NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

Technical Manual for the SAM 

Physical Trough Model 

Michael J. Wagner and Paul Gilman

Technical Report

NREL/TP-5500-51825

June 2011
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Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

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Contract No. DE-AC36-08GO28308

Technical Manual for the SAM 

Physical Trough Model 

Michael J. Wagner and Paul Gilman

Prepared under Task No. SS10.1110

Technical Report

NREL/TP-5500-51825

June 2011
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Preface

This technical manual was developed by the authors to document the engineering principles

and methodology underlying the Physical Trough model in the System Advisor Model

(Version 2011.5.4). The Physical Trough model was designed to provide performance analysis

capability for the parabolic trough concentrating solar power technology using system

geometry and physical properties as input. This allows the modeler to assess performance

without (always) requiring pre-existing empirical formulations for the system that may not be

available for all systems of interest.

This model is intended to supplement existing trough modeling tools such as the System

Advisor Empirical Trough model that - as the name implies - approaches system performance

modeling from a “top-down” empirical perspective. Users may find different modeling

approaches useful throughout the various stages of the feasibility analysis and project

development process. An empirical modeling approach works best for systems with

well-known performance curves, or when detailed subsystem models are available and can be

correlated and rolled into simplified relationships.

Alternatively, the first-principles or “physical” modeling approach can be better suited for

systems where performance is unknown a priori. Accordingly, the authors emphasize that both

modeling approaches are equally viable, although empirical and physical models answer

fundamentally different questions and should be used within their inherent constraints. The

authors also note that this document only presents one possible model formulation for

parabolic troughs; many alternative and equally valid modeling solutions exist, and the

approach selected here does not necessarily represent the best possible one.

A complex engineering system such as the parabolic trough technology requires

correspondingly complex performance modeling tools, and with complexity comes an

opportunity for misunderstanding or misuse on the part of the modeler. The authors have

developed this technical manual to provide model transparency for the SAM user and to

document the motivation and justification for the various performance equations used in the

Physical Trough model. This manual presents equations, formulations, coding logic, and

system descriptions as they appear in the performance model, although some “translation” is

required between computer code and textual discussion. All major subsystems are discussed in

detail, including the solar field, power cycle, thermal storage, fossil backup, and control

interface.

This document is intended as a companion to the System Advisor help material, and the

authors recommend that new users become familiar with the Help material in addition to

reviewing this document. The content is catered to technical users with an engineering or

physics background who are interested in understanding the System Advisor Physical Trough

model on a more fundamental level. Non-technical users or those new to concentrating solar

may find this manual instructive in becoming familiar with the engineering principles behind

these systems.

iv
Acknowledgments

The authors acknowledge the invaluable contribution of the System Advisor team for their

feedback and discussion throughout the model development and documentation process.

Special thanks go to Nate Blair (NREL) for technical guidance, and to Aron Dobos and Steve

Janzou (NREL) for their software integration support. The document benefited significantly

from the thorough review and input of Mark Mehos (NREL), Ty Neises (NREL), and Craig

Turchi (NREL).

The authors thank Chuck Kutscher (NREL), Frank Burkholder (NREL), Greg Kolb (Sandia),

Cliff Ho (Sandia), David Kearney (Kearney & Associates), Mike Erbes (Enginomix), Hank

Price (Abengoa Solar), Diego Arias (Abengoa Solar), and Bruce Kelly (Abengoa Solar), all of

whose insight, guidance, and previous work were valuable in the model formulation process.

Finally, the authors acknowledge Tobias Hirsch (DLR), Markus Eck (DLR), Jurgen Dersch  ̈

(DLR), Jan Fabian Feldhoff (DLR), Daniel Benitez (Flabeg), and the SolarPACES Task I team

for enlightening discussion on parabolic trough systems and modeling methodologies.

Mike Wagner

National Renewable Energy Laboratory

v
Table of Contents

1 Introduction 1

1.1 TRNSYS simulation framework ................................................................ 1

1.2 Variable Names and Abbreviations............................................................. 1

1.3 Background and Modeling Approach.......................................................... 6

2 Solar Field 7

2.1 Energy balance ...................................................................................... 7

2.1.1 Nodal Energy Balance .................................................................. 8

2.2 Solar Field Control ................................................................................. 12

2.2.1 Collector defocusing..................................................................... 14

2.2.2 Field freeze protection .................................................................. 16

2.2.3 Accounting for transient effects....................................................... 16

2.3 Collector Assembly and Field Optics.......................................................... 19

2.3.1 Determining an average efficiency value ........................................... 26

2.4 Receivers (HCE’s) .................................................................................. 27

2.4.1 Modeling approach....................................................................... 28

2.4.2 Model formulation ....................................................................... 31

2.5 Piping model......................................................................................... 39

2.5.1 Piping diameter selection............................................................... 40

2.5.2 Piping layout and components ........................................................ 46

2.5.3 Performance calculations ............................................................... 48

3 The Power Cycle 52

3.1 General description................................................................................. 52

3.1.1 Modeling Approach...................................................................... 53

3.1.2 “Basis” Rankine cycle................................................................... 54

3.2 Regression model ................................................................................... 55

3.2.1 Regression modeling concepts ........................................................ 56

3.2.2 Off-design response framework....................................................... 57

3.2.3 Parameter normalization ................................................................ 57

3.2.4 Experimental design ..................................................................... 58

3.2.5 Formulating a model..................................................................... 58

3.2.6 Other calculated output ................................................................. 61

3.3 Heat rejection ........................................................................................ 63

3.3.1 Performance considerations............................................................ 64

3.3.2 Condenser performance limitations .................................................. 66

3.3.3 Wet cooling model ....................................................................... 68

3.3.4 Dry cooling model ....................................................................... 72

3.3.5 Parallel hybrid cooling model ......................................................... 73

3.4 Power cycle implementation ..................................................................... 76

3.4.1 Normal operation ......................................................................... 76

3.4.2 Shutdown and cold startup ............................................................. 77

3.4.3 Standby operation ........................................................................ 78

vi
3.4.4 Warm restart conditions................................................................. 79

4 Thermal storage and plant control 80

4.1 Thermal storage ..................................................................................... 80

4.1.1 TES Sizing ................................................................................. 81

4.1.2 Model formulation ....................................................................... 82

4.1.3 Storage freeze protection ............................................................... 85

4.2 Indirect storage heat exchanger ................................................................. 86

4.2.1 Heat exchanger sizing ................................................................... 87

4.2.2 Heat exchanger performance model ................................................. 88

4.3 Auxiliary heater ..................................................................................... 90

4.4 Plant control.......................................................................................... 91

4.4.1 Controller background and approach ................................................ 91

4.4.2 Available mass flow...................................................................... 92

4.4.3 Converting to thermal energy.......................................................... 93

4.4.4 Control logic............................................................................... 95

4.4.5 Operating modes.......................................................................... 97

4.4.6 Controller iteration ....................................................................... 101

5 Parasitic Losses 105

5.1 Losses modeled in System Advisor ............................................................ 105

5.2 Parasitic loss calculations......................................................................... 106

5.2.1 SCA drives ................................................................................. 106

5.2.2 Power cycle HTF pumps................................................................ 106

5.2.3 Storage HTF pump....................................................................... 106

5.2.4 Fixed parasitic losses .................................................................... 107

5.2.5 Balance of plant parasitics ............................................................. 107

5.2.6 Auxiliary heater operation.............................................................. 107

5.3 Practical considerations ........................................................................... 108

5.4 Net production at design .......................................................................... 108

A Glossary of Terms 112

vii
List of Tables

1 Abbreviations used in this document .......................................................... 2

2 Variable naming conventions .................................................................... 3

3 Variable subscript conventions .................................................................. 5

4 General definitions for each fixed optical loss term ........................................ 26

5 Inputs to the receiver model...................................................................... 30

6 Values of the mean free path between collisions of a molecule for free molecular

convection ............................................................................................ 32

7 Selection of coefficients C and m for Zhukauskas’ correlation based on the

Reynolds number at D3............................................................................ 33

8 Bracket geometry and material properties assumed by System Advisor for conductive heat loss calculations .................................................................... 37

9 Pipe sizing schedules used in the trough model. The wall thickness and schedule

are selected to match a pressure rating of 25 bar. ........................................... 45

10 The assumed piping lengths for the steam generator and pumping system. .......... 46

11 Variable assignments for the piping equipment by domain. Referenced equations are denoted with square brackets. The pressure drop coefficients for the

various piping components are defined in the kΔP column................................ 47

12 Design-point conditions for the basis Rankine cycle. The conditions were selected to match common parabolic trough operating conditions. ....................... 55

13 The experimental design for characterizing power block performance. The experiment is adapted from a three-level full-factorial design. ............................. 59

14 Summary of inputs to the wet cooling model. ............................................... 68

15 Property functions used by the wet cooling model. ........................................ 69

16 Summary of inputs to the dry cooling model. ............................................... 72

17 Double-check criteria for the converged power cycle solution under normal operation. ................................................................................................ 77

18 A summary of the parasitic losses accounted for by System Advisor ................. 105

List of Figures

1 One possible field arrangement, where the field is broken up into two header

sections. Each loop in this illustration contains 8 individual SCA’s, and each

portion of the header is connected to two loops - one on the top and one on the

bottom of the image. ............................................................................... 7

2 The nodal structure of the loop is shown (left) where each SCA in the loop is

an autonomous node. This framework allows multiple receiver/collector types -

shown as A and B (center) - and user-specified defocusing schemes (right). ........ 8

3 Energy balance for the receivers in an SCA. The enclosed box represents the

control volume, encompassing the HTF within the absorber tubes and piping

for a single SCA..................................................................................... 8

viii
4 Hypothetical situation where neither the final temperature, nor the average of

initial and final temperatures gives an accurate representation of the temperature

over the time step. This situation can be encountered in dynamic systems like

CSP plants. ........................................................................................... 17

5 The trough includes both a collector to reflect light and a receiver to absorb and

transport heat......................................................................................... 19

6 The angle between the solar irradiation and the normal vector to the collector

aperture plane. ....................................................................................... 20

7 Focal length geometry for calculating the average focal length. ........................ 23

8 Two adjacent collector rows may shadow each other if the tracking angle is

sufficiently severe. The shadowing is subject to the collector aperture width, the

row spacing (centerline to centerline), and the tracking angle of the collectors. .... 24

9 A heat balance for the modeled receiver. Heat transfer in the radial direction

(left to right) is considered, while circumferential and axial transfer is not. ......... 28

10 The thermal resistance network for the “intact” receiver model shown in Figure

9. Energy is absorbed at T3 and T4−5........................................................... 29

11 The thermal resistance network for the “broken glass” receiver model. Energy

is absorbed on the absorber tube surface at T3 and heat is exchanged directly

with the sky and ambient temperatures. ....................................................... 29

12 An illustration of the flow diverting from the header into individual field loops.

In this example, two loops extract HTF per section. The diameter of the header

is adjusted to maintain optimal flow velocities. ............................................. 41

13 Three possible field configurations modeled by System Advisor. Two (top left),

four (bottom left), and six (right) field subsections are shown........................... 42

14 Numbering scheme and notable mass flow rates for the subfield and runner piping. 44

15 The solution for Eq.[2.108] at X = 0 is shown for three unique Re values. .......... 49

16 Control volume near the power cycle condenser. The power cycle control volume bisects the condenser to include the steam flow but not the cooling flow. ...... 53

17 Schematic for simplified basis Rankine cycle ............................................... 54

18 Data showing how the power and heat output varies with condenser pressure

[bar] at three distinct levels of HTF inlet temperature. .................................... 60

19 Data showing how the power and heat output varies with HTF temperature at

three distinct levels of HTF mass flow rate (top), and the associated interaction

effects (bottom)...................................................................................... 61

20 A simplified Rankine cycle diagram. Heat is added to the cycle by the solar

field via a heat exchanger. Heat is removed from the system via a wet cooling

tower. Power output is equal to the heat addition minus the heat rejection. .......... 64

21 Dry and wet bulb temperatures for five summer days in an arid climate. Peak

temperature differences approach 20◦C during the hottest hours of the day.......... 65

22 The impact of a 20◦C temperature change on condenser pressure and cycle performance. ............................................................................................. 66

23 Steam velocity as a function of steam pressure through ductwork in a plant

designed for a minimum condenser pressure of 0.068 bar ............................... 67

ix
24 An illustration of the various temperature rises that influence the condenser pressure for a wet-cooled system. These include the ambient wet-bulb temperature,

the approach temperature, the cooling water temperature rise, and the heat exchanger hot side temperature difference....................................................... 70

25 An illustration of the various temperature rises that influence the condenser pressure for a dry-cooled system. These include the ambient dry-bulb temperature,

the air temperature rise, and the heat exchanger hot side temperature difference. .. 73

26 An illustration of the parallel wet/dry cooling configuration is shown. The ACC

(left) and wet system (right) share the heat rejection load. ............................... 74

27 The trough power plant subsystems include the solar field, piping, thermal storage, auxiliary heating, and the power cycle. ................................................. 80

28 A schematic of the variable-volume tank model. Fluid level varies with differences in the inlet and outlet mass flow rate. The outlet temperature is equal to

the average of the fully mixed tank. ............................................................ 82

29 Conceptual representation of the tank control volume..................................... 83

30 The indirect storage heat exchanger is defined by the hot and cold side approach

temperatures. A sketch of the physical system arrangement (left) corresponds

to a plot of the temperatures in the heat exchanger as a function of position (right). 86

31 The control flow diagram for the physical trough model.................................. 103

32 The net and gross power over a period of four days........................................ 109

x
1 Introduction

This manual defines and documents the conventions, methodology, and information flow

associated with the System Advisor Model (SAM) physical trough model in the System

Advisor Model [10]. A detailed overview of each subsystem is provided, with the

engineering/mathematical basis laid out where appropriate. While this documentation contains

general information on the model structure, the beginning modeler may find portions of the

content challenging. For an introductory overview on how to use the model, please refer to the

System Advisor help system. Definitions of technical terms (denoted with italic text) are

provided in the Glossary of Terms at the end of this document (page 112).

The System Advisor Model provides a consistent framework for analyzing and comparing

power system costs and performance across the range of solar technologies and markets, from

photovoltaic (PV) systems for residential and commercial markets to concentrating solar

power and large PV systems for utility markets.

System Advisor is based on an hourly simulation engine that interacts with performance, cost,

and finance models to calculate energy output, energy costs, and cash flows. The software can

also account for the effect of incentives on cash flows. System Advisor’s spreadsheet interface

allows for exchanging data with external models developed in Microsoft® Excel. Most of

System Advisor’s inputs can be used as parametric variables for sensitivity studies to

investigate impacts of variations in performance, cost, and financial parameters on model

results.

1.1 TRNSYS simulation framework

System Advisor models system performance using the TRNSYS1 software developed at the

University of Wisconsin combined with customized components. TRNSYS is a validated,

time-series simulation program that can simulate the performance of photovoltaic,

concentrating solar power, water heating systems, and other renewable energy systems using

hourly resource data. TRNSYS is integrated into System Advisor so there is no need to install

TRNSYS software or be familiar with its use to run Solar Advisor.

The default source code folder \SAM\<version number>\exelib\trnsys\source contains

the FORTRAN code for each TRNSYS module: sam_mw_trough_Type250.f90 (solar field),

sam_mw_trough_Type251.f90 (storage and dispatch), sam_mw_pt_Type224.f90 (power

cycle), and the shared HTF property subroutines sam_mw_pt_Type229.f90 and

sam_mw_pt_propmod.

1.2 Variable Names and Abbreviations

The following tables contain information and conventions used in this document. Table 1 lists

the abbreviations, Table 2 lists variable naming conventions, and Table 3 list commonly used

variable subscript conventions.

1For more information on the TRNSYS software, refer to documentation from the University of

Wisconsin at sel.me.wisc.edu/trnsys/default.htm

1
Table 1: Abbreviations used in this document

Abbreviation Description

ACC Air-cooled condenser

CSP Concentrating Solar Power

DNI Direct-normal irradiation

HTF Heat transfer fluid

IAM Incidence angle modifier

IOCop Inlet/outlet/cross-over pipe

ITD Initial temperature difference

LHV Lower heating value

NREL National Renewable Energy Laboratory

SAM System Advisor Model

TES Thermal energy storage

TOU Time of use

2
Table 2: Variable naming conventions

Name Description Units Units Abbrv.

A Area meters squared m2

c, cp Specific heat Joules per kilogram Kelvin J

kgK

C Calculation coefficient varies varies

C ̇ Thermal capacity rate Watts per Kelvin W

K

D Diameter meters m

E Energy Joules J

g Gravitation constant meters per sq. second m

s2

f Fraction none none

f r Friction factor none none

h Enthalpy Joules per kilogram J

kg

H Height meters m

i, j Counting indexes none none

Ibn Direct solar irradiation Watts per sq. meter W

m2

k Thermal conductivity Watts per meter-Kelvin W

m·K

L Length meters m

m Mass kilograms kg

m ̇ Mass flow rate kilograms per second kg

s

(mc) Thermal inertia Joules per Kelvin J

K

N Integer number none none

Nu Nusselt number none none

P Pressure Pascals Pa

Pr Prandtl number none none

q ̇ Heat transfer rate Watts W

Q ̇ Cycle heat absorption rate Watts W

r Ratio none none

R Radius meters m

Rˆ Thermal resistance Watts per Kelvin W

K

Ra Rayleigh number none none

Re Reynolds number none none

s Entropy Joules per kilogram Kelvin J

kg

T Temperature Kelvin K

T Average temperature Kelvin K

t Time seconds s

u Specific internal energy Joules per kilogram J

kg

U Internal energy Joules J

UA Thermal conductance Watts per Kelvin W

K

v Velocity meters per second m

s

V Volume meters cubed m3

W ̇ Cycle power output Watts W

Continued...

3
Continued...

Name Description Units Units Abbrv.

α Absorptance none none

β Volumetric expansion coef. inverse Kelvin 1

K

Δ Change in value none none

γ Thermal loss coefficient Watts per sq. meter Kelvin W

m2·K

γsol Solar azimuth degrees ◦

δ Solar declination degrees ◦

η Efficiency none none

θ Aperture incidence angle degrees ◦

θe/θz Solar elevation/zenith degrees ◦

ε Emittance / Effectiveness none none

μ Dynamic viscosity Pascal-seconds Pa ·s

ν Kinematic viscosity sq. meters per second m2

s

π Pi none none

ρ Density kilograms per cubic meter kg

m3

τ Transmittance none none

φ Latitude degrees ◦

χ Thermodynamic quality none none

ψ Longitude degrees ◦

ω Hour angle degrees ◦

4
Table 3: Variable subscript conventions

0 Previous value / initial value

1..9 Item n in a sequence

1tank Single tank

A Available

abs Absorbed

ad j Adjusted

air Ambient air

amb Ambient

ap Aperture

approach Approach temperature

aux Auxiliary (fossil)

ave Average

bal Balance-of-plant

bd Blowdown

boil Steam boiler

brac Bracket

c Cold

cap Capacity

calc Calculated

chg Charging

col Collector

cond Conduction / condenser

conv Convection

cs Cross-sectional

cw Cooling water

cycle Power cycle

db Dry-bulb

def Defocus

dem Demand

des Design point

dis Discharging

drift Condenser drift

dump Dumped energy

duty Heat exchanger duty

e Electric

env Envelope

f Focal

f an Cooling fan

fin Final

f sec Field sections

f uel Fossil fuel

g Guess

gross Gross electric

f p Freeze protection

h Hot

hdr Header

hdrsec Header section

hgrp Header section groups

hl Heat loss

hsec Header sections

ht f Heat transfer fluid

hyd Hydraulic

hx Heat exchanger

in Inlet

inc Incident

loop All SCA’s in a loop

LHV Lower heating value

m Mirror

min Minimum

max Maximum

ND Non-dimensional, normalized

net Net electric

opt Optical

out Outlet

p Pump

par Parasitic

pb Power block (or power cycle)

pm Per meter basis

rad Radiation

rec Receiver

re j Rejected

run Runner pipe

runsec Runner section

s Isentropic

sby Standby

sca Solar collector assembly

set point Design point or setpoint

s f Solar field

sol Solar

spacing Row spacing

st Steam

start Startup

sys System

tes Thermal energy storage

th Thermal

tot Total

tou Time of use

track Collector tracking

trans Transient

w Water

wb Wet-bulb

↑ / ↓ Increase / Decrease

5
1.3 Background and Modeling Approach

The physical trough model characterizes a parabolic trough CSP plant by deriving

performance equations from first principles of heat transfer and thermodynamics where

possible. In practice, this means that empirical “curve-fit” relationships are eliminated to the

degree that is practical for the type of modeling analysis done in System Advisor. The primary

benefit of this approach is the added flexibility in changing system parameters and component

properties at a fundamental level (i.e. absorber emissivity, glass thickness, etc.) and simulating

their impact on overall system performance. With this increased flexibility come a few

drawbacks; these include the addition of multiple layers of modeling uncertainty and the

increased opportunity for divergence in the results from a real system. While an empirical

model can produce high accuracy over the range of parameters used in its development, such

models cannot provide predictive performance outside these ranges. Both the physical and

empirical trough models are included in System Advisor to allow for comparisons between the

physical and empirical modeling approaches.

Besides fulfilling the goal of deriving system performance from first principles, the physical

modeling approach achieves several other objectives: the model includes transient effects

related to the thermal capacity of the HTF in the field piping, headers, and the balance of the

plant; it allows for more flexible field component specification, including multiple receiver and

collector types within a single loop; it maintains a reasonably short run-time allowing for

parametric and statistical analyses; and it makes use of existing models where possible.

Previously existing subsystem models that are adapted and incorporated into the physical

model include the collector model from System Advisor’s empirical trough model, the receiver

heat loss model [8], a field piping pressure drop model [14], and the power cycle performance

model [20] originally developed for System Advisor’s power tower CSP system model. These

models are discussed in more detail in dedicated sections of this report.

6
2 Solar Field

The solar field is the heat-collecting portion of the plant. It consists of one or more loops of

solar collector assemblies (SCA’s), with each loop laid out in parallel. A common header pipe

provides each loop with an equal flow rate of heat transfer fluid (HTF), and a second header

collects the hot HTF to return it either directly to the power cycle for power generation or to

the thermal energy storage system for use at a later time. To minimize pumping pressure

losses, the field is typically divided into multiple sections, each section with its own header set,

and the power cycle is situated near the middle of the field. Figure 1 shows one possible plant

layout where two header sections are used for 20 total loops.

Figure 1: One possible field arrangement, where the field is broken up into two header sections.

Each loop in this illustration contains 8 individual SCA’s, and each portion of the header is

connected to two loops - one on the top and one on the bottom of the image.

2.1 Energy balance

Within each loop, a number of SCA’s are used to incrementally heat the HTF to the design

outlet temperature. Each SCA is composed of a number of parabolic collectors and their

receivers2 in series that share a single common tracking drive. In this model, the SCA serves

as the lowest level of discretization. Each SCA is treated as an independent calculation node

within the loop, and the absorbed energy, losses, temperature, pressure drop, and other

performance values are calculated independently for each SCA. This allows each SCA to

impact performance separately and potentially allows each SCA to contain different receiver

and/or collector attributes. System Advisor allows the user to specify unique geometry and

performance characteristics for each SCA in the loop up to a limit of four unique

configurations. The order in which the SCA’s are defocused in the loop during high-flux

conditions can also be modified under this framework. Figure 2 illustrates these principles.

2The term “receiver” in this model is interchangeable with “heat collection element” (or HCE) that

has been used in other models

7
Figure 2: The nodal structure of the loop is shown (left) where each SCA in the loop is an

autonomous node. This framework allows multiple receiver/collector types - shown as A and B

(center) - and user-specified defocusing schemes (right).

2.1.1 Nodal Energy Balance

A typical steady-state receiver model determines the temperature rise across the node by

considering the absorbed energy, the mass flow rate of HTF through the receiver, and the

specific heat of the HTF. This energy balance for node i is represented in Eq.[2.1].

ΔTi = Tout,i −Tin,i = q ̇abs

m ̇ ht f cht f

(2.1)

However, in the case of the parabolic trough technology, the thermal inertia associated with the

energy state of the node can impact performance to the extent that a steady-state model is

insufficient; thus transient terms must be included. The most significant transient effect in the

solar field is the thermal mass of the HTF in the headers and in the receiver piping, so we need

to consider the change in energy of the HTF in deriving the energy balance equations. We’ll

start with the energy balance for a single SCA node drawn in Figure 3.

Figure 3: Energy balance for the receivers in an SCA. The enclosed box represents the control

volume, encompassing the HTF within the absorber tubes and piping for a single SCA.

8
The energy balance above shows a receiver tube with an inlet flow, outlet flow, absorbed

energy (net absorption is positive, net loss is negative), and an internal energy term. The q ̇ heat

flows are functions of the mass flow rate (m ̇ ht f) which is constant across the boundary, and the

temperatures Tin and Tout. The average nodal temperature T is equal to the average of the inlet

and outlet temperatures of the calculation node, since the temperature rise across the node is

assumed to be linear.

The internal energy term ∂U

∂t represents the change in energy of the node as a function of time

t. Equivalently, this term can be expressed as:

∂U

∂t = 

m cht f + (mc)bal,sca L

 ∂T

∂t (2.2)

Here, m is the mass of the HTF contained in the node, L is the length of a single SCA, and cht f

is the specific heat of the HTF. An additional thermal inertia term (mc)bal,sca is included to

account for the thermal mass of piping, joints, insulation, and other SCA components that

thermally cycle with the HTF. The (mc)bal,sca L term is dimensionally equivalent to the HTF

capacitance term m· cht f , even though it is input into the model as a single value. This

convention eliminates the need for specific knowledge of either the exact mass or specific heat

of the additional thermal inertia while still allowing the user to account for the inertia effect.

This term is dimensionally defined to represent the amount of thermal energy per meter of

collector length required to raise the temperature of the node one degree K. The input units are

Wt−h/m−K, and the value is converted within the code to units of J/m−K.

Note that in terms of the mathematical formulation, the HTF and material properties are

assumed to be constant. However, when evaluating system performance, material and HTF

thermal properties are evaluated as functions of temperature (and pressure, if applicable).

The total energy balance in the control volume is:

q ̇in +q ̇abs = ∂U

∂t

+q ̇out (2.3)

The inlet and outlet heat flows can be expressed as:

q ̇in −q ̇out = m ̇ ht f cht f (Tin −Tout)

= 2 m ̇ ht f cht f 

Tin −T

 (2.4)

Substituting this equation and the definition for the internal energy term into the energy

balance equation, then solving for the first differential, the result is:

∂T

∂t = 2 m ̇ ht f(Tin −T) +q ̇absm cht f + (mc)bal,sca

(2.5)

9
This is a linear first order differential equation, which has the general solution:

T = q ̇abs

2 m ̇ ht f · cht f

+C1 exp− 2 m ̇ ht f cht f

m cht f + (mc)bal,sca

Δt



+Tin (2.6)

This equation has an unknown constant C1 that can be determined by enforcing a boundary

condition. In this situation, we know that the average nodal temperature T = T0 at the

beginning of the time step when t = 0, and we define T0 to be the temperature T at the end of

the previous time step (the average temperature is T = (Tout +Tin)/2). Thus, solving for the

unknown constant C1:

T0 = T|t=0

= q ̇abs

2 ·m ̇ ht f cht f

+C1e0 +Tin

C1 = T0 − q ̇abs

2 ·m ̇ ht f cht f

−Tin (2.7)

Finally, we substitute the constant into the general solution to find the final equation for the

outlet temperature from each SCA shown in Eq.[2.8].

