INTERNATIONAL SORPTION HEAT PUMP CONFERENCE June 22-24, 2005; Denver, CO, USA ISHPC-093-2005

WORKING FLUID SELECTION THROUGH PARAMETER ESTIMATION Laura A. Schaefer Mechanical Engineering Department, University of Pittsburgh [email protected] Samuel V. Shelton Woodruff School of Mechanical Engineering, Georgia Institute of Technology ABSTRACT This paper looks at the use of computational techniques to evaluate potential working fluid triplets (a refrigerant, a pressure-equalizing fluid, and an absorbent) for a singlepressure absorption cycle. The cycle was modeled using two separate property models: (1) a corresponding states/ideal solution model, and (2) a Patel-Teja/Panagiotopoulos and Reid model. The first model was used to predict which working fluid parameters would increase the COP. The specific heats, critical temperatures, critical pressures, acentric factors, and molecular masses were each varied, and both the magnitude and the trend of the COP change were quantified. Based on the results of the first model, and after accounting for adverse chemical combinations, one absorbing fluid, five pressure equalizing fluids, and five refrigerants, for a total of fifteen possible fluid triplets, were examined at three temperature conditions suitable for gas water heating. For each of the triplets, the second model based on the Patel-Teja equation of state was used to more accurately predict the behavior the cycle. Using either ammonia-water-butane or ammonia-water-propane as the working fluids, the Einstein cycle has a heating COP of 1.5 for the first operating configuration, 1.88 for the second configuration, and 1.76 for the third. Keywords: Working Fluid, Absorption, Equation of State, COP

for each working fluid, as well as decreasing computational complexity. This complexity is an important consideration, since even a simple model demonstrates nonlinear property behavior and component relations. The basic model chosen will in turn determine the specific relevant working fluid properties. After performing an individual or multivariate optimization of those properties, actual working fluids should be matched as closely as possible to the “ideal” fluids. The behavior of the alternative working fluids can then be more accurately modeled using a higher-order equation of state and fluid mixing rules. In this paper, a corresponding states/ideal solution property model was chosen for the property search, yielding the parameters of constant-pressure specific heat, critical temperature, critical pressure, acentric factor, and molecular mass for each working fluid. The more accurate modeling of actual working fluids was accomplished by using the Patel-Teja equation of state and Panagiotopoulos and Reid mixing rules. For the work presented in this paper, the objective was to maximize the heating coefficient of performance for a specific single pressure absorption cycle. This type of cycle was chosen because in order to eliminate the need for any work input, single pressure absorption cycles add a third working fluid and replace the mechanical pump with a bubble pump. (A bubble pump can move fluids across a difference in height simply by using a heat input.) The use of three working fluids adds further choices to the parameter optimization and search. Furthermore, for the particular cycle investigated in this study (the Einstein cycle), the three working fluids interact only as fluid pairs throughout the various components, which simplifies the overall cycle analysis. It should be noted that the objective should be carefully considered when optimizing cycle performance through variation of working fluid parameters. One application may be better served by maximizing the generator temperature, while another may need a set range of evaporator temperatures. It is also possible that multiple objectives could be specified, and include secondary (or nonmathematical) considerations such as elimination of hydrocarbons or other fluids.

INTRODUCTION In the field of vapor compression, the need to replace first chlorofluorocarbons and then hydrochlorofluorocarbons forced the refrigeration industry to examine the ideal properties of various cycles and their working fluids. However, because the working fluids used in many absorption cycles are already considered to be generally environmentally friendly, the same level of scrutiny has not necessarily been applied. Within a given cycle, the various properties of the working fluids can have multiple and conflicting effects for each component. Therefore, it is unlikely that random experimentation and replacement will necessarily yield an optimal result. In order to quantify the role of each working fluid, its basic relevant parameters should be identified, as well as the importance of those parameters throughout the entire cycle, rather than just in the individual components. To accomplish this, a complete cycle should first be modeled using a number of fluid interaction simplifications. These simplifications reduce the number of relevant parameters

CYCLE DESCRIPTION In 1928, Albert Einstein and Leo Szilard patented a single pressure absorption cycle. Unlike the more commonly-known Platen and Munters cycle, however, the Einstein cycle uses a pressure-equalizing absorbate fluid rather than an inert gas. In

1

law equations were also applied to each component to evaluate the individual and overall rates of entropy generation. When operating in a heating mode, the coefficient of performance of the Einstein cycle was calculated as: & Q COPh = ca (1) & Q gen

