ON THE DESIGN OF ENVIRONMENTALLY BENIGN

REFRIGERANT MIXTURES: A MATHEMATICAL

PROGRAMMING APPROACH

Amit Duvedi and L. E. K. Achenie∗

Department Of Chemical Engineering, U-222

University Of Connecticut, Storrs, CT 06269

(In Computers & Chemical Engineering)

May, 1996

Copyright Luke Achenie

∗ Author to whom all correspondence should be addressed.

(Phone : (860) 486 4020, Fax : (860) 486 2959, Email : achenie@eng2.uconn.edu)

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Abstract

Chlorofluorocarbon (CFC) based refrigerants (such as CFC12) have found widespread

uses in home refrigerators and automotive air conditioners primarily due to their nontoxic,

nonflammable nature and their high overall thermodynamic efficiency. However, CFC and

hydrofluorocarbon (HCFC) refrigerants with intermediate to high ozone depletion potentials

(ODP's) will be banned during the next two decades. The outcome of replacing CFC's in the

vapor recompression cycle and various other processes is vital to several industries. Feasible

solutions appear to include mixtures of hydrofluorocarbons (HFC's) which have the potential for

matching thermodynamic properties of current working fluids while meeting several criteria for

ozone depletion potential, flammability, toxicity, materials compatibility and cost.

In this paper, a proof of concept study is made to show that mathematical programming

can effectively be used to identify a small set of alternative refrigerant mixtures which can then

be evaluated experimentally. In the mathematical programming model, binary variables specify

the type and number of single component refrigerants that exist in a mixture and continuous

variables specify the mixture properties in addition to the proportion in which the single

component refrigerants are blended to form the mixture. The environmental issue has partially

been addressed by incorporating the Ozone Depletion Potential (ODP), a quantitative measure of

the ozone depleting capability of a compound, into the mathematical programming framework.

The results indicate the viability of the approach.

1. INTRODUCTION

Fully halogenated chlorofluorocarbon (CFC) refrigerants have extremely adverse

environmental effects in that they deplete atmospheric ozone. Legislation (Steed, 1989,

Zurer, 1992) restricting future production of CFC's has stimulated development of

suitable environmentally safe substitutes for these fluids. Several hydrohalomethanes and

hydrohaloethanes (e.g. HCFC22 and HFC134a) have emerged as potential short and

intermediate-range CFC replacements (Spauschus, 1991). Several factors must be

considered in developing feasible CFC replacements. The primary objective is to find

fluids having low ozone depletion potentials with thermodynamic and transport

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properties similar to CFC's 11 and 12 for example. Several pure halohydrocarbons have

been exhaustively studied (Kubota, 1988, McLinden, 1989, Baehr, 1991, Chynoweth,

1992) with candidates identified and some of these fluids in current use (e.g., HCFC141b

for CFC11). Mixtures, however, offer much greater flexibility in matching multiple

thermodynamic properties. For example, the HFC32-HFC134a system is of particular

interest as a replacement for HCFC22 and R502 in air-conditioning and refrigeration

equipment.

Identifying appropriate refrigerant mixtures that have specific attributes, such as

low ODP, is expensive and time consuming especially since the choice of single

component refrigerants and the proportion in which these will occur in the mixture is a

very large combinatorial problem. Clearly, a systematic procedure is needed for

refrigerant mixture design. This paper describes a systems engineering approach (via

mathematical programming) to systematically identify refrigerant mixtures with a high

potential as CFC substitutes.

The paper starts with the development of a mixed integer nonlinear programming

(MINLP) formulation of the refrigerant mixture design problem. Subsequently, a brief

description of the estimation techniques employed to predict the properties of refrigerant

mixtures is presented. The formulation is then tested on a simple case-study.

