## POWER LAW MODEL

The derivation requires two generalisations: a pressure potential versus flow relationship, and a system pressure drop versus flow relationship. A very simple but useful assumption for the relationship between pressure potential and volume flow is:

pp = KPVm (1)

 Figure 1: Pressure potential vs. volume flow

where Kp = ppref/Vrefm is determined at a reference point (Vref, pref) near the optimum, and m is a negative exponent. Fig. 1 shows pressure potential lines for a few values of m. Note that if m = 0, then pp = Kp, denoting a constant pressure potential.

A useful assumption for the system pressure drop in incompressible flow is: Pl = K|_Vn

 (2)

where n will typically be 2 when system pressure drop is dominated by minor losses, and closer to 1.75 when the pressure drop is dominated by Reynolds number dependent wall friction losses (White, 2003). The solid line in Fig. 2 represents the system loss curve.

Note that the effect of the variation of density with temperature rise through the system is disregarded, but may be included in the choice of K and n in the vicinity of each operating point. The turbine pressure drop is then: pt = pp — pL = pp — KL V (3)

 Figure 2: Pressure potential and pressure loss vs. volume
 flow, and fluid power (shaded area)

Since the change in density across a solar chimney turbine is typically small (Apt < 2 %) we can regard the air flowing through the turbine as incompressible, i. e. the fluid power is equal to the product of the volume flow and total pressure drop across the turbine:

P = pt V = (pp — Kl Vn) V (4)

The shaded area in Fig. 2 represents the fluid power. The power generation rate of the turbine depends not only on the characteristics of the flow system it is part of, but also on those of the turbine itself. In the present paper, however, we assume that the turbine efficiency does not vary appreciably with changes in flow rate, or, if it does, the variation in turbine losses may be accounted for in the system pressure losses.