For each run, the temperatures of the generator, condenserabsorber, and evaporator were specified and the mass flow rate entering the evaporator was normalized. The enthalpies and entropies at each state point were found through simultaneous solution of the governing equations. BASIC FLUID PROPERTY AND MIXTURE MODEL The Principle of Corresponding States As a first order simplification, the behavior of each individual component was assumed to follow the principle of corresponding states. The three-parameter principle of corresponding states asserts that the reduced pressure of any substance is solely a function of that substance’s reduced temperature, volume, and compressibility (which represents the molecular variability between substances). To calculate the critical compressibility (Zc), the critical volume (vc) must be known. Unfortunately, since the differential compressibility, (∂v/∂P)T, approaches infinity at the critical point, the critical volume cannot be accurately measured. As an alternative to the critical compressibility, Pitzer et al. suggested a different third parameter that they termed the acentric factor [9]: ω = log 10 (Prsat ) Simple Fluid at T = 0.7 − log 10 (Prsat ) T = 0.7 (2) = −1 − log 10 (Prsat ) T = 0.7 The acentric factor measures the deviation of intermolecular potential from a simple fluid, which is defined as one with a spherical shape and an inverse sixth power potential. The acentric factor can therefore be determined from the critical temperature and pressure and from one saturation pressure measurement. With three parameters (Pc, Tc, and ω), the principle of corresponding states becomes highly accurate for subcooled and superheated nonpolar and slightly polar substances. While the accuracy is lessened in the saturated region and near the critical point, the three-parameter model still provides a reasonable approximation. Lee and Kesler expanded the work of Pitzer et al. to provide increased accuracy and to include a wider range of temperatures [10]. They found that the compressibility factor of any fluid is a function of the compressibility of a simple fluid (Z(0)), the compressibility of a reference fluid (Z(r), n-octane), and the acentric factor, where Z(0) and Z(r) are functions of Tr and Pr: ω Z = Z (0 ) + ( r ) ( Z ( r ) − Z ( 0 ) ) (3) ω Through a combination of experimental data and the modified Benedict-Webb-Rubin equation of state, the reduced compressibilities (Zr = Z/Zc) for both the simple and reference cases were found to conform to:

Figure 1. The Einstein Cycle the original Einstein cycle, butane is the refrigerant, water is the absorbent, and ammonia is the pressure-equalizing fluid. Figure 1 is a general diagram of the Einstein cycle. The cycle shown in this figure has been modified slightly from Einstein's original configuration: various heat exchangers, such as the internal generator solution heat exchanger, have been added in order to raise the efficiency. While a general overview of the cycle’s operation is given below, more detailed descriptions of the Einstein cycle can be found in [1-8]. At point 1, liquid refrigerant leaves the condenserabsorber, passes through a precooler, and arrives in the evaporator at point 2. The pressure-equalizing fluid enters the evaporator as a vapor at point 4b. The refrigerant and pressureequalizing fluid combine and leave the evaporator at point 3. This mixture becomes superheated in the precooler, and is bubbled into the condenser-absorber at point 6. Liquid absorbing fluid from the generator is also sprayed into the condenser-absorber (point 11). Heat is rejected to the environment, and the absorbing fluid absorbs the pressureequalizing vapor from the mixture entering at point 6. The refrigerant vapor and the remaining pressure-equalizing vapor are then cooled until they condense. The refrigerant is siphoned into the evaporator (as noted above), and the pressureequalizing/absorbing fluid mixture leaves the condenserabsorber at point 7. The mixture stream exchanges heat with the liquid absorbing fluid that is to be sprayed into the condenserabsorber, and enters the generator at 8. In the generator, heat is applied to separate the absorbing fluid from the pressureequalizing fluid. The pressure-equalizing vapor leaves the generator at point 5, and passes through the precooler to point 4. To remove most of the pressure-equalizing fluid that remains in the liquid mixture, that mixture is pumped through a bubble pump to a reservoir (point 9). Any residual pressure-equalizing vapor (point 9g) is then combined with the pressure-equalizing vapor leaving the precooler, and enters the evaporator at point 4b. The liquid absorbing fluid exchanges heat with the entering mixture stream, and leaves the generator at point 10. To model the cycle, the conservation of mass, species, and energy equations were developed for each component. Second

r

r

2

r

Z r = 1+

c4 B C D + + + Vr Vr2 Vr5 Tr3 Vr2

 γ β + 2  V r 

  γ  exp −   V2 r  

  (4)  

 H − H *  ( r )  H − H *  (0 )    −   (7)  R u Tc   R u Tc   Using fundamental property relations, the enthalpy departure function can be related to the Gibbs energy, which is in turn related to the compressibility, such that: P ∂Z H − H*   dP (8) (constant T) = −T   0  ∂T  R u Tc P P Given equation 8, the enthalpy departure functions can be found for both the simple and the reference fluids. These departure functions are incorporated into equation 7 to find the overall enthalpy departure function. The ideal enthalpy (H*) is found using ideal gas (dh = cpdT) and incompressible solution theory, so the actual enthalpy difference across any two state points can then be calculated. H − H*  H − H*  =  R u Tc  R u Tc 

where b2 b3 b4 c c d − 2 − 3 , C = c 1 − 2 + 33 , D = d 1 + 2 , and Tr Tr Tr Tr Tr Tr the constants for both fluids are given in Table 1. The reduced compressibility can also be stated as: PV Zr = r r (5) Tr To find Z(0), the simple fluid coefficients listed in Table 1 are used in equation 4. That equation is set equal to equation 5, a reduced temperature and pressure are chosen, and the reduced volume is found. This is not an actual reduced volume, but rather a pseudo-reduced volume. Once Vr(0) is calculated, equation 4 is used to find Z(0). This process is then repeated using the reference fluid coefficients to find Vr(r) and Z(r). In the saturated region, there are three real roots that satisfy equations 4 and 5. Therefore, the state of the substance must be known to determine which root to utilize. The definition of the acentric factor and the principle of corresponding states dictates that the reduced saturation pressure of a substance is a function of Tr and ω such that [11]: r (6) ln(Psat ) = ∫ (0) + ω ∫ (1) B = b1 −