2. MINLP MODEL FOR MIXTURES

An MINLP model is used to design refrigerant mixtures with desired target

properties. Initially, an appropriate set (basis set) of single component refrigerants is

chosen to be used in the design. Subsequently, the refrigerant mixture design problem is

formulated as an MINLP in which discrete (binary integer) variables denote which single

component refrigerants make up the mixture, and continuous variables represent mole

fractions and other mixture properties. More formally, we pose the mixture design

problem as:

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xMax p(x)

subject to:

A x′ = b (1)

C w = d (2)

E x′ - F w ≤ g (3)

x - h(x′) = 0 (4)

where

x′ ⊂ x

xl ≤ x ≤ xu

w : binary integer

x : continuous

Variables

w is a vector of binary (0-1) variables which form a special class of integer or

discrete variables. These variables specify the “type” of refrigerants that exist in the

selected refrigerant mixture. Thus, if the j-th component of w (i.e. wj ) takes a value of 1,

then the j-th refrigerant (from the basis set) exists in the mixture. Alternatively, a value

of 0 implies the absence of the j-th refrigerant.

x is a vector of all the continuous variables in the problem formulation. x′ is a

subset of x and it represents the mole fractions of each single component refrigerant

(from the basis set) in the refrigerant mixture. Thus x′j represents the mole fraction of the

j-th refrigerant in the refrigerant mixture. In addition to x′, x includes all the physical

properties of interest, namely the saturated liquid (or vapor) mixture pressure in the

evaporator (or condenser), the saturated vapor temperature in the condenser, the

compressor displacement and the ozone depletion potential.

Objective function

p is a quantitative measure of the performance of the chosen refrigerant mixture.

The compressor displacement xCD is a good measure of the performance of a refrigerant

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(single component or mixture) in the cycle, and the ozone depletion potential xODP

accounts for the environmental damage caused by certain compounds by depleting the

earth’s ozone layer. Therefore, for the refrigerant mixture design problem, an

appropriate performance measure is a function of xCD and xODP as follows :

p = fxn( xCD, xODP) (5)

Pressure ratio, discharge temperature, capacity and COP sensitivity to cycle variability

also need to be reflected in the performance criteria for a more realistic cycle. However,

in this proof of concept study, we will use a simpler model.

Constraints

There are four main types of constraints in the MINLP formulation of the mixture

design problem, namely :

(i) Additivity constraint (A x ′ = b):

A specific form of a constraint of this type ensures that the mole fractions of all

the single component refrigerants that make up the mixture add up to 1 as follows:

i∑ x′i = 1.0 (6)

Here the summation is performed over all the single component refrigerants in the basis

set.

(ii) Integer constraints (C w = d):

This type of linear constraint involves only the integer variables and can be used

to impose restrictions on the type and the number of the single component refrigerants

that make up the refrigerant mixture. One such constraint is:

i

m

=∑

1 wi = NC (7)

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where NC is the total number of single component refrigerants allowed in the mixture and

m is the total number of single component refrigerants that are in the basis set. The

constraint as posed forces the mixture to be composed of exactly NC single component

refrigerants. To allow for up to NC single component refrigerants, Equation (7) ought to

be replaced with i

m

=∑

1 wi ≤ NC. In this paper only binary mixtures will be considered (NC

= 2) due to the limited availability of experimental data on refrigerant mixtures.

In addition to Equation (7), other types of integer constraints can be imposed to

express restrictions on the type of refrigerants that exist together in a mixture. If

refrigerant A (e.g. HFC32) and refrigerant B (e.g. HFC134a) should not exist together in

the mixture, then the following constraint can be imposed:

wA + wB ≤ 1 (8)

where wA and wB are the binary variables corresponding to refrigerant A and refrigerant B

respectively. For example, HFC32 which has been shown to have negligible ODP is

flammable and must be mixed with non-flammable single component refrigerants

(Chynoweth, 1992).

(iii) Mixed Constraints (E x ′ - F w ≤ g):

These constraints provide a link between the integer and the continuous variables.

These constraints are linear in both the integer and the continuous variables. A particular

type of this constraint results from the fact that the mole fractions should be non-zero

only for those refrigerants that constitute the refrigerant mixture. Thus, the mole fraction

should obviously be zero for all those refrigerants that do not exist in the mixture. This

constraint can be expressed as:

wi *ε ≤ x′i ≤ wi (1 - ε ) ∀ i = 1,2,.....m (9)

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Thus, whenever wi is 0 (that is, the refrigerant “i” does not exist in the mixture),

Equation (9) will drive x′i to 0. ε is a small non-zero parameter (chosen to be 0.001)

used to ensure that the mixture is truly an NC - component mixture since in this paper we

have restricted ourselves to designing for such a mixture. If in a globally optimal

mixture, the mole fraction of a particular single component refrigerant is ε, then we will

need to consider designing for less than an NC - component mixture.

