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Theoretical calculations of the steady-state heat losses through a slab-on- ground floor
Building and Environment, Vol. 23, No. 1, pp. I I 17, 1988. Printed in Great Britain.
0360-1323/88 $3.00 +0.00 Pergamon Journals Ltd.
Theoretical Calculations of the SteadyState Heat Losses Through a Slab-onGround Floor A. E. DELSANTE* A new analytical expression for the two-dimensional steady-state heat loss from a slab-on-ground floor is derived as a function of the floor width, wall thickness, ground conductivity, and surface film conductance. Two approximate forms, which are easier to calculate, are also given. Analogous approximate expressions are obtained for the three-dimensional case. Comparisons with previous theoretical work generally show good agreement, except for very small floors, for which the expressions derived here predict somewhat lower heat losses.
derivation is open to criticism. First, the temperature gradients were calculated assuming a wall of zero thickness, placed at the mid-point of the real wall, and replacing the real wall. The consequent divergence of the rate of heat flow at the base of the zero-thickness wall was avoided by integrating only up to the beginning of the finite-thickness wall. Second, the surface film resistances were assumed to be zero. The first of these criticisms was met by Delsante et al. [4], who used Fourier transform theory to derive explicit analytical expressions for the steady-state floor heat loss (and hence the U-value) in two and three dimensions, taking into account the finite wall thickness, but not the surface film resistance. This work will be described more fully in the next section. Other workers have taken into account the wall thickness and the surface film resistance, but have not derived analytical expressions for the ground heat loss. For example, Billington [5] used a network analyser to investigate two-dimensional heat flow for three values of the floor breadth, with fixed values of the wall thickness, ground conductivity, and surface film resistances ; and Muncey and Spencer [6] used a Fourier series technique to investigate three-dimensional heat flow, taking into account the wall thickness and surface film resistances. Their results were presented in graphical form. The work of Kusuda and Bean [7] should also be noted: they used a Green's function approach to calculate time-dependent three-dimensional heat flow, but without taking into account the wall thickness or surface film resistances. In this paper, the work of Delsante et al. is extended to provide an exact analytical expression for the steadystate heat loss from a slab-on-ground floor, using twodimensional geometry and taking into account the wall thickness and the surface film resistances. Two approximate simpler forms of this expression are given and evaluated, and the expressions are compared with previous theoretical and experimental work. The three-dimensional analogues of the approximate two-dimensional expressions are also given and evaluated.
INTRODUCTION THE DERIVATION of an analytical expression for the heat loss from an uninsulated slab-on-ground floor, even in the steady state, has for a long time been one of the more difficult problems in the calculation of the overall thermal transmittance of a building. This is because the heat flow from the floor slab into the ground must be assumed to be three-dimensional (or at least two-dimensional), instead of one-dimensional as is usually assumed elsewhere in the building fabric. Although finite-difference and finite-element methods can be applied to the problem with some success (see e.g. [1]), and can certainly deal with geometries and variations in thermal properties that are much more complex than could be attempted with an analytical approach, it is still highly desirable to have a simple, analytical expression for the steady-state heat loss from a ground floor as a function of the most important variables determining this loss, such as the thermal conductivity of the soil, the floor dimensions, the wall thickness, and the surface air film resistances. One of the first attempts to provide such an expression was that of Macey [2], who considered a floor of infinite length (i.e. two-dimensional heat flow) laid on a ground with the same thermal conductivity as the floor. A correction factor for rectangular floors was derived, resulting in the following expression for the U-value of the floor and ground : 2kB
U = - 7za - artanh (2a/(2a + v)),
(1)
where k is the soil thermal conductivity, B is a correction factor for rectangular floors (ranging from 1.6 for a square floor to 1.0 for an infinitely long floor), a is half the breadth (lesser dimension) of the floor, and v is the wall thickness. Equation (1) is used to derive the U-values of solid floors given in the CIBS Guide [3]. However, its * CSIRO Division of Building Research, PO Box 56, Highett, Vic. 3190, Australia. 11
A. E. Delsante
12
C A L C U L A T I O N OF THE FLOOR U-VALUE The first assumption to be made is that the thermal conductivities of concrete and soil are equal, so that the floor and the ground can be treated as a uniform semiinfinite solid, z ~> 0, where z is the vertical distance into the ground. This is a reasonable assumption for typical soils. The geometry of the problem is shown in Fig. 1, which shows the bulk fluid temperature above the floor as a function of distance along the floor and ground. The floor extends from - a to a, and walls of thickness v are placed at a and - a . The bulk fluid temperature O(x), where x is the distance along the surface, is assumed to change linearly over the wall thickness, from one (indoors) to zero (outdoors). No assumption is made regarding the soil temperature at any depth, which is therefore determined by O(x). In practice this means that the soil temperature at large depths will be close to the mean annual outdoor air temperature, which accords with observation [5]. Let the soil conductivity be k, and the surface air film conductance be H. In practice the indoor value of H will differ from the outdoor value, and the latter is very difficult to estimate, as it depends greatly on the type of ground surface surrounding the building. In this analysis, H is assumed to be independent of x, and an average of the indoor and outdoor values of the surface resistances will be used to determine H for numerical calculations. Let T(x, z) be the temperature field in the ground, and let h = H/k. The bulk fluid temperature is given by [ l , Ixl ~ a:
where i = x / ( - 1 ) , and y(w) is to be found from the boundary conditions. Applying (4) to (5) gives
~
(Io)l + h) exp ( - io)x)g(o)) do) = hO(x).
(6)
oo
The total heat flow into the ground from the indoor region Ixl ~< a is given by
f° 0_r
dp=-k,,,~ dz := dx. 0
Using (5), this gives Io)l 4~ = ~k f~:~ ~-g(o))[exp ( - io)a)- exp (i~oa)] do), which, if written as =
k r ~ sgn o)qo)[+ h)g(o)) x [exp ( - io)a) - e x p (io)a)] do),
where sgn o) ( = [o)l/o)) is the sign of 09, can be combined with (6) to give 4) = ~
{F.[sgn o)/(Io)l +h)]*O(a) -F_,[sgno)/(lo)l+h)]*0(-a)}
(7)
where * denotes the convolution operation, and F, denotes a Fourier transform with respect to t: for any function y(o)),
l
O(x) = t0, Ixl/> a + v ;
y(o))exp (io)O do).
F,b,(o))] =
!
((a+v-lxi)/v, a <~Ixl <, a+v.
ao
In the steady state, T(x, z) satisfies V2T(x, z) = O;
(2)
T(x, oo) = O; and
(3)
Performing the Fourier transforms and convolutions in (7) yields the final expression for the total heat flow into the ground from the indoor region : 4~ = ~ {In [(2a + v)/v] + (2a/v) In [(2a + v)/2a]
c~T
= z~
h( T(x, 0)-- O(x)).
(4)
1
0
+ ~ [ f ( 2 a h ) - f((2a + v)h) + f ( v h ) - n/2]},
(8)
The solution of (2) and (3) is where f is a function defined in [8] as r ( x , z ) = 2n 1
f' exp(-i~ox)exp(-zlo)l)g(o))do), (5)
f(x) = Ci(x) sin x - si(x) cos x, where
si(x)=
t ~ sint dt -
J.~
t
O(xl
and o~cos t Ci(x) = - .Ix t dt.
