AlllOb 034355

NBSIR 85-3201

Validation Test of An Earth

Contact Heat Transfer Algorithm

Reference

NBS

PUBLICATIONS

George N. Walton

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Building Physics Division

Gaithersburg, MD 20899

October 1985

Prepared for: Solar Buildings Technology Division

u-qq of Solar Heat Technology

100

• U56

85-3201

1985

apartment of Energy

gton, DC 20585

V

NBSIR 85-3201

VALIDATION TESTS OF AN EARTH

CONTACT HEAT TRANSFER ALGORITHM

NATIONAL BUREAU OF STANDARDS

LIBRARY

LPC

George N. Walton

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Building Physics Division

Gaithersburg, MD 20899

October 1985

Prepared for: Solar Buildings Technology Division

Office of Solar Heat Technology

U.S. Department of Energy

Washington, DC 20585

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
.
ABSTRACT

Experimental tests and numerical calculations are performed to determine

the suitability of including a simplified earth contact heat transfer

algorithm in building energy analysis computer simulations. Reasonable

agreement is shown between the finite difference test program and the

simplified method. There is very good agreement between the floor surface

temperature of the NBS Passive Solar Test Facility and the temperature

predicted by the test program. These results indicate that simplifications

based upon the specific configuration can give good results for a simple

calculation of the annual earth contact heat flux. There may be

considerable differences in the hourly and daily heat flux values when the

actual weather departs from the annual harmonic approximation of the simple

method. A procedure is developed for using two-dimensional (2-D)

simulation to closely approximate the full 3-D effects of heat transfer

from rectangular basements and slabs. The 2-D method can be applied to

other numerical procedures for which use a 2-D method for computing earth

contact heat flux. It is recommended that any a Igor ithm us ed in a large

building energy analysis computer program include the capability of

modeling the specific geometry under consideration instead of (or in

addition to) using tables of coefficients which have been generated for a

manual method. The algorithm should also include the ability to handle the

same weather data that is used in the energy analysis.

1
TABLE OF CONTENTS

Page

1. Introduction 1

2. Methods 2

2.1 The Mitalas Ground Heat Transfer Algorithm 2

2.2 The Finite Difference Test Program 4

2.3 The Passive Solar Test Facility 5

3. Results and Discussion 7

3.1 Comparison to Mitalas 7

3.2 Comparison to Measured Data 7

3.3 Comparison of 2-D and 3-D Models 9

4. Summary and Recommendations 12

5. Acknowledgments 12

6. References 13

ii
LIST OF TABLES

Page

Table 1. Comparison of Mitalas (S, V, , t, C) and computer

(S, V, t) basement heat transfer factors

Table 2. Comparison of 3-D and 2-D calculations for a basement

configuration, heat flux values in Btu/hr ft^ (x 3.15

gives W/n/)

Table 3. Comparison of 3-D and 2-D calculations for a slab

configuration, heat flux values in Btu/hrft^ (x 3.15

gives W/m2

)

15

16

17

in
LIST OF FIGURES

Page

Figure 1. The Mitalas Basement Model 18

Figure 2. A sample input for the 2-D finite difference

conduction computer program 19

Figure 3. A sample graphic output of the steady state

temperature field 20

Figure 4. Floor plan of the NBS Passive Solar Test Facility

(Ground heat transfer tests are in cell 3.) 21

Figure 5. Longitudinal section of Cell #3 22

Figure 6. Floor surface temperature sensor locations in Cell 3 . 23

Figure 7. Floor surface temperatures near the south wall 24

Figure 8. Floor surface temperatures near the north wall ...... 25

Figure 9. Basic ground configuration for 2-D and 3-D

conduction tests 26

Figure 10. 2-D approximation of a rectangular basement or slab .. 27

IV
1. INTRODUCTION

Earth contact heat transfer for basement and slab-on-grade floors is a major component of the heat balance in many buildings, particularly

residences, and therefore significantly affects the heating and cooling

energy consumption. This earth contact heat transfer is particularly

important in mild climate zones. Most exiting building energy analysis

programs estimate this earth contact heat transfer in very crude manners

[1, 2, 3] even though they have very detailed calculations for many of the

other heat transfer phenomena in buildings.

Detailed analysis of ground heat transfer is very difficult. It requires

two- and in some case three-dimensional heat transfer analysis which,

although not conceptually difficult, usually requires very time consuming

calculations. The time scale of ground heat transfer is much longer than

all other building phenomena, and it is difficult to combine both long and

short time scale phenomena efficiently. The greatest difficulties occur

with the ground thermal properties which are not constant and are often

unknown at a particular site. Moisture is the primary factor in changing

soil properties. It is dependent on local conditions of rainfall,

drainage, and groundwater level, and it has a profound effect during

freezing and thawing cycles.

Several approaches have been tried in developing tools for estimating earth

contact heat transfer. A mathematically analytic method is used in

reference [4] which solves a slab-on-grade configuration with constant

thermal properties. The highly simplified geometry is characteristic of

most analytic methods. Two-dimensional finite element or finite difference

conduction calculations are often employed [5, 6, 7]. These numerical

methods allow, but do not always use, more complex configurations and

variable thermal properties. Another approach is to establish correlations

based on purely experimental measurements of the ground heat flux as is

done in [8] for basements. In each of the above cases, the method is used

to develop a simplified approach (each of which is different from the

others) allowing manual calculation of annual and/or peak heat loss from

basements or slabs.

A recently published report by Mitalas [9] describes another technique for

estimating earth contact heat transfer. It employes finite element

calculations to establish sets of "conduction shape factors" for various

earth contact configurations. Three dimensional effects are accounted for

by "corner factors". Heat fluxes are computed monthly for the several

sections of the earth contact surface. The final simplified calculation

method provides enough detail in the time and spatial description of the

heat transfer to be included as a subroutine in one of the more detailed

building energy analysis programs. Although not all configurations are, or can be, included in a set of tables, the procedure exists for developing

building specific shape factors.

This report will discuss tests of the Mitalas method using both numerical

simulations and experimental data from a building with a slab-on-grade

floor.

