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NATL INST OF STANDARDS & TECH RI.C.

A1 11 0251 6992

Fang, Jin B/Mlnimum life cycle cost heat QC100 .U56 NO.86-3381 1986 V19 C.1 NBS-P Minimum Life Cycle Cost Heat

Losses for Shallow Trench

Underground Heat Distribution

Systems

Jin B. Fang

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Building Physics Division

Gaithersburg, MD 20899

May 1986

Prepared for; Tri-Service Building Materials Committee

Headquarters, U.S. Army Corps of Engineers

Washington, DC 20314-1000

U.S. Navy, Naval Facilities Engineering Command

Alexandria, VA 22332-2300

QC Air Force, Air Force Engineering and Services Center

100 all Air Force Base, FL 32403-6001

.056

86-3381

1986

C. 2

R3S

RESEARCH

INFORMATION CENTER

NBSIR 86-3381

MINIMUM LIFE CYCLE COST HEAT

LOSSES FOR SHALLOW TRENCH

UNDERGROUND HEAT DISTRIBUTION

SYSTEMS

Jin B. Fang

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Building Physics Division

Gaithersburg, MD 20899

May 1986

Prepared for: Tri-Service Building Materials Committee

Headquarters, U.S. Army Corps of Engineers

Washington, DC 20314-1000

U.S. Navy, Naval Facilities Engineering Command

Alexandria, VA 22332-2300

U.S. Air Force, Air Force Engineering and Services Center

Tyndall Air Force Base, FL 32403-6001

U.S. DEPARTMENT OF COMMERCE. Malcolm Baldrige. Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director

ABSTRACT

The rates of heat loss from two underground insulated pipes installed in a

shallow trench were calculated using a finite element method computer

program to solve a two-dimensional steady-state heat conduction problem.

The results of pipe heat loss study based on a specified ground temperature

condition are summarized for a range of pipe insulation thicknesses,

shallow trench sizes, and pipe fluid temperatures. Methods of determining

the heat loss associated with the minimum life-cycle cost and the

corresponding economic insulation thickness for shallow trench heat

distribution systems are presented. Life-cycle costing analysis was

performed for two insulated pipes in a concrete trench to determine the

cost of construction, annual energy cost associated with pipe heat loss,

and yearly operating and maintenance costs. Based on this economic

analysis,

thickness

supply and

Keywords:

the least life-cycle cost heat loss and the optimum insulation

were determined for specified fluid temperatures in the heat

the return lines.

finite element method, fuel energy cost, heat loss, life-cycle

cost analysis, shallow trench, underground heat distribution

system.

1
TABLE OF CONTENTS

Page

1. Introduction 1

2. System Heat Loss Calculations 1

2.1 The Equations Used for Numerical Solution 1

2.2 Description of Computer Program 3

2.3 Preparation of Input Data 4

2.4 Sample Calculations 6

3. Equivalent Energy Cost 7

4. Material and Installation Costs 8

5. Maintenance and Repair Costs 8

6. Life-Cycle Cost 9

7. Minimum Life-Cycle Heat Loss 10

8. Conclusions 10

9 . Acknowledgment 11

10. References 11

Appendix A. The Input Data Files and the Outputs from the

Computer Program on Sample Calculations 23

Appendix B. A Listing of the Computer Simulation Program 28

ii
LIST OF TABLES

Table 1.

Table 2.

Table 3

Figure 1

F igure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Dimensions of Concrete Shallow Trench Systems for Various

Thicknesses of Pipe Insulation

Calculated Results

Underground Pipes at

210 F).

of Heat Losses from Two 150 mm (6-in.)

Water Temperatures of 196 and 99 C (385 and

. Calculated Results of Heat Losses from Two 150 mm (6-in.)

Underground Pipes at Water Temperatures of 171 and 143 C (340 and

290 F).

LIST OF FIGURES

. Concrete Shallow Trench Underground Heat Distribution System

. Finite Element Design for the Outer Boundary Earth and Concrete

Trench Walls and Cover

. Finite Element Design for Air Space Surrounding the Pipes in

Concrete Trench

. Finite Element Design for the Pipe Insulation and the Outer

Surfaces of the Pipes

. Material and Installation Costs of Two 150 mm (6-in). Pipe

Shallow Trench Systems as a Function of Pipe Insulation

Thickness

. Life-Cycle Cost Analysis for a Shallow Trench System

A Plot of the Life-Cycle Cost Versus the Heat Loss from

Underground Pipes at 171 and 143 C (340 and 290 F)

A Plot of the Life-Cycle Cost Versus the Heat Loss from

Underground Pipes at 196 and 99 C (385 and 210 F)

iii
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1. Introduction

Underground piping systems distribute hot and chilled water or steam

through a network of insulated pipes to serve military installations for

space heating and cooling, and industrial purposes. These pipe-lines can

be directly buried in the ground, installed in a concrete shallow trench, or placed above the ground. Recent studies [1,2] compared the difference

in life cycle costs between direct buried and shallow trench heat

distribution systems based on the results of field surveys of existing

systems, and estimates of capital cost, maintenance and operating expenses.

Based on both studies, it is difficult to draw any conclusions on cost

comparison between these two systems, since their findings are discrepant

with each other, especially for systems with large diameter pipes.

However, continued construction of concrete shallow trench systems as an alternative to the direct buried conduit systems is anticipated due to

their reduced maintenance and repair costs although they are minor

components of the life-cycle cost.

Engineering projects involving substantial expenditures require

economically sound investment decisions based on choosing the best

alternative. Economic analysis plays an important role in virtually every

engineering project, and can identify and evaluate the economic outcome of

a proposed project. Life cycle costing is a useful tool in which the

individual costs are determined for the expected life of the project, and

the total cost is compared with other alternatives. This approach provides

the financial justification for selecting the final design and size for a new underground heat distribution system.

It is well known that the heat loss from piping decreases with an increase

in insulation thickness, however a point is reached when further reduction

in heat loss becomes uneconomical. In this period of high fuel cost, the

conservation of energy through the use of optimal insulation thickness

corresponding to the minimum life cycle cost heat loss has obvious benefits

to reduce fuel energy and maintenance expenditures in operating the heat

distribution system.

This report describes the methodologies and procedures used to estimate the

energy cost associated with the pipe heat loss and the minimum life cycle

cost heat loss from various underground shallow trench heat distribution

systems. It presents cost data for materials and installation for

acquisition, and maintenance and repair costs for some selected trench

systems. The heat loss data are prepared for various thicknesses of pipe

insulation, for different sizes of shallow trench systems, and for a range

of pipe fluid temperatures.

2. System Heat Loss Calculations

2.1 The Equations Used for Numerical Solution

Finite element methods have been developed to a high level of refinement

and have made a great impact in structural mechanics. Such methods are

also applied to steady and transient heat conduction problems [3-5].

1
The heat loss per unit length from a shallow trench or a loose-fill

insulation underground heat distribution system can be calculated using the

computer simulation program developed recently by Kusuda [6]. This program

applies the finite element method to a steady-state, two-dimensional

shallow trench system consisting of a heat supply and a return pipe having

different fluid temperatures for the pipe sizes shown in Figure 1. The

program contains a predesigned finite-element mesh with 130 triangular

elements and 80 nodal points. Figure 2 shows the triangular element mesh

for the outer boundary earth region, and for the concrete trench walls and

cover. Figure 3 shows the mesh for the air space between the insulated

pipes and the trench walls, and Figure 4 illustrates the mesh for the pipe

insulation. Prescribed boundary temperatures include the nodal points

around the outer surfaces of the circular pipes (nodes 49 through 64 in

Figure 4) and the perimeter of the earth zone (nodes 65 through 80 in

Figure 2). The remaining nodal point temperatures are calculated by

solving the system of simultaneous equations using the Gaussian elimination

method

.