Although none of the properties of a mixture may show critical behavior (where

liquid and vapor phases are identical), nearby critical points (of individual components)

can lead to numerical problems and large errors, since the properties of mixtures change

rapidly near them (Morrison, 1985). Subsequently, we impose the constraint that the

highest temperature in the cycle at which mixture properties are evaluated, is less than the

lowest critical temperature exhibited by the constituent single component refrigerants.

The saturated vapor mixture temperature Tcsv (xTcsv in the model) is the highest

temperature in the cycle at which we need to evaluate mixture properties such as

enthalpy, pressure and specific volume. This constraint can be expressed as:

xTcsv + UT wi ≤ [Tci + UT - 20] ∀ i = 1,2,.....m (10)

UT = Tcd (11)

where Tcd is the average condensing temperature, Tci is the critical temperature of the i-

th refrigerant, Tcsv (xTcsv in the model) is the saturated vapor temperature in the condenser

and UT is set equal to a large value such as Tcd to make the constraint trivial whenever

wi is 0 (i.e. i-th single component refrigerant is absent from the mixture). Whenever wi

equals 1, inequality (10) ensures that Tcsv is at least 20oK lower than the critical

temperature of the refrigerant “i” which is chosen in the mixture. The value of 20oK

provides a good measure of the minimum difference between the individual critical

temperatures and the temperature at which VLE calculations are performed in order to

avoid large inaccuracies in the property prediction.

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(iv) Physical property and process constraints (x - h(x ′ ) = 0):

In general, these constraints relate the mole fractions x′ to the target physical

properties (i.e. the appropriate components of x). As mentioned before, the properties of

interest are (a) compressor displacement, (b) mixture evaporator pressure, (c) mixture

condenser pressure, (d) saturated vapor temperature in the condenser and (e) ozone

depletion potential.

Constraints on physical property values are expressed as bounds. In the case

study considered, constraints on the evaporator and the condenser pressures (Joback,

1989) have been included. The lowest pressure in the cycle should be greater than the

atmospheric pressure (Dossat, 1981). This reduces the possibility of air and moisture

leaking into the system. A high system pressure increases the size, weight and cost of

equipment (Dossat, 1981). A pressure ratio of 10 is considered to be the maximum for a

refrigeration cycle (Perry, 1984). Thus the mixture pressure Pvp(Te) at the evaporating

temperature and the condensing pressure Pvp(Tcd) at the condensing temperature satisfy

Pvp ( Te ) ≥ 1.4 bar (12)

Pvp ( Tcd ) ≤ 14 bar (13)

For a more practical design, materials and lubricants compatibility, toxicity,

flammability, stability and miscibility of the refrigerant mixture ought to be incorporated

into the mathematical programming model. For this to be possible correlations for these

attributes will need to be developed perhaps using black box approaches such as neural

networks, or semi-empirical and fundamental approaches. At any rate, the basic structure

of the model does not change with these suggested additions.

3. TARGET PROPERTY ESTIMATION FOR MIXTURES

The target properties for the design of refrigerant mixtures that we have

considered are: (1) mixture pressure (Pvp), (2) compressor displacement (CD), (3) ozone

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depletion potential (ODP), and (4) saturated vapor temperature in the condenser (Tcsv).

An equation of state based approach has been employed to evaluate the first two

properties, whereas a simple linear mixing rule has been used to estimate the ODP of the

mixture from the ODP values of the individual components.

Here we have employed the Patel-Teja equation of state (Patel et. al., 1982, Lee

et. al., 1992) which predicts the equilibrium phase properties of refrigerant mixtures

reasonably well. This equation of state involves an adjustable binary interaction

parameter (kaij), which has been expressed in terms of individual component acentric

factors and critical compressibility factors (Lee et. al., 1992) to extend its application to

general multicomponent mixtures. The equation of state requires single component

critical properties, acentric factors, molecular weights and normal boiling points; thus,

target property estimation for mixtures is based directly on experimental data (which are

available) and hence the level of uncertainty in the predictions is less.