I
? -a-v
--0
We note that the limit h ~ ~ immediately recovers the result for a surface temperature boundary condition, ¢(h 0
0
g+ V
2k {in [(2a + v)/v] + (2a/v) in [(2a + v)/2a]}, ---, oo) -- -~
mX
(9) Fig. 1. Bulk fluid temperature O(x) as a function of the distance along the floor and ground. The floor extends from - a to a; walls of thickness v are placed at a and - a.
given by Delsante et al. [4], and the limit h ~ 0 gives ~b = 0 as expected. Note that (9), when divided by 2a to
Theoretical Calculations o f Steady-State Heat Losses give the U-value, is very similar to Macey's expression (1) for two-dimensional heat flow (B = 1), which can be re-written as U = k In [(4a + v)/v].
(1O)
~a
It is easy to show that (9) always yields a slightly higher U-value than does (10). We also note that the limit v --* 0 may be taken in (8), since the divergence in the heat flux that would occur with a surface temperature boundary condition is prevented by the presence of a surface air film. The result is
13
changes from its indoor value to its outdoor value over a distance greater than the actual wall thickness. This distance will be called the effective wall thickness, v*. If a general expression for v* could be found, the U-value of the ground in this approach would be given by
kD* , na
U2 -
(15)
where D* is defined by (13), with v replaced by v*. In an attempt to find an expression for v*, equations (11) and (15) were evaluated for the following ranges of the relevant parameters :
~b(v ~ 0) = ~ [ln (2ah) + 7 - Ci (2ah) cos (2ah) k: 0.5-2.0 W m 1 K - t ; -
si (2ah) sin (2ah)],
where ? is Euler's constant. Equation (8) may be simplified by noting that if2a >> v (i.e. the building width is much greater than the wall thickness, which is always true), then the term f(2ah) -f[(2a+v)h] is very small {typically, f(2ah)=O.Ol, f[(2a+v)h] = 0.0098}, and may be neglected. In this approximation the U-value of the ground is therefore
H : 5.0-30.0 W m - 2 K - I ; v: 0.1-0.6 m and 2a : 2.0-30.0 m. (Typical values are k = 1.0, H = 10.0, v = 0.2, and 2a = 10.) It was found that if
k U = na {In [(2a + v)/vl + (2a/v) In [(2a + v)/2a] + [ f ( v h ) - n/Z]/vh}.
f
(11)
This expression will be used for subsequent calculations.
SIMPLIFIED
FLOOR
1.74"~
v* = v[ l +~ff-),
U-VALUE,S
Although (11) is simple in form, the calculation of f(x) can be a little awkward, especially ifx < 1. In this section, two alternative simpler expressions for the U-value will be derived and compared with (11). The first approach is to note that (9), which is valid for a surface temperature boundary condition, can be used to obtain the apparent thermal resistance Ra of the ground from the indoor surface to the outdoor surface as
RG = ~a/kD,
(12)
D = In [(2a + v)/v] + (2a/v) In [(2a + v)/2a].
(13)
where
Since the thermal resistance of the surface air film is given by l/hk, the total thermal resistance from indoor air to outdoor air, Rr, is given by
(16)
the errors in using (15) instead of (11) were minimized, with a maximum error of - 1 7 % ; however, the maximum error occurred at the extremes of the parameter ranges, namely H/k = 2.5, v = 0.1, and 2a = 30.0, which is somewhat unrealistic. For more realistic values, the errors were 1% or less, as shown in Table 1, which gives percentage errors for H/k = 10. We may therefore conclude that (15) is an excellent approximation for the ground U-value, for all likely values of the relevant parameters ; it is also much easier to evaluate than (11). Similarly, the approximation given by (14) was tested against (11) for the same parameter ranges. Here the agreement with (11) was not as good as was found with (15), with a maximum error of - 51%, again at the same extremes of the ranges. Again, however, for more realistic values the agreement was quite good, as is shown in Table 2, which gives percentage errors for h = H/k = 10. In both approaches, the percentage errors decrease as H/k increases. Since (15) is the more accurate of the two, and just as easy to evaluate, it will be used in the comparisons that follow.
Rr = Ra+ 2/hk, so that the U-value for this approach, denoted by U~, is given by U~ = 1/Rr, or
kDh Ui = nah+ 2D"
(14)
The approximation involved in deriving (14) is the assumption that the apparent thermal resistance of the ground can be added in series to the surface film resistances, in the same way as is done in other parts of the building where one-dimensional heat flow is assumed. For the second approach, we note that the presence of the surface air film means that the surface temperature
Table 1. Percentage errors in calculating the floor U-value arising from the use of an effective wall thickness [equation (15)] instead of (11), for h = 10 m Actual wall thickness (m)
Floor width (m) 2
6
10
14
18
22
26
30
0.1 0.2 0.3 0.4 0.5 0.6
0.6 0.0 --0.3 --0.5 --0.7 -0.8
1.1 0.7 0.5 0.4 0.3 0.2
1.I 0.7 0.6 0.4 0.4 0.3
1.1 0.7 0.6 0.5 0.4 0.3
1.0 0.7 0.6 0.5 0.4 0.4
1.0 1.0 0.7 0.7 0.6 0.6 0.50.5 0.4 0.4 0.4 0.4
1.0 0.7 0.5 0.5 0.4 0.4
A. E. Delsante
14
Table 2. Percentage errors in calculating the floor U-value arising from using (14) instead of (1 I), forh = 10m Actual wall thickness (m) 0.1 0.2 0.3 0.4 0.5 0.6
Floor width (m) .... 2 -4.2 0.1 1.8 2.6 3.0 3.3
6
10
14
-10.9 5.4 -3.1 -1.8 -1.0 -0.4
-12.3 -6.8 --4.4 --3.0 --2.2 -1.6
-12.9 -7.4 --5.0 -3.6 -2.7 2.I
COMPARISON WITH OTHER THEORETICAL AND EXPERIMENTAL WORK Equation (15) will first be compared with the work of BiUington [5], who used the network analyser method to investigate two-dimensional heat flow from solid floors with breadths of 250, 150, and 90 inches, a wall thickness of 10 inches, and a ground conductivity of 10 B T U in. per sq. ft h °F (1.44 W m - t K t). The indoor temperature was fixed at 60°F, and U-values given graphically for ground temperatures at 20 ft of 30, 36, 42, 48, 54, and 60°F, and indoor-outdoor temperature differences ranging from 0 to 30°F. The actual values used for the surface film resistances were not stated, and it assumed that those given in the CIBS guide [3] are appropriate. It is clear that Billington's boundary conditions are very similar to those used here, except for the imposition of a constant ground temperature at 20 ft depth. This difference must be taken into account before a fair comparison can be made. The following approximate approach was taken. Consider the situation of Fig. 1, but with O(x) now the surface temperature (to simplify the calculation). Then the methods used to derive (5) can be used to obtain
T ( x , z ) - - a f x' exp (_~.~u) 7~U~0
b/x
cos/~u{cos u-cos [(1 + 6)u]} du,
where c~ = z/a, ~ = x/a, and 6 = v/a. Making the further approximation via << 1, we obtain 1
T(x, z) ~ - artan [2~/(~ 2 + fl 2 _ 1)] + s, where s=0,~2+fi2~>l; s = 1, ~2+32 < I. If To is the outdoor and Ti the indoor air temperature, then the ground temperature Tg will be T9 = (T~- T 0 )/1 ~ a r t a n ( 2 ~ / ( ~ 2+/~ 2-
1))+s)
+ To.