1
2 . METHODS

2.1 THE MITALAS GROUND HEAT TRANSFER ALGORITHM

The Mitalas method is based on a finite element calculation which

determines the conduction shape factors from several segments of a building's earth contact surface. These shape factors are used for

computing earth contact heat transfer in much the same way as the U-value

is used in computing one-dimensional heat transfer through walls. Figure 1 shows the six elements involved in heat loss from a basement: the outside

environment, the basement wall and floor below grade, the ground surface,

the lower thermal boundary, and the conducting soil between the basement,

the ground surface, and the lower thermal boundary. The model assumes that

the lower thermal boundary of the soil is at a constant temperature equal

to the mean ground temperature and that the two vertical boundaries are

adiabatic

The walls are divided into five segments:

A1 = inside surface area of the basement wall above grade,

A2 = upper inside surface strip (0.6 m wide) of the wall below grade,

A3 = lower inside surface area of the wall below grade,

A4 = inside surface area of a floor strip 1 m wide adjacent to the

wall,

A5 = inside surface area of the remainder of the floor.

The wall below grade and the floor are both divided into two segments

because calculations show significant variations in the heat flux over the

entire basement wall and floor. A slab-on-grade configuration would

consist of segments Al, A4, and A5.

The total instantaneous heat loss from the basement is equal to the sum of

the heat losses through all segments. The heat loss in each segment is

given by the area of the segment times the instantaneous heat flux. The

instantaneous heat flux in segment n, Qn

(t), is approximated by

1n (t) = <U,n + Vo ‘ 8in< £) (1)

where q a n

= annual mean value of qn

(t),

qVjD = amplitude of the first harmonic of the heat flux variation,

a) = angular velocity of the first harmonic,

t - time.

The two components of qn

(t) in eq (1) can be expressed as

<la,n (l:) = Sn

' (0B

and

" V

Vn(t) = Vn

' Qn' 9v

‘ 8 in < t+ Atn

)

)

( 2 )

(3)

2
where S Q

= shape factor for the steady-state heat loss component,

0g = basement air temperature,

0q = ground surface temperature averaged over time and area,

V

Q

= shape factor for the periodic heat loss,

a n

= amplitude attenuation factor,

0 y =amplitude of the first harmonic of the ground surface

temperature

,

Atn

= time lag of the heat flux relative to the surface temperature.

The shape factor, S n

, represents the overall conductance between the

basement interior surface segment n (including the surface heat transfer

coefficient) and the two boundaries: the ground surface and the deep

ground plane. That is, S n

= S n g + S n , where S n g is conductance to the

ground surface and Sn

is conductance to the hypothetical lower boundary

plane which is at mean ground temperature. The shape factor, V , represents the overall conductance for periodic heat flow between the

basement interior surface segment n and the ground surface. Therefore, V

Q

equals S n g. Note that the shape factors differ from a conventional shape

factor which is dependent only on geometry and is independent of thermal

conductivity.

Basement insulation is handled by modifying the shape factors according to

Sn(R) = l/(an

+ bn

* R) (4)

V

n

(R) = l/(cn

dn

* R) (5)

where S n(R) and Vn(R) = S and V for a basement insulated with a thermal

resistance of R.

an* kn » cn > dn

= constants specific to the basement insulation

configuration

.

The coefficients for Sn and Vn for many basement insulation configurations

and two different soil conductivity values are given in table 2 of [9].

These coefficients are based on interior surface heat flux profiles

calculated by finite-element numerical methods for heat conduction. The

attentuation factor, a n , and the time-lag factor, Atn

, are based on a sine

wave variation of the ground surface temperature. Experimental studies

have indicated that the variation in basement heat loss can be adequately

described by monthly mean values of the basement heat loss. Therefore, the

Mitalas method uses a time increment of one month with an angular velocity

of 30 degrees per month. In order to account for the three-dimensional

nature of the basement heat loss, a set of corner factors, Cn

, was derived

based on two levels of basement insulation.

Several factors cannot be taken directly into account by this model:

~ the time variation in temperature and level of groundwater,

3
- the flow of rain or into the soil,

- the spatial variation of ground surface temperature around a basement due

to solar effects, adjacent buildings, ground cover, and snowcover,

- the difference in thermal properties of backfill and undisturbed soil and

of soil above and below the freezing plane.

When the permeability of the soil as a porous medium saturated with ground

water is high enough, in addition to the thermal conduction estimates one has to take into account the contribution due to thermal convection.

Poulikakos and Bejan [10] indicate that convection is not negligible when

Ra^' ^ >> 1, where Ra^ is the Darcy-modified Rayleigh number and X is the

horizontal length scale of the building. This relation serves as one

constraint on the validity of the pure conduction models mentioned in this

paper

2.2 THE FINITE DIFFERENCE TEST PROGRAM

In order to test geometries and soil properties not explicitly tabulated by

Mitalas, a simple transient heat conduction analysis program was written.

This program uses the lumped parameter method because the simple

rectangular geometries of basement and slab-on-grade configurations do not

require the greater geometric power of the more complex finite element

method. Versions of the program were developed for two-dimensional

rectangular and cylindrical coordinates and three-dimensional rectangular

coordinates. Steady state conduction is solved by Gauss-Seidel iteration.

Transient conduction is solved by the standard explicit method. This was chosen instead of an implicit method because of ease of programming and the

minimum grid dimensions give a stable solution with time steps no shorter

than one-half hour. The experimental comparison data uses a one hour time

step, so the simulation cannot use a longer one.

The program allows variable spacing in the finite difference grid which

improves accuracy by allowing more detail where the heat flux lines diverge

rapidly. Each grid element may be of a different material to describe

complex material geometries within the rectangular gird. This freedom of

description is simplified by a semi-graphic input as shown in figure 2.

This sample input includes the width and height of each grid element, the

names (W, C, G, D) and thermal properties of each material, convection

coefficients and temperatures, the arrangement of the materials in the

grid, the locations where conductive fluxes are computed, and the control

values for the output sketches. A sample output page is shown in figure 3 where a simple graphic output of the steady state temperature field allows

the user to determine if the results are reasonable. Other output gives

the values for conductive heat flux across each of the previously defined

areas

In addition to a simple steady test which produced exact results, the

program was tested against a known solution [11] for a semi-infinite solid

with periodic boundary conditions which closely resembles heat transfer

into undisturbed earth. When the surface of the semi-infinite region has a prescribed periodic temperature of the form

T

o = Tom

cos(2irt/t0

) ( 6 )
where t = time,

t = period for one full cycle,

Tom = one half the total amplitude of the temperature swing.

The resulting temperature as a function of time and depth, x, into the

region is given by

T = Tomexp (-x/rr/at

0

) cos (2^t/t Q - x'/ n/at

0

) (7)

and the heat flux at the surface at time t is given by

Q

= kTom cos(2^t/t0

-Tr/4) (8)

where k = thermal conductivity in the region,

a = thermal diffusivity.

