The basic parabolic trough STEGE process has a large area of parabolic reflectors that concentrate insolation to absorber pipes through which
a heat transfer fluid (typically) flows to receive the energy. The pipes have a selective surface and are surrounded by a clear glass concentric tube to minimize heat transfer to the surroundings, similar to the cover used in a flat plate solar energy collector.
The heat transfer fluid can be a synthetic oil, like Therminol VP1, whose properties are given in Appendix B, that can withstand the high temperatures required to establish an efficient process. Yet, the development of new heat transfer fluids is an active area of research and in some cases molten salts are being considered. The preferred properties of these materials mean that they have a vastly reduced vapor pressure as a result of a high boiling temperature that does not limit the process. If water were used, high pressure steam would be directly generated in the absorber pipes placing some engineering constraints on the system.
A schematic of the process is shown in Fig. 9.6, demonstrating the two separate parts: the solar field which recirculates the heat transfer fluid and the Rankine cycle which has water as the working fluid to generate electricity. The solar field has Ns parabolic reflector units in series, called a loop as will be evident below, and there are Np of these series systems in parallel for a total of Nu = Ns x Np units.
The units placed in series will be used to heat the fluid up to its maximum temperature (i. e. from Tcold to Thot) while the units in parallel are required to increase the overall heat content that can be supplied to the Rankine cycle. The total rate of heat transfer is Qhe, which occurs within the heat exchanger, and is required to heat the water in the Rankine cycle. Of course, Qhe = Qb for the Rankine cycle, discussed above. The schematic has the two streams operating countercurrently in the heat exchanger, meaning that they travel in opposite directions. This mode of operation is fairly efficient in promoting heat transfer since the overall thermal driving force, the temperature difference between the two fluids, remains larger than for cocurrent operation. Cocurrent operation is used in cases where it is desired to reduce the average hot fluid temperature. This is useful if this fluid is temperature sensitive, however, a larger heat transfer area is required that will increase the cost. Suffice to say that the heat exchanger will be taken as a known quantity here and that it operates at 100% efficiency, allowing all the energy garnered within the solar field to be transferred to the water in the Rankine cycle.
The design of this plant begins with knowledge on the amount of electricity that one wants to generate Wnet which will be 30 MW for the example given in this section, a value for some operational units and enough power for a small town or village. Then other details of the cycle are assumed, as discussed below in Example 9.2, to ultimately produce
Qhe = Qb.
Modeling of the solar field is similar to that for the flat plate solar energy collector. The operating line comes from the FLOT while the design line is determined through detailed heat transfer calculations for the fluid flowing in the pipes. Of course, the operating point is at the intersection of the two lines. There is a difference between the design










Fig. 9.8 An idealized version of the SEGS VI. There are Ns = 192 units in series, each unit is 4 m long, which makes up a loop. There are Np = 50 loops that operate in parallel making the total number of units Ns x Np = 9600. The aperture W is 5.0 m and so the total aperture area is 192,000 m2 which is slightly greater than the area used in reality, 188,000 m2. The electrical power generated by this power plant is 30 MW. The total mass flow rate of the heat transfer fluid is mf and the mass flow rate in each loop is mо = mf /Np.
the operating line (eqn (9.7)) for the entire solar field can be written in two forms:
Q he = m f Cpf AT and
Q he = Np Ns [Cr aPn (0) – (lout ] A (9.9b)
which is the mass flow rate in each loop of Ns units.
The inherent assumption we will make in using eqn (9.9b) is that the absorber pipe temperature will be constant along its length. This sets qout (see eqn (9.12) below) and everything in the equation is known except Ns, after Np is determined. The proper, detailed way to determine the operating point of the system is to use eqn (9.7) as the operating line together with eqn (9.18), shown below, as the design line on each 4 m long unit and calculate the absorber pipe temperature as well as the heat transfer fluid temperature rise on that unit. The temperature rise is added to the entry temperature of the first unit and is taken as the entry temperature to the second unit and the operating point again
determined. This process is repeated until the heat transfer fluid reaches the required temperature and one can calculate Ns (once Np is known).