ω + (r) ω

∫

Ideal Solution Theory In the previous subsection, the means for calculating the enthalpies of the individual fluids was presented. Next, these fluids must be combined using simplified equilibrium conditions and mixing rules. The most basic statement of equilibrium for a mixture in a vapor and a liquid phase is that the fugacities of the individual substances must be equal: fiV = fiL (9) This form of the fugacity relationship, however, does not present an explicit representation of the mole fractions, temperature, and pressure. To provide a more useful form of this equation, some simplifying assumptions can be made, which result in a reduction to Raoult's Law [12]: yi P = xi Pisat (10) where yi is the vapor mole fraction, and xi is the liquid mole fraction of component i. Once the mole fractions and the individual component enthalpies are known, the enthalpy of a vapor or a liquid mixture can be calculated. Based on the Gibbs energy of an ideal gas, the enthalpy of an ideal gas mixture can be derived:

∫ ( 0) = 5.92714 - 6.09648/Tr - 1.28862 ln (Tr ) + 0.16934 Tr6 , and

∫ (1) = 15.2518 -15.6875/Tr - 13.4721 ln (Tr ) + 0.43577 Tr6 . Therefore, when Pr > Prsat, the root that corresponds to the vapor state should be used, and when Pr < Prsat, the root that corresponds to the liquid state should be chosen. Once the compressibility is known, many other properties can be calculated by using the three-parameter corresponding states model. These properties include departure functions for enthalpy, entropy, isochoric specific heat, and isobaric specific heat. A departure function is the difference between an actual property and the value of that property if it behaved as an ideal gas at the same temperature and pressure. As a brief example, the enthalpy departure function will be presented, where H* is the ideal enthalpy and H is the actual enthalpy on a molar basis:

n

H ig = ∑ y i H igi

(11)

i

The ideal liquid solution enthalpy is analogous to the ideal gas mixture enthalpy [Ref]: n

Table 1. Constants for Calculating Compressibility Constant b1 b2 b3 b4 c1 c2 c3 c4 d1 x 10-4 d2 x 10-4 β γ

(0)

H ideal solution = ∑ x i H i

(12)

i

Simple Fluid (0) Reference Fluid (r) 0.1181193 0.2026579 0.265729 0.331511 0.15479 0.027655 0.030323 0.203488 0.0236744 0.0313385 0.0186984 0.0503618 0 0.016901 0.042724 0.041577 0.155488 0.48736 0.623689 0.0740336 0.65392 1.226 0.060167 0.03754

Using the molecular mass of each substance, equations 11 and 12 can also be restated in terms of specific properties and mass fractions such that: n

h ig = ∑ y m ,i h igi

(13)

i

and n

h ideal solution = ∑ x m,i h i

(14)

i

Therefore, based on the principle of corresponding states and ideal solution theory, the enthalpy (or entropy) at any point in the cycle can be found solely as a function of the critical temperature, critical pressure, acentric factor, constant-pressure specific heat, and molecular mass of the three working fluids.

3

1.75

1.73

1.73

1.71

1.71

COP h

COP h

1.75

1.69

1.67

1.65 1.0

1.69

1.67

1.5

2.0

2.5

3.0

3.5

1.65 415

4.0

418

421

423

T c,re frg

c pl,re frg 1.75

1.73

1.73

1.71

1.71

COP h

COP h

1.75

1.69

1.69

1.67

1.67

1.65 4.8

5.2

5.7

6.1

6.6

1.65 401

7.0

406

411

416

c pl,p e

421

426

429

432

425

430

435

440

675

683

692

700

T c,pe 1.75

1.73

1.73

1.71

1.71

COP h

COP h

1.75

1.69

1.69

1.67

1.67

1.65 3.1

3.5

3.9

4.4

4.8

1.65 632

5.2

c p l,abs

641

649

658

666

T c,abs

Figure 2. COP versus cp BASIC PARAMETER SEARCH RESULTS AND FLUID SELECTION Searching for Optimum Properties Using the conservation of mass, species, and energy equations developed for the cycle and the corresponding states and ideal solution thermodynamic property model, a computer program was created to describe the system behavior. In order to produce conditions suitable for domestic and commercial water heating, the temperature of the condenser was set at 325 K (125°F) and the evaporator temperature was specified to be 295 K (72°F). For the base case of the originally proposed ammonia water-butane mixture, the generator temperature that produced the highest heating coefficient of performance was found to be 495 K. However, the COP was only slightly degraded by lowering the generator temperature to

Figure 3. COP versus Tc 425 K, which provided more stability to the solver when the fluid parameters were varied. Initially, a full-scale nonlinear optimization of the cycle model was attempted, in which all the parameters were varied simultaneously. Unfortunately, this technique repeatedly caused the solver to become unstable before a solution could be found. To avoid this problem, each fluid property was instead individually varied, and the effect on the COP was noted. The starting point for each parameter variation was the default value from the ammonia-water-butane triplet. Figure 2 shows the effect of varying the liquid specific heat at constant pressure for the refrigerant (()refrg), pressure-equalizing fluid (()pe), and absorbing fluid(()abs). In each graph, the dashed line denotes the base case COP. Figure 3 shows the effect of varying the critical temperature. Similar trends were observed

4

Table 2. Parameter Variation for Increased COP Refrigerant

Absorbing Fluid

Increase (+) Increase (+) Decrease (+) Increase (++) Increase (++)

Decrease (+) Increase (+) Increase (++) Decrease (++) Decrease (+++)

Decrease (++) Decrease (+) Decrease (+) Increase (++) Increase (++)

