The optimal design of refrigerant mixtures for a two-evaporator refrigeration system

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The Optimal Design of Refrigerant Mixtures for a Two-Evaporator Refrigeration System Nachiket Churl Luke E.K. Achenie* Department of Chemical Engineering, U-222 University of Connecticut Storrs, CT 06269 Abstract

and, b) Isobaric phase change operations do not occur at constant temperatures. Rather, there is a range of The use of liquid-phase mixtures is common in a num- temperatures over which the mixture exists in a twober of industrial applications. Mixtures have certain phase state. advantages over individual components, and these The extent to which the vapor and liquid phase have been exploited advantageously. For the case compositions differ and the temperature range over of refrigerants, a number of azeotropic mixtures are which phase change occurs depends on the VLE propused in practise. The use of non-azeotropic refriger- erties of the mixture. For the case of refrigerants, ant mixtures, however, is not as common. This paper some of the advantages of using mixtures over singlelooks at the design of refrigerant mixtures for a refrig- component fluids are as follows : eration cycle consisting of two evaporators operating 1. Since there is a range of temperatures over which at two different temperatures. Such a cycle has been phase change occurs, there is a gradual change proposed for use in, for example, a commercial refrigin the fluid temperature as it changes phase over erator having a separate freezer section. the length of an evaporator or condenser. The Compared to the single evaporator cycle operfluid with which the refrigerant exchanges heat ating at the freezer temperature and circulating the also experiences a change in temperature since it cold air from the freezer to the rest of the refrigeradoes not undergo phase change. This results in tor, the double-evaporator cycle can give higher effitemperature profiles that are parallel to and near ciencies since cooling is not done at a lower tempereach other. This increases efficiency by reducing ature than what is required. Also, the use of nonthermodynamic irreversibility during heat transazeotropic mixtures has a number of advantages over fer. single component refrigerants and azeotropic mixtures. A mathematical programming approach devel2. For refrigeration cycles, a change in operating oped earlier is applied to the double-evaporator cycle conditions (such as the ambient temperature) to obtain refrigerant mixtures that maximize cooling. results in a change in the desired refrigeration capacity. It is not possible to change the refrigeration capacity for a single component refrig1 Introduction erant since its composition is fixed. For a nonazeotropic refrigerant mixture this can be done Mixtures have the potential for giving a good mix by removing some quantity of one of the mixture of properties unattainable by individual components. components. This is because mixture properties are not always a linear interpolation of the properties of its con- In this paper we begin with the description of a genstituents. For example, mixed solvents in associa- eral mixture design methodology. Subsequently, a tive systems can give equilibrium distribution coeffi- two-evaporator refrigeration cycle is described, and cients and separation factors substantially different the mixture design formulation applied to it to obfrom what would be predicted by simple interpo- tain optimal mixtures. lation (Munson and King, 1984). For the case of non-azeotropic mixtures, the differences in thermodyMixture Design namic properties lead to a number of potentially ad- 2 vantageous situations. Non-azeotropic mixtures beA general strategy for forming a refrigerant mixture have differently from pure fluids in two significant involves the steps listed below and shown in Figure 1. ways : a) During a phase change operation, the comThis has been described in detail in an earlier paper positions of the liquid and vapor phases are different, (Duvedi and Achenie, 1996). *Author to whom all correspondence should be ad1. Choose an initial set of single component refrigdressed. Phone 860-486-2756, Fax 860-486-2959, Email achenie~eng2.uconn.edu erants. $349

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2. Select physical properties which must be set at prespecified levels, such as compatibility with polymers and lubricants, volatility, flammability or toxicity level. 3. Select a performance index, p, upon which refrigerants are evaluated (e.g. ozone depletion potential, coefficient of performance, corrosive tendency, economics, safety, pressure ratio, discharge temperature, capacity). 4. Design the refrigerant mixture. In this paper this is done by formulating and solving a mathematical program. 5. Design for a series of good refrigerant mixtures which may be used for experimental evaluation. A series of mixtures is obtained by eliminating all previous solutions from the search space. This can be done by introducing an integer cut, which is a mathematical way of eliminating a binary solution from the search space. 6. Verify experimentally the predicted performance characteristics.