Using minimum grid spacings of 0.3 m (1.0 ft) and 0.15 m (0.5 ft), the

error in peak heat flux at the surface was found to be 2.9 percent and the error in the phase shift of that peak was 2.2 percent. The uncertainty in

knowledge of actual soil thermal properties would usually lead to larger

errors

2.3 THE PASSIVE SOLAR TEST FACILITY

In 1980, the Department of Energy established a Program Area Plan for

systematic performance evaluation of passive/hybrid heating and cooling

systems [12]. This plan defines three levels of performance monitoring

procedures which have been designated, in decreasing order of

sophistication, as Class A, B, and C. Class C monitoring involves simple

hand-recorded measurements and surveys of occupant reactions. Class A and

Class B level monitoring involve fixed instrumentation and data acquisition

equipment. Class B monitoring provides limited detailed data (about 20

sensors per building) from occupied buildings for field testing and

statistical evaluation of passive systems and buildings. Class A level

monitoring provides carefully measured detailed data (about 200 sensors per

building) under controlled conditions for use in (1) detailed building

energy analysis and mode 1 /a lgorithm validation, and (2) performance

characterization of various passive subsystems.

The National Bureau of Standards (NBS), under the sponsorship of DoE, has

constructioned a full-scale four-cell passive solar test facility for the

purpose of acquiring and distributing Class A level performance data for

various passive subsystems. The test facility contains several types of

generic passive solar features such as a direct gain system, a collector

storage wall (Trombe wall), and clerestory windows. The building is

described in detail in the Instrumentation and Site Handbook [13]. A brief

description of the building components and instrumentation relative to

analyzing earth contact heat transfer follows.

The NBS Passive Solar Test Facility is located in Gaithersburg, Maryland,

on an open field with no shading from the surroundings. The building is a rectangular, one-story, s lab-on-grade, frame structure with the long axis

5
running east to west. The floor plan is shown in figure 4. There are four

cells of equal floor area and an entry area adjoining cell #1 on the west

end of the building. Each cell has a floor area of 30.1 m^ (324 ft^) an a total volume of 88.4 m^ (3122 ft^).

The floor, a slab-on-grade construction, was designed to be of 102 mm (4

in) of concrete over an equal thickness of gravel on compacted soil.

However, the core samples taken from the slab revealed that the slab

thickness varies from 120 to 150 mm (4.8 to 6.0 in). The core samples also

indicate that the gravel beneath the slab has settled in places, leaving an air gap of up to 30 mm (1.2 in). The measured thermal properties of the

slab are a conductivity of 1.42 W/m°C (0.82 Btu/hr*f t°F), a density of 2240

kg/m^ (140 lb/ft^), and a specific heat of 837 J/k°C (.20 Btu/ lb°F). The

exterior walls of cell 3 are of conventional frame construction with an average U-value of 0.36 W/nr°C (0.6 Btu/ft^*hr°F). The roof has an average

U-value of 0.18 W/m^°C (.031 Btu/f t^*hr°F). The windows have a U-value of

3.52 W/m^°C (.62 Btu/f t^*hr°F). The clerestory was covered with an insulating shutter during the tests. A total of 457 sensors are installed

in and around the building. There are 26 sensors outside the building and

6, 173, 26, and 226 in cells 1 through 4 respectively. There are 20

thermocouples used to monitor the floor surface temperature in cell 3.

Figure 6 schematically shows the location of these sensors. The air

temperature in cell 3 is monitored at two locations with unshielded

thermocouples and averaged. The maximum overall uncertainty in the

temperature measurements is estimated to be _+ 0.5 C.

Two short duration performance monitoring experiments were conducted: the

first for the period of January 16 to February 13, 1984; and the second

from February 17 to March 12, 1984. Results from cells 2 and 4, the

collector storage wall and direct gain cells are reported in [14].

Because of the long phase shifts involved in ground heat transfer, weather

data are also needed for up to one year prior to the actual test period.

This data was obtained from the standard weather data recorded at the

Washington, D.C. airport. Comparison between the local weather data and

the airport data during the test period indicate that the temperature at

the airport averages about 2°C (3°F) warmer than at the Gaithersburg site.

So the prior year temperature data has been modified by this amount before

being used with the ground heat transfer test program.

6
3. RESULTS AND DISCUSSION

3.1 COMPARISON TO MITALAS

A further test of the two-dimensional finite difference heat conduction

program is obtained by using it to compute the conduction shape factors for

two basement configurations which are included in Table 2 of the Mitalas

report [9]. The basement geometry is shown in Figure 1. The upper soil

has a conductivity of 0.8 W/m°K (.46 Btu/hr*ft°F) and the lower soil has a conductivity of 0.9 W/m°K (.52 Btu/hr*f t°F). The soil thermal diffusivity

is 6.5 x 10”^ m^/s (0.025 ft^/hr). A conductivity of 1.73 W/m°K (1.0

Btu/hr , ft°F) and diffusivity of 0.036 is used for the concrete of the

basement to the outside air is important, especially for the top basement

wall segment. The results of this test are shown in Table 1 where the

values reported by Mitalas are compared with those computed by the finite

difference program.

The first basement configuration is uninsulated. The steady-state shape

factors, S, are in good agreement for the second and fifth segments, and

they differ by -12 percent and -43 percent for the third and fourth

segments, respectively. The dynamic shape factors, V, differ by 26

percent, 15 percent, 14 percent, and -40 percent for segments 2 through 5,

respectively. The phase shifts, t, measured in months, agree to within

0.6 month. The computed value of V are based on the reported values of , the amplitude attenuation coefficient, which is an average for all

configurations and is reported with one digit accuracy. This accuracy,

together with the uncertainty in soil diffusivity, indicates a reasonable

agreement of two sets of values. In every case, the correct trend for the

computed values is obtained.

The second configuration includes insulation on the upper basement wall

segment. This insulation significantly reduces the heat loss through the

second segment. The values for S differ by -5 percent, -9 percent, -17

percent, and -22 percent for segments 2 through 5 respectively. The values

of V differ by +19 percent, +28 percent, +20 percent, and +40 percent. The

phase shifts agree to within 0.6 month. Again, the percentage differences

are rather large, but all the same trends appear in the reported and the

computed data.

3.2 COMPARISON TO MEASURED DATA

Cell #3 of the Passive Solar Test Facility provides over 1100 hours of data

for the room air temperature and 20 floor surface temperatures. Figures 7 and 8 show 108 hours of temperature data for the four sensors near the

south and north walls respectively. Note that the two figures are

different implying that the heat transfer from the floor is not

symmetrical. This can reasonably be attributed to differences in

construction details and to the solar heating of the earth and foundation

at the south of the building while the north side is always in shade.