Tgen = Tgen,optimum

1.75

COPh

cp,l M Pc Tc ω

PressureEqualizing Fluid

1.80

Tgen > Tgen,optimum

1.70 Tgen < Tgen,optimum

for variations in the critical pressure and acentric factor, while the molecular mass was found to have a lesser effect on the COP. Table 2 summarizes the manner in which each of the parameters must be changed in order to increase the COP, with the magnitudes of the variations denoted by “+” signs. In addition to testing the effects of parameter deviation, the contribution of the generator temperature was also examined. As the generator temperature was lowered from its optimum, the trend for each parameter remained the same, but the magnitude of the COP improvement or degradation increased. Identical behavior occurred when the generator temperature was raised above the optimum. This is illustrated for the critical temperature of the absorbing fluid in Figure 4. One potential difficulty with this method of optimization, however, was that the effects of the parameters might be interrelated. In fact, once the parameters were matched to actual fluids, this was indeed found to be the case. Constraints that specify the maximum and minimum temperature differences between the evaporator and the condenser-absorber must be met, and the effect of varying the refrigerant parameters on the pressure-equalizing fluid parameters (and vice-versa) explored. These difficulties are explained in more detail in the following Selection of Fluid Triplets subsection.

1.65 630

640

650

660

670

Tc,absorb

680

690

700

Figure 4. COP versus Absorbing Fluid Critical Temperature for Varying Generator Temperature list of alternatives was narrowed so that only formaldehyde, hydrogen chloride, methyl amine, methanol, and ethanol were considered. Refrigerant. As with the pressure-equalizing fluid, there were a number of alternate fluids found for butane. Initially, 18 fluids were considered: vinyl fluoride, hydrogen bromide, vinylacetylene, hydrogen sulfide, 2-butyne, propylene, propane, methyl chloride, pentane, nitrous oxide, acetylene, ethyl chloride, chlorodifluoromethane, dimethyl ether, ketene, isobutane, methyl acetylene, and ethyl fluoride. As a secondary screen, toxicity was again considered, and certain choices, such as methyl chloride, were eliminated. Selection of Fluid Triplets As noted in the Introduction, the Patel-Teja equation of state and Panagiotopoulos and Reid mixing rules were selected to model the actual fluid behavior. In order to accurately do so, binary interaction parameters needed to be found for each absorbing fluid/pressure-equalizing fluid and refrigerant/ pressure-equalizing fluid pair. These interaction parameters are found through correlations with experimental equilibrium data. The availability of this experimental data further narrowed the choices as to which alternatives could be considered. For

Matching the Ideal Properties to Actual Fluids Three databases were used to gather property information on a total of 612 fluids [13-19]. For each fluid type (refrigerant, pressure-equalizing fluid, and absorbing fluid), the parameters were ranked in order of their greatest effect on the COP, and the best matches were determined. Absorbing Fluid. Einstein and Szilard used water as their absorbing fluid. The only viable alternative to water that was found in the fluid search was hydrazine (H4N2). Hydrazine has the potential to raise the system’s COP since it has a higher critical temperature and a lower critical pressure and liquid heat capacity than water. Unfortunately, hydrazine also has a higher molecular weight and a lower acentric factor. These variations combine to actually lower the coefficient of performance. Since water has unique properties, water was kept as the absorbing fluid for alternative mixtures. Pressure-Equalizing Fluid. In contrast to the previous case, a number of alternatives were found for ammonia. These are listed in Table 3, where the values of the parameters for ammonia are also given. However, in addition to satisfying the parameter requirements, the toxicity and environmental impact of the alternatives were also examined [20]. For example, nitrogen dioxide initially appears to be a viable alternative, but it is actually a pollutant that is extremely harmful to humans and the environment. Including this sort of consideration, the

Table 3. Pressure-Equalizing Fluids Component Ammonia Formaldehyde Methyl Amine Hydrogen Chloride Sulfur Dioxide Nitrogen Dioxide Methyl Bromide Sulfur Trioxide Methanol Ethanol

5

M

Tc (K) Pc (bar)

ω

cp

 kJ     kg ⋅ K 

17.03 405.69 30.03 408.04 31.06 430.00

112.80 65.90 74.58

0.253 0.282 0.275

5.272 2.382 1.774

36.46 324.69

83.10

0.132

2.112

64.06 430.79

78.84

0.245

1.369

46.01 431.19

101.33

0.851

3.25

94.94 467.04

80.00

0.192

0.857

80.06 490.89 32.04 512.68 46.07 513.96

82.10 80.97 61.48

0.424 0.564 0.645

3.224 2.842 2.861

that equation, the above constraint can be stated mathematically as: 0) 1) (15) Pc, refrg exp ∫ (refrg + ω ∫ (refrg = Pc, pe exp ∫ (pr0) + ω ∫ (pe1) where

Propane Ammonia

[

Propylene Butane

Pentane

Water

Tcond / abs − Tevap ≥ Tlift . Since methanol and ethanol do not meet this criterion, they were eliminated from consideration. The effect of each of the remaining fluid triplets on the system performance could then be examined using the complex cycle model based on the PatelTeja equation of state and Panagiotopoulos and Reid mixing rules.

