When a cover was placed on a flat plate solar energy collector it was found that the device became more thermally efficient despite the fact that the cover admitted less insolation. The gauge of the insolation reaching and absorbed by the absorber plate was || or ||, which is a complicated calculation requiring detailed knowledge of the cover and absorber material properties. In spite this, consideration of the product was given in Chapter 8; now the analysis becomes more challenging with a curved surface such as the envelope. In addition, the reflection optics, that is the reflection of insolation from the parabolic mirror to the absorber pipe, must also be considered to find an effective jar|. Much has been done on this topic and can be found in the references at the end of this chapter. Here we will write |ат| as apTe, where ap is the absorbance of the absorber pipe and Te is the transmittance of the envelope (exp(-aed), where ae is the envelope absorptivity and d is the envelope thickness, [Do – Dj]/2). One can always use this product as an empirical correction factor and a value of 0.7 may be appropriate after all the various factors are considered. Reference to Table 9.4 shows that apTe = 0.94, which will overestimate the performance of the absorber pipe system. Regardless, this value will be used and results derived through its use can, of course, be rectified by using other values, such as 0.7 mentioned above.

An envelope modifies the rate of heat transfer out of the system and is very effective at this. There will be series heat transfer as heat is lost by the absorber pipe to the envelope then from the envelope to the surrounding atmosphere. In addition there will be parallel heat transfer processes occurring through radiant and conductive and convective heat transfer from the absorber pipe and envelope.

The absorber pipe will lose heat to the surrounding glass envelope through radiative and conductive or convective heat transfer. Which mechanism dominates depends on the pressure in the annulus, as will become clear below. Similarly to what was done in the previous chapter the rate of heat transfer for a unit of length L from the pipe to the envelope Qpe is written

where kpe is an effective thermal conductivity for the gas in the annulus between the absorber pipe and the glass envelope, Di, the inner diameter of the glass envelope, Te, the envelope temperature (the inner and outer part of the envelope are considered to be at the same temperature) and ee, the emissivity of the envelope. The first term on the right-hand side of the equation is simple conductive heat transfer through an annulus, while the second term is radiative heat transfer across the same annulus where the area A is that for the absorber pipe nDL (compare to eqn

(8.31a) for radiative heat transfer between two parallel, flat plates). This radiative heat transfer term will not be derived here and can be found in most undergraduate heat transfer texts.

The effective thermal conductivity is a complicated function of pressure. At very low pressure, say below 100 Pa, one has free molecular conduction and

k Г1 11 / D

——— = 1 + bx — + — ln I — I P < 100 Pa (9.24)

kpe L D Di D)

where k, is the gas thermal conductivity at standard temperature and pressure (i. e. 0 °C and 1 atm.), b, an interaction coefficient and X, the mean free path between molecular collisions in the gas, which was mentioned in note 3 in Section 5.1. Parameters required for eqn (9.24) and below are given in Table 9.5. The remarkable fact is that for air, and most gases in general, k, as well as other physical properties, are not a large function of pressure. They are a function of temperature (except Cp to some degree) and so the pressure dependence of kpe is solely through the pressure dependence of x.

The mean free path is written here as (assuming ideal gas behavior)

kB Tm /2псРР

with kB being Boltzmann’s constant, Tm, the mean temperature in the annulus (Tm = [Tp + Te]/2), d, the molecular (collision) diameter of the gas molecule and P, pressure in the annulus.

The interaction coefficient is given by

b = [2 – a] x [9y – 5] 2a[y + 1]

where a is the thermal accommodation coefficient7 and 7 is the ratio of the constant pressure to constant volume heat capacity. An ideal gas would have 7 = 5/3 while the ratio for air is 1.4.

Although one may expect air to be the gas to leak into the annulus, it is in fact Hydrogen which can cause difficulties. Hydrogen permeates through the absorber pipe after formation through degradation of the heat transfer fluid. Since Hydrogen has a thermal conductivity approximately 6-7 times greater than air, even a small pressure of pure Hydrogen can affect performance. Thus, the molecular diameter, accommodation coefficient, interaction coefficient and heat capacity ratio are given in Table 9.5 for both air and Hydrogen. Hydrogen gas can be taken out of the annulus by evacuation or the use of a ‘getter’ material placed within the annulus to absorb Hydrogen.