During the entire test period the temperatures on the south side of the

floor average .44°C (.79°F) warmer than those on the north side.

Since cell #3 is located near the center for the building, the heat

conduction is modeled by the two-dimensional planar finite difference

Program. The two sets of thermocouples near the walls provide the

7
temperatures of narrow strips along the walls while the other twelve

thermocouples provide an average temperature for the remainder of the room. The narrow strips comprise 1.3 percent, 1.3 percent, 2.7 percent, and 3.6

percent of the floor area in sequence from the wall to the center of the

room for both north and south sides. The central area accounts for 81

percent of the total floor area. The grid structure of the simulation is

set up to give heat fluxes and temperatures at points corresponding to the

measured area.

The initial computer simulation uses the building thermal properties given

in section 2.2. The soil assumed to have values of .90 W/m°K (.52

Btuhr*ft°K) for conductivity and 6.5 x 10”^ m^/s (0.025 ft^/hr) for thermal

diffusivity. This simulation gives a temperature in the central floor

region which is only 0.08°C (.14°F) higher than the measured values. The

root-mean-square error is 0.20°C (,36°F). This is equivalent to an error

of 0.5 W/m^ (.16 Btu/ft^) in the heat flux to the ground assuming the floor

convection coefficient is 6.13 W/m^ (1.08 Btu/hr'f t^°F). However, the

predicted temperatures in the thin stripes near the walls average .74°C

(1.3°F) too high which corresponds to a heat flux error of 4.4 W/m^ (1.4

Btu/hr’ft^). Combining the central and edge areas gives a total error of

1.2 W/m^ (0.39 Btu/hr'ft^) in the predicted ground heat flux. The average

error in the rate of energy loss from the floor is 37 w (127 Btu/hr).

From the initial simulation, several changes in the modeling assumptions

have been made in order to get the best fit to the measured data. The

higher error in the floor edge temperatures than in the center temperature

means that ground thermal conductivity cannot account for the entire error. Additional heat losses have to be occurring at the edge of the floor. The

simulation is improved by the following assumptions. The thermal

conductivity of the ground is increased to 1.07 W/m°K (0.62 Btu/hr*f t°F).

This value is the average of the values used by Mitalas to represent wet

and dry soil. The lowest segment of the wall is modeled as if it were

solid wood, which represents the wall's base plate, instead of using*

average thermal properties as in the initial simulation. The insulation on the outside edge of the slab is assumed to have degraded about 50 percent

in R-value due to exposure to moisture and poor contact with the wall. And

instead of modeling simple convection from the floor to room air, the model

was revised to include this component plus radiation to the other room

surfaces. These changes should be more accurate models of the heat

transfer in the room. Another factor which could cause increased heat loss

at the slab edge is infiltration through the point where the wall and the

slab meet. This is commonly a significant source of infiltration in homes.

The infiltration effect has not been modeled and is, therefore, included to

some extent in the other two mechanisms for increased edge loss. These

changes produce a simulation where the temperature in the central portion

of the floor is in error by only 0.02°C (0.04°F) over the entire simulation

period and the temperatures of the strips near the walls averaged only

0.23°C (0.43°F) high. These values correspond to a total error of 0.35

W/m^ (0.11 Btu/hr*ft^) in the average heat flux into the ground. The

average error in the rate of energy loss from the entire floor of cell #3

is only 10.6 W (36. Btu/hr).

The area averaged measured temperature of the floor is 17.8°C (64.1°F)

during the entire test period. The average room air temperature is 20.7°C

(69.2°F). Therefore, the total heat flux to the ground is computed to be

8
17.5 W/m2 (5.54 Btu/hr‘ft 2

). This corresponds to a total energy loss rate

of 527 W (1790 Btu/hr). The relative error in heat flux to the ground in

the initial simulation is 7.1 percent and in final simulation it is only

2.0 percent. This very good agreement indicates that shape factors

generated for the specific configuration would give good annual results for

a simple calculation of the heat flux. The difference for hourly or daily

heat flux values could be large because of the simplification of the ground

surface temperature to a simple harmonic function. The effects of brief

periods when the ambient temperature is much above or below the

corresponding harmonic value would not be modeled.

3.3 COMPARISON OF 2-D AND 3-D MODELS

The above comparisons to the Mitalas coefficients and the experimental data

have involved two-dimensional (2-D) planar ground heat transfer. The data

obtained from cell #3 of the Solar Test Facility cannot be used to test

three-dimensional (3-D) effects because the measurements occur near the

center of the building where 3-D effects are minimal. Therefore, a set of

computer simulations described below analyze the importance of 3-D heat

conduction into the ground.

A simple rectangular basement geometry is chosen as the base case. Figure

9 shows the finite difference grid used for the heat conduction simulation

in two dimensions. Similar grids are used in the extension to three

dimensions. The simulation uses variable grid spacing with a minimum grid

spacing of 0.3 m (1.0 ft). There is no heat transfer through the above

ground portion of the wall. The left and right boundaries are abiabatic —

not heat flux crosses them. This is equivalent to having another identical

basement 23.2 m (76 ft) from the basement shown. The deep ground plane is

15.24 m (50 ft) below the ground surface and is maintained at the annual

average ground surface temperature. The ground, including the basement

wall, has uniform thermal properties with a conductivity of 1.56 W/m 2o K

(0.90 Btu/hr*f

t

2o F) and a value of 6.5 x 10~ 2 m

2 /s (0.025 ft 2 /hr) for the

thermal diffusivity. The heat flux into the ground is reported for six

areas on the basement wall and floor. Al, A2, A3 are strips 0.61 m (2.0

ft) wide from the top to the bottom of the wall. A4 and A5 are also 0.61 m

(2.0 ft) wide strips on the floor; A4 is adjacent to the wall; A5 is inside

A4. A6 is the remaining area of the floor — the part in the center.

Table 2 presents the results of a series of 2-D and 3-D steady-state heat

transfer simulations. The heat fluxes are presented for each area, Al

through A6, and the total average heat flux for the entire wall and floor

surface of the basement. Case 1 is the results for the 2-D planar

simulation of 288 nodes as shown in figure 9. In case 2 a uniform grid

size of 0.3 m (1.0 ft) is used instead of the variable grid. The uniform

grid with 2670 nodes gives a total heat flux 0.9 percent greater than case

1. In case 3 several of the grid elements adjacent to the basement wall

and floor are divided to give a size of 0.15 m (0.5 ft). Using smaller

grid elements near the walls gives a total heat flux 2.0 percent greater

than case 1. Most of the difference is in the top wall segment — Al.