Propylene Butane Propane

Methanol

Propylene

ADVANCED CYCLE MODEL Equation of State The Patel-Teja equation of state is given as [21]: RT a P= − (16) v − b v( v + b ) + c ( v − b ) In equation 16, b and c are constants, and a is a function of temperature for each individual substance. Their values are found through examination of the critical point isotherm inflection. The value of c is straightforward: R T  c = Ω c  u c  , where Ω c = 1 − 3ζ c (17)  Pc  Determining b is slightly more complicated: R T  b = Ω b  u c  (18)  Pc 

Butane Propane Ethanol

]

and

Butane Propane

Hydrogen Chloride

[

T   Tevap  0) 1)  ∫ (refrg = Fn cond / abs  , ∫ (pe0) , ∫ (pe1) = Fn , ∫ (refrg   T T   c, refrg   c, pe 

Pentane Methyl Amine

]

Propylene Isobutane

Figure 5. Alternative Fluid Triplets example, while there were data available for waterformaldehyde vapor-liquid equilibrium, no data could be found for formaldehyde and any of the refrigerant fluid alternatives. Therefore, formaldehyde was eliminated as a pressureequalizing fluid alternative. The various possible matches were extensively explored, which resulted in fourteen alternative fluid triplets. These are outlined in Figure 5, which also includes the base case triplet of ammonia-water-butane. Unfortunately, once the vapor-liquid equilibrium behavior of the fluid pairs was modeled, some of the alternatives had to be eliminated. The optimum system pressure has been found to be that which allows the refrigerant to begin to condense at its partial pressure and the condenser-absorber temperature. At that pressure, the minimum evaporator temperature is then either the saturation temperature of the pressure-equalizing fluid or the minimum boiling azeotrope of the fluid pair, whichever is lower. For a water heating application, the temperature of the condenser-absorber should be 325 K and the maximum temperature of the evaporator should be approximately 295 K. Although the ideal model optimization indicated that using either methanol or ethanol as the pressure-equalizing fluid would increase the COP, it was found that the triplets containing those fluids could not produce a suitably low evaporator temperature for a condenser-absorber temperature of 325 K. Therefore, in addition to matching actual fluids to parameter trends, parameter boundaries must also be given. A constraint was added that the saturation temperature of the pressureequalizing fluid must be less than or equal to the condenserabsorber temperature minus the lift at the system pressure. Equation 6 from the corresponding states theory relates the saturation temperature and pressure to the acentric factor. Using

The value of Ωb is the smallest positive root of: Ω 3b + (2 − 3ζ c )Ω 2b + 3ζ c2 Ω b − ζ 3c = 0 Finally, as mentioned above, a is dependent on the temperature:  R 2T2  a = Ω a  u c α (19)  P   c  While Ωa is a constant: Ω a = 3ζ 2c + 3(1 − 2ζ c )Ω b + Ω 2b + 1 − 3ζ c

a is a function of the reduced temperature.

[ (

α = 1 + F 1 − TR

)]

2

Both ζc and F are empirically-determined constants which can be expressed as functions of the acentric factor for nonpolar and weakly-polar fluids: ζ c = 0.329032 − 0.076799ω + 0.0211947ω 2 F = 0.452413 + 1.30982ω − 0.295937ω2

In addition to providing the pressure, volume, and temperature of the state of a substance, the Patel-Teja equation of state can also be used to determine the enthalpy and entropy. Following a similar derivation to that given for the principle of corresponding states, the enthalpy departure function is: H − H* 1  ∂a  Z+M (20) = Z −1− − a  ln T R uT 2R u TN  ∂T  Z+Q

6

−

Table 4. Binary Interaction Parameters

1

 ( b + c) 2  2 P b+c  where and  , N = bc +  M= − N ,   2 R T 2   u   P  b+c  Q= + N  . The differential ∂a/∂T is found by taking  RuT  2 

Pair Ammonia-Water Ammonia-Butane Ammonia-Pentane Ammonia-Propylene Ammonia-Propane Methyl Amine-Water Methyl Amine-Butane Methyl Amine-Pentane HCl-Water HCl-Propane HCl-Propylene HCl-Butane

the derivative with respect to the temperature of equation 19. Mixture Properties In addition to predicting the behavior of individual substances, the Patel-Teja equation of state can also be used to model mixtures. To do so, mixture parameters (am, bm, and cm) must be used in equation 16 rather than the individual substance parameters. The mixture parameters are a function of the component parameters and mole fractions: a m = ∑ ∑ x i x j a ij , b m = ∑ x i b i , and c m = ∑ x ic i . i

i

j

i

The values of bi and ci for each component can be found using equations 17 and 18. Determining aij is more complicated, and is a function of the individual component parameters and one or more binary interaction coefficients. In this study, the Panagiotopoulos and Reid rule was used to find aij: a ij = [1 − k ij + x i (k ij − k ji )] a i a j

+

(21)