Basis Set of

where ~ is binary integer (0, 1) and x is continuous with lower and upper bounds, xlo and xhi. A, C, E and F are matrices of appropriate dimensions while b, d, g, h, xto, Xhi are vectors of appropriate dimensions. In the model for refrigerant mixtures, the first step involves identification of a basis set of single component refrigerants; each of these refrigerant can potentially be included in the final mixture. Subsequently, the mixture design problem is formulated as a mixed integer nonlinear program (MINLP) in which binary integer variables denote which single component refrigerants make up the mixture, and continuous variables represent mole fractions and other mixture properties. For mixtures, ff from Equation 1 is a vector of binary variables that specify the "type" of refrigerants that exist in the designed refrigerant mixture. Thus, if the jth component of ~ (i.e., ~j) takes a value of 1, then the jth refrigerant (from the basis set) exists in the mixture. Alternately, a value of 0 implies absence of the jth refrigerant in the mixture. &j is the mole fraction of the jth single component refrigerant in the mixture. &, a subset of x, is a vector of all &j. In Equation 1, x also includes all the physical properties of interest.

3

individualrefrigerants[ $ I

SpecifyPhysicalProperties[

I

g(x).< 0

3.1

I

DefinePerformanceCriterion[

I

min p(x,y)

Designforbest RefrigerantMixture I

ExperimentalVerification

Figure 1: Flowchart for Refrigerant Mixture Design A general mathematical programming model for mixture design with performance objective p(x, Z)) can be posed as (Duvedi and Achenie, 1996): min p(x, ~)

X,y subject to : Ax C~ E x - F~

(1)

= b < d <__ 0

(2)

g(x)

<

o

(5)

h(~)

_< 0

(6)

z

(7)

Xto <_

< zhi

(3) (4)

A Two-Evaporator ation Cycle

Refriger-

Description of the Cycle

For refrigeration applications where cooling is required at two different temperatures, such as a domestic dual-compartment refrigerator, different schemes can be used to attain the different cooling temperatures. The method that is commonly used involves cooling the freezer region to the desired temperature, and then circulating cold air from the freezer to the rest of the refrigerator compartment. This approach is not very efficient since cooling is done at a lower temperature than what is desired. A more efficient approach involves using two evaporators operating at different temperatures as shown in Figure 2 (Stoecker, 1985). This system not only possesses two evaporators, but also incorporates two refrigerant-to-refrigerant heat exchangers. While these heat exchangers may not be always effective in improving the operating efficiency for a single refrigerant, they are beneficial when using refrigerant mixtures (Stoecker and Walukus, 1981). This refrigeration cycle was studied experimentally by Stoecker (1985) who showed that a 50% mixture of R12/Rl14 gives better performance than R12 alone. In this cycle, the refrigerant mixture leaving the condenser (Position 7) is assumed to be saturated liquid. This is then cooled through two heat exchangers to reach position 9 as sub-cooled liquid. Upon passing through an expansion valve, a part of the mixture evaporates as a result of which position 1 is at a very low temperature. After gaining heat in the two evap-

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~

C

l HeatExchanger1

l HeatExchanger2

,®

ompmssor 5 4

HighTemperature --~rator LT__J o~ 3 2

LowTemperature ---~rator

1

Figure 2: Double Evaporator Refrigeration Cycle orators and two heat exchangers, the mixture enters the compressor as superheated vapor. The starting set for mixture formation consists of 43 individual refrigerants used by the REFPROP subroutines (Huber et al., 1996). These subroutines are used for property prediction of single component refrigerants and refrigerant mixtures. 3.2

Mathematical

Variables : For this system, it is convenient to use the temperatures at various positions as variables. For this case study T7 and 7"4 are specified, and the refrigerant mixtures leaving the condenser (position 7) and the high-temperature evaporator (position 4) are assumed to be saturated liquid and saturated vapor, respectively. The pressures at these positions are then calculated by Bubble Point and Dew Point calculations. The pressure at position 4 (PL) prevails in positions 1 to 5, while the pressure at position 7 (PH) prevails in positions 6 to 9, i.e., there is no pressure drop across the heat exchangers, the condenser or the evaporators. Thus the continuous variables are & and T1 ... Tg. The discrete variables ~ indicate presence or absence of individual refrigerants.

