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Subsections


MODELING THE CYCLE

Property Models

In order to thermodynamically model the cycle, the properties of the working fluids have to be modeled. Reliable data exist on the pure substances ammonia, butane, and water, however, properties of mixtures of these substances is less frequent. In the Einstein cycle, properties of the mixtures of ammonia and butane as well as ammonia and water are necessary.

While ammonia-water data are readily available, ammonia-butane data are quite rare. A literature search did produce one paper with P-T-x data on the ammonia-butane system, unfortunately, the range of pressures was much too high for direct use (Kay & Fisch, 1958). In order to create a relatively simple model to better understand the cycle's operation and performance, an ideal gas mixture was assumed for the vapor and an ideal solution in the liquid.

The Ideal Gas Mixture

The simplest model of a vapor mixture is the ideal gas mixture. First recall the ideal gas law for n moles of an ideal gas mixture occupying a total volume Vt at temperature T:
(2-1)

For a mixture, the partial pressure of a species k occupying the same total volume with nk moles is:

(2-2)

Dividing equation 2-1 by equation 2-2 yields:

(2-3)

where yk is the mole fraction of species k in the gas mixture. Recall an ideal gas is a model gas comprised of imaginary molecules of zero volume that do not interact. Because of this non-interaction, each chemical species in the mixture has its own private properties which are uninfluenced by the presence of other species. This is the basis of Gibb's theorem (Smith & Van Ness, 1994):

For example, the ideal gas enthalpy of a system comprised of several different chemical species is:
(2-4)

In other words, the enthalpy change of mixing is zero. Other properties that are independent of pressure can be written similarly.

While the enthalpy change of mixing is zero for the ideal gas mixture, the entropy change of mixing must be some positive quantity since mixing is irreversible. Recall for an ideal gas:

(2-5)

Since the entropy change of an ideal gas depends on pressure, the change of entropy of a species k at a constant temperature, T, is:

(2-6)

Integrating from the partial pressure pk to the total pressure P yields:

(2-7)

So, according to Gibb's theorem, the entropy of an ideal gas mixture at a fixed temperature would be:

(2-8)

Rearranging the above equation as

(2-9)

yields the entropy change of mixing for ideal gases. Since 1/yk > 1, this change is always positive and in agreement with the second law.

The Ideal Solution

Like the ideal gas mixture, an idea solution model requires only pure substance property data on the substances which comprise the mixture. The only information necessary on the mixture is its composition. Mathematically, the ideal solution model is very simple due to the following assumption: Mathematical interpretation of the above assumption results in the following equations.
(2-10)
(2-11)

Similar to the ideal gas mixture, the entropy change of mixing for the ideal solution is:

(2-12)

and can be shown via methods of statistical thermodynamics (Pitzer, 1995). The total entropy of an ideal solution is then:

(2-13)
 

Vapor-Liquid Equilibrium

In the Einstein cycle, the ammonia-butane and ammonia-water systems appear as a vapor mixture and a liquid mixture together in vapor-liquid equilibrium (VLE). In order to couple the ideal gas and ideal solution models, a criterion for VLE is necessary. This criterion is the equivalency of the fugacity for a particular species in the vapor and the liquid.
(2-14)

For the liquid phase, two simplifying assumptions are made:

    1. Assume the fugacity of component i at T and P of the system equals the fugacity of saturated i (liquid or vapor) at the same T, and its corresponding saturation pressure, Pisat:
(2-15)
 
    2. Next, assuming that pure saturated vapor i at T and Pisat behaves as an ideal gas the fugacity of saturated i is Pisat, and equation 2-15 becomes
(2-16)

The only assumption for the vapor phase is that each pure gas component behaves as an ideal gas at T and P, so that

(2-17)

Substituting equations 2-16 and 2-17 into equation 2-14 yields Raoult's Law:

(2-18)

For a two component system, equation 2-18 together with:

(2-19)
(2-20)

forms a set of four equations in four unknowns. At a given T and P, this set of equations can be solved to determine the VLE compositions in each phase (Pisat depends only on T).

Thermodynamic Modeling

The Evaporator

Each component of the cycle will be modeled individually before being assembled into a complete working model. The first component considered is the evaporator (Figure 2-1). Sub-cooled butane weak in ammonia enters the evaporator from the pre-cooler at state point 2 and falls as a thin film down the wall of the evaporator. During its fall, it encounters ammonia-butane vapor rising. Chemical and thermal equilibrium of the counter-flowing streams may not be realized until the bottom of the evaporator.
Figure 2-1: Evaporator Model Schematic

During the fall, the liquid film absorbs ammonia from the rising vapor while its butane is evaporated. Some of the heat of vaporization of this butane provides the cooling for the heat of absorption of the ammonia. Also, the falling film surrounds a tube containing superheated ammonia from the generator entering at state point 4 so as to keep it in thermal equilibrium, further cooling it before it reaches the bottom of the evaporator. The evaporating butane from the film also provides for this load.