The limiting value of kpe at higher pressures is k that occurs when X decreases in value, and this happens at a reasonably low pressure; for air this is at approximately 30 Pa, which is quite small. Increasing the pressure even further results in natural convection to increase kpe, similarly to that seen in the flat plate solar hot water heater, and the Rayleigh number Ra becomes important to determine kpe

with Ra* given by

where Ra was defined in eqn (8.34) and is written in present variables as

Ra = gXpC ATpeLpe (9.27c)

where g is the gravitational constant, x, the thermal expansion coefficient, p, the density, Cp, the heat capacity, k, the thermal conductivity at the temperature of interest, g, the viscosity, ATpe, the temperature difference between the pipe and envelope (in our case this will be Tp – Te) and Lpe, the gap between the pipe envelope ([Di – D]/2). As before all gas properties will be determined at the mean temperature [Tp + Te]/2. We note here that kpe in this regime of natural convection is not strictly correct, the assumption of using a thermal conductivity rather than a heat transfer coefficient is certainly not conceptually true. Yet, the changeover from conduction to convection necessitates the use of this nomenclature, where kpe should be referred to as an effective thermal conductivity, at least when natural convection effects occur.

The Prandtl number Pr for air can be assumed to be constant over the temperature range 200 0 C-400 0 C and equal to 0.68, while the other variables such as k, Cp, x and g are not a (large) function of pressure and are only that of temperature. One can determine x with the ideal gas law (x = 1/T) as well as p = PM/RT, where M is the molecular weight and R, the gas constant. The other variables can be determined with the relation in Appendix B or in Table 9.3.

The effective thermal conductivity will be taken as the maximum of either eqn (9.24) or (9.27) so

kpe = MAX(eqn (9.24), eqn (9.27)) (9.28)

The result in determining kpe for the standard conditions listed in Table 9.4 is shown in Fig. 9.10. The air pressure in the annular region between the absorber pipe and envelope was varied between 10-1 and 105 Pa. The amazing property of gases is that the thermal conductivity is relatively constant over an extended pressure range. Molecular conduction dominates until pressures of order 104 Pa are approached. Thereafter, depending on the temperature driving force, natural convection can occur to promote more efficient heat transfer. Obviously this is undesirable.

Now, just as with the flat plate solar energy collector, the rate of energy transfer from the system can be determined. The heat transfer

rate from the absorber pipe to the envelope, Qpe, is given in eqn (9.23), and must equal that from the envelope to the surrounding atmosphere, Qea, given by eqn (9.12), with the variables updated. This is because they occur in series. Also, each rate, i. e. Qpe and Qea, has two heat transfer rates that occur in parallel too. So, we can write the rate of heat transfer out of the system as

where the subscript ea represents envelope-to-atmosphere, A is the absorber pipe area noted above and Ae is the envelope area nDoL. The heat transfer coefficient hea is determined with eqn (9.13), as with the bare absorber pipe above. Note how Qout must be considered, rather than qout, since the area of the absorber pipe is different to that of the envelope, in addition to the manner in which kpe is defined.

The absorber pipe temperature Tp will be assumed to be constant for the entire system and the surrounding air Ta and sky Tsky temperatures will be known, thus, eqns (9.29) are used to find the envelope temperature Te. As mentioned above, the envelope material, glass, represents little thermal resistance relative to other components and so the assumption of the temperature being constant across the envelope thickness introduces little error.

The operating line for the 4 m-long unit can be written, after reference to eqns (9.7) and (9.21), as

Qо = – Ns [Cr Іат||PD(0)A – Qout] operating line (9.30)

lp

which is similar to eqn (9.21), except that |ат| и apTe is included.

In addition to the absorbance-transmission product introducing some complexity to the analysis, heat transfer from the system is also more complicated than from the bare absorber pipe, as can be seen in eqn (9.29). The following example is included to estimate the rate of heat transfer from the absorber pipe when an envelope is included, before consideration of a full power plant, including an envelope surrounding the absorber pipe, is examined.

Determine the rate of heat transfer from the absorber pipe of the standard system given in Table 9.4 when an envelope is included. Assume the air pressure in the annulus is 5 Pa (и 0.04 torr).

Firstly, the absorber pipe temperature is assumed to be equal to 400 °C and to not vary along the length of an entire loop. The surrounding air temperature is 25 °C and so one must use eqn (9.29) and change Te until Qpe is equal to Qea, which is easily accomplished with a numerical equation solver. Results for manually changing Te are shown in Table

9.6 and the envelope temperature is found to be 60.1 °C.

Some intermediate results required to find Qout, which is 1521 W, are that the thermal conductivity of air at atmospheric pressure at the mean temperature in the annulus (Tm = 230.1 °C) is 0.04041 W/m-K. Since the air pressure is 5 Pa in the annulus one determines that kpe = 0.03881 W/m-K, which is approximately 4% lower. The manufacturer’s specification for the annulus pressure in the SEGS VI STEGE is apparently 0.013 Pa or 10-4 torr. Should this be the case, then kpe = 0.002396 W/m-K and much smaller. This reduces Qout to 830.0 W and Te = 44.1 °C, which is low enough that one could touch it without damage to a finger! It is amazing that having a glass envelope which is only about 20 mm from an absorber pipe at a temperature of 400 °C could be touched. In addition, the rate of heat transfer has also been reduced by almost a factor of almost two, should the high vacuum be used relative to a pressure of 5 Pa.

