Cycle Thermodynamic Performance
As seen in Figure 5-1, pure butane at a pressure of 4 bar condenses at 315 K. The addition of ammonia allows the mixture to boil as low as 266 K, therefore, the base case system pressure was chosen to be 4 bar. At 4 bar, the minimum temperature at which the ammonia-butane mixture can boil is the three phase flash temperature, as discussed in Chapter III. The system pressure fixes this temperature so this temperature is known as soon as a system pressure is chosen. Therefore, the 4 bar ammonia butane three phase flash temperature, 266 K, is used for the base case evaporator temperature.
To determine a generator temperature, the behavior of the ammonia water mixture at the system pressure, as shown in Figure 5-2, is necessary. To generate ammonia vapor, the nearly 50/50 ammonia-water mixture flowing from the condenser/absorber at 315 K is heated, driving off ammonia water vapor. Heating the ammonia-water to 375 K reduces the mass concentration of ammonia in the liquid to under 0.2 and doesn't generate too much water vapor. Therefore, the base case generator temperature will be 375 K.
Figure 5-1 shows that at a fixed system pressure, the characteristics of the ammonia-butane mixture constrains the evaporator and condenser/absorber temperatures to a minimum and a maximum respectively. In this study, these temperatures are often used for calculations since they are the maximum temperature hill up which the cycle can pump heat. When these temperatures are used for a calculation, it will henceforth be referred to as the "maximum lift" condition where lift is the difference between the condenser/absorber and evaporator temperatures.
For the base case, the pinch points for both heat exchangers were assumed zero. Finally, all mass flow rates were normalized to the mass flow rate into the bubble pump since this flow rate can be controlled by heat input to the bubble pump. With ammonia-water-butane fluids and the evaporator, condenser/absorber, and generator temperatures, zero pinch points, and the mass flow of the bubble pump, the refrigeration cycle model base case is fully specified.
To simultaneously solve the large set of nonlinear equations in both the refrigeration cycle thermodynamic model and the Patel-Teja cubic equation of state property model, the software Engineering Equation Solver was used (Klein, 1997).
Table 5-1 provides the important results of this base case while Appendix B provides a complete listing of both the program and solution. Note that the heat transfer rates are per unit mass flow rate into the bubble pump. For example, if mbp = 1 kg/s, then Qgenerator = 709 kJ/s, and m1 = 0.38 kg/s.
A parametric study was carried out to determine the effect of varying; 1) the system pressure, 2) the temperatures in the evaporator, condenser/absorber, and generator, and, 3) the pinch points.
Next, the pressure is varied for a fixed lift condition when the condenser/absorber temperature is set to 315 K, and the evaporator temperature is set to 275 K. As seen in Figure 5-5, increasing pressure causes a decrease in the COP. At a system pressure of 5.5 bar, 275 K is the minimum evaporator temperature, but in the condenser/absorber, 315 K is well below the maximum condenser/absorber temperature of 327 K. At a system pressure of 4 bar, 315 K is the maximum condenser/absorber temperature, however, 275 K is well above the minimum evaporator temperature of 266 K. Operating at a condenser/absorber temperature below the maximum causes a performance increase over operating at the maximum lift condition, however, operating with an elevated evaporator temperature increases performance more, as seen in Figures 5-6 and 5-7.
As seen in Figure 5-6, when the condenser/absorber temperature is varied from the base case, the COP increases as the temperature of the evaporator is approached. As seen in Figure 5-7, when the evaporator temperature is varied from the base case, the COP increases as the temperature of the condenser/absorber is approached. As the lift decreases, the cycle expectedly operates more efficiently. However, increasing the evaporator temperature increases the COP more than decreasing the condenser/absorber temperature. This effect is why the COP increases with reducing pressure in Figure 5-5.
