Optimization of Power Solar Dish-Stirling: Induced Effects of Heat Source Temperature and Working Fluid Temperature in Hot Side

Veiw figure View Figure

Figure 2 is a schematic diagram of a Stirling heat engine cycle with finite-time heat transfer and regenerative heat losses. As shown in Figure 3, Stirling cycle consisted of four processes. Process 1-2 is an isothermal process, in which the working fluid after compressing at constant temperature, and rejected heat to the heat sink at low temperature, Therefore, the temperature of heat sink is increased to . Then the working fluid crosses over the regenerator and warms up to in an isochoric process 2-3. In process 3-4, the working fluid is expanded in a constant temperature process and receives heat from the heat source in which its temperature is reduced from into . Last process (4-1), is an isochoric cooling process, where the regenerator absorbs heat from the working fluid.

Figure 2. Schematic diagram of the Stirling heat engine cycle

Download as

Veiw figure View Figure

Figure 3. T-S diagram of a Stirling engine cycle

Download as

Veiw figure View Figure

In a real cycle it is impractical to have an ideal heat transfer in the regenerator in which entire amount of absorbed heat (in the process 4-1) is transferred to the working fluid in the heating mode (process 2-3). Therefore a heat transfer loss denoted by is occurred in the regenerator. In addition, a conduction heat transfer between the heat source and the heat sink namely as thermal bridge loss () must be considered.

3. Thermodynamic Analysis of the System

The actual useful heat gain of the dish collector, considering conduction, convection and radiation losses is given by ^{[25, 39]} as follows:

(1)

where is the direct solar flux intensity,is the collector aperture area, is the collector optical efficiency, is the absorber area, is conduction/convection heat transfer coefficient, is the average absorber temperature, is the ambient temperature,is an emissivity factor of the collector, is the Stefan’s constant.

The thermal efficiency of the dish collector is obtained as ^{[25, 39]}:

(2)

3.1. Regenerative Heat Losses in the Regenerator

It is important to mention that there also exists a finite heat transfer in the regenerative heat transfer () is given by ^{[39, 40, 41, 42]}:

(3)

is the heat loss during the two regenerative processes in the cycle. By the following relationship is obtained ^{[39, 40, 41]}:

(4)

is the mass of the working fluid in mole, is the specific heat capacity of the working fluid in the regenerative processes in terms of mole, is the effectiveness of regenerator, and are the working fluid temperatures in the hot space and cold space, respectively.

Owing to the influence of irreversibility of the finite-rate heat transfer, the time of the regenerative processes is not negligible in comparison to that of the two isothermal processes ^{[37, 38, 39]}. In order to calculate the time of the regenerative processes, one assumes that the temperature of the working fluid in the regenerative processes as a function of time is given by ^{[37, 38, 39]}:

(5)

whereis the proportionality constant which is independent of the temperature difference and dependent only on the property of the regenerative material, called regenerative time constant and the sign belong to the heating (i= 1) and cooling (i= 2) processes respectively ^{[37, 38, 39]}.

(6)

(7)

3.2. The Amounts of Heat Released by the Heat Source and Absorbed by the Heat Sink

The heat released between heat source and working fluid (), the heat absorbed between the working fluid and the heat sink () is obtained as follows:

(8)

(9)

On the other hand

(10)

(11)

where and are the heat capacitance rate of external fluids in the heat source and heat sink, respectively.

(12)

(13)

whereand are the effectiveness’s of the high and low temperature heat exchangers, respectively.

where

The cyclic period

Using Eqs. (3)-(11), we get that the cyclic period is:

(14)

3.3. The Conductive Thermal Bridging Losses from Heat Source to the Heat Sink

This value is proportional to the average temperature difference of the heat source and heat sink and the cycle time it is obtained as follows:

(15)

(15a)

(15b)

We have ^{[41, 42]}

(15c)

(15b)

Thus, using Eqs.(15a)-(15d), we have

(16)

The net heat released from the heat source and the net heat absorbed by the heat sink are obtained as follows:

(17)

(18)

Considering the cyclic period of the Stirling engine, the output power, the thermal efficiency and entropy production of the engine are given by:

(19)

(20)

(21)

Substituting Eqs. (3)- (14) into Eqs. (19) and (20) we have,

(22)

(23)

With simplification and considering and we obtain:

(24)

(25)

For the sake of convenience, a new parameter is introduced into Eqs. (24) and (25), then we have:

(26)

(27)

For the optimization for maximum Power, Eq. (26) must be differentiated from and put it equal to zero:

(28)

(29)

where

(29a)

(29b)

(29c)

(29d)

(29e)

(29f)

(29g)

(29h)

(29i)

(29j)

With Placement in and , maximum power and its corresponding thermal efficiency are calculated as follows:

(30)

(31)

The maximum thermal efficiency of the entire solar-dish Stirling engine is a product of the thermal efficiency of the collector and the optimal thermal efficiency of the Stirling engine ^[39]. Namely:

(32)

Therefore by substituting of Eqs. (2) And (31) into Eq. (32), we have the following expression for thermal efficiency of the entire solar-dish Stirling engine:

(33)

4. Numerical Results and Discussion

In order to evaluate the effect of the heat source working fluid inlet temperature (), heat capacitance rates (), the effectiveness of the regenerator (), effectiveness of the heat exchangers () and the heat leak coefficient () on the powered Stirling heat engine system, all the other parameters will be kept constant as ,,,,,, ,,,,,,,,,,.