For i = 1,Nsca:

Tout,i = 2 Ti −Tin,i

Tout,i = q ̇abs,i

m ̇ ht f cht f,i

+Tin,i

+2 ·



T0,i − q ̇abs,i

2 ·m ̇ ht f cht f,i

−Tin,i



exp −2 m ̇ ht f cht f,i Δt

mi cht f,i +mci,bal,sca Li



(2.8)

This equation is applied to each node i in the loop, where Tin,i is equal to the outlet temperature

of the previous node in the loop, Tout,i−1. Since the calculated temperature for each node

depends on both the inlet temperature of the previous node and the node temperature from the

previous time step, these values must be established as boundary conditions. The temperature

of the node at the previous time step is stored from time step to time step, and the inlet

temperature is set equal to the outlet temperature of the previous node for each but the first

node in the loop to satisfy these requirements. The HTF mass of each node is calculated as a

function of the receiver piping volume and the local HTF density.

For i = 1,Nsca:

mi = ρht f,i Li Acs,i (2.9)

The inlet temperature at the first node, representing the inlet of the entire field, requires

additional consideration. In a similar derivation process as the one described in Figure 3, a

“system” temperature is calculated for both the hot and cold sides of the solar field. The cold

10
system temperature is used as the node #1 inlet temperature, and the hot system temperature is

used as the effective solar field outlet temperature. These values incorporate the thermal inertia

associated with the header and balance-of-plant HTF mass. Under steady-state conditions, the

loop inlet HTF temperature equals either:

• the power cycle outlet temperature

• the storage loop outlet temperature

• a mass-flow weighted average of the storage and power cycle outlet temperatures

• or the solar field outlet temperature (during nighttime recirculation)

depending on the control situation. However, directly using any of these outlet temperatures as

the loop inlet value is inaccurate because it fails to account for the thermal inertia of the

header. If we include thermal inertia as a transient effect, the derived equation for loop inlet

temperature (denoted Tsys,c) is shown in Eq.[2.10].

Tsys,c = (Tsys,c,0 −Ts f,in) exp

⎡

⎣− m ̇ ht f

Vc · ρc + (mc)bal,c

cc

Δt

⎤

⎦+Ts f,in (2.10)

Here, the cold header temperature from the last time step is Tsys,c,0, the volume in the cold

header and the runner pipe is given by Vc, and cold fluid density is ρc. Analogously, the hot

system outlet temperature combines loop outlet flow, the header and runner pipe volumes, and

the state of the system at the last time step.

Tsys,h = (Tsys,h,0 −T∗

loop,out) exp

⎡

⎣− m ̇ ht f

Vh · ρh + (mc)bal,h

ch

Δt

⎤

⎦+T∗

loop,out (2.11)

Here, T∗

loop,out represents the heat-loss adjusted loop outlet temperature calculated in Eq.

[2.26]. The capacity term (mc)bal,h (or (mc)bal,c for the cold header) is used to account for any

thermal inertia that isn’t included in the HTF volume calculations. Varying this term

effectively adds or subtracts internal energy capacity to the HTF in the system; its best use is as

an empirical adjustment factor to help match observed plant performance. This variable is

defined in terms of the thermal power (kilowatt hours) per gross electricity capacity

(Megawatts) needed to raise the header temperature one degree Celsius. The (mc)bal,h term is

applied specifically to the temperature calculation for the hot header, while (mc)bal,c describes

only the additional cold header thermal inertia.

To illustrate this concept, consider the following example. A hypothetical 100 MWe (gross)

trough plant with a 35% conversion efficiency is observed starting up from a cold overnight

temperature of 140◦C. The system temperature increases a total of 150◦C to 290◦C after 1

hour of running the solar field at full load. A modeler predicts from known HTF inventory that

the plant should have reached its startup temperature in only 30 minutes. Thus, the (mc)bal

11
terms must together account for an half of the full-load hour that the plant took to start up.

Assuming both coefficients are equal, the hot-side coefficient would be calculated as follows:

(mc)bal,h = 0.5

2 · 0.35 · 100,000[kWe]100[MWe] · 150[K]

(2.12)

(mc)bal,h = 4.8

 kW hr

MWecap ·K



Realistically, the value calculated here will be lower once the thermal inertia of the solar field

piping, insulation, and joints are considered. This calculation is only provided as a

demonstration of the procedure for determining an empirical thermal inertia term, and the user

should carefully select their input to model their desired system.

Section 2.1 Summary

• The model calculates solar field temperatures and corresponding performance values independently at each SCA specified in the loop.

• Each SCA can have a different associated collector or receiver, up to four in total.

• The performance of one loop is representative of each loop in the field.

• The model determines the solar field inlet temperature by accounting for the return flow

from storage/power cycle/field outlet and the volume of the cold header and piping.

• The model determines the solar field outlet temperature by accounting for the hot flow from

the collection elements, the volume of the hot header and piping, and it adds user-specified

thermal capacity terms that impact the transient behavior of the system.

2.2 Solar Field Control

CSP plants convert direct normal irradiation (DNI) from the sun into thermal energy and

eventually into electricity or some other useful product. DNI can vary significantly over a

relatively short period of time, and the solar field in a trough plant must be designed to handle

these variations. The solar field control algorithm uses user-specified input to make operational

decisions based on the DNI resource level, ambient temperature, presence of thermal storage,

etc. The minimum HTF mass flow rate, the maximum HTF mass flow rate, the HTF outlet

temperature, defocus state, and the temperature of each node relative to the freeze-protection

set point are parameters that are monitored and enforced by the field controller.

The field logic employs an “ideal” control strategy. This means that the HTF mass flow rate is

controlled to allow the loop outlet temperature to meet the design point value when possible.

Receiver heat loss and surface temperature are both functions of the HTF temperature, and

12
HTF temperature in turn is a function of HTF mass flow rate, so the HTF temperature must be

calculated iteratively. System Advisor uses successive substitution, where an initial estimate

for the field mass flow rate is provided along with guessed temperature values, then the mass

flow rate is recalculated and adjusted until the outlet temperature converges to the design value.

If the mass flow rate is calculated to fall outside of the acceptable range specified by the user,

special control handling is required. The minimum and maximum allowable loop flow rates

are specified indirectly by the user on the System Advisor Solar Field page through an

allowable HTF velocity range. Eq.[2.13] shows how the minimum (vht f,min) and the maximum

(vht f,max) HTF velocities are converted to mass flow rates.

m ̇ ht f,min = vht f,min ρht f,c π

Dmin

2

2

m ̇ ht f,max = vht f,max ρht f,h π

Dmin

2

2

(2.13)

Here, ρht f represents the HTF density at the cold (c) and hot (h) design temperatures, and

Dmin is the minimum diameter receiver tube that is present in the system.

The solar field mass flow rate is constantly adjusted within the upper and lower flow limits to

maintain the desired HTF outlet temperature. Several methods of calculation are possible for

determining the mass flow rate. System Advisor uses a customized numerical solution

algorithm that considers the rate of change in the mass flow rate and HTF outlet temperature

variables from iteration to iteration. Numerical techniques generally require an initial guess

value, and this is the case for System Advisor. The initial mass flow guess is shown in Eq.2.14.

m ̇ ht f,guess = ∑Nsca

i=1 q ̇i,abs Nloops

cht f,ave (Tloop,out,des −Ts f,in) (2.14)

Note that the average specific heat value refers to the integral average over the length of the

collector loop, the HTF outlet temperature is the solar field design outlet temperature, and the

inlet temperature is equal to the value provided by the power block or fluid source. Subsequent

iterations depart from this energy balance basis by using the mass flow and temperature values

from the previous iteration (m ̇

ht f and Tloop,out, respectively), and the values from the current

iteration (m ̇ ht f and Tloop,out) to predict the mass flow for the next iteration (m ̇ ∗

ht f). This

calculation is shown in Eq.[2.15].

m ̇ ∗

ht f = m ̇ ht f + (m ̇ ht f −m ̇ht f)·



Tloop,out,des −Tloop,out

Tloop,out −T

loop,out 


(2.15)

13
where:

−0.75 |m ̇ ht f | ≤ 

(m ̇ ht f −m ̇

ht f)·



Tloop,out,des −Tloop,out

Tloop,out −T

loop,out 
 ≤ 0.75 |m ̇ ht f | (2.16)

The limits of [−.75,+.75] in Eq. [2.16] were selected to promote model convergence and do

not reflect any physical limit on the bounds of the mass flow rate. The limits apply from

iteration-to-iteration within a single time step, while time-step-to-time-step variations in the

mass flow rate may exceed these limits.

2.2.1 Collector defocusing

If the calculated mass flow rate through the loop falls below the minimum allowable flow rate

(m ̇ ht f,min), then the code resets the flow rate to be equal to this minimum value and recalculates

the outlet temperature of the loop, assuming a fixed flow rate. Thus, in conditions where the

minimum criterion is not met, the outlet temperature from the field will fall below the design

value.

In the opposite case, the solar field may absorb so much energy that the mass flow rate

required to maintain the design outlet temperature exceeds the maximum specified value. Once

the mass flow rate reaches an upper limit, the only way to avoid over-temperature HTF (and a

dangerous situation) is to reduce the amount of absorbed energy by defocusing collectors. The

plant controller can defocus collectors either when the amount of energy produced by the solar

field exceeds what can be consumed in the power cycle and/or storage, or when the converged

mass flow rate exceeds the value calculated in Eq.[2.13]. For the latter case, the logic considers

the total amount of absorbed energy in each SCA in the field and defocuses the SCA’s until the

total absorbed energy falls below the threshold required by the maximum mass flow rate.

SCA’s are defocused using one of three user-selected schemes.

Full SCA Defocusing

The first option allows the user to specify an order in which each SCA in a loop should be

defocused. During defocusing periods, the SCA’s will be completely defocused in successive

order until the total absorbed power falls below the imposed limit. Note that thermal losses

incurred by the defocused SCA(s) are still accounted for in all energy calculations. Eq.[2.17]

shows how the code determines the number of collectors to defocus. The total effective

thermal output power qs f,eff is reduced by the amount absorbed in each collector according to

defocus order until the maximum allowable thermal output qs f,limit is met.The array items A[i]

indicate the order of defocusing requested by the user, and the subscripts abs, hl, and inc refer

to the absorbed thermal energy, thermal heat losses, and irradiation on each SCA, respectively.

For i = 1,Nsca:

And while q ̇s f,eff > q ̇s f,limit and i <= Nsca:

q ̇s f,eff = q ̇s f,eff −maxq ̇A[i],abs −q ̇A[i],hl

,0

 (2.17)

14
Then, for j = 1,i:

q ̇i,inc = 0

This option is activated by clearing the Allow partial defocusing check box on the Solar Field

page.

Partial Sequenced Defocusing

The second option also allows sequenced defocusing of SCA’s, but instead of completely

defocusing the SCA’s, partial defocusing is allowed. Sometimes called feathering, this practice

is implemented in real plants by modulating the tracking angle to partially shift the receiver out

of the reflected radiation, thus reducing the intercept factor. This control scheme sequentially

defocuses the SCA’s according to the order specified on the Solar Field page, but allows one

SCA to be defocused partially. The defocusing calculations in this mode require three steps:

first, the number of affected SCA’s is determined, second, the SCA’s that are wholly unneeded

are completely defocused, and third, the final defocused SCA is partially modulated to match

the desired thermal output.

For i = 1,Nsca:

And while q ̇s f,eff > q ̇s f,limit and i <= Nsca:

q ̇s f,eff = q ̇s f,eff −maxq ̇A[i],abs −q ̇A[i],hl

,0

 (2.18)

Then, for j = 1,i−1:

q ̇i,inc = 0

Finally, for j = i:

q ̇i,inc = q ̇i,inc

q ̇s f,eff −∑Nsca

j=i+1 q ̇i,abs

q ̇i,abs

(2.19)

This option is selected by enabling the Allow partial defocusing check box and selecting the

Sequenced option from the drop down menu.

Partial Simultaneous Defocusing

The final option implements defocusing by simultaneously partially defocusing all of the

SCA’s in the loop. Because all SCA’s are defocused equally, the defocus order specified in the

loop configuration control on the Solar Field page is disregarded. Eq.[2.20] shows the simple

scaling method used for each SCA.

For i = 1,Nsca:

q ̇i,inc = q ̇i,inc

q ̇s f,limit

q ̇s f,eff

(2.20)

Select this method by enabling the Allow partial defocusing check box and choosing the

Simultaneous option in the drop down menu.

15
Note that the defocusing and control algorithms are iterative; consequently, the variables

appearing on both sides of the equations from Eq.[2.17] through Eq.[2.20] are provided with

initial values and are iteratively modified until the solution converges.

The model tracks and reports the total defocused energy (also called “dumped” energy). This

value measures the amount of incident energy that is not allowed to reach the collector as a

result of defocusing, and the reported value includes dumped energy caused by both the field

controller and the general plant controller. Dumped energy is approximated by considering the

product of the total aperture area (Aap,tot) collector optical efficiency during the time step

(ηopt), the solar irradiation (Ibn) and the fraction of defocused SCA’s during the time step

(ηdef). Thermal losses are not included in the dumped energy calculation.

q ̇dump = Aap,tot Ibn ηopt ηdef (2.21)

2.2.2 Field freeze protection

During times of extended shutdown or cool nighttime temperatures, the heat transfer fluid in

the solar field may cool to an unacceptably low temperature. To avoid this situation, electric

heat trace equipment can provide supplemental heat to the HTF in the solar field. System

Advisor models this situation by enforcing the minimum field HTF temperature specified as

the freeze protection temperature on the Solar Field page. The temperatures in each node of the

loop and in the headers are monitored. If the temperature falls below the minimum allowable

value, heat is added to the system to maintain the temperature at the minimum value. This

energy is tracked and reported as a parasitic loss. The magnitude of freeze protection energy is

calculated for each of the i nodes in the loop of Nsca collector assemblies in Eq.[2.22].

Ef p =

Nsca

∑

i=1



(Tf p −Tht f,ave,i) Acs,i Li ρht f cht f +q ̇i,hl·Nloops (2.22)

The total freeze protection energy Ef p can be divided by the time step to determine the rate of

energy consumption. In Eq.[2.22] the average node temperature is represented by Tht f,ave,i, the

receiver tube cross-sectional area is Acs,i, the length of the receiver tube for the node is Li, and

density and specific heat properties of the HTF are used. Likewise, the freeze protection

energy required for the headers and runner piping is calculated in the following general form.

Ef p,hdr = 

Tf p −Thdr

·(Vhdr +Vrun) ρ cp +q ̇hl,hdr +q ̇hl,run (2.23)

Electric heat tracing is assumed for the freeze protection parasitic.

2.2.3 Accounting for transient effects

Unlike traditional utility-scale power plants, CSP systems are subject to frequent and

significant temporal fluctuations in the thermal resource. Fossil or nuclear plants will spend

most of their lifetime operating near design conditions, but CSP plants vary significantly in

output over relatively small time periods. Thus, the impact of transient effects may become

16
significant in the overall performance of the plant. Simply including transient terms in the

formulation of the system model (see Eq.[2.8] for example) may not be sufficient to adequately

account for their impact. Instead, averaged equations that account for plant behavior over the

duration of the time step are required.

Numerical simulations rely on an approximation that assumes that a continuous variable

surface can be discretized into finite homogeneous sections. This is the case for the calculation

of loop temperature depicted in Figure 2 where the collector loop is discretized according to

the number of SCA’s in the loop. Likewise, the time variable t that is assumed to be continuous

in the model formulation must be discretized for the numerical simulations performed by

System Advisor. The challenge for CSP simulations that rely on hourly weather information is

that the discretized time step is often much longer than what is required to change the state of

the CSP system. The state of the system at the end of the 1-hour time step may not really

resemble conditions through most of the hour. The 1-hour time step is long enough that

calculating the solar field mass flow based on the final system temperature, or even using the

average temperature, would lead to significant inaccuracies. Figure 4 illustrates this difficulty.

Figure 4: Hypothetical situation where neither the final temperature, nor the average of initial and

final temperatures gives an accurate representation of the temperature over the time step. This

situation can be encountered in dynamic systems like CSP plants.

To remedy this situation, System Advisor adjusts the total absorbed energy by subtracting the

amount of energy that contributed to changing the energy state of the plant (i.e. the “transient

energy”). The transient energy is calculated in Eq.[2.24].

Etrans = 

Vhdr,h ρht f,h cht f,h + (mc)bal,h



(Tsys,h −Tsys,h,0)

+

Vhdr,c ρht f,c cht f,c + (mc)bal,c



(Tsys,c −Tsys,c,0) (2.24)

+

Nsca

∑

i=1



Acs,i Li ρht f,i cht f,i +Li (mc)bal,sca,i



·(T ̄i −T ̄i,0)·Nloops

Equation [2.24] consists of three parts; the first and second terms measure the energy that

contributes to changing the “hot” and “cold” parts of the system, respectively (i.e. the header

17
HTF volume and piping between the solar field and the power cycle), while the third term

measures the energy contributing to changing the temperature of the HTF, piping, and

insulation in the collector loops.

Included in the thermal inertia calculations are the hot and cold header volume plus

supplemental runner pipe (Vhdr) and the user-specified thermal mass terms (mc)bal,h and

(mc)bal,c described previously. The collector thermal inertia term represents the sum of HTF

mass for all collectors in all loops, and is increased by the length-specific thermal inertia term

(mc)bal,sca,i also discussed previously. The energy state is relative to the temperature from the

previous time step - (Tsys,h,0) for the hot header and (Tsys,c,0) for the cold header.

The energy value calculated in Eq.[2.24] is subtracted from the total absorbed energy in the

solar field to determine the total available energy3 from the solar field during the given time

step. The mass flow rate is recalculated based on the energy available from the field.4

m ̇ A,s f = q ̇A,s f

cht f,ave (Tsys,h −Ts f,in) (2.25)

Where:

q ̇A,s f = q ̇s f − Etrans

Δt

The temperature Ts f,in represents the inlet HTF temperature to the solar field and Tsys,h is the

return temperature to the power cycle calculated in Eq.[2.11]. The return temperature

calculation presented in Eq. [2.11] uses the heat-loss adjusted loop outlet temperature rather

than the temperature at the immediate outlet of the field loops. Thermal losses from the header

and runner piping are accounted for as part of the loop outlet temperature by using the

user-specified piping loss coefficient λhl from the Parasitics page and the total transport piping

surface area Apipe,tot calculated by the model.

T∗

loop,out = Tloop,out − λhl Apipe,tot 

Tloop,out −Tamb

m ̇ ht f cht f,ave

(2.26)

3Note that the term “available” is not used in a thermodynamic sense, but simply to indicate that a

portion of the absorbed energy from the field is not available for use in the power cycle or storage because

of internal energy changes within the field.

4In practical terms, the mass flow and thermal energy values calculated in Eq.’s [2.25, 2.26] represent

the effective achieved values over the time step. These values are reported to the power block, control

algorithm, and thermal storage system for subsequent calculation. These are also the values that are

reported in the System Advisor hourly performance results.

18
Section 2.2 Summary

• The solar field controller limits the mass flow rate of HTF through each loop by defocusing

collectors in over-temperature conditions or by overriding the outlet temperature requirement in low-resource conditions.

• Collector defocusing is achieved by either completely defocusing the SCA’s in the sequence

specified by the user, or by partially defocusing the SCA’s in sequence or simultaneously.

• Freeze protection is applied according to the minimum allowable temperature specified by

the user.

• Transient effects are applied by recalculating the mass flow rate from the field according to

absorbed energy minus energy that goes into changing the energy state of the HTF.

• Piping thermal losses are determined based on the hot outlet temperature from the solar

field.

2.3 Collector Assembly and Field Optics

The collector model and optical calculations used in the physical trough model are based on

the collector model in System Advisor’s empirical trough model. System Advisor defines the

collector as the portion of the solar field that reflects irradiation to the receiver. This equipment

is distinct from the receiver component that consists of an evacuated glass envelope and tube

receiver, as shown in Figure 5. The optical calculations for the collector model extend to the

point of determining the magnitude of solar flux that is incident on the receiver.

Figure 5: The trough includes both a collector to reflect light and a receiver to absorb and

transport heat.

To determine the flux incident on the receiver, we must consider both constant optical losses

and variable optical losses that change with solar position. The total irradiation on the field is a

function of the equivalent aperture area of all of the collectors in the field, the strength of the

solar insolation, and the angle at which the irradiation strikes the aperture plane. The

equivalent aperture area refers to the total reflective area of the collectors that is projected on

the plane of the collector aperture. This area is distinct from the curved reflective surface. Area

that is lost due to gaps between mirror modules or non-reflective structural components is not

included in the aperture area value. Thus, though the structure of the collector may occupy

100m lengthwise and 5m across the aperture, for example, the total reflective aperture area

19
may be somewhat less than 100×5 = 500m2 after spaces, gaps, and structural area are

accounted for.

When the solar irradiation is not normal to the plane of the collector aperture, losses are

incurred that scale with the acuteness of incidence angle. The incidence angle θ is equal to the

difference in angles between the normal to the aperture plane and the solar irradiation. This is

depicted in Figure 6.

Figure 6: The angle between the solar irradiation and the normal vector to the collector aperture

plane.

The incidence angle is a function of solar position and the collector tracking angle. To find the

solar position for a particular plant at latitude φ and longitude ψ, we first calculate the solar

time tsol [5].

B = (day−1) 360

365

EOT = 9.2 (0.000075+0.001868 cosB−0.032077 sinB

−0.014615 cos(2B)−0.04089 sin(2B))

shi ft = zone · 15−ψ

tsol = hour + shi ft

15 +

EOT

60 (2.27)

EOT is an equation of time that determines the deviation in local time from solar time as a

function of the day of the year, shi ft represents the fixed time-shift in minutes due to the

difference between the standard longitude for the time zone (zone, negative zones to the west

and positive to the east of GMT+0) and the longitude at the location of interest. The day of the

year (day) and the hour of the day (hour, from 0 to 24) are also required. The time of day is

converted into an hour angle (ω) in Eq.[2.28].

ω = (tsol −12)· 15◦ (2.28)

Since the solar position is impacted by the tilt of Earth’s axis, the declination angle δ must also

20
be determined.

δ = 23.45◦ ·sin

360(284+day)

365 

(2.29)

Finally, the solar azimuth (γsol) and the solar elevation (θe) angles are calculated [5].

θe = sin−1 (sin(δ) sin(φ) +cos(φ) cos(δ) cos(ω)) (2.30)

θz = 90◦ −θe

γsol = sign(ω)









cos−1

cos(θz) sin(φ)−sin(δ)

sin(θz) cos(φ)









(2.31)

The trough collector is capable of single-axis tracking about the lengthwise axis. This axis

may be oriented in any compass direction, though it typically aligns in either the North-South

or the East-West direction. The collector tracks the solar position in such a way that the angle

between the aperture plane normal and the solar irradiation is minimized. The tracking angle

ωcol is calculated in Eq.[2.32], where the collector orientation with an azimuth angle (γcol) and

a tilt angle (θcol) that is positive when the portion of the field closet to the equator is tilted up

[19].

ωcol = tan−1

 cos(θe) sin(γsol −γcol)

sin(θe −θcol) +sin(θcol) cos(θe) (1−cos(γsol −γcol))

(2.32)

All of the information needed to calculate θ has now been determined. Thus:

θ = cos−1



1−[cos(θe −θcol)−cos(θcol) cos(θe) (1−cos(γsol −γcol))]2 (2.33)

The total radiation incident on the solar field is equal to the available beam-normal irradiation

(Ibn) times the total aperture area, but is scaled by the cosine of θ. Thus, this optical derate is

referred to as “cosine loss” and it is the primary variable loss mechanism for the solar field.

Several other solar position losses are modeled, including spillage of reflected radiation off the

end of the collector row5, shadowing from one row to another, energy lost before deploying in

the morning and after stowing in the evening while solar energy is still available, and an

incident-angle modifier that accounts for all remaining position-dependent losses.

End spillage losses

During hours when incoming solar radiation is not directly normal to the collector aperture,

some radiation is reflected off the end of the collector that doesn’t reach the receiver. The end

loss from each collector can be partially recovered by the adjacent collector, and this energy is

calculated as the “end gain” in Eq. [2.35]. While all collectors incur some end loss, the

collectors that benefit from end gain depend on the solar position as well as the position of the

5A collector loop contains two rows of collectors. See glossary definition for more information.

21
collector within the loop.

For i = 1,Nsca

ηendLoss,i = 1− Lf,ave,i tan(θ)

Li

+ηendGain,i (2.34)

Where:

ηendGain,i = Lf,ave,i tan(θ)−Lsca,gap,i

Li

(2.35)

The ηendGain,i term is equal to zero in the following circumstances:

1. The sun is in the southern sky6 and the collector in question is the southernmost of the

collector row (e.g. i = 1 or i = Nsca)

2. The sun is in the northern sky and the collector in question is the northernmost of the

collector row (e.g. i = Nsca

2 	 or i = Nsca2 	+1)

Li is the net total collector length, Lsca,gap,i is the piping distance separating each SCA within a

single row, and Lf,ave,i is the average surface-to-focus path length from the System Advisor

Collectors page. Note that the latter value is not the focal length of the parabola at the vertex,

but instead is the total averaged value the light reflecting from the parabolic surface must travel

to reach the focus for aperture-normal incidence. Often, only the focal length at the vertex is

readily available, but the averaged value can be calculated using an integral approach, so long

as the aperture width (w) is also known.

Beginning with the equation of a parabola y = x2

4a where a is the focal length at the vertex, we

can express the distance traveled from any point (x,y) on the parabola to the focus at (0,a).

This arrangement is illustrated in Figure 7. Using the Pythagorean theorem with sides of

length x and y−a, the distance at point (x,y) is:

Lf,(x,y) =



x2 + (y−a)2

=



x2 +

 x2

4a −a

2

The average focal length is then the integral of Lf,(x,y) with respect to x over the aperture width

6Assuming a field in the northern hemisphere.

22
Figure 7: Focal length geometry for calculating the average focal length.

(−w

2 ..+ w2 ) and divided by the total aperture width.

Lf,ave =

+w

2

−w

2



x2 +

x2

4a −a

2

w

dx

=





4a2 +w

2

2

2

a2 · 12a2 +w

2

2

12(4a2 +w

2

2

) (2.36)

Evaluation of the definite integral results in Eq.[2.36] above, and provides a simple expression

for average focal length as a function of the focal distance and aperture width.

Row shadowing

Shadowing between rows generally occurs at extreme solar positions (i.e. dusk or dawn) when

the shadow cast by a collector closer to the sun obscures a portion of an adjacent collector.

Figure 8 shows the geometry associated with row shadowing.

23
Figure 8: Two adjacent collector rows may shadow each other if the tracking angle is sufficiently

severe. The shadowing is subject to the collector aperture width, the row spacing (centerline to

centerline), and the tracking angle of the collectors.

The shadowing effect is derived by considering the geometry of the two adjacent collector

rows. Relating the two rows as shown in Figure 8, a right triangle is drawn with a hypotenuse

equal to the centerline-to-centerline row spacing Lspacing and short side equal to the

non-shadowed aperture width, wa. To determine the fraction of active collector aperture, we

first calculate the length of the non-shadowed aperture.

wa = cos(ωcol) Lspacing (2.37)

The shadowing efficiency is equal to the ratio of the non-shadowed aperture to the total

aperture width, w, as shown in Eq. [2.38]. The total shadowing efficiency is limited to the

range of values between 0.5 and 1.0. If the shadowing efficiency is less than 0.5, the solar field

is unlikely to operate successfully, so the total optical efficiency is set to zero and the

simulation progresses to the next time step.

ηshadow = wa

w = |cos(ωcol)|

Lspacing

w

(2.38)

Stow and Deploy Angles

The user can enforce limits on when the solar field will track the sun. This is given in terms of

the solar elevation angle, and can be specified for the deploy event and stow event separately.