This result indicates that small grid elements are useful at the point of

highest heat flux but do not significantly change the results in a low flux

region. However, the change is relatively small and the purpose of the

following simulations is comparative only, so the minimum grid spacing in

the remaining 2-D and 3-D simulations is 0.3 m (1.0 ft).

9
Cases 4-3D, 5-3D, 6-3D, and 7-3D present the results for rectangular

basements using rectangular three-dimensional simulation with more than

6000 nodes. All four basements are 1.83 m (6.0 ft) deep. Case 4-3D is a square basement 8.53 m (28 ft) on a side. Its total heat flux is 50

percent greater than for case 1. Case 5-3D is a rectangular basement 8.53

m (28 ft) by 17.07 m (56 ft) which are reasonable dimensions for a single

family residence. Its total heat flux is 29 percent greater than case 1. Case 6-3D is an elongated rectangular basement 8.53 m (28 ft) wide by 34.14

m (112 ft) long which has a heat flux 15.5 percent greater than case 1. Case 7-3D is a very large square 34.14 m (112 ft) on a side which has a total heat flux 30.7 percent less than case 1. These four cases show both

the large effect of the 3-D nature of earth contact heat conduction and the

importance of the overall size of the basement on the average heat flux.

Note that case 7-3D has heat fluxes in segments A1 through A5 which are

very similar to the 2-D model of case 1, and the main difference occurs in

the center of the floor. This case cannot be accurately modeled by

coefficients which have been generated for a much different basement width

such as the values published in [9].

The problem with 3-D simulations is that the calculations are very time

consuming. A cube shaped region divided into a N grid points in each

direction requires a computation time proportional to N . A 2-D simulation

requires only computation time. The simulations above used N=20 which

is a minimal value for achieving reasonable accuracy. Therefore the 3-D

solution requires about 20 times as much computation time as the 2-D

solution. 3-D simulations require about 20 minutes on the NBS mainframe

computer. There is another way to model heat transfer into a three

dimensional region with the same run time as a 2-D simulation: the 2-D

simulation using cylindrical (rather than rectangular) coordinates. But

cylindrical coordinates can only be applied to an axi-symmetr ic

configuration. Simulations with a 2-D cylindrical coordinate program and

the geometry of case 1 are found to give higher heat fluxes than the 3-D

simulation values. The 2-D planar simulation of case 1 gives lower heat

fluxes than the 3-D simulations for basements of the same width (cases 4,

5, and 6). This suggests the possibility of combining the 2-D planar and

cylindrical simulations in some manner to give results near the 3-D values

with much less computational effort.

The geometric interpretation of the best combination procedure is shown in

figure 10. A basement with a rectangular floor plan, as shown in the upper

figure, is converted to a basement with circular ends, the lower figure.

The circular portion is modeled with cylindrical coordinates and the

straight section with rectangular coordinates. The geometry is set up to

conserve the wall area which is proportional to the perimeter length, P,

and the floor area. A, of the basement. The diameter, a, of the circular

portion is given by

a = (P - /p 2 - 4ttA)/tt (9)

and the length, b, of the straight portion by

b = (p - TT a )/2 (10)

The depth of the basement is unchanged in this transformation. The heat

fluxes from the planar and cylindrical models are combined with proportions

10
2b and a, respectively, to give the total heat flux from the basement.

The idea of a 2-D approximation for 3-D ground heat transfer has been

applied before [15], but in that model the basement is assumed to have a cylindrical shape with the floor area held constant. The 2-D approximation

developed in this report is much more accurate.

Results using the combined 2-D planar and cylindrical model are presented

in Table 2 cases 4-2D, 5-2D, 6-2D, and 7-2D for comparison to the 3-D

results. The error in total heat flux is less than 1.5 percent in all

cases and the heat fluxes in the individual segments are never in error by

more than 2.5 percent. The remaining cases in Table 2 probe the conditions

where the 2-D approximation looses accuracy.

In case 8 the distance from the edge of the basement to the outer adiabatic

boundary is reduced to 4.9 m (16 ft) which causes the 2-D approximation to

over estimate the total average heat flux 1.7 percent. This is not

significantly different from the original value in case 5 so it is

concluded that interbasement distances greater than about 9 m (30 ft) can be handled by the 2-D approximation. In case 9 the distance to the lower

constant temperature surface is reduced to 7.9 m (26 ft) and is found to

have no effect on the accuracy relative to case 5 although the total

average heat flux is increased by 8.5 percent. In case 10 the soil

conductivity is reduced to .90 Wm°K (.52 Btu/hr*ft°F) and is also found to

not change the relative accuracy of the 2-D approximation. Cases 11 and 12

are an H-shaped and an x-shaped basement respectively. In these cases

equations (9) and (10) are applied using the perimeter and area which

results in the equivalent of a long rectangle. This does not reproduce the

primary aspects of the basement geometry, in particular, the degree to

which the center of the floor tends to be a long distance from the ground

surface as is shown by the increasing errors going from segment A1 to A6.

The 2-D approximation for 3-D earth cpntact heat transfer is also

appropriate for rectangular slab-on-grade configurations. The slab

geometry is the same as the basement geometry of figure 9 except the floor

is level with the ground surface. There are no wall segments Al, A2, or

A3. The results of the simulations are presented in Table 3. Case 1-2D is

the result for a 2-D planar simulation. Case 2-3D is a square slab 8.53 m

(28 ft) on a side. Its total heat flux is 50 percent greater than case 1. Case 3-3D is a rectangular slab 8.53 m (28 ft) by 17.07 m (56 ft). Its

total heat flux is 30 percent greater than case 1. Case 4-3D is an elongated rectangular slab 8.53 m (28 ft) wide by 34.14 m (112 ft) long

which has a flux 15 percent greater than case 1. Case 5-3D is a very large

square 34.14 m (112 ft) on a side which has a total heat flux 38 percent

less than case 1. These cases again show the importance of 3-D heat

conduction and overall size on the average heat flux. The 2-D

approximation to the 3-D model gives results that are accurate to within

1.7 percent for the total average flux in all four cases. The greatest

heat flux errors in any given segment occur in the centers of the square

s labs

.