The procedure advocated here is conservative by assuming a value for the absorber pipe temperature (Tp) and will be justified later. The temperature Tp is set to be the maximum at which the heat transfer fluid can be used, its degradation temperature. This is typically of order 400 °C at least for contemporary organic oils. The temperature rise is set so that Tcold is not too low or the viscosity of the fluid will become too large and will increase the pressure drop in the loop. Operational STEGEs have AT equal to 100 °C thereby making Tcold = 300 °C since That ~ 400 °C. Having a smaller AT will reduce the pressure drop as well as subsequent pumping costs. If AT = 200 °C rather than 100 °C then pumping costs will increase by 1020%, which is significant.
The conservative nature in assuming Tp = 400 °C is that the radiant heat loss will be larger than if the pipe temperature is allowed to gradually increase along the loop. This will occur as more solar energy is absorbed by the pipe and given to the heat transfer fluid. The error in the constant Tp assumption is only 1015%, which is acceptable to the level of the design we wish to accomplish here (see Example 9.4).
A temperature rise in the solar field of 100 °C will be assumed and sets the heat transfer fluid mass flow rate
Q he
Cpf AT
after using eqn (9.9a). Remember Qhe is known since this must be the heat supply to run the Rankine cycle.
As mentioned above, the absorber pipe temperature Tp will be as large as possible, a temperature just at the organic oil’s degradation temperature, and for Therminol VP1 this is 400 °C. Other commercial products have similar degradation temperatures and Tp should be assumed accordingly. Knowing this temperature allows the heat transfer rate from the absorber pipe to the atmosphere, qout, to be calculated from
qout = hpa [Tp – Ta] + ep&S [Tp – Tsky] (9.12)
since the convective and radiative heat transfer rates operate in parallel. Here hpa is the convective heat transfer coefficient for heat transfer from the absorber pipe to the air, Ta, the air temperature, ep, the (selective surface) absorber pipe emissivity and Tsky, the effective sky temperature for radiative heat transfer calculations (see note 2 in Section 8.1).
The convective heat transfer coefficient hpa can be estimated with a standard correlation for flow of air past a cylinder,
Nu = 0.35 + 0.56Re052 (9.13)
where Nu and Re are the Nusselt and Reynolds number, respectively, given by Nu = hpaD/kair and Re = pairvwD/pair. The parameters are: kair is the thermal conductivity of air, pair, the density of air, pair, the viscosity of air and vw, the wind velocity. As we have done before, the
physical constants for air will be determined at the mean temperature Tm = [Tp + Ta]/2. Since the mean temperature is so high for this system, a glass envelope is not used, the correlations given in Appendix B are not valid and the ones in Table 9.3 should be used.
Now one can determine Ns since everything is known in eqn (9.9b), except Ns and Np, and will be found from
Q he
Np [CraPD (0) – <iout A
Of course the value of Np is one parameter that is not easily assumed, however, its value is determined by the amount of energy required by the Rankine power cycle and economic and pressure drop considerations. If it is too small then the flow rate in each loop is large, as is the pressure drop, which increases pumping costs. The pressure drop ДP can be determined from the definition of the friction factor f given in eqn (8.14) to be
where Dp is the inner diameter of the absorber pipe. The flow will be found to be turbulent in the absorber pipe and so the friction factor is fairly insensitive to the Reynolds number, see eqn (8.15b), and of order 3 x 102 for typical conditions within a loop. So, one can determine the variables to which the pressure drop is most sensitive by using eqns (9.10), (9.11), (9.14) and (9.15) to arrive at
where ~ means ‘scales as,’ see note 3 in Section 7.1. The dependence of the pressure drop on Np is large, as it is with Qhe, while it is less sensitive to ДТ. If Np is halved from 50, the value used for SEGS VI, the pressure drop is expected to increase by a factor of 8, quite large. In fact, a detailed calculation shows it will increase by a factor of 6.7, demonstrating that the simple relation in eqn (9.16) is a reasonable one. Note, increasing the heat load on the solar field Qhe similarly increases the pressure drop and doubling it also increases it by a factor of 6.7, again substantial. Thus, one must perform a true optimization for the process; an increase in Np will reduce ДР and the associated operational costs versus an increase in Np that will increase the capital cost of the power plant. This type of design is beyond the scope of this monograph.