Given a stow angle ωstow and deploy angle ωdeploy that correspond to the solar elevation angle,

24
the time of stow and deploy can be calculated using the same relationship.

tstow/deploy = tnoon + sign(tan(180◦ −ω))

15 · cos−1

⎛

⎝

c1c2 +



c2

1 −c22 +1

c2

1 +1

⎞

⎠ (2.39)

where :

c1 = cos(φ)

tan(ω)

c2 = −tan(δ)sin(φ)

tan(ω)

For the stow or deploy time, ω = ωstow or ω = ωdeploy, respectively. Once the deploy and stow

times have been calculated, the model calculates the fraction of the time step during with the

system can operate.

For the deploy hour:

ftrack = tΔt,end −tdeploy

Δt (2.40)

For the stow hour:

ftrack = tΔt,start −tstow

Δt (2.41)

The times at the beginning and end of the time step are given as tΔt,start and tΔt,end,

respectively. The fraction is limited by the model to fall between 0 and 1 during

non-startup/shutdown time steps.

Incident Angle Modifier

The incident angle modifier ηIAM is a derate factor that accounts for collector aperture

foreshortening, glass envelope transmittance, selective surface absorption, and any other losses

that are a function of solar position. The incident angle modifier factor is calculated using an

empirical formula of the form in Eq. [2.42]. The default coefficients for this equation were

derived from the field tests of the SEGS LS2 collectors [4]. Equation coefficients a0, a1, and

a2 are specified as inputs on the Collectors page.

ηIAM = a0 +a1

θ

cosθ +a2

θ2

cosθ (2.42)

(θ in radians)

Constant optical derate factors

The trough collector model captures optical efficiency with losses that are a function of solar

position and with fixed losses that are applied as constant multipliers. Fixed losses include

tracking error, geometry defects, mirror reflectance, mirror soiling, and general error not

captured by the other items. Because the model multiplies the loss factors together to calculate

25
Table 4: General definitions for each fixed optical loss term

Error description Term Definition

Tracking error ηtrack

Inability of the collector to perfectly orient along the tracking angle; twisting of the collector about the lengthwise axis

Geometry defects ηgeo

Poor alignment of the mirror modules; deviation in the position of

the receiver tube from the optical focus; warping or discontinuities

along the reflective surface

Mirror reflectance ρm

Specular reflectance within a cone angle defined by the collector

and receiver geometry

Mirror soiling ηsoil

Dirt or soiling on the reflective surface that prevents irradiation

from reflecting to the receiver

General error ηgen Any effect not captured within the previous categories

an overall loss factor, the value of each individual loss factor is not as important as the value of

the product of all of the losses. Table 4 describes the physical effect each loss factor is

intended to represent.

The total optical efficiency is defined in Eq.[2.43], and we finally calculate the total radiative

energy incident on the solar field in Eq.[2.44].

ηopt(θ,ωcol) = ηendLoss(θ) ηshadow(ωcol) ηIAM(θ) ηtrack ηgeo ρm ηsoil ηgen (2.43)

q ̇inc,s f = Ibn Aap,tot ηopt(θ,ωcol) (2.44)

The incident energy may be adjusted, depending on the stow and deploy times calculated

above. For situations where the deploy or stow event occurs part way through a time step, the

collector efficiency is reduced by the fraction of time step not in operation. For example, if the

solar field deploys at 7:45am (solar time), the fraction of the time step spent tracking (ftrack) is

multiplied by the total collector optical efficiency. In this case, ftrack = 1−45/60, or 0.25.

2.3.1 Determining an average efficiency value

System Advisor allows the user to assign multiple collector types and geometries within the

same SCA loop (see the Solar Field page). Since each collector variation may have a unique

optical efficiency, the reported collector efficiency in the simulation output is equal to the

weighted average of all of the collectors used in the loop. System Advisor calculates the

weighted optical efficiency by determining the total aperture area of the loop, then multiplying

each collector’s efficiency value by it’s relative share of the loop aperture. Mathematically, this

is represented in Eq.[2.45] for i collector assemblies in the loop, each with unique area Acol,i

and efficiency ηcol,i.

ηcol,ave =

Nsca

∑

i=1

ηcol,i · Acol,i

Aloop,tot

(2.45)

26
Section 2.3 Summary

• The collector field model determines the incoming solar flux by considering weather data,

plant location, solar position, and derate values that are both constants and functions of

solar/collector position.

• Variable losses modeled are end spillage, row shadowing, stow and deploy events, and the

incident angle modifier.

• Constant loss fractions account for tracking error, geometry defects, mirror reflectance,

mirror soiling, and general error.

2.4 Receivers (HCE’s)

The receiver formulation used in the physical trough model uses a 1-dimensional model

developed in [8]. Forristall’s work uses the Engineering Equation Solver (EES) package [15]

that is designed to evaluate complex systems of equations using an iterative approach. The

system of equations describe the relationship between temperature and heat loss: The surface

temperature of the absorber tube is a function of the heat absorption, while convective and

radiative losses are strong functions of surface temperature. Convective loss is directly

proportional to the difference between receiver surface and ambient temperatures, and

radiation loss is proportional to temperature difference to the fourth power. The performance

of the receiver can’t be accurately modeled using simple explicit relationships. Instead, System

Advisor uses implicit equations, solving iteratively with successive substitution until the

solution converges.

The receiver is modeled as a 1-dimensional energy flow. Only the temperature gradient in the

radial direction is assumed to be significant - axial and circumferential heat transfer are

neglected. Figure 9 presents a diagram of one quarter of the receiver in cross-section. Each

temperature T1−5 is calculated by the model using an energy balance and

temperature-dependent loss coefficients. The receiver geometry is specified by the user with

radii R1−4.

27
Figure 9: A heat balance for the modeled receiver. Heat transfer in the radial direction (left to

right) is considered, while circumferential and axial transfer is not.

Concentrated irradiative flux from the collector passes through the transparent glass envelope

(R3−4), and a portion of that flux is absorbed by the glass. The absorbed flux is determined

from the absorption fraction specified by the user as Envelope Absorptance (αenv) on the

Receivers page and influences the calculated temperature of the glass. The flux that passes

through the glass envelope reaches the absorber tube at R2. Note that the fraction of energy

passing through the glass envelope is specified by the Envelope Transmittance value on the

Receivers page, and need not equal the compliment of the absorptance value. This is because

absorptance by the glass is only one of several possible loss mechanisms. Others include

reflective loss and light refraction where incoming rays are bent away from the absorber.

During operation, the heated surface at R2 drives thermal energy through the absorber wall

(R1−2) and into the cooler HTF. Thermal losses from the absorber surface occur via convection

and radiation exchange with the glass envelope. The glass envelope is in turn exposed to

ambient air. Figure 10 shows the heat transfer network, conceptualized as a set of thermal

resistances in series and parallel. This is analogous to an electrical resistance network where

thermal energy represents current, thermal resistance represents electrical resistance, and

temperature drop is equivalent to voltage drop. Incidentally, the same resistance formulae

apply to thermal and electrical networks.

2.4.1 Modeling approach

The nodal nature of the collector loop was discussed in Section 2.1 (see Figure 2 on page 8).

To summarize, each node corresponds to an assembly of individual receiver elements and

collector modules. As HTF passes through the loop, it gradually warms until it reaches the

design-point field outlet temperature at the end of the last SCA. The gradual warming of HTF

over the length of the loop corresponds to a trend of decreasing thermal efficiency, since

28
Figure 10: The thermal resistance network for the “intact” receiver model shown in Figure 9.

Energy is absorbed at T3 and T4−5.

Figure 11: The thermal resistance network for the “broken glass” receiver model. Energy is

absorbed on the absorber tube surface at T3 and heat is exchanged directly with the sky and

ambient temperatures.

receiver performance is inversely related to receiver temperature. Variability in receiver

performance within a loop can be significant, so the receiver model is applied individually for

each node in the loop. This system is solved iteratively to determine the mass flow rate that is

required to meet the design outlet temperature, as discussed in Section 2.2.

The receiver model uses information about the HTF temperature, receiver geometry, ambient

conditions, and incoming solar flux to determine the performance of the receiver.

Conceptually, the solar field can be dissected into four different models:

1. the collector model

2. the receiver model

3. the piping model, and

4. the HTF model.

This distinction is particularly noteworthy for the receiver and HTF models. The HTF model

calculates the HTF temperature throughout the loop based on absorbed energy and mass flow

rate. The receiver model calculates the thermal performance of the receiver given an HTF

temperature and other information. Thus, from the perspective of the receiver model, the HTF

29
temperature (T1) is an input value even though T1 is closely tied to the results calculated by the

receiver model. Other specified values for the model are summarized in Table 5.

Table 5: Inputs to the receiver model

Item Description

T1 HTF inlet temperature

m ̇ ht f HTF mass flow rate

Tamb Ambient temperature

Tsky Effective sky temperature

vwind Wind velocity at the receiver surface

pamb Ambient pressure

q ̇inc,i Incident radiation at node i

Acs Cross-sectional area of the absorber tube

D2 Absorber tube internal diameter

D3 Absorber tube external diameter

D4 Glass envelope internal diameter

D5 Glass envelope external diameter

Dp Internal flow plug diameter

ε3 Absorber surface emittance

ε4 Glass surface emittance

αabs Absorber surface absorptance

αenv Glass envelope absorptance

ηcol,i Collector optical efficiency at node i

τenv Glass envelope transmittance

Pa Annulus pressure

- Annulus gas type

- HTF type

- Absorber material

For any solver using successive substitution, initial guess values must also be provided. The

guess values for the receiver model are initially calculated based on the HTF temperature

provided to the model, and depend on the condition of the receiver envelope. Temperature

guesses for the absorber tube and glass envelope must be provided. Eq.[2.46] shows the initial

settings for the temperatures for intact receivers, and Eq.[2.47] shows the settings for receivers

with broken glass.

T2 = T1 +2◦

C

T3 = T2 +5◦

C (2.46)

T4 = T3 −0.8 ·(T3 −Tamb)

T5 = T4 −2◦

C

T2 = T1 +2◦

C

T3 = T2 +5◦

C (2.47)

T5 = T4 = Tamb

30
Once guess values have been calculated, subsequent calls to the subroutine use the converged

values from the previous call as the new guess values. However, several conditions may trigger

recalculation of the guess values using Eq.’s[2.46] and [2.47]:

• The difference between the last T1 and the current T1 is greater than 50◦C

• The minimum value in the group T1−5 is less than the current Tsky value

• Any temperature from the last call returned as invalid (Not a Number error)

2.4.2 Model formulation

The first step in determining receiver heat loss is to calculate the thermal resistance between

the outer absorber tube and the inner glass envelope surfaces. Both convection and radiation

contribute to the total heat transfer, though convection between the two surfaces is very small

for intact receivers. Convection becomes significant in cases where the vacuum is lost due to

broken glass or where hydrogen from the HTF has diffused through the absorber tube wall into

the annulus.

Convection from the absorber

Convection may occur either between the absorber and the inner glass surface or directly to the

ambient air in the case that the envelope is broken. The convection subroutine handles both

situations. First, for intact receivers, the annulus gas properties are evaluated at the average

temperature T34. Convection from R3 to R4 can be generally expressed in terms of thermal

resistance Rˆ34,conv as7:

q ̇34,conv = T3 −T4

Rˆ34,conv

(2.48)

Where:

Rˆ34,conv = 1

γ34,conv π D3

The receiver model calculates natural internal convection using the modified Raithby and

Hollands correlation [2] (for more information on the convective algorithms, see [8] pages

11-14). The calculation for annular natural convection begins with determining the Rayleigh

number at diameter D3 using Eq.[2.49].

RaD3 = g β34 |T3 −T4|D3

3

α34 ν34

(2.49)

The volumetric expansion coefficient β34, the thermal diffusivity constant α34, and the

kinematic viscosity ν34 of the annular gas are all evaluated at the averaged temperature T34.

7The subscript “34” (or other two-number subscripts in this section) denote properties, temperatures,

or other quantities that describe the intermediate step between surface 3 and surface 4 in the resistance

model. In practical terms, “34” can be thought of as the subscript that denotes the thermal interaction of

surface 3 and surface 4. This model describes the continuous substance between surfaces 3 and 4 using

thermal properties that are evaluated at the average of T3 and T4 (T34).

31
Using Prandlt number Pr34 = ν34/α34, we calculate the heat transfer due to natural convection

in the annulus and the associated heat transfer coefficient.

q ̇34,conv = 2.425 k34

T3 −T4



1+ D3

D4

0.6

1.25  Pr34RaD3

0.861+Pr340.25

(2.50)

γ34,conv = q ̇34,conv

π D3 (T3 −T4) (2.51)

For very low annular pressures, the molecular density drops below the physical limit for

establishing convective currents; instead, free molecular heat transfer relationships more

appropriately describe convective heat loss. The receiver model handles this by using the

largest convective loss predicted by either annular natural convection or free molecular heat

transfer. Eq.[2.52] shows the steps for calculating free molecular heat transfer.

Λ = C1 ×10−20 · T34

Pa · ζ2

Γ = cp,34

cv,34

b = 9Γ−5

2Γ+2

γ34,conv = k34

D3

2 logD4D3



+ b Λ

100 D3D4 +1

 (2.52)

q ̇34,conv = π D3 γ34,conv (T3 −T4)

In the calculation for Λ, C1 is a constant 2.331×10−20 mmHg·cm3

K , ζ is the free-molecular

collision distance shown in Table 6 [8], and Pa is the annulus pressure in torr.

Table 6: Values of the mean free path between collisions of a molecule for free molecular

convection

Annulus Gas ζ [cm]

Air 3.53×10−8

Hydrogen 2.4×10−8

Argon 3.8×10−8

The annular convection calculations assume that the receiver’s glass envelope is intact.

However, the glass envelope sometimes breaks due to impact or excessive thermal cycling.

Thermal loss from a broken-glass receiver is significantly higher than for an intact receiver,

and the loss must be modeled differently. System Advisor provides specialized calculations for

broken-glass receivers and further divides the heat transfer relationships applied based on

ambient wind speed.

32
If the ambient wind speed is very low (less than 0.1 m/s), then the Nusselt number is calculated

using the Churchill & Chu correlation for a long isothermal horizontal cylinder [13], where the

fluid properties are determined at the averaged temperature T36.

Nu =

⎡

⎢

⎢

⎢

⎣

0.60+0.387 ·Ra0.1667

D3



1+



0.559

Pr36 0.56250.2963

⎤

⎥

⎥

⎥

⎦

2

(2.53)

The convection coefficient calculated in Eq.[2.54] is then used to determine the total

convective heat transfer.

γ34,conv = Nu

k36

D3

(2.54)

q ̇34,conv = γ34,conv π D3 (T3 −T6)

If the ambient wind speed is above 0.1 m/s, thermal properties are required for both the air in

contact with the absorber surface at T3 and the ambient air at T6. In this case, the Nusselt

number is calculated using Zhukauskas’ correlation for external forced convection [13].

Nu = C Rem

D3 Prn

6

Pr6

Pr3

0.25

(2.55)

Where:

ReD3 = v6 D3

ν6

The coefficients m, n, and C are selected according to the Prandtl number and the Reynolds

number as shown in Table 7. For Pr ≤ 10, n = .37, otherwise n = .36.

Table 7: Selection of coefficients C and m for Zhukauskas’ correlation based on the Reynolds

number at D3.

Reynolds Number Range C m

0 ≤ ReD3 < 40 0.75 0.4

40 ≤ ReD3 < 1000 0.51 0.5

1000 ≤ ReD3 < 2×105 0.26 0.6

2×105 ≤ ReD3 < 106 0.076 0.7

To summarize the absorber convection calculations:

1. Convection loss is determined based on the condition of the receiver (whether the glass

envelope is intact or broken).

33
2. If the glass is intact, the natural convection coefficient and the molecular diffusion

convection coefficient are both calculated and compared, with the larger of the two

selected for use in the loss equation.

3. If the envelope is not intact, direct convection to ambient is calculated based on whether

the wind speed is below or above 0.1 m/s.

No matter the method used to calculate the convective loss coefficient from the absorber, the

thermal resistance due to convection is expressed as follows.

Rˆ34,conv = 1

γ34,conv π D3

(2.56)

Radiation from the absorber

Radiation loss from the absorber tube to the surroundings is the most significant contributor to

heat loss for intact collectors. Two alternate equations are used for calculating radiative loss

depending on whether the glass envelope is intact. For intact receivers, radiative exchange is

between the absorber surface at D3 and the inner envelope surface at D4, as shown in Eq.[2.57].

γ34,rad = σ(T2

3 +T24 ) T3 +T4

1

ε3 + D3D4

 1

ε4 −1

 (2.57)

q ̇34,rad = π D3 γ34,rad (T3 −T4)

If the receiver envelope is broken, radiation exchange occurs directly between the absorber

surface and the ambient surroundings at temperature Tsky.

q ̇34,rad = σ γ34,rad ε3 π D3 (T4

3 −T4

sky) (2.58)

Where:

γ34,rad = q ̇34,rad

π D3 (T3 −T7)

The radiation thermal resistance Rˆ34,conv is calculated with the same arrangement used in

Eq.[2.56] above. The total thermal resistance between the absorber surface and the glass

envelope is expressed as two parallel thermal resistances and applies only for intact receivers.

Rˆ34,tot =

 1

Rˆ34,conv

+

1

Rˆ34,rad −1

(2.59)

34
Conduction across the glass envelope

Conduction across the intact glass envelope is modeled using the formula for radial resistance

in a cylinder [17], assuming the thermal conductance k45 = 1.04 W

m−K .

Rˆ45,cond =

logD5

D4



2πk45

(2.60)

Convection and radiation loss from the envelope

Like convection from the absorber to ambient air in the case of broken glass receivers,

convection loss from the glass envelope occurs between a cylindrical object and free-stream

air. Thus, the equations developed for convective loss from broken receivers can be reused

here. For wind speeds of less than 0.1 m/s, the Churchill and Chu correlation in

Eqs.[2.53-2.54] is used, otherwise Eq.[2.55] is used. Air properties are evaluated at T5 and T6

rather than at T3 and T6.

Radiative loss is calculated using Eq.[2.58], substituting T5 for T3. The corresponding thermal

resistances are:

Rˆ56,conv = 1

γ56,conv π D5

(2.61)

Rˆ57,rad = 1

γ57,rad π D5

(2.62)

Heat flows and temperatures

Thermal energy is absorbed into the resistance network at two locations, as illustrated in Figure

10 above. The absorber tube receives most of the thermal energy incident on the receiver at

D3, but a small portion of the incoming energy is also absorbed in the glass envelope at D5.

The energy absorbed by absorber tube i is a function of the energy concentrated by the

collector q ̇inc,i, the envelope transmittance τenv, and the absorber surface absorptance αabs.

q ̇abs,i = q ̇inc,i ηopt,i τenv αabs (2.63)

The energy absorbed by the envelope is:

q ̇abs,env,i = q ̇inc,i ηopt,i αenv (2.64)

We estimate the heat transfer from the absorber surface to the inner envelope surface using the

temperature guess values in Eq.[2.46] for T3 and T4. This heat flow (q ̇34,tot) is increased by the

energy absorbed within the envelope. For simplicity, the energy is assumed to be absorbed just

after the inner envelope wall between D4 and D5 so that the thermal energy must travel the

entire distance between the envelope walls. Thus, the thermal energy conducted across the

35
wall is expressed as:

q ̇45,cond = q ̇34,tot +q ̇abs,env (2.65)

The outer envelope surface temperature is then the inner surface temperature minus the heat

flow conducted scaled by the thermal resistance.

T5 = T4 −q ̇45,cond Rˆ45,cond (2.66)

Thermal losses from the envelope to ambient are now redetermined for this iteration with the

newly recalculated glass envelope outer surface temperature T5.

q ̇56,conv = T5 −Tamb

Rˆ56,conv

(2.67)

q ̇57,rad = T5 −Tsky

Rˆ57,rad

This fully defines the thermal resistance network from the outer absorber surface to the

surroundings. If we consider the temperature profile of an intact receiver during operation, the

figurative “top of the temperature hill” is located at the boundary between the absorber surface

and the inner annulus (D3). Heat always flows “downhill”; assuming we know the temperature

at D3, we can now recalculate the magnitude of the heat flow from the absorber to ambient.

This quantity defines the receiver heat loss. This can be done analytically by expressing the

total heat loss in terms of the equivalent thermal resistance between D3 and ambient, and the

total temperature difference between T3 and the ambient temperatures Tamb and Tsky. The heat

conducted across the glass envelope is equal to the heat flow across the annulus (q ̇34,tot) plus

the heat absorbed directly by the envelope. This total heat flow is then radiated/convected to

ambient, though the practical receiver heat loss is only equal to the heat transferred away from

the absorber tube across the annulus, and the envelope energy absorption is accounted for an

an optical loss. Thus:

q ̇hl = q ̇34,tot = q ̇45,cond = 

q ̇56,conv +q ̇57,rad −q ̇abs,env

= ΔThl

Rˆ3,amb

(2.68)

A practical implementation of this equation is easily derived by applying resistance network

rules to the section of Figure 10 between T3 and the ambient temperatures. Eq.[2.68] is then

equivalently:

q ̇hl = (T3 −Tamb) Rˆ57,rad + (T3 −Tsky) Rˆ56,conv −q ̇abs,env ΩRˆ

Rˆ34,tot Rˆ57,rad +Rˆ34,tot Rˆ56,conv +ΩRˆ

(2.69)

36
Where:

ΩRˆ = Rˆ56,conv Rˆ57,rad +Rˆ45,cond Rˆ57,rad +Rˆ45,cond Rˆ56,conv

This equation is somewhat simplified as the envelope resistances drop out in the case that the

receiver glass is removed and the absorber surface is in direct thermal communication with the

ambient.

q ̇hl = (T3 −T6) Rˆ34,conv + (T3 −T7) Rˆ34,rad (2.70)

Having considered convective and radiative losses from the receiver for both intact and broken

receivers, one remaining heat loss mechanism must be accounted for. This is conductive loss

from the support brackets in contact with the receiver ends. Radiation and convection heat

losses tend to overwhelm bracket conduction heat loss, so the System Advisor interface

excludes inputs related to this loss. System Advisor hard-codes the required inputs for this

calculation to reduce the number of user input variables. Table [8] enumerates the bracket

geometry and material properties assumed by System Advisor.

Table 8: Bracket geometry and material properties assumed by System Advisor for conductive

heat loss calculations

Item Assumed Value Units

Effective bracket perimeter 20.32 cm

Effective bracket diameter 5.08 cm

Minimum bracket cross sectional area 1.6129×10−4 m2

Conduction coefficient (carbon steel, 600K) 48.0 W

m−K

Effective bracket base temperature T3 −10 ◦C

Single receiver length 4.06 m

Bracket heat loss is estimated by assuming that the bracket convects heat to the surroundings at

a driving temperature difference of Tbrac −Tamb, where the average bracket temperature Tbrac is

defined as:

Tbrac = 1

2



Tbase +Tamb

2 +Tamb

(2.71)

The base temperature Tbase is the effective bracket base temperature from Table 8. Like

convection from the outer glass envelope, convection from the bracket is subdivided into

still-air convection (velocity less than 0.1 m/s) and forced convection. The Nusselt number for

natural convection is given in Eq.[2.53] as shown above, except using Rayleigh and Prandtl

numbers corresponding to the average bracket temperature. The convective heat loss

coefficient is then scaled by the effective bracket diameter.

γbrac = Nubrac

kbrac

Dbrac

(2.72)

37
Forced convection from the bracket is calculated analogously to the procedure described in

Eq.[2.55] and Table 7, but using bracket temperatures. The bracket convective heat loss

calculation uses either the coefficient for forced convection or for natural convection,

depending on the ambient wind speed. This coefficient γbrac is used to calculate the heat loss

as shown in Eq.[2.73] for a given bracket perimeter perbrac, cross-sectional area Abrac,cs, and

single receiver length Lrec.

q ̇brac = #γbrac perbrac kbrac Abrac,cs Tbase −T6

Lrec 

(2.73)

The total heat loss from the receiver is the sum of convection, radiation, and bracket

conduction losses. The energy flow into the HTF q ̇12,conv is then equal to the energy absorbed

by the absorber tube minus the heat loss from the receiver. The heat loss value is reduced by

the amount of heat absorbed in the glass envelope that also must be rejected; this avoids

double-counting the heat loss from the absorber tube.

q ̇12,conv = q ̇23,cond = q ̇abs −q ̇hl (2.74)

Using the calculated value for heat flow into the HTF and the guessed average HTF

temperature T1, we calculate the temperature drop from the inner absorber tube surface to the

HTF due to the internal convective resistance. The convection correlations require fluid

properties evaluated at the inner wall temperature and the bulk HTF temperature, and are

divided into several cases: for HTF flow velocities greater than 0.1 m/s, the Nusselt number is

calculated for either annular or round-pipe flow, and for higher velocities the Nusselt number is

determined using a modified Gnielinski correlation [11]. For low velocities with annular flow:

Nu12 = 41.402 D5

R −109.702 D4R +104.570 D3R −42.979 D2R

+ 7.686 DR +4.411 (2.75)

Where:

DR = DP

D2

and for low velocities with normal pipe flow, the Nusselt number is fixed where Nu12 = 4.36.

The inner wall temperature for low-flow conditions is then:

T2 = T1 + q ̇12,conv Dhyd

π D2 k1 Nu12

(2.76)

Any significant HTF velocity drives the Nusselt number sufficiently high that the inner wall

temperature approaches the bulk HTF temperature. This will be the case for most situations

where a minimum flow is enforced in the field during off-sun conditions.

The final step in solving the receiver model is to recalculate the absorber tube outer surface

38
temperature, T3. Recall that this value was initially guessed, but now enough information is

available to calculate this temperature and update the guess for iteration. T3 is found this time

as a function of the inner wall temperature T2 and the heat flow by conduction across the tube

wall.

T3 = T2 + q ̇23,cond

2 π k23

log D3

D2

(2.77)

The calculation process beginning with Eq.[2.65] and ending with Eq.[2.77] is repeated

iteratively until convergence is achieved in temperatures T3 and T4 with respect to their

previous iteration guess values T3g and T4g. Convergence is obtained when the error calculated

in Eq.[2.78] falls below a tolerance of 0.05 or when the number of iterations exceeds 4.8

err =

T4g −T4

T4

2

+

T3g −T3

T3

2

(2.78)

Section 2.4 Summary

• The receiver model uses a 1-dimensional formulation, where heat flows uniformly in the

radial direction.

• The receiver can account for a broken glass envelope, and for either air, argon, or hydrogen

gas in the receiver annulus.

• The model is analogous to a thermal resistance network, with heat transfer occurring in

both parallel and series flow .

• Guess values are used to iteratively solve the temperature and heat flow distribution in the

receiver.

• Model convergence is achieved when temperatures T3 and T4 remain acceptably constant

between iterations.

2.5 Piping model

The largest parasitic loss for a trough plant is the electricity consumed by the solar field HTF

pumps. Since pumping power scales proportionally with the HTF pressure drop across the

solar field and with the HTF mass flow rate, accurately capturing both of these values is

important in characterizing the total plant performance. System Advisor takes a moderately

detailed approach in the design and performance modeling of the solar field piping. The piping

model in System Advisor is derived directly from work presented in [14].

The piping model in System Advisor accounts for the pressure drop associated with a variety

of field components, including:

8These tolerance criteria were selected based on a sensitivity analysis that balanced model speed with

annual output precision.