Data (which are not shown) for the dynamic heat transfer during the year

indicate the same accuracy for the 2-D approximation compared to the full

3-D model as is shown for the steady-state results. This includes both the

accuracy of the maximum and minimum annual heat flux values and the phase

shifts for the various segments. Therefore, it is concluded that the 2-D

11
approximation is good for estimating the 3-D heat conduction from

rectangular basement and slab-on-grade configurations. The idea of a 2-D

approximation for the 3-D effects could be applied in the development of

simplified ground heat transfer models or be used directly in detailed

building energy analysis simulations.

4. SUMMARY AND RECOMMENDATIONS

This report has shown reasonable agreement between the finite difference

test program and the Mitalas method. There is very good agreement between

the measured floor surface temperatures of the NBS Passive Solar Test

Facility and the temperatures predicted by the test program. These results

indicate that shape factors generated for the specific configuration can give good results for a simple calculation of the annual earth contact heat

flux. There may be considerable differences in the hourly and daily heat

flux values when the actual weather departs from the annual harmonic

approximation of the simple method.

A procedure was developed for using 2-D simulation to closely approximate

the full 3-D effects of heat transfer from rectangular basements and slabs.

The 2-D method requires much less computer time than a 3-D method. The 2-D

method can be applied to other numerical procedures for which use a 2-D

method for computing earth contact heat flux.

There is a need to incorporate a better ground conduction algorithm into

several building energy analysis programs. Since these programs are

computer simulations, the algorithm should include the capability of

modeling the specific geometry under consideration instead of (or in

addition to) using tables of coefficients which have been generated for a manual method. A ground conduction computer algorithm could also include

the same weather such as Mitalas' single harmonic function for the earth

surface temperature. Interfacing the ground conduction algorithm may still

involve problems with time scale matching or generating a true simultaneous

solution. Such a ground conduction algorithm should be incorporated into

TARP as soon as possible to serve as a demonstration for its use in other

building energy analysis programs.

5 . ACKNOWLEDGMENTS

This work is sponsored by the Solar Buildings Division, Office of Solar

Heat Technology, U.S. Department of Energy, Washington, D.C. 20585, as part

of the passive solar energy experimental systems research program.

The author wishes to express his gratitude to Dr. Bal Mahajan for his work

with the passive solar test facility and assistance in interpreting the

measurements

.

12
6 . REFERENCES

[1] Kusuda, T., "NBSLD, the Computer Program for Heating and Cooling Loads

in Bu i 1 d ing s ," Nat iona 1 Bureau of Standards (U.S.) Building Science

Series 69, July 1976.

[2] Hit 1 1 e , D. C., "BLAST, the Building Loads Analysis and System

Thermodynamics Programs," U.S. Army Construction Engineering Research

Laboratory Report TR-E-153, June 1979.

[3] Walton, G. N., "Thermal Analysis Research Program Reference Manual,"

National Bureau of Standards (U.S.) NBSIR 83-2655, March 1983.

[4] Kusuda, T., Piet, 0., and Bean, J. W., "Annual Variation of Temperature

Field and Heat Transfer Under Heated Ground Surfaces (S lab-on-Grade

Floor Heat Loss Calculations)," National Bureau of Standards (U.S.)

Building Science Series 1565, June 1983.

[5] Wang, F. S., "Mathematical Modeling and Computer Simulation of

Insulation Systems in Below Grade Application," ASHRAE SP 28,

ASHRAE/DoE Conference on Thermal Performance of the Exterior Envelopes

of Buildings. Orlando, Florida, December 1979.

[6] Akridge, J. M., and Poulos, J. F., "The Decremented Average GroundTemperature Method for Predicting the Thermal Performance of

Underground Walls," ASHRAE Transactions. Vol. 86, Pt. 2A, 1983.

[7] Yard, D. C., Morton-Gibson, M., and Mitchell, J. W., "Simplified

Dimensionless Relations for Heat Loss from Basements," ASHRAE

Transactions . Vol. 90, Pt. IB, 1984.

[8] Swinton, M. C. and Platts, R. E., Engineering Method for Estimating

Annual Basement Heat Loads and Insulation Performance," ASHRAE

Transactions . Vol. 87, Pt. 2, 1981.

[9] Mitalas, G., "Calculation of Basement Heat Loss," ASHRAE Transactions , Vol. 89, Pt. IB, 1983.

[10] Poulikakos, D. and Bejan, A., "Penetrative Convection in Porous Medium

Bounded by a Horizontal Wall with Hot and Cold Spots," Intemat iona 1 Journal of Heat and Mass Transfer. Vol. 27, 1984, pp. 1749-1758.

[11] Eckert, E. R.G. and Drake, R. M., Analysis of Heat and Mass Transfer , McGraw-Hill, 1972.

[12] ''Program Area Plan: Performance Evaluation of Passive/Hybrid Solar

Heating and Cooling Systems," Solar Energy Research Institute Report

SERI/PR-721-788, October 1980, Golden, CO.

[13] Mahajan, B. M., "National Bureau of Standards Passive Solar Test

Facility - Instrumentation and Site Handbook," National Bureau of

Standards (U.S.) NBSIR 84-2911, August 1984.

13
[14] Mahajan, B. M., "Short Duration Winter-Time Performance of Different

Passive Solar Systems," National Bureau of Standards (U.S.) NBSIR 84-

2930, August 1984.

[15] Davies, G. R., "Thermal Analysis of Earth Covered Building,"

Proceedings of the Fourth National Passive Solar Conference . Kansas

City, MO, October 3-5, 1979.

14
Configuration

Soil \ ^Concrete

Computed values

R=1 .55

Computed values

Variable Segment

a

II cs n = 3 n = 4 n = 5

s 1.9 0.74 0.42 0.17

V 1.9 0.65 0.24 0.05

<J 0.9 0.7 0.4 0.3

At 0 -1 -2 -4

C 0 1.0 2.6 0.5

S 1.93 0.62 0.33 0.14

V 2.40 0.75 0.28 0.03

At 0 **.5 -2.6 -3.5

S 0.37 0.96 0.41 0.18

V 0.36 0.89 0.25 0.05

<7 0.9 0.7 0.4 0.3

At 0 -1 -2 -4

C 0 1.0 2.6 0.5

S 0.35 0.87 0.34 0.14

V 0.43 1.14 0.30 0.03

At 0 -0.4 -1.6 -3.5

Table 1. Comparison of Mitalas (S, V, a, At> C) and computed (S, V, At)