At the bottom of the evaporator (state points 3, 5, and 6) the evaporator load Qe is transferred to the evaporator at a constant temperature Te. This is accomplished by completely evaporating the butane arriving in the falling film by bubbling pure ammonia vapor into it. The ammonia vapor reduces the partial pressure of the vapor on the butane thus allowing it to evaporate and take in heat. The ammonia-butane vapor mixture then rises to interact with the falling film before exiting at state point 7. The following assumptions are made in the evaporator:
 
 

Two control volumes were necessary in the evaporator, one around the entire component and one around the lower portion. For the overall control volume (CV1), the sub-cooled liquid mixture consisting nearly entirely of butane with little ammonia arriving from the pre-cooler at state point 2. Pure ammonia vapor also arrives from the pre-cooler at state point 4. A saturated vapor mixture of ammonia-butane exits the evaporator at state point 7.

To account for mass and energy flow through CV1, two equations of mass conservation and one equation of energy conservation are necessary.

(2-21)
(2-22)
(2-23)

Equation 2-21 accounts for the all the mass flowing into and out of the evaporator. Equation 2-22 accounts for the ammonia mass flowing into and out of the evaporator, and equation 2-23 is the conservation of energy for CV1.

A second control volume, CV2, is used for the lower portion of the evaporator. Three similar equations can be written:

(2-24)
(2-25)
(2-26)

In CV2, ammonia flowing from state point 4 is further cooled due to heat exchange with the liquid butane-ammonia film falling down the wall of the evaporator. The stream flowing in at state point 2 is in VLE with state point 6 by the time it reaches state point 3. All the streams flowing into and out of CV2 are in thermal equilibrium at Te.

The Pre-Cooler

The pre-cooler resides above the evaporator (Figure 2-1), and cools the two fluid streams entering the evaporator with the relatively cool ammonia-butane vapor leaving the evaporator. Thus, it is a three stream heat exchanger requiring two pinches. For this study, zero pinches were assumed. The conservation of energy is also written from a control volume surrounding the well insulated pre-cooler.
(2-27)
(2-28)
(2-29)
 

The Condenser/Absorber

Like the evaporator the condenser/absorber cannot be modeled in its entirety, and internal information is necessary to effectively model the condenser/absorber (Figure 2-2). The goal of the condenser absorber is to remove the ammonia from the butane and condense the butane. This model accomplishes that goal in the following manner.
Figure 2-2: Condenser Model Schematic

To remove the ammonia, a stream of sub-cooled water weak in ammonia arrives from the generator at state point 12 and falls as a film down the left side of the absorber side of the condenser absorber. Meanwhile, the superheated ammonia-butane vapor leaving the pre-cooler bubbles up through the bottom of the absorber and rises counter-flow to the falling film. During this counter-flow process, the ammonia weak stream entering at 12 absorbs ammonia vapor from the counter-flow stream entering at 8. The concentration of the ammonia-water mixture leaving at state point 9 is constrained by the partial pressure of the superheated ammonia vapor entering at state point 8. The heat of absorption must be rejected in order to maintain steady state operation.

At the top of the absorber side of the condenser/absorber, the mostly butane vapor passes to the condenser side of the condenser/absorber. Here, the heat must be rejected to steadily condense the butane vapor. At the bottom of the condenser side (state point 1), all of the ammonia-butane entering at the top as a vapor must leave as a liquid. This constrains the liquid concentrations at state point one to equal the vapor concentrations between the absorber and condenser. Since the pressure at both points is the same, the temperature at state point 1 is determined from the concentrations and the pressure.

For a control volume surrounding the entire condenser/absorber, the conservation of mass and energy are as follows:

(2-30)
(2-31)

A species mass balance is not required for the condenser since both the generator and the evaporator fully determine the system.

The Generator

In the generator (Figure 2-3) ammonia from the ammonia-water stream leaving the condenser is liberated via the addition of heat and returned to the evaporator. This incoming ammonia-water stream enters at state point 9 and falls as a thin film down the side of the generator. The first addition of heat comes from an internal heat exchanger which dumps heat unnecessary from the weak ammonia-water stream on its way back to the condenser to pick up more ammonia. Next, heat is added from an external thermal source.
 