39
• “Runner” piping from the power block to and from the solar field headers

• Hot and cold headers

• Receiver tube piping

• Pipe expansions and contractions

• Elbows (long, medium, standard)

• Valves (gate, globe, check, and control)

• Ball joint assemblies

The piping model keeps track of the total fluid volume and surface area of the piping,

excepting the surface area of the internal receiver tubing. The model does not account for

varying insulation thickness, but instead applies an area- and temperature-specific heat loss

coefficient to determine the total thermal energy loss from piping.

2.5.1 Piping diameter selection

Full inventory of the field geometry requires a long list of inputs; specifying information on the

positions, quantities, and pressure drops for all possible runner and header piping, expansions

and contractions, elbows, valves, and ball joints would quickly overwhelm both the System

Advisor interface and the person using the model. To avoid this, System Advisor makes a

number of simplifying assumptions about the placement and quantity of piping components.

The user-specified information includes the number of subfield sections, the piping thermal

loss coefficient, thermal inertia terms (mc)bal,h, (mc)bal,c, and (mc)bal,sca, and the min/max

HTF velocities in the header at design. All other information is assumed or calculated by the

model.

The header pipes are used to distribute cold HTF from the power cycle to each of the individual

loop inlets and then collect the hot HTF from each loop and return it for use in the power cycle

or storage. Optimal header design accounts for the mass of HTF flowing through the header

becoming successively smaller as it moves farther away from the power cycle. Flow through

each loop must be balanced throughout the field to maintain constant hot outlet temperatures.

Part of the balancing effort requires that the header diameter match the mass of HTF flowing

through it such that the flow velocity is relatively constant. System Advisor calculates the

header diameter for each loop section to ensure that the minimum and maximum flow

velocities provided by the user are maintained at the design condition. Figure 12 illustrates

how the mass flow and header diameter decrease as flow is distributed to the field loops.

System Advisor handles the header sizing calculations by restricting the HTF velocity at

design to the range specified by the user on the Solar Field page. This requires an estimate of

the HTF mass flow rate in the solar field at design, as expressed in Eq.[2.79].

m ̇s f,des = (Aloop Ibn −q ̇hl,loop) Nloops ·ηopt,des

cht f,ave (Ts f,out,des −Ts f,in,des) (2.79)

40
Figure 12: An illustration of the flow diverting from the header into individual field loops. In this

example, two loops extract HTF per section. The diameter of the header is adjusted to maintain

optimal flow velocities.

The optical efficiency at design (ηopt,des) is calculated using the solar position at noon on the

summer solstice according to the collector model presented in Section 2.3. The loop aperture

area Aloop is equal to the sum of the aperture of each collector assembly in the loop, as defined

on the Collectors page.

Aloop =

Nsca

∑

i=1

Acol,i (2.80)

The estimated heat loss is the weighted sum of the heat loss estimate values for each SCA in

the loop. Heat loss from each SCA is weighted according to the ratio of the SCA receiver

length to the total loop receiver length.

q ̇hl,loop =

Nsca

∑

i=1

q ̇hl,des,i

Li

Lloop,tot

(2.81)

The design-point HTF mass flow through each individual header is determined using the total

solar field mass flow rate from Eq.[2.79]. The number of field subsections is specified by the

user. Each field section is assumed to have two symmetric subsections; consequently, the user

can select any even number of subfield sections on the Solar Field page in System Advisor.

The number of field sections (Nf sec) determines the overall header piping layout. The first

three possible selections are illustrated in Figure 13.

System Advisor determines the number of loop extraction points per header by dividing the

total number of loops by two times the number of field sections (two loops extract at a single

point, per Figure 12).9

Nhsec =

$ Nloops

2 ·Nf sec%

(2.82)

9Note that the mathematical functions for “floor” and “ceiling” are expressed in this document as  	

and 
 , respectively.

41
Figure 13: Three possible field configurations modeled by System Advisor. Two (top left), four

(bottom left), and six (right) field subsections are shown.

Four different mass flow rates characterize the solar field. These are (1) the total solar field

mass flow rate from Eq.[2.79], (2) the mass flow rate in the runner pipe that feeds the header

sections, (3) the mass flow rate in each individual header, and (4) the mass flow rate in a single

loop. The single-header mass flow rate (item 3) is defined as:

m ̇ hdr = 2 ·m ̇run

Nf sec

(2.83)

Runner piping

The runner pipe diameter is sized to match the velocity limits specified by the user according

to the total mass flow rate carried by the runner piping at design. The runner piping is broken

conceptually into multiple sections, as follows:

1. The first section is the piping in and around the power block. The main runner pipes -

the ones extending from the power cycle to the North and to the South - each carry one

half of the total solar field mass flow rate.

m ̇runsec,1 = m ̇s f,des

2 (2.84)

2. The second section includes the runner piping between the boundary of the power cycle

out to the first set of subfield headers. Depending on the solar field configuration, the

mass flow rate carried through this section may be either half of the total mass flow rate,

or half of the flow less the amount sent to immediately adjacent subfields as in the case

of the six-subsection arrangement in Figure 13. The general formula for the section 2

runner piping mass flow rate is shown in Eq. [2.85]. Note that for a field with only two

subfields, there is no second runner piping section since the power cycle piping directly

42
feeds the subfield headers.10

m ̇runsec,2 = m ̇s f,des

2



1− Nf sec (mod 4)

Nf sec 

(2.85)

3. For solar fields with more than 6 subsections, subsequent runner piping sections scale

down in diameter as the mass flow rate decreases. The mass flow rate of each runner

section k is equal to the mass flow rate from the previous section less the mass flow

diverted to its subfields.

For k = 2,Nrunsec:

m ̇ runsec,k = m ̇ runsec,k−1 −2 ·m ̇ hdr (2.86)

The number of runner sections Nrunsec is dependent on the number of subfields specified

by the user on the Solar Field page, and is calculated as shown in Eq. [2.87]. Note that

the additional section represents the fixed piping length in and around the power cycle

that connects to the solar field.

Nrunsec = 

Nf,sec

4 	+1 (2.87)

The runner piping diameter is sized according to the design-point mass flow rate for each

section and the maximum HTF velocity limit specified on the Solar Field page.

For k = 1,Nrunsec:

Drunsec,k =



4 ·m ̇ runsec,k

ρave vmax π (2.88)

Runner piping lengths are determined according to solar field geometry and subfield layout.

Length calculations are discussed in detail on page 48.

Subfield header piping

Like the runner piping diameter calculations, the subfield header design must balance the need

for a varying pipe diameter with practical complexity limitations. Using a different header

diameter for each of the Nhsec loop sections along the header may optimize performance, but

the marginal benefits of consistent flow velocity are likely outweighed by the cost of

assembling the piping. Thus, System Advisor only adjusts the header diameter for a section if

the flow velocity falls outside of the specified range, and limits the total number of unique

header diameters to 10.

Each unique header section i is assigned to a header diameter group j as depicted in Figure 14.

The diameter of the first header section is calculated using a mass flow rate equal to m ̇ hdrsec, j,0,

10The mathematical operator “a (mod b)” represents the modulo function, whereby the remainder of

a divided by the argument b is obtained. For example, 5 (mod 2) = 1.

43
Figure 14: Numbering scheme and notable mass flow rates for the subfield and runner piping.

the flow rate at the header group j inlet. Using this pipe diameter, the minimum allowable HTF

mass flow rate is calculated for the lower velocity limit at group j.

m ̇ min, j = ρave vmin π

D2

hdr, j

4 (2.89)

The same diameter Dhdr, j is applied to the next Ndiv, j header divisions in each group j among

Nhgrp groups before being recalculated, but the section i mass flow rate is constantly adjusted

to be incrementally smaller as flow is diverted into the loops.

For j = 1,Nhgrp:

Ndiv, j =

&

m ̇ hdrsec, j,0 −m ̇ min, j

2 m ̇loop '

(2.90)

And for i = 1,Ndiv, j:

m ̇ hdrsec, j,i = m ̇ hdrsec, j,i−1 −2 m ̇loop (2.91)

The diameter for each group is calculated according to Eq.[2.92], and a table of header

diameters is written as an output file11.

Dhdr, j =



4 ·m ̇ hdr, j,0

ρave vmax π (2.92)

The header diameters are restricted to common piping schedule diameters and thicknesses.

11From System Advisor, you can access the working directory that contains the output file by pressing

SHIFT+F9

44
The possible pipe schedule selections are shown in Table 9 [14]. When selecting the best pipe

schedule, the code looks for the smallest pipe diameter that meets the flow velocity restriction.

Often, the realized pipe schedule is somewhat larger than the idealized piping diameter.

Table 9: Pipe sizing schedules used in the trough model. The wall thickness and schedule are

selected to match a pressure rating of 25 bar.

Nominal

pipe size

(in)

Schedule

Internal

diameter

(in)

Internal

diameter

(m)

Wall

thickness

(m)

2.5 A 2.71 0.0688 0.0013

3 A 3.33 0.0847 0.0016

4 B 4.26 0.1082 0.0021

6 B 6.36 0.1615 0.0031

8 C 8.13 0.2064 0.0040

10 D 10.25 0.2604 0.0051

12 C 12.25 0.3112 0.0060

14 C 13.38 0.3398 0.0066

16 C 15.38 0.3906 0.0076

18 D 17.25 0.4382 0.0085

20 C 19.25 0.4890 0.0095

22 C 21.00 0.5334 0.0104

24 C 23.00 0.5842 0.0113

26 C 25.00 0.6350 0.0123

28 D 26.75 0.6795 0.0132

30 D 28.75 0.7303 0.0142

32 D 30.75 0.7811 0.0152

34 E 32.62 0.8286 0.0161

36 E 34.50 0.8763 0.0170

42 D* 40.50 1.0287 0.0200

48 D* 46.00 1.1684 0.0227

54 D* 52.00 1.3208 0.0256

60 D* 58.00 1.4732 0.0286

66 D* 64.00 1.6256 0.0316

72 D* 70.00 1.7780 0.0345

*Schedule D was the thickest pipe wall available in these diameters

Piping HTF volume

In addition to the header and runner solar field piping, System Advisor calculates the HTF

volume associated with the power block steam generator system (SGS) and pump piping. This

HTF volume is used in the thermal inertia calculations of the power block during warm-up and

transients. System Advisor specifically models the following components:

1. Individual pump suction line from the suction header to the pump inlet

2. Individual pump discharge line from the pump discharge to the discharge header

3. Pump discharge header

4. Collector field outlet header to the expansion vessel or thermal storage tank

45
5. Steam generator supply header

6. Inter-steam-generator piping

7. Steam generator exit header to expansion vessel or to thermal storage

Several assumptions are used to determine the HTF volume of each of these piping lines. The

design-point HTF velocity is assumed to be 1.85 m/s [14], and the system is assumed to have

three pumps, each capable of producing 50% of the design-point flow rate. The line lengths are

fixed and do not scale with power block or solar field size, though the piping diameter does

scale. Table 10 shows the assumed piping length for each line.

Table 10: The assumed piping lengths for the steam generator and pumping system.

Line Description Length (m)

1 Pump suction header to pump inlet 45

2 Pump discharge to discharge header 45

3 Pump discharge header 100

4 Collector field to expansion vessel/TES 120

5 Steam generator supply header 80

6 Inter-steam-generator piping 120

7 Steam generator exit to exp. vessel/TES 80

Sizing for the individual piping lines depends on the HTF volumetric flow rate. The

design-point solar field mass flow rate m ̇s f,des is related to the design-point power block mass

flow rate by the solar multiple, as shown in Eq.[2.93]. The volumetric flow rate is equal to the

mass flow rate divided by the HTF density.

m ̇ pb,des = m ̇ s f,des

SM (2.93)

Piping lines 1-2 in Table 10 each carry a volumetric flow of 1

2 ·  ̇

Vs f , piping lines 3-4 each carry

the full  ̇

Vs f , and piping lines 5-7 carry  ̇V pb.

2.5.2 Piping layout and components

The remaining pipe sizing calculations are an aggregation of simple estimates based on the

total solar field size and selected configuration. The piping model is organized into four

convenient domains, each of which individually calls the main pressure drop calculation

algorithm. These domains are the field inlet/outlet/cross-over piping (IOCop), the receiver

assemblies and interspersed connective piping, the hot and cold runner pipe, and the hot and

cold headers. The valve and joint component assignments for each domain are shown in Table

11.

The length-of-pipe equations referenced in Table 11 are shown in Eq.’s[2.94-2.97], and the

number of elbows for the runner pipe is calculated in Eq.[2.99]. The number of expansions and

46
Table 11: Variable assignments for the piping equipment by domain. Referenced equations are

denoted with square brackets. The pressure drop coefficients for the various piping components

are defined in the kΔP column.

Runners Headers

Item kΔP IOCop Receivers Hot Cold Hot Cold

Length of pipe Eq. - [2.94] [2.95] [2.96] [2.96] [2.97] [2.97]

Eval. mass flow - m ̇loop m ̇loop m ̇ run m ̇ run m ̇ hdr m ̇ hdr

Eval. temperature - Ts f,ave T ̄i Ts f,out Ts f,in Ts f,out Ts f,in

Tube diameter - D2 D2 Drun Drun Dhdr,i Dhdr,i

Expansions 0.50 0 0 [2.98] [2.98] Nhgrp 0

Contractions 0.50 0 0 0 0 0 Nhgrp

Standard elbows 0.9 2 10/loop 0 0 0 0

Medium elbows 0.75 0 0 0 0 0 0

Long elbows 0.6 0 0 [2.99] [2.99] 1 1

Gate valves 0.19 2 0 1 1 0 0

Globe valves 10 0 0 0 0 0 0

Check valves 2.5 0 0 0 0 0 0

Loop weldolets 1.8 2 0 0 0 0 0

Loop control valves 10 1 0 0 0 0 0

Ball joint assemblies 8.69 0 3+Nsca 0 0 0 0

contractions in the hot and cold runner piping depends on the number of subfield sections in

the solar field. Assuming the runner piping steps down/up in diameter as it moves from the

innermost subfields to those farther away from the power block, an expansion/contraction is

applied for each step in runner piping diameter. This quantity is calculated in Eq.[2.98].

LIOCop = 40[m] +Lspacing (2.94)

Lloop,tot =

Nsca

∑

i=1

(Li +Lsca,gap) (2.95)

Lrun,tot =

Nrunsec

∑

i=1

Lrunsec,i (2.96)

Lhdr =

Nhsec

∑

i=1

2 ·(Lspacing +4.275[m]) (2.97)

Nrun,exp =

&

Nf sec

4

'

(2.98)

Nelbow,L = max& Lrun

70[m]

+.5

'

· 4,8



(2.99)

The constant additional piping length of 4.275m in Eq.[2.97] represents the normalized piping

length per loop required for thermal expansion/contraction compensation [14]. Thermal

cycling of the piping causes the piping to expand or contract axially such that short diverting

47
bends must be interspersed in the field layout to prevent joint separation or buckling.

The runner piping length in Eq.[2.96] is a function of the layout of the field subsections.

Considering again Figure 13 on page 42, note that the runner piping length for the field

configuration with four field subsections is roughly half of what the field configuration with six

field subsections is. For fields with more than four subfields, the runner piping length is a

combination of the shortened run from the power block to the first set of field subsections and

the full lengths spanning the remainder of the field subsections. In addition to the variable

lengths, the runner piping calculations assume a fixed length of 50 m to account for piping in

and around the power block. Eq.’s[2.100-2.103] show the length calculations for runner piping

sections as they extend from the power block. Each length represents the distance in one

direction from the power block: a mirrored counterpart is assumed on the opposite side, and an

equivalent length is assumed to account for both hot and cold piping.

Lrunsec,1 = 50[m] (2.100)

With a “base” runner piping length:

Lrun,base = 2 Lspacing +

Lloop,tot

2 (2.101)

if Nf sec/2 is an even number:

Lrunsec,2 = Lrun,base (2.102)

otherwise:

Lrunsec,2 = 2 · Lrun,base (2.103)

All subsequent (i > 2) runner piping lengths are equal to 2 · Lrun,base. To illustrate the runner

pipe length calculations, consider a plant with eight subsections. The power block is located in

the center of the solar field between the 3-4 and 5-6 subsections. The runner pipe will have

three unique diameters, decreasing in size as it moves away from the power block. The first

diameter is sized to carry one-half of the total solar field HTF flow (the other half goes to the

subfields on the other side of the power block). The length of this piping is a fixed 50 m. The

second section will be a shortened length as described in Eq.[2.102] and will also carry one

half of the HTF flow to the first two field subsections. The final runner pipe section will be of

the length described in Eq.[2.103] and will carry one fourth of the total HTF flow. Return

piping for the hot HTF will follow the same pattern in reverse.

2.5.3 Performance calculations

The equations for the piping equipment use a coefficient kΔP to scale the pressure drop

according to a particular component’s function and geometry. Pressure drop is calculated using

one of three methods according to the type of component. First, pressure drop through a pipe

section is calculated using the Reynolds number and friction factor to determine the head loss

and pressure difference.

ΔPpipe = hlpm ρ g Lpipe (2.104)

48
Where head loss per meter (m/m) is:

hlpm = fr v2

ave

2 Dpipe g

(2.105)

The mean fluid velocity vave scales with the mass flow rate divided by the pipe cross-sectional

area.

vave = m ̇

ρ π D2

4

(2.106)

The friction factor fr is a function of the pipe surface relative roughness and the Reynolds

number (Re). For Re < 2750, the friction factor is simply 64/Re, though larger Reynolds

numbers are more common in trough systems. The friction factor solution in turbulent regimes

requires an iterative solution of an implicit equation, as presented in [9]. The general

expression to be solved is:

fr = 1

X2 (2.107)

where X can only be expressed as an argument for an implicit equation Fx.

Fx = X +2 log10 

Rough

3.7 +2.51 XRe

(2.108)

A solution for X is found when Fx is equal to zero. Since no analytical solution exists for X,

the code iterates until the condition Fx = 0 is satisfied. The function Fx for three possible Re

values as a function of X are plotted in Figure 15.

5.0 5.5 6.0 6.5 7.0 7.5  X 32

1

0

1

2

3

Fx

Re=8,000

Re=20,000

Re=50,000

Figure 15: The solution for Eq.[2.108] at X = 0 is shown for three unique Re values.

49
The second and third methods for pressure calculation apply to the piping equipment rather

than the pipe itself, and these require the use of the pressure coefficient kΔP from Table 11. For

pipe expansions (D ↑) and contractions (D ↓), the pressure drop calculation is given in

Eq.[2.109].

ΔPD↑ = ΔPD↓ = 1

2 kΔP ρ v2

ave(Nhgrp −1) (2.109)

Lastly, System Advisor calculates the pressure drop across the elbows, valves, ball joints, and

weldolets.

ΔPfit = kΔP

D

fr hlpm ρ g Nfit (2.110)

The pressure drop for each piping domain is equal to the sum of pipe, expansion/contraction,

and fitting pressure drops allocated to each column of Table 11. Receiver pressure drop is

equal to the sum of the pressure drops across each collector assembly in the loop. The total

solar field pressure drop is the sum of all the pressure drops from the four piping domains,

though the effective pressure drop through the runner and header sections (calculated using

Eq.’s [2.104, 2.109, 2.110] above) must be averaged according to the mass flow rate passing

through each subsection, as shown in Eq.’s [2.112] and [2.113].

ΔPs f = ΔPloop +ΔPhdr,hot +ΔPhdr,cold +ΔPrun,hot +ΔPrun,cold +ΔPIOCop (2.111)

Where:

ΔPhdr,cold/hot =

Nhgrp

∑

j=1

Ndiv, j

∑

i=1



ΔPpipe, j,i +

ΔP↑/↓,i +ΔPfit,i

Ndiv, j



·

2 m ̇ hdrsec, j,i

m ̇ hdr 

(2.112)

ΔPrun,cold/hot =

Nrunsec

∑

i=1



ΔPpipe,i +ΔP↑/↓,i



·

m ̇ runsec,i

m ̇ hdr

(2.113)

Solar field HTF pumps are assumed to operate continuously over the full range of mass flows

between the minimum and maximum rates specified by the user on the Solar Field page. The

pumps are located along the cold runner where material stress and HTF density are minimized.

The total solar field mechanical pumping power is equal to the pressure drop times the

volumetric flow rate, and a pump efficiency value is used to convert mechanical power into an

electrical parasitic.

W ̇ p,s f = ΔPs f

ηp

m ̇ s f

ρhdr,cold

(2.114)

50
Section 2.5 Summary

• Pressure drop in the inlet/outlet/cross-over pipe, headers, runner piping, and receivers are

all modeled.

• The piping model assumes distributions of expansions, contractions, valves, joints, and

other components based on the field size and geometry to account for pressure drop effects.

• The piping model designs the SGS, pump piping, header, and runner piping diameters to

match the velocity requirements provided by the user.

• The thermal inertia calculations account for pump and SGS piping volume, HTF volume in

runner piping, headers, and the collector loop, and thermal inertia terms (mc)bal,h, (mc)bal,c,

and (mc)bal,sca.

51
3 The Power Cycle

The power cycle subsystem contains the equipment required to convert thermal energy from

the solar field into useful mechanical or electrical energy. For utility-scale CSP systems, this

most often entails a conventional steam Rankine cycle and electric generator, though a number

of other approaches are possible. For example, CSP systems can generate power with a

stand-alone power cycle unit or can be integrated as a combined cycle into a fossil-fired plant

to offset fuel use [3]. Electricity is not always the end goal, as some work has been done on

incorporating the field thermal output directly into various industrial applications [1]. Even

steam as the working fluid is not prerequisite since organic Rankine cycles have been proposed

for lower-temperature operations [16]. But because the scope of System Advisor as an analysis

tool for CSP focuses on utility-scale electricity generation, the current model caters to steam

Rankine cycles with electricity generation.

3.1 General description

Steam Rankine cycle technology is widely used in utility-scale power generation and is well

understood for fossil-fuel, nuclear, and biomass applications. The superheated steam Rankine

cycle also represents the most common large-scale power generation option currently

available. However, while traditional plants typically operate in a narrow range near their

design point, CSP plant operating conditions can vary widely depending on solar resource

availability, storage capacity, and ambient conditions. Analysis of annual electricity production

for CSP is thus highly affected by the quality of off-design performance characterization.

A Rankine cycle can incorporate a relatively broad group of equipment configurations that are

all tied together by the evaporation, energy extraction, condensation, and pressurization

thermodynamic cycle common to the Rankine design. Parabolic trough designs seek to

optimize system efficiency under the thermodynamic constraints of the technology by

maximizing the field outlet temperature and power cycle efficiency, and by minimizing field

thermal losses. The HTF temperature at the field outlet is bound primarily by HTF stability, so

maximum HTF temperatures for oil troughs typically range between 370◦C and 410◦C. Thus,

similar Rankine cycle configurations are typically encountered in different CSP systems.

The definition of the control volume around the power cycle not only can change from system

to system as equipment changes, but also from modeler to modeler based on analytical

convenience. Still, several components are frequently found within the power cycle analytical

control volume; these include the turbine(s), condenser, condensate pump, feedwater pump(s),

deaerator(s), feedwater heater(s), preheater heat exchanger, boiler, superheater heat exchanger,

and reheat heat exchanger(s). Depending on the heat rejection technology, some models will

include the cooling tower or air-cooled condenser in this control volume as well.

The power cycle in this model uses a control volume that surrounds all of the components

listed above with the exception of the heat rejection equipment. The power cycle modeled in

System Advisor uses a Rankine cycle with feedwater heating and superheated steam, and it

allows selection of either fixed or floating turbine inlet pressure on the Power Cycle page. Heat

rejection can occur with either a wet-cooled condenser, air-cooled condenser, or wet/dry

52
parallel cooling system.

A control volume complication arises at the condenser. By allowing the user to choose

between air-cooled and wet-cooled heat rejection, the condenser cooling stream can be either

air or water, and the power cycle model can’t easily handle both stream types. To keep the

power cycle model simple and flexible, the control volume is drawn to exclude the condenser

cooling stream, but includes the steam flow as shown in Figure 16. Thus, the information flow

across the control volume boundary includes the heat rejection load (q ̇re j) as an output and

steam condensing pressure as an input.

The power cycle works closely with the heat rejection model to determine the steam

condensing temperature and load. While these are separate models conceptually, they often are

forced to iterate together to converge on a solution. The heat rejection technology models are

discussed in detail in Section 3.3.

Figure 16: Control volume near the power cycle condenser. The power cycle control volume

bisects the condenser to include the steam flow but not the cooling flow.

3.1.1 Modeling Approach

The ultimate goal of the power cycle model is to accurately characterize off-design

performance while providing enough flexibility to handle typical steam Rankine cycle designs.

Detailed process modeling software packages often provide this capability but often require

extensive setup and long run-times, presenting practical challenges for implementation in

System Advisor’s TRNSYS framework12. Instead of incorporating a detailed model directly

into TRNSYS, process-simulation software is used to construct a representative detailed cycle,

and the output is converted into an off-design performance response surface. System Advisor

uses the well-known “design of experiments” statistical approach [24] to characterize variable

dependencies and generate the response surfaces. The approach used in Solar Advisor was

originally described in [20] and expanded in [21]. A more detailed description of this

technique is presented below in Section 3.2.

12Refer to page 1 for a description of TRNSYS

53
3.1.2 “Basis” Rankine cycle

The basis cycle is a detailed process model of a representative 10 MWe Rankine cycle

developed in EES [15] whose simulated design-point calculations and performance output

forms the basis for the regression model. In the modeled cycle, feedwater is heated in open

(mixed stream) feedwater heaters with two intermediate turbine extractions - once for high

pressure and once for low pressure, and the steam generation equipment consists of a

preheater, boiler, and superheater. A “technology non-specific” (i.e. energy balance) condenser

is used for heat rejection, and this equipment can be linked to either a wet cooling tower or can

be specified as an air-cooled condenser. The model accommodates either fixed pressure at the

turbine inlet or floating pressure as a function of inlet temperature and load. Figure 17 shows

an equipment schematic for this cycle.

Figure 17: Schematic for simplified basis Rankine cycle

The design point temperatures and pressures for this power cycle model match representative

parabolic trough plant operating conditions, though the scale of the modeled system (10 MW)

is somewhat smaller than most operating systems. Table 12 on page 55 lists the applicable

design-point conditions. System Advisor scales the model by using normalized performance

curves and energy balance calculations as discussed in detail below.

The purpose of a basis Rankine cycle is to capture the impact of off-design operating

conditions on the heat absorption and power generation of the power cycle. Overall cycle

thermodynamic efficiency is often increased with the addition of multiple feed-water heating

stages and with a reheat heat exchanger, and while this added complexity improves the

54
Table 12: Design-point conditions for the basis Rankine cycle. The conditions were selected to

match common parabolic trough operating conditions.

Item Value Units

HTF inlet temperature 393 ◦C

HTF outlet temperature 293 ◦C

Steam temperature at turbine inlet 373 ◦C

Boiler steam temperature 311.1 ◦C

Boiler pressure 100 bar

Condenser pressure 0.085 bar

Steam extraction fraction, high pressure 0.13 -

Steam extraction pressure, high pressure 23.9 bar

Steam extraction fraction, low pressure 0.16 -

Steam extraction pressure, low pressure 2.9 bar

Turbine isentropic efficiencies (all) 0.7 -

Pump isentropic efficiencies (all) .695 -

Turbine mechanical power 10.0 MW

Calculated heat exchanger sizes:

Preheater size (UA) 267.6 kW

K

Boiler size (UA) 691.1 kW

K

Superheater size (UA) 115.6 kW

K

design-point efficiency, the behavior of the plant in off-design conditions relative to the design

point stays approximately the same as for a simpler cycle. Therefore, the importance of the

values listed in Table 12 lies only in their relative relationships. Although the design point of

the basis power cycle is 10 MW, the behavior exhibited by this power cycle in off-design

conditions applies to any other power level.

