J. P Pretorius, Dept. Mech. Eng. Univ. of Stellenbosch, South Africa D. G. Kroger, Dept. Mech. Eng. Univ. of Stellenbosch, South Africa J. D. Buys, Dept. Math. Univ. of Stellenbosch, South Africa T. W. Von Backstrom, Dept. Mech. Eng. Univ. of Stellenbosch, South Africa

The performance of large scale solar tower power plants are evaluated. The investi­gation focuses on a reference plant with a 4000 m diameter glass collector roof and a 1500 m high, 160 m diameter tower. A numerical simulation model solves the relevant draught and conservation equations using specified meteorological input data for a particular location in South Africa. It is shown that plant power output varies consid­erably with the time of day as well as monthly. The dependency of the power output on collector diameter and tower height is illustrated, while showing that greater power production can be facilitated by optimizing the solar collector roof shape and height.

1 Overview

A solar tower power plant, illustrated schematically in figure 1, consists of a transparent circular collector supported relatively low above the ground surface. Central to the collector is a high tower with one or more turbo-generators located at its base. Solar radiation penetrates the collector roof and heats the ground beneath, which in turn heats the adjacent air, causing it to rise through the central tower which drives the turbine and consequently generates electricity. Performance measurements from a 50kW prototype solar tower power plant plant built in Manzanares, Spain in 1982 proved that the solar tower concept is technically reliable ((Haaf etal., 1983; Haaf, 1984)) and potentially economically viable (Schlaich, 1994). The first relatively detailed published analyses of the performance of such a plant were presented by Gannon and Von Backstrom (2000) and Kroger and Buys (2001). More recent publications are by Gannon and Von Backstrom (2003) and Bernardes et al. (2003).

This study determines the performance characteristics of a reference solar tower power plant, as specified in the appendix of this document, for the meteorological conditions given in tables 1 and 2.

2 Analysis

2.1 Collector

Relevant conservation equations are derived for a defined elementary control volume in the collector of the solar tower power plant. Since changes in the dynamics of the air stream are relatively slow, transient terms in the conservation equations are found to be negligible.

2.1.1 Continuity

Assuming purely radial airflow in the collector, the steady state collector continuity equation is [15]

2.1.2 Momentum

The simplified steady state momentum equation applicable to a collector control volume is

There exists a developing flow region near the collector inlet which becomes fully developed fairly rapidly. This developing flow region was investigated by Kroger and Buys (1999), but is not considered in this study. For a fully developed flow assumption in the collector, according to Kroger and Buys (1999) the roof shear stress is given by тг = 0.02[(p°8v 1-8ц°-2)/(Н0-2)] while the ground shear stress is given by тд = 0.014875 pv2(eg/2H)0254[1.75 (ц/pvtg)°51+1].

The collector roof height at a specific radius is evaluated according to H = H2 (r2/r)b. Sup­ports are arranged under the collector roof at specified radial and tangential pitches. For an annular control volume (360°control volume between two collector radii), the support drag force per unit radial length is given by Fsupports = ^ FsrD/Ar where ^ FsrD is the sum of all the support drag forces in the specific control volume. The drag force enforced on the air by all the supports at a specific collector radius is given by FsrD = (CsD m2ds rb-1) / (4npP(H2 rbb).

2.1.3 Energy Collector roof

The steady state energy balance for a transparent collector roof is

aeb ihb + aed ihd + qgr = qra + qrs + qrh (3)

Approximating the collector roof as a horizontal surface, the total solar radiation incident on the roof is Ih = Ihb + Ihd. The respective beam and diffuse effective absorptivities are determined by aeb = (ay, b + a±, b)/2 and aed = (ay, d + a±, d)/2, where the respective parallel and perpendicular polarization components are evaluated according to a =

Ta)] / (1 — PTx). Relations for the roof interface reflectivity are given by Modest (1993) as

Ру = [tan2(0i — 02)] / [tan2(0i + 02)j and p± = [sin2(01 — 02)j / [sin2(01 + 02)j. According to Duffie and Beckman (1991), an equivalent incidence angle of 0! = 60° should be used for diffuse solar radiative calculations. The refractive angle 02 is determined using Snell’s law.

Modest (1993) gives a relation for the transmissivity due to the absorptance as та = exp[-(Cetr)/cos 02]. The radiation heat flux from the ground to the collector roof is given by qgr = o(Tf — Tf / (1 /eg + 1 /er — 1). The convection heat flux from the collector roof to the ambient air is qra = hra(Tr — Ta), where hra is evaluated according to the very approxi­mate equation hra = 5.7 + 3.8vw (Duffie and Beckman, 1991). The radiative heat flux from the roof to the sky is qrs = erc(T4 — Tfy), where Tsky = 0.0552 T)5 according to Swin — bank (1963). The convection heat flux from the collector roof to the air in the collector is qrh = hrh(Tr — T), where hrh is determined using Gnielinski’s equation for fully developed turbulent flow: hrh = {[(f/8)(Re — 1000)Pr] / [1 + 12.7(f/8)1/2(Pr2/3 — 1)]}(k/dh). Gnielin — ski’s equation alone will tend to underestimate the convection heat transfer rate since it does not make provision for effects due to natural convection. For smooth surfaces the friction factor is given by f = (1.82 log10 Re — 1.64)-2 (Kroger, 2004). For rough surfaces, Haa — land (1983) recommends f = 0.3086 [log10(6.9/Re + (e/(3.75dh))1-11)]-2 for t/dh > 10-4 and f = 2.7778 {log10[(7.7/Re)3 + (e/(3.75dh))3-33]}-2 for cases where t/dh < 10-4.