The undisturbed earth temperature used as the prescribed temperature

boundary conditions, is a function of time and depth below the ground

surface and can be determined by the following eqution [7]:

V

w j w

2a ) sin [2it (t-3)/12 - 2ct ] where T = the monthly average earth temperature, oC (oF)

= the annual average earth temperture of the site, °C (of)

Xu = the annual amplitude of the monthly average temperature

°C (OF)

y

= depth from the ground surface, m (ft) w = angular frequency of the annual cycle, rad/h

a = thermal diffusivity of the soil, m^/h (ft^/h)

t = the elapsed time from January, in months

( 1 )

eye le

,

The rate of heat flow by natural convection through the airspace bounded by

the trench walls and the outer surfaces of the insulated piping can be

approximated by an equivalent heat conduction of the form [8,9]:

Q/A = kg (Th - Tc^/I^

where Q

= the average heat flow rate, W (Btu/h)

A = the mean cross-sectional area of the enclosed air layer, m'^ (f t2)

kg = the effective thermal conductivity of the enclosed air layer,

W/m-K (Btu/h*ff°F)

Tjj = the temperature of the hot surface, °C (°F)

Tg = the temperature of the cold surface, °C (°F)

L = the thickness of air layer, m (ft)

An effective thermal conductivity is used to modify the simple conduction

solution to account for convection. The effect of radiant exchange between

the pipes and trench walls may be negligible due to low emissivity of

aluminum jacket surface. The ratio of the effective to the actual thermal

conductivity of the enclosed airspace, kg/l^> is a function of the Rayleigh

number based on the characteristic dimension (the thickness) of the air

2
layer and on the temperature difference of the hot and cold surfaces [8].

The effective thermal conductivity is also a function of the product of the

convective heat transfer coefficient and the characteristic dimension. It

is noted that if the natural convection is suppressed and the heat is

transferred through the air layer only by heat conduction, the effective to

the actual thermal conductivity ratio is unity. No experimental data or correlations are available on natural convection in air confined between a pair of heated pipes and cooler enclosure walls. However, correlations for

free convection through plane or cylindrical air layers in enclosures from

a hot to a cold surface is the closest configuration available [8,9].

Convective heat transfer between the ground surface in the vicinity of the

shallow trench and the ambient air is dependent upon the local climate and

surface conditions. The surface resistance to convective transfer may be

considered as being contained in an equivalent soil layer. The thickness

of this fictitious layer represents the amount by which the earth boundary

is extended to account for the effect of the film resistance due to

convection, and can be determined by

where = the thickness of the equivalent soil layer, m (ft)

kQ = the thermal conductivity of the soil, W/m*K (Btu/h*ft*

h = surface heat transfer coefficient, W/m^’K (Btu/h*ft^’°

°F)

F)

The heat loss rate per unit length of piping is obtained using the

following equation along with the calculated value of average temperature

drop across a circular cylindrical shell of pipe insulation layers:

2^ki(li-To)

InCr^/r^)

(4)

where q

kT

= the heat loss rate per unit length of the insulated underground

pipe, W/m (Btu/h*ft)

= the thermal conductivity of insulation materials, W/m*K

(Btu/h*ff°F)

= outside radius of the insulation layer, m (ft)

= inside radius of the insulation layer, m (ft)

and Tq = the surface temperature of the insulation layer at inner

and outer radii r^ and r^, respectively, °C (°F)

For an insulated piping system, the surface film resistance between the hot

water and the pipe, and the thermal resistance of the pipe wall can

generally be ignored in comparison to the thermal resistance of the

insulat ion

.

2.2 Description of the Computer Program

The computer program, which is written in FORTRAN language, consists of a main program (FEUHDS) and seven subroutines called PIPE2, TGO, TWOPIP,

SOILK, TGXX, SOLVP and PIPEHL. Input data are read in by the main program,

and subroutines PIPE2 and TGO. Output is provided by the main, PIPE2,

3
TWOPIP, and PIPEHL programs. In addition to handling portions of input

data, the main program coordinates and performs most of the calculations.

Subroutines PIPE2 is used to calculate rectangular coordinates for each

nodal point based on the shallow trench and piping geometry, while

subroutine TGO calculates the average undisturbed earth temperatures at

various depths for the month of interest. Subprogram TWOPIP called from

the main program determines the rate of heat loss from two insulated buried

pipes to the underground. Subroutine SOILK provides soil thermal

conductivity values for various earth temperatures by linear interpolation

of a set of soil thermal conductivity versus temperature data, and TGXX

furnishes the external boundary temperatures of the outer earth region

surrounding the shallow trench system. Subroutine SOLVP is used to solve

system of simultaneous equations by Gauss elimination method, and PIPEHL

computes the temperature drops across the pipe insulation layers and the

heat loss rates for both underground pipes. A listing of the source code

of this computer program is given in Appendix B.

2,3 Preparation of Input Data

Two input data files, DATAl and DATA2, are created prior to execution of

the computer simulation program FEUHDS. The DATAl file shown in Appendix

A.l contains data for run control parameters, the month of interest, the

thermal conductivity and dimensions of the trench walls, convection

conditions for the trench enclosed air space, the pipe fluid temperatures,

the thermal conductivities and dimensions of the carrier pipes and

insulation layers, the thermal properties and dimensions of the earth

region surrounding the trench system, and the annual average temperature

and amplitude of the monthly temperature cycle for the site involved. The

DATA2 file shown in Appendix A. 2 consists of data for the finite-elements

including the element number, the node numbers for its three vertices, and

the surface convection coefficients and ambient temperatures for the sides

of each element that experiences convection loss.

Description of Variables ;

The computer program requires that the numerical values of all input

variables are in engineering units.

MREPT Number of iterations to be performed for soil temperature and

moisture effect analysis

ITREN An index, ITREN =1 A shallow trench system

=0 A loose-fill insulation system

MONTH Month for which the heat loss is to be determined

ICALB An integer, ICALB = 1 Nodal coordinates will be printed out

=0 No print-out

KTCT Thermal conductivity of trench walls, ( Btu * in/h * f

t ^ ’°F)

TRTK Thickness of the trench wall, (inches)

KASP Equivalent therma

underground pipes in

1 conductivity of air surrounding

the shallow trench, (Btu*in/h’ft ‘°F)

the

4
T1,T2 Fluid temperatures of pipe numbers 1 and 2, respectively, (°F)

KII Thermal conductivity of pipe insulation materials,

(Btu-in/h-ft^*°F)

KIG Thermal conductivity of the soil in the vicinity of the shallow

trench, (Btu* in/h *ft^ *°F)

PIl ,PI2 Outside diameters of pipe numbers 1 and 2, respectively, (inches)

THIl,

THI2

Thicknesses of the thermal insulation layers for pipe numbers 1 and 2, respectively, (inches)

B1 ,B2 Depths from ground surface to the centers of pipe numbers 1 and

2, respectively, (ft)

S Separation distance between the centers of the pipes, (ft)

TG Monthly average earth temperature, (°F)

WW Width of the earth region surrounding the shallow trench system,

(ft)

HH Thickness of the equivalent soil layer to account for the surface

convective transfer effect, (ft)

HY Depth of the earth region underneath the shallow trench system,

(ft)

D Thickness of the trench cover, (ft)

A Total width of the trench system, (ft)

B Height of the trench wall, (ft)

AO Annual average outdoor temperature of the site, (°F)

BO Annual amplitude of the monthly average temperature cycle of the

site, (°F)

DIFF Thermal diffusivity of the soil in the vicinity of the trench

system, (ft^/h)

N The element number, dimensionless

I,J,K The node numbers of three vertices of an element

C Thermal conductivity of an element, (Btu ' in/h ’f t^ *°F)

IXCB An index, IXCB = 1 The element has convection boundary

=0 No convection boundary

HIJjHJK,

HKI

Convective heat transfer coefficients for sides IJ, JK and KI of

a triangular element, (Btu/h’f t^*°F)

5
TIJjTJK, Ambient temperatures for sides IJ, JK and KI of an element, (°F)

TKI

The input format of these aforementioned variables which are stored in data

files DATAl and DATA2, are respectively:

(1) DATAl File

Record No. Variables Format

1 MREPT, ITREN (15, IX, 14)

2 MONTH, ICALB (215)

3 KTCT, TRTK (2F10.4)

4 KASP (F10.5)

5 Tl, T2 (2F10.3)

6 KII, KIG (2F10.4)

7 PIl, PI2 (2F10.4)

8 THU, THI2 (2F10.4)

9 Bl, B2 (2F10.4)

10 S, TG (2F10.4)

11 WW, HH, HY (3F10.4)

12 D, A, B (3F10.4)

13 AO, BO, DIFF (3F10.4)

DATA2 File

Record No. Variables Format

1 N, I, J, K, C, IXCB (4I5,F10.4

(Use record No. 2 if IXCB = 1)

2 HU, HJK, HKI, TIJ, (6F10.4)

TJK, TKI

(Repeat record numbers 1 and 2 for a total of 130

elements

)

2.4 Sample Calculations

Two nominal 150 mm (6-inch) steel pipes are laid side by side in a 1.24 m

(4.08 ft) wide by 0.89 m (2.92 ft) high concrete shallow trench having 152

mm (6 in.) thick trench cover and walls. Figure 1 shows a typical high

temperature hot water underground system consisting of a heat supply and a return pipe in a concrete trench with the top cover placed flush with

ground level. The hot water pipe temperture is 196°C (385°F) and the

return pipe temperature is 99°C (210°F). The centers of both pipes are

located 1.30 ft (0.40 m) beneath the ground surface. The pipes are insulated with an aluminum jacket and 89 mm (3.5 in.) thick calcium

silicate, allowing 102 mm (4 in.) of separation between surfaces of

adjacent pipe insulation. The concrete trench system is surrounded by

earth having a thermal conductivity of 2.16 W/m’K (15 Btu*in/h*f f^'^’F) and an annual average temperature of 13°C (56°F). The system heat loss

calculations were carried out on an IBM personal computer. The input data

6
files for the sample case and the outputs from the computer program are

listed in Appendix A.