In the design cycle model, the values at the top of Table 12 are set as inputs. The design model

then calculates heat exchanger sizes (expressed as thermal conductance, or UA) and heat

transfer into the cycle (Q ̇ des) according to the condenser pressure (Pc) and desired power

output (W ̇ des). But in the performance cycle, the heat exchanger sizes are fixed while power

output and heat absorption are calculated as outputs.

3.2 Regression model

A detailed performance model like the one described above provides a great deal of accuracy

and capability. By modeling the power cycle using the EES program (or any other process

simulation software), we ensure that the behavior of the power cycle at part-load conditions

matches closely with what we might expect from a real plant. Detailed models calculate steam

conditions at each position throughout the power cycle and can model subsystem performance

in detail. Using this modeling approach in System Advisor presents several obstacles.

First, the Rankine power cycle contains half a dozen major components - at a minimum - and

most typically contain more. Each component may require complicated sub-models to predict

behavior, and most components operate at or near the saturation point for steam. The mix of

55
latent heat transfer, complex sub-models, and steam property instability can be difficult to

handle when the model is expressed in general terms. Secondly, System Advisor performance

simulations require several thousand individual hourly runs to complete, and the iterative

TRNSYS solver can require a total number of iterations orders of magnitude larger. System

Advisor’s simplified power cycle modeling approach minimizes model run times to allow for

the fast calculations required for the type of modeling System Advisor is designed for.

One common method for modeling simplified cycle performance uses one or more polynomial

curves that express power output, cycle efficiency, or some other useful metric as a function of

cycle thermal load. This approach works well when one operating condition varies, but it can

fail when multiple conditions deviate simultaneously from their design point. For example, if

the HTF mass flow rate falls below its design point value, cycle behavior can be captured

adequately by a part-load curve. But if HTF mass flow rate and ambient temperature both

drop, the resulting cycle performance may not be described by simply multiplying the effects

of each individual part-load efficiency curve. In reality, cycle performance is subject not only

to the disassociated main effects of HTF inlet temperature, mass flow rate, and condenser

pressure (a direct byproduct of ambient temperature), but also to the impacts of interaction

between these variables.

The System Advisor power cycle model uses a statistical “design of experiments” approach

[24] to characterize the behavior of the power cycle under a variety of operating conditions.

The method used here is originally described in [20], and a more thorough methodology is

developed in [21]. The procedure for developing a regression model from more detailed

performance calculations is summarized as follows:

• Practical limits on the range of the three independent variables are identified. The

variables are (A) HTF inlet temperature, (B) Condenser pressure, and (C) HTF mass

flow rate

• Parametric runs evaluate the power output (W ̇ ) and cycle heat absorption (Q ̇) over the

full range of inputs. Table 13 shows the experimental design that is used to generate the

required information.

• The information generated by parametric runs in detailed modeling software is

non-dimensionalized.

• Non-dimensional information is analyzed to determine the main effects and effect

interactions.

• These effects are consolidated and applied in the code.

3.2.1 Regression modeling concepts

Instead of relying on part-load polynomial curves or lookup tables to predict plant

performance in off-design conditions, this methodology uses a structured

design-of-experiments approach [24] to characterize plant behavior using detailed process

modeling software, then extracts output dependence from that information within the

framework of statistical effects and interactions.

56
To illustrate this idea, consider the input and output of interest in a Rankine power cycle

model. For CSP plants, the modeler is primarily interested in the total power output, thermal

absorption from the heat transfer fluid (HTF), HTF return temperature, and cooling system

load. These in turn depend on inputs like HTF inlet temperature, turbine back-pressure, HTF

mass flow rate, and others. If the HTF flow rate varies exclusively, a simple part-load curve

might adequately capture the performance impact. However, accurate results are unlikely if the

inlet HTF temperature and mass flow rate both vary simultaneously and the two independent

effect functions are overlaid. The proposed approach not only captures the impact of varying a

single input, but also captures competing interaction effects introduced by concurrent

off-design variation.

3.2.2 Off-design response framework

The ultimate goal of this method is to convert a complex and detailed power block model into

a descriptive simplified model. To accomplish this task, a thorough information set is required

that describes the cycle’s off-design performance as a function of operating conditions, and a

detailed process simulation software tool like IPSEproTM[7], GateCycleTM, or

ThermoFlowTM[12] provides this information. One purpose of this methodology is to

minimize the modeling effort that is required to develop a thorough system performance map,

and this is achieved through strategic selection of the simulation parameters. The response

framework provides a design for the analytical experiment that meets these criteria.

The first step in constructing the response framework is establishing which quantities impact

cycle performance (model inputs) and which are required to fully describe the observed

performance (model outputs). Cycle performance is fundamentally expressed in terms of

power output (W ̇ ) and heat absorption (Q ̇); the relationship between the independent variables

and these two values allows the calculation of secondary information including cycle

efficiency, heat rejection load, and HTF temperature drop. For any given cycle design,

performance is a function of three independent variables13: HTF inlet temperature (Tin),

condenser steam pressure (Pc), and HTF mass flow rate (m ̇). Deviation among these three

variables from their design-point values adversely impacts the heat-to-power conversion

efficiency of the cycle.

3.2.3 Parameter normalization

The input and output variables are readily expressed in normalized terms, and the regression

methodology presented below requires this generalized state. The power output, heat input,

and HTF mass flow rate are simply divided by their design-point values such that they are

equal to 1.0 when they match the design rate. The two exceptions to this approach are the

condenser pressure and the inlet HTF temperature. Condenser pressure does not generally

scale up or down as the size of the power block varies, so it is left in dimensional form.

Inlet HTF temperature is normalized with respect to the superheater temperature rise. For

cycles with superheat, most of the heat transfer from the HTF to the steam working fluid

13This statement assumes that the turbine inlet pressure is either fixed or varies systematically with

HTF inlet temperature.

57
occurs in boiling and superheat. Steam evaporates at a constant temperature for a given boiler

pressure, and this temperature represents the effective minimum temperature at which the HTF

could enter the power block and still provide heat flow into the steam. Therefore, the boiler

steam temperature (not the HTF outlet temperature) is considered to be the zero-value for

normalized HTF inlet temperature.

Tˆ

in = Tin −Tboil

Tin,des −Tboil

(3.1)

3.2.4 Experimental design

The three independent variables are varied parametrically within the response framework to

obtain the required performance information. Each independent variable has a direct impact on

performance, but simultaneous variation of the independent variables can also induce

secondary impacts that adversely affect accuracy. To measure these “interaction” effects, the

analysis includes parametric runs where the inputs are evaluated concurrently at design and

off-design conditions.

Table 13 provides the experimental design framework for this methodology. Each run in the

table represents a parametric set of simulations where the variable indicated with “” is varied

stepwise from the lower to upper bounds. The other two variables are held fixed at either their

lower bound (-), upper bound (+), or design-point value (0). Runs 2, 5, and 8 measure the

primary effects of one of the three independent variables while the other two are held at their

design-point value. The remainder of runs 1-9 aid in measuring the impact of two-variable

interaction (denoted AB), and runs 10-13 measure the impact of the three-variable interaction

(which is often negligible and has been neglected in this paper). The upper and lower bounds

are selected based on practical operation limits of the power cycle. For example, if the

minimum turn-down state for the modeled cycle is 30% of design (i.e. a 70% reduction in

operation), the lower bound on m ̇ should be 0.3×m ̇ des.

3.2.5 Formulating a model

The output of the runs in Table 13 is used to generate a model that can express both heat

absorption and power output as functions of the three independent variables and their

interactions. The intermediate result of each run is a table that relates heat absorption or power

output to a particular arrangement of Tin, Pc, and m ̇. For example, Run #1 will produce a pair

of functions (one for power output and a second for heat absorption, both expressed as

functions of Tin) that vary with HTF inlet temperature at a fixed lower-bound value for HTF

mass flow rate. Substituting the general output variable Y for Q ̇ and W ̇ :

YRun1 = f



Tin,m ̇ (−)



(3.2)

Figure 18 corresponds to runs 4-6 in Table 13 and shows how power (left) and heat (right)

output vary as a function of parameterizing Pc at three distinct levels of Tin. The central lines in

Figure 18 represent the situation where only Pc is varied and both Tin and m ̇ are left at their

58
Run Tin Pc m ̇ Measures

1 0 - f(m ̇Tin)

2 0 0 f(Tin)

3 0 + f(m ̇Tin)

4 - 0 f(TinPc)

5 0 0 f(Pc)

6 + 0 f(TinPc)

7 0 - f(Pcm ̇)

8 0 0 f(m ̇)

9 0 + f(Pcm ̇)

10 - - f(TinPcm ̇)

11 - + f(TinPcm ̇)

12 + - f(TinPcm ̇)

13 + + f(TinPcm ̇)

Table 13: The experimental design for characterizing power block performance. The experiment

is adapted from a three-level full-factorial design.

design-point values (run #5).

This methodology asserts that the upper and lower curves observed in Figure 18 can be

approximated by multiplying the main effects of YME(Tin) and YME(Pc), where the two-variable

interaction effect YINT (TinPc) scales the impact of the second main effect. Since the purpose

of the interaction effect is to adjust the magnitude of main effect’s impact on the output, the

interaction effect is applied only to the difference between the calculated main effect and the

basis normalized value (i.e. YME −1). Since all effect functions in the set Y are normalized

about 1, the main effect must first be shifted from 1 to 0 before being multiplied by the

interaction effect. The product is then readjusted to 1.0 as shown in Eq.[3.3]. In general terms,

the calculated output variable Y∗ is determined for any set of inputs Xi with the relationship:

Y∗ =

nX

∏

i=1



(YME(Xi)−1)·YINT XjXi



+1

 (3.3)

In Eq.[3.3], Xj represents the input variable that interacts with Xi. For example, in Table 13, m ̇

interacts with Tin, and this effect is measured in runs #1-3. In this example, Xj ≡ m ̇ and

Xi ≡ Tin.

Interaction effects measure the relative difference between the calculated value that is

predicted by multiplying only main effects and the actual value observed from the detailed

modeling simulation. Mathematically speaking, the interaction is the ratio of values (Observed

- Main Effect)/(Predicted - Main Effect), as shown in Eq.[3.4] (the Yˆ notation indicates a data

point obtained from the detailed model as opposed to the function family Y within the

regression model). The interaction equation can be formulated for either the lower or upper

59
0.00 0.05 0.10 0.15 0.20 0.25

Condenser pressure [bar] (Pc )

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Normalized Power Output (  ̇W)

Tin at (+)

Tin at (0)

Tin at (-)

0.00 0.05 0.10 0.15 0.20 0.25

Condenser pressure [bar] (Pc )

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Normalized Heat Input (  ̇

Q)

Tin at (+)

Tin at (0)

Tin at (-)

Figure 18: Data showing how the power and heat output varies with condenser pressure [bar] at

three distinct levels of HTF inlet temperature.

bound runs (±), assuming either the lower or upper bound of Xj is used.

Y(±)

INT 

X(±)

j Xi



= Yˆ (±)

(Xi)−YME(Xi)

YME(Xi)



YME(X(±)

j )−1

 (3.4)

The interaction calculations provide a function that indicates how variation in one independent

variable influences the effect of another varying independent variable. The relative impact of

an interaction effect applies consistently to the main effect that it’s adjusting. In other words, if

two independent variables Xi and Xj simultaneously deviate from their design, the magnitude

of the interaction effect that is applied to the main effect is only a function of the value of the

first independent variable Xi. Consequently, interaction effects that are calculated for the lower

and upper limit values for a particular effect (i.e. runs #1 and #3 for run #2) both represent the

same interaction effect and can be averaged to estimate a better interaction curve as shown in

60
Eq.[3.5].

YINT 

XjXi



=

Y(+)

INT 

X(+)

j Xi



+Y(−)

INT 

X(−)

j Xi



2 (3.5)

Note that the resulting expression for YINT (XjXi) contains arguments for Xj and Xi; however,

the function is only truly dependent on Xi, retaining Xj in the nomenclature merely as a

reference to the two interacting variables. Figure 19 shows the heat absorption and power

output as a function of HTF temperature for three mass flow rates and the associated

interaction effect curves. This shows that even for cases where the output appears to be linearly

related to the independent variables, interaction effects can have a significant relative impact.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

HTF Inlet Temperature (Tin)

0.0

0.5

1.0

1.5

2.0

Power output (  ̇W)

m ̇ at (-)

m ̇ at (0)

m ̇ at (+)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

HTF Inlet Temperature (Tin)

0.0

0.5

1.0

1.5

2.0

Heat absorption (  ̇

Q)

m ̇ at (-)

m ̇ at (0)

m ̇ at (+)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

HTF inlet temperature (Tin)

0.99

1.00

1.01

1.02

1.03

1.04

1.05

1.06

Interaction value

(m ̇ |Tin) for power output

(m ̇ |Tin) for heat abs.

Figure 19: Data showing how the power and heat output varies with HTF temperature at three

distinct levels of HTF mass flow rate (top), and the associated interaction effects (bottom).

3.2.6 Other calculated output

Since the power cycle model regresses only power output and heat input as functions of HTF

temperature, condenser pressure, and mass flow rate, the model uses another set of equations

to calculate other performance values. The calculation of power cycle efficiency is

straightforward. Thermodynamic power cycle efficiency is determined by simply dividing the

61
work output by the required heat input.

ηcycle = W ̇

Q ̇ (3.6)

The outlet HTF temperature from the power cycle is calculated by considering the HTF mass

flow rate, the heat input, the inlet temperature, and the specific heat of the HTF. The specific

heat is assumed to vary linearly over the range of the inlet and outlet temperatures. Thus, an

average specific heat value is used and is recalculated throughout the simulation.

Tht f,out = Tht f,in − Q ̇

m ̇ ht f cht f,ave

(3.7)

The power cycle model does not directly model the performance of the heat rejection

equipment. Instead, it supplies information to an external heat rejection model on the

magnitude of the rejection load. This value is equal to the difference between work output and

heat absorption, or alternatively:

q ̇re j = 

1−ηcycle

Q ̇ (3.8)

Minerals, scales, and other solids accumulate in the power cycle equipment [6] during

operation. Excessive buildup of these particulates detrimentally affects heat exchanger

performance and can damage the turbine blades and pumps. This situation is avoided by

extracting a fraction of the steam flow (called “blowdown”) and replacing it with fresh,

demineralized water. Additional water and steam is lost through other mechanisms, including

leaks in the pressurized flow circuit, entrainment of steam with non-condensibles in the

deaeration process, and evaporation from the wet cooling system (see Section 3.3). All of

these water losses must be made up to maintain a constant level in the plant. System Advisor

models the steam working fluid replacement rate (excluding wet-cooled evaporation) by

calculating an approximate steam mass flow rate during operation and replacing a fixed

fraction with fresh water.

The steam blowdown mass flow rate is calculated by Eq.[3.9], where the blowdown fraction is

given by fbd and the total enthalpy of heat addition across the heat exchangers is Δhheat.

m ̇ st,bd = W f  ̇ bd

ηcycle Δhheat

(3.9)

The model assumes a default blowdown fraction of 2.0%, which falls in the range

recommended by [6], but this fraction can be adjusted by the user. The enthalpy change across

the heat exchangers is estimated by assuming that the temperature drop from the hot inlet HTF

to the turbine inlet steam is 25◦C, and by assuming that the temperature of the feedwater at the

62
inlet of the preheater is 100◦C below the boiling temperature. Thus:

Δhheat = hst,hot −hst,cold (3.10)

Where:

hst,hot = enthal py

T = Tht f,in −25◦C,P = Pboil

hst,cold = enthal py(P = Pboil,χ = 0)−100◦

C·



4.91 kJ

kg.C



The specific heat value of 4.91 kJ

kg.C is selected based on the integrated average of the 100◦C

temperature range below the boiling temperature for a boiler pressure of 100 bar.

Section 3.2 Summary

• The power cycle uses a regression model to capture the off-design performance behavior of

a basis power cycle modeled in EES.

• The regression approach captures the impact of HTF inlet temperature, condenser pressure,

and HTF mass flow rate on Power output and Heat input.

• The regression methodology accurately models part-load behavior by accounting for main

effects and interactions between simultaneously varying inputs.

• The model implemented in System Advisor achieves fast run-times and good adherence to

the basis model performance.

• The regressed heat and power values are used to calculate cycle efficiency and HTF outlet

temperature.

• The power cycle models blowdown water use by estimating the steam mass flow rate at

each time step.

3.3 Heat rejection

Discussion in Section 3 has focused thus far on modeling the response of heat and power as

functions of three independent variables: (A) HTF inlet temperature, (B) condenser pressure,

and (C) HTF mass flow. A and C are determined directly by the solar field (or the controller)

and are inputs to the power cycle model, but the condenser pressure is more closely affiliated

with the power cycle than the solar field. Unlike the HTF temperature and mass flow rate that

can be varied more directly, condenser pressure is dependent on external phenomena like

ambient temperature, cooling technology, and cooling load.

Why is condenser pressure incorporated into the model as an independent variable? Recall

from Figure 16 and the surrounding discussion that the power cycle analysis control volume

excludes the condenser because of the complication of multiple possible heat rejection

63
streams. This means that the condenser and all of the associated heat rejection equipment must

be modeled outside of the power cycle itself. The interface between the power cycle model and

the heat rejection model requires passing the heat rejection load q ̇re j from the power cycle to

the cooling model and passing the achieved condenser pressure pcond from the cooling model

to the power cycle. A high level of attention to heat rejection is important because both the

thermodynamic performance and the total plant water consumption depend on how heat is

rejected from the power cycle.

3.3.1 Performance considerations

The thermodynamic cycle can take many forms - the Rankine steam power cycle is just one

manifestation. All of these cycles have one operating principle in common: heat energy

flowing from a hot reservoir to a cold one can be used to generate mechanical power (see

Figure 20). In the Rankine cycle, the hot reservoir is the process that provides heat to the cycle.

For CSP, this is the solar field, but in other technologies it may be a coal-fired boiler or a

nuclear reactor. The cold reservoir may be a large or moving body of water, an evaporation

pond, or ambient air.

Figure 20: A simplified Rankine cycle diagram. Heat is added to the cycle by the solar field via a

heat exchanger. Heat is removed from the system via a wet cooling tower. Power output is equal

to the heat addition minus the heat rejection.

The efficiency of the power cycle depends on the “quality” of the thermal reservoirs. A hotter

heat source or a colder heat sink both increase thermodynamic efficiency. Even a seemingly

small change in the quality of either reservoir can impact cycle efficiency in a way that gives

one design a competitive advantage over another. This is especially true of CSP plants, where

the heat source is the most expensive portion of the initial capital investment. The two cooling

technologies available to nearly all CSP plants are wet cooling and dry cooling. These

technologies lie on opposite ends of the spectrum in terms of both performance and water use

[23], and these are the technologies that System Advisor models.

64
Both wet and dry cooling use ambient air as the ultimate cold thermal reservoir, but differ in

the mechanism of heat transfer between the cycle and air. Wet cooling systems use a deluge of

water to remove heat through evaporation; thus the temperature of the cold reservoir is driven

by the wet-bulb air temperature. Dry cooling systems transfer heat directly from the steam

working fluid to air using a sensible-heat process. This technique is limited by the dry-bulb

temperature of air, which can be significantly higher than the wet bulb temperature, especially

in arid regions where CSP is most desirable. Figure 21 shows the significant difference

between dry and wet-bulb temperature for five summer days in an arid climate.

Figure 21: Dry and wet bulb temperatures for five summer days in an arid climate. Peak

temperature differences approach 20◦C during the hottest hours of the day.

As steam exits the turbine, it is a mixture of saturated steam and a small amount of liquid

water. In this partially condensed state, the temperature and pressure of the steam are directly

linked such that one property fixes the other. Whether the system is wet or dry cooled, the

temperature achieved by the cooling technology will set the condenser pressure, and this value

is then used as an input to the power cycle model. Thus, the ambient temperature, condenser

pressure, and overall cycle thermodynamic performance are directly connected. Consider the

performance impact of the 20◦C temperature difference between wet and dry technologies

suggested in Figure 21. Superimposed on a performance plot in Figure 22, this temperature

difference shows how the condenser pressure and subsequent cycle performance are adversely

impacted.

65
Figure 22: The impact of a 20◦C temperature change on condenser pressure and cycle

performance.

The disadvantage of achieving improved performance with wet cooling systems is the

significant amount of water that these systems lose through evaporation. Furthermore, a typical

100MW wet-cooled plant with a 40% capacity factor can consume nearly 1.4 cubic kilometers

of water per year, of which 94% is due to evaporation [23]. Dry cooling uses virtually no

water, and thus may be the only option for project locations where water is limited. System

Advisor models both the water use and performance impacts of the different cooling

technologies. System Advisor also models an emerging cooling technology that places a wet

cooling system in parallel with an dry cooling system with the purpose of reducing the

temperature rise of the cooling air across the dry system. All three of these cooling options are

discussed in detail below.

3.3.2 Condenser performance limitations

The wet, dry, and hybrid cooling systems all require the use of a condensing heat exchanger to

convert high quality steam from the turbine to feedwater that is pumped back through the heat

exchanger system. Steam exits the turbine under vacuum conditions and must be piped via

ducting to the condenser equipment. Transport of steam at low pressures can be problematic,

since the pressure drop between the turbine outlet and the condenser inlet is proportional to the

steam velocity through the ducting. Rule-of-thumb guidelines require that low-pressure steam

velocity through ductwork be limited to less than 30 m/s (approximately 100 ft/s). Figure 23

shows how steam velocity varies with condensing pressure for a plant that has been designed

for a minimum pressure of 2 inHg14 (0.068 bar). The selected design corresponds to a 110

MWe gross power cycle with four steam ducts with diameters of 4.1m each.

14Inches of mercury. 1 inch of mercury corresponds to 0.03386 bar at 15.5◦C.

66
0.00 0.05 0.10 0.15 0.20 0.25 0.30

Steam Pressure [bar]

0

50

100

150

200

Steam Velocity [m/s]

Figure 23: Steam velocity as a function of steam pressure through ductwork in a plant designed

for a minimum condenser pressure of 0.068 bar

In order to accommodate steam pressures below the 0.067 bar limitation shown above, the

plant designer must either increase the ducting diameter or include additional ducts in parallel.

This quickly becomes economically and technically impracticable; instead, the system is

designed to maintain suction pressure with acceptable steam velocity and pressure drop

through the ducting. Since higher turbine outlet pressure detracts from overall power cycle

efficiency, this practice balances duct cost and cycle performance. System Advisor

incorporates a user-specified minimum condenser pressure (on the Power Cycle page). The

default value of 1.25 inHg applies to wet cooling systems. A higher value of 2.0 inHg is

commonly used for air cooling and hybrid cooling systems.

In order to maintain the condenser pressure above the minimum requirement, the heat rejection

equipment must partially shut down. By incrementally reducing the heat rejection capacity of

the system, the temperature rise of the cooling water (or cooling air stream, in the case of dry

cooling) increases thereby increasing the condenser temperature and pressure. System Advisor

incrementally reduces the cooling system capacity according to the “Cooling system part load

levels” (Nhr,pl) input on the Power Cycle page. During time steps when the calculated

condenser pressure falls below the minimum value, System Advisor will reduce the cooling

capacity by the fraction 1/Nhr,pl until the condenser pressure is restored to an acceptable level.

Parasitic pumping/fan loads are also reduced in proportion with the partial operation.

Occasionally, the ambient and power cycle operating conditions cause the condenser pressure

to fall below the minimum level even when the cooling equipment operates at its minimum

level. System Advisor models this situation by assuming that the cooling equipment is capable

of operating in an idealized part-load mode where the condenser pressure is set equal to the

minimum value. The parasitic loads and other performance parameters are then recalculated

assuming operation at this minimum pressure value. This approximation is sufficient so long

67
as this situation is encountered infrequently throughout the year.

3.3.3 Wet cooling model

The wet cooling model calculates the condenser pressure, parasitic load, and water use for a

forced-draft wet cooling system common in Rankine cycle power plants. The model requires a

number of parameters and inputs to calculate performance, and these are provided either

directly from the user, from values available to the power cycle model, or from assumed values

coded into the model that cannot be changed by the user. These items are summarized in Table

14.

Table 14: Summary of inputs to the wet cooling model.

Variable Description Units Value

User-supplied inputs

ΔTcw,des The cooling water temperature rise across the condenser ◦C 10

ΔTapproach The cooling water approach temperature ◦C 5

Inherited inputs

W ̇ des Power output at design MW -

ηdes Power cycle efficiency at design - -

Tdb Dry bulb temperature ◦C -

Twb Wet bulb temperature ◦C -

Pamb Atmospheric pressure Pa -

Assumed values

ΔTout Temperature difference at the hot side of the condenser ◦C 3

fdrift Drift loss fraction - 0.001

fctbd Cooling tower blowdown fraction - 0.003

ΔPcw Circulating water pressure drop bar 0.37

ηpump Cooling water pump mechanical efficiency - 0.75

ηpump,s Cooling water pump isentropic efficiency - 0.80

ηf an Fan mechanical efficiency - 0.75

ηf an,s Fan isentropic efficiency - 0.80

rp, f an Fan pressure ratio - 1.0025

rmass, f an Fan air-flow to water-flow mass ratio - 1.01

In addition to these input values, the wet cooling model makes use of a number of hard-coded

property curves. These are presented in Table 15.

The wet cooling model assumes that Tapproach entered by the user and ΔTout assumed by the

model are constant throughout the performance calculations. Wet cooling systems operate by

circulating water through a condenser heat exchanger to remove heat from the steam flow. This

circulating water then passes through the wet cooling tower where heat is removed by direct

evaporation. The cooling water temperature rise (ΔTcw,des) is a function of the heat rejection

68
Table 15: Property functions used by the wet cooling model.

Units Input Expression

Enthalpy of evaporation near atmospheric pressures

J

kg P [Pa] 2.362 · 106 −1.355 ·P+3.085 · 10−6 ·P2

Specific heat of saturated liquid water

J

kg.K P [Pa] 4170.5+5.381 · 10−4 ·P−7.734 · 10−10 ·P2

Saturation pressure of steam

Pa T [K] 1123.1−19.64 · T +4.426 · T2 −0.039 · T3 +9.655 · 10−4 · T4

Enthalpy of water at liquid saturation near ambient pressures

J

kg P [Pa] 2.296 · 105 +2.785 ·P−1.112 · 10−5 ·P2 +2.12 · 10−11 ·P3

Entropy of water at liquid saturation near ambient pressures

J

kg.K P [Pa] 779.99+7.916 · 10−3 ·P−3.330 · 10−8 ·P2 +6.386 · 10−14 ·P3

Density of water at liquid saturation near ambient pressures

kg

m3 P [Pa] 984.1−3.071 · 10−4 ·P+5.323 · 10−10 ·P2

Enthalpy of air as a function of temperature at ambient pressure

J

kg T [◦C] 2.735 · 105 +1002.9 · T +0.0327 · T2

load and the mass flow rate of the circulating water as shown in Eq.[3.11].

ΔTcw,des = q ̇re j

m ̇ cw cp,cw

(3.11)

The mass flow rate during operation is restricted by the capabilities of the circulating pumps.

The modeled system assumes two fixed-speed pumps, each sharing 50% of the load. Thus, the

operating strategy allows for either 100% or 50% mass flow conditions. The cooling water

temperature rise necessarily varies proportionally with the heat rejection load. If the load falls

below 50% of the design value, the circulating water mass flow rate is reduced to 50%. Design

conditions are determined as follows:

q ̇re j,des = W ̇ des  1

ηcycle,des

−1



(3.12)

m ̇ cw,des = q ̇re j,des

cp,cw ΔTcw,des

(3.13)

As previously mentioned, the mass flow rate is a discrete value equal to either 100% or 50% of

full-load.

q ̇re j ≥ q ̇re j,des

2 ⇒ m ̇ cw = m ̇ cw,des (3.14)

q ̇re j < q ̇re j,des

2 ⇒ m ̇ cw = m ̇ cw,des2

69
Once the operating mass flow rate has been determined, the actual temperature rise across the

condenser is calculated.