Estimation of mixture pressure

Vapor-liquid equilibrium (VLE) calculations are necessary to estimate the

evaporating pressure Pvp(Te) and the condensing pressure Pvp(Tcd). Since non-azeotropic

mixtures exhibit non-isothermal (non-flat) temperature profiles in the evaporator and the

condenser, the conventional meaning of the terms “evaporating temperature” and

“condensing temperature” do not apply and will have to be modified. With this in mind,

the following assumptions are made:

(a) The condensing temperature (Tcd) is taken to be the average of Tcsl (saturated liquid

mixture temperature ) and Tcsv, (saturated vapor mixture temperature ) at the condenser

pressure which is assumed constant throughout the condenser. Superheated vapor mixture

enters the condenser. However, the temperature drop due to de-superheating in the

condenser is assumed to be small and not considered while defining the average

condensing temperature.

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(b) The evaporating temperature (Te) is the average of the temperatures Tesl (saturated

liquid mixture temperature ) and Tesv, (saturated vapor mixture temperature ). Tesl and Tesv

are evaluated at the mixture composition and the evaporator pressure which is assumed

constant throughout the evaporator. The mixture entering the evaporator is not a

saturated liquid at the mixture composition. It is, however, assumed to be close to

saturated liquid conditions in order to justify the use of Tesl in the definition of the

average evaporating temperature.

(c) The mixture leaving the condenser is saturated liquid at the mixture composition.

(d) The mixture leaving the evaporator is saturated vapor at the mixture composition.

Different non-azeotropic mixtures exhibit a different temperature rise during evaporation.

Assuming a fixed evaporator duty and heat exchange area, the superheat temperature at

the evaporator exit is fixed. Hence, it is quite possible that some non-azeotropic mixtures

will exhibit a saturated vapor temperature greater than this superheat temperature, thus

implying that the mixture is in the two phase region (instead of being superheated) at the

evaporator exit. These arguments lead to the simplifying assumption made here.

Let zi represent the composition of the i-th single component refrigerant in a

single phase mixture (for example superheated vapor). Likewise let xi and yi be the

compositions of the liquid and vapor phases of a two-phase mixture. Let Psl and Psv

represent the saturated liquid and the saturated vapor mixture pressures (at the mixture

composition) respectively, and let Tsl and Tsv be the corresponding temperatures. When

the mixture is a saturated liquid at the mixture composition then xi = zi. Similarly, when

it is a saturated vapor at the mixture composition, then y i = zi.

For a given T sl, Psl can be obtained by performing a bubble point VLE calculation

(Tsl and xi = zi are known). Similarly, Psv can be obtained by performing a dew point

VLE calculation (Tsv and y i = zi are known), where Tsv is calculated from Tavg and Tsl

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using Tavg = (Tsl +Tsv)/2. T avg is the average evaporating (or condensing) temperature in

the evaporator (or condenser). Since the pressure is assumed to remain essentially

constant throughout the evaporator (or condenser), Tsl has to satisfy:

f(Tsl) = Psl (Tsl) - Psv (Tsl) = 0 (14)

that is Psl = Psv = Pvp . Solution of Equation (14) will yield the mixture pressure. The

same procedure is employed for both the evaporator and the condenser. It should be

noted that VLE calculations are performed at saturated liquid (at the mixture

composition) conditions in the evaporator even though such a point does not exist in the

evaporator (see Figure 1). The average evaporating temperature is defined as the mean of

the saturated vapor and the saturated liquid mixture temperatures (at the mixture

composition) and thus the calculations are being performed so as to be consistent with

this definition.

A quasi-Newton scheme was used to solve the above equation using Raoult’s law

to obtain initial guesses. The solution of Equation (14) yields the value of another target

property, namely, the saturated vapor temperature (Tcsv) in the condenser. It equals Tsv in

the condenser, thus:

Tcsv = Tsv (15)

Estimation of Compressor displacement

Once the condensing and evaporating pressures and the corresponding saturated

liquid and saturated vapor pressures are known, other properties such as enthalpy,

entropy and specific volume can be found using the Patel-Teja equation of state. The

theoretical compressor displacement (CD [=] m3/min.ton) per ton of capacity for a

refrigerant is calculated as (Perry, 1984):

CD = Wf Vg (16)

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where, the weight flow (Wf [=] kg/min) per ton of capacity is given by:

Wf = 200 x 104 / (9.47831 RE) (17)

Here, the refrigerating effect (RE [=] J/kg) is evaluated as:

RE = Hvs - Hlc (18)

where, Hvs is the enthalpy of vapor leaving the evaporator and Hlc is the enthalpy of

liquid leaving the condenser in the refrigeration cycle. Vg is the specific volume (m3/kg)

of the suction vapor entering the compressor.