Constraints

For a typical refrigeration application, the specified parameters usually include the temperatures at which cooling is required, and the temperature at which the condenser can lose heat. For this refrigeration cycle, T4 and 7"7 are assumed to be specified) Since mixtures undergo a change of phase over a range of temperatures at a given pressure, i.e., the phase change is not isothermal, the entry and exit temperatures at the two evaporators will not be the same. Hence T3 will be less than T4, and T1 will be less than T2. The lowest temperature in the system is T1, which is obtained by passing the refrigerant at position 9 through an expansion valve. The temperature at point 9 (7"9) has to be suitably low to be able to achieve the desired cooling in the low-temperature evaporator. Since T7 is fixed, a certain amount of heat has to be removed from the stream before it reaches position 9. This is done with the help of two heat exchangers between whom the heat load is distributed. In theory, there is no upper limit to temperatures at points 5 and 6 (7'5 and Te). However, due to limitations of the prediction method used, it is difficult to obtain thermodynamic information if the fluid is in the supercritical state. To get around this problem we introduce constraints to ensure that the refrigerant mixture is not too close to critical points (Equation 20).

Constraints : An analysis of Figure 2 gives a number of process constraints that have to be applied to make the refrigeration cycle feasible. Points 1 to 5 in the cycle are at a lower pressure than points 6 to 9. The lowest temperature exists at point 1, and the temperature rises in steps till point 6 which has the highest temperature. On the other side of the cycle, temperatures drop from point 7 to point 9. These constraints are reflected in Equations 8 to 15. Enthalpy changes in the two streams passing through a heat exchanger are the same (neglecting heat losses). This gives rise to Equations 16 and 17 for the two heat exchangers. Equation 18 is applied to obtain isentropic compression, while Equation 19 models an isenthalpic expansion valve.

7"1 ___ T~

(8)

T2 _< 7"3 T3 _ T4 7"4 _< 7"5

(9) (10) (11)

T5 Tr Ts

<_ Ts

(12) (13) (14) (15)

//7 -/-/8

=

/-/5- H4

(16)

Hs - H9

=

Hs - H2

(17)

&

=

&

(18)

H1

=

H9

T9

_< T6 <_ T6 _< Tv

Tcsv+U~)i <_ Tci+U-IO Vi

(19) (20)

Variable B o u n d s : Since the cycle temperatures (T1...T g) are variables, it is possible to assign appropriate upper and lower bounds such that some of the constraints above get satisfied automatically. In the bounds given below, Te is the temperature at the exit of the high-temperature evaporator (T4), Tcd is the temperatures at the condenser's exit (TT), and AT,n,~ is the minimum approach temperature in the two heat exchangers. The lower limits for 7"1, T2 & tTi is the temperature at position i in the refrigeration T3 are fixed by considering the likely lowest tempercycle. ature in the cycle, and the upper limit of T6 is the

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likely upper limit for critical points.

Te

< < _< _

Te Tcd Ted T~ + ATmi,~ Te + AT,,in

<_ <_ <_ < <_

150 150 150

<_ Te

(21)

T2 _< Te

TI

(22)

T3 _< Te

(23)

T4 _< T~

(24)

T5 T6 T7 Ts T9

(25) (26)

<_ <_ <_ < <_

T e d - A T , ran 400 Tcd Ted Tcd

(27) (28) (29)

With these variable bounds, the following conclusions can be made : Equation 10 is redundant since the upper limit of Tz corresponds to the lower limit of T4 (Equations 23 and 24). Similarly, Equation 11 is redundant because of Equations 24 and 25, and Equation 13 can be eliminated in view of Equations 26 and 27. The bounds of Equations 27 and 28 make it unnecessary to have Equation 14. Equation 8 and 9 have to be retained since they cannot be eliminated by assigning appropriate limits. Once Equation 9 is satisfied, Equation 15 is automatically satisfied since the enthalpy change between positions 8 and 9 is the same as that between 2 and 3 (Equation 17). Since position 6 is at a higher pressure than 5, Equation 18 will force T6 to be higher than Tb, removing the need for Equation 12. Thus, Equations 8, 9, 16, 17, 18, 19 are necessary to describe the problem. The MINLP was solved using the Augmented Penalty - Outer Approximation - Equality Relaxation (AP/OA/ER) algorithm proposed by Viswanathan and Grossman (1990). This algorithm involves solution of a series of Non-Linear Programs (NLPs) and Mixed Integer Linear Programs (MILPs) till the optimal point is attained. This approach, however, only gives locally optimal solutions. We are currently investigating an interval arithmetic approach for obtaining globally optimal solutions.

