Effect of spatial variation of soil thermal properties on slab-on-ground heat transfer

Building and Environmew, Copyright 0 1996 Elsevier

Pergamon

Vol. 31, No. I, pp. 51-57, 1996 Science Ltd. All rights reserved. Printed in Great Britain 036%1323/96 S15.oO+O.CCI

0360-1323(95)00026-7

Effect of Spatial Variation of Soil Thermal Properties on Slab-on-Ground Heat Transfer MONCEF

KRARTI*

(Accepted 2 March 1995) A time-varying solution is developed to determine the heat transfer between an insulated slab-ongradefloor and non-homogeneousground. The soiljust beneath the slab has thermalproperties that are different from those of the ground surrounding the slab foundation. Using the Interzone Temperature Profile Estimation (ITPE) technique, the soil temperaturefieldand the total slab heat loss are presented and analyzedfor both winter and summer conditions. In particular, the effect of soil thermal conductivity on the total slab heat loss is discussed in detail.

transfer patterns. This zone of influence varies with the size of the foundation and its insulation characteristics and with soil thermal properties. While data about the building and its foundation are readily available, information about soil thermal properties is often very difficult to obtain. In foundation heat transfer, soil thermal conductivity is the most influential parameter but is unfortunately the most elusive parameter to provide a reasonable estimation for. Several researchers [l-4] have shown that soil thermal conductivity is influenced by spatial and temporal variations of factors such as soil type, soil density, soil moisture content, and soil surface vegetation. For instance, DeVries [4] found that sandy soil thermal conductivity can vary by more than a factor of eight between dry conditions and 50% saturation. Thus, the thermal conductivity of soil surrounding a building foundation is significantly affected by rainfall. In cold climates, the soil thermal conductivity changes when the soil freezes. The thermal conductivity of water increases by four times when it changes phase from liquid to solid. Therefore, soils under frozen conditions have typically higher thermal conductivity than soils under unfrozen conditions. The vast majority of the existing ground-coupled heat transfer calculation methods assume no spatial or temporal variations for soil thermal properties [5,6]. Among the existing methods for foundation heat loss calculation, only the Mitalas method [7] allows for some spatial variation of soil thermal properties. In his method, Mitalas considers an upper and lower layer of the ground medium. Each layer is characterized by a constant thermal conductivity. The Mitalas method does not allow for lateral variation of soil thermal properties. Rainfall or frost can cause lateral variation of soil thermal properties, particularly between the protected ground below the foundation and the exposed soil surrounding the structure. Backfill is another source of lateral variation of soil thermal properties. Typically, the backfill around

NOMENCLATURE building half width [m] water table depth [m] soil specific heat [w kg-’ K-‘1 Fourier coefficients interior perimeter insulation location [m] exterior perimeter insulation length [m] ratio h/k, [m-*1

overall slab heat transfer conductance including insulation and surface coefficient [w mm2Km’] soil thermal conductivity in zone (I) [w m-’ K-‘1 soil thermal conductivity in zone (II) m m-’ Km’] soil thermal conductivity ratio (k,/ktt) total slab heat losses [w m-‘1 soil thermal diffusivity ratio (~t/~tt) time [s] soil temperature [K] building air temperature [K] soil surface temperature [K] water table temperature [K] space coordinates [m] Greek symbols coefficients defined in equation coefficients defined in equation eigenvalues complex eigenvalues soil thermal diffusivity in zone soil thermal diffusivity in zone soil density [kg m-‘1 complex temperature amplitude angular frequency [rad s-‘1

(10) (12)

(I) [m* s-‘1 (II) [m’ s-l] (1 + i) [K]

Subscripts I, II zone (I), zone (II)

1. INTRODUCTION

THE ground adjacent to a building foundation is the most important zone of soil affecting the foundation heat

*Joint Colorado,

Center for Energy Management, Boulder, CO 80309, U.S.A.

