TW von Backstrom and TP Fluri

Department of Mechanical Engineering, University of Stellenbosch
Private Bag X1, Matieland 7602, South Africa, Tel: +27 (0)21 808 4267
Fax: +27 (0)21 808 4958, E-mail: twvb@sun. ac. za

Abstract — Main features of a solar chimney power plant are a circular greenhouse type collector and a tall chimney at its centre. Air flowing radially inwards under the collector roof heats up and enters the chimney after passing through a turbo­generator. The objective of the study was to investigate analytically the validity and applicability of the assumption that, for maximum fluid power, the optimum ratio of turbine pressure drop to pressure potential (available system pressure difference) is 2/3. An initial power law model assumes that pressure potential is proportional to volume flow to the power m, where m is typically a negative number between 0 and -1, and that the system pressure drop is proportional to the power n, where typically n = 2. 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 when m = 0, implying a constant pressure potential, independent of flow rate. 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. A more comprehensive optimization scheme, incorporating the basic collector model of Schlaich in the analysis, shows that the power law approach is sound and conservative. It is shown that the constant pressure potential assumption (m = 0) may lead to appreciable under estimation of the performance of a solar chimney power plant, when compared to the analyses presented in the paper. More important is that both these analyses predict that maximum fluid power is available at much lower flow rate and much higher turbine pressure drop than predicted by the constant pressure potential assumption. 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 analysis and in control algorithm design. The analyses may also serve to set up test cases for more comprehensive plant models.

INTRODUCTION

In order to design a flow system containing a turbine for maximum power production and to run it at maximum power, engineers need to find the optimal pressure drop across the turbine as a fraction of the total available system pressure difference. The design flow rate through the system determines the size and cost of the plant flow passages as well as the size, design and cost of the turbine. In the design phase some iterative algorithm may suffice to find the optimum, but a simple analytical method would be more convenient in a control algorithm. It could also serve to set up test cases for more comprehensive methods.

Many solar chimney investigators have made the assumption that the optimum ratio of pt/pp is 2/3, (Haaf et al., 1983; Lautenschlager et al., 1984; Mullett, 1987; Schlaich, 1995). In more detailed calculations Schlaich (1995) apparently used an optimum value of pt/pp = 0.82 as evident from the values of pt and pp reported in tables. Hedderwick (2001)
presented graphs showing values around 0.7. Von Backstrom and Gannon (2000) used the 2/3 assumption only for optimization at constant available pressure difference, but Gannon and Von Backstrom (2000) employed an optimization procedure under conditions of constant solar irradiation. Schlaich et al. (2003) reported a pt/pp value of about 0.80, while Bernardes et al (2003) reported a value of as high as 0.97. The wide variation in values warrants further investigation.

The question is the existence or not of a relevant optimum pt/pp in solar chimney power plants, and how to determine it. Even under conditions of constant solar irradiation the pressure potential of a solar chimney plant is not fixed but is a function of the air temperature rise in the collector, which varies with flow rate.

The first objective of this paper applies to any general process where the pressure potential is not constant and the system pressure drop is not necessarily proportional to the flow rate to the power 2.0. The objective is to derive simple, generally applicable equations for the determination of the volume flow for maximum fluid power (MFP) and the associated ratio of turbine total pressure drop to pressure potential. The second objective applies to solar chimney power plants. It is to derive equations for finding the optimum flow rate and pt/pp conditions as dependent on the relevant design and operating conditions of the plant, using a simple solar collector model.