ΔTcw = q ̇re j

m ̇ cw cp,cw

(3.15)

Finally, the condensing temperature is calculated as the difference between the ambient

wet-bulb temperature and the sum of the various temperature rises. The condenser pressure is

determined using the property relations presented in Table 15. Figure 24 illustrates the various

temperature rises that impact the condenser pressure.

Figure 24: An illustration of the various temperature rises that influence the condenser pressure

for a wet-cooled system. These include the ambient wet-bulb temperature, the approach

temperature, the cooling water temperature rise, and the heat exchanger hot side temperature

difference.

Tcond = Twb +ΔTapproach +ΔTcw +ΔTout (3.16)

Condenser pressure is the most important performance metric provided by the cooling model.

However, two other outputs are of interest; these are parasitic power consumption and water

use. The parasitic load for a forced-draft cooling tower is driven by the circulating water pump

power and the fan power to maintain airflow through the system. System Advisor models both

of these parasitics.

The pumping power is calculated by using a simple constant isentropic efficiency. The

enthalpy and entropy of the cooling water at the pump inlet are obtained using property

relations in Table 15. Temporarily assuming no irreversibilities, the pump outlet enthalpy is:

hcw,out,s = ΔPcw

ρcw

+hcw,in (3.17)

70
The actual enthalpy at the pump outlet is calculated by applying the isentropic efficiency.

hcw,out = hcw,in +

hcw,out,s −hcw,in

ηpump,s

(3.18)

The total power consumed by the cooling water pump is calculated in Eq.[3.19].

W ̇ cw,pump = (hcw,out −hcw,in) m ̇ cw

ηpump

(3.19)

The wet cooling model calculates fan power with the assumption that the ratio of air mass flow

to cooling water mass flow is constant according to the fraction rmass, f an from Table 14. Thus,

parasitic fan power is reduced during part load when the mass flow rate of cooling water is

reduced. The moist air moved by the fan is at an effective temperature somewhere between the

ambient wet and dry-bulb temperatures plus the approach temperature. This value is

approximated as:

Tf an,in = Tdb +Twb +ΔTapproach

2 (3.20)

Fan power is proportional to the enthalpy increase across the fan. This enthalpy rise can be

approximated by assuming that ambient air behaves as an ideal gas. In this case, the air

temperature at the outlet of the fan is calculated assuming no thermodynamic irreversibilities

(the temperatures are absolute), where R = 8314

28.97  Jkg.K



is the gas constant for air.

Tf an,out,s = Tf an,in r

 R cp,air 

p, f an (3.21)

The isentropic enthalpy at the fan outlet is determined directly as a function of this outlet

temperature from the relations in Table 15, and the actual outlet enthalpy is calculated by

considering the isentropic fan efficiency.

hf an,out = hf an,in +

hf an,out,s −hf an,in

ηf an,s

(3.22)

Finally, we calculate fan power and total cooling parasitic power as:

W ̇ f an =



hf an,out −hf an,in

m ̇ air

ηf an

(3.23)

W ̇ cool [MW] = 

W ̇ cw,pump +W ̇ f an

·



1 MW

106 W



(3.24)

The cooling tower water use is a straightforward calculation that accounts for evaporative loss,

71
drift loss, and cooling tower blowdown. The latter two calculations are dependent on the

cooling water mass flow rate, while the former only depends on the heat rejection load and the

enthalpy of evaporation, which is a function of ambient pressure.

m ̇ evap = q ̇re j

Δhevap

m ̇ drift = fdrift m ̇ cw

m ̇ ctbd = fctbd m ̇ cw

m ̇ water = m ̇ evap +m ̇ drift +m ̇ ctbd (3.25)

3.3.4 Dry cooling model

The dry cooling model makes use of the user-supplied inputs and assumed values that are

summarized in Table 16. Property relations are borrowed from the wet cooling model.

Table 16: Summary of inputs to the dry cooling model.

Variable Description Units Value

User-supplied inputs

TITD,des The initial temperature difference (steam-to-ambient) ◦C 16

rp,cond The condenser air pressure ratio - 1.0028

Inherited inputs

W ̇ des Power output at design MW -

ηdes Power cycle efficiency at design - -

Tdb Dry bulb temperature ◦C -

Pamb Atmospheric pressure Pa -

Assumed values

ΔTout Temperature difference at the hot side of the condenser ◦C 3

ηf an,s Fan isentropic efficiency - 0.80

ηf an Fan mechanical efficiency - 0.94

cp,air Specific heat of air J

kg.K 1005

Like the wet cooling model, the cooling air flow generated by the fans is limited to either

100% or 50% flow relative to the design value. The design air mass flow rate is calculated in

Eq.[3.26].

m ̇ air,des = q ̇re j,des

cp,air 

TITD,des −ΔTout (3.26)

Since dry cooling systems don’t consume appreciable amounts of water for use in cooling, the

only metrics of interest are the condenser pressure and the parasitic fan power. Condenser

pressure is determined by the condenser temperature, which is simply equal to the dry-bulb

temperature plus the actual initial temperature difference(ITD) as illustrated in Figure 25. The

ITD is the sum of the temperature rise of the air across the condenser and the hot-side

72
temperature difference.

Tcond = Tdb +TITD

where :

TITD = ΔTout + q ̇re j

m ̇ air cp,air

(3.27)

Figure 25: An illustration of the various temperature rises that influence the condenser pressure

for a dry-cooled system. These include the ambient dry-bulb temperature, the air temperature

rise, and the heat exchanger hot side temperature difference.

The relationship for parasitic fan power is analogous to the one previously developed for the

wet cooling fan, except the fan inlet temperature is equal to the ambient dry-bulb temperature.

Tf an,out,s = Tf an,in ·r

 R cp,air 

p,cond (3.28)

hf an,out = hf an,in +

hf an,out,s −hf an,in

ηf an,s

(3.29)

W ̇ f an [MW] =



hf an,out −hf an,in

m ̇ air

ηf an

·



1 MW

106 W



(3.30)

3.3.5 Parallel hybrid cooling model

System Advisor models parallel hybrid cooling in addition to the wet and dry systems

discussed above. This configuration places an undersized wet cooling system in parallel with a

standard (or slightly undersized) ACC. An illustration of this concept is shown in Figure 26.

The hybrid cooling configuration has been identified in previous work [23] [22] as a candidate

technology that both reduces power plant water use and mitigates the negative performance

impact of switching from wet to dry cooling. The technology is based on the premise that air

cooling provides sufficient cooling capability during the majority of the plant’s operating

73
Figure 26: An illustration of the parallel wet/dry cooling configuration is shown. The ACC (left)

and wet system (right) share the heat rejection load.

hours. However, performance for a dry cooling system is most negatively impacted during

summer afternoon hours when both the ambient temperature and electricity sales revenue are

highest. During these summer hours, the temperature rise of the cooling air stream across the

ACC is relatively high, and placing a wet cooling system in parallel to share the heat rejection

load reduces this temperature rise and improves cycle thermodynamic efficiency.

Unlike the exclusively wet cooling system whose performance tracks the wet bulb

temperature, the hybrid system is forced to track the dry bulb temperature. Since the wet and

dry cooling systems are interconnected, the condensing steam pressure is equal in both

systems. If the steam pressure (and associated temperature) were to theoretically fall below the

dry bulb temperature, thermal energy would begin flowing from the ambient air into the ACC,

thus defeating the purpose of incorporating the dry cooling system. Therefore, hybrid system

thermodynamic performance falls between the lower bound of a non-ideal dry cooling system,

and the upper bound of an ideal dry cooling system (that is, an ACC that achieves a steam

temperature equal to the ambient dry bulb temperature).

The distribution of the heat rejection load between the wet and dry cooling systems is specified

by the user on the Power Cycle page using the Hybrid Dispatch fractions. Each of the input

values corresponds to the particular TOD period specified by the user on the Thermal Storage

page. For each time step during the simulation, the wet cooling load in the hybrid system will

equal the total cooling load times the associated user-specified fraction. Thus, the total heat

rejection load q ̇re j in the air cooled system is shown in Eq.[3.31] and the wet cooled system is

shown in Eq.[3.32]. The fraction of wet cooling during the current timestep is denoted by fwc.

The valid range for the wet cooling fraction is 0 ≤ fwc < 1.

q ̇re j,air = (1− fwc) q ̇re j (3.31)

q ̇re j,wc = fwc q ̇re j (3.32)

74
The performance equations for the hybrid cooling configuration are nearly identical to the

equations for the individual technologies presented above. The primary difference lies in the

heat rejection load that each subsystem is required to accommodate and the change in this

value relative to the design point. For a hybrid cooling system, the ACC is sized to

accommodate the full heat rejection load throughout the year15. The wet cooling system is

sized to match it’s largest required heat rejection load throughout the year. For example, if the

user provides hybrid fractions of 0.25, 0.5, and 0.75 for three of the available hybrid fractions,

then the wet cooling system will be sized to provide 0.75 (75%) of the total heat rejection load.

System Advisor calculates the condenser pressure for the hybrid system by first calculating the

performance of each cooling system individually, then using the larger of the two condenser

pressures as the actual achieved pressure.

Pc = max[Pc,wc,Pc,ACC] (3.33)

Like the wet and dry technology models, the hybrid system models part-load behavior by

assuming that the cooling equipment turns off incrementally when the condenser pressure falls

below the minimum value. The part-load behavior is applied equally to the wet and dry

systems (if both are operating) except in the case that the calculated wet cooling condenser

pressure exceeds the dry cooling pressure. In this situation, System Advisor incrementally

shuts down only the wet system until the minimum condenser pressure is satisfied. This

control mechanism is implemented to reduce water use when possible.

Section 3.3 Summary

• The condenser and all heat rejection equipment are modeled separately from the power

cycle

• Performance varies significantly as the ambient temperature varies; thus dry cooling suffers

a greater performance penalty compared to wet cooling. Hybrid cooling performance falls

between wet and dry cooling.

• The wet cooling model calculates condenser pressure, water use, and parasitic pumping/fan

power. The dry cooling model calculates condenser pressure and parasitic fan power. The

hybrid cooling model calculates performance for both systems in parallel.

• All cooling models assume that part-load heat rejection allows for finite reductions in heat

rejection capacity.

• The condenser calculations enforce a minimum condenser pressure according to user input.

15This statement is true in a general sense. However, [22] shows that an undersized ACC is optimal

in some utility markets. Consequently, the ACC sizing and ITD design value should be items that are

optimized by the plant designer for each project.

75
3.4 Power cycle implementation

The power cycle has three possible main operating modes; these include complete shut down,

“standby” operation, and normal operation. The normal operation mode is further broken

down into mass-demand and power-demand modes. The operating modes are controlled

externally by the plant control algorithm, and the conditions under which each mode is

activated are discussed in detail in Section 4. Each of these modes is unique in the way inputs

and outputs are handled, so each is discussed in detail in this section.

3.4.1 Normal operation

The “normal operation” mode encompasses all situations where the power cycle is producing

power. Depending on the operating conditions, it may be advantageous to either require that

the power cycle produce a target power and require the corresponding HTF mass flow rate, or

alternatively a HTF mass flow rate may be provided and the power output calculated. System

Advisor automatically determines the best option based on the ability of the solar field, storage

system, and/or the auxiliary heater to fully meet the design-point thermal power for the power

cycle.

In the target-power situation, the power cycle model uses the time-of-use adjusted16 target

power value W ̇ dem to determine a guess-value for the HTF mass flow rate. This initial guess is

recalculated at each time step to correspond to possible changes in the demand power value.

m ̇ ht f,re f = Q ̇des

cht f ·



Tht f,hot,des −Tht f,cold,re f  ·

W ̇ dem

W ̇ des

(3.34)

Where:

W ̇ dem = W ̇ des ftou,pb (3.35)

Using the guess mass flow rate and the provided HTF inlet temperature and condenser pressure

(which is concurrently iterated), the model calculates the power output and heat absorption

using the regression model described in Section 3.2. Since the cooling models determine

condenser pressure as a function of ambient temperature and heat rejection load, and since heat

rejection load may change during power cycle iteration, the cooling model must be called

initially to estimate performance and be re-called during each iteration. The convergence error

is based on the relative difference between the calculated cycle power output and the desired

output.

err = W ̇ calc −W ̇ dem

W ̇ dem

(3.36)

If the convergence error is greater than the lower limit of err = 10−6, the mass flow rate is

subsequently adjusted according to Eq.[3.37], and the iterative solution continues. This

adjustment equation was selected based on empirical observations of calculation speed and

16For a more detailed discussion of TOU power cycle adjustments, see page 94

76
output stability. The maximum number of power cycle iterations is set to 100.

ad j = 1+

W ̇ dem −W ̇ calc

3 ·W ̇ dem

(3.37)

Once the iterative mass flow solution is found, this information is passed to the controller

component which iterates to find an agreeable system solution. If no iterative solution can be

found, or if the solution returned by the power cycle fails to satisfy all of the bounds in Table

17, power output and cycle efficiency are set to zero, the HTF outlet temperature is set to the

inlet temperature, and the demand mass flow rate is set to the reference mass flow rate.

Table 17: Double-check criteria for the converged power cycle solution under normal operation.

ηcycle ≤ 1

ηcycle > 0

Tht f,cold ≤ Tht f,hot

Tht f,cold ≥ Tht f,cold,des −50

The target-mass flow strategy is the more straightforward approach. This mode is used when

insufficient thermal energy is provided by the solar field, storage system, and/or the auxiliary

heater to meet the full-load thermal requirement of the power cycle. In this operating mode,

the plant controller provides a particular HTF mass flow rate to the power cycle. The power

cycle responds by calculating power output in a once-through manner - no internal iteration is

required.

3.4.2 Shutdown and cold startup

The second mode of operation for the power cycle involves a complete shutdown of the

system. The more obvious features of this mode include setting power output, cycle efficiency,

HTF demand mass flow rate, water use, and parasitic loads for the power cycle to zero. During

shutdown, the HTF outlet temperature is equal to the HTF inlet temperature since no thermal

energy is consumed by the power cycle and the thermal losses from the power cycle equipment

are assumed to be negligible.

Once the power cycle enters shutdown, cold startup of the turbine requires the full time and

energy specified by the user. Startup can occur once a thermal resource becomes available.

Cycle startup is constrained by a time duration and a thermal fraction. The startup time

requirement is checked during each time step, and the startup algorithm can handle both

startup times that are longer and shorter than the time step duration. For longer startup times,

the remaining startup time (Δtstart,remain) is tracked as a ledger balance. So long as the

remaining startup time exceeds the duration of the time step (Δt), the cycle remains completely

shut down.

If the remaining startup time (or initial startup time) is less than the duration of the time step,

or if the remaining startup energy falls below the startup energy requirement, then the power

77
production from the cycle is reduced during the time step. System Advisor calculates the

startup energy requirement by using the energy fraction specified by the user (fstart), as shown

in Eq.[3.38].

Estart = fstart

W ̇ des

ηcycle,des Δt (3.38)

Once thermal energy is available for use in the power block, it contributes to fulfilling the

startup energy requirement. The remaining startup energy at the end of the time step

(Estart,remain) is equal to the difference between the startup energy at the beginning of the time

step (Estart,remain0) and the total amount of startup energy that is “used” during the time step.

The startup energy used is equal to:

Estart,used = min W ̇ des

ηcycle,des

Δt,Estart,remain0



(3.39)

The remaining startup energy at the end of the time step is calculated by subtracting the used

startup energy, and the final remaining startup energy value is limited to be greater than or

equal to zero.

Estart,remain = max

Estart,remain0 −Estart,used,0

 (3.40)

The power cycle power output and the required mass flow rate both require adjustment during

startup. However, they are adjusted differently since the power cycle requires HTF mass flow

during startup even while it does not produce usable power. The adjustment method is shown

in Eq.’s[3.41 - 3.42], where the unadjusted values are denoted with the superscript “◦”.

W ̇ cycle = W ̇ ◦

cycle max

⎡

⎣min

Δtstart,remain

Δt ,1



,

Estart,used

W ̇ des

ηdes

Δt

⎤

⎦ (3.41)

m ̇ pb = m ̇ ◦

pb

⎛

⎝1−max

⎡

⎣min

Δtstart,remain

Δt ,1



− Estart,used

W ̇ des

ηdes

Δt

⎤

⎦,0

⎞

⎠ (3.42)

3.4.3 Standby operation

The amount of time needed to start up the power cycle from a cold state is an important

consideration in CSP systems. For certain systems, it may be advantageous to use energy from

thermal storage to maintain the power cycle in a warmed and ready state rather than starting up

from a cold shutdown state. Unlike cold startup, System Advisor models a restart from standby

mode as instantaneous, and reinitializing power cycle operation doesn’t require waiting for the

startup time or energy fraction. System Advisor models this situation by providing a “standby”

option for the power cycle. This mode is defined by a maximum standby time duration (ΔTsby)

and a standby fraction (fsby). The time duration limits the number of consecutive hours that the

system can operate in standby mode. The standby fraction defines the thermal energy required

to maintain the power cycle in a ready state, where the fraction is the ratio of standby thermal

78
demand to full-load design point demand.

The mass flow rate of HTF needed to maintain standby operation is calculated based on the

temperature of the HTF at the power cycle inlet. The HTF is assumed to be returned to the heat

source at the design HTF outlet temperature. Thus, the thermal load required is:

q ̇sby = W ̇ des

ηcycle,des

fsby (3.43)

and the associated required mass flow rate is:

m ̇ sby = q ̇sby

cht f (Tht f,hot −Tht f,cold,des) (3.44)

Power output, cycle efficiency, cooling water use, and power cycle parasitics are set to zero in

this mode.

3.4.4 Warm restart conditions

When encountering significant weather transients, the trough plant may need to temporarily

stop electricity production and take the turbine and generator offline. Once the solar resource

rebounds, electricity production may not resume immediately as the turbine inlet steam

conditions and generator synchronization must be reestablished. This delay is modeled by

System Advisor by enforcing a minimum restart time when the turbine trips. The user can

specify this value on the Power Cycle page in the “Minimum power block restart time” field.

When the turbine trips in the model, the control code enforces this minimum restart time

before resuming electricity generation. Thermal energy captured during this time interval is

used to either reheat the solar field or charge thermal storage (if available).

Section 3.4 Summary

• The power cycle has three operating modes: normal operation, standby, and shutdown.

• Normal operation can use mass flow rate or power output as an independent variable.

• Standby mode allows quick startup but consumes thermal energy; shutdown mode requires

full startup time and energy.

79
4 Thermal storage and plant control

Utility-scale CSP systems are unique in that they are both more complex and more flexible

than other renewable energy systems in how energy is produced, utilized, and stored.

State-of-the-art trough plants include not just a solar field and power cycle but also a thermal

storage subsystem that can be designed to dispatch anywhere from a few minutes to a day’s

worth of thermal energy. Some systems also include a fossil-fuel heater that can provide

supplemental energy when solar-derived energy is unavailable. Decisions about when and in

what proportions each of these subsystems should be utilized can have a significant impact on

project profitability.

Figure 27: The trough power plant subsystems include the solar field, piping, thermal storage,

auxiliary heating, and the power cycle.

A dispatch schedule is just one aspect of plant control. Practical limits on system performance

are also required to maintain component health. This includes considerations for high wind

speeds, peak power production limitations, minimum fluid levels in the storage tanks, and

minimum fluid temperatures in storage. When all of these contingencies are amalgamated with

proper dispatch control, the need for a detailed and inclusive control strategy is apparent.

Section 4 describes the trough plant control strategy and the subsystems that are directly

modeled within the controller code.

Though the balance-of-plant subsystems are generally autonomous like the solar field or power

cycle, their relative simplicity lend them to be easily implemented in the control code. This is

done as a practical consideration; consequently, the thermal storage tanks, field-to-storage heat

exchanger, and auxiliary fossil heater are included in the balance-of-plant control volume.

4.1 Thermal storage

A utility-scale electric plant that is subject to intermittent outages during weather transients

(like centralized PV and wind turbine farms) can create operation challenges for grids with

high penetrations of renewable energy [18]. CSP is unique among renewable technologies in

80
its ability to cost-effectively store energy in a thermal energy storage (TES) system. Storage

allows power production during temporary weather transients, shifting the operating hours to

match peak demand, generally increasing the capacity factor of the plant, or supplying

low-level heat to plant processes that require it (like maintaining the power cycle in a standby

mode).

System Advisor models thermal storage for a two-tank system; that is, two tanks each are

capable of holding the entire HTF volume for thermal storage. One tank is dedicated to storing

the hot fluid while the other holds the exhausted cold fluid. The tanks are assumed to be fully

thermally mixed so that no stratification occurs17.

4.1.1 TES Sizing

Thermal storage capacity is conventionally expressed in System Advisor in terms of equivalent

full-load hours of TES. The magnitude of this value (“Full load hours of TES” on the Thermal

Storage page) indicates the number of hours that thermal storage can supply energy to operate

the power cycle at its full design point output. The realized number of storage hours is usually

somewhat less than the number specified, since thermal losses and unavailable storage volume

(see below) are not included in the sizing calculation. The total TES thermal capacity is equal

to the design-point power cycle thermal requirement times the total number of desired storage

hours (Δttes). The design-point cycle efficiency ηcycle,des is constant and equal to the value

specified on the Power Cycle page.

Etes = W ̇ des Δttes

ηcycle,des

(4.1)

System Advisor calculates the actual volume of HTF required to match this energy content

using average material property values for the hot and cold tank design temperatures.

Vtes = Etes

ρtes,ave ctes,ave fhx (Ts f,out −Ts f,in) (4.2)

The design temperature difference is equal to the hot solar field outlet temperature (Ts f,out)

minus the cold inlet temperature (Ts f,in) times the heat exchanger derate factor (fhx). This

factor is equal to the ratio of the realized temperature difference on the storage side of the heat

exchanger to the solar field temperature difference. Alternatively, this ratio can be expressed in

terms of the user-specified heat exchanger hot and cold side approach temperatures that are

specified on the System Advisor Thermal Storage page (Thx,top and Thx,bot, respectively). See

Figure 30 on page 86 for a graphical illustration of these temperatures.

fhx = 1− ΔThx,top +ΔThx,bot

Ts f,out −Ts f,in

(4.3)

17Thermocline storage is another TES option that makes use of thermally stratified HTF in a single

tank. This technology model is not currently included in System Advisor.

81
Tank geometry impacts not only the total area exposed to ambient air through which heat

transfer can occur, but also determines the amount of unusable volume at the bottom of the

tank (the “heel height”). System Advisor allows the user to specify a number of hot-cold tank

pairs using the Parallel tank pairs input on the Thermal Storage page. System Advisor

calculates the tank diameter required to contain the total storage volume, assuming that all

storage tanks have the same geometry and meet both the user-specified requirements from the

Thermal Storage page.

Vtes,1tank = Vtes

npairs

(4.4)

D1tank = 2



Vtes,1tank

π H1tank

(4.5)

4.1.2 Model formulation

Both the hot and cold tanks use the same tank model18 to simulate their behavior, although the

inputs and outputs for each are managed separately. The basic premise for the storage tank is

that the tank HTF volume varies according to difference between the inlet and outlet mass flow

rate. Figure 28 shows a schematic of the tank model.

Figure 28: A schematic of the variable-volume tank model. Fluid level varies with differences in

the inlet and outlet mass flow rate. The outlet temperature is equal to the average of the fully

mixed tank.

Several tank calculations are functions of time and inlet/outlet conditions. Each call to the tank

18The tank model is based on a similar methodology to the variable-volume tank model (Type 39) in

the standard TRNSYS library.

82
model requires information about not only what is happening in the system during the current

time step, but at the end of the previous time step. The controller keeps track of the state of the

hot and cold tank from time step to time step and passes this information to the tank

(specifically, the mass and temperature of the HTF in each tank are tracked). The mass of HTF

in the tank at the end of the current time step is equal to:

mfin = m0 +Δt (m ̇in −m ̇ out) (4.6)

Calculating the temperature at the end of the current time step is not as elementary as the mass

balance in Eq.[4.6]. Since the HTF temperature varies over the time step, thermal losses from

the tank also vary. When the tank energy balance is expressed analytically, the resulting

differential equation contains several variables that are functions of time, and this complicates

the solution for final HTF temperature. The tank can be conceptualized as the simplified

control volume in Figure 29.

Figure 29: Conceptual representation of the tank control volume.

Equation [4.7] expresses the energy balance on the tank control volume according to the first

law of thermodynamics.

∂(u(t)m(t))

∂t = −q ̇hl,tank(t) +m ̇in h(Tin)−m ̇ out h(T(t)) (4.7)

The internal energy u(t), mass m(t), and heat loss q ̇hl,tank(t) are all functions of time. The

internal energy is equal to the specific heat of the HTF in the tank times the HTF temperature

at time t with respect to some arbitrary reference temperature Tre f . The mass in the tank can be

calculated by adding the initial mass to the time multiplied by the difference in inlet and outlet

mass flow rates. Finally, the heat loss is equal to the heat transfer coefficient (UA) times the

temperature difference between the HTF at time t and the ambient temperature for the given

time step. Like internal energy, enthalpy is also a function of temperature for an

incompressible fluid. Since the specific heat remains approximately constant under nearly all

practical conditions over the duration of a single time step, the enthalpy equation can be

83
expressed as shown in Eq.[4.11].

u(t) = cht f (T(t)−Tre f) (4.8)

m(t) = m0 +t (m ̇in −m ̇ out) (4.9)

q ̇hl,tank(t) = UA (T(t)−Tamb) (4.10)

h(t) = cht f (T(t)−Tre f) (4.11)

These equations are substituted into Eq.[4.7], and the resulting ordinary differential equation

(ODE) expresses the tank temperature as a function of time.

cht f (m ̇in −m ̇ out) (−Tre f +T(t)) +cht f (m0 + (m ̇in −m ̇ out) t)

∂T

∂t

= cht f m ̇in (Tin −Tre f)−UA (−Tamb +T(t))−cht f m ̇ out(−Tre f +T(t)) (4.12)

This linear ODE can be solved for temperature. The resulting equation is a function of time,

and the reference temperature Tre f cancels out in calculation.

T(t) = A2 +

cht f(m0 +A1 t)





−A0

A1



·C (4.13)

Where:

A0 = m ̇in +

UA

cht f

A1 = m ̇in −m ̇ out

A2 = m ̇in Tin +Tamb

UA

cht f

A0

The constant C is solved by enforcing the initial boundary condition, where the tank

temperature T(0) = T0.

T(t) = A2 +

1

A0



(cht f m0)



−A0

A1

 

cht f(m0 +A1 t)





−A0

A1



A3



(4.14)

Where:

A3 = cht f m ̇in (T0 −Tin) +UA (T0 −Tamb)

The final HTF temperature at the end of the time step is calculated by replacing t with the time

step value Δt. We are also interested in the average temperature over the whole time step since

this information is used as the temperature output from the model. The expression for average

temperature as a function of time is derived by integrating Eq.[4.14] with respect to t, then

dividing by the duration of the time step. The constant of integration is evaluated by enforcing

84
the boundary condition where Tave(0) = T0.

