FUNCTION fq_cond_bracket(T_3, i) 

$COMMON L_HCE, Number_HCE, L_aperture, T_6, T_0, P_6, v_6, DELTAL, imax, g 

" effective bracket perimeter for convection heat transfer"

 P_brac := 0.2032 "[m]"

" effective bracket diameter (2 x 1in) "

 D_brac := 0.0508 "[m]"

" minimum bracket cross-sectional area for conduction heat transfer"

 A_cs_brac := 0.00016129 "[m^2]"

" conduction coefficient for carbon steel at 600 K"

 k_brac := 48 "[W/m-K]"

" effective bracket base temperature"

 T_base := T_3 - 10 "[C]"

" estimate average bracket temperature "

 T_brac := (T_base + T_6) / 3 "[C]"
136 

" estimate film temperature for support bracket "

 T_brac6 := (T_brac + T_6) /2 "[C]"

" convection coefficient with and without wind"

 If (v_6 <= 0.1) Then 

 

 MU_brac6 := viscosity(AIR, T=T_brac6) "[N-s/m^2]"

 Rho_brac6 := Density(AIR, T=T_brac6, P=P_6) "[kg/m^3]"

 Cp_brac6 := CP(AIR, T=T_brac6) "[kJ/kg-K]"

 k_brac6 := conductivity(AIR, T=T_brac6) "[W/m-K]"

 NU_brac6 := MU_brac6 / Rho_brac6 "[m^2/s]"

 Alpha_brac6 := k_brac6 / (Cp_brac6 * Rho_brac6 * 1000) "[m^2/s]"

 Beta_brac6 := 1 / (T_brac6 + T_0) "[1/K]"

 Ra_Dbrac := g * Beta_brac6 * ABS(T_brac - T_6) * (D_brac)^3 / (Alpha_brac6 * NU_brac6) 

 " Warning Statement if following Nusselt Number correlation is used out of recommended 

range "

 If (Ra_Dbrac <= 10^(-5)) or (Ra_Dbrac >= 10^12) Then CALL WARNING('The result may not 

be accurate, since 10^(-5) < Ra_Dbrac < 10^12 does not hold. See Function fq_cond_bracket. 

Ra_Dbrac = XXXA1', Ra_Dbrac) 

 " Churchill and Chu correlation for natural convection from a long isothermal horizontal cylinder 

"

 Pr_brac6 := NU_brac6 / Alpha_brac6 

 Nu#_bar := (0.60 + (0.387 * Ra_Dbrac^(0.1667)) / (1 + (0.559 / Pr_brac6)^(0.5625))^(0.2963) 

)^2 

 h_brac6 := Nu#_bar * k_brac6 / D_brac "[W/m^2-K]"

 

 Else 

 

 " Thermophysical Properties for air "

 MU_brac := viscosity(AIR, T=T_brac) "[N-s/m^2]"

 MU_6 := viscosity(AIR, T=T_6) "[N-s/m^2]"

 Rho_6 := Density(AIR, T=T_6, P=P_6) "[kg/m^3]"

 Rho_brac := Density(AIR, T=T_brac, P=P_6) "[kg/m^3]"

 k_brac := conductivity(AIR, T=T_brac) "[W/m-K]"

 k_6 := conductivity(AIR, T=T_6) "[W/m-K]"

 k_brac6 := conductivity(AIR, T=T_brac6) "[W/m-K]"

 Cp_brac := CP(AIR, T=T_brac) "[kJ/kg-K]"

 Cp_6 := CP(AIR, T=T_6) "[kJ/kg-K]"

 NU_6 := MU_6 / Rho_6 "[m^2/s]"

 NU_brac := MU_brac / Rho_brac "[m^2/s]"

 Alpha_brac := k_brac / (Cp_brac * Rho_brac * 1000) "[m^2/s]"

 Alpha_6 := k_6 / (Cp_6 * Rho_6 * 1000) "[m^2/s]"