P e r f o r m a n c e O b j e c t i v e : The performance objective is to maximize cooling in the two evaporators. Applying equal weights to cooling in the two evaporators, the objective function to be minimized becomes : p = - ( ( H 2 - H1) + (Ha - / / 3 ) )

(30)

Here, Hi is the enthalpy at the i th position in the refrigeration cycle. Numerous computational runs were performed with different starting points. Some of the refrigerant mixtures obtained with Te = 273 K, Tea = 300 K, and ATtain = 5 degrees K are listed in Table 1. Cooling effects are expressed in terms of kilojoules of heat per mole of refrigerant circulated through the cycle. Sample computational times required for a series of iterations without the termination criterion are given

in Table 2. These numbers refer to the CPU time in seconds for the program running on an IBM SP/2 computer. The refrigerant mixture that gives the greatest cooling consists of refrigerants 10 (Rl13) and 43 (RE170). Most of the cooling effect is concentrated in the high-temperature evaporator. The second-best refrigerant mixture is 19 (R141b) and 25 (R290). In this case the entire cooling effect is confined to the low-temperature evaporator. There is no enthalpy change as the refrigerant passes through the hightemperature evaporator. A similar situation exists for mixtures 4 and 5. Refrigerant mixtures that result in cooling through both evaporators include mixtures 1,3, 6 and 7. For applications in which separate evaporating temperatures are required, mixtures 2, 4 and 5 cannot be used since they eliminate one evaporator. The evaporator temperatures obtained for the mixtures is given in Table 3. As expected for mixtures, sliding temperatures are observed in the evaporators. If the evaporators are cross-current heat exchangers and if the minimum approach temperature is maintained at 10 degrees K, the lowest temperatures attainable at the two evaporators for mixture 3 will be 251.3 K and 281.6 K. For comparison, all 43 refrigerants from the REFPROP database were used to determine singlecomponent cooling effects. The highest cooling effect obtained from an individual refrigerant was only 0.61 k J/tool for refrigerant 5 (R14). Even after relaxing constraint 19, the highest cooling effect obtained was 20.7 kJ/mol for refrigerant 19 (R141b). This is less than some of the mixture cooling effects listed in Table 1. Thus we see that mixtures give superior performance compared to individual refrigerants, and that the approach used here is able to identify such mixtures. The refrigeration problem can be posed in a different way : for a fixed cooling effect in one evaporator, maximize cooling in the other evaporator. For example, if the cooling effect in the low-temperature evaporator is fixed at 10 kJ per mole of circulated refrigerant, cooling effect in the high-temperature evaporator can be maximized. For this problem, one additional constraint has to be added to those used before, namely : //2 - HI = 10 kJ/mol (31) Here, the objective function to be minimized is

p = - ( H 4 -/-/3)

(32)

The results for this problem are given in Table 4. The best mixture for this case consists of refrigerants 14 (R123a) and 41 (R1270). Its cooling effect in the high-temperature evaporator is 14.89 kJ/mol, giving a combined cooling effect of 24.89 kJ/mol. It is observed that this and other mixtures in this list do not appear in Table 1 inspite of having total cooling effects higher than some of the mixtures in that table. This points to non-convexities in the search space, and that the best mixtures in the two tables are local minima.

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4

Conclusion

As shown experimentally by Stoecker (1985), refrigerant mixtures have the potential of giving better performance than individual refrigerants. The use of an optimization method for obtaining the best mixture is therefore of much practical use. The dual-evaporator refrigeration cycle is ideally suited for applications that require cooling at two different temperatures. The mixture design methodology discussed here is able to obtain optimal refrigerant mixtures for the two-evaporator refrigeration cycle. We are currently developing a global optimization algorithm based on interval arithmetic in order to ensure global optimality. One way of checking for the ability of an optimization method to obtain good mixtures is by comparing its results with all possible mixtures obtained by enumeration. Enumeration techniques, however, are computationally intensive, even for small problem sizes. For example, if there are 43 starting components and we consider only binary mixtures, there are 1806 possible combinations. If we consider discrete points for ~ with a step size of 0.01, the number of combinations increases to 180,600. Further, the temperature variables can be similarly discretized, with the result that the number of combinations rises to the order of millions. It is to be noted that property prediction is computationally intensive, and estimating a number of properties for several million possible mixtures is not a very efficient method for getting the best mixture. An enumeration scheme was used to obtain the computational effort required to determine properties for all possible binary mixtures with ~ discretized with a step size of 0.01. The temperature variables were fixed for this test case, and hence a few process constraints were violated for some of the mixtures. Even for this low level of discretization, the computational time was of the order of one hour on the IBM SP/2 computer. In light of the foregoing, a complete enumeration including the temperature variables will require several hours of computational effort. For the MINLP approach proposed here, the time required to identify a locally optimal solution depends on the starting point, and is usually of the order of tens to hundreds of seconds. Thus the optimization scheme outlined in this paper is seen to be superior in terms of computational effort required to identify optimal mixtures.