The steady – state heat loss without an envelope was 7905 W for the 4 m-long absorber pipe. However, including the envelope reduces the heat loss to 1521 W, a more than a five times reduction in energy loss! An envelope certainly reduces the heat loss and is worth its cost.

Calculation of kpe shown in Fig. 9.10 shows a large increase in the effective heat transfer coefficient at larger pressures due to natural convection. Since this is such a large pressure, и 104 Pa, this effect does not have to be considered since the unit would most likely be at a condition that is not operational. However, if kpe is doubled, as is the size of the natural convection effect, then Qpe increases to 2208 W and Te = 76.0 °C. Both have considerable increases in their value and significantly affect the power plant design. For example, the total aperture area increases by almost 5% after the full design is completed (two more units are required in a loop to have the heat transfer fluid achieve the operational temperature).

A final comment about the air pressure assumed in the annulus. The manufacturer’s specification for the annulus pressure is 0.013 Pa or 10-4 torr. However, there is evidence that the pressure is above this, as the envelope temperature is greater than expected (see Price et al., ‘Field Survey of Parabolic Trough Receiver Thermal Performance’, NREL/CP – 550-39459). This was determined by measuring the envelope temperature relative to the heat transfer fluid temperature (see Exercise 9.9). Furthermore, there was some evidence that Hydrogen gas was in some of the annuli which tremendously increased the rate of heat transfer and the envelope temperature. So, the standard conditions assumed for the STEGE in this chapter have a pressure of 5 Pa which will produce a conservative estimate of the power plant design.

The design of the solar field having a glass envelope surrounding the absorber pipe is similar to the bare absorber pipe considered in the previous section. Here eqn (9.14) is re-written with the absorbance- transmittance product \ar|| written as apTe

where Qout must be determined according to Example 9.5. All the variables in the equation have been defined above, however, the definition of A is reinforced to be nDL, where D and L are the absorber pipe unit outer diameter (70 mm) and length (4 m), respectively.

A drawback in assuming that the absorber pipe has a constant temperature along its total length can be seen upon inspection of the above equation. It diverges when CrapTePd (0)A = Qout. This will occur if Pd (0) becomes too small and there is more heat transfer out of the system than into it. Of course, this is the result of our simplified model and in reality an infinite number of units in series will not have to be used, yet, Ns is expected to increase rapidly when Pd(0) becomes too low. For example, if the heat collection element has only the bare absorber pipe and no envelope then the heat transfer fluid will have increased in temperature by only about 15 0C in 500 m for the standard conditions if the insolation is reduced to 500 W/m2. Thus, low values of Pd (0) do indeed affect performance and certainly one should have a system design that includes an envelope around the absorber pipe.

This effect is represented graphically in Fig. 9.11 (see Example 9.6 below). The absorber pipe temperature Tp was assumed to be constant along the absorber pipe length at 400 0C and the loop was sized so the heat transfer fluid achieved a temperature of 395 0C. It is clear that if an envelope is not used then the aperture area Aap is given by

Aap = Ns Np WL (9.32)

Here W is the width of the parabolic trough (see Fig. 9.5) and assumed to be 5 m for the standard conditions, so, each unit has an aperture area of 20 m2. The aperture area diverges at a relatively small value of Pd (0) and is due to the very high heat transfer rate from the system. This is only due to the assumption of a constant absorber pipe temperature, as mentioned above. If the more detailed calculation is performed the area does not uniquely diverge, it does go to very large values though.

The calculations required to produce Fig. 9.11 are straightforward after the heat transfer rate out of the system Qout is determined as described in Example 9.5. If Qout is ignored though one can find the minimum aperture area Aap, min by rearranging eqn (9.31),

which is a reasonable equation to begin the design of the system especially when an envelope is used.

Efficiency has not been discussed and can be determined with an equation similar to eqn (9.2). In this case, though, we have to include all the energy harvested by the Sun and so it is given by

_ Q he

Vsf _ PD (0)Aap

where nsf is the efficiency of the solar field. The maximum value of nsf is apre and is limited only by the optical properties of the system. However, the efficiency of the Rankine cycle has not been taken into account for an overall efficiency. Equations (9.2) and (9.34) are combined to yield the overall system efficiency psf-R,

Using the result given in Example 9.5 determine the number of units in a loop, the total aperture area and solar field and overall efficiencies for a solar field designed to deliver 30 MW of net power. In this case use an envelope and the standard conditions given in Table 9.4. Assume that the air pressure in the annulus is 5 Pa (& 0.04 torr).