Estimation of Ozone depletion potential

Compounds with a minimum of one chlorine, bromine or iodine group are known

to have a non-zero ozone depletion potential (ODP). However, ODP data is scarce and

limited to a few halogenated compounds. Prediction techniques are needed to estimate

the ODP of compounds for which empirical data is unavailable. For

hydrochlorofluorocarbons (HCFC’s) with one or two carbon atoms, the ODP can be

quantitatively expressed as (Nimitz, 1992):

ODP = 0.05013 nCl 1.510 exp(-3.858 / τ ) (19)

where the tropospheric lifetime τ is given by :

τ = 0.787(M / nH ) exp(-2.060nα -Cl - 4.282n2C + 1.359nβ -F + 0.926nβ -Cl ) (20)

Here nα -Cl represents the number of α chlorine atoms present. The α designation

implies that a substituent is attached to the same carbon atom as the hydrogen atom being

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considered; β means the substituent is attached to the carbon atom adjacent to the carbon

atom bonded to the hydrogen. nH represents the number of abstractable hydrogen atoms.

Finally, the subscript 2C indicates a term to be added only if two carbon atoms are

present.

The ODP values for a number of chlorofluorocarbons (CFC’s) with one or two

carbon atoms have been reported (Solomon, 1992, Solomon, 1992a, Lea, 1991, Stamm,

1989). Using these values, the following general correlations for ODP of CFC’s with one

or two carbon atoms have been formulated (Duvedi and Achenie, 1996):

ODP=0.585602nCl-0.0035 exp(M/238.563) ...one carbon (21)

ODP=0.0949956nCl-0.0404477 exp(M/83.7953) ...two carbon (22)

where M is molecular weight. ODP values have also been reported for some other types

of halogenated compounds (see Appendix). The ozone depletion potential for the

refrigerant mixture is estimated from the ODP values of the individual components using

a simple linear mixing rule.

ODP = i∑ zw(i) ODP(i) (23)

zw(i) = z(i) Mw(i) / i∑ z(i) Mw(i) (24)

where z(i) is the mole fraction, zw(i) is the weight fraction and Mw(i) is the molecular

weight of the i-th component.

4. SOLUTION OF MINLP FOR MIXTURES

The locally optimal augmented penalty outer approximation (AP/OA) algorithm

(Viswanathan et. al., 1990) is employed to solve the MINLP. This algorithm involves

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solving a finite sequence of nonlinear programming (NLP) and mixed integer linear

programming (MILP) problems. In the NLP subproblems, derivative information is

obtained by perturbation. The flowchart for the algorithm is given in Figure 2.

A relaxation of the MINLP is initially solved to provide an initial guess for the

MILP-NLP sequence. Hence, even a non-optimal solution to the relaxed problem can

serve as a reasonably good initial guess for the MILP-NLP iterations and yield a locally

optimal solution to the MINLP. This was observed while solving the case-study to be

considered in the next section, when the relaxed problem solution, though not optimal,

was used to obtain an optimal solution for the MINLP. Due to the non-convexity of the

problem, however, there is no guarantee that this solution is the global optimum.

As will be seen from the results in the next section, the locally optimal solutions

are identified within a few MILP-NLP iterations. This implies that the solution of the

relaxed problem provides a very good initial point for the MILP-NLP sequence.

5. CASE STUDY

In this case study, only binary mixtures have been considered due to the limited

availability of experimental data and the ease of vapor liquid equilibrium (VLE)

calculations. Most of the common refrigerants are included in the basis set of single

component refrigerants because these compounds are known to have properties suitable

for refrigerants. This basis set excludes compounds for which the fundamental properties

such as the critical temperature, pressure and volume, the acentric factor and the normal

boiling point are unavailable. This will make the search for the “optimal” mixtures

restrictive, but this exclusion is necessary to avoid large inaccuracies in the estimates of

the target properties which are derived from these fundamental properties. Therefore 21

single component refrigerants (see Table 1) have been considered in the basis set. These

include the most commonly used refrigerants. One particular refrigeration cycle is

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considered in which the condensing, evaporating and superheat temperatures are given by

(a) Te = 40oF, (b) Ts = 40oF and (c) Tcd = 100oF.