University

of

51

M. Krarti

52

the foundation is covered with low permeability soil, a membrane beneath the top layer of soil, or a hard surface (i.e. asphalt or concrete) to divert surface run-off away from the foundation and thus provide appropriate drainage. In the present paper, the effect on foundation heat transfer of lateral variation of soil thermal properties around the building foundation is investigated. Specifically, this paper presents steady-periodic solution to the heat conduction problem in a non-homogeneous soil under a slab-on-ground floor with horizontal insulation. A water table effect is considered in this model. The soil temperature field is obtained and analyzed using the Interzone Temperature Profile Estimation (ITPE) technique [S-14]. The effect of the insulation placement on total slab heat losses is discussed for various insulation levels and soil thermal conductivities.

2. FORMULATION

OF THE

PROBLEM

Figure 1 shows a model of a slab-on-ground floor with horizontal insulation. The ground just beneath the slab has a different thermal conductivity, k,, from that of the exposed ground surrounding the foundation, k,,. A water table at a depth b has a constant temperature T,. The air above the floor is kept at the temperature T,, while the soil surface is at temperature T, which varies with time. The time-dependent heat conduction equation in a non isotropic medium is given by the following equation :

V3(r,

t) = d T(r, t),

where V is the Laplacian operator and K, is the soil thermal diffusivity with two possible values: K, in zone (I) and K[[ in zone (II). The soil thermal diffusivity ratio Rd = KJK, provides an insight on the variation of soil thermal properties from zone (I) to zone (II). In the particular case R, = 1, the soil is isotropic with uniform thermal properties. In steady-periodic conditions, the solution T(r, t) of equation (1) can be found by applying the complex temperature technique. T(r, t) is then in the form T(r, t) = Tss(r) + Re [T,(r) e’““].

The coordinates x and y are denoted by the vector space r [i.e. r = (x, y)]. In equation (I), r is time, k, is the soil thermal conductivity, pS is soil density, and C, is soil specific heat. The values of k,, p., and C, are assumed constant in each zone [i.e. in zone (I) and in zone (II)]. Therefore, the heat conduction equation (1) can be reduced in each zone to a Fourier equation

(3)

In the above equation, 7’,, is the mean of the periodic temperature variation over one cycle (i.e. one year) and T, is the complex amplitude of the annual temperature fluctuations. A real amplitude and a phase shift can be obtained by taking the modulus and the argument of the complex value of T,, respectively. Both T,,(r) and T,(r) can be deduced from a complex temperature solution Q(r) of the following Helmholtz equation as explained in [9] : V0(r) = &9(r),

(4)

with 6,=

J(> *.

K,

(1)

(2)

(5)

Note that 6, can take two values : 6, in zone (I) or a,, in zone (II), depending on the value of the soil thermal diffusivity, K,. The complex temperature amplitude @(x,y) for an insulated slab-on-ground floor configuration shown in Fig. 1 is subject to the following equation :

ve = s;e

(6)

with

-

0=8,

fory=b

Q = 0,

fory = Oand 1x1 > a

ae

- = H(B-&(x)) aY

fory = Oand 1x1 < a,

where H is the ratio of the equivalent air-insulationslab-soil conductance to the soil thermal conductivity in zone (I) [i.e. H = h/k,]. Without loss of generality, the water temperature can be set to zero (i.e. 0, = 0). Figure 1 shows that the surfaces IxI = a divide the ground medium into three zones. Because of the symmetry around the axis x = 0, the temperature 0(x, y) needs to be determined only in zones (I) and (II). Let f(y) be the temperature profile along the surfaces 1x1 = a. In zone (I), the solution 0,(x, y) of equation (6) is

e,(x,y)

Fig. 1. Slab-on-grade floor with horizontal insulation and nonhomogeneous ground.