(17)
For T~ = 60°F, Tg was calculated for various values of To for z = 20 ft and x = 0 (i.e. under the centre of the slab), or x = a (i.e. under the wall), and an average of the two T o values was taken. When this average was close to one of the earth temperatures used by Billington (36,
18 13.1 7.7 --5.3 -3.9 -3.[ -2.4
22
26
30
-13.2 -7.9 --5.5 -4.1 -3.3 --2.6
-13.3 -8.0 -5.6 -4.3 .... 3.4 2.8
~ 13.3 -8.0 5.7 -4.4 3.5 2.9
42, 48, or 54°F), that value of To was used to obtain the floor U-value from the curves given by Billington. The results of this procedure are given in Table 3. It is noteworthy that the U-values in the fourth column of Table 3 are largely independent of the i n d o o r - o u t d o o r temperature difference. For comparison with the U-value given by (15), it is necessary to determine an average value for H. The CIBS guide [3] recommends an indoor surface resistance of 0.14 m 2 K W - ~for floors with heat flow down. Because of the roughness and degree of exposure of the ground surface, the outdoor surface resistance can be expected to be very small: a value of 0.02 m 2 K W ~ will be assumed, giving an average value of 12.5 W m 2 K ~for H. Converting all dimensions to SI units, we have v = 0 . 2 5 4 m, h = 8 . 6 8 m ', c h = 2 . 2 0 , and therefore, using (16), v* = 0.454 m. Table 4 compares the resultant U-values from (15) with those from the last column of Table 3 (note the conversion to SI units). The agreement is quite good, except for the smallest floor. The differences may well be caused by the simplified treatment of the surface resistances in (15), and the differences in the earth temperature boundary conditions. Table 4 also includes values obtained with Macey's equation [equation (1) with B = I]. For the three floors under consideration, (1) agrees with (15) to better than 9%. In the experimental work of [9], a number of floors, all 6 x 12.ft, were laid side by side, with the shorter edges exposed to outdoors, and the floors insulated from each other so that the heat flow could be considered to be approximately two-dimensional. The walls were approximately 6 inches thick, and the mean i n d o o r - o u t d o o r temperature difference was 40"F over the measurement period (27 January-27 February). Unfortunately no estimate of the ground conductivity or surface resistances is given, so values of k = 1 . 4 4 W m ~ K ~ and H = 12.5 W m a K ~will be used, as before, in (15). In the similar experimental arrangement of [10], a number of floors were monitored for three months, starting in January; all were 5 ft 8 in. x 11 ft 8 in., with walls 4.38 in. thick ; floors B and J were closest to the simplified conditions considered here. Only one short edge was exposed to outdoors, the other being protected by a corridor. As pointed out by Billington [5], this arrangement makes the value of 2a rather uncertain, but for this comparison we shall assume that 2a is 2 x I 1 ft 8 in. Again, k and H w e r e not given, and the values given above will be used in (15).
Theoretical Calculations o f S t e a d y - S t a t e H e a t Losses
15
Table 3. Results of calculations of floor U-values using Billington's method [5], under conditions corresponding to the boundary conditions used in this paper
Floor width (in.) 250 250 250 250 250 150 150 150 150 150 90 90 90 90 90
Selected outdoor temp. (°F)
Average earth temp. at 20 ft [from eqn. (17)] corresponding to given outdoor temperature (°F)
Floor U-value from Billington [5] corresponding to given outdoor and earth temperature (BTU ft- 2 h ~°F- 1)
27 35 43 52
36.4 42.3 47.9 54.3
0.100 0.104 0.106 0.107
31 38 46 53
36.4 42.1 48.6 54.3
0.151 0.152 0.154 0.154
33 40 46 53
36.1 42.3 47.6 53.8
0.224 0.229 0.230 0.232
Av. U-value for each floor (BTU ft- 2 h- ~°F- J)
0.104
0.153
0.229
The indoor temperature 7",.is always 60°F. The earth temperature at 20 ft is close to one of the earth temperatures used in [5]. Table 4. Comparison of floor U-values from equation (15) with those of Billington [5] (see last column of Table 3) and Macey [2] [equation (1)] Floor U-values Floor width (m) 6.35 3.81 2.29
Billington
Macey
Eq. (15)
0.591 0.869 1.300
0.568 0.829 1.187
0.531 0.767 1.088
(Wm 2K-I) ( W m - 2 K *) (Wm ZK-I)
Heat losses were measured for a large number of 24-hour periods over the three months, and plotted against the average indoor-outdoor temperature difference for each period. However, a 24-hour period is too short to use as a basis for deducing a floor U-value. The best that can be done is to estimate an average temperature difference for all the periods, and deduce an average heat loss and hence a U-value. F o r floor B, the average temperature difference over the periods for which heat losses were measured was approximately 39°F, while for floor J it was approximately 47°F. The results of [9] and [10] are compared with (15) in Table 5, and the agreement appears to be good.
However, any agreement, or lack of it, with either set of measurements must be viewed with considerable caution, because the measurements were taken over periods much shorter than one year. The steady-state theory developed here, and that of Macey, can only be used to predict long-term (i.e. annual) average heat losses, using annual average indoor and outdoor temperatures. Since the monthly average heat losses will be out of phase with the monthly average temperatures (as shown clearly by the measurements of ref. [11]), the U-value cannot be used to predict monthly average losses from the monthly i n d o o r outdoor temperature difference. An analytical method for predicting such losses wilt be the subject of a future paper.
THREE-DIMENSIONAL CONSIDERATIONS F o r a rectangular slab of length 2a, width 2b, and wall thickness v, with a surface temperature boundary condition, [4] gives U= ~
k
[a In (4a/v) + b In (4b/v)
+ 2(a 2 + b 2)~/2 _ a - b - b In {[(a 2+ b 2) ~/2+ b]/a}
-aln{[(a2+b2)~/2+a]/b}]
Table 5. Comparison of floor U-values from the experimental work of Dill et al. and Bareither et al. with equation (15) (a ground conductivity of k = 1.4 W m - l K - l was assumed) Floor U-values Floor width (m) 3.66 3.56 3.56
Temperature difference (K) (°F) 22.2 21.7 26.1
40 39 47
Dill et al. Bareither et al. Floor 1 Floor B Floor J Eq. (15) ( W m - 2 K -j) ( W m - 2 K -l) ( W m - 2 K -j) ( W m - Z K -l) 0.769 ---
-0.607 --
--0.520
0.850 0.536 0.536
(18)
16
,4. E. Delsante
as an a p p r o x i m a t e form, valid for the limits v/a << 1, v/b << 1. The exact expression is too lengthy to quote
m ~ K ~ were used, so these values will be used in the comparison. As before, H = 12.5 W m ~ K ' will be used as an average value. Table 6 shows the c o m p a r i s o n ; note that the exact form o f ( 1 8 ) was used in calculating the first two columns if t?/a >~ O. 1 or t,/b >~O. 1. T a k e n as a whole, Table 6 shows satisfactory agreement between the four methods of calculating the floor U-value, particularly in view of the errors involved in using the ref. [6] graphs, and the sensitivity of equations (18) or (19) to the choice of t f when the floor dimensions are small. Finally, a c o m p a r i s o n with some measured data will be attempted. Spooner [11] has measured the floor heat loss from a small house, and the i n d o o r and o u t d o o r temperatures, for a period of over two years. The internal dimensions of the floor were 3.7 x 7.3 m, and the floor was insulated with 25 m m polystyrene b o a r d s topped with 8-10 m m of glass-fibre-reinforced cement. According to Spooner, the thermal resistance of this insulation was 0.93 m : K W '. Using an average surface film c o n d u c t a n c e of H = 12.5 W m ~ K ~ (i,e. average film resistance o f 0.08 m 2 K W *), the value of R i + R j to be used in (19), with Ri i n c o r p o r a t i n g the insulation, will be 1.09 m ~ K W ~. In a previous report [12], Spooner gave