For this case study, we consider a particular form of Equation 5 (the objective

function) as a trade off between two objectives:

p = -Wt1 xCD - Wt2 xODP

where Wt1 and Wt2 are weights. Equal weights are assigned to the compressor

displacement and the ozone depletion potential in the objective function. To initialize the

algorithm, the relaxed problem (in which all the binary variables are assumed to be

continuous) is solved and a hypothetical mixture containing fractional values for the

binary variables is obtained as the solution. This solution has no physical significance.

CH2F2 /CH3Cl is eventually identified as the local optimum (see Viswanathan et. al.,

1990, for an explanation of the criteria for a local optimum). Table 2 lists the solutions

and their property values.

Finally, let us consider another refrigeration cycle specified by: (a) Evaporator

temperature (Te) = 30oF, (b) Superheat temperature (Ts) = 30oF and (c) Condenser

temperature (T cd) = 110oF. Equal weights are assigned to the compressor displacement

and the ozone depletion potential in the objective function. The results are listed in Table

3.

Table 4 compares the properties of the locally optimal solution obtained in Table

2 to both the estimated and experimental values of the properties of the refrigerant

mixture R22/R11 (CHClF2/CCl3F, Radermacher, 1989) at the same conditions. This

mixture has a concentration of 60% by weight (70.44% mole percent) of R22. The

experimental values are obtained from the pressure-enthalpy diagram for this refrigerant

mixture (Radermacher, 1989). It can be seen that the locally optimal refrigerant mixture

(CH2F2[34.4% mole]/CH3Cl[65.6% mole]) is better in terms of a lower compressor

displacement in addition to a lower ozone depletion potential.

Finally, the properties of the locally optimal refrigerant mixture in Table 3 are

compared to those obtained for a few available single component refrigerants under

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similar conditions (see Table 5). The experimental values of the compressor

displacement are the values that are calculated using experimental values of enthalpies

and specific volumes. The experimental objective function is the value obtained

from the experimental values of the compressor displacement and the ozone

depletion potential. It can be seen that the mixture properties are superior to those of the

single component refrigerants for the given process conditions and constraints. CH2F2 is

a better refrigerant in terms of a low compressor displacement and a zero ozone

depletion potential. However, its vapor pressure at the condensing temperature is too

high and it violates the upper bound of 14.0 bars. This “undesirable attribute” of CH2F2

is overcome in the mixture of CH3Cl and CH2F2.

6. CONCLUSIONS AND FUTURE WORK

Refrigerant mixtures offer greater flexibility in the search for efficient and

environmentally friendly refrigerants. Identifying appropriate refrigerant mixtures that

have specific attributes, such as low ODP, is expensive and time consuming, especially

since the choice of single component refrigerants and the proportion in which these will

occur in the mixture is a large combinatorial problem even if we restrict ourselves to a

finite set of values for the mole fractions.

The MINLP based approach for refrigerant mixtures, developed in this paper is

systematic and efficient. It yields refrigerant mixtures with properties better than those

of the single component refrigerants included in the basis set. Environmental aspects

have also been incorporated. Due to the non-convexity of the formulation, the global

optimum cannot be guaranteed when using the AP/OA algorithm for solving the MINLP

model. However, current advances in global optimization (see for example Floudas et.

al., 1993) can be employed to obtain globally optimal solutions to the MINLP

formulation. As the problem size increases, there will be a need to consider

decomposition and parallel branch and bound algorithms (Pekny et. al., 1992). Numerical

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difficulties associated with vapor-liquid equilibrium calculations around critical points

can be alleviated with complex domain techniques (see for example Sridhar et. al.,

1995).

Sensitivity of the analysis to uncertainties in the physical property estimation is

very important but falls outside the scope of this paper. These will need to be addressed

in the future through sensitivity analysis.

