Steady-state analysis of the convective loop

Solar Energy, Vol. 26, pp. 461--463. 1981 Printed in Great Britain.

0038-092X/81/050461--03502.0010 Pergamon Press Ltd.

TECHNICAL NOTE Steady-state analysis of the convective loop ALVIN O. CONVERSE Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, U.S.A.

(Received 27 May 1980; revison accepted 8 December 1980) I. INTRODUCTION Although restricted by the steady-state assumption, this analysis demonstrates the short-run importance of ground coupling in the performance of a convective loop building. The purpose of the analysis is to evaluate the ratio of the heat loss from the living space to the loss that would occur if the loop were omitted and the same amount of insulation were used in the shell of the building. It provides a partial answer to Shurcliff's question, "Why bother with the loop?"[1]. A fuller answer would obviously have to consider the loop's ability to accept insolation and store energy in the structure and ground; it would involve a transient analysis. The analysis presented here does, however, point out the importance of partial loop flows, such as between the sunspace and basement, and exposes a mechanism whereby the cool ground can assist in reducing losses from a warm house. 2. ANALYSIS Consider the schematic diagram presented in Fig. 1. The loop temperature, T4, is assumed to be constant through the loop whenever the loop is colder than the ground, thus neglectin;~ resistance to heat transfer within the loop itself when there is natural convection between the loop and basement. This, then, is a limiting-case analysis which provides an upper bound estimate of the loop contribution. The steady-state heat balance on the loop yields the following equations.

A3U3(T3- T4)

-

Heat flow into loop from living space walls

is evaluated in an iterative manner. First, eqn (t) is solved for T4 with 8 set equal to 1. If the resulting 7'4> T2, 8 is set = 0 and T4 is reevaluated from eqn (1). For the case when the loop is present, the heat loss from the living space is given by the following equation

Q, =

A3U3(T3

Loss from upper part of shell

3. RESULTS For the parameter values presented in Fig. 1, the ratio, QI/Q2, and T~ are presented as a function of R~ in Fig. 2. As R= is varied from 0 to 24, R2 is varied so that RI+R~=4.42(Wlm2-C) -~ (24(Btu/hr-ft 2- F) -~) for all the results presented in Fig. 2.

(l) (l)

Thus, the insulation thickness is held constant and the variation in Qt/Q2 is due entirely by the presence of the loop. When there is no insulation in the outer wall, R~ =0, the system acts as a no-loop upper shell with a hole into the basement from the outside. The ratio Q1/Q2 exceeds 1 because the basement is colder than it would be in a single wall building with no leak into the basement. As R~ increases, Qm/Q2 at first

~

TI • - 6 . 7 = C (2OIF) 1.0 RI

I

r4 R~

T4

///

t

T~,- 194°C (67"F1

0.75

L

I"

(1.64 Btu/h-ft2-F)

307m 2 (3300 f t 2 )

T4

] A 4 " l l | l t ~ (120O f t t') U4=0.47 W/nt2c (.083 Btulh-ft2F) Iuz-ga2 w ~ c A 2 • ~23m2

0.50

#

. 20 (6el u.

Ld a~

I0 (SO)

o.a

o

(2400 ft 2)

0

0

0!25

0180

0!75

t.O

0 (321

4 . 2 2 (W/m2C) "1 , {24 (Btu / h - f t 2 F ) -I )

RI//(R I 4- R2)

Fig. I. Schematic diagram of convective loop.

Fig. 2. Effect of loop placement on performance. 461

uJ a. tu

0.25

//

T 2 - 1 2 8 " C (55=F) R I -t- R 2 "

Loss from floor

where Ri = l/U, and R2 = I/U3.

~[A2U2(T2-T4)+A4U4(T3-T4)]=O

The switchin¢ function, 8, is used so that the loop is thermally coupled to the ground only when T4< 7"2, i.e. when natural convection would be expected because the upper chamber is colder than the lower chamber. In the computation algorithm, 8

(3)

T,)+ A4U4(T3- T2) (4)

Loss from upper part of shell

Heat flow into loop from ground and floor. Effective only when 6 = 1

(2)

Loss from floor

Q, =(A3 + A,)/2 [ R t ~ ] ( T 3 -

A,U,(T4-TO +

10 when T2>T4 when T2 ~< T4"

- 8) T2].