Figure 2-3: Generator Model Schematic
 
 
The heat added to the falling film generates ammonia vapor with a low concentration of water in it which exits at state point 13. The remaining solution at the bottom of the generator (state point 11) has much less ammonia in it, however, the heat it contains is too useful to be dumped in the condenser. Instead, the hot water stream at state point 11 is allowed to exchange heat with the strong ammonia stream entering at state point 9 in the previously mentioned internal heat exchanger. After its useful heat is removed, the weak ammonia stream flows to the condenser.

For ease of analysis, the internal heat exchanger in the generator has an assumed zero pinch at state point 9. The temperatures at the other end of the heat exchanger will not be equal so the external heat added, Qg, will be added at a varying temperature Tg. Also, the bubble pump present in Einstein's patent drawing will be neglected in this model.

Only one control volume is necessary to analyze the generator. Mass and energy balances about this control volume (equations 2-32 through 2-34) provide the remaining equations for the complete cycle model.

(2-32)
ammonia balance (2-33)
(2-34)

The complete cycle is shown in Figure 2-4.
 

Figure 2-4: Einstein Refrigeration Cycle Model Schematic
 

Computer Model

After completing the property and thermodynamic models, they were entered into the commercially available software Engineering Equation Solver (EES, Klein, 1997). EES is quite useful for this type of problem for several reasons: The EES computer model is printed in Appendix A. Appendix A also contains a printout of the solution for the cycle for a base set of operating parameters.

Second Law Analysis

The Einstein cycle, while quite clever, is full of mixing, separating, and heat transfer processes. Since these are all irreversibilities , the COP for this cycle is expected to be much less than that of an ideal cycle. With a second law analysis, the ideal COP for the cycle can be calculated. Furthermore, the irreversibility of each component and its effect on the ideal COP can be calculated separately.

With the equations modeling the mixtures in the cycle as well as the various thermodynamic processes, the model can now be used to calculate the cycle's heat transfer rates and performance coefficients for given constraints. Furthermore, the entropy of each state can now be calculated. With these values, a detailed second law analysis could be performed, but will be left for further studies.

Still, the best possible COP for such a cycle, or its ideal COP, is of interest. This cycle is a heat pump operating between three temperature reservoirs: the evaporator, the condenser/absorber, and the generator. In Figure 2-5, these reservoirs are represented by a box labeled with the reservoir's temperature. The control volume is drawn so as to cut into each reservoir and heat transferred from the reservoir is assumed to occur at a zero temperature difference at the temperature of the reservoir (i.e. with zero irreversiblities). The control volume represents a three temperature heat pump with heat being added from the generator and the evaporator and removed from the condenser/absorber. Writing the first law of thermodynamics for this control volume:

(2-33)
 
  Again for the control volume, the second law is:
(2-34)

Note, the term Tg,s represents the entropic average temperature. Since the heat transfer in the generator occurs at a varying temperature, the reservoir temperature must vary identically in order to maintain reversible heat transfer. The entropic average temperature accounts for this variation and lies somewhere between the high and low temperatures experienced during the heat transfer process.

(2-35)

This can be seen in Figure 2-3. Since state points 9 and 12 are at a zero pinch and state point 11 is fixed, the fluid flowing out of the internal heat exchanger, T9i, must be at some point below T11. So, the heat transferred into the generator is transferred at an increasing temperature from T9i to T11.

Since there is no work, the COP for this cycle will be the heat transferred to the evaporator divided by the heat required by the cycle, namely the heat input to the generator. The above two equations can be combined so as to eliminate the heat transferred from the condenser.

(2-36)

Next, rearrangement provides the equation for the COP in terms of the reservoir temperatures, the entropy generation, and the generator heat transfer.

(2-37)

In equation 2-37, the first term is the ideal COP, which is more easily seen by setting Sgen equal to zero. Since Sgen is actually the sum of entropy generations in the cycle, the detrimental effect of each process on the ideal COP could be determined. In this manner, the most destructive process could be pinpointed. Again, this analysis will be completed in future studies.

In Appendix A, the base case results for the cycle are shown. For the base case of Te = 275 K, Tc = 315 K, Tg,s = 360 K, and P = 4.5 bar, the ideal COP and the entropy generation for the entire cycle are calculated. Using equations 2-37 and 2-34, the ideal COP is 0.8568 and the entropy generation for the cycle is 0.2471. Running the base case numbers through equation 2-37 returns the actual COP of 0.3475 which agrees exactly with the COP calculated via the first law.

The temperature leaving the internal heat exchanger, T9i, was calculated at 344.7 K in the base case. The entropic average temperature, Tg,s, was calculated at 360 K which is between T11 = 375 K and T9i.

Figure 2-5: Second Law Analysis Diagram