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Cycle Thermodynamic Performance


The previous three chapters provide sufficient information for a complete thermodynamic model of the Einstein refrigeration cycle. This model is now used to determine the cycle's thermodynamic performance. For example, what effect does the generator temperature have on the cycle's efficiency? The cycle's performance will first be evaluated for a base case at a system pressure of 4 bars. Next, the system pressure, generator temperature, condenser/absorber temperature, and evaporator temperature will be varied to determine their effect on the cycle's performance. Next, a direct second law analysis will be performed on the cycle to determine the irreversibilities of its processes. In this chapter, the cycle will use ammonia, water, and butane as its working fluids. Chapter VI will discuss the cycle's performance using alternative fluids. Finally, the conceptual demonstration prototype of the Einstein refrigeration cycle which was built, charged with ammonia-i-butane-water as determined by the thermodynamic model, and successfully operated, will be discussed.

Base Case Performance, The Maximum Lift, and Variable Performance

The base case for the Einstein refrigeration cycle was chosen to be ammonia-water-butane at a system pressure of 4 bar, a condenser/absorber temperature of 315 K, an evaporator temperature of 266 K, and a generator temperature of 375 K. First, a suitable temperature for the condenser/absorber was chosen to be around 110 F, or 315 K so that heat could be rejected to ambient air. Next, the VLLE behavior of the ammonia-butane mixture, shown in Figure 5-1, was studied to determine the base case system pressure and evaporator temperature.

Figure 5-1: T-x-x-y Diagram for Ammonia-Butane at P = 4 bar

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.

Figure 5-2: T-x-y Diagram for Ammonia-Water at P = 4 bar

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.

Table 5-1: Base Case Results for System Pressure = 4 bars

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.

Variation of System Pressure and Generator, Condenser, and Evaporator Temperatures

The system pressure was varied while holding the generator temperature constant and operating at the maximum lift condition between the evaporator and condenser/absorber. Figure 5-3 shows a plot of the resulting maximum lift for varying system pressure. As the system pressure increases the lift also increases. This is due to the increase in the difference between the saturation temperature of pure butane and the three phase flash temperature of the mixture with the increase in pressure, shown in Figure 5-4.

Figure 5-3: Maximum Lift vs. System Pressure

Figure 5-4: Condenser/Absorber, Evaporator Temperatures vs. System Pressure

Figure 5-5: COP vs. System Pressure Tcondenser = 315 K , Tevaporator = 275 K

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.

Figure 5-6: Coefficient of Performance vs. Condenser Temperature

Figure 5-7: Coefficient of Performance vs. Evaporator Temperature

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.

Figure 5-8: Coefficient of Performance vs. Generator Temperature

Variation of Finite Heat Exchanger Pinch Points

In the base case, all three pinch points were set to zero. They were varied to show the effect of heat exchanger performance on the system performance. Figure 5-9 shows the effect of varying each of the three pinch points individually and then varying all three together. Pinch 2,3 refers to the temperature difference between states 2 and 3 on Figure 3-1.

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.

Figure 5-9: Coefficient of Performance vs. Pinch Point

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.

Process Second Law Analysis

In Chapter I, the first and second laws were combined to provide equation 1-4 for the reversible COPc, rev of a three temperature reservior heat pump in terms of only the reservoir temperatures. Given any three temperature reservoirs, equation 1-4 predicts the best possible COP of a heat pump operating between them. Due to irreversiblilities such as fluid mixing and heat transfer across a finite temperature difference, the COPc, rev is degraded to the actual COPc. Using the concept of direct second law analysis developed by George Alefeld (Alefeld, 1990), it is possible to calculate the amount by which each process in a system of interest degrades its reversible performance.

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.


Figure 5-10: Einstein Refrigeration Cycle


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:

Table 5-2: Degrading of Reversible COP

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.

Conceptual Demonstration Prototype

After creating such a complete analytical model of the Einstein refrigeration cycle, a conceptual demonstration prototype was constructed for the purpose of demonstrating the principle of the cycle. As shown in Figures 5-10 through 5-12, the prototype was constructed from sections of stainless steel tubing TIG welded together. After completion, the prototype was charged with masses of ammonia, water, and iso-butane calculated from the thermodynamic model for a system pressure of 45 psi, a condenser/absorber temperature of 70 F, and an evaporator temperature of 30 F. The masses were; 0.337 kg of i-butane, 0.21 kg of ammonia, and 0.636 kg of water.

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.

Figure 5-11: Conceptual Demonstration Prototype

Figure 5-12: Conceptual Demonstration Prototype: Bubble Pump, Generator, Condenser/Absorber, and Controls

Figure 5-13: Conceptual Demonstration Prototype: Condenser/Absorber and Evaporator

Bubble Pump Performance | Contents | Alternative Working Fluids Performance