Finally, we are indebted to the reviewers who have suggested ways of making the

mathematical programming model more practical for industrial use. More specifically,

they have suggested that materials and lubricants compatibility, toxicity, flammability,

stability and miscibility of the refrigerant mixture ought to be incorporated into the

mathematical programming model. The general consensus is that pressure ratio,

discharge temperature, capacity and COP sensitivity to cycle variability also need to be

reflected in the performance criteria. Incorporation of these issues do not change the

general structure of the model. We are well aware of these issues and we have been

collaborating with a group at one of the local Universities to incorporate some of these

issues into the mathematical programming framework and also to carry out experimental

verification.

7. ACKNOWLEDGMENTS

The authors wish to acknowledge partial support for this work under NSF grant

CTS-9211691. We also wish to thank Prof. Art Gow of the University of New Haven for

helping bring the refrigerant mixture problem to the second author’s attention.

8. NOMENCLATURE

CD = compressor displacement

COP = coefficient of performance

ODP = ozone depletion potential

ε = a parameter

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Hlc = enthalpy of liquid mixture leaving the condenser

Hvs = enthalpy of vapor mixture leaving the evaporator

kaij= binary interaction parameter between single component refrigerants “i” and “j”

m = total number of single component refrigerants considered in the basis set

Mwi = molecular weight of i-th single component refrigerant

w = vector of 0-1 integer variables

Nc = number of single components that exist in the mixture

nα-Cl = number of α chlorine atoms

nβ-Cl = number of β chlorine atoms

nβ-F = number of β fluorine atoms

nCl = number of chlorine atoms

nH = number of abstractable hydrogen atoms

wj = i-th binary variable

p = performance index

Pc = critical pressure

Psl = saturated liquid mixture pressure

Psv = saturated vapor mixture pressure

Pvp = vapor pressure

Pvp(Tcd) = condenser pressure

Pvp(Te) = evaporator pressure

RE = refrigerating effect

τ = tropospheric lifetime

T = temperature

Tavg = average evaporating/condensing temperature

Tb = normal boiling point

Tc = critical temperature

Tci = critical temperature of i-th single component refrigerant

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Tcd = condensing temperature

Tcsl = saturated liquid mixture temperature in the condenser

Tcsv = saturated vapor mixture temperature in the condenser

Te = evaporating temperature

Tesl = saturated liquid mixture temperature in the evaporator

Tesv = saturated vapor mixture temperature in the evaporator

Ts = superheat temperature

Tsl = saturated liquid mixture temperature

Tsv = saturated vapor mixture temperature

UT = parameter

Vc = critical volume

Vg = specific volume of the suction vapor mixture entering the compressor

ω = acentric factor

Wt1 & Wt2 = weights in the objective function

x = vector of all continuous variables

xl = vector of lower bound on x

xu = vector of upper bound on x

x′ = vector of mole fractions

xCD = compressor displacement

xODP = ozone depletion potential

xTcsv= saturated vapor mixture temperature in the condenser

x′i = liquid mole fraction of i-th single component refrigerant in the mixture

yi = vapor mole fraction of i-th single component refrigerant in the mixture

zi = mole fraction of i-th single component refrigerant in the mixture

zwi = mass fraction of single component “i”

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21

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22

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12. APPENDIX

Reported ODP values of some halogenated compounds:

Compound ODP

CH3Br 0.561

CHBrF2 1.102

CF2ClBr 4.101

CF3Br 12.501

C2HF4Br 0.302

C3HF5Cl2 0.042

C2F4Br2 6.101

[1: Solomon et al (1992, 1992a); 2: Senecal (1992)]

24

A B C

Te

Tsl

Tsv

A: saturated liquid B: evaporator inlet (two-phase) C: evaporator exit (saturated vapor)y-axis = temperature x-axis = axial length along evaporator

FIGURE 1: Mixture Temperature Profile in the Evaporator

Relaxed NLP

integersoln ?

Obj fndecrease?