In order to maintain some clearances between the insulated pipes and

adjoining trench walls and floor as specified in the design guide [10], the

concrete trench has different overall dimensions to accommodate various

thicknesses of pipe insulation, as listed in Table 1. The effects of the

thickness of pipe insulation on the heat loss of the trench system for two

sets of pipe temperatures are shown in Tables 2 and 3. The water

temperatures of the heat supply and return lines are 196°C (385°F) and 99°C

(210°F), respectively for the heat loss calculations shown in Table 2, and

171°C (340°F) and 143°C (290°F) in Table 3. As expected, the results

indicate that the total heat loss from the underground piping system

decreases with an increase in insulation thickness due to the increased

thermal resistance for radial heat conduction. For a given insulation

thickness, the rate of heat loss from a single insulated pipe increases

with increasing pipe temperature since the convection loss to the trench

air is directly proportional to the temperature difference between the

outer surface of the pipe and the surrounding air.

3. Equivalent Energy Cost

The energy cost per unit length of the shallow trench system is calculated

using the computed heat losses from the piping system obtained from the

computer program, and the equivalent fuel energy cost adjusted for future

escalation in energy prices over the life of the system. The equation used

for calculating the equivalent energy cost (EEC) or the cost of lost energy

per annum, is as follows:

EEC = 8760 * Q

* EC * UPW* (5)

where Q

= system heat loss rate, which is equal to the sum of the heat

loss rates of two pipes, (Btu/h*ft)

EC = equivalent fuel energy cost, ($/MBtu)

= 100 (FC/EF), in which FC is fuel cost, ($/MBtu), and EF is plant

^ fuel conversion efficiency (percent)

UPW” = the modified uniform present worth factor adjusted for future

escalation in fuel prices.

In this sample case of life-cycle cost calculations, the fuel to heat

conversion efficiency at the plant is assumed to be 100%, and the price of

fuel is at $5.00 per million Btu. In reality, the plant fuel conversion

efficiency is dependent upon the type of boiler used, and ranges between 50

and 90 percent.

The energy cost is assumed to escalate in accordance with the data on annual inflation rate of natural gas as projected by the U.S. Army Corps of

Engineers [1]:

Type of Fuel 1985-1990 1990-1995 1995-2000 2000-2009

Natural gas 5.13 8.81 3.91 4.69

Distillate fuel 6.02 5.98 3.91 4.69

Residual fuel 6.50 6.06 3.91 4.69

Coal 1.85 3.04 3.04 3.04

7
The Modified Uniform Present Worth factor (UPW ) is calculated based on a 10% discount rate for a 25 year life for the underground system, and the

escalation rates in natural gas price projected by the U.S. Army Corps of

Engineers for the periods covering from 1985 to 2009, using the following

equation [ 11 ]:

UPW = z (—) + (7—) ^ z (r-r^

j=]^

+ (r-ri) ^ ^ z

1 +d

1 +e

1 +d

I.N1 1+e

j=l

1 + 0'

1 +d

%

Zj=l

1 +d 1 +d

l+^k j 1 +ev ,^+®k^N. , ( )^ = ( ) [1 - ( 7-

7

-) k

1 +d d-e^. 1 +d

1 +e,

1 +d

( 6 )

= the length of the period for a given escalation rate in a given

period, (year)

d = the discount rate

= the rate of escalation in each of period

The salvage value of a shallow trench heat distribution system after 25

years of service is assumed to be zero. With these economic assumptions,

• • • "ic • the modified uniform present worth factor, UPW for the sample case is

found to be 15.8 percent.

4. Material and Installation Costs

The capital cost including material and installation costs for constructing

a shallow trench underground heat distribution systems can be calculated

using cost data and the estimate procedures given in references [1,12-14].

In order to maintain clearance between the outer surfaces of the pipe

insulation and the adjoining trench walls and floor described previously,

the trench size varies to accommodate different insulation thicknesses, in

accordance with the dimensions listed in Table 1. The costs of materials

and installation for constructing concrete shallow trench systems

consisting of two 150 mm ( 6 -in.) steel pipes with calcium silicate

insulation encased in aluminum jacket are estimated and shown as a function

of insulation thickness in Figure 5. As anticipated, both the material and

installation costs rise with an increase in insulation thickness.

5. Maintenance and Repair Costs

An annual maintenance cost of $1,000 per mile for a typical shallow trench

system, which was derived from the results of a recent survey of field

installations [1], was used for the life-cycle cost analysis. The total

present value of these annually recurring costs was calculated by

multiplying this maintenance cost by 9.077, the uniform present worth (UPW)

8
factor for a 10% discount rate over a 25 year system life. This was

determined to be $5.64 per meter ($1.72 per foot) for the shallow trench

system.

Based on the field survey results, the calculations for the present value

of the repair costs were carried out with an assumption that these

expenditures occur during years 12 through 25 [1]. The present value of

the nonannually recurring repair costs was approximately $0.30 per meter

($0.09 per foot), and was neglected compared to the system installation and

the routine maintenance costs.

6. Life-Cycle Cost

Life cycle cost analysis can be used to determine the total expenses

associated with various design alternatives. These expenses would include

the cost of acquisition, year ly maintenance and repair cost, and yearly

fuel energy cost with the adjustment for future escalation in fuel prices.

The total life-cycle cost of an underground heat distribution system can be

expressed as follows:

LCC = MC + IC + EEC + MRC (7)

where MC = material cost, ($/ft of the underground system)

IC = installation cost, ($/ft of the system)

EEC = equivalent energy cost including the adjustment for future

escalation in energy prices over the life of the system, ($/ft

of the system)

MRC = maintenance and repair costs, ($/ft of the system)

Evaluation of the effect of varying insulation thickness for the

underground piping system on the construction and fuel energy costs

associated with various insulation thicknesses over the economic life of an underground heat distribution system was performed. Figure 6 shows the

effect of pipe insulation thickness on the material and installation costs,

equivalent energy cost, and total life-cycle cost for two 150 mm (6-in.)

pipes installed in a concrete shallow trench based on a 171°C (340°F)

supply and 143°C (290°F) return temper atures. In order to provide the

specified clearance between the outer surfaces of pipe insulation and the

adjoining trench walls and floor, the overall dimensions of the trench

system and the relative location of the underground pipes were varied with

the insulation thickness according to dimensions given in Table 1. An

equivalent fuel energy cost of $5.00 per million Btu was assumed for energy

cost calculations along with the use of the material and installation cost

data shown in Figure 5. Figure 6 shows that the life-cycle cost is minimal

at an insulation thickness of 89 mm (3.5 inches). This is the most cost

effective thickness for insulating the underground piping.

The calculated economic thickness should be compared to the thickness

required to prevent moisture condensation since condensation occurs when

the temperature of cold pipe surface is below the dewpoint temperature of

the surrounding trench air. However, an insulation system with a protective jacket can minimize moisture penetration if all jacket joints

and seams are correctly sealed.

9
7. Minimum Life-Cycle Heat Loss

A plot of the costs of material and installation, fuel energy cost, and

total life-cycle cost of a concrete shallow trench containing two 150 mm

(6-in.) insulated pipes versus the heat loss from these pipes is given in

Figure 7, which is a replot of Figure 6. At high values of the system heat

loss shown in Figure 7, the annual cost of material and installation is

low, but the annual equivalent energy cost is high. Decreasing the heat

loss by increasing the thickness of pipe insulation reduces the fuel energy

cost but adds to the material and installation costs. At a certain value

of pipe heat loss, the sum of the costs of material and installation, and

the fuel energy cost will be a minimum as indicated by the life-cycle cost

curve. Beyond the minimum, the life-cycle cost curve rises because the

increased costs of material and installation due to additional insulation

thickness is no longer offset by the reduced cost of the system heat loss.