Tave(t) = T0 +A4 +

1

A0



t (cht f m ̇in Tin +Tamb UA) +Eq.[4.14]

 (4.15)

Where:

A4 = cht f m0 (cht f m ̇in (T0 −Tin)+(T0 −Tamb) UA)

(cht f m ̇in +UA)(cht f m ̇ out +UA)

Total heat loss is calculated implicitly in Eq.s[4.13 - 4.15] as a function of time, but is

expressed more directly as a function of the average HTF temperature. This heat loss

coefficient (UA) captures both the area-specific heat transfer coefficient specified on the

Thermal Storage page and the total surface area of all the tanks.

q ̇tank,hl = UA·(Tave −Tamb) (4.16)

Where:

UA = γtank  Vtes

H · npairs

+π D1tank H



npairs

4.1.3 Storage freeze protection

The tank model includes the option for freeze protection. The minimum allowable fluid

temperature is equal to the “Tank heater set point” specified by the user on the Thermal

Storage page. During time steps where the fluid temperature falls below this value, heat is

added to the fluid until either the minimum temperature is maintained, or until the maximum

heating rate is achieved. Heat may be required either because of heat loss directly from the

volume in the tank, or because of below-temperature flow out of the tank. Heat addition is

calculated based on the volume and temperature of the HTF at the very end of the time step.

q ̇ht,vol = cht f V fin (Tset point −Tfin) (4.17)

The temperature of the outgoing flow must also be maintained at the minimum allowable

temperature.

q ̇ht, f low = cht f m ̇ out (Tset point −Tfin) (4.18)

Provided that the total tank heat trace heating rate q ̇ht,tot is less than the maximum value, the

average and final tank temperatures are recalculated.

Tfin = T0 +Δt q ̇ht,vol

cht f ρht f V fin

(4.19)

85
Section 4.1 Summary

• System Advisor models storage with a 2-tank system. This can be either direct or indirect

depending on the selection of HTF types.

• Storage sizing is computed based on equivalent full-load hours, and accounts for heat exchanger availability losses.

• Storage tank performance is modeled to include varying thermal losses throughout the time

step.

• Both storage tanks include the option for freeze protection heating. The heating load is

considered an electric parasitic.

4.2 Indirect storage heat exchanger

Most parabolic trough systems use oil as the heat transfer fluid in the solar field. However, the

cost of this fluid can be prohibitively high for the large volumes of the TES system. A less

expensive approach uses oil in the field and an alternative HTF like molten nitrate salt in the

storage system. So-called “indirect” storage requires an intermediate heat exchanger between

the oil in the solar field and the salt in storage. System Advisor models this heat exchanger

with the effectiveness-NTU method [17] for a counter-flow arrangement (shorthand ε−NTU).

System Advisor automatically determines the size of the heat exchanger with information

provided by the user and with calculated design-point values when the user-specified storage

fluid is different from the solar field HTF. A necessary consequence of heat transfer is the

presence of a temperature drop on both sides of the heat exchanger between the heat source

fluid and the heat sink fluid. Figure 30 shows how the hot and cold side approach temperatures

relate to the physical system.

Figure 30: The indirect storage heat exchanger is defined by the hot and cold side approach

temperatures. A sketch of the physical system arrangement (left) corresponds to a plot of the

temperatures in the heat exchanger as a function of position (right).

86
4.2.1 Heat exchanger sizing

The temperature drops across the heat exchanger and its total duty determine the heat

exchanger’s heat transfer size (UA). Heat exchanger duty corresponds to the difference

between the power cycle’s thermal energy absorption rating (W ̇ des/ηdes) and the thermal energy

produced by the solar field at design. The solar multiple (SM) represents the ratio of solar field

to power cycle thermal capacities at design, thus:

q ̇duty = q ̇s f,des −Q ̇des = (SM −1) Q ̇ des (4.20)

The design calculations assume a charging cycle where the solar-field-side HTF is the hot

stream and the storage-side HTF is the cold stream. As indicated in Figure 30, the approach

temperature differences are equal to the difference between the hot and cold streams at

common sides of the heat exchanger. In other words, the “top” approach temperature is

defined as the temperature difference between the hot stream inlet and the cold stream outlet,

and the “bottom” approach is the hot stream outlet minus the cold stream inlet.

ΔThx,top = Thot,in −Tcold,out (4.21)

ΔThx,bot = Thot,out −Tcold,in (4.22)

The mass flow rates of the hot and cold streams at design are calculated from the known

temperature drop/rise and the heat transfer duty. HTF specific heat is evaluated at the average

stream temperature.

m ̇ hx,hot = q ̇duty

chot (Thot,in −Thot,out) (4.23)

m ̇ hx,cold = q ̇duty

ccold (Tcold,out −Tcold,in) (4.24)

The ε−NTU method makes use of the capacitance rates (defined as the mass flow times the

specific heat capacity) for the hot and cold streams. The relationship of interest is between the

maximum and minimum capacitance rates, since the minimum rate sets the ceiling on the ideal

heat exchanger performance as shown in Eq.[4.25]. The capacitance ratio is then calculated in

Eq[4.26].

q ̇hx,max = C ̇

min (Thot,in −Tcold,in) (4.25)

CR = C ̇

min

C ̇

max

(4.26)

Heat exchanger effectiveness (ε) quantifies the ratio of actual heat transfer to the maximum

possible at design. The non-dimensional size of the heat exchanger is given in terms of the

effectiveness value using semi-empirical relationships specific to each heat exchanger

configuration. Eq.[4.27] gives the relationship for the counter-flow heat exchanger that is

87
modeled in System Advisor.

NTU =

log

1−ε·CR

1−ε



1−CR

(4.27)

Eq.[4.27] requires practical limits for the argument of the log function and for the

denominator. If the capacitance ratio happens to be equal to 1.0, NTU is instead calculated

with Eq.[4.28], according to [17].

NTU = ε

1−ε (4.28)

In the situation that the argument of the log function is less than 1, the simulation returns with

an error. The user must correct this situation by adjusting the approach temperatures until a

suitable heat exchanger size is achieved.

The dimensional size of the heat exchanger (UA) is equal to the product of the

non-dimensional size and the minimum capacitance rate. This value is used during hourly

simulation calculations to determine heat exchanger performance.

UA = NTU ·C ̇

min (4.29)

4.2.2 Heat exchanger performance model

The heat exchanger can operate in one of two modes depending on whether storage is charging

or discharging. Both modes are designed to calculate the storage-side mass flow rate required

given a mass flow rate on the solar field side. Both modes also maintain the specified approach

temperatures on both sides of the heat exchanger.

The solar field produces a particular amount of thermal energy during operation according to

the performance limitations of the field components. The power cycle has priority in

consuming the amount of thermal energy that it requires to match the desired power cycle

output, and the balance goes to storage. Consequently, the process of charging storage in an

indirect system entails passing the balance of the unused HTF through the heat exchanger and

simultaneously pumping the cold storage HTF through the heat exchanger to extract thermal

energy from the solar field HTF. The remaining unconstrained variable after calculating the

approach temperatures, solar field-side mass flow rate, and the UA is the storage-side mass

flow rate.

Storage charge mode

During storage charging, heat transfer across the heat exchanger is calculated based on the

known inlet temperature and approach temperatures. The solar field provides the thermal

energy source - or the hot stream - and storage provides the thermal sink stream. The stream

inlet/outlet temperatures reflect this arrangement.

Thot,out = Tcold,in +ΔThx,cold (4.30)

Tcold,out = Thot,in −ΔThx,hot (4.31)

88
The heat exchanger duty is calculated:

q ̇duty = m ̇ hx,hot chot (Thot,in −Thot,out) (4.32)

The storage charge mass flow rate (from cold tank to hot tank) is then determined to match the

required temperature difference and the heat transfer duty.

m ̇ hx,cold = q ̇duty

ccold (Tcold,out −Tcold,in) (4.33)

Specifying the approach temperatures, mass flow rate, and UA, may lead to a situation where

more heat transfer is calculated than is physically possible (i.e. ε > 1). System Advisor checks

for this condition and returns a warning so that the set-point approach temperatures can be

adjusted for future simulations. The effectiveness is calculated during performance for a

counter-flow heat exchanger [17].

ε = 1−e−NTU (1−CR)

1−CR (−NTU (1−CR)) (4.34)

If the capacitance ratio CR happens to equal 1.0, the appropriate equation is instead:

ε = NTU

1+NTU (4.35)

Storage discharge mode

During storage discharge, storage becomes the thermal resource and the solar field (and

eventually the power cycle) side is the thermal sink. However, storage does not provide an

arbitrary amount of thermal energy to the power cycle; rather, the power cycle requires a

particular thermal power to produce the required electrical output. Even though the thermal

roles of the two streams have been reversed, in this case the controlling role of the power cycle

dictates that the heat exchanger performance be restricted by the cold flow.

Whereas the cold mass flow rate was the dependent variable calculated in Eq.[4.33], here the

hot mass flow rate is calculated.

m ̇ hot = q ̇duty

chot (Thot,in −Thot,out) (4.36)

The temperature relationships provided in Eq.’s[4.30] and [4.31] still apply, except that now

the hot stream refers to storage and the cold to the solar field side. Therefore, the achieved

HTF temperature on the solar field side during discharge is less than the original solar field

outlet charging temperature by 2×ΔThx,hot.

89
Section 4.2 Summary

• System Advisor models the heat exchanger between the solar field and storage in indirect

systems using the ε−NTU method.

• Heat exchanger sizing occurs before the initial time step and uses design values provided

by the user. The duty corresponds to the solar multiple.

• The heat exchanger model assumes a counter-flow arrangement.

• Heat exchanger performance is modeled in either a storage charge or storage discharge

mode. Both situations use the collector-side conditions to control the flow rate on the storage side.

4.3 Auxiliary heater

A fossil-fired auxiliary heater is included in some systems to supply thermal energy during

times of no solar resource or when storage cannot fully meet the required load. Solar advisor

models a simple fossil-fuel burning auxiliary heater that generates heat for use in power

production. The auxiliary heater is controlled by the dispatch controls on the Thermal Storage

page. It is automatically limited to a maximum heating rate equal to the power cycle design

thermal input. Control is discussed in more detail in Section 4.4.

When the aux heater is in operation, energy is added to the HTF flowing through the heater

until the HTF reaches the setpoint temperature or until the heater reaches the maximum heat

production rate. The setpoint temperature is a user-specified input on the Thermal Storage

page. The achieved outlet temperature is calculated as the minimum of either the setpoint

temperature or the temperature achieved by using the maximum heating rate.

Taux,out = min

Taux,set,Tin + q ̇aux,max

m ̇ aux,ht f cht f 

(4.37)

The fluid’s specific heat is evaluated at the HTF temperature at the heater inlet. The total

thermal energy transfer to the fluid is proportional to the achieved temperature rise in the HTF.

q ̇aux = m ̇ aux,ht f cht f (Taux,out −Tin) (4.38)

Fuel usage is an important metric in the performance of the auxiliary heater. The energy

content of the fuel usage is measured in the units MMBTU. Assuming thermal losses from the

heater are negligible, the fuel usage relates to the total energy usage by the heating value. Solar

advisor uses the lower heating value (LHV) efficiency to estimate fuel energy content.

Ef uel = q ̇aux

ηLHV

Δt ×



3.412×10−6 MMBTU

W −hr 

(4.39)

90
4.4 Plant control

The plant controller is the hub of calculation activity in the parabolic trough model. The

controller ensures that the system operates to meet user-specified requirements, thermal energy

requirements of the power cycle, and the energy resource available from the solar field, thermal

storage, and auxiliary heater. The controller also determines how and when the solar field is

used. For example, if TES is full and the power cycle is running at full capacity, the controller

must detect the situation and adjust the solar field by forcing a reduction in total thermal output

to avoid overheating the HTF, which would have disastrous consequences in a real system.

4.4.1 Controller background and approach

Though several metrics can be used in controlling plant operation, System Advisor’s primary

metric for controlling the trough system is the HTF mass flow rate through the system. Mass

flow must always balance regardless of the temperature or energy considerations. Thus, the

mass flow rate is the most important control metric employed in this control strategy. HTF

temperature is also important; temperatures outside of the acceptable range for the equipment

or for the fluid itself can damage equipment and have other negative consequences. Mass flow

rates are controlled to avoid undesirable temperatures in the system. The last control metric is

thermal energy flow. This is a secondary quantity based on mass flow, temperature, and

specific heat that is useful in allocating mass flow appropriately. The thermal energy flow and

HTF temperature are used together to distribute mass flow to ensure that the system

requirements and power cycle demand are satisfied.

The general control strategy approach satisfies power cycle demand by making use of

resources in a prioritized order. A series of logical statements are used (as discussed below) to

determine whether the power cycle demand can be met with only the solar field, with the solar

field and with TES, or with the solar field, TES, and the auxiliary heater. Resources to meet

power cycle demand are always used in order: (1) solar field, (2) thermal storage, (3) auxiliary

heater. The final combination of contributing resources can be any permutation of these three

options. However, when the allocation decisions are made, the availability of each resource is

taken into account, including user-enforced limitations.

Two examples may help illustrate the control approach:

First, consider a plant with thermal storage but without any backup auxiliary

fossil heater. Consistent solar resource during the morning hours has allowed a

substantial charge to build up in thermal storage. During these hours, power

cycle demand is fully met by energy produced in the solar field. The controller

allocates enough flow from the field to the power cycle to meet the demand, and

the leftover flow is allocated to thermal storage. Suddenly, a cloud passes over

the field, reducing its output to 50% of the power cycle requirement. The

controller first allocates all of the solar field flow to the power cycle, but finds

that this is insufficient to fully meet demand. It then calculates the available

thermal energy in storage, and finds that enough is available to supplement the

solar field flow and reach full power output.

Second, consider a plant with storage and an auxiliary fossil heater.

91
Re-examining the scenario in the first example, we will still find that thermal

storage fills the demand when the cloud passes over the field. But this time, the

cloud cover persists and storage is slowly depleted as the solar field can only

produce part of the required thermal energy. During the time step when storage

is about to be fully depleted, the power cycle demand cannot be fully met by the

combination of the solar field and thermal storage. The auxiliary heater is

called on during this time step to supplement the load that isn’t met by storage

or the solar field.

4.4.2 Available mass flow

Making best use of the solar field, storage, and auxiliary heat requires knowledge of the

thermal energy available from each in any given time step. For thermal storage, we need to

know both the quantity of flow available for discharging and the available unfilled volume for

charging the tank(s). Since mass flow is the primary control metric, we first need to determine

the available mass flow from the three sources if they were to each fully operate during the

current time step. The available solar field mass flow is simply the value that is provided to the

controller by calculations in the solar field. Recall from Section 2.2 that the solar field controls

mass flow to meet the outlet temperature setpoint.

Thermal storage is not so simple. Depending on the mode of operation, storage may be

charging or discharging, so the controller needs to know the maximum charge and discharge

mass flow rates (representing the rates if the tanks were to completely deplete their contents

during the time step). The availability of storage is also limited by the minimum tank volume

level and by the user-specified dispatch fractions referred to as time-of-use (TOU) controls on

the Thermal Storage page.

The storage TOU fractions are applied on a volume basis. However, the HTF volume does not

remain constant from time step to time step even when no flow enters or leaves storage

because thermal losses change the specific volume of the fluid by changing the temperature.

The controller keeps track of the HTF in storage by passing the HTF mass (rather than

volume) from time step to time step.

Disregarding the TOU limitations, the available volumes in the hot and cold tanks at the

beginning of the time step are then:

Vhot = Vhot,0 −Vtot

Hmin

Htank

(4.40)

Vcold = Vcold,0 −Vtot

Hmin

Htank

(4.41)

The HTF volume from the previous time step V(hot/cold),0 is equal to the stored mass value

divided by the density of the storage HTF evaluated at the hot or cold tank temperature,

respectively. The discharge mass flow rate depends on the available HTF in the hot tank after

the TOU restrictions are enforced. The TOU period is determined by the user-specified control

array on the Thermal Storage page and is assigned according to hour of the day, day of the

92
week (weekday or weekend), month of the year, and the presence of the solar resource. The

TOU fraction (ftou,tes) is applied only to the “active volume” of HTF in the tank - that is, the

volume that is between the minimum tank fluid level and the corresponding maximum level.

This is represented in Eq.[4.42].

Vactive =



1−2

Hmin

Htank 

·Vtot (4.42)

The mass available for discharge is calculated as follows.

m ̇ A,dis = 

Vhot −Vactive ftou,tes ρht f,hot

Δt (4.43)

Unlike discharging TES, charging TES is not subject to the dispatch fractions; only the

minimum fluid level dictates the amount of charging that is possible. Therefore, the available

storage charging mass flow rate is:

m ̇ A,chg = Vcold

ρht f,cold

Δt (4.44)

Equations [4.43] and [4.44] describe the mass flow rate that is available on the storage side of

the system. For direct storage systems, the flow available to the power cycle and the flow

available from storage are equivalent. But for indirect systems, the available storage flow rates

do not provide enough information on the flow that is actually achieved across the heat

exchanger. Thus, initial calls to the heat exchanger model are required to establish the actual

available storage mass flow rate.

The flow rate available from the auxiliary heater depends on the capacity of the heater and on

the inlet temperature of the HTF. The flow rate is also scaled to the fossil fill fraction (ftou,aux)

controls on the Thermal Storage page. This is expressed as:

m ̇ A,aux = Q ̇ des

cht f,aux (Taux,set −Tpb,out)

· ftou,aux (4.45)

The total available mass flow rate is the sum of all three possible sources.

m ̇ A,tot = m ̇ A,s f +m ̇ A,dis +m ̇ A,aux (4.46)

4.4.3 Converting to thermal energy

Mass flow rate control is the proximate goal, but the ultimate purpose for controlling the plant

is to meet the thermal load required by the power cycle. Mass flow rates only provide part of

the equation in determining whether the stream entering the power cycle has enough thermal

energy. This means that the mass flows and temperatures must be converted into thermal

93
energy in order to be useful in control decisions.

The first step in this conversion involves determining the specific heat values of the HTF in the

different streams so that the heat content can be accurately assessed. Specific heat varies with

temperature for most HTF’s, so System Advisor’s ability to accurately characterize this value

is important for model convergence. Eq. [4.47] provides the temperature values on which the

specific heat calculations are based for the solar field, TES discharge, TES charge, and

auxiliary heater streams.

cht f,s f = cp

HTF,

Ts f,out +Ts f,in

2



cht f,dis = cp

HTF,

Ttes,hot +Tpb,out

2



(4.47)

cht f,chg = cp

HTF,

Ts f,out +Ttes,cold

2



cht f,aux = cp

HTF,

Taux,set +Tpb,out

2



The value Ttes,hot is the temperature of the collector-side HTF that has been heated and is

leaving storage. This temperature is equal to either the heat exchanger outlet temperature or

the hot tank temperature, depending on the system configuration. Likewise, Ttes,cold represents

the temperature of the collector-side HTF that has been exhausted from charging storage and is

returning to the solar field. This temperature can either be equal to the heat exchanger outlet

temperature for indirect systems or the cold TES tank temperature for direct systems.

Using the specific heat values, the mass flow rates, and the temperature drops for each stream,

the thermal energies are calculated.

q ̇A,s f = m ̇ A,s f cht f,s f (Ts f,out −Ts f,in)

q ̇A,dis = m ̇ A,dis cht f,dis (Ttes,hot −Tpb,out) (4.48)

q ̇A,chg = m ̇ A,chg cht f,chg (Ts f,out −Ttes,cold)

q ̇A,aux = m ̇ A,aux cht f,aux (Taux,set −Tpb,out)

Any of these energy flows can be used to satisfy the power cycle demand that is calculated

according to the design thermal energy load (see Eq.[3.6]) and the turbine dispatch fraction

control (ftou,pb) entered on the Thermal Storage page.

Q ̇dem = Q ̇ des ftou,pb (4.49)

94
4.4.4 Control logic

The main controller logic tree19 makes use of the thermal energy calculations to select the

operating mode. The following discussion moves step by step through each control decision,

eventually considering the options for each statement. The logic first determines whether the

field can fully supply the power cycle demand.

q ̇A,s f < Q ̇ dem (4.50)

If the statement in Eq.[4.50] is true, then the solar field cannot supply all of the required

thermal energy and the other thermal sources must be considered. The energy from thermal

storage discharge and the auxiliary heater are added to the total energy subsequently until the

total equals the demanded amount. First, for thermal storage, we check if the available energy

meets demand:

q ̇A,s f +q ̇A,dis < Q ̇ dem (4.51)

If the requirement is not met, then we know that the actual mass flow rates for the solar field

and TES discharge are equal to their maximum available flow rates. We can also infer that

because the flow of thermal energy from the solar field is not sufficient to meet demand, the

realized storage charging mass flow rate is zero.

m ̇ s f = m ̇ A,s f (4.52)

m ̇ dis = m ̇ A,dis (4.53)

m ̇ chg = 0 (4.54)

Continuing in the situation where solar field mass flow and thermal storage do not meet the

required load, the logic next determines whether the auxiliary heater can provide the remaining

energy.

q ̇A,s f +q ̇A,dis +q ̇A,aux < Q ̇ dem (4.55)

The auxiliary heater has two modes of operation that can be selected by the user on the

Thermal Storage page under the “Fossil dispatch mode” drop-down. The first mode -

minimum backup level - controls the auxiliary heater to maintain thermal production at the

“Fossil fill fraction” specified in the dispatch control section. Fossil backup will only be

provided when the solar field and thermal storage energy contributions fall below the Fossil fill

fraction level (with respect to the design-point operation). The second mode - supplemental

operation - forces the auxiliary heater to supply the difference in thermal energy between the

design-point requirement and the solar field and TES contributions. When the combined solar

field and TES contributions fall below the design-point requirement, the auxiliary heater

produces energy up to the design point requirement while maintaining a total auxiliary

contribution less than the Fossil fill fraction provided by the user.

19The reader may find the control process flow diagram presented in Figure 31 on page 103 useful

while reading this section.

95
If the auxiliary heater, TES, and the solar field cannot meet the demand load, then the auxiliary

mass flow is also set equal to the maximum available flow. Now that all possible resources are

accounted for, the total mass flow rate, heat input and power cycle inlet temperature can be

calculated.

m ̇ aux = m ̇ A,aux (4.56)

q ̇tot = q ̇s f +q ̇dis +q ̇aux

m ̇tot = m ̇ s f +m ̇ dis +m ̇ aux (4.57)

Tpb,in = m ̇s f Ts f,out +m ̇ dis Ttes,hot +m ̇ aux Taux,set

m ̇tot

Moving back up the logic hierarchy, we must consider the alternatives to Eq.’s[4.51] and [4.55]

being true. First, if Eq.[4.55] is false and the auxiliary heater is able to provide sufficient

supplemental energy, then the auxiliary heater runs at part load according to the required

balance of energy. In minimum backup level mode, Eq.’s[4.56] and [4.57] are instead:

q ̇aux = Q ̇dem · ftou,aux −(q ̇s f +q ̇dis)

m ̇ aux = q ̇aux

cht f,aux (Taux,set −Tpb,out) (4.58)

q ̇tot = Q ̇dem (4.59)

In supplemental operation mode, the auxiliary heat equation becomes:

q ̇aux = min

Q ̇ dem −(q ̇s f +q ̇dis),Q ̇ dem · ftou,aux (4.60)

Moving up again, we consider the alternative to Eq.[4.51] where thermal storage and the solar

field are able to provide enough energy to the power cycle. Now the auxiliary heater mass flow

and thermal energy contributions aren’t needed, but the TES discharge rates require attention.

q ̇tot = Q ̇ dem (4.61)

q ̇tes,dis = Q ̇ dem −q ̇A,s f

m ̇tes,dis = q ̇tes,dis

cht f,dis (Ttes,hot −Tpb,out) (4.62)

Equations [4.51] through [4.62] all apply to the situation where the solar field is unable to

produce the required thermal energy for the power cycle. Two operating modes apply to this

situation: mode 1 handles the case where the total thermal energy produced by all resources

falls below the cycle cutoff fraction fcutoff (i.e. the minimum allowable thermal load handled

by the power cycle), and mode 2 accommodates normal part-load operation. The mode is

96
selected according to Eq.[4.63], where “true” corresponds to mode 1.

q ̇tot < Q ̇ des fcutoff (4.63)

The final step back up the hierarchy reconsiders Eq.[4.50]. If this statement is false, then the

solar field is able to supply all of the required power cycle load, and the logic must only decide

whether the balance of energy from the solar field not going to the power cycle can be placed

in thermal storage. This is expressed in Eq.[4.64]. If the statement is true, then operating mode

3 is selected; otherwise, mode 4 is used.

q ̇A,s f −Q ̇dem > q ̇A,chg (4.64)

4.4.5 Operating modes

The four operating modes can be generally described as follows:

1. Operation below the minimum turbine cutout fraction

2. Part-load operation above the minimum turbine cutout fraction and below the design

point power

3. Over-design operation where more energy is produced than can be used by the power

cycle and/or storage

4. Over-design operation where all of the energy can be used in the power cycle and/or

storage

The operating modes determine the defocusing methodology. Since defocusing is a highly

implicit process affecting solar field energy collection and production, thermal losses, and

other relationships, System Advisor employs an iterative defocusing process. Each operation

mode defocuses in largely the same way, but each has nuances that are tuned to the operating

conditions typically encountered within the mode. At the beginning of each time step (first

iteration), the field defocusing value fdef is set to 1.0, and subsequent iterations modify the

initial value as conditions warrant. The defocusing equations do not always require that fdef

remain in the physical range between 0 and 1; however, limitations in the solar field

calculations ensure that the defocusing fraction does not have deleterious effect on system

performance calculations.

The iterative process for field defocusing can cause problems in convergence when field

conditions during the current iteration differ significantly from conditions in the previous time

step. To aid convergence, the defocusing calculations are scaled by a “relaxation parameter”

nc. The value of this coefficient was selected to be a function of the time step duration based

on sensitivity analyses comparing total code run-time and the number of non-convergence

instances. For a time step in seconds, the relaxation parameter is defined as:

nc = 0.9− 3600[s]−Δt

7200[s] (4.65)

97
Occasionally, the model conditions approach a border condition between operating modes and

subsequent iterations result in constant switching between operating modes. The difference in

calculated performance between operating modes is often very slight when performance falls

on mode borders, so System Advisor enforces “sticky mode” operation. After 8 iterations

within the same time step, the controller sticks with the operation mode selected on the 8th

iteration.

Mode 1

The first operation mode handles all situations where the total energy produced by the solar

field, TES, and auxiliary heater falls below the power cycle cutoff fraction. This mode is most

typically used in conditions where the auxiliary heater does not provide energy, where thermal

storage is non-existent or exhausted, and where the solar field is producing a small amount of

charged HTF. Within Mode 1, the power cycle can remain in one of two states: (1) cold

shutdown, or (2) hot standby.

Hot standby mode allows the power cycle to remain in a “startup ready” state such that the

cycle can resume producing power without applying a startup penalty once the thermal

resource becomes available. However, standby operation requires a supply of thermal energy

from either the solar field or from thermal storage. The control algorithm attempts to operate

the power cycle in standby mode by default, but the following criteria must all be met for

standby operation:

• The solar elevation angle is greater than zero

• The total time in standby is less than or equal to the limit specified on the Power Cycle

page

• The sum of thermal energy available from thermal storage and the solar field meets or

exceeds the standby thermal energy requirement, shown in Eq. [4.66]

q ̇sby = fsby Q ̇des (4.66)

• The power cycle was operating in one of Modes 2-4 in the previous time step, or the

cycle was operating in Mode 1 - Standby during the previous time step

In standby mode, the mass flow rate from thermal storage (if available) is equal to the balance

of the flow from the solar field and the requirement for standby operation. The following

relationships apply:

m ̇ dis = max q ̇sby −q ̇s f

cht f,dis (Ttes,hot −Tpb,out)

,0



(4.67)

Tpb,in = Ts f,out m ̇ s f +Ttes,hot m ̇ dis

m ̇ s f +m ̇ dis

(4.68)

Depending on the control parameters specified by the user, it’s also possible that the solar field

may produce more thermal energy than the power block requires for standby operation, but

98
less energy than is required for power generation. In this specific case, the solar field may need

to partially defocus. Eq.’s [4.70] and [4.72] below describe the defocusing algorithm.