= i zjisinvnyz n--l

I.n

53

Spatial Variation of Soil Thermal Properties while, in zone (II), the temperature 8,, (x, y) is given by 0 f&(x, y) = f +f sin v,y < (1 - e-v~~.~(lXl-d)) n--l 1

where &P = -

Bi,, = -

n7I

~

b

=

v;,*= J(vn2+a:) ;

W-1)

2an

”

P#+

P;,, coth &b)

2(- lYI*,vn

WV,,:+~;)(ff+~L;,pcoth A,&4 ’

The system of equations (10) and (12) is solved by truncating the sum to a finite number of terms, N. A linear system of 2N equations with 2N unknowns (C, and f,, p = 1, 2,. . , N) is obtained and is solved by the Gauss-Jordan elimination method.

where v,=--;

(- l)PHBi

4I.n = J(d + #I)

A,, = Join’+m; AI,” = JCL4+m.

3. SOIL TEMPERATURE DISTRIBUTION

C, and fn are Fourier coefficients to be determined. The continuity of the heat flux at the surface 1x1= a gives the condition

Figure 2 shows the annual average soil temperature distribution beneath a slab of width 2a = 10 m. Two horizontal insulation configurations are presented in Fig. 2, as follows.

(9)

(a) The slab is well insulated with an R-value of 10 m2 K/W (i.e. H = 0.1 m-l). (b) The slab is moderately insulated with an R-value of 1 m2 K/W (i.e. H = 1.0 m-i).

or, introducing K = k&i

the soil thermal

conductivity

ratio

Multiplying the above equation by sin vPy and integrating over [0, b] yields an expression of the form of

(10)

Lx,’=

Kvpes v;~,,[Kv;,,~ + v;,~ tad

8L.P= -

(vi,,41

2(- l)“PnVp a[Kv;,,~+v;., tanh (v~,p)l@~,~~+v~) .

The third boundary condition of equation (6)

-a4

ayy=.= H(X)(41-

ei)

(11)

gives the following expression :

Multiplying this equation by cos KX and integrating over the interval [-a, a] yields

The building above the slab is kept at a constant temperature Ti = 20°C. The soil surface temperature varies with time as T, = 15 -8 (“C) cos (it) with w = 241 yr = 1.992 lo-’ rad/s, and t = 0 corresponds to January 15, when the soil surface is assumed to be at its minimum temperature. A water table at a depth b = 5 m below the soil surface is assumed to have a constant temperature T, = 10°C. Two scenarios for the soil thermal properties are illustrated in Fig. 2. In the first scenario (K = 1), the soil is assumed to be isotropic (k, = k,, = 1.0 W/m K), while in the second scenario (K = 2), the exposed soil is assumed to have higher thermal conductivity (kII = 2.0 W/m K) than the soil just beneath the slab (k, = 1.OW/m K). As illustrated in Fig. 2a for the well insulated slab configuration, the annual average soil temperature just beneath the slab is lower for the homogeneous ground case (i.e. K = 1) than for the non-homogeneous ground case (i.e. K = 2). However, for the moderately insulated slab configuration illustrated in Fig. 2b, the soil temperature is slightly higher for the homogeneous ground. This result indicates that the effect of the variation of soil thermal properties depends closely on the insulation level of the foundation. In general, the slab floor is in thermal interaction with both the water table and the soil surface outside the building. As the soil thermal conductivity increases in zone (II), the thermal interaction between the slab and the soil surface becomes more significant. In other words, the influence of the soil surface conditions on the ground-coupled heat transfer increases as the soil thermal conductivity in zone (II) becomes larger than that in zone (I). Consequently : (i) When the foundation is only moderately insulated, the temperature of the slab surface remains warmer (varying between 18°C in the slab center and 16°C in the slab edges) than that of the soil surface (T, = 15°C). As a result of increasing soil thermal conductivity in zone (II) and thus of increasing the

M. Krarti

54 (a) 0

ANNUAL SOIL TEMPERATURE H=O.lm-1

Dlstawe

Fig. 2. Annual

(bl

PROFILES

ANNUAL SOIL TEMPERATURE

PROFILES

15c

fromSlab center,m

soil temperature

Distance

isotherms moderately

beneath insulated

(a) a well insulated slab (H = 1.0 m-l).