For the corresponding case when the loop is not present,

Heat loss from loop to ambient

8=

- ,~T4 - ( 1

- T,) + A,U,[T3

Technical Note

462

Table 1. The effect of loop position on performance (parameters are as given in Fig. 1) Heat Loss From: R1/(RI + R2)

Loop Temp. °C

0.02 0.104

1 8.4

0.187 0.27

Living Space Living Space, Q~ + Ground, Q~ W w

ql/q2 1.039 0.655

2332 1470

26746 10506

10.2

0.58

1303

6543

11.0

0.565

1268

4756

0.354

11.5

0.576

1293

3739

0.437

11.9

0.606

1360

3085

0.52

12L2

0.656

1472

2629

0.604

12.4

0.733

1645

2296

0.687

12.7

0.853

1914

2044

0.77

13.4

1.0

2243

2243

0.854

15.6

1.0

2243

2243

0.937

17.8

1.0

2243

2243

Table 2. The effect of ambient temperature on system performance. Ground temi)erature = 12.8°C(55°F) Ri = R2 = 2.11 (W/m e - °C)-I (12(Btu/hr- ft2°F)-~). Other parameters as shown in Fig. I Heat Loss from: Ambient Temp. °C

LoopTemp. Living Space, Q1 °C W

Living Space + Ground, Q3 W

QI/Q2

-28.9

10.7

1485

5745

-23.4

11.0

1427

4988

0.404 0.436

-17.8

11.3

1370

4231

~.477

-12.3

11.7

1313

3474

0.532

-6.7

12.0

1255

2717

0.608

-1.2

12.4

1198

1961

0.722

4.4

12.7

1140

1204

0.907

10.0

14.7

853

853

1.0

15.5

17.5

450

450

1.0

Table 3. The effect of increased ground temperature on system performance as a function of ambient temperature. Ground temperature=18.3°C (65°F), RI=R2=2.11(W/m2-°C)-~ (12)Btu/hr- ft 2 - °F)-I). Other parameters as shown in Fig. 1 Heat Loss From: Ambient Temp. °C

Loop Temp.

Livin~lSpace ,

°C

W

Living Space

+ Ground, OR W

ql/q2

-28.9

15.5

663

6444

0.187

-23.4

15.8

606

5688

0.193

-17.8

16.2

548

4931

0.201

-12.3

16.5

491

4174

0.211

-6.7 -1.2

16.8 17.2

434 376

3417 2660

0.225 0.248

4.4

17.5

319

1903

0.286

10.0

17.9

261

1146

0.367

15.5

18.2

204

389

0.659

decreases because the ground heats the loop causing T4 to increase as shown in Fig. 2, and the living space heat loss to decrease. As R1 increases still further, QI/Q~ increases because R2 is decreasing and T4, as shown in Fig. 2, is increasing very slowly. Finally, when RI is such that T~ = T2, the ground coupling ceases, and QJQ2 = 1, regardless of T4. The values of Q, that correspond to Fig. 2 are presented in

Table 1. Values for Q3 = Q~+ 8A2U2(T2- T4), the total heat loss rates from living space and ground, are also presented. Note that the reduction in the living space heat loss is "paid for" by increased heat loss from the ground. For example, when (RI/RI + R2)= 0.27, and Qi is at its minimum value, 1268 W, Q3 is 4756 W. Hence, the ground loss is 3488 W. However, at RJ(R~ +R~) =0.437, O~ is still low. 1360W. and

Technical Note yet Q3 is down to 3085 W, making the ground loss 1725, about half of the value obtained when QJQ2 is minimum. For the results presented in Tables 2 and 3, Rt is set = R2 = 2.11(W/m2-°C) -t, (12(Btu/hr-°F-ft2)-b. Note that the contribution of the loop is greater when the ambient temperature is lower. This makes the concept attractive in colder regions if the ground temperature can be maintained. The importance of ground temperature can be seen by comparing Tables 2 and 3. It is hoped that this rather restricted analysis provides some understanding of the importance of heat transfer from the ground to the upper portions of the shell. It is important to note that in this analysis the ground temperature is fixed. The ability of the ground surface to be recharged, either from above from solarheated air or from below, must, of course, be included in a more complete analysis.

463

Q3 rate of heat loss from living space plus ground, W Tj temperature /3/ heat transfer coefficient, W/m 2 - C

Subscripts i= 1 2 3 4 ./= I 2 3 4

outer shell ground-loop inner shell floor ambient ground living space loop

Greek ~$ a switch defined by eqn (2)

NOMENCLATURE A~ area across which heat flow, m2 Q~ rate of heat loss from living space with loop, W Q2 rate of heat loss from living space without loop, W

RE~..RENCF_~ 1. Shurcliff, W. A. Superinsulated Houses and Double-Envelop Houses. Shurcliff, Cambridge, Mass. (Mar. 1980).