basement heat transfer factors

15
heat fluxes at the indicated basement surfaces

case W L total Al A2 A3 A4 A5 A6

1-2D 1 — 1.625 4.423 2.963 2.622 1.769 1.049 0.686

2-2D 1

- 1.640 4.432 2.978 2.640 1.778 1.078 0.698

3-2D 1 - 1.658 4.702 2.982 2.642 1.782 1.043 0.685

4-3D 1 1 2.446 4.651 3.282 3.088 2.265 1.361 0.956

4- 2D 2.479 4.716 3.343 3.132 2.266 1.383 0.969

+1.3% +1.4% +1.9% +1.4% +0.0% +1.6% +1.4%

5-3D 1 2 2.095 4.575 3.176 2.933 2.090 1.240 0.814

5-2D 2.120 4.621 3.220 2.966 2.094 1.260 0.831

+1.2% +1.0% +1.4% +1.1% +0.2% +1.6% +2.1%

6-3D 1 4 1.877 4.514 3.091 2.809 1.958 1.157 0.746

6-2D 1.893 4.542 3.118 2.829 1.961 1.170 0.758

+0.9% +0.6% +0.9% +0.7% +0.2% +1.1% +1.6%

7-3D 4 4 1.124 4.447 3.035 2.718 1.848 1.075 0.503

7 -2D 1.138 4.497 3.055 2.733 1.847 1.081 0.515

+1.3% +1.1% +0.7% +0.6 % -0.1% +0.6% +2.4%

8-3D 1 2 2.032 4.555 3.129 2.837 1.977 1.163 0.762

8- 2D 2.066 4.604 3.181 2.886 2.000 1.194 0.785

+1.7% +1.1% +1.7% +1.7% +0.2% +2.7% +3.0%

9-3D 1 2 2.273 4.602 3.243 3.092 2.348 1.466 1.048

9- 2D 2.304 4.648 3.289 3.129 2.356 1.490 1.078

+1.4% +1.0% +1.4% +1.2% +0.3% +1.6% +2.9%

10-3D 1 2 1.277 2.924 1.935 1.785 1.257 0.716 0.468

10-2D 1.293 2.953 1.963 1.807 1.260 0.727 0.478

+ 1.3% +1.0% +1.4% +1.2% +0.2% +1.5% +2.1%

11-3D H 1.688 4.376 2.936 2.616 1.794 1.040 0.651

11-2D 1.782 4.493 3.058 2.751 1.887 1.121 0.725

+5.6% +2.7% +4.2% +5.2% +5.2% +7.8% +11.4%

12-3D X 1.734 4.378 2.949 2.646 1.836 1.062 0.654

12-2D 1.843 4.522 3.092 2.794 1.929 1.149 0.743

+6.3% +3.3% +4.8% +5.6% +5.1% +8.2% +13.6%

Table 2. Comparison of 3-D and 2-D calculations for a basement configuration,

heat flux values in Btu/hrft^ (x 3.15 gives W/m^).

W and H indicate the relative width and length of the basement.

("H" and "X" are non-rectangular shapes.)

lb
heat fluxes at the

indicated slab surfaces

case W L total A4 A5 A6

1-2D 1 - 1.365 3.540 1.663 0.871

2-3D 1 1 2.055 4.099 2.036 1.236

2-2D 2.066 4.088 2.081 1.297

+0.5% -0.3% +2.2% +4.9%

3-3D 1 2 1.780 3.902 1.891 1.040

3-2D 1.809 3.893 1.921 1.081

+1.6% -0.2% +1.6% +3.8%

4-3D 1 4 1.573 3.753 1.791 0.950

4- 2D 1.591 3.747 1.810 0.973

+1.1% +0.2% +1.1% +2.4%

5-3D 4 4 0.848 3.619 1.683 0.557

5-2D 0.863 3.607 1.688 0.575

+1.7% -0.3% +0.3% +3.2%

Table 3. Comparison of 3-D and 2-D calculations for a slab configuration,

heat flux values in Btu/hrft^ (x 3.15 gives W/m^).

17
14m

ISOTHERMAL BOUNOARY AT MEAN GROUND TEMPERATURE

Figure 1 The Mitalas Basement Model
SOLAR TEST FACILITY

38 27 0.5 405 4

$OUT DX=1., 3*2., 4*1.,. 875, 3*. 5, 4*. 25, 5*. 3333, 2*. 5, 3*1.

$DOWN DY=1 .0 ,3*. 5 ,3*. 3333 ,3*. 4167 , .6667 ,2*1,1 .0833 ,3*2

W

0.0959 13.655 .24

C

0.82 140. .20

G

0.834 55. .40

D

0.62 60. .347

4. 1000. 1.46 1.08

67 .7 0.

57 .6 0. 0.

OZZZZZZZZZZZZZZZZWAAAAAAAAAAAAAAAAAAAO

OZZZZZZZZZZZZZZZZWAAAAAAAAAAAAAAAAAAAO

OZZZZZZZZZZZZZZZZWAAAAAAAAAAAAAAAAAAAO

OCCCCCCCCCCCCCCCCCAAAAAAAAAAAAAAAAAAAO

OGGGGGGGGGGGGGGGCCAAAAAAAAAAAAAAAAAAAO

ODDDDDDDDDDDDDDDCCDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDCCDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDCCDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDCCDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDCCCCDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

ODDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDO

} run control input

, 4*2., 3*3., 4*4., 1. $END

.,3*3., 3*4., 4*5. $END

} material name

} thermal properties

} material name

} thermal properties

} material name

} thermal properties

} material name

} thermal properties

} convection values

} room air temp.

} outside air temp.

} graphic input

} of slab/ground

} configuration

$DATA R=3 .2 ,XX=14.0416 ,Y1=1 .5 ,Y2=3 .75 $END > insulation

$DATA R=3 .2 ,YY=3 .75 ,X1=14.0416 ,X2=15 .0415 $END

$DATA F=1 .0 ,YY=1 .5 ,Xl=13 .125 ,X2=13 .7083 $END } report heat fluxe

$DATA F=2 .0 ,YY=1 .5 ,X1=12 .87 5 ,X2=13 .375 $END } on these surfaces

$DATA F=3.0,YY=1.5,X1=12.375,X2=12.875 $END

$DATA F=4.0,YY=1.5,X1=10.875,X2=12.375 $END

$DATA F=5.0,YY=1 .5 ,X1=0 .0 ,X2=10.875 $END

$DATA $END

0. 40. .5 0. 46. .83333 } specifications for

15 105 195 285 00000000 } graphic output

Figure 2. A sample input for the 2-D finite difference conduction computer

program.