here, but will be used when these limits are not valid. Introducing a surface resistance to the three-dimensional problem unfortunately results in some intractable integrals, so that it has not been possible to derive the threedimensional analogue of (11). Therefore, to take into account surface resistances, it is necessary to modify (18), either by using the effective wall thickness, given by (16), or by adding the surface resistances explicitly. To use the effective wall thickness method, (18) is used with t, replaced by v*, defined by (16). To add the surface resistances explicitly, let R, and Rt be the i n d o o r and o u t d o o r surface resistances respectively. Then the air-toair U-value of the floor, U,, is given by U,, = U/(1 + U(&+R/)),
(19)
where U is given by (181. The two m e t h o d s will be compared with the theoretical work of ref. [6], who used a Fourier series technique to develop a graphical m e t h o d for calculating the U-values of floors of any shape, a n d with the values tabulated in the CIBS guide [3], which are derived from (1). In the latter reference, a wall thickness of v = 0.3 m a n d a ground conductivity of 1.4 W
Table 6. Floor U-values as given by equation (19); equation (18) with an effective wall thickness ; Muncey and Spencer [6] ; and the CIBS guide [3] Floor U-values
Eqn. (19) ( W m 2K i)
Eqn. (18) with effective wall thickness ( W m ~ 2 K i)
Muncey and Spencer ( W m 2K L)
CIBS guide ( W m ~K '1
100 x 100 100 x 60 100 x40 100 x 20
0.106 0.134 0.167 0.255
0.099 0.125 0.156 0.239
0.108" 0.133' 0.166" 0.253
0.10 0.12 0.15 0.24
60×60 60x40 60 x 20 60 x 10
0.160 0.191 0.276 0.419
0.149 0.178 0.259 0.397
0.165 0.189 0.277 0.418
0. f5 0.17 0.26 0.41
40 x 40 40 x 20 40 x 10 40 x 6
0.220 0.301 0.438 0.597
0.206 0.282 0.416 0.579
0.224 (I.310 0.423 0.600
0.2 I (/.28 0.43 0.59
20 x 20 20 × 10 20 x 6 20 × 4
I),371 0.496 0.647 0.8(}1
0,350 0.472 0.632 0.795
0,380 0,515 0.660 0.833
(}.36 0.48 0.64 0.82
10 × 10 10 x 6 10 x 4 10 × 2
0N) 1 0.741 0.881 1.208
0,576 0.731 0.884 1.264
0.633 0.761 0.923 1.300"
0.62 0.74 0.90 1.3 I
6x6 6x4 6×2
0.854 0.979 1.280
0.855 0.996 1.355
0.890* 1.023" 1.375"
091 1.03 1.411
4×4 4x2
1.090 1.362
1.126 1.462
1.154* *
t .22 1.52
2x2
1.576
1.751
*
[.96
Floor dimensions (m)
A wall thickness of 0.3 m, ground conductivity of 1.4 W m ~ K ', and average surface H of 12.5 W m 2 K ' were used. Values marked with an asterisk are unreliable or unobtainable from the Muncey and Spencer graphs because of difficulties in extrapolating their curves.
Theoretical Calculations o f Steady-State Heat Losses
details of the wall construction, from which a thickness of about 0.33 m can be deduced. No information is given about the ground conductivity, so a value of 1.4 W m K - ~ will be assumed. Equation (19) can then be used to give a U-value of 0.497 W m - 2 K - t. The measured average indoor-outdoor temperature difference for the period February 1980-January 1982 was 8.1 K, and hence the predicted average heat loss for this period is Q = 0.497 x 8.1 x 7.3 x 3.7 = 108.7 w . The measured average heat loss over this period was 73.1 W, a discrepancy of 49%. However, it should be noted that in May 1980 the external wall insulation (50 mm polystyrene boards) was extended down to the wall footings (a distance of some 500 mm). An estimate of the effect of this insulation may be made by using the work of Landman and Delsante [13] who investigated the effect of vertical insulation into the ground on the steady-state floor losses in two dimensions. Taking the breadth of the floor as 3.7 m, and the insulation conductivity as 0.036 W m - ~K - t, the reduction factor is found from Fig. 2 of [13] to be 0.67. The revised predicted value is therefore 0.67 x 108.7 = 72.8 W. Such good agreement must be considered coincidental to some extent, since the assumption of two-dimensional heat flow is not a good one for this floor, and, more importantly, the calculated heat loss is strongly dependent on the assumed value of the ground conductivity.
CONCLUSION This paper has presented some new analytical expressions for the steady-state heat loss from a slabon-ground floor in two and three dimensions. Where
possible comparisons have been made with other theoretical and experimental work. For two-dimensional heat flow, an explicit expression for the floor U-value has been given in terms of the building width, wall thickness, ground conductivity and average surface film conductance. A very good approximation, which is much easier to calculate, has also been obtained by introducing the concept of an effective wall thickness. Agreement with other theoretical work is reasonably good; agreement with two experimental studies is of dubious value, because of the short periods over which the measurements were made. For three-dimensional flow, a previously-derived expression for the floor U-value, which did not take into account the surface resistances, has been adapted in two ways to take them into account. Again, either expression is very easy to calculate, and agrees quite well with other theoretical work, except for very small floors. For such floors, it must be borne in mind that the results are sensitive to the treatment of the wall thickness (which is no longer very small compared with the floor dimensions) and the values of the surface film resistances (which will become significant in comparison with the resistance of the ground, which decreases as the floor dimensions decrease). Comparison with the only available experimental study of three-dimensional heat flow resulted in varying degrees of agreement, depending on assumptions made about soil conductivity and the effect of the installed vertical edge insulation. Although the agreement between the results presented here and previous theoretical work is generally good, there is clearly still a need for more experimental studies to check the theory.
Acknowledgement--The author gratefully acknowledges the assistance of D. C. Spooner in supplying his experimental heat loss data.
REFERENCES 1. G.P. Mitalas, Calculation of basement heat loss. ASHRAE Trans. 89, Part 1,420-437 (1983). 2. H.H. Macey, Heat loss through a solid floor. J. Inst. Fuel22, 369-371 (1949). 3. CIBS Guide A3, Thermal properties of building structures. The Chartered Institution of Building Services (1980). 4. A.E. Delsante, A. N. Stokes and P. J. Walsh, Application of Fourier transforms to periodic heat flow into the ground under a building. Int. J. Heat Mass Transfer 26, 121-132 (1983). 5. N.S. Billington, Heat loss through solid ground floors. IHVEJ. 19, 351-372 (1951). 6. R.W. Muncey and J. W. Spencer, Heat flow into the ground under a house. Energy Conservation in Heating, Cooling and Ventilating Buildings, Vol. 2, 649-660. Hemisphere Publishing Corp., Washington, D.C. (1978). 7. T. Kusuda and J. W. Bean, Simplified methods for determining seasonal heat loss from uninsulated slab-on-ground floors. ASHRAE Trans. 90, Part IB, 611-632 (1984). 8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Dover Publications Inc., New York (1972). 9. R.S. Dill, W. C. Robinson and H. E. Robinson, Measurements of heat losses from slab floors. U.S. Dept. of Commerce, Nat. Bureau of Standards Report No. BMS 103 (1945). 10. H . D . Bareither, A. N. Fleming and B. E. Alberty, Temperature and heat-loss characteristic of concrete floors laid on the ground. Univ. of lllinois, Small Home Council--Building Research Council, Res. Report 48-I (1948). 1I. D.C. Spooner, Heat loss measurements through an insulated domestic ground floor. Build. Set. Eng. Res. and Tech. 3, 147-151 (1982). 12. D.C. Spooner, Heat losses from an unoccupied house. Cement and Concrete Association, Technical Report 549 (1982). 13. K.A. Landman and A. E. Delsante, Steady state heat losses from a building floor slab with vertical edge insulation--II. Bldg. Envir. 22, 49 (1987).