 Re_Dbrac := v_6 * D_brac / NU_6 

 Pr_brac := NU_brac / Alpha_brac 

 Pr_6 := NU_6 / Alpha_6 

 " Warning Statements if following Nusselt Correlation is used out of range "

 If (Re_Dbrac <= 1) or (Re_Dbrac >= 10^6) Then CALL WARNING('The result may not be 

accurate, since 1 < Re_Dbrac < 10^6 does not hold. See Function fq_cond_bracket. Re_Dbrac = 

XXXA1', Re_Dbrac) 

 If (Pr_6 <= 0.7) or (Pr_6 >= 500) Then CALL WARNING('The result may not be accurate, 

since 0.7 < Pr_6 < 500 does not hold. See Function fq_cond_bracket. Pr_6 = XXXA1', Pr_6) 
137 

 " Coefficients for external forced convection Nusselt Number correlation (Zhukauskas's 

correlation) "

 If (Pr_6 <= 10) Then 

 n := 0.37 

 Else 

 n := 0.36 

 EndIf 

 If (Re_Dbrac < 40) Then 

 C := 0.75 

 m := 0.4 

 Else 

 If (40 <= Re_Dbrac) and (Re_Dbrac< 10^3) Then 

 C := 0.51 

 m := 0.5 

 Else 

 If (10^3 <= Re_Dbrac) and (Re_Dbrac < 2*10^5) Then 

 C := 0.26 

 m := 0.6 

 Else 

 If (2*10^5 <= Re_Dbrac) and (Re_Dbrac < 10^6) Then 

 C := 0.076 

 m := 0.7 

 EndIf 

 EndIf 

 EndIf 

 EndIf 

 " Zhukauskas's correlation for external forced convection flow normal to an isothermal cylinder 

"

 Nu#_bar := C * (Re_Dbrac)^m * (Pr_6)^n * (Pr_6 / Pr_brac)^(0.25) 

 h_brac6 := Nu#_bar * k_brac6 / D_brac "[W/m^2-K]"

 

 EndIf 

 

" number of HCE support brackets for each HCE segment "

 If (DELTAL <= L_HCE) Then 

 index_1 := ROUND(i * DELTAL / L_HCE) 

 index_2 := index_1 * L_HCE 

 

 If (((i - 1) * DELTAL) <= index_2) AND (index_2 <= (i * DELTAL)) Then 

 n := 1 

 Else 

 n := 0 

 EndIf 

 Else 

 n := ROUND(DELTAL / L_HCE) 

 EndIf 

 

 If (i = 1) OR (i = imax) Then 

 n := n + 1 
138 

 EndIf 

 

 fq_cond_bracket := n * SQRT(h_brac6 * P_brac * k_brac * A_cs_brac) * (T_base - T_6) "[W]"

END 

"****************************************************************************************************************** 

 FUNCTION fHeatLoss: Heat loss term for temperature out equation 

******************************************************************************************************************"

FUNCTION fHeatLoss(q_34rad, q_34conv, q_56conv, q_57rad) 

$COMMON GlazingIntact$ 

 If (GlazingIntact$ = 'No') Then 

 fHeatLoss := q_34conv + q_34rad "[W/m]"

 Else 

 fHeatLoss := q_56conv + q_57rad "[W/m]"

 EndIf 

END 

"****************************************************************************************************************** 

 Constants and conversions 

******************************************************************************************************************"

"Stefan-Boltzmann constant"

sigma = 5.67E-8 "[W/m^2-K^4]"

" Used to convert temperature from C to K "

T_0 = 273.15 "[C]"

" Gravitational constant "

g = 9.81 "[m/s^2]"

" Wind speed conversion "

v_6 = v_6mph * CONVERT(mph, m/s) "[m/s]"

" Glass envelope conductance "

K_45 = 1.04 "[W/m-K]"

" Absorber pipe inside surface equivalent roughness factor "

e = 1.5E-6 "[m]"

"****************************************************************************************************************** 