Symbols ATtain P

Ps PL Qes Q~L

&

Minimum HX approach temperature Enthalpy at i th position, kJ/mol Performance Index Condenser Pressure, kPa Evaporator Pressure, kPa Cooling effect, high-temp, evaporator Cooling effect, low-temp, evaporator Entropy at i th position, kJ/mol K

Tcd Tc,, T~ Ti U x ~i Yi

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Condenser Temperature, K Critical temperature, i th component, K Highest temperature in condenser, K Evaporator Temperature, K Temperature at i th position, K A sufficiently large number Continuous Variables Mole fraction, i th component Presence of i th component, binary

References [1] Duvedi, A. and Achenie, L. E. K. On the Design of Environmentally Benign Refrigerant Mixtures : A Mathematical Programming Approach. Comp. FJ ChE., 1996, in print. [2] Huber, M., Gallagher J., McLinden M. and Morrison G. NIST Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP). U.S. Department of Commerce, National Institute of Standards and Technology, Gaithersburg, MD, 5.0 edition, 1996. [3] Munson, C. L. and King, C. J. Factors Influencing Solvent Selection for Extraction of Ethanol from Aqueous Solutions. I ~ E C Process Design and Development, 26:109-115, 1984. [4] Stoecker, W. F. Internal Performance of a Refrigerant Mixture in a Two-Evaporator Refrigerator. ASHRAE Transactions, 91B:241-249, 1985. [5] Stoecker, W. F. and Walukas, D. J. Conserving Energy in Domestic Refrigerators through the use of Refrigerant Mixtures. ASHRAE, 87(3):279291, 1981. [6] Viswanathan, J. and Grossmann, I. E. A Combined Penalty Function and Outer Approximation Method for MINLP Optimization. Comp. ChE., 14:769-782, 1990.

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1 (0.90s) 11 (0.817) 11 (0.821) 9 (0.755)

Comp 2 (X2) 43 25 16 12 25 12 19

(0.100) (0.055) (0.158) (0.092) (0.183) (0.178) (0.245)

-p [kJ/mol] 26.45 24.80 22.85 22.81 19.26 19.21 18.58

QeL [kJ/mol] 00.96 24.80 21.79 22.81 19.26 09.92 14.18

[kJ/moll 25.49 1.06

9.29 4.40

[kPa] 143.5 179.4 406.5 218.2 375.8 375.2 1345.1

[kPa] 17.0 30.1 48.1 44.5 103.0 102.7 109.6

Table 1: Best Refrigerant Mixtures (1) 1 : R l l , 9 : R32, 10 : Rl13, 11 : Rl14, 12 : Rl15, 16 : R125, 19 : R141b, 25 : R290, 43 : RE170

Itn. MILP NLP

1 0.94 9.86

2 0.83 237.32

3 0.94 358.74

4 0.81 1040.36

5 1.05 118.42

6 1.14 36.63

7 1.02 26.54

Table 2: Computation times in seconds

[ No. ] T1 [K] [ T2 [K] ]Ts [K] I T4 [K] I 1 2 3 4 5 6 7

244.4 247.4 241.3 254.9 260.8 261.4 236.4

245.5 272.0 270.6 272.0 272.0 266.9 261.8

246.5 271.6 267.9 265.1

273.0 273.0

273.0 273.0

Table 3: Evaporator Temperatures

No. I Comp 1 (Xt)

14 (0.931) 14 (0.851) 9 (0.140) 1 (0.868) 36 (0.909)

Comp 2 (X2)

41 43 36 30 41

(0.069) (0.149) (0.860) (0.132) (0.091)

- P = QeH I T1 [kJ/mol] I [K] 14.89 251.1 14.26 256.9 12.70 256.8 12.69 257.7 12.68 256.6

Table 4: Best Refrigerant Mixtures (2) 1 : R l l , 9 : R32, 14 : R123a, 30 : R245cb, 36 : RE134, 41 : R1270, 43 : RE170

7"3 [K] 260.5 264.1 264.5 265.0 264.4

8 0.95 2.96

9 0.87 26.77