Cold shutdown is the second option within Mode 1. This option is enacted any time the

standby criteria are not met. Since the power cycle cannot produce power in this mode, energy

from the solar field must either be diverted to charge storage if it’s available, otherwise the

solar field must be defocused.

m ̇ dis = 0

m ̇ pb = 0

Tpb,in = Tpb,out

m ̇ chg = minm ̇ A,chg,m ̇ s f  (4.69)

For systems with thermal storage, partial defocusing may be required to ensure that TES is not

overcharged. The defocus variable fdef indicates the fraction of the solar field that is in

operation. This variable is used by the controller to adjust solar field performance when

needed. Since the defocusing algorithm is iterative, the defocusing value from the previous

time step is modified based on the current system conditions. The defocus calculation for

Mode 1 for systems with thermal storage is then:

fdef = fdef,0 ·

m ̇ chg

m ̇ s f nc

(4.70)

For systems without storage, the field is effectively completely defocused by setting the solar

field mass flow rate m ̇ s f and auxiliary heater power to zero. While operating in Mode 1, the

power block is turned off. Any restrictions on turbine restart time can be satisfied by

maintaining Mode 1 operation for a duration of at least the minimum restart time.

Mode 2

Mode 2 applies when the thermal resource is less than the design-point thermal load but is

greater than the turbine cutoff fraction. In this mode, the field is fully focused and the turbine

operates in part load mass flow-basis mode. The power cycle mass flow rate and inlet

temperature are equal to the values calculated in Eq.[4.57].

On occasion, the model enforces sticky-mode operation in Mode 2, and subsequent iterations

determine that the solar field must be defocused. Defocusing calculations are provided within

Mode 2 for this reason, and one of two separate equations is used depending on whether the

user has selected Partial Defocusing as an option on the Solar Field page. If partial defocusing

is allowed, the field will defocus as follows:

fdef = min

fdef,0 ·

q ̇A,chg +Q ̇ dem

q ̇A,s f nc

,1.0



(4.71)

99
For whole-SCA defocusing:

fdef = max

fdef,0 ·Nsca q ̇A,chg +Q ̇ dem

q ̇A,s f nc

· 1

Nsca

, fdef,0



(4.72)

Mode 3

The third mode accommodates cases where the solar field produces more energy than can be

used in the power cycle or by charging storage. This situation may be caused in systems with

storage by a fully (or nearly fully) charged TES system, or in systems without storage by an

overproduction of energy from the solar field with respect to the power cycle capacity. The

charging mass flow rate is determined in Eq.[4.73]. Note that for systems without storage, the

equation still applies, but the available charging energy rate q ̇A,chg is equal to zero.

m ̇ chg = q ̇A,chg

cht f,s f (Ts f,out −Ttes,cold) (4.73)

Because the solar field is overproducing energy, the field must be partially defocused. The

defocus fraction is first calculated as the ratio of available consumption streams to production

streams. Like Mode 2, two separate equations are used depending on whether the user allows

partial defocusing. Partial defocusing uses the following calculation:

fdef = fdef,0 ·

q ̇A,chg +Q ̇dem

q ̇A,s f nc

(4.74)

For whole-SCA defocusing:

fdef = fdef,0 ·Nsca q ̇A,chg +Q ̇dem

q ̇A,s f nc

· 1

Nsca

(4.75)

The mass flow rate to the power cycle is determined according to the thermal demand, and the

power cycle inlet temperature is equal to the solar field outlet temperature, since that is the sole

thermal resource in this mode.

m ̇ pb = Q ̇dem

cht f,s f (Tpb,in −Tpb,out) (4.76)

Tpb,in = Ts f,out

Mode 4

The final operation mode applies exclusively to systems with thermal storage, and handles the

case where the solar field overproduces energy with respect to power cycle demand, but where

all of the excess thermal energy can be diverted to TES. Here, the TES charging mass flow rate

is equal to the difference between available thermal power from the solar field and demand

100
from the power cycle.

m ̇ chg = min

q ̇A,s f −Q ̇ dem , q ̇A,chg

cht f,s f (Ts f,out −Ttes,cold) (4.77)

The power cycle mass flow rate is calculated to match demand, and the power cycle inlet

temperature is equal to the solar field HTF outlet temperature since neither storage nor the

auxiliary heater contribute to the thermal resource in this mode.

m ̇ pb = Q ̇dem

cht f,s f (Ts f,out −Tpb,out) (4.78)

Tpb,in = Ts f,out

Like Mode 2, it is possible that sticky-mode operation can result in a situation where partial

field defocusing is required in Mode 4. The defocusing equations used here are identical to

Mode 2 Eq.’s[4.71] and [4.72].

4.4.6 Controller iteration

The controller calculates the solar field inlet temperature based on the performance of the

various plant subsystems, including the solar field. However, the solar field performance is

also directly impacted by the solar field inlet temperature. A poorly calculated solar field inlet

temperature can cause problems in finding the best allocation of HTF flow to the power cycle,

solar field, storage, and auxiliary heater. To overcome this challenge, the control code

recalculates the solar field inlet temperature and iterates until the temperature and mass flow

allocations converge. The field inlet temperature is determined by a weighted average of the

power cycle mass flow and the TES charge mass flow.

Ts f,in = m ̇ pb Tpb,out +m ̇ chg Ttes,cold

m ̇ pb +m ̇ chg

(4.79)

For cases where the solar field is not in operation (i.e. m ̇s f = 0), the solar field inlet

temperature is equal to the lesser of the solar field outlet temperature and the user-specified

startup temperature.

Convergence is achieved when either the convergence error doesn’t change from iteration to

iteration, or when the convergence error itself falls below the specified tolerance (0.0001).

Convergence error is calculated by comparing the mass flow and temperature values in thermal

storage, the power cycle, and the solar field.

err =



m ̇ chg +m ̇ pb −(m ̇ dis +m ̇ s f)

m ̇s f 2

+



Ts f,in,0 −Ts f,in

Ts f,in 2

(4.80)

The solar field inlet temperature from the previous iteration (subscript 0) is compared to the

current calculation to find convergence. In some conditions, convergence may be obtained

101
even though the mass flow rate portion of the error equation does not approach zero. This

situation is handled by calculating the change in error from iteration to iteration. If Δerr falls

below a tolerance of 0.001, the control code is assumed to have converged and calculation

moves on. The equation for this calculation is:

Δerr = err0 −err

err

(4.81)

One special situation does not require iteration. If the defocus control calculated in Eq.’s[4.74]

or [4.70] changes from the last iteration, the code updates the solar field inlet temperature

guess value and continues without further iteration, and the solar field is allowed to recalculate

performance. This action prevents the defocus control from modifying the operating field

fraction and then continuing iteration without accounting for the new field performance.

If unexpected conditions are encountered during the simulation, iteration can occasionally loop

indefinitely. The controller limits the number of possible iterations to 20 to keep the simulation

from “hanging”. If the number of iterations exceeds this limit, a warning message is passed to

TRNSYS, and calculation proceeds with the most recently calculated values. Warning

messages do not automatically terminate a simulation, but the entire simulation will abort after

a maximum number of warning messages have been passed to TRNSYS. Since convergence

errors typically occur during time steps with low thermal resource, annual performance is not

significantly affected by non-convergence.

After the control code has converged on the appropriate field inlet temperature, the storage

heat exchanger flows, tank conditions, and auxiliary heater performance must be calculated.

The heat exchanger model is called according to whether TES is charging or discharging. If

TES is a direct system, no heat exchanger is required.

The thermal storage tank model is called for both hot and cold storage using input/output flows

and temperatures according to the values determined by the heat exchanger model. Another

iteration loop is required for the storage model since the tank temperatures used throughout the

controller calculations refer to the average HTF temperature over the current time step. If the

calculated tank temperatures do not match the estimated tank temperatures used in the control

code, the estimated values are updated and the entire control code is re-evaluated.

Convergence is achieved when the error falls below 1×10−5 or after the 9th iteration, where

tank convergence error is given as:

errtank =



Ttes,hot,0 −Ttes,hot

Ttes,hot 2

+



Ttes,cold,0 −Ttes,cold

Ttes,cold 2

(4.82)

Figure 31 presents the control flow diagram for this model.

102
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103
Section 4.4 Summary

• The plant controller uses stream temperature and thermal energy to allocate HTF mass flow

in the plant.

• The control algorithm first makes use of energy from the solar field, then thermal storage,

and finally from the auxiliary heater in providing energy to the power cycle.

• The controller monitors user-specified requirements and energy demand from the power

cycle to manage use of the solar field, thermal storage, and the auxiliary heater.

• Four controller operating modes capture the full range of plant behavior.

• Field defocusing is handled by the plant controller.

• The control code requires iteration to solve for the correct solar field inlet temperature and

the hot and cold tank temperatures.

104
5 Parasitic Losses

All thermal power conversion systems include devices that consume energy produced by the

power cycle. A number of critical components require electrical energy for power including

pumps, fans, site lighting, and tracking devices. For plants that produce electricity as the final

product, this energy can be siphoned off from the plant generator output before it reaches the

grid. Though not applicable for the System Advisor trough model, plants that produce only

thermal energy or mechanical power, or plants that are not actively producing power must

acquire electricity from external sources.

Parasitic losses have a major impact on the overall plant performance. A trough CSP plant can

lose anywhere between 10-15% of the gross electric output to parasitic losses. Accurately

accounting for these losses both improves model accuracy and allows the modeler to optimize

the design of the parasitic systems. For example, the parasitic pumping power required to

move the HTF through the solar field is a strong function of piping diameter; by characterizing

the parasitic field pumping losses, we can begin to understand the relative impact of changing

the receiver diameter on plant performance.

5.1 Losses modeled in System Advisor

System Advisor uses several different approaches to model parasitic losses. These include the

detailed approach described for the piping model in Section 2.5 and the power cycle parasitics

in Section 3.3 as well as the more general coefficient-based model explained here. The model

directly accounts for ten individual parasitic loads, not including the power cycle pumps that

are implicit in the regression model. Table 18 lists each of these items, the code block that

calculates the loss, and a summary of the modeling approach.

Table 18: A summary of the parasitic losses accounted for by System Advisor

Loss Subsystem Modeling Approach

1. SCA drives & electronics Solar field Coefficient-based calculation

2. Solar field HTF pumps Solar field Detailed performance model

3. Piping freeze protection Solar field Detailed performance model

4. Power cycle HTF pump Controller Coefficient-based calculation

5. Storage HTF pump Controller Coefficient-based calculation

6. Fixed parasitic losses Controller Constant fractional loss

7. Balance of plant parasitics Controller Polynomial curve with coefficients

8. Auxiliary heater operation Controller Polynomial curve with coefficients

9. Heat rejection equipment Power cycle Detailed performance model

10. Storage heat trace heater Thermal storage Detailed performance model

Detailed performance-based models characterize losses based on system geometry. Whether

it’s piping freeze protection (Sec. 2.4), field HTF pumps (Sec. 2.5), power cycle heat rejection

(Sec. 3.3) or thermal storage heaters (Sec. 4.1), each of these parasitic losses are calculated as

part of the performance of their respective subsystems.

105
5.2 Parasitic loss calculations

The remainder of the parasitic loss calculations take a simpler approach, formulating the loss

in terms of system capacity or performance. Each coefficient- or equation-based parasitic loss

is presented in the following discussion.

5.2.1 SCA drives

SCA drives are part of the collector tracking system in the solar field. The drives are assumed

to operate at a constant power during all hours of solar field operation, although this may not

be the case in a real plant. This parasitic loss accounts for any electrical load associated with

the collector assemblies during daylight hours, including the tracking motors, electronic

control equipment, and possibly mirror washing equipment. The total load is a function of the

number of collectors in the solar field, where each collector is assigned the user-specified

power rating CW ̇ ,track from the Parasitics page.20

W ̇track = Nsca Nloops CW ̇ ,track : q ̇s f > 0 (5.1)

5.2.2 Power cycle HTF pumps

Real trough plants use a single set of HTF pumps to move fluid through the solar field, thermal

storage (on the solar field side), and the power cycle loop. However, System Advisor

conceptually separates the parasitic load required for the solar field from the load for pumping

HTF through the power cycle and thermal storage. This separation is necessary because almost

no information is available concerning the geometry and sizing of the power cycle heat

exchangers or the thermal storage heat exchanger, whereas enough information is provided in

the model to accurately calculate a pressure drop across the solar field. For that reason, a

simplified coefficient-based approach is used for the HTF mass that passes through the power

cycle and thermal storage.

W ̇ p,pb&tes = Cp,pb&tes (m ̇ pb +m ̇tes,s f) (5.2)

The user-specified pump power coefficient Cp,pb&tes is expressed in units of kilojoules per kg

of mass passing through either the power cycle steam generators or through the thermal

storage heat exchanger on the solar field side. Alternately, the units for this coefficient can be

expressed as the ratio of the pumping power requirement to the mass flow rate requirement, or

J

s/kg

s . In indirect storage systems, the HTF mass flow rate m ̇tes,s f refers only to the mass flow

rate on the solar field side that is distinct from the mass flow rate into and out of the tanks

themselves.

5.2.3 Storage HTF pump

The storage HTF pump parasitic is calculated using the same methodology as the power cycle

HTF pump. The user-specified coefficient Cp,tes determines the power requirement to move

HTF from one tank to another. For indirect systems, this means moving HTF from one tank

through the field-to-TES heat exchanger. Direct storage systems can charge and discharge

20Mathematical operator “:” = “such that”

106
storage without a heat exchanger, so the dedicated TES-side pumping power is set to zero in

the model.

W ̇ p,tes = Cp,tes m ̇tes : Indirect TES (5.3)

W ̇ p,tes = 0 : Direct TES

5.2.4 Fixed parasitic losses

Utility-scale CSP projects require maintenance staff and support facilities on site to keep the

plant operational. Building loads, lighting, and other loads must be accounted for in the net

electricity output of the plant. These loads are incurred for all hours of the day over the

lifetime of the plant, and System Advisor approximates their sum as a fixed fractional loss of

the total gross design output based on the user-specified ”Fraction of rated gross power

consumed at all times” variable on the Parasitics page.

W ̇ fixed = Cfixed W ̇ des (5.4)

5.2.5 Balance of plant parasitics

The balance of plant parasitic load is a polynomial equation that provides a parasitic load as a

function of energy production. This load does not account for any specific set of parasitics, but

captures any loss that scales as a function of the plant energy production. The equation is

specifically a function of the energy delivered to the power cycle and the design power output.

The user-specified coefficients and factors for this equation are labeled as “Balance of plant

parasitic” variables on the Parasitics page. Three coefficients Cbal,0−2 are used to define the

shape of the equation, and a multiplying factor fbal and an adjustment factor fad j set and scale

the magnitude of the parasitic loss.

W ̇ bal = W ̇ des fbal fad j 

Cbal,0 +Cbal,1

Q ̇

Q ̇des

+Cbal,2

 Q ̇

Q ̇ des2




(5.5)

5.2.6 Auxiliary heater operation

Operating the auxiliary fossil heater can consume a moderate electrical load. This parasitic

loss is modeled using the same approach as the balance of plant losses, except that the

equation is based on the load supplied to the auxiliary heater.

W ̇ aux = W ̇ des faux fad j 

Caux,0 +Caux,1

q ̇aux

Q ̇des

+Caux,2

 q ̇aux

Q ̇ des2




(5.6)

Thermal loss, etc.

The Parasitics page in System Advisor also includes a piping thermal loss coefficient. This

loss, like other thermal losses, is not a true parasitic loss since it doesn’t directly draw energy

from the total output. Piping thermal loss is modeled in the solar field using Eq.[2.26] in

107
Section 2.2.

5.3 Practical considerations

Net energy output during any given time step is equal to the gross power produced by the

power cycle minus the sum of all active parasitic loads, as shown in Eq.[5.7]. All financial

equations in System Advisor including LCOE, and PPA price use the total-annual net energy

value in their calculations.

W ̇ net = W ̇ −

W ̇track +W ̇ p,pb&tes +W ̇ p,tes +W ̇ fixed +W ̇ bal +W ̇ aux (5.7)

While the net electricity output is certainly an important metric of performance, some users

may also require specific knowledge about how parasitic loads are met. Since no power plant

(not even fossil or nuclear plants) can operate at 100% capacity, electricity for parasitic power

consumption during non-production hours must sometimes be purchased from the grid.

Parasitic energy during production hours is “free” - it comes from the plant’s generated

electricity and doesn’t need to be purchased. Therefore, a distinction between “online” and

“offline” parasitics is important, and may impact the profitability of a project.

The online and offline parasitics can be calculated from the hourly output file using only the

net and gross electrical output columns. The parasitic load is always equal to the difference

between gross and net output, and offline parasitics are tallied during hours when the gross

output is equal to zero, otherwise the parasitics are considered online.

W ̇ par,online = W ̇ −W ̇ net : W ̇ > 0 (5.8)

W ̇ par,offline = W ̇ net : W ̇ = 0 (5.9)

5.4 Net production at design

Most electricity production systems are described in terms of how much power will be

delivered to the load when operating at design conditions. This characterization makes sense

for plants that operate at steady state over most of their lifetime. However, CSP power

production can change significantly depending on the mode of operation. Since solar field

HTF pumps are the largest parasitic load, the plant will produce somewhat less energy when

the pumps are operating at full load. Figure 32 illustrates how the net electric output varies

depending on the time of day, while the gross generator output stays constant.

108
Figure 32: The net and gross power over a period of four days.

If we are to estimate the net power output “at design”, we first need to establish which parasitic

loads are active during the design state. However, it is rarely the case that all parasitic devices

will operate at full-load concurrently. Accounting for all parasitics at once can lead to a

complicated and confusing result. Rather than estimate design-point net power output, System

Advisor uses an average conversion ratio approach, shown in Eq.[5.10]. The gross-to-net

conversion ratio indicates the relative difference between the net electricity that reaches the

grid and the gross power cycle output. This approach accounts for parasitics on an

annual-average basis regardless of whether the plant operates at “design” or not. The authors

believe that this is the best and simplest way to accurately characterize the impact of parasitic

losses.

rnet/gross =

8760

∑

i=1

W ̇ net,i

W ̇i

·

 1 year

8760 hours

(5.10)

109
References

[1] Martin J. Atkins, Michael R.W. Walmsley, and Andrew S. Morrison. Integration of solar

thermal for improved energy efficiency in low-temperature-pinch industrial processes.

Energy, 35:1867–1873, 2010.

[2] A. Bejan. Convection Heat Transfer. John Wiley and Sons, New York, New York, USA,

2nd edition, 1995.

[3] Jurgen Dersch, Michael Geyer, Ulf Herrmann, Scott A. Jones, Bruce Kelly, Rainer  ̈

Kistner, Winfried Ortmanns, Robert Pitz-Paal, and Henry Price. Trough integration into

power plants: A study on the performance and economy of integrated solar combined

cycle systems. Energy, 29:947–959, 2004.

[4] V. Dudley, G. Kolb, M. Sloan, and D. Kearney. SEGS LS2 solar collector test results.

Technical Report SAND94-1884, Sandia National Laboratories, Albuquerque, New

Mexico, USA, 1994.

[5] J. Duffie and W. Beckman. Solar Engineering of Thermal Processes. John Wiley and

Sons, Hoboken, New Jersey, USA, 3rd edition, 2006.

[6] M. El Wakil. Powerplant Technology. McGraw-Hill Book Company, 1984.

[7] M. Erbes, J. Rheinlander, and S. Bergmann. Technical and economic performance of  ̈

parabolic trough solar power plants - a computational tool for plant feasibility studies.

Las Vegas, Nevada, USA, March 2008. Proceedings of the SolarPACES International

Symposium.

[8] R. Forristall. Heat transfer analysis and modeling of a parabolic trough solar receiver

implemented in engineering equation solver. Technical Report NREL/TP-550-34169,

National Renewable Energy Laboratory, Golden, Colorado, USA, 2003.

[9] R.W. Fox, A.T. McDonald, and P.J. Pritchard. Introduction to Fluid Mechanics. John

Wiley and Sons, Inc, Hoboken, New Jersey, USA, sixth edition, 2006.

[10] P. Gilman. Solar Advisor Model User Manual. Technical report, National Renewable

Energy Laboratory, Golden, Colorado (USA), April 2010.

[11] V. Gnielinski. New equations for heat and mass transfer in turbulent pipe and channel

flow. International Chemical Engineering, 16:359, 1976.

[12] P. Griffin, K. Huschka, and G. Morin. Software for design, simulation, and cost

estimation of solar thermal power and heat cycles. Berlin, Germany, 2009. Proceedings

of the SolarPACES International Symposium.

[13] F. Incropera and D. DeWitt. Fundamentals of Heat and Mass Transfer. Wiley, 5th

edition, 2001.

110
[14] B. Kelly and D. Kearney. Parabolic trough solar system piping model: Final report.

Technical Report NREL/SR-550-40165, National Renewable Energy Laboratory,

Golden, Colorado, USA, 2006.

[15] S. Klein. Engineering Equation Solver (EES) for Microsoft Windows Operating System:

Commercial and Professional Versions. Online at www.fchart.com, 2010.

[16] A. McMahan. Design and optimization of organic rankine cycle solar-thermal power

plants. Master’s thesis, University of Wisconsin, Madison, Wisconsin, USA, 2006.

[17] G. Nellis and S. Klein. Heat Transfer. Cambridge University Press, New York, New

York, USA, 2009.

[18] R. Piwko, K. Clark, L. Freeman, G. Jordan, and N. Miller. Western wind and solar

integration study: Executive summary. Subcontract Report SR-550-47781, National

Renewable Energy Laboratory, 1617 Cole Blvd., Golden, CO (USA), May 2010.

[19] W. Stine and M. Geyer. Power from the Sun. Published online at

www.powerfromthesun.net, 2001.

[20] M. Wagner. Simulation and predictive performance modeling of utility-scale central

receiver system power plants. Master’s thesis, University of Wisconsin, Madison,

Wisconsin, USA, 2008.

[21] M. Wagner. Methodology for constructing reduced-order powerblock performance

models for csp applications. Perpignon, France, 2010. Proceedings of the SolarPACES

International Symposium, SolarPACES.

[22] M. Wagner and C. Kutscher. Assessing the impact of heat rejection technology on CSP

plant revenue. Perpignon, France, 2010. Proceedings of the SolarPACES International

Symposium, SolarPACES.

[23] M. Wagner and C. Kutscher. The impact of hybrid wet/dry cooling on concentrating solar

power plant performance. Phoenix, Arizona, USA, 2010. Proceedings of the 4th

International Conference on Energy Sustainability, ASME.

[24] C.F.J. Wu and M. Hamada. Experiments: Planning, Analysis, and Parameter Design

Optimization. Wiley, 2000.

111
A Glossary of Terms

Absorptance An optical property that quantifies the tendency of a material to behave opaquely

to radiation.

Annulus The region in a receiver between the metal absorber tube and the glass envelope.

Aperture The rectangle containing the opening of the collector; equivalently, the area within

the plane that intersects the bounds of the parabola, extending from edge to edge

and along the axial direction.

Azimuth The angle formed between a cardinal direction (typically North) and the Sun when

a line connecting the observer and the Sun is projected onto a plane parallel with

the surface of the Earth where the observer stands.

Bellows A portion of the receiver that is designed to accommodate thermal expansion and

contraction as operating conditions vary. In some cases, the bellows design may

reduce the active absorbing area of the receiver.

Blowdown The fraction of steam mass flow in the power cycle that is extracted and replaced

with fresh, treated water to mitigate the buildup of solids in power cycle equipment.

Collector The assembly that contains the parabolic reflective surface and the structural

support.

Collector loop See Loop.

Collector row See Row.

Control volume An imaginary boundary around a process, component, or group of components

that defines what is considered within an analysis. Selection of this boundary also

defines what information flows into and out of the model.

Elevation angle The smallest angle between the plane of the horizon and the Sun in the sky as

viewed by an observer on the surface of the Earth. Angle range is from 0 to 90

degrees.

Emmitance An optical property that quantifies the tendency of a material to emit radiation. This

value is often highly dependent on the temperature of the material.

Envelope A term for the transparent glass tube surrounding the metal absorber tube in a receiver. Gas within the envelope is evacuated to a near-vacuum state to minimize

thermal loss.

Feathering A term sometimes used to refer to the intentional partial reduction in component

performance by mechanical modulation. The term derives from wind turbine technology where turbine blades are sometimes partially rotated to reduce excessive

wind force.

Focal length In the case of the vertex focal length: The distance between the vertex of the theoretical parabola describing the shape of the collector surface and the optical focus

of the parabola. In the case of the average focal length: The average distance between the surface of the collector and the optical focus. Refer to section 2.2 for

derivation of this value.

HCE See Receiver.

112
Header The section of pipe through which the HTF is evenly distributed to and collected

from each loop in the solar field.

Interaction effect A statistical quantity that describes the impact of simultaneous variation between

two or more independent variables (IV) on a given output metric. This quantity

describes how IV A affects the output if IV B also varies.

Loop A set of SCAs in series, with a source at the cold HTF header and a terminal at the

hot HTF header. The HTF for each loop enters at the field design inlet temperature

and exits at the field design outlet temperature. A solar field may contain any

number of individual loops in parallel.

Main effect A statistical quantity that describes the impact of variation in an independent variable on a given output metric. Compare with interaction effects

Parasitic A device or subsystem that consumes electrical or mechanical energy as part of its

normal operation, thereby reducing the total amount of electricity that is produced

by the plant.

Power block The power generation equipment for the system. Among other components, this includes the HTF-to-steam heat exchangers, the steam turbines, the electric generator,

the condenser equipment, and the feedwater heat exchangers. The thermodynamic

cycle is a Rankine cycle for this model.

Receiver (Also known as the heat collection element, or HCE) The linear component that

runs along the axial focus of the collector to absorb heat and transfer it to a thermal

fluid. Typically, this component consists of a metallic absorber tube concentrically

contained within an evacuated glass tube.

Row A set of SCA’s in series extending in a single direction away from or returning to

the header. The number of SCA’s in a row is usually equal to half of the number of

SCA’s in a loop. A loop is always composed of two collector rows (out and return).

Runner piping The runner piping (sometimes referred to as the “to-section header piping”) is the

large-diameter piping that carries HTF between the solar field subsections and the

power cycle.

SCA Solar collector assembly. An assembly of multiple collectors and receivers in series, all driven by a single tracking drive device.

Temperature, drybulbThe steady-state temperature of an object that is exposed to the ambient air and

shielded from moisture and radiation.

Temperature, wetbulbThe wet-bulb temperature of air represents the steady-state temperature of an object

from which water is actively evaporating. This temperature depends on the relative

humidity of the ambient air and is limited to the range between the dry-bulb temperature and the dew-point temperature.

Torr A unit of measure for pressure. 1 torr equals 1 mmHg by definition, or 0.13 kPa.  ̄

Transient Any effect whose impact is subject to time. Most typically, an effect that causes the

input of a ”control volume” to not be equal to the output. One example is a storage

tank where mass accumulates causing the inlet mass flow rate to differ from the

outlet mass flow rate.

113
Transmittance An optical property that quantifies the tendency of a material to behave transparently towards radiation.

Zenith The compliment of the Elevation Angle. Angle range is from 90 degrees at the

horizon to 0 degrees vertically in the sky.

114