 Optical properties 

******************************************************************************************************************"

" Glass envelope absorbtivity "

Alpha_env = .02 

" Calls procedure that determines optical properties " 

CALL pSelectiveCoatingProperties(imax, T_3[1..imax]: EPSILON_3[1..imax], Alpha_abs, 

TAU_envelope) 

" Calls procedure that determines optical efficiencies at the glass envelope and absorber "

CALL pOpticalEfficiency(HCEtype$: OptEff_env, OptEff_abs) 
139 

" Inner and outer glass envelope surface emissivities (pyrex)"

EPSILON_4 = 0.86 

EPSILON_5 = 0.86 

" Incident angle modifier from test data for SEGS LS-2 Receiver "

K = COS(THETA) + 0.000884 * THETA - 0.00005369 * (THETA)^2 

"****************************************************************************************************************** 

 Heat collector element size 

******************************************************************************************************************"

" 'SNL AZTRAK Tracker' plug diameter"

D_p = 0.0508 "[m]"

" Inner and outer absorber pipe diameters "

D_2 = 0.066 "[m]"

D_3 = 0.070 "[m]"

" Calls procedure for determining HCE dimensions based on 'HCE Type'"

CALL pHCEdimensions(HCEtype$: D_4, D_5, A_aperture, L_aperture, L_HCE, Number_HCE) 

"****************************************************************************************************************** 

 Ambient conditions 

******************************************************************************************************************"

" Effective sky temperature estimated as 8 C below ambient "

T_7 = T_6 - 8 "[C]"

" Converts ambient pressure from 0.83 atm to kPa, ambient pressure is treated as a constant "

P_6 = 0.83 * CONVERT(atm, kPa) "[kPa]"

" Standard ambient air temperature "

T_std = 25 "[C]"

" Incoming solar irradiance per aperture length "

q_i = I_b * A_aperture / L_aperture "[W/m]"

"****************************************************************************************************************** 

 Indexes 

******************************************************************************************************************"

" Iteration stepsize along aperture length "

DELTAL = (L_aperture)/(imax) "[m]"

"****************************************************************************************************************** 

 Hydraulic diameter 

******************************************************************************************************************"

" Calls function to calculate HTF hydraulic diameter "

D_h = fD_h(TestType$) 

"****************************************************************************************************************** 

 Heat transfer fluid flow rates 

******************************************************************************************************************"

" Calls function to calculate HTF cross-section flow area"
140 

A_cs = fA_cs(D_2, D_p) "[m^2]"

" Converts HTF flow rate from gpm to m^3/s "

v_1volm = v_1volg* CONVERT(gpm, m^3/s) "[m^3/s]"

" Calculates HTF velocity "

v_1inlet = v_1volm / (A_cs) "[m/s]"

" Mass flow rate (conserved) "

m_dot = v_1inlet * RHO_1inlet * A_cs "[kg/s]"

RHO_1inlet = INTERPOLATE(Fluid$,'RHO','T',T=T_1inlet) "[kg/m^3]"

"****************************************************************************************************************** 

******************************************************************************************************************* 

******************************************************************************************************************* 

 Do-loops for temperatures along aperture length 

******************************************************************************************************************* 

******************************************************************************************************************* 

******************************************************************************************************************"

" Do-Loop to set the inlet temperature and velocity of each increment to the previous increment's 

outlet temperature and velocity"

T_1in[1] = T_1inlet "[C]"

v_1in[1] = v_1inlet "[m/s]"

Duplicate i = 1, (imax-1) 

T_1in[i+1] = T_1out[i] "[C]"

v_1in[i+1] = v_1out[i] "[m/s]"

End 

" Do-Loop to conduct an energy balance on each increment that determines outlet temperature, heat 

loss, and efficiency "

Duplicate i = 1,imax 

v_1ave[i] = (v_1in[i] + v_1out[i])/2 "[m/s]"

" HTF Densities "