BA~ 2 3 ~ 1 - B
17
0360-1323/88 $3.00 +0.00 Pergamon Journals Ltd.
Theoretical Calculations of the SteadyState Heat Losses Through a Slab-onGround Floor A. E. DELSANTE* A new analytical expression for the two-dimensional steady-state heat loss from a slab-on-ground floor is derived as a function of the floor width, wall thickness, ground conductivity, and surface film conductance. Two approximate forms, which are easier to calculate, are also given. Analogous approximate expressions are obtained for the three-dimensional case. Comparisons with previous theoretical work generally show good agreement, except for very small floors, for which the expressions derived here predict somewhat lower heat losses.
derivation is open to criticism. First, the temperature gradients were calculated assuming a wall of zero thickness, placed at the mid-point of the real wall, and replacing the real wall. The consequent divergence of the rate of heat flow at the base of the zero-thickness wall was avoided by integrating only up to the beginning of the finite-thickness wall. Second, the surface film resistances were assumed to be zero. The first of these criticisms was met by Delsante et al. [4], who used Fourier transform theory to derive explicit analytical expressions for the steady-state floor heat loss (and hence the U-value) in two and three dimensions, taking into account the finite wall thickness, but not the surface film resistance. This work will be described more fully in the next section. Other workers have taken into account the wall thickness and the surface film resistance, but have not derived analytical expressions for the ground heat loss. For example, Billington [5] used a network analyser to investigate two-dimensional heat flow for three values of the floor breadth, with fixed values of the wall thickness, ground conductivity, and surface film resistances ; and Muncey and Spencer [6] used a Fourier series technique to investigate three-dimensional heat flow, taking into account the wall thickness and surface film resistances. Their results were presented in graphical form. The work of Kusuda and Bean [7] should also be noted: they used a Green's function approach to calculate time-dependent three-dimensional heat flow, but without taking into account the wall thickness or surface film resistances. In this paper, the work of Delsante et al. is extended to provide an exact analytical expression for the steadystate heat loss from a slab-on-ground floor, using twodimensional geometry and taking into account the wall thickness and the surface film resistances. Two approximate simpler forms of this expression are given and evaluated, and the expressions are compared with previous theoretical and experimental work. The three-dimensional analogues of the approximate two-dimensional expressions are also given and evaluated.
INTRODUCTION THE DERIVATION of an analytical expression for the heat loss from an uninsulated slab-on-ground floor, even in the steady state, has for a long time been one of the more difficult problems in the calculation of the overall thermal transmittance of a building. This is because the heat flow from the floor slab into the ground must be assumed to be three-dimensional (or at least two-dimensional), instead of one-dimensional as is usually assumed elsewhere in the building fabric. Although finite-difference and finite-element methods can be applied to the problem with some success (see e.g. [1]), and can certainly deal with geometries and variations in thermal properties that are much more complex than could be attempted with an analytical approach, it is still highly desirable to have a simple, analytical expression for the steady-state heat loss from a ground floor as a function of the most important variables determining this loss, such as the thermal conductivity of the soil, the floor dimensions, the wall thickness, and the surface air film resistances. One of the first attempts to provide such an expression was that of Macey [2], who considered a floor of infinite length (i.e. two-dimensional heat flow) laid on a ground with the same thermal conductivity as the floor. A correction factor for rectangular floors was derived, resulting in the following expression for the U-value of the floor and ground : 2kB
U = - 7za - artanh (2a/(2a + v)),
(1)
where k is the soil thermal conductivity, B is a correction factor for rectangular floors (ranging from 1.6 for a square floor to 1.0 for an infinitely long floor), a is half the breadth (lesser dimension) of the floor, and v is the wall thickness. Equation (1) is used to derive the U-values of solid floors given in the CIBS Guide [3]. However, its * CSIRO Division of Building Research, PO Box 56, Highett, Vic. 3190, Australia. 11
A. E. Delsante
12
C A L C U L A T I O N OF THE FLOOR U-VALUE The first assumption to be made is that the thermal conductivities of concrete and soil are equal, so that the floor and the ground can be treated as a uniform semiinfinite solid, z ~> 0, where z is the vertical distance into the ground. This is a reasonable assumption for typical soils. The geometry of the problem is shown in Fig. 1, which shows the bulk fluid temperature above the floor as a function of distance along the floor and ground. The floor extends from - a to a, and walls of thickness v are placed at a and - a . The bulk fluid temperature O(x), where x is the distance along the surface, is assumed to change linearly over the wall thickness, from one (indoors) to zero (outdoors). No assumption is made regarding the soil temperature at any depth, which is therefore determined by O(x). In practice this means that the soil temperature at large depths will be close to the mean annual outdoor air temperature, which accords with observation [5]. Let the soil conductivity be k, and the surface air film conductance be H. In practice the indoor value of H will differ from the outdoor value, and the latter is very difficult to estimate, as it depends greatly on the type of ground surface surrounding the building. In this analysis, H is assumed to be independent of x, and an average of the indoor and outdoor values of the surface resistances will be used to determine H for numerical calculations. Let T(x, z) be the temperature field in the ground, and let h = H/k. The bulk fluid temperature is given by [ l , Ixl ~ a:
where i = x / ( - 1 ) , and y(w) is to be found from the boundary conditions. Applying (4) to (5) gives
~
(Io)l + h) exp ( - io)x)g(o)) do) = hO(x).
(6)
oo
The total heat flow into the ground from the indoor region Ixl ~< a is given by
f° 0_r
dp=-k,,,~ dz := dx. 0
Using (5), this gives Io)l 4~ = ~k f~:~ ~-g(o))[exp ( - io)a)- exp (i~oa)] do), which, if written as =
k r ~ sgn o)qo)[+ h)g(o)) x [exp ( - io)a) - e x p (io)a)] do),
where sgn o) ( = [o)l/o)) is the sign of 09, can be combined with (6) to give 4) = ~
{F.[sgn o)/(Io)l +h)]*O(a) -F_,[sgno)/(lo)l+h)]*0(-a)}
(7)
where * denotes the convolution operation, and F, denotes a Fourier transform with respect to t: for any function y(o)),
l
O(x) = t0, Ixl/> a + v ;
y(o))exp (io)O do).
F,b,(o))] =
!
((a+v-lxi)/v, a <~Ixl <, a+v.
ao
In the steady state, T(x, z) satisfies V2T(x, z) = O;
(2)
T(x, oo) = O; and
(3)
Performing the Fourier transforms and convolutions in (7) yields the final expression for the total heat flow into the ground from the indoor region : 4~ = ~ {In [(2a + v)/v] + (2a/v) In [(2a + v)/2a]
c~T
= z~
h( T(x, 0)-- O(x)).
(4)
1
0
+ ~ [ f ( 2 a h ) - f((2a + v)h) + f ( v h ) - n/2]},
(8)
The solution of (2) and (3) is where f is a function defined in [8] as r ( x , z ) = 2n 1
f' exp(-i~ox)exp(-zlo)l)g(o))do), (5)
f(x) = Ci(x) sin x - si(x) cos x, where
si(x)=
t ~ sint dt -
J.~
t
O(xl
and o~cos t Ci(x) = - .Ix t dt.
I
? -a-v
--0
We note that the limit h ~ ~ immediately recovers the result for a surface temperature boundary condition, ¢(h 0
0
g+ V
2k {in [(2a + v)/v] + (2a/v) in [(2a + v)/2a]}, ---, oo) -- -~
mX
(9) Fig. 1. Bulk fluid temperature O(x) as a function of the distance along the floor and ground. The floor extends from - a to a; walls of thickness v are placed at a and - a.
given by Delsante et al. [4], and the limit h ~ 0 gives ~b = 0 as expected. Note that (9), when divided by 2a to
Theoretical Calculations o f Steady-State Heat Losses give the U-value, is very similar to Macey's expression (1) for two-dimensional heat flow (B = 1), which can be re-written as U = k In [(4a + v)/v].