19
DAY= 0 TA= 57.60 TZ= 67.70 TG= 57.60

KKKKKKKKKKKKKKKKKKKKKKKKKKKGAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

KKKKKKKKKKKKKKKKKKKKKKKKKKKHAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

JJJJJJJJJJJJJJJJJJJJJJ I I I IHGAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

JJJJJJJJJJJJJJ I I I I I I I I IHHGGFBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

IIIIIIIIIIIIIIIIIIIHHHHGGGFFDCCBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

IIIIIIIIIIIIIIHHHHHHHGGGFFFEDDCCCBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

IIIIIIHHHHHHHHHHHHHGGGGFFFEEDDDCCCBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

HHHHHHHHHHHHHHHGGGGGGFFFFEEEDDDCCCCBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

HHHHHHHHHHHGGGGGGGGGFFFFEEEEDDDCCCCCBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

GGGGGGGGGGGGGGGGGFFFFFFEEEEDDDDDCCCCCBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

GGGGGGGGGGGGGGFFFFFFFFEEEEEDDDDDCCCCCCBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

GGGGGGGGGGGFFFFFFFFFEEEEEEDDDDDDCCCCCCBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

GGGFFFFFFFFFFFFFFFEEEEEEEDDDDDDDCCCCCCCBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAA

FFFFFFFFFFFFFFFFEEEEEEEEDDDDDDDCCCCCCCCBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAA

FFFFFFFFFFFFFEEEEEEEEEEDDDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAA

FFFFFFFFFEEEEEEEEEEEEEDDDDDDDDCCCCCCCCCCBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAA

EEEEEEEEEEEEEEEEEEEEDDDDDDDDDDCCCCCCCCCCBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAA

EEEEEEEEEEEEEEEEEEDDDDDDDDDDDCCCCCCCCCCCBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAA

EEEEEEEEEEEEEEEEDDDDDDDDDDDDCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAA

EEEEEEEEEEEEEDDDDDDDDDDDDDDCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAA

EEEEEEEEDDDDDDDDDDDDDDDDDDCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAA

DDDDDDDDDDDDDDDDDDDDDDDDCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAA

DDDDDDDDDDDDDDDDDDDDDDDCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAA

DDDDDDDDDDDDDDDDDDDDDCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAA

DDDDDDDDDDDDDDDDDDCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAA

DDDDDDDDDDDDDDDCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAA

DDDDDDDDDDCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAA

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAA

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAA

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAA

CCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAA

CCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAA

CCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAA

CCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAA

CCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAA

CCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

BBBBBBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

Figure 3. A sample graphic output of the steady state temperature field.

Each capital letter is a range 1 degree F warmer than the letter

preceeding it (i.e. A = 57 .6F to 58. 6F, B = 58. 6F to 59. 6F, ...)

20
C JC JD • A 4-> *3 aj 0) •4-J u

pmm l/> w t/> to u •*- *T3 •*- V*- L. TD -C L.

-4-» Qj 13 U OJ U 1/3 jC -—» ^3 c w wo T3

-r- u c

OJ -C 3 3

SA 4~> u «> O

L. c U

U •#- L. C7> •i TD o

c o a» 3 C — .

-C Z •*- *»— **-

21

Figure

4.

Floor

plan

of

the

NBS

Passive

Solar

Test

Facility

(Ground

heat

transfer

tests

are

in

cell

3.)
22

Figure

5.

Longitudinal

section

of

Cell

#3
All dimensions in inches

T488•—T489 T490 T49

T480

T492

86

T48

T493

81

T482 T483*0

T494 T495

T

48

-T-«.

48

81 41

Figure 6. Floor surface temperature sensor locations in Cell 3

23
CM

CM

CO

o

<N

CO

o

CO

05

CO

CM

1^

o

CO

CO

CO

CO

CM

CM

0)V43

00 •H

(Oo) 3UfUVH3dlAI31

24

HOUR

AFTER

0.00

A.M.

1-16-84
OJ

(Oo) 3dmva3d^31

00

0)

Vj3

OO

•Eb

25

HOUR

AFTER

0.00

A.M.

1-16-84
50'

Figure 9.

1

' -14' 38'

A A 1 , ^ / O AO.

A 6

| A3;

T -A. . A. 5

Basic ground configuration for 2-D and 3-D conduction tests

26
Same perimeter P

Same area A

a = (P-/p 2 -47rA)/7r

b = (P-7ra)/2

Figure 10. 2-D approximation of a rectangular basement or slab

27

> 0*m NBS-i 14 A (REV 1 1-84)

U.S. OEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instruction s)

1. PUBLICATION OR

REPORT NO.

NBSIR 85-3201

2. Performing Organ. Report No. 3. Publ ication Date

OCTOBER 1985

4

.

TITLE AND SUBTITLE

5. Validation Tests of an Earth Contact Heat Transfer Algorithm

AUTHOR(S)

George N. Walton

6.

PERFORMING ORGANIZATION (If joint or other than NBS. see in struct/on s)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MO 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

9.

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

Solar Buildings Technology Division

Office of Solar Heat Technology

U.S. Department of Energy

Washington, DC 20585

10.

SUPPLEMENTARY NOTES

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

1

1.

A BSTRACT ( A 200-word or less factual summary o f most significant information. If document includes a signi fican t bibliography or literature survey, mention it here)

Experimental tests and numerical calculations are performed to determine the suitabili^

of including a simplified earth contact heat transfer algorithm in building energy

analysis computer simulations. Reasonable agreement is shown between the finite difference test program and the simplified method. There is very good agreement between

the floor surface temperature of the NBS Passive Solar Test Facility and the temperatur.

predicted by the test program. These results indicate that simplifications based upon

the specific configuration can give good results for a simple calculation of the annu harmonic approximation of the simple method. A procedure is developed for using twodimensional (2-D) simulation to closely approximate the full 3-D effects of heat transfer from rectangular basements and slabs. The 2-D method can be applied to other

numerical procedures for which use a 2-D method for computing earth contact heat flux.

It is recommended that any algorithm used in a large building energy analysis computer

program include the capability of modeling the specific geometry under consideration

instead of (or in addition to) using tables of coefficients which have been generated

for a manual method. The algorithm should also include the ability to handle the same weather data that is used in the energy analysis.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by sem,co-a- Earth contact heat transfer; building energy analysis, computer simulation;

ground conduction; basement heat loss, slab-on-grade heat loss

13. AVAILABILITY

[_£_) Unlimited

| | For Official Distribution, Do Not Release to NTIS

"3| Order From Superintendent of Documents, U.S. Government Printing Office, Washington, DC

20402.

[x] 0rder From National Technical Information Service (NTIS), Springfield, VA 22161

14. NO. OF

PRINTED PAGES

34

15. P r

$9.95

USCOmm OC 13 «CG*