RHO_1out[i] = INTERPOLATE(Fluid$,'RHO','T',T=T_1out[i]) "[kg/m^3]"

RHO_1ave[i] = INTERPOLATE(Fluid$,'RHO','T',T=T_1ave[i]) "[kg/m^3]"

MU_1ave[i] = INTERPOLATE(Fluid$,'MU','T',T=T_1ave[i]) "[kg/m-s]"

Re_D2ave[i] = RHO_1ave[i] * v_1ave[i] * D_2 / MU_1ave[i] 

" friction factor, f, relation "

1 / SQRT(f[i]) = -2.0 * LOG10( e / (D_2 * 3.7) + 2.51 / (Re_D2ave[i] * SQRT(f[i]))) 

"****************************************************************************************************************** 

 HTF velocities 

******************************************************************************************************************"

" Outlet velocity of each increment "

v_1out[i] = m_dot / (RHO_1out[i] * A_cs) "[m/s]"

"****************************************************************************************************************** 
141 

 Heat loss estimate through HCE support bracket 

******************************************************************************************************************"

q_cond_bracket[i] = fq_cond_bracket(T_3[i], i) "[W]"

"****************************************************************************************************************** 

 HTF temperatures 

******************************************************************************************************************"

" Outlet HTF temperature for each increment "

HeatLoss[i] = fHeatLoss(q_34rad[i], q_34conv[i], q_56conv[i], q_57rad[i]) "[W/m]"

T_1out[i] = ((q_5SolAbs + q_3SolAbs - HeatLoss[i] )* DELTAL - q_cond_bracket[i]) / (m_dot * 

Cp_1ave[i]) + ((v_1in[i])^2 - (v_1out[i])^2)/2 / Cp_1ave[i] + T_1in[i] "[C]"

" Average HTF temperature for each increment "

T_1ave[i] = (T_1in[i] + T_1out[i])/2 "[C]"

DELTAP[i] = ( f[i] * (DELTAL / D_2) * (m_dot/A_cs)^2 ) /(2*RHO_1ave[i]) "[Pa]"

"****************************************************************************************************************** 

 q_12conv[i] Convective heat transfer rate between the heat transfer fluid and absorber 

******************************************************************************************************************"

CALL Pq_12conv(T_1ave[i], v_1ave[i], T_2[i], Fluid$: q_12conv[i], Cp_1ave[i]) 

"****************************************************************************************************************** 

 q_23cond[i] Conduction heat transfer rate through the absorber 

******************************************************************************************************************"

" Absorber conductance, temperature and material type dependent "

k_23[i] = fk_23(T_2[i], T_3[i]) "[W/m-K]"

q_23cond[i] = 2 * PI * k_23[i] * (T_3[i] - T_2[i]) / LN(D_3 / D_2) "[W/m]"

"****************************************************************************************************************** 

 q_34conv[i] Convective heat transfer rate between the absorber pipe and glazing 

******************************************************************************************************************"

q_34conv[i] = fq_34conv(T_3[i], T_4[i], AnnulusGas$) "[W/m]"

"****************************************************************************************************************** 

 q_34rad[i] Radiation heat transfer rate between the absorber surface and glazing inner surface 

******************************************************************************************************************"

q_34rad[i] = fq_34rad(T_3[i], T_4[i], i) "[W/m]"

"****************************************************************************************************************** 

 q_45cond[i] Conduction heat transfer rate through the glass envelope 

******************************************************************************************************************"

q_45cond[i] = 2 * PI * K_45 * (T_4[i] - T_5[i]) / LN(D_5 / D_4) "[W/m]"

"****************************************************************************************************************** 

 q_56conv[i] Convective heat transfer rate from the glazing to the atmosphere 

******************************************************************************************************************"

q_56conv[i] = fq_56conv(T_5[i], T_6) "[W/m]"
142 

"****************************************************************************************************************** 

 q_57rad[i] Radiation heat transfer rate between the glazing outer surface and the sky 

******************************************************************************************************************"

q_57rad[i] = fq_57rad(T_5[i], T_7) "[W/m]"

