Consider steady state operation of a solar chimney power plant. The solar collector consists of a transparent deck over a floor that receives solar energy. The collector floor

transfers energy to the air flowing over it at the same rate at which it receives energy from the sun, namely at a G Acoll, where a is the effective absorption coefficient of the collector. The air in the collector loses heat through the collector deck at the same rate that the collector deck loses heat to the environment, namely at pATAcoll, where p is an adjusted heat transfer coefficient that allows for radiation and convection losses and the fact that the temperature difference between deck and environment increases from 0 at the outer edge of the collector to AT at the chimney entrance. The real situation is more complex, but this simple model employed by Schlaich (1995) may be used to derive an approximate expression for the collector temperature rise and the exponent m of the analysis above. Find the air temperature rise by equating energy entering and leaving the collector:

Qfloor — Qdeck — Qcfe

a G Acoll PAT Acoll V pcollcpAT

(11)

AT —___ a G Acoll____

V pcollcp + P Acoll

If the ambient temperature at ground height is T0, the collector exit (chimney inlet) density is pcoll and assuming parallel temperature profiles inside and outside the chimney a chimney of height Hc will generate a hydrostatic pressure potential, pp:

AT

pp — pcoll g Hc

1 0

pcoll g Hc____ a G Acoll (12)

T0 V pcollcp + в Acoll

This equation is of the form: pp —

C1

C2 V + C3

(13)

It can be shown that for a relationship of the form p = A Vm, m depends only on the local value of the function and the local value of its gradient, and is given by:

m —-

dpp pp

“dV V (14)

For a function of another form, for example Eq. (13), an equivalent m can be calculated at any point, since m depends only on the coordinates of the point and its local gradient:

— dpp V

dV pp

‘ C1C2

(2 V + C3 )2

V(c2 V + C3)

C2V

(2 V + C3;

Back substitute for C2 and C3 and multiply the denominator and numerator by AT:

V p collc p AT

m = _

(V p coiicp AT + p Aeon AT) = _ V pcollCpAT a G A coll

Qfloor Q

deck

Q

ncfe

floor

(16)

= _ a G A coll _ в A coll AT a G Acoll

Here ncfe is the net rate at which heat is absorbed by the air between the inlet and exit of the collector, expressed as fraction of the rate of heat transfer from the floor to the air. We shall call it the collector floor-to-exit efficiency since it is a measure of how efficiently the collector transfers heat from its floor to the air leaving the collector. Schlaich (1995) writes the standard collector efficiency for his collector model as:

PAT (17)

ncoll =«_ g

The collector transfer efficiency can be written similarly by dividing out aGAcoll in Eq. (16):

ncfe 1

PAT

aG

= n coll (18)

a

It is remarkable that in the case of the simplified solar chimney collector model, the exponent m turns out to be simply the negative of the collector floor-to-exit efficiency. The immediate implications are the following:

• m must have a value between 0 and -1

• for n = 2, the optimum pt/pp is between 2/3 and 1

• the optimal pt/pp ratio is 2/3 only if the collector efficiency equals zero.