fromSlab

slab (H = 0.1 m-‘)

Center. m

and (b) a

influence of soil surface temperature, the temperature of the ground medium decreases as shown in Fig. 2b. (ii) When the foundation is well insulated, the slab surface temperature decreases to about 14°C and is therefore lower than the soil surface temperature T, = 15°C. As the soil thermal conductivity in zone (II) becomes large, the earth temperature increases due to the increased influence of the warm soil surface (Fig. 2a).

plex amplitude of the total slab heat losses is obtained from equation (8)

Figure 3a and b show the soil temperature distribution beneath the well insulated slab (i.e. H = 0.1 rn-‘) during respectively summer and winter for the following two scenarios.

where

The time variation of total slab heat losses is then obtained using an expression similar to equation (3)

Q(t) = Q,, + Re(Qt efw’),

l

l

(9

The homogeneous ground with properties (i.e. K = 1 and R, = (ii) The non-homogeneous ground thermal conductivity (K = 2) diffusivity (& = 2) in zone (II).

uniform thermal 1). with higher soil and soil thermal

During the summertime, the soil surface temperature is higher than the slab temperature and therefore the increased soil thermal properties in zone (II) tend to increase the ground temperature as illustrated in Fig. 3b. In the wintertime, the ground temperature is generally warmer for the non-homogeneous case (i.e. K = 2) than for the homogeneous case (i.e. K = 1). Note that the soil is more sensitive to outdoor temperature variation in the non-homogeneous case. As illustrated in Fig. 3a, the homogeneous ground has a warm spot (the area inside the isotherm 13°C) in zone (II) where the low value of soil thermal diffusivity results in low soil penetration depth. The soil penetration depth is characteristic of how deep in the earth the soil surface temperature fluctuations are practically eliminated.

4. TOTAL SLAB HEAT LOSSES The slab heat loss is obtained by integrating the temperature gradient along the slab surface. First, the com-

(14)

Qss is the steady-state total slab heat losses and is obtained from expression (13) by replacing 6, by zero. Qt is the complex amplitude of the total slab heat losses and is directly obtained from Q of equation (13).

Figure 4 shows monthly four cases, as follows.

total slab heat loss profiles for

(i> Homogeneous

soil with uniform thermal properties (K= 1 and Rd = 1). (ii) Zone (I) is more conductive than zone (II) (K = 0.5 and R,, = 1). (iii) Zone (I) is less conductive than zone (II) (K = 2 and Rd = 1). (iv) Zone (II) has higher thermal conductivity and thermal diffusivity values than zone (I) (K = 2 and Rd = 2). The slab configuration of Fig. 4 has the same geometrical and thermal characteristics as the slab used in Figs 2 and 3. As indicated in Fig. 4, the total slab heat loss is significantly affected by the variation of the soil thermal properties. When zone (II) has the lower thermal conductivity value (i.e. the case of K = 0.5), the slab loses less heat during winter months and gains less heat during the summer months. The low soil thermal conductivity in zone (II) isolates the slab from the seasonal fluctuations of the soil surface temperature. However, the opposite behavior occurs when zone (II) has the higher

55

Spatial Variation of Soil Thermal Properties (b)

(a)

WINTER SOIL TEMPERATURE

SUMMER SOIL TEMPERATURE HdJ.1 m-1

PROFILES

PROFILES 23c

0

Fig. 3. Soil temperature

isotherms beneath a well insulated slab-on-grade wintertime and (b) summertime.

MONTHLY SLAB HEAT LOSS

(H = 0.1 m-‘) during (4

TOTAL SLAB HEAT LOSS

H-O.1 m-l 20

40

../,

,,,...

H-O.1 m-l .I ,,,,,,

...,.,,/

,,.