In the Figure 4 different system parameters against the heat source working fluid inlet temperature in the different heat capacities rate of the heat source are plotted. And in these figures by increasing the heat source working fluid temperature, maximum output power, Stirling thermal efficiency and entropy generation will be raised. Also the overall thermal efficiency of the dish system reaches a peak at about 900K and after that decreases. In Addition, at the given heat source working fluid inlet temperature all parameters (the maximum output power, Stirling thermal efficiency, entropy generation and thermal efficiency of the dish system) have increased with increasing of the heat capacities rate of the heat source.

Figure 4. Variation of (a) maximum output power, (b) Stirling thermal efficiency, (c) thermal efficiency of the dish system and (d) entropy generation with heat source working fluid inlet temperature at a different heat capacities rate of heat source

Download as

Veiw figure View Figure

In the Figure 5, all discussed parameter in Figure 4 are illustrated for the heat capacities rate of the heat sink. Surprisingly all parameters show the same trend against the heat source working fluid inlet temperature. Furthermore at given heat source working fluid inlet temperature, all mentioned parameters increase with increasing the of the heat capacities rate of the heat sink.

Figure 5. Variation of maximum (a) output power, (b) Stirling thermal efficiency, (c) thermal efficiency of the dish system and (d) entropy generation with heat source working fluid inlet temperature in the different heat capacities rate of heat sink

Download as

Veiw figure View Figure

From Figure 6 it can be seen that, the maximum output power, the thermal efficiency and entropy generation increase considerably with increasing of the heat source working fluid inlet temperature at various values of the hot side heat exchanger effectiveness (0.4-0.6). But the thermal efficiency of the Stirling dish system in the temperature of about 1000K reaches a maximum and then decreases.

Figure 6. Effects of heat source working fluid inlet temperature and effectiveness of the hot side heat exchanger on the maximum (a) output power, (b) Stirling thermal efficiency,(c)thermal efficiency of the dish system and (d) entropy generation

Download as

Veiw figure View Figure

Also in the Figure 7 the similar demonstrated parameters in the Figure 6 are plotted against the heat source working fluid inlet temperature at the different effectiveness of the cold side heat exchanger. As it can be seen overall trends are the same as Figure 6.

Figure 7. Effects of heat source working fluid inlet temperature and effectiveness of the cold side heat exchanger on the (a) maximum output power, (b) Stirling thermal efficiency, (c) thermal efficiency of the dish system and (d) entropy generation

Download as

Veiw figure View Figure

In the Figure 8 different system parameters against the heat sink working fluid inlet temperature in the different heat capacities rate of the heat source are shown. As can be seen, in all graphs with increasing the heat sink working fluid temperature, maximum output power, Stirling thermal efficiency, dish system efficiency and entropy generation decrease. In Addition, at the given heat sink working fluid inlet temperature all four discussed parameters are increased with increasing in the heat capacities rate of the heat source.

Figure 8. Variation of (a) maximum output power, (b) Stirling thermal efficiency, (c) thermal efficiency of the dish system and (d) entropy generation and heat sink working fluid inlet temperature at a different heat capacities rate of heat source

Download as

Veiw figure View Figure

From Figure 9 all parameters in Figure 8 are illustrated for the heat capacities rate of the heat sink. Similarly all parameters express same manner versus the heat sink working fluid inlet temperature. Also at given heat sink working fluid inlet temperature, all parameters increase with increasing the in the heat capacities rate of the heat sink.

Figure 9. Variation of (a) maximum output power, (b) Stirling thermal efficiency, (c) thermal efficiency of the dish system and (d) entropy generation and heat sink working fluid inlet temperature at different heat capacities rate of heat sink

Download as

Veiw figure View Figure

From Figure 10 it can be observed that, the maximum output power, the thermal efficiency, the dish system efficiency and entropy generation decrease with increasing of the heat sink working fluid inlet temperature at different values of the hot side heat exchanger effectiveness. Moreover, at given heat sink working fluid inlet temperature, all parameters increase by increasing of the hot side heat exchanger effectiveness from 0.4 to 0.8.

Figure 10. Variation of (a) maximum output power, (b) Stirling thermal efficiency, (c) thermal efficiency of the dish system and (d) entropy generation with heat sink working fluid inlet temperature at the different effectiveness of the hot side heat exchanger

Download as

Veiw figure View Figure

In the Figure 11 the similar considered parameters in the Figure 10 are scheming against the heat sink working fluid inlet temperature at the different effectiveness of the cold side heat exchanger. As it can be pictured overall trends are the same as Figure 10.

Figure 11. Variation of (a) output power, (b) Stirling thermal efficiency, (c) thermal efficiency of the dish system and (d) entropy generation with heat sink working fluid inlet temperature at the different effectiveness of the cold side heat exchanger

Download as

Veiw figure View Figure