As illustrated in the Figure 7, the minimum life-cycle cost was reached at

a system heat loss of 146 W/m (152 Btu/h*ft). The insulation thickness

corresponding to this minimum life-cycle cost heat loss is 89 mm (3.5

inches)

.

A plot similar to Figure 7 is shown in Figure 8 where the heat supply pipe

transported 196°C (385°F) hot water and the return pipe carrying 99°C

(210°F) hot water. In Figure 8, the total life-cycle cost of the trench

underground system involved can be represented by a "U" shaped curve. In

this sample case, the life-cycle cost curve has a minimum that occurs at

the pipe heat loss rate of 151 W/m (157 Btu/h*ft). The thickness of

insulation layers at this minimum life-cycle cost heat loss is found to be

76 mm (3 inches).

8. Conclusions

The calculation of heat loss from a pair of insulated piping system

installed in a shallow trench underground heat distribution system has been

performed using a computer simulation program based on the finite element

method. General formulation of the relevant equations of heat flow and

boundary conditions for a two dimensional, steady heat conduction problems

are presented. The computational scheme and the input data required for

executing the simulation program are described, and the output from the

computer program presented for a sample case.

Life-cycle costing analysis was performed for a typical concrete shallow

trench system containing two nominal 152 mm (6-inch) insulated pipes with

various insulation thicknesses. It was demonstrated that the least-cost

heat loss from the underground pipes or the most economic insulation

thickness can be determined for given fluid temperatures based on the

results of the cost analysis. The calculated economic insulation thickness

should be sufficient to maintain the surface temperature of the insulation

above the dewpoint of the surrounding trench air preventing moisture

condensation. The procedures for calculating the equivalent energy cost

due to the pipe heat loss, and determining the minimum life-cycle cost heat

loss for an underground trench system are presented.

10
9

.

Acknowledgment

This investigation was conducted under the Tri-Service Building Materials

Investigational Program and was jointly sponsored by the Headquarters, U.S.

Army Corps of Engineers; U.S. Navy, Naval Facilities Engineering Command;

and the U.S. Air Force, Air Force Engineering and Services Center. The

author would like to thank Dr. T. Kusuda of NBS for his assistance and

permission in using the finite element computer simulation program for heat

loss calculations.

10.

References

1. Pan Am World Services, Inc., "Heat Distribution Systems Life Cycle

Cost Analysis - Comparison Between Direct Buried and Shallow Trench

Systems," Report 130319, July 1985.

2. Parsons Corporation, "Life Cycle Cost Analysis - Comparison Between

Direct Buried Conduit and Shallow Concrete Trench Underground Heat

Distribution Systems," February, 1986.

3. Segerlind, L. J., 'Applied Finite Element Analysis,' John Wiley and

Sons, New York, 1976.

4. Zienkiewicz, 0. C. and Morgan, K., 'Finite Elements and

Approximation,' John Wiley and Sons, New York, 1983.

5. Torrance, K. E., "Numerical Methods in Heat Transfer," in 'Handbook of

Heat Transfer Fundamentals,' Second Edition, edited by Rohsenow, W.

M. , Hartnett, J. P. and Ganic, E. N., McGraw-Hill, New York,- 1985.

6. Kusuda, T., "FEMTKE - Finite Element Method Computer Program for

Underground Heat Distribution System Heat Loss Calculations," a report

for Tri-Service Committee on Underground Heat Distribution Systems,

January 1984.

7. Kusuda, T., and Achenbach, P. R., "Earth Temperature and Thermal

Diffusivity at Selected Stations in the United States," ASHRAE

Transactions, Vol. 71, Part I, pp. 61-75, (1965).

8. Eckert, E. R. G. and Drake, R. M., 'Analysis of Heat and Mass

Transfer,' McGraw-Hill, New York, 1972.

9. Holman, J. P., 'Heat Transfer," Fourth Edition, McGraw-Hill, New York,

1976.

10. U.S. Army Corps of Engineers, Guide Specifications for Military

Construction, 'Heat Distribution Systems Outside of Buildings Concrete

Shallow Trench Systems,' CEGS-15709, November 1983.

11. Ruegg, R. T., "Life-Cycle Costing Manual for the Federal Energy

Management Programs," National Bureau of Standards Handbook 135,

December 1980.

12. Means Mechanical Cost Data, 7th Annual Edition, Robert Snow Means Co.,

Kingston, MA, 1984.

11
13. U.S. Department of Energy, 'Economic Thickness of Industrial

Insulation,' The Fairmont Press Inc., Atlanta, GA, 1983.

14. Building Construction Cost Data 1985, 43rd Annual Edition, Robert Snow

Means Co., Kingston, MA, 1985.

1 2
Table 1

Dimensions of Concrete Shallow Trench Systems

for Different Thickness of Pipe Insulation

Pipe

Insulation

Thickness

( inch) A B C D E F G H I

1.0 28 19 40 31 6 6 8 6 11

1.5 30 20 42 32 6 6 8 6 12

2.0 32 20 44 32 6 6 8 6 12

2.5 34 21 46 33 6 6 9 6 12

3.0 36 22 48 34 6 6 9 6 13

3.5 38 23 50 35 6 6 10 6 13

4.0 40 24 52 36 6 6 10 6 14

5.0 44 26 56 38 6 6 11 6 15

6.0 48 28 60 40 6 6 12 6 16

Note

:

1 . Refer to Figure 1 for descript ions of symbols A to I.

2. All dimensions in inches, 1 inch = 2.54 cm

13
Table 2

Calculated Results of Heat Losses from Two 1 Underground Pipes Inside a Shallow Trench

Temperatures of 196 and 99 C (385 and

50 mm (6

for Pip

210 F)

Insulation Heat Loss Rate (Btu/h* ft)

Thickness

( inch) Pine No. 1 Pipe No. 2 Total

1.0 238 94 332

1.5 177 74 251

2.0 144 62 206

2.5 124 54 178

3.0 109 48 157

3.5 98 44 142

4.0 90 40 130

5.0 79 35 114

6.0 70 32 102

Table 3

Calculated Results of Heat Losses from Two 150 mm (6-

Underground Pipes Inside a Shallow Trench for Pipi

Temperatures of 171 and 143 C (340 and 290 F)

Insulat ion Heat Loss Rate (Btu/h« ft)

Thickness

( inch) P ipe No. 1 Pipe No. 2 Total

1.0 199 157 356

1 .5 150 120 270

2.0 122 99 221

2.5 105 85 190

3.0 93 75 168

3.5 84 68 152

4.0 77 63 140

5.0 67 55 122

6.0 60 49 109

14
Ground surface

Figure 1. Concrete Shallow Trench Underground Heat Distribution System

1 5

Two

6-in.

hot

water

pipes

with

calcium

silicate

insulation/aluminum

jacket
I

n

h».

CM

N.

O

r«.

a>

(O

Figure 2. Finite Element Design for the Outer Boundary Earth and Concr< tr

Trench Walls and Cover

1 6

MM

—

M

—

4-

MM
CO

CQ

Figure 3. Finite Element Design for Air Space Surrounding the Pipes in

Concrete Trench

17
F igure 4. Finite Element Design

Surfaces of the Pipes

for the Pipe Insulation and the Outer

18
MATERIAL

OR

INSTALLATION

COST,

$/ft.

0 5

cm

10 15

800

600

400

200

0

0 1 2 3 4 5 6

PIPE INSULATION THICKNESS, in.

Figure 5. Material and Installation Costs of Two 6-in. Pipe Shallow Trench

Systems as a Function of Pipe Insulation Thickness

19

$/m
COST,

$/ft.

OF

SYSTEM

500 -

400 -

300 -

200 -

100 -

0

F igu

cm

5 10 15

1 1 1 1 1 rJ I I I I L

1 2 3 4 5 6

PIPE INSULATION THICKNESS, in

5 6. Life-Cycle Cost Analysis for a Shallow Trench System

- 1500

- 1200

- 900

- 600

- 300

0

20

$/m
COST,

$/ft.

OF

SYSTEM

500

400

300

200

100

0

F igure

W/m

100 200 300 —I

1 1

400

1500

1200

900

600

300

__i I I 1 I 0

100 200 300 400

TOTAL HEAT LOSS, Btu/h ft

7. A Plot of the Life-Cycle Cost Versus the Heat Loss from

Underground Pipes at 340 and 290 Degree F

$/m
COST.

$/ft.