Here the value of Np will be determined heuristically rather than having to perform a complete power plant design that would also have to include economic considerations. The SEGS VI STEGE has 1.67 loops/MW while more recent STEGEs in the Mojave Desert have 1.78 loops/MW and 1.85 loops/MW for the SEGS VIII and SEGS IX, respectively. Given these values one can assume a value of 1.75 loops/MW or





















Given the 30 MW Rankine cycle in Example 9.2, design the solar field required to generate the steam in the power plant. By design it is meant to find mf, Np and Ns. Use the standard conditions in Table 9.4 and the heuristic for Np given in eqn (9.17). This solar field will be close to the size of the SEGS VI STEGE in the Mojave Desert.
The first parameter to be determined is Np. The heuristic given in eqn
(9.17)
is used to find, to the nearest integer and rounding up, Np = 53. Now Ns can be found from eqn (9.14),
however, qout is not known a priori. This is found from
(lout = hpa [Tp – Ta] + epPS [Tp – Tpky] (9.12)
since Tp will be assumed to be equal to 400 °C. Everything is known in this equation except hpa, the heat transfer coefficient between the absorber pipe and the surrounding air. This can be determined from eqn (9.13),
Nu = 0.35 + 0.56Де0’52 (9.13)
The mean temperature of the pipe and air is 212.5 °C and care must be taken to use the correlations in Table 9.3 for the physical properties of air. Since the wind velocity is 3 m/s one finds Re = 5960 and Nu = 51.8, making hpa = 29.0 W/m2K, which is quite large. Inserting numbers in eqn (9.12), the rate of heat transfer from the system is found to be q_out = 1.26 x 104 W/m2K.
One can determine Ns from eqn (9.14). Before this is done the net rate of heat transfer to the fluid is calculated by CroPd (0) – q_out to be 8.99 x 103 W/m2. So, even though the amount of heat coming into the system is large, CroPd(0) = 2.16 x 104 W/m2, there is a large rate of heat transfer out of the system, resulting in a relatively small net rate of heat transfer to the heat transfer fluid. This is obviously due to the fact that an envelope is not used around the absorber pipe and is a similar circumstance to the flat plate solar hot water heater without a cover.
Putting in all the numbers results in Ns = 186, close to that used in SEGS VI, which has approximately 200 in series. The total number of units is Nu = Ns x Np = 9858 and since the aperture area of each unit is 20 m2 the total aperture of the solar field is 1.97 x 105 m2, while that for SEGS VI is 1.88 x 105 m2.
Table 9.4 Standard conditions used in designing a solar thermal energy generated electricity power plant in this chapter. See Figures 9.5, 9.6 and 9.7 for definitions of some variables. The insolation was assumed to be 1000 W/m2, similar to the AM1.5G spectrum, allowing ready comparison to solar photovoltaic devices and unless otherwise stated this is direct beam insolation. The parabolic trough unit aperture is the aperture of the solar collector assembly or SCA which is the device that holds the parabolic reflector and tilts the reflector to have the aperture normal to the direct beam. Each SCA holds a number of heat collection elements or HCEs which are the absorber tubes and are called the parabolic trough unit length in the table.
Insolation (Bter)
Heat (energy) required (Qb)
Air temperature (Ta)
Wind velocity (vw)
Rankine cycle
Pump inlet water pressure (Pa)
Water state at pump entrance Heat exchanger exit temperature (Tc)
Heat exchanger exit pressure (Pc)
Solar field
Heat transfer fluid temperature rise (ДТ) Absorber pipe temperature (Tp)
Temperature difference (ST = Tp – Thot) Concentration ratio (Cr)
Focal length (f)
Parabolic trough unit aperture (W)
Parabolic trough unit length (L)
Absorber pipe inner diameter (Dp)
Absorber pipe outer diameter (D)
Envelope inner diameter (Dj)
Envelope outer diameter (Do)
Absorber pipe short wavelength absorbance (ap) Absorber pipe long wavelength emissivity (ep) Envelope absorptivity (ae)
Envelope emissivity (ee)
Heat transfer fluid
One should determine if heat transfer is a limiting process in a loop. The mass flow rate of the heat transfer fluid, mf, from eqn (9.11) is found first, which is written as mf = Qhe/Cpf AT. The heat load on the heat exchanger (or Qb) is 78.0 MW from Example 9.2, while the temperature rise for the fluid AT will be и 100 °C. All that needs to be determined is the the heat capacity for Therminol VP1, which is given in Appendix B as a function of temperature.