1 (b' c 2 + b 2 c'− b' bc − c' cb) QD

 D' ( −b 2 − c 2 − 6bc + b' b + 3b' c + 3bc'+ c' c) − ln( Z) Q 

D = v 2 + vb + vc − bc 1  2v + b + c − − Q  −  D' = (−Q) 2 ln  2v + b + c + − Q    a ' = −2∑ ∑ y j y k (1 − k jk ) a ja k − 3∑ ∑ y 2j y k ( k jk − k kj ) a k a j j

k

j

Reference

-0.264 0.283 0.283 0.205 0.29 -0.205 0.123 0.123 -1.4 0.15 0.142 0.15

-0.294 0.128 0.128 0.1 0.13 -0.423 0.057 0.057 -0.18 0.058 0.025 0.058

18 23 Butane: 23 Propane: 23 23 24 18 18 25 26 Propane: 26 27

ADVANCED MODEL RESULTS Once again, using the conservation of mass, species, and energy equations developed for the cycle and the Patel-Teja and Panagiotopoulos and Reid thermodynamic property model a computer program was created to describe the system behavior with a high degree of accuracy. Equation of state coefficients (F, Ωa, etc.) were calculated for each substance [6, 21], and the binary interaction coefficients were determined for each fluid pair using experimental data. The kij and kji values for each fluid pair are given in Table 4. The source for each pair’s experimental data is listed in the Reference column (a refrigerant listed in that column indicates that the interaction parameters were extrapolated from that data). It was found that the Patel-Teja equation of state provided excellent correlation for a range of mixtures, including azeotropic pairs, zeotropic pairs, and vapor-liquid-liquid equilibrium points. Once the binary interaction parameters were determined, the equation of state coefficients and entropy and enthalpy departure functions were used to find the cycle COP for the alternate fluid triplets. As explained earlier, the system pressure for each triplet is set such that the partial pressure of the refrigerant in the condenser-absorber is equal to the refrigerant’s saturation pressure at the temperature of the condenser-absorber. The desired evaporator temperature ranges from 295 K to 306 K, which may or may not be the minimum possible temperature. For each triplet, the generator temperature was varied to find the maximum COP. Three temperature configurations suitable for water heating were examined. The results for the first configuration, a condenser-absorber temperature of 325 K and evaporator temperature of 295 K, are given in Table 5. The second

Q = −b 2 − 6bc − c 2

where

kji

The enthalpy departure function for each component in the mixture is given by equation 20. However, since a, b, and c are now mixture parameters, to determine ∂am/∂T an expression must be found for ∂aij/∂T from equations 19 and 21: ∂a ij Ω a ,i R 2 Tc2,i Ω a , j R 2 Tc2, j  Fi Tc, j α j TR , j − Fj Tc,i α i TR ,i    = ∂T Pc,i Pc, j   2Tc,i Tc, j TR ,i TR , j   The departure functions can then be used to determine the enthalpy (and entropy) of the mixture at any point in the cycle.

The parameters kij and kji are the binary interaction coefficients, and are determined by minimizing the difference between experimentally and analytically determined equilibrium points. The computationally determined equilibrium points are found by using the fugacity. For mixtures, in addition to vaporliquid equilibrium, liquid-liquid equilibrium (fiL1 = fiL2) and vapor-liquid-liquid equilibrium (fiL1 = fiL2= fiV) can also exist. The expression for the fugacity of a component in a mixture is more complicated than the expression for the fugacity of a single substance. For component i, the fugacity is [21-22]: f y   v  b + b' a ' D' a  v 2 2 ln i i  = ln + + +  QD (b + c + 6bc P v b v b ' RT RT − −      + b' b + 3b' c + 3bc'+c' c) +

kij

k

+ 2∑ y i y j ( k ij − k ji ) a i a j j

+ ∑ y j[ 2 −k ij − k ji + y j ( k ij − k ji ) a i a j j

b' = b i − b c' = c i − c

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Table 5. COP for Tca = 325 K, T evap = 295 K Triplet

COP

Psys (bar)

Ammonia-Butane-Water Ammonia-Propane-Water Methyl Amine-Pentane-Water H Chloride-Propane-Water H Chloride-Propylene-Water H Chloride-Butane-Water Ammonia-Pentane-Water

1.51 1.49 1.44 1.41 1.39 1.39 1.38

5.25 18.3 1.7 18 21.7 5.25 1.78

Table 6. COP for Tca = 325 K, T evap = 306 K

configuration’s (Tca = 325 K, Tevap = 306 K) results are in Table 6, and the results for the third configuration (Tca = 316 K, Tevap = 295 K) are in Table 7.

Effect of State Point Properties For the first set of operating temperatures, the ammoniawater-butane and ammonia-water-pentane triplets display the most disparate COPs. There are various reasons for this degradation in the COP, as can be seen in Table 8, where the evaporator cooling capacity has been set to 439.2 kW. The pentane case requires a much larger mass flow rate of ammonia leaving the generator and entering the evaporator. As a result of this, more ammonia must be absorbed by the water stream in the condenser-absorber, and thereby desorbed in the generator. Furthermore, the decreased pressure in the pentane case means that the concentration of ammonia is much higher in the liquid stream entering the bubble pump. For that case, the bubble pump must receive nearly three times as much heat transfer in order to remove the remaining ammonia and circulate the liquid water. In fact, twenty-five percent of the heat that is applied to the generator goes to the bubble pump. Previously, the bubble pump heat input had been treated as a minor factor in the efficiency, but this is obviously no longer

Psys (bar)

Ammonia-Propane-Water Ammonia-Butane-Water Methyl Amine-Pentane-Water H Chloride-Propylene-Water H Chloride-Propane-Water Ammonia-Propylene-Water Ammonia-Pentane-Water H Chloride-Butane-Water Methyl Amine-Butane-Water

1.76 1.75 1.7 1.62 1.61 1.61 1.59 1.56 1.53

15.3 4.2 1.3 17.8 15 17.6 1.4 4.2 4.2

Psys (bar)