(1O)
~a
It is easy to show that (9) always yields a slightly higher U-value than does (10). We also note that the limit v --* 0 may be taken in (8), since the divergence in the heat flux that would occur with a surface temperature boundary condition is prevented by the presence of a surface air film. The result is
13
changes from its indoor value to its outdoor value over a distance greater than the actual wall thickness. This distance will be called the effective wall thickness, v*. If a general expression for v* could be found, the U-value of the ground in this approach would be given by
kD* , na
U2 -
(15)
where D* is defined by (13), with v replaced by v*. In an attempt to find an expression for v*, equations (11) and (15) were evaluated for the following ranges of the relevant parameters :
~b(v ~ 0) = ~ [ln (2ah) + 7 - Ci (2ah) cos (2ah) k: 0.5-2.0 W m 1 K - t ; -
si (2ah) sin (2ah)],
where ? is Euler's constant. Equation (8) may be simplified by noting that if2a >> v (i.e. the building width is much greater than the wall thickness, which is always true), then the term f(2ah) -f[(2a+v)h] is very small {typically, f(2ah)=O.Ol, f[(2a+v)h] = 0.0098}, and may be neglected. In this approximation the U-value of the ground is therefore
H : 5.0-30.0 W m - 2 K - I ; v: 0.1-0.6 m and 2a : 2.0-30.0 m. (Typical values are k = 1.0, H = 10.0, v = 0.2, and 2a = 10.) It was found that if
k U = na {In [(2a + v)/vl + (2a/v) In [(2a + v)/2a] + [ f ( v h ) - n/Z]/vh}.
f
(11)
This expression will be used for subsequent calculations.
SIMPLIFIED
FLOOR
1.74"~
v* = v[ l +~ff-),
U-VALUE,S
Although (11) is simple in form, the calculation of f(x) can be a little awkward, especially ifx < 1. In this section, two alternative simpler expressions for the U-value will be derived and compared with (11). The first approach is to note that (9), which is valid for a surface temperature boundary condition, can be used to obtain the apparent thermal resistance Ra of the ground from the indoor surface to the outdoor surface as
RG = ~a/kD,
(12)
D = In [(2a + v)/v] + (2a/v) In [(2a + v)/2a].
(13)
where
Since the thermal resistance of the surface air film is given by l/hk, the total thermal resistance from indoor air to outdoor air, Rr, is given by
(16)
the errors in using (15) instead of (11) were minimized, with a maximum error of - 1 7 % ; however, the maximum error occurred at the extremes of the parameter ranges, namely H/k = 2.5, v = 0.1, and 2a = 30.0, which is somewhat unrealistic. For more realistic values, the errors were 1% or less, as shown in Table 1, which gives percentage errors for H/k = 10. We may therefore conclude that (15) is an excellent approximation for the ground U-value, for all likely values of the relevant parameters ; it is also much easier to evaluate than (11). Similarly, the approximation given by (14) was tested against (11) for the same parameter ranges. Here the agreement with (11) was not as good as was found with (15), with a maximum error of - 51%, again at the same extremes of the ranges. Again, however, for more realistic values the agreement was quite good, as is shown in Table 2, which gives percentage errors for h = H/k = 10. In both approaches, the percentage errors decrease as H/k increases. Since (15) is the more accurate of the two, and just as easy to evaluate, it will be used in the comparisons that follow.
Rr = Ra+ 2/hk, so that the U-value for this approach, denoted by U~, is given by U~ = 1/Rr, or
kDh Ui = nah+ 2D"
(14)
The approximation involved in deriving (14) is the assumption that the apparent thermal resistance of the ground can be added in series to the surface film resistances, in the same way as is done in other parts of the building where one-dimensional heat flow is assumed. For the second approach, we note that the presence of the surface air film means that the surface temperature
Table 1. Percentage errors in calculating the floor U-value arising from the use of an effective wall thickness [equation (15)] instead of (11), for h = 10 m Actual wall thickness (m)
Floor width (m) 2
6
10
14
18
22
26
30
0.1 0.2 0.3 0.4 0.5 0.6
0.6 0.0 --0.3 --0.5 --0.7 -0.8
1.1 0.7 0.5 0.4 0.3 0.2
1.I 0.7 0.6 0.4 0.4 0.3
1.1 0.7 0.6 0.5 0.4 0.3
1.0 0.7 0.6 0.5 0.4 0.4
1.0 1.0 0.7 0.7 0.6 0.6 0.50.5 0.4 0.4 0.4 0.4
1.0 0.7 0.5 0.5 0.4 0.4
A. E. Delsante
14
Table 2. Percentage errors in calculating the floor U-value arising from using (14) instead of (1 I), forh = 10m Actual wall thickness (m) 0.1 0.2 0.3 0.4 0.5 0.6
Floor width (m) .... 2 -4.2 0.1 1.8 2.6 3.0 3.3
6
10
14
-10.9 5.4 -3.1 -1.8 -1.0 -0.4
-12.3 -6.8 --4.4 --3.0 --2.2 -1.6
-12.9 -7.4 --5.0 -3.6 -2.7 2.I
COMPARISON WITH OTHER THEORETICAL AND EXPERIMENTAL WORK Equation (15) will first be compared with the work of BiUington [5], who used the network analyser method to investigate two-dimensional heat flow from solid floors with breadths of 250, 150, and 90 inches, a wall thickness of 10 inches, and a ground conductivity of 10 B T U in. per sq. ft h °F (1.44 W m - t K t). The indoor temperature was fixed at 60°F, and U-values given graphically for ground temperatures at 20 ft of 30, 36, 42, 48, 54, and 60°F, and indoor-outdoor temperature differences ranging from 0 to 30°F. The actual values used for the surface film resistances were not stated, and it assumed that those given in the CIBS guide [3] are appropriate. It is clear that Billington's boundary conditions are very similar to those used here, except for the imposition of a constant ground temperature at 20 ft depth. This difference must be taken into account before a fair comparison can be made. The following approximate approach was taken. Consider the situation of Fig. 1, but with O(x) now the surface temperature (to simplify the calculation). Then the methods used to derive (5) can be used to obtain
T ( x , z ) - - a f x' exp (_~.~u) 7~U~0
b/x
cos/~u{cos u-cos [(1 + 6)u]} du,
where c~ = z/a, ~ = x/a, and 6 = v/a. Making the further approximation via << 1, we obtain 1
T(x, z) ~ - artan [2~/(~ 2 + fl 2 _ 1)] + s, where s=0,~2+fi2~>l; s = 1, ~2+32 < I. If To is the outdoor and Ti the indoor air temperature, then the ground temperature Tg will be T9 = (T~- T 0 )/1 ~ a r t a n ( 2 ~ / ( ~ 2+/~ 2-
1))+s)
+ To.