The exit temperature from a given loop is Thot which is ST below the absorber pipe temperature Tp. This takes into account any heat losses, such as to the SCA structure itself, that are not accounted for, as well as reduction in the heat transfer due to fouling the inside of the absorber pipe. Thus, the mean temperature is Tm = [Thot + Tcold]/2 = 345 °C and Cpf is 2465 J/kgK. Combining the above results in mf = 316.5 kg/s, quite large!
This allows mo to be determined from eqn (9.10), which is 5.971 kg/s. The Reynolds number can be found for the heat transfer fluid (Re = [4/n] x [m0/pDp]) after calculating the viscosity p at Tm for Therminol VP1 to be 6.13 x 105 Pas. The friction factor is calculated with eqn (8.15b) to be 2.82 x 103. Now the Nusselt number Nu can be determined with eqn (8.13) and is equal to 2.25x 103, quite large too!
The heat transfer coefficient hf is now calculated to be 2.97 x 103 W/m2K after the thermal conductivity for the fluid is determined at Tm (0.0873 W/mK). This makes the absorber number Ab equal to 31.2 for the 186 units in series. Clearly, based on eqn (9.18) heat transfer is not a limiting factor.
Although the absorber pipe temperature is assumed constant to be at 400 °C this is not a poor assumption and greatly simplifies the design. The next example is given to gauge the goodness of this assumption and a more detailed calculation is given.
Determine how accurate the assumption of a constant absorber pipe temperature is in Example 9.3.
Demonstrating this is not conceptually difficult, one must apply eqn (9.9b) on each unit, starting with the first where the cold heat transfer fluid enters. This equation is rewritten as
Qo = Щг = [CroPd(0) – qout] A operating line (9.21)
Np
for a single 4 m long unit. The definition of the rate of heat transfer required for each loop Q0 is evident from the equation. The rate of heat transfer out of the unit is given by eqn (9.12) and is
qout = hpa [Tp – Ta] + ep&S [Tp – Tsky]
The design line for a unit is given in eqn (9.18) and is rewritten as
where Tin and Tout are the inlet and outlet temperatures to/from the unit. The initial unit’s inlet temperature will be Tcold and the final unit’s outlet temperature will be Thot. The heat transfer fluid’s physical properties will be determined at the mean temperature for the given unit, Tm = [Tin + Tout]/2. The heat transfer coefficient will be found from the Nusselt number correlation given by eqn (8.13).
Everything is known in the above equations except Tp and Tout for each unit. One starts with the first unit and then continues to the next and soforth. The result of this calculation is given in Fig. 9.9 assuming each unit to be 4 m long. The cold heat transfer fluid enters the first unit at 295.0 0C and the absorber pipe achieves a temperature of 300.2 0C while there is a 0.81 0C fluid temperature rise to make Tout = 295.8 0C. The second unit now has an inlet temperature of 295.8 0C and so on. The upper graph in Fig. 9.9 shows this calculation, which requires about 166 units to have Tout = 400 0C.
The lower graph in the figure shows the effect of unit length on the total loop length to achieve a final outlet temperature of 400 0C. Various unit lengths L were assumed to discretize the system more and more coarsely until only one unit of length 744 m was assumed, which is the loop length in the previous example. Of course for this unit length the absorber pipe temperature was also 400 0C for its entire length.
Remarkably, assuming unit lengths of 480 m made little difference and a total loop length of 664 ± 3 m was required. As mentioned above, having only one unit in the loop makes the overall heat loss from the system greater, requiring a longer loop length. The heat transfer rate out of the system qout was 1.26 x 104 W/m2 using a single unit with a length of 744 m in the loop, compared to 1.02 x 104 W/m2 if 166 x 4 m long units were placed in series and the absorber pipe temperature was allowed to gradually increase. Interestingly, the result of this calculation is that it is only 12% longer than that from the more detailed calculation making this a reasonable approximation.
These examples demonstrate an approximate way to design a solar thermal energy generated electricity system. However, to keep the analysis as simple as possible, the effect of an envelope around the absorber pipe was not considered. This complicates the analysis in the same manner that a cover did for the flat plate solar hot water heater, an additional criterion for equilibrium must be established. Most power plants have envelopes and the effect of this additional element in the STEGE is discussed in the next section.