..I....

...

15

10

c

3

5

4

0

VK=l *i* -K=2 -K-2

K4.5

R_d=l R-d.1 R-h., R-d=2

g 0 a

-5

-10

-15

-20

ok, 0

““”

“’

1

“”

2

““““““‘-“” 3

4

K-ratio

Fig. 4. Effect of soil thermal properties on the monthly variation of total slab heat loss/gain.

thermal conductivity value (cases of K = 2). In these cases of non-homogeneous ground, the slab is more affected by the soil surface conditions than in the case of homogeneous ground. Indeed, the slab loses more heat during winter, and gains more heat during summer. Note that when the soil thermal diffusivity is also higher in zone (II)-the case of K = 2 and Rd = 2-the amplitude of the slab heat loss/gain increases further due to the increase of the penetration depth of the soil surface temperature fluctuations. Figure 5 shows the effect of the soil thermal conductivity K-ratio on the total slab heat losses, characterized by its mean, annual amplitude, and phase lag. The slab is assumed to be well insulated (i.e. H = 0.1 m-‘). Figure 5 mdicates that an increase in the K-ratio reduces both the annual mean and the phase lag of the total slab heat loss/gain. However, the same increase of the K-ratio results in an increase of the amplitude of the annual variation of the slab heat loss/gain. These results are due to the significant influence of zone (II) thermal conductivity on enhancing the heat transfer between the slab and the soil surface. As k,, increases, more heat transfer occurs between the soil surface and the slab. In periodic conditions, this increase in thermal interaction

Fig. 5. Variation of the total slab heat loss annual mean, amplitude, and phase lag as a function of the K-ratio. ANNUAL TOTAL SLAB HEAT LOSS H=O.l m-l

O-+Ts.15oC;Tw=lOoC Ts=Tw- 10 OC

Fig. 6. Variation of the total slab heat loss annual mean as a function of the K-ratio and soil surface temperature.

results in an increase of heat loss/gain amplitude and in a decrease of the phase lag between the soil surface temperature and the slab heat loss/gain variation. However, in steady-state conditions, the increased thermal interaction between the soil surface and the well

56

M. Krarti

insulated slab results in a decrease of the mean value of the annual total heat loss. This particular result is the consequence of the low earth temperature just beneath the slab (see Fig, 2a). Therefore, an increased interaction between the slab and the ‘warm’ soil surface results in less heat loss from the slab. Figure 6 indicates clearly the importance of the soil surface temperature in determining the effect of the Kratio on the total slab heat loss. When the annual soil surface temperature is reduced from T, = lS”C-as assumed in Fig. S-to T, = lO”C-same as the water table temperature-the behavior of the slab heat loss as a function of the K-ratio has completely changed. Indeed, for the case T, = lO”C, the annual mean of slab heat loss increases with the K-ratio. This behavior can be understood again by the fact that the interaction between soil surface and the slab increases with the K-ratio. In the case of T, = 10°C the earth temperature along the slab surface is still higher than the soil surface temperature. Thus, enhancing heat transfer between the slab and the soil surface results in an increase of slab heat loss. The effect of both the insulation level and the K-ratio on the annual mean of total heat loss is illustrated in Fig. 7. The annual mean of the soil surface temperature is again T, = 15°C. Figure 7 shows that the total slab heat loss variation as a function of the K-ratio depends on the insulation level of the slab, as follows

(4 For low insulation H = h/k,),

levels (i.e. high values of the slab heat loss increases with the K-

ratio. high insulation levels (i.e. low values of H), the slab heat loss decreases with increasing K-ratio.

(b) For

Figure 7 indicates also that a critical insulation level exists for which the slab heat loss is not affected by the K-ratio. For the slab configuration treated in Fig. 7, this insulation level corresponds to H = 0.8 mm’ or about a thermal insulation of an R-value of 1.25 m2 K/W.