(17)
For T~ = 60°F, Tg was calculated for various values of To for z = 20 ft and x = 0 (i.e. under the centre of the slab), or x = a (i.e. under the wall), and an average of the two T o values was taken. When this average was close to one of the earth temperatures used by Billington (36,
18 13.1 7.7 --5.3 -3.9 -3.[ -2.4
22
26
30
-13.2 -7.9 --5.5 -4.1 -3.3 --2.6
-13.3 -8.0 -5.6 -4.3 .... 3.4 2.8
~ 13.3 -8.0 5.7 -4.4 3.5 2.9
42, 48, or 54°F), that value of To was used to obtain the floor U-value from the curves given by Billington. The results of this procedure are given in Table 3. It is noteworthy that the U-values in the fourth column of Table 3 are largely independent of the i n d o o r - o u t d o o r temperature difference. For comparison with the U-value given by (15), it is necessary to determine an average value for H. The CIBS guide [3] recommends an indoor surface resistance of 0.14 m 2 K W - ~for floors with heat flow down. Because of the roughness and degree of exposure of the ground surface, the outdoor surface resistance can be expected to be very small: a value of 0.02 m 2 K W ~ will be assumed, giving an average value of 12.5 W m 2 K ~for H. Converting all dimensions to SI units, we have v = 0 . 2 5 4 m, h = 8 . 6 8 m ', c h = 2 . 2 0 , and therefore, using (16), v* = 0.454 m. Table 4 compares the resultant U-values from (15) with those from the last column of Table 3 (note the conversion to SI units). The agreement is quite good, except for the smallest floor. The differences may well be caused by the simplified treatment of the surface resistances in (15), and the differences in the earth temperature boundary conditions. Table 4 also includes values obtained with Macey's equation [equation (1) with B = I]. For the three floors under consideration, (1) agrees with (15) to better than 9%. In the experimental work of [9], a number of floors, all 6 x 12.ft, were laid side by side, with the shorter edges exposed to outdoors, and the floors insulated from each other so that the heat flow could be considered to be approximately two-dimensional. The walls were approximately 6 inches thick, and the mean i n d o o r - o u t d o o r temperature difference was 40"F over the measurement period (27 January-27 February). Unfortunately no estimate of the ground conductivity or surface resistances is given, so values of k = 1 . 4 4 W m ~ K ~ and H = 12.5 W m a K ~will be used, as before, in (15). In the similar experimental arrangement of [10], a number of floors were monitored for three months, starting in January; all were 5 ft 8 in. x 11 ft 8 in., with walls 4.38 in. thick ; floors B and J were closest to the simplified conditions considered here. Only one short edge was exposed to outdoors, the other being protected by a corridor. As pointed out by Billington [5], this arrangement makes the value of 2a rather uncertain, but for this comparison we shall assume that 2a is 2 x I 1 ft 8 in. Again, k and H w e r e not given, and the values given above will be used in (15).
Theoretical Calculations o f S t e a d y - S t a t e H e a t Losses
15
Table 3. Results of calculations of floor U-values using Billington's method [5], under conditions corresponding to the boundary conditions used in this paper
Floor width (in.) 250 250 250 250 250 150 150 150 150 150 90 90 90 90 90
Selected outdoor temp. (°F)
Average earth temp. at 20 ft [from eqn. (17)] corresponding to given outdoor temperature (°F)
Floor U-value from Billington [5] corresponding to given outdoor and earth temperature (BTU ft- 2 h ~°F- 1)
27 35 43 52
36.4 42.3 47.9 54.3
0.100 0.104 0.106 0.107
31 38 46 53
36.4 42.1 48.6 54.3
0.151 0.152 0.154 0.154
33 40 46 53
36.1 42.3 47.6 53.8
0.224 0.229 0.230 0.232
Av. U-value for each floor (BTU ft- 2 h- ~°F- J)
0.104
0.153
0.229
The indoor temperature 7",.is always 60°F. The earth temperature at 20 ft is close to one of the earth temperatures used in [5]. Table 4. Comparison of floor U-values from equation (15) with those of Billington [5] (see last column of Table 3) and Macey [2] [equation (1)] Floor U-values Floor width (m) 6.35 3.81 2.29
Billington
Macey
Eq. (15)
0.591 0.869 1.300
0.568 0.829 1.187
0.531 0.767 1.088
(Wm 2K-I) ( W m - 2 K *) (Wm ZK-I)
Heat losses were measured for a large number of 24-hour periods over the three months, and plotted against the average indoor-outdoor temperature difference for each period. However, a 24-hour period is too short to use as a basis for deducing a floor U-value. The best that can be done is to estimate an average temperature difference for all the periods, and deduce an average heat loss and hence a U-value. F o r floor B, the average temperature difference over the periods for which heat losses were measured was approximately 39°F, while for floor J it was approximately 47°F. The results of [9] and [10] are compared with (15) in Table 5, and the agreement appears to be good.
However, any agreement, or lack of it, with either set of measurements must be viewed with considerable caution, because the measurements were taken over periods much shorter than one year. The steady-state theory developed here, and that of Macey, can only be used to predict long-term (i.e. annual) average heat losses, using annual average indoor and outdoor temperatures. Since the monthly average heat losses will be out of phase with the monthly average temperatures (as shown clearly by the measurements of ref. [11]), the U-value cannot be used to predict monthly average losses from the monthly i n d o o r outdoor temperature difference. An analytical method for predicting such losses wilt be the subject of a future paper.
THREE-DIMENSIONAL CONSIDERATIONS F o r a rectangular slab of length 2a, width 2b, and wall thickness v, with a surface temperature boundary condition, [4] gives U= ~
k
[a In (4a/v) + b In (4b/v)
+ 2(a 2 + b 2)~/2 _ a - b - b In {[(a 2+ b 2) ~/2+ b]/a}
-aln{[(a2+b2)~/2+a]/b}]
Table 5. Comparison of floor U-values from the experimental work of Dill et al. and Bareither et al. with equation (15) (a ground conductivity of k = 1.4 W m - l K - l was assumed) Floor U-values Floor width (m) 3.66 3.56 3.56
Temperature difference (K) (°F) 22.2 21.7 26.1
40 39 47
Dill et al. Bareither et al. Floor 1 Floor B Floor J Eq. (15) ( W m - 2 K -j) ( W m - 2 K -l) ( W m - 2 K -j) ( W m - Z K -l) 0.769 ---
-0.607 --
--0.520
0.850 0.536 0.536
(18)
16
,4. E. Delsante
as an a p p r o x i m a t e form, valid for the limits v/a << 1, v/b << 1. The exact expression is too lengthy to quote
m ~ K ~ were used, so these values will be used in the comparison. As before, H = 12.5 W m ~ K ' will be used as an average value. Table 6 shows the c o m p a r i s o n ; note that the exact form o f ( 1 8 ) was used in calculating the first two columns if t?/a >~ O. 1 or t,/b >~O. 1. T a k e n as a whole, Table 6 shows satisfactory agreement between the four methods of calculating the floor U-value, particularly in view of the errors involved in using the ref. [6] graphs, and the sensitivity of equations (18) or (19) to the choice of t f when the floor dimensions are small. Finally, a c o m p a r i s o n with some measured data will be attempted. Spooner [11] has measured the floor heat loss from a small house, and the i n d o o r and o u t d o o r temperatures, for a period of over two years. The internal dimensions of the floor were 3.7 x 7.3 m, and the floor was insulated with 25 m m polystyrene b o a r d s topped with 8-10 m m of glass-fibre-reinforced cement. According to Spooner, the thermal resistance of this insulation was 0.93 m : K W '. Using an average surface film c o n d u c t a n c e of H = 12.5 W m ~ K ~ (i,e. average film resistance o f 0.08 m 2 K W *), the value of R i + R j to be used in (19), with Ri i n c o r p o r a t i n g the insulation, will be 1.09 m ~ K W ~. In a previous report [12], Spooner gave