Next, the generator's temperature is varied while the other parameters remain fixed at the base case. Figure 5-8 shows the variation of COP with generator temperature. There is a maximum COP of 0.17 at a generator temperature of 373 K, but the variation of COP around this optimum generator temperature is small. As the generator temperature increases, the arriving liquid must be heated through a higher temperature difference. Also, at higher generator temperatures, the heat required in the bubble pump increases since the fluid's water composition is higher. However, with lower generator temperature, less ammonia is driven from the water before it is returned to the condenser/absorber. At 373 K, these effects are balanced to produce the highest COP.
As pinch 2,3 is varied, the COPc is unaffected. In the evaporator pre-cooler, the largest thermal mass flow rate is the ammonia-butane vapor stream which cools both the incoming liquid butane and vapor ammonia. When pinch 2,3 is increased while holding pinch 6,1 constant, the temperature of the exiting ammonia vapor must decrease, and the COPc remains the same. Thus, the governing pinch in the pre-cooler is pinch 6,1. Increasing pinch 6,1 reduces the COP as expected since reducing the exiting temperature of this stream reduces the cooling available for the other two streams.
The performance of the generator's internal heat exchanger also affects the performance of the overall cycle. Increasing the pinch between streams 7 and 9 reduces the regenerative effect thereby decreasing the COP of the cycle. Increasing all three pinches also decreases the COP of the cycle. However, when the three pinch points are 5 K, the COP is only reduced by 10 percent.
For the Einstein refrigerator, direct second law analysis begins by applying the first and second law to the control volume shown in Figure 5-10.
In the above equation, represents the entropy generation of a process, k, inside the control volume of Figure 5-10. The evaporator temperature, Tevaporator, is assumed to be the lowest temperature occurring in the evaporator. Likewise, the condenser temperature, Tcondenser, is assumed to be the lowest temperature occurring in the condenser. The generator temperature, Tgenerator, is assumed to be the highest temperature occurring in the generator. Now, equation 5-2 is multiplied by Tgenerator and subtracted from equation 5-1. After some rearrangement, the following equation is produced:
The first term in square brackets on the right side in equation 5-3 is the COPc, rev also given by equation 1-4. Equation 5-3 may thus be rewritten as follows:
Equation 5-5 conveniently shows how much the entropy generation of each process, k, inside the control volume of Figure 5-8 degrades the reversible COPc to the actual COPc. This allows the most performance destructive process to be identified.
Utilizing equation 5-5, the following degradations were calculated for the processes in the base case:
For the ideal process, the degradation is zero resulting in the reversible COP for the base case of 0.8646. Including the degradations of all the processes results in the actual COP of 0.17. The largest degradation is contributed by the generator due to heat transfer across the inherently large temperature difference. In the generator's internal heat exchanger, the liquid entering the generator from the condenser/absorber at 315 K is heated to around 320 K. The maximum temperature of the generator is 375 K which is the assumed temperature at which the heat crosses the control volume. Heat flow across the large temperature difference (375 K to 320 K) causes a high entropy generation to occur in the generator.
The mixing of ammonia and butane in the evaporator causes its degradation to be relatively high as well. Likewise, mixing occurring in the condenser/absorber causes the third largest degradation to the COP. The bubble pump and the pre-cooler contribute relatively minor degradations.
The generator and bubble pump were heated with 300 and 200 watt clamp on tube heaters respectively. Each was separately controlled by a dimmer switch. The condenser/absorber was cooled with tap water flowing through an 8 foot long coil of ¼ inch stainless steel tubing welded into the condenser/absorber. A pressure gauge placed on the condenser/absorber monitored the pressure of the cycle.
Before completion of the prototype, the bubble pump was tested with water in order to determine an estimate for the liquid flow rate for a given heat input.
Supplying between 150 and 250 watts to the generator and between 50 and 70 watts to the bubble pump while cooling the condenser/absorber with 70 F tap water causes the evaporator to maintain temperatures as low as 28 F on a steady state basis. The prototype was operated unsupervised for as long as 24 hours at a time and well over 200 hours total. The prototype is noiseless during operation. This demonstrates for the first time the practical viability of the Einstein refrigeration cycle.