OF

SYSTEM

100

W/m

300 400

500

400

300

200

100

0

F igur

200 —I 1 r

__l I I i_

100 200 300 400

TOTAL HEAT LOSS, Btu/h.ft

8. A Plot of the Life-Cycle Cost Versus the Heat Loss

Underground Pipes at 385 and 210 Degree F

1500

1200

900

600

300

0

from

22

$/m
APPENDIX A. The Input Data Files and the Outputs

from the Computer Program on Sample Calculations

A.l A Listing of DATAl Input File

1 . 1 1 2

9.7

46.0

6.0

385.0 210.0

0.44 15.0

6.625 6.625

3.50 3.50

1.30 1.30

1.385 56.0

10.0 0.417 zp.o

0.50 3.08 1.917

56.0 0.00 0.025

23
A. 2

12

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

A Listing of DATA2 Input File

1 16 2 .06

16 17 2 .06

2 17 18 .06

2 18 3 .06

3 18 19 .06

3 19 20 .06

3 20 4 .06

4 20 21 .06

4 21 6 .06

4 6 5 .06

14 13 12 .06

14 12 29 .06

29 12 28 .06

28 12 11 .06

28 11 27 .06

27 11 26 .06

26 11 10 .06

26 10 25 .06

25 10 8 .06

8 10 9 .06

16 32 17 .06

16 15 32 .06

32 15 31 .06

31 15 30 .06

15 14 30 .06

30 14 29 .06

6 21 22 .06

6 22 7 .06

22 23 7 .06

23 24 7 .06

7 24 8 .06

24 25 8 .06

17 32 18 .06

18 32 34 .06

18 34 35 .06

18 35 19 .06

19 35 20 .06

20 35 36 .06

20 36 22 .06

20 22 21 .06

30 29 28 .06

30 28 48 .06

48 28 47 .06

47 28 27 .06

47 27 26 .06

47 26 46 .06

46 26 24 .06

24 26 25 .06

32 33 34 .06

32 31 33 .06

31 40 33 .06

31 42 40 .06

31 41 42 .06

31 30 41 .06

30 48 41 .06

2A
36

22

37

38

23

23

45

40

42

39

39

33

33

34

35

35

35

36

37

37

37

38

39

56

40

33

33

41

41

42

43

43

43

44

45

45

45

46

47

47

47

41

41

65

66

66

66

66

67

67

68

68

68

78

78

37 22 .06

37 23 .06

38 23 .06

44 23 .06

44 45 .06

45 24 .06

46 24 .06

42 39 .06

43 39 .06

43 44 .06

44 38 .06

49 50 .06

50 34 .06

50 35 .06

50 51 .06

51 52 .06

52 36 .06

52 37 .06

52 53 .06

53 54 .06

54 38 .06

54 39 .06

54 55 .06

39 55 .06

39 56 . .06

40 56 .06

56 49 .06

57 58 .06

58 42 .06

58 43 .06

58 59 .06

59 60 .06

60 44 .06

60 45 .06

60 61 .06

61 62 .06

62 46 .06

62 47 .06

62 63 .06

63 64 .06

64 48 .06

48 64 .06

64 57 .06

80 66 .06

80 1 .06 1 2 .06 2 3 .06 3 67 . 06 3 4 .06 4 68 .06 4 5 .06 5 70 . 06

70 69 .06

77 76 .06

76 13 .06

25
Ill

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

13 76 12 .06

12 76 11 .06

11 76 75 .06

11 75 10 .06

10 75 74 .06

10 74 9 .06

9 74 72 .06

72 74 73 .06

80 16 1 .06

80 79 16 .06

16 79 15 .06

15 79 14 .06

79 78 14 .06

14 78 13 .06

5 6 70 .06

70 6 71 .06

6 7 71 .06

71 7 8 .06

71 8 72 .06

8 9 72 .06

26
i.3 The Outputs from the Computer Program

T1 T2 Kl KG D1 D2

J05.OO 210.00 .44 15.00 6.63 6.63

THl TH2 DPI DP2 S TG

3.50 3.50 1.30 1.30 1.38 56.00

m HH HY MONTH

10.00 .42 20.00 1 W H D A B m HH HY

4.08 2.92 .50 3.08 1.92 10.00 .42 20.00

XCl YCl XC2 YC2

1.347 1.717 2.733 1.717

D1 D2 S TH KI KG T1 T2 Q1 Q2 DT

6. 63 6.63 1 .38 3. 50 .44 15.00 385. 210. 95.32 41.26 136.58 m) ,M= 1 ,NX

.00 .00 .00 .00 .00 1.02 2.04 3.06 4.08 4.08

4.08 4.08 4.08 3.06 2.04 1.02 .50 .50 .50 .50

.50 1.27 2.04 2.81 3.58 3.58 3.58 3.58 3.58 2.81

2.04 1.27 1.35 .95 .78 .95 1.35 1.75 1.92 1.75

2.73 2.33 2. 16 2.33 2.73 3. 13 3.30 3. 13 1.35 1. 15

1.07 1. 15 1.35 1.54 1.62 1.54 2.73 2.54 2.46 2.54

2.73 2.93 3.01 2.93 - 10.00 -10.00 -10.00 -10. 00 -10.00 .00

i 2.04 4.08 14.08 14.08 14.08 14.08 14.08 4.08 2.04 .00

' Y(M) ,M*1 ,NY

' .42 1.15 1.88 2.60 3.33 3.33 3.33 3.33 3.33 2.60

, 1.88 1. 15 . 42 .42 .42 .42 .92 1.40 1.88 2.35

! 2.83 2.83 2.83 2.83 2.83 2.35 1.88 1 . 40 .92 .92

.92 .92 1. 15 1.32 1.72 2. 12 2.28 2. 12 1.72 1.32

1.15 1.32 1.72 2. 12 2.28 2. 12 1.72 1.32 1.44 1.52

' 1.72 1.91 1.99 1.91 1.72 1.52 1.44 1.52 1.72 1.91

1.99 1.91 1.72 1.52 .00 .42 3.33 13.33 23.33 23.33

23.33 23.33 23.33 13.33 3.33 .42 .00 .00 .00 .00

MONTH • • 1 TEMPERATURE ARRAY

5B.37 62.45 64.72 66.07 66. 10 67.21 67.32 66.62 65.21 64.92

63.43 61.33 57.96 60.98 62.00 62.30 66.73 70.92 72.83 72. 16

1^.43 73.86 74. 17 71.70 67.64 69. 14 68.85 67. 17 64.36 69.80

73.82 73.29 75.06 73.41 74.32 75.30 76.77 78. 13 79. 45 76.92

71.34 75.20 78.79 76.59 73.65 71.21 69.56 69. 11 385. 00 385.00

585. 00 385.00 385.00 385.00 385.00 385.00 210.00 210.00 210.00 210.00

JIO.OO 210.00 210.00 210.00 56.00 56.00 56.00 56.00 56.00 56.00

! 56.00 56.00 56.00 56.00 56.00 56.00 56.00 56. 00 56.00 56.00

i AVERAGE TEMPERATURE DROP ACROSS INSULATION

' Tl« 308.83 T2= 136.82 DEG F

i heat losses from underground pipes I

i'qIs 98.67 Q2= 43.71 QT= 142.39 BTU/H-FT

1

1

I

i

I

KP

.566

27
APPENDIX B. A Listing of the Computer Simulation Program

PROGRAM FEUHDS

* THIS IS A PROGRAM FOR ANALYSIS OF UNDERGROUND SYSTEM HEAT LOSS

* BASED ON FINITE ELEMENT METHOD

* SUBROUTINES CALLED: PI PE2 , TWOPI P , TGO , S 01 LK , TGXX , S OLVP , AND PIPEHL

* INPUT DATA FILES: DATAl AND DATA2

* OUTPUT DATA FILE: PLTDTA

* X(I): X-COORDINATE OF NODAL POINT I, IN FT

* Y(I): Y-COORDINATE OF NORAL POINT I, IN FT

* I,J,K NODAL POINTS OF ELEMENT M

* M ELEMENT INDEX

* C THERMAL CONDUCTIVITY, BTU-I N/HR/FT**2 /F

* L THICKNESS OF THE ELEMENT, FT

C WHEN TEMPERATURE ( T( I ), I »MZ , NX ) ARE KNOWN MZ.NE.O.

DIMENSION Q(130),T(130),X(130),Y(130),KK(130,130)

DIMENSION IZ(130) , JZ(130) ,KZ(130) ,AS(130) ,B2IZ(130) ,B3IZ( 130) ,B2JZ

&(130) ,B2KZ(130) ,B3JZ(130) ,B3KZ(130)

DIMENSION CC(130) ,TGX(12,5) ,QQ(130)

COMMON/PP/TPl ,TP2 ,KI ,KG,D1 ,D2 ,TH1 ,TH2 ,DP1 , DP 2 , S , TG , WW , HH , BY

,

& MONTH

REAL L,KK,KI ,KG,KIX,KTCT,KASP

TRTK»0

.