here, but will be used when these limits are not valid. Introducing a surface resistance to the three-dimensional problem unfortunately results in some intractable integrals, so that it has not been possible to derive the threedimensional analogue of (11). Therefore, to take into account surface resistances, it is necessary to modify (18), either by using the effective wall thickness, given by (16), or by adding the surface resistances explicitly. To use the effective wall thickness method, (18) is used with t, replaced by v*, defined by (16). To add the surface resistances explicitly, let R, and Rt be the i n d o o r and o u t d o o r surface resistances respectively. Then the air-toair U-value of the floor, U,, is given by U,, = U/(1 + U(&+R/)),
(19)
where U is given by (181. The two m e t h o d s will be compared with the theoretical work of ref. [6], who used a Fourier series technique to develop a graphical m e t h o d for calculating the U-values of floors of any shape, a n d with the values tabulated in the CIBS guide [3], which are derived from (1). In the latter reference, a wall thickness of v = 0.3 m a n d a ground conductivity of 1.4 W
Table 6. Floor U-values as given by equation (19); equation (18) with an effective wall thickness ; Muncey and Spencer [6] ; and the CIBS guide [3] Floor U-values
Eqn. (19) ( W m 2K i)
Eqn. (18) with effective wall thickness ( W m ~ 2 K i)
Muncey and Spencer ( W m 2K L)
CIBS guide ( W m ~K '1
100 x 100 100 x 60 100 x40 100 x 20
0.106 0.134 0.167 0.255
0.099 0.125 0.156 0.239
0.108" 0.133' 0.166" 0.253
0.10 0.12 0.15 0.24
60×60 60x40 60 x 20 60 x 10
0.160 0.191 0.276 0.419
0.149 0.178 0.259 0.397
0.165 0.189 0.277 0.418
0. f5 0.17 0.26 0.41
40 x 40 40 x 20 40 x 10 40 x 6
0.220 0.301 0.438 0.597
0.206 0.282 0.416 0.579
0.224 (I.310 0.423 0.600
0.2 I (/.28 0.43 0.59
20 x 20 20 × 10 20 x 6 20 × 4
I),371 0.496 0.647 0.8(}1
0,350 0.472 0.632 0.795
0,380 0,515 0.660 0.833
(}.36 0.48 0.64 0.82
10 × 10 10 x 6 10 x 4 10 × 2
0N) 1 0.741 0.881 1.208
0,576 0.731 0.884 1.264
0.633 0.761 0.923 1.300"
0.62 0.74 0.90 1.3 I
6x6 6x4 6×2
0.854 0.979 1.280
0.855 0.996 1.355
0.890* 1.023" 1.375"
091 1.03 1.411
4×4 4x2
1.090 1.362
1.126 1.462
1.154* *
t .22 1.52
2x2
1.576
1.751
*
[.96
Floor dimensions (m)
A wall thickness of 0.3 m, ground conductivity of 1.4 W m ~ K ', and average surface H of 12.5 W m 2 K ' were used. Values marked with an asterisk are unreliable or unobtainable from the Muncey and Spencer graphs because of difficulties in extrapolating their curves.
Theoretical Calculations o f Steady-State Heat Losses
details of the wall construction, from which a thickness of about 0.33 m can be deduced. No information is given about the ground conductivity, so a value of 1.4 W m K - ~ will be assumed. Equation (19) can then be used to give a U-value of 0.497 W m - 2 K - t. The measured average indoor-outdoor temperature difference for the period February 1980-January 1982 was 8.1 K, and hence the predicted average heat loss for this period is Q = 0.497 x 8.1 x 7.3 x 3.7 = 108.7 w . The measured average heat loss over this period was 73.1 W, a discrepancy of 49%. However, it should be noted that in May 1980 the external wall insulation (50 mm polystyrene boards) was extended down to the wall footings (a distance of some 500 mm). An estimate of the effect of this insulation may be made by using the work of Landman and Delsante [13] who investigated the effect of vertical insulation into the ground on the steady-state floor losses in two dimensions. Taking the breadth of the floor as 3.7 m, and the insulation conductivity as 0.036 W m - ~K - t, the reduction factor is found from Fig. 2 of [13] to be 0.67. The revised predicted value is therefore 0.67 x 108.7 = 72.8 W. Such good agreement must be considered coincidental to some extent, since the assumption of two-dimensional heat flow is not a good one for this floor, and, more importantly, the calculated heat loss is strongly dependent on the assumed value of the ground conductivity.
CONCLUSION This paper has presented some new analytical expressions for the steady-state heat loss from a slabon-ground floor in two and three dimensions. Where
possible comparisons have been made with other theoretical and experimental work. For two-dimensional heat flow, an explicit expression for the floor U-value has been given in terms of the building width, wall thickness, ground conductivity and average surface film conductance. A very good approximation, which is much easier to calculate, has also been obtained by introducing the concept of an effective wall thickness. Agreement with other theoretical work is reasonably good; agreement with two experimental studies is of dubious value, because of the short periods over which the measurements were made. For three-dimensional flow, a previously-derived expression for the floor U-value, which did not take into account the surface resistances, has been adapted in two ways to take them into account. Again, either expression is very easy to calculate, and agrees quite well with other theoretical work, except for very small floors. For such floors, it must be borne in mind that the results are sensitive to the treatment of the wall thickness (which is no longer very small compared with the floor dimensions) and the values of the surface film resistances (which will become significant in comparison with the resistance of the ground, which decreases as the floor dimensions decrease). Comparison with the only available experimental study of three-dimensional heat flow resulted in varying degrees of agreement, depending on assumptions made about soil conductivity and the effect of the installed vertical edge insulation. Although the agreement between the results presented here and previous theoretical work is generally good, there is clearly still a need for more experimental studies to check the theory.
Acknowledgement--The author gratefully acknowledges the assistance of D. C. Spooner in supplying his experimental heat loss data.
REFERENCES 1. G.P. Mitalas, Calculation of basement heat loss. ASHRAE Trans. 89, Part 1,420-437 (1983). 2. H.H. Macey, Heat loss through a solid floor. J. Inst. Fuel22, 369-371 (1949). 3. CIBS Guide A3, Thermal properties of building structures. The Chartered Institution of Building Services (1980). 4. A.E. Delsante, A. N. Stokes and P. J. Walsh, Application of Fourier transforms to periodic heat flow into the ground under a building. Int. J. Heat Mass Transfer 26, 121-132 (1983). 5. N.S. Billington, Heat loss through solid ground floors. IHVEJ. 19, 351-372 (1951). 6. R.W. Muncey and J. W. Spencer, Heat flow into the ground under a house. Energy Conservation in Heating, Cooling and Ventilating Buildings, Vol. 2, 649-660. Hemisphere Publishing Corp., Washington, D.C. (1978). 7. T. Kusuda and J. W. Bean, Simplified methods for determining seasonal heat loss from uninsulated slab-on-ground floors. ASHRAE Trans. 90, Part IB, 611-632 (1984). 8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Dover Publications Inc., New York (1972). 9. R.S. Dill, W. C. Robinson and H. E. Robinson, Measurements of heat losses from slab floors. U.S. Dept. of Commerce, Nat. Bureau of Standards Report No. BMS 103 (1945). 10. H . D . Bareither, A. N. Fleming and B. E. Alberty, Temperature and heat-loss characteristic of concrete floors laid on the ground. Univ. of lllinois, Small Home Council--Building Research Council, Res. Report 48-I (1948). 1I. D.C. Spooner, Heat loss measurements through an insulated domestic ground floor. Build. Set. Eng. Res. and Tech. 3, 147-151 (1982). 12. D.C. Spooner, Heat losses from an unoccupied house. Cement and Concrete Association, Technical Report 549 (1982). 13. K.A. Landman and A. E. Delsante, Steady state heat losses from a building floor slab with vertical edge insulation--II. Bldg. Envir. 22, 49 (1987).
BA~ 2 3 ~ 1 - B
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