Ground

The energy balance at the ground surface (z = 0) is

The respective beam and diffuse effective transmissivities are determined by Teb = (ту, b + т±,b)/2 and Ted = (ту, d + т±,d)/2, where the respective parallel and perpendicular polariza­tion components are evaluated according to т = [(1 — р)2та] / (1 — р2тО). From Duffie and Beckman (1991), the beam and diffuse transmittance-absorptance products of equation (4) are determined by (теag) = теag / [1 — (1 — ag)pd], where pd may be estimated using the equation by Duffie and Beckman (1991): pd = т^ — Ted. The convection heat flux from the ground surface to the air in the collector is qgh = hgh(Tg — T), where hgh is determined using the previously mentioned Gnielinski equation. Assuming constant ground properties, the energy balance below the ground surface (z > 0) is

At a considerable depth in the ground the temperature gradient becomes zero and the boundary condition dTg/dz « 0 is valid.

Collector air

When only regarding terms of significant order, the steady state energy balance for the air in a collector control volume becomes

qrh + qgh = pvH —{Cp T)

2.2 Tower

Relevant conservation equations are derived for a defined elementary control volume in the tower of the solar tower power plant.

2.2.1 Continuity

The tower steady state continuity equation is

d

(p, W )=0 (7)

2.2.2 Momentum

The simplified steady state momentum equation applicable to a tower control volume is

According to White (1999), the tower wall shear stress is determined by Tt = (ft ptvf)/8. The total bracing wheel drag force per unit tower height, based on the tower inlet dynamic pressure, is given by Fbw = [AtKbw (Ptiv%)/2 nbw ] / H.

2.2.3 Energy

When only regarding terms of significant order, the steady state energy balance for the air in the tower is

d / d

P, W g^(CPt + dZ (P,* gz) = 0

2.3 Power

The power generated by the turbine is P = ntgApturbVavg, where the pressure drop across the turbine is calculated by the draught equation:

Apturb = Ap — (Ар, + Apcoii + Apturb, І + Ap, + Apto + Apdyn) (10)

The driving force or potential that causes air to flow through the solar tower power plant is due to a pressure difference between a column of cold air outside and a column of hot air inside the tower. Assuming a dry adiabatic lapse rate (DALR) for the air outside and inside the tower, we find the drive potential from relations by Kroger (2004): Ap = pi{1 — [(1 — 0.00975 (Ht/T|)) / (1 — 0.00975 (Ht/T5))]3-5}, where the numbered subscripts refer to the positions in figure 1. Assuming a constant mass flow rate and constant specific heat capacity over the turbine, the temperature drop across the turbine can be expressed as T5 = T4 — (ApturbVavg )/(mCp).

The collector inlet pressure drop between the essentially stagnant air at 1 and the inlet at 2 (see figure 1) is Ap, = (pi — p2) = K (p2v|)/2 + (p2v|)/2. The pressure drop in the collector, Apcoli, caused by accelerating radial airflow, roof and ground friction and roof support drag forces are all incorporated in the collector momentum equation (equation (2)). The pressure drop over the turbine inlet is Apturb, i = p3 — p4) = Kturb, i (P4 V42)/2 + (p4 V42)/2 — (P3 V32)/2. The accelerating tower airflow, the inside tower wall friction and the internal bracing wheel drag forces result in a pressure drop over the tower height, Apt, and are incorporated in the tower momentum equation (equation (8)). The air exiting the tower experiences a pressure differential due to the shape of the tower outlet, and is expressed as Apto = Kto (p6vf)/2. Employing a relation by Kroger (2004), during relatively quiet (no significant ambient winds) periods, the tower outlet loss coefficient is approximated as Kto = -0.28Fr-1 + 0.04Fr-15,
where FrD is determined (see figure 1) by FrD = (m/A6)2 / [p6(P7 — p6)gdt]. The dynamic tower outlet loss is Apdyn = (p6vf)/2.

The analysis performed in this study does not differ significantly in approach to the paper by Bernardes et al. (2003). However, there are some notable differences: this study always assumes fully developed turbulent flow in the collector, takes into account the temperature change across the turbine and employs a quasi-steady state solution procedure. A draught equation calculates the pressure drop across the turbine for conditions determined during a specific iteration. At each time step, for the given environmental conditions, the mass flow rate through the system is optimized to produce a maximum plant power output. The study by Bernardes et al. (2003) selects a particular ratio of pressure drop over the turbine to total pressure difference (0.9) which governs the mass flow rate, pressures and temperatures throughout the plant. This paper does not specify such a ratio, but allows the optimized mass flow rate to govern the pressures and temperatures through the system.