Ammonia-Butane-Water Ammonia-Propane-Water Methyl Amine-Pentane-Water H Chloride-Propane-Water H Chloride-Propylene-Water H Chloride-Butane-Water Ammonia-Pentane-Water Methyl Amine-Butane-Water Ammonia-Propylene-Water

1.88 1.87 1.75 1.68 1.66 1.66 1.63 1.61 1.52

5.25 18.3 1.7 18 21.7 5.25 1.78 5.2 21.7

Corresponding States/Ideal Solution Model Parameters To test the validity of the parameter trends indicated by the corresponding states/ideal solution model, four theoretical fluids have been created as examples. Three of the fluids are a Table 8. Comparison of Refrigerant Performance

Table 7. COP for Tca = 316 K, T evap = 295 K COP

COP

true. The higher bubble pump heat transfer rate can be traced to the ammonia concentration differences between the ammoniabutane and ammonia-pentane cases. For the ammonia-pentane mixture, the relative difference in concentration between the ammonia-water mixture entering the generator and the ammonia-water mixture entering the bubble pump dictates that more water must be pumped to provide the given ammonia requirement. Similar results occur in both alternative temperature configurations. The effect of changing the pressure-equalizing fluid can be seen by examining the water-methyl amine-pentane and waterammonia-pentane triplets for the first set of operating temperatures (Table 9). Again, the evaporator load is set to 439.2 kW. In this instance, the pressures and generator temperatures are comparable, but the methyl amine case requires nearly twice as much pressure-equalizing fluid vapor to be produced in the generator to meet the refrigerant’s evaporator partial pressure requirements. This stems from the difference in molecular weight between ammonia and methyl amine. To provide the same evaporator partial pressure, more mass of pressure-equalizing fluid is required since the molecular weight of methyl amine is nearly twice that of ammonia. Once again, though, a driving force in lowering the COP is the amount of heat added to the bubble pump. The bubble pump for the ammonia-pentane mixture requires approximately twice as much heat transfer as for the methyl amine-pentane mixture. This parasitic pumping power has a large influence on the COP.

ANALYSIS Unfortunately, for this particular cycle, none of the alternate triplets provided a better coefficient of performance. For all three cases, the water-ammonia-propane COP is close to the water-ammonia-butane COP, with differences so small that they can be neglected. To analyze this result, the relationship of the COP to the various states throughout the cycle will be explored. The validity of the corresponding states/ideal solution model will also be reexamined.

Triplet

Triplet

8

& bubpump Q & gen,act Q

Pair

COP

P

Ammonia-Butane Ammonia-Pentane

1.51 1.38

5.25 1.78

Pair

& bubp m & pe,5 m

Ammonia-Butane Ammonia-Pentane

1 0.465 1.465 2.931 0.509 3.44

88.5 258.2 & 7 m

779.1 912.8 & refrg,3 m

1.293 1.264

Table 9. Pressure-Equalizing Fluid Performance Pair

COP

Methyl Amine-Pentane Ammonia-Pentane Pair M. Amine-Pentane Ammonia-Pentane

1.44 1.38

& bubp m 2.035 2.931

P 1.7 1.78 & pe,5 m 1.003 0.509

Table 10. Theoretical Fluids’ Effect on the COP

& bubpump Q & gen,act Q

140.4 258.2 & 7 m 3.04 3.44

Base Fluid Parameter Alteration Butane Butane Butane Ammonia

863.6 912.8 & refrg,3 m 1.265 1.264

Tc to 415 from 425.2 Pc to 50 from 37.9 w to 0.10 from 0.199 M to 30 from 17

COP

COPincrease

1.52648 1.52824 1.5278 1.50905

0.02028 0.02204 0.0216 0.00285

triplet for the Einstein cycle, ammonia-water-propane and methyl amine-water-pentane are acceptable alternatives that do not significantly lower the COP. Furthermore, three hypothetical refrigerants and one hypothetical pressureequalizing fluid were also modeled using the Patel-Teja equation of state. The hypothetical fluids each increased the COP, but to a somewhat lesser degree than that predicted by the corresponding states/ideal solution model.

hypothetical alkane with properties similar to butane. One parameter has been changed for each of the fluids to a value that was predicted to increase the COP. The fourth fluid is an ammonia alternative with a different molecular mass. Appropriate changes were made to each cycle model to reflect the new parameter values, and the COP was calculated for a condenser temperature of 325 K and an evaporator temperature of 295 K. Each of the changes and the effect on the COP is summarized in Table 10. In each case, the COP was increased as predicted. Unfortunately, these properties cannot be matched to actual fluids.