PI-4.*ATAN( 1 .

)

71 CONTINUE

OPEN (8 ,FILE='DATA1'

)

OPEN (6 ,FILE»'PRN'

)

OPEN (9 ,FILE«'DATA2'

)

OPEN dO,FILE«='PLTDTA' , STATUS NEW' ) C READ IN MREPT AND ITREN

READ (8,200 ,ERR-2000) MREPT, ITREN

200 FORMAT (15, IX, 14)

72 MIl-32

MI2=99

IFdTREN.EQ. 0) GO TO 50

MI1«=66

MI2«99

50 CONTINUE

C READ MONTH OF INTEREST AND ICALB

READ (8,250) MONTH, ICALB

250 FORMAT (215)

IFdTREN.NE. 1 ) GO TO 73

C READ THE THERMAL CONDUCTIVITY (IN BTU-I N . /H-FT**2 - DEG F) AND OF

C THICKNESS (IN INCHES) OF CONCRETE WALL FOR SHALLOW TRENCH SYSTEM.

READ (8,260) KTCT,TRTK

260 FORMAT (2F10.4)

C READ EQUIVALENT THERMAL CONDUCTIVITY OF AIR SPACE SURROUNDING THE

C PIPES IN SHALLOW TRENCH, IN BTU-I N . /H-FT** 2 - DEG F

READ (8,265) KASP

265 FORMAT (FlO.5)

73 CONTINUE

CALL PIPE2(X,Y, ITREN, TRTK)

CALL TWOPIPd, ITREN)

NX-=80

NY“80

MZ«49

MX=130

28
IF(ICALB.NE.2) GO TO 1050

WRITE(10,15) (X(I ) ,I«1 ,NX)

15 FORMATC10F7.2)

WRITE(10,15) (Y(I ) ,I«1 ,NY)

1050 IFCICALB.EQ. 0) GO TO 80

WRITE(6 ,20)

20 FORMAT(' X (M ) ,M = 1 , NX'

)

WRITE(6,15) (X(I) ,I«=1 ,NX)

WRITE(6 ,21

)

21 FORMATC' Y(M) ,M=1 ,NY'

)

WRITE(6,15) (Y(I ) ,1=1 ,NY)

80 CONTINUE

CALL TGO(TGX,PI ,Y)

IFCICALB.EQ. 1) WRITE(6,16)

16 FORMATC' M I J X C'

)

L»1 . DO 1080 IREPT=1 ,MREPT

DO 4 1=1, NX

QCI)“0.

DO 4 J=1 ,NY

4 KKCI,J)=0.

DO 5 M=1,MX

IFCIREPT.GT. 1 ) GO TO 1090

HIJ=0.

HJK=0.

HKI=0.

TIJ=0

.

TJK=0.

TKI=0

C READING IN THE ELEMENT NUMBER AND ITS NODAL POINTS AND THERMAL

C CONDUCTIVITY , AND INDEX OF CONVECTION BOUNDARY

READC9,500) N,I ,J,K,C,IXCB

500 FORMAT C 41 5 , F 1 0 . 4 , 1 5

)

C=KG/12.

IFCM.GT.MI1 .AND. M.LT.MI2) C=KI/12.

IFCITREN.EQ. 1 .AND.M.LE.32) C=KTCT/12

.

IFCITREN.EQ. 1 .AND.M.GT.32.AND.M.LT.67)C=KASP/12

.

IZCM)»I

JZCM)=J

kzCm)-k

ccCm)=c

GO TO 1070

1090 I-IZCM)

j-jzCm)

k=kzCm)

c=ccCm)

IFCM.GT.mi1 .AND.M.LT.MI2) GO TO 1070

TM=CtCI )+TC J)+TCK) ) /3

.

CALL S0ILKCTM,CZ)

C«CZ*KG/C1.1*12)

1070 XI=XCl)

XJ=XCJ)

XK=XCK)

yi-yCi

)

yj=yC J)

29
n

a

YK=Y(K)

CXX»=C

CXY-0

.

CYX=0.

CYY=C

B2I-YJ-YK

B3I=XK-XJ

B2J»YK-YI

B3J«=XI-XK

B2K=YI-YJ

B3K=XJ-XI

S“0 .5*(XJ*B2J+XI*B2I+XK*B2K)

S>=ABS(S)

A2=S*2.

AS(M)»A2

B2I=B2I/A2

B3I*=B3I/A2

B2J«=B2J/A2

B3J-'B3J/A2

B2K=B2K/A2

B3K«=B3K/A2

B2IZ(M)»B2I

B3IZ(M)»=B3I

B2JZ(M)=B2J

B3JZ(M)»B3J

B2KZ(M)=B2K

B3KZ(M)-B3K

BII«S*L*(B2I*B2I*CXX+B2I*B3I*CXY+B3I*B2I*CYX+B3I*B3I*CYY)

BIJ-S*L*(B2I*B2J*CXX+B2I*B3J*CXY+B3I*B2J*CYX+B3I^B3J*CYY

)

BIK=S*L*(B2I*B2K*CXX+B2I*B3K*CXY+B3I*B2K*CYX+B3I*B3K*CYY

BJI-S*L*(B2J*B2I*CXX+B2J*B3I*CXY+B3J*B2I*CYX+B3J*B3I*CYY)

BJJ=S*L*(B2J*B2J*CXX+B2J*B3J*CXY+B3J*B2J*CYX+B3J*B3J*CYY)

BJK*=S*L*(B2J*B2K*CXX+B2J*B3K*CXY+B3J*B2K*CYX+B3J*B3K*CYY)

BKI«=S*L*(B2K*B2I*CXX + B2K*B3I*CXY + B3K*B2I*CYX + B3K*B3I*CYY)

BKJ«S*L*(B2K*B2J*CXX+B2K*B3J*CXY+B3K*B2J*CYX+B3K*B3 J*CYY

)

BKK*=S*L*(B2K*B2K*CXX + B2K*B3K*CXY + B3K*B2K*CYX + B3K*B3K*CYY ) KK(I ,I )=KK(I ,I )+BII

KK(I , J)=KK(I , J)+BIJ

KK(I ,K)«=KK(I ,K ) + BIK

KK( J ,I )=KK( J ,I )+BJI

KK( J , J)-KK( J , J)+BJJ

KK( J ,K)«KK( J ,K)+BJK

KK(K,I)=KK(K,I )+BKI

KK(K, J)«KK(K, J)+BKJ

KK(K,K)*KK(K,K)+BKK

C ACCOUNT FOR CONVECTION ON BOUNDARY

IFCIXCB ,EQ. 0) GO TO 800

READING IN CONVECTIVE HEAT TRANSFER COEFFICIENTS AND AMBIENT

TEMPERATURES FOR THREE BOUNDARY SEGMENTS

READ(9 ,700) HIJ ,HJK ,HKI ,TIJ ,TJK ,TKI

700 FORMATC6F10.4)

HHIJ=HIJ*L*SQRT( (X(I)-X( J) )**2+(Y(I )-Y( J ) )**2)/6

.

HHJK=HJK*L*SQRT( (X(J)-X(K))**2+(Y(J)-Y(K))**2)/6.

HHKI-HKI*L*SQRT( (X(K)-X(I))**2+(Y(K)-Y(I))**2)/6.

KK(I ,I )=HHIJ*2.+HHKI*2.+KK(I ,I

)

30
KK(I , J)»HHI J+KK(I , J)

KK(I ,K)=HHKI+KK(I ,K)

KK( J ,I )“HHIJ+KK( J ,I

)

KK( J ,J)«HHIJ*2 + HHJK*2.+KK( J ,J)

KK( J ,K)-HHJK+KK( J ,K)

KK(K ,I )=HHKI+KK(K ,I

)

KK ( K , J

) -HHJK+KK ( K , J

)

KK(K ,K)*=HHJK*2.+HHKI*2.+KK(K ,K)

HHIJ»TIJ*3 .*HHI

J

HHJK=TJK*3

HHKI=TKI*3 ,*HHKI

Q(I ) »Q(I )+HHIJ+HHKI

Q( J)»Q(J)+HHIJ+HHJK

Q(K)«Q(K)+HHJK+HHKI

800 IFCICALB.EQ. 1) WRITE(6,9) M,I,J,K,C

IFCICALB .EQ. 2) WRITE(10,9)

9 F0RMAT(4I5 ,F10.4)

5 CONTINUE

MZZ=MZ-1

MZ8=MZ+7

DO 900 I-MZ,MZ8

T(I )-TPl

II«=I + 8

T(II)-TP2

900 CONTINUE

MMZ16*=MZ+16

DO 1080 MJ-MONTH, MONTH

CALL TGXX(T ,TGX,MJ)

DO 77 I»1,MZZ

SUM«=0 . DO 78 J»MZ,NX

78 SUM«SUM+KK(I , J)*T(J)

77 QQ(I)«Q(I)-SUM

WRITE(6,10) MJ

10 FORMATC ' MONTH 15)

IFCICALB.EQ. 1 ) WRITE(6,6)

6 FORMAT (6X,'QQ ARRAY')

IFCICALB.EQ. 1 ) WRITEC6.7) C QQ C I )

, I =1 , NX

)

7 FORMAT C10F7.2)

NXX*=NX

CALL SOLVPCMZZ,MZ,KK,QQ,T,130)

WRITEC6 , 14)

14 FORMATC' TEMPERATURE ARRAY')

WRITEC6,7) CTCI ) ,1=1 ,NX)

IFCICALB.EQ. 2) WRITEC10,7) C T C I )

, I - 1 , NX

)

Rl=Dl/24.