"****************************************************************************************************************** 

 One dimensional (Radial) model 

******************************************************************************************************************"

CALL Pq_45cond(q_34conv[i], q_34rad[i]: q_45cond[i]) "[W/m]"

CALL Pq_56conv(q_45cond[i], q_5SolAbs, q_57rad[i]: q_56conv[i]) "[W/m]"

q_12conv[i] = q_23cond[i] "[W/m]"

q_3SolAbs - q_23cond[i] - q_34conv[i] - q_34rad[i] - q_cond_bracket[i] / DELTAL = 0 "[W/m]"

End 

"****************************************************************************************************************** 

******************************************************************************************************************* 

******************************************************************************************************************"

" Outlet temperature set equal to the final increment outlet temperature "

T_1outlet = T_1out[imax] "[C]"

v_1outlet = v_1out[imax] "[m/s]"

DELTAP_total = SUM(DELTAP[j], j=1, imax) "[Pa]"

"****************************************************************************************************************** 

 Average HTF Temperature 

******************************************************************************************************************"

T_ave = AVERAGE(T_1ave[1..imax]) "[C]"

"****************************************************************************************************************** 

 Effective optical efficiency and optical loss 

******************************************************************************************************************"

EffectOptEff = (OptEff_abs * Alpha_abs + OptEff_env * Alpha_env) 

Eta_EffectOptEff = EffectOptEff * 100 "[%]"

q_OptLoss = q_i * (1-EffectOptEff) "[W/m]"

"****************************************************************************************************************** 

 ETA_Col Collector efficiency 

******************************************************************************************************************"

" Total HTF convection heat gain equals summation of heat gain for each increment "

q_heat_gain = SUM((q_12conv[j] * DELTAL ), j=1,imax) / L_aperture "[W/m]"

ETA_Col = fETA_Col(q_heat_gain, q_i) * 100 "[%]"

"****************************************************************************************************************** 

 q_3SolAbs Solar flux on absorber pipe 
143 

******************************************************************************************************************"

q_3SolAbs = q_i * OptEff_abs * Alpha_abs "[W/m]"

"****************************************************************************************************************** 

 q_5SolAbs Solar Flux on glazing Envelope 

******************************************************************************************************************"

q_5SolAbs = fq_5SolAbs(q_i) "[W/m]"

"****************************************************************************************************************** 

 HCE heat losses 

******************************************************************************************************************"

q_cond_bracket_total = SUM(q_cond_bracket[j], j=1, imax) "[W]"

q_cond_bracket_L = q_cond_bracket_total / L_aperture "[W/m]"

q_HeatLoss_ApertureLength = SUM((HeatLoss[j] * DELTAL), j=1, imax) / L_aperture + 

q_cond_bracket_L "[W/m]"

q_HeatLoss_HCEarea = q_HeatLoss_ApertureLength/(PI*D_5) "[W/m^2]"

q_HeatLoss_ColArea = (q_HeatLoss_ApertureLength * L_aperture)/A_aperture "[W/m^2]"
144 

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Heat Transfer Analysis and Modeling of a Parabolic Trough Solar Receiver 

Implemented in Engineering Equation Solver 

6. AUTHOR(S) 

 R. Forristall 

5. FUNDING NUMBERS

CP032000 

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National Renewable Energy Laboratory 

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13. ABSTRACT (Maximum 200 words)

This report describes the development, validation, and use of a heat transfer model implemented in Engineering Equation 

Solver. The model determines the performance of a parabolic trough solar collector’s linear receiver, also called a heat 

collector element. All heat transfer and thermodynamic equations, optical properties, and parameters used in the model are 

discussed. The modeling assumptions and limitations are also discussed, along with recommendations for model 

improvement. 

15. NUMBER OF PAGES 14. SUBJECT TERMS 

heat transfer; engineering equation solver; parabolic trough; solar collector; heat 

collector element 16. PRICE CODE

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