REFERENCES [1] Einstein, A. and Szilard, L., 1930, Refrigeration, Appl. U.S. Patent: 16 Dec. 1927; Priority: Germany, 16 Dec. 1926. [2] Alefeld, G., 1980, “Einstein as Inventor,” Physics Today, pp. 9–13. [3] Delano, A. D., 1998, Design Analysis of the Einstein Refrigeration Cycle, Ph.D. thesis, Georgia Institute of Technology. [4] Shelton, S. V., Delano, A., and Schaefer, L. A., 1999, “Design Analysis of the Einstein Refrigeration Cycle,” Proceedings, Renewable & Advanced Energy Systems for the 21st Century, CD Transactions, RAES-04. [5] Shelton, S. V., Delano, A., and Schaefer, L. A., 1999, “Second Law Study of the Einstein Refrigeration Cycle,” Proceedings, Renewable & Advanced Energy Systems for the 21st Century, CD Transactions, SLA-02. [6] Schaefer, L. A., 2000, Single Pressure Absorption Heat Pump Analysis, Ph.D. Dissertation, Georgia Tech. [7] Follin, J. W. and Yu, K., 1980, “Energy Conversion and Storage Techniques: Evaluating the Einstein Refrigerator,” Technical report, Johns Hopkins Applied Physics Laboratory. [8] Rojey, A., 1984, Process for Cold and/or Heat Production with Use of Carbon Dioxide and a Condensable Fluid, U.S. Patent: 4,448,031. [9] Pitzer, K., Lippmann, D., Curl, R., Huggins, C., and Petersen, D., 1955, “The volumetric and thermodynamic properties of fluids - II. Compressibility factor, vapor pressure and entropy of vaporization,” Journal of the American Chemical Society, 77(13): 3433–3440. [10] Lee, B. and Kesler, M., 1975, “A generalized thermodynamic correlation based on three-parameter corresponding states,” AIChE Journal, 21(3): 510–527. [11] Wong, Y., Cheng, S., and Groeneveld, D., 1990, “Generalized thermodynamic and transport properties evaluation for nonpolar fluids,” Heat Transfer Engineering, 11(1): 60–72. [12] Prausnitz, J., Lichtenthaler, R., and Azevedo, E., 1986, Molecular thermodynamics of fluid-phase equilibria, 2nd ed., Prentice-Hall, Englewood Cliffs, N.J. [13] Reid, R., Prausnitz, J., and Poling, B., 1987, The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York.

CONCLUSIONS An example single pressure absorption cycle (the Einstein cycle) was modeled using two separate property models in order to quantify the influence of the various working fluid properties. The first property model used departure functions calculated from the three-parameter theory of corresponding states to find the properties of individual fluids. Once the properties of the individual fluids were calculated, the properties of the fluid mixtures were found by utilizing Raoult’s Law and ideal solution mixing rules. The corresponding states/ideal solution model requires only five characteristic fluid parameters per fluid to model the system behavior. These are the critical temperature, critical pressure, acentric factor, molecular weight, and the specific heat at constant pressure. By varying these parameters, trends were observed that could increase the COP. Differences in the molecular weight of each of the fluids caused only small to negligible changes in the COP. Varying the acentric factor and the critical temperature resulted in the largest increase in the COP, with the refrigerant acentric factor producing the largest effect. Lowering the refrigerant acentric factor from 0.20 to 0.08 should increase the COP by 6%. Based on the results of the corresponding states/ideal solution model, alternative fluids were sought for the three Einstein cycle working fluids. No viable alternative was found for water, the absorbing fluid. For the pressure-equalizing fluid, methyl amine and hydrogen chloride were selected for study as replacements for ammonia. Propane, propylene, and pentane were chosen as refrigerant alternatives. Experimental equilibrium data were found in the literature for each of these alternative mixtures. To model the actual cycle behavior, the Patel-Teja equation of state and Panagiotopoulos and Reid mixing rules were used. Three operating conditions were examined that would be suitable for domestic water heating. While it appears possible that ammonia-water-butane may be the ideal fluid

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[14] The Physical Properties Database, 2004, Syracuse Research Corporation. [15] Muste, M., 2004, Fluid Physical Properties and Constants, IIHR-Hydroscience & Engineering, University of Iowa. [16] NIST, 2004, Chemistry WebBook, http://webbook.nist.gov/chemistry/. [17] Cranium: Component Software for Physical Property Estimation, 2004, Molecular Knowledge Systems. [18] Gmehling, J., Onken, U. and Arlt., W., 1977, Vaporliquid equilibrium data collection. [19] Chu, J., Wang, S., Levy, S., and Paul, R., 1956, Vapor-Liquid Equlibrium Data, J.W. Edwards, Publisher, Inc., Ann Arbor, MI. [20] Environmental Defense Scorecard, 2000, About the Chemicals, http://www.scorecard.org/chemical-profiles/. [21] Patel, N. and Teja, A., 1982, “A new cubic equation of state for fluids and fluid mixtures,” Chemical Engineering Science, 37: 463–473. [22] Smith, V. S., 1995, Solid-Fluid Equilibria in Natural Gas Systems, Ph.D. Thesis, Georgia Institute of Technology. [23] Wilding, W., Giles, N., and Wilson, L., 1996, “Phase equilibrium measurements on nine binary mixtures,” Journal of Chemical Engineering Data, 41: 1239–1251. [24] Stumm, F., Heintz, A., and Lichtenthaler, R., 1993, “Experimental data and modeling of VLE of the ternary system carbon dioxide + water + methylamine at 313, 333, and 353 K,” Fluid Phase Equilibria, 91: 331–348. [25] Brandani, S., Brandani, V., and Giacomo, G., 1994. “Vapor-liquid equilibrium calculation of the system waterhydrogen chloride,” Fluid Phase Equilibria, 92: 67–73. [26] Ashley, J. and Brown, G., 1972, “Vapor-liquid phase equilibria: Hydrogen chloride-ethane,” Chemical Engineering Symposium Series, 150: 129–136. [27] Ottenweller, J., Holloway, C., and Weinrich, W., 1943, “Liquid-vapor equilibrium compositions in HCl-nButane system,” Industrial and Engineering Chemistry, 35: 207–209.

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