R2=D2/24.

TH1X=TH1/12.

TH2X»TH2/12

.

IS1=49

IS2-57

KIX=KI/12.

CALL PIPEHLCT.RI , R2 , THlX , TH2X , I S 1 , I S 2 , K IX , 8 , QTX

)

1080 CONTINUE

GOTO 2010

31
2000 WRITE (6,2005)

2005 FORMAT (IX, 'THERE ARE SOME ERRORS IN DATA')

2010 STOP

END

SUBROUTINE TGOCTGX ,PI ,Y ) * THIS SUBROUTINE CALCULATES THE UNDISTURBED EARTH TEMPERATURES

* AT VARIOUS DEPTHS

DIMENSION TGX(12,5),Y(1)

* READING IN THE ANUAL AVERAGE TEMPERATURE AND AMPLITUDE OF THE

* MONTHLY NORMAL TEMPERATURE CYCLE OF THE SITE, IN DEG F, AND

* THERMAL DIFFUSIVITY OF SOIL, IN FT**2/H.

READ (8,300) AO,BO,DIFF

300 FORMAT (3F10.4)

W-2. PI/12.

WZ-2.*PI/(8760*DIFF*2)

ZZ-SQRT(WZ)

DO 1 1-1,12

DO 1 J»l,5

Z-ZZ*Y(64+J)

1 TGXd ,J)-AO + BO*EXP(-Z)*SIN(W*(I-3)-Z)

RETURN

END

32
no

o

o

CIO

no

o

to

n

o

SUBROUTINE PIPE2 (X,Y ,ITREN ,TRTK)

* THIS SUBROUTINE GENERATES X AND Y -COORDINATES OF NODAL POINTS FOR

* THE TWO PIPE SYSTEM

REAL XII ,KIG,KI ,KG

DIMENSION X(1),Y(1)

COMMON /PP/Tl ,T2 ,KII ,KIG,PI1 ,PI 2 , THI 1 , THI 2 , B 1

, B 2 , S , TG

,

& WW,HH,HY, MONTH

C READ TEMPERATURE OF PIPES 1 AND 2, IN DEG F

READ (8,300) T1,T2

300 FORMAT (2F10.3)

READ THERMAL CONDUCTIVITY OF THERMAL INSULATION AND SOIL,

RESPECTIVELY, IN BTU-IN . /H-FT**2 - DEG F

READ (8,310) KII,KIG

10 FORMAT (2F10.4)

READING IN THE OUTSIDE DIAMETERS OF STEEL PIPES 1 AND 2, IN INCHES

READ (8,310) PI1,PI2

READING IN THE THICKNESS OF THERMAL INSULATION USED FOR PIPES 1 AND 2, RESPECTIVELY, IN INCHES

READ (8,310) THI1,THI2

READ THE DEPTHS FROM GROUND SURFACE TO THE CENTERS OF PIPES 1 AND

2, RESPECTIVELY, IN FT.

READ (8,310) B1,B2

READING IN THE SEPARATION DISTANCE (IN FT.) OF THE PIPE CENTERS,

AND THE AVERAGE EARTH TEMPERATURE, IN DEG F.

READ (8,310) S,TG

READ IN THE WIDTH AND DEPTHS OF EARTH REGIONS SURROUNDING THE

UNDERGROUND SYSTEM, IN FT.

READ (8,315) WW,HH,HY

315 FORMAT (3F10.4)

WRITE(6 ,44) T1 ,T2 ,KII ,KIG,PI1 ,PI2

44 FORMAT(' T1 T2 KI KG D1 D2'/6F7.2)

WRITE(6,55) THIl ,THI2 ,B1 ,B2 ,S,TG

55 FORMAT(' THl TH2 DPI DP2 S TG'/6F7.2)

WRITE(6 ,66) WW,HH,HY, MONTH

66 FORMAT (' WW HH HY MONTH '/3F7.2,I7)

C READ IN THE DEPTH OF EARTH COVER, THE WIDTH AND HEIGHT OF TRENCH

C OR LOOSE-FILL.

READ (8,315) D,A,B

W»2*A

H-2 .*B+D

Pl-PIl/12 . R1«P1*0 .

5

P2»PI2/12 . R2»P2*0 . KI'KII/12 . KG=KIG/12 . IFdTREN.NE. 1 ) GO TO 67

W=A+2*TRTK/12

H=B+2*TRTK/12

67 WRITE(6,22) W , H , D , A , B , WW , HH , HY

B1=B1+HH

B2«B2+HH

22 FORMAT(' W H D A B WW HH HY

&'/ ,8F7 .2)

PI-4.*ATAN( 1 .

)

33
THl-THIl/12

.

TH2=THI2/12

DO 1 1-1,4

X(I )»0

.

Y(I )/4.+HH

1 4-1+4

I8-I+8

I12-I+12

I16-I+16

120 =1+20

I24-I+24

I28-I+28

Il-I-l

X(l4)-W*Il/4.

Y(I4)-H+HH

X(I8)=W

Y(l8)-H-H*Il/4.+HH

X(ll2)-W-Il*W/4.

Y(I12)»HH

X(I16)-(W-A)*0,5

Y(ll6)-D+Il*B/4.+HH

X(I20)“0.5*(W-A)+Il*A/4.

Y(I20)-D+B+HH

X(I24)»(W+A)*0.5

Y(l24)-(D+B)-Il*B/4.+HH

X(I28)-0.5*(W+A)-Il*A/4.

Y(I28)=D+HH

1 CONTINUE

XC1-W*0.5-S*0.5

YCl-Bl

XC2-0.5*(W+S)

YC2-B2

WRITE(6 ,77 )XC1 ,YC1 ,XC2 ,YC2

77 FORMAK' XCl YCl XC2 YC2'/4F7.3)

DO 2 1-1,8

Q-2.*PI*(I-l)/8.

Il-I

K1-32+I1

K2-40+I1

K3-48+I1

K4-56+I1

X(K3)-XC1-0.5*P1*SIN(Q)

Y(K3)-YC1-P1*C0S(Q)*0,5

X(K4)-XC2-0 . 5*P2*SIN(Q)

Y(K4)-YC2-0 . 5*P2*COS (Q)

X(K1 )«XC1-(TH1 + 0.5*P1 )*SIN(Q)

Y(Kl)-YCl-(THl+0. 5*P1 )*COS (Q)

X(K2)=XC2-(TH2+0 .5*P2 )*SIN(Q)

Y(K2)-YC2-(TH2+0 . 5*P2)*C0S(Q)

2 CONTINUE

X(65)--WW

X(77 )-W+WW

DO 3 1-1,4

II-65+I

JJ-77-I

34
X(II )«=X(65)

X(JJ)«X(77)

3 CONTINUE

X(70)-X(5)

X(80)=X(70)

X(71)-=X(7)

X(72)»X(9)

X(78)«X(72)

X(79)=X(71)

DO 5 1=77,80

5 Y(I)-0.

Y(65)=0.

Y(66)»Y(1

)

Y(76)-Y(66)

DO 6 1-69,73

6 Y(I)»HH+H+HY

Y(67)=Y(5)

Y(75)-Y(67)

Y(68)=(Y(5)+Y(70))*0.5

Y(74)-Y(68)

RETURN

END

35
SUBROUTINE TWOPIP (I KEPT , ITREN

)

* THIS SUBROUTINE DETERMINES TWO PIPE HEAT LOSS TO UNDERGROUND

COMMON /PP/Tl ,T2,ZKI ,ZKSI ,DA1 ,DA2 ,THI1 ,THI2 ,D1 ,D2 ,AX,TG

& ,WW,HH,HY, MONTH

PI-4.*ATAN(1.