ANNUAL TOTAL SLAB HEAT LOSS E”ect01K-ratio

/ 30

Hd.0

m-1

tbs.0

m-1

H=2.0 m-1

H=l .o In-, H-0.8 m-1 tb0.6

m-1

H=O 4 m-1

01,.

0

“’

““” 1

”

“” 2

““““”

“’

3

4

.-..,I

5

K-ratio

Fig. 7. Variation of the total slab heat loss annual mean as a function of the K-ratio and slab insulation level.

5. CONCLUSIONS The ITPE technique has been applied to develop the steady-periodic solution of the heat conduction equation for non-homogeneous ground above a slab with horizontal insulation. The effects of the soil thermal properties on the annual variation of the earth temperature field and on the foundation heat loss/gain are analyzed. In particular, it is found that low soil thermal conductivity around the foundation perimeter results generally in a reduction of the annual amplitude of total slab heat loss. However, the annual mean of well insulated slab heat loss can increase depending on the soil surface temperature even when the earth around the foundation has low thermal conductivity values. For moderately insulated or uninsulated slabs, the annual mean of the foundation heat loss typically decreases when low thermal conductivity soils are used around the slab perimeter.

REFERENCES 1. M. S. Kersten, The Thermal Conductivity of Soils. Bulletin 28, Engineering Experiment Station, University of Minnesota, Minneapolis, MN (1949). 2. L. A. Salomone and J. I. Marlowe, Soil and rock classification according to thermal conductivity: design of ground-coupled heat pump systems. EPRI Report No. CU-6482, Chantilly, VA (1989). 3. T. P. Bligh and E. A. Smith, Thermal conductivity measurements of soils in field and laboratory using a thermal conductivity probe. U.S. DOE Report No. DOE/DE-ACOS-80sf-11508 (1983). 4. D. A. DeVries, Thermal properties of soils, in Physics of Plant Environment (Edited by W. R. Van Wijk), Chapter 7, North-Holland, Amsterdam (1966). 5. R. L. Sterling and G. D. Meixel, Review of underground heat transfer research, in Earth Sheltered Performance and Evaluation Proceedings, Second Technical Conference (Edited by L. L. Boyer), pp. 67-74, Oklahoma State University, OK (1981). 6. D. E. Claridge, Design methods for earth-contact heat transfer, in Progress in Solar Energy (Edited by K. Boer), American Solar Energy Society, Boulder, CO (1987). 7. G. P. Mitalas, Calculation of below-grade residential heat loss : low-rise residential building. ASHRAE Transactions93, Pt 1, 1112-1121 (1987). 8. M. Krarti, D. E. Claridge and J. F. Kreider, The ITPE technique applied to steady-state groundcoupling problems. International Journal of Heat and Mass Transfer 31, 1885-1898 (1988). 9. M. Krarti, D. E. Claridge and J. F. Kreider, ITPE method applications to time-varying two-dimensional ground-coupling problems. International Journal of Heat and Mass Transfer 31, 189991911 (1988). 10. M. Krarti, Steady-state heat transfer beneath partially insulated slab-on-grade floor. International Journal of Heat and Mass Transfer 32,961-969 (1989). 11. M. Krarti, D. E. Claridge and J. F. Kreider, The ITPE method applied to time-varying threedimensional ground-coupling problems. Journal of Heat Transfer 112 (4), 849-856 (1990).

Spatial Variation of Soil Thermal Properties 12. M. Krarti and D. E. Claridge, two-dimensional heat transfer from earth-sheltered buildings. Journal of Solar Energy Engineering 112 (l), 43-50 (1990). 13. M. Krarti, Heat transfer from vertically insulated slab-on-grade floor. Znternational Journal of Heat and Muss Trunsfer 36 (5), 1175-l 184 (1993). 14. M. Krarti, Steady-state heat transfer from horizontally insulated slab-on-grade floor. Znternationul Journal of Heat and Mass Transfer 36 (5), 1167-l 174 (1993).

57