Firstly, the absorber pipe temperature is assumed to be equal to 400 °C and not to vary along the length of a loop. The number of parallel loops is determined from eqn (9.17), so one has Np _ 53 for 30 MW of power.

Now, the rate of heat loss to the surroundings was found in Example 9.5 to be Qout _ 8.319 x 102 W while the amount coming in CRapTePD (0)A _ 1.696 x 104 W, allowing one to calculate Ns with eqn (9.31) to be 91 and so the total number of units is Nu _ Ns x Np _ 4823, which is approximately half the number required if no envelope were used (Nu _ 9858). Clearly the envelope makes a large impact on the performance. Since each unit has an aperture of 20 m2 the total aperture area is 9.631 x 104 m2.

This aperture area can be compared to the minimum using eqn (9.33) to find Aap, min _ 9.184 x 104 m2, which is only 5% less than the more rigorously determined area. This is due to the fact that the envelope significantly decreases the rate of heat transfer from the absorber pipe. Thus, with an envelope in place one can reasonably estimate the aperture area due to the reduced rate of heat transfer out of the system.

As mentioned earlier in this chapter, the SEGS VI power plant has an aperture area of 1.88 x 105 m2, which is approximately twice as large as we have calculated. Reference to Fig. 9.11 shows that the aperture area depends on the amount of radiation received by the absorber, as expected. The standard conditions have PD (0) _ 1000 W/m2 so this technology can be compared to photovoltaic devices that are tested with

an AM1.5G spectrum whose total power is equal to this. However, if PD(0) is reduced to 500 W/m2 then the area is increased to 2.032 x 105 m2 and nearer to that actually used. It could be that the engineers who designed SEGS VI actually designed the system for a lower solar power density and so our result for the lower PD (0) may represent the true design value. Finally, Aap, min is still only 10% lower than the more rigorously calculated value for this lower insolation level, again suggesting that for quick estimates eqn (9.33) is good enough, especially when an envelope is used.

The efficiencies can be calculated with eqns (9.34) and (9.35) to be nsf = 80.99% and nsf-R = 31.20%. These are too high compared to data from the SEGS VI power plant, which are approximately half of these values (depending on the time of year). Again, using a lower apparent absorbance for the absorber pipe can lower the efficiency and lower values are determined. Furthermore, we have not included other heat transfer terms which can be important, such as heat loss to the structure that holds the heat collection element (i. e. the absorber pipe) which acts as a thermal sink, clearly undesirable and which will reduce the efficiency.

Finally we discuss heat transfer effects since the number of units in series Ns is 91 rather than 186. Since Ns is reduced by effectively a factor of two, will heat transfer be a limiting factor for the envelope case? The answer is no, since the absorber number Ab is still quite large and equal to 15.2. Thus, heat transfer is certainly not a limiting factor and one merely requires the loop to be long enough to absorb enough insolation to increase the temperature of the heat transfer fluid to the operating level.

Making electric power from the Sun was considered in this chapter. This is not direct generation of electricity as with a photovoltaic module, rather it is production of electricity using a contemporary Rankine cycle with energy supplied by the Sun. A difference to the flat plate solar energy collector is required and that difference is concentration of the solar energy. In order to manufacture the higher temperature needed to operate the Rankine cycle, concentration optics are needed. This imposes extra engineering and operational costs on this technology.

However, the cost of the fuel is nothing for solar thermal energy generated electricity, so, it should have an operational advantage over a contemporary coal-fired power plant. The challenge is that the temperatures that can be achieved are lower than when using coal as an energy source and so the efficiency of solar thermal energy generated electricity is less. In addition, inclusion of a heat transfer fluid, to lower the effect of fluctuations in solar radiation, adds additional cost to the maintenance and capital expenditures. However, depending on one’s goals this technology can certainly be a viable option.

No mention has been made of the condenser required to operate the

Rankine cycle. At the beginning of this chapter there was a discussion about the immense amount of energy that is rejected to the environment, see Table 9.1. This is certainly a challenge to the operation of these power plants and water used in traditional condensers or even make-up water required when cooling towers are used can limit where these plants are built. Water should be available. Many solar thermal power plants use air cooled condensers though. Although these are less efficient than a water cooled condenser they do eliminate the water requirement. So, the efficiency of these power plants will further suffer, yet, as mentioned above, the fuel is free and minimal carbon dioxide emissions occur.