(n + 1)KlV"(v Pcollcp +P Acoll)2)= 0

If n = 2, Eq. (24) is a fourth order polynomial for which an analytical solution procedure exists. Otherwise it has to be solved numerically for Vmfp Coll. It will be more instructive, however to compare power versus flow graphs for the constant m and variable collector efficiency approaches. Fig. 3 shows that for a 100 MW test case from Schlaich (1995), Vmfp pl and Vmfp Coll are quite similar, but both differ substantially from V*, the flow at which the turbine pressure drop is 2/3 of the pressure potential. It also shows that use of the 2/3 rule seriously over estimates the maximum flow at which the plant produces any power at all. This flow has a large effect on the turbine runaway speed. Table 2 summarizes similar comparisons for the data from Schlaich for several test cases.

Typically Vmfp pl and Vmfp Coll are between 67 and 62 % of V*, and the corresponding optimal turbine pressure drops are between 173 and 200 % of the values associated with V*. It is encouraging to see that the simple power law model predicts the maximum power flow within one percentage point compared to the MFP Coll. Model in all the test cases and is pessimistic in the prediction of the maximum fluid power value, and optimistic in the prediction of turbine pressure drop.

CONCLUSIONS

The study developed two analyses for finding the optimal ratio of turbine pressure drop to available pressure drop in a solar chimney power plant for maximum fluid power. In the first part the system pressure potential is assumed to be proportional to Vm where V is the volume flow and m a negative exponent, and the system pressure loss is proportional to Vn where typically n = 2. Simple analytical solutions were found for the optimum ratio of pt/pp and for the flow associated with it. This ratio is not 2/3 as used in simplified analyses, but depends on the relationship between available pressure drop and volume flow, and on the relationship between system pressure loss and volume flow. The analysis shows that the optimum turbine pressure drop as fraction of the pressure potential is (n-m)/(n+1), which is equal to 2/3 only if m = 0 (i. e. constant pressure potential, independent of volume flow) and n = 2. Consideration of a basic collector model proposed by Schlaich led to the conclusion that the value of m is equal to the negative of the collector floor-to-exit heat transfer efficiency.

The basic collector model is sensitive the effect of volume flow on the collector efficiency. Its introduction into the analysis indicated that the power law model is conservative in its prediction of maximum fluid power produced by the plant, and in the magnitude of the flow reduction required to achieve this. It was shown that the constant pressure potential assumption may lead to appreciable under estimation of the performance of a solar chimney power plant, when compared to the model using a basic model for the solar collector. More important is that both analyses developed in the paper predict that maximum fluid power is available at much lower flow rate and much higher turbine pressure drop than the constant pressure potential assumption predicts. Thus, the constant pressure potential assumption may lead to overestimating the size of the flow passages in the plant, and designing a turbine with inadequate stall margin and excessive runaway speed margin. The derived equations may be useful in the initial estimation of plant performance, in plant performance analyses and in control algorithm design. The analyses may also serve to set up test cases for more comprehensive plant models.