)

Xl=2 .*PI

Rl=DAl/24.

R2-DA2/24.

TH1X*=THI1/12 . TH2»THI2/12

.

ZK1-=ZKI/12.

ZR2>»ZK1

D1=D1+HH

D2-D2+HH

ZKS=ZKSI/12.

WRITE(6 ,6)

DO 5 I-1,IREPT

2 TH1-=TH1X + 0.1*(I-1)

TH2-TH1

A-R1+R2+TH1+TH2+0.05

THI1-TH1*12

.

IF(ITREN.EQ.l) A«AX

C1«X1*ZK1/L0G( (Rl+THl )/Rl

)

C2-Xl*ZK2/LOG( (R2+TH2)/R2)

PI 1»1 .+C1/ (Xl*ZKS)*LOG( (2*D1 )/ (Rl+THl )

)

P12-C2/(X1*ZKS)*L0G( ( A*A+ ( D 1 +D2 ) **2 ) / ( A+A+ ( D

1 -D2 ) *+2 )

) *0 .

5

P21-C1/(X1*ZKS)*L0G( (A+A+(D1+D2)**2)/ (A+A+(Dl-D2)++2) )*0 .

5

P22-1 .+C2/(X1*ZKS)*L0G( (2+D2)/ (R2+TH2)

)

DEL«P12*P21-P11*P22

ZKP1»C1*(P12-P22)/DEL

ZKP2»C2*(P21-P11 )/DEL

TP1»(P12*T2-P22*T1)/ (P12-P22)

TP2»(P21*T1-P11*T2)/(P21-P11

)

Q1-=ZKP1*(TP1-TG)

Q2«ZKP2*(TP2-TG)

QT«=Q1+Q2

TM=(T1+T2)*0 .

5

ZK=(QT)/ (TM-TG)

WRITEC6 ,3) DAI ,DA2 ,A,THI1 ,ZKI ,ZKSI , T 1 , T2 , Q 1 , Q2 , QT , ZK

3 FORMAT(6F6.2,2F6.0,3F7.2,F6.3)

6 FORMATC' D1 D2 S TH KI KG Tl T2 Ql

& Q2 QT KP')

5 CONTINUE

RETURN

END

36
STJBROUTIHE SOILK(T,ZK)

* THIS ROUTINE DETERMINES THERMAL CONDUCTIVITY OF SOIL

* AS A FUNCTION OF TEMPERATURE

REAL E(14)

DIMENSION TZ(14)

DATA K/ 1.1, 1.1, 1.1, 1.0, 0.4, 0.31, 0.25, 0.19, 0.15, 0.11, 0.09, 0.07,

& 0.05,0.05/

DO 1 1-1,14

1 TX(I)-50.+(I-l)*25.

IF(T.GT.TX(1 )) GO TO 7

ZK-1 .

1

GO TO 5

7 IF(T.LT.TX(14) ) GO TO 6

ZK-0.05

GO TO 5

6 DO 2 1-1,13

Tl-T-TX(I)

IF(Tl.NE.O) GO TO 3

ZK-K(I)

GO TO 5

3 CONTINUE

T2-T-TX(I+1)

IF(T2.NE.O.) GO TO 4

ZK-K(I+1

)

GO TO 5

4 CONTINUE

P-T1*T2

IF(P.GT.O) GO TO 2

ZK-K(I+l)+T2*(X(I+l)-X(l))/25.

GO TO 5

2 CONTINUE

5 RETURN

END

SUBROUTINE TGXX (T , TGX , MONTH

)

* THIS SUBROUTINE PROVIDES OUTER BOUNDARY TEMPERATURES OF EARTH REGION

DIMENSION T(l) ,TGX(12,5)

T(65)-TGX(MONTH,l)

DO 1 1-77,80

T(I)-T(65)

1 CONTINUE

T(66)-TGX(MONTH,2)

T(76)-T(66)

T(67)-TGX(MONTH,3)

T(75)-T(67)

T(68)-TGX(MONTH,4)

T(74)-T(68)

T(69)-TGX(MONTH,5)

DO 2 1-70,73

2 T(I)-T(69)

RETURN

END

37
SUBROUTINE SOLVE (M , N , C ,D ,X , I)

* THIS SUBROUTINE SOLVES THE SIMULTANEOUS EQUATIONS

DIMENSION A(100,101),C(I,1),D(1),X(1)

DO 10 IX=1,M

DO 10 IY=1,M

10 A(IX,IY)«C(IX,IY)

DO 20 12 = 1,

M

20 A(I2,N)=D(I2)

L = 1 30 AA=A(L,L)

DO 40 K=L,N

40 A(L,K)=A(L,K)/AA

DO 60 K=1,M

IF(K.EQ.L) GO TO 60

AA«-A(K,L)

DO 50 IA=L,N

50 A(K,IA)-A(K,IA)+AA*A(L,IA)

60 CONTINUE

L»L + 1 IF(L.LE.M) GO TO 30

DO 70 IP-1,

M

70 X(IP)-A(IP,N)

RETURN

END

38
SUBROUTINE PIPEHL ( T , Rl , R2 , THl ,TH2 ,IS1 ,IS2 ,ZKS ,N1 ,QT)

* THIS SUBROUTINE CALCULATES THE AVERAGE TEMPERATURE DROPS ACROSS THE

* PIPE INSULATIONS AND THE RATES OF HEAT LOSS FROM THE UNDERGROUND

* PIPES IN TRENCH SYSTEM

DIMENSION T(l)

PI»4.*ATAN(1 .

)

SUMl-O

.

SUM2=0

DO 1 I*=1,N1

K1=IS1+(I-1

)

K2=IS2+(I-1

K3=K1-16

K4»K2-16

SDM1»SUM1+T(K1)-T(K3)

SUM2«SUM2+T(K2)-T(K4)

1 CONTINUE

T1»SUM1/N1

T2=SUM2/N1

WRITE(6,3) T1,T2

3 formatC//' average temperature drop across insulation'

& //' Tl- ',F10.2,' T2= ',F10.2,' DEG F')

Q1=ZKS*2 .*PI*Tl/LOG( (Rl+THl )/Rl)

Q2-ZKS*2 .*PI*T2/L0G( (R2+TH2) /R2)

QT-Q1+Q2

WRITE(6,2) Q1,Q2,QT

2 FORMAT(3X,'HEAT LOSSES FROM UNDERGROUND PIPES Q1*=',F10.2,

& ' Q2-',F10.2,' QT=',F10.2,' BTU/H-FT')

RETURN

END

39
NBS'lUA (REV. 2-80 ) U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Dace BIBLIOGRAPHIC DATA

SHEET (See instruct/ons)

REPORT NO.

86/3381 MAY 1986

4. TITLE AND SUBTITLE

Minimum Life Cycle Cost Heat Losses for Shallow Trench

Underground Heat Distribution Systems

5. AUTHOR(S)

Jin B. Fang

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8 . Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State. ZIP)

Headquarters, U.S. Army Corps of Engineers, Washington, DC 20314-2300

U.S. Navy, Naval Facilities Engineers Command, Alexandria, VA 22322-2300

U.S. Air Force, Air Force Engineering and Services Center, Tyndall Air Force Base,

FL 32403-6001

10. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The rates of heat loss from two underground insulated pipes installed in a shallow

trench were calculated using a computer program developed based on the application

of the finite element method to solution of two-dimensional steady heat conduction

problems. The calculated results of pipe heat loss under a specified groud temperature condition are summarized for a range of pipe insulation thickness, different sizes of

shallow trench, and various pipe fluid temperatures. Methods of determining the

minimum life—cycle cost heat loss and the corresponding economic insulation thickness

for shallow trench heat distribution systems are presented. Life-cycle costing analysis was performed for two insulated pipes in a concrete trench to determine

the cost of construction, annual energy cost associated with pipe heat loss, and

yearly operating and maintenance costs. Based on this economic analysis, the least

lifs~cycle cost heat loss and the optimum insulation thickness were determined for

specified fluid temperatures in the heat supply and the return lines.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capi tali ze only proper names; and separate key words by semicolons.

finite element method; fuel energy cost; heat loss; life-cycle cost analysis;

shallow trench; underground heat distribution system

13. AVAILABILITY

I yI Unlimited

I I For Official Distribution. Do Not Release to NTIS

I I Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

Iy 1 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PACES

45

15. Price$9 . 9 ‘,