Y

Stop with optimal solution

Initial Guess

Stop with optimal solutionY

N

MILP [ Master ]

NLP subproblem

N

FIGURE 2 : Flowchart for the AP/OA algorithm for a maximization problem

25

TABLE 1 Basis set of single component refrigerants for mixture design[

(Nimitz et al., 1992, Reid et al., 1987, ASHRAE Handbook, 1993)

Units: Tc [=] oK; Pc [=] bar; Vc [=] cm3/gmol; Tb [=] oK

Ref

# Chemical Formula

Critical

Temp

Tc

Critical

Press

Pc

Critical

Vol

Vc

Boiling

point

Tb

Acentric

Factor

ω

Ozone

Dp Pot

ODP

A C Cl3 F (R 11) 471.2 44.1 247.8 296.9 0.189 1.037

B C Cl2 F2 (R 12) 385.0 41.4 216.7 245.2 0.204 1.0

C C Cl F3 (R 13) 302.0 38.7 180.4 193.2 0.198 0.907

D C Br F3 (R 13B1) 340.2 39.7 195.9 215.3 0.171 10.0

E C H Cl F2 (R 22) 369.3 49.7 165.6 232.4 0.221 0.05

F C H F3 (R 23) 299.3 48.6 132.7 191.0 0.260 0.0

G C H2 F2 (R 32) 351.6 58.3 120.8 221.5 0.271 0.0

H C Cl2 F C Cl F2 (R 113) 487.3 34.1 325.5 320.8 0.256 0.8

I C Cl F2 C Cl F2 (R 114) 418.9 32.6 293.8 276.2 0.246 1.0

J C H Cl2 C F3 (R 123) 456.8 36.7 277.8 301.0 0.281 0.02

K C H Cl F C F3 (R 124) 395.6 36.3 246.2 261.1 0.284 0.035

L C H F2 C F3 (R 125) 339.2 35.9 209.8 224.9 0.303 0.0

M C H2 F C F3 (R 134a) 374.2 40.6 198.7 247.1 0.324 0.0

N C H3 C Cl F2 (R 142b) 409.6 43.3 231.0 263.4 0.251 0.038

O C H3 C H F2 (R 152a) 386.7 45.0 181.0 248.2 0.256 0.0

P C H Cl2 F (R 21) 451.6 51.8 196.4 282.1 0.210 0.005

Q C H2 Cl2 (R 30) 510.0 63.0 169.7 313.0 0.199 0.0005

R C H3 Cl (R 40) 416.3 67.0 138.9 249.1 0.153 0.002

S C Cl F2 C F3 (R 115) 353.2 32.3 251.8 235.2 0.279 0.6

T C3 Cl2 F6 (R 216) 453.2 27.6 175.7 303.2 0.237 1.291

U C H3 C H2 F 375.3 50.2 169.0 235.5 0.215 0.0

26

TABLE 2: Iteration sequence (Maximization problem).

Wt1 = 1.0, Wt2 = 1.0. Te = 40oF, Ts = 40oF, Tcd = 100oF (Units: Pvp [=] bar; CD [=]

m3/(min)(ton))

Mixture type

Comp1/Comp2

Composition

Comp1/Comp2

Obj

Function

CD ODP

Pvp

(Te)

Pvp

(Tcd)

** ** 0.04720 0.04563 0.00157 5.86 14.0

CH2F2

+

CH3Cl

0.344/0.656 -0.05300 0.05170 0.00130 5.45 14.0

CHF2CF3

+

CH3Cl

0.486/0.514 -0.05769 0.05708 0.00061 5.46 14.0

** : Relaxed problem solution: (F)0.164(R)0.836

TABLE 3: Iteration sequence (Maximization problem).

Wt1 = 1.0, Wt2 = 1.0. Te = 30oF, Ts = 30oF, Tcd = 110oF (Units: Pvp [=] bar; CD [=]

m3/(min)(ton))

Mixture type

Comp1/Comp2

Composition

Comp1/Comp2

Obj

Function

CD ODP

Pvp

(Te)

Pvp

(Tcd)

** ** -0.0638 0.0603 0.0035 4.01 14.0

CH2F2

+

CH3Cl

0.227/0.773 -0.0733 0.0718 0.0015 3.92 14.0

CH3Cl

+

CH3CH2F

0.239/0.761 -0.0803 0.0798 0.0005 3.90 14.0

**: (F)0.3834(P)0.6165

27

TABLE 4: Comparison of properties of R32 (0.344)/R40(0.656) and R22(0.7044)/R11(0.2956).

Wt1 = 1.0, Wt2 = 1.0. Te = 40oF, Ts = 40oF, Tcd = 100oF (Units: Pvp [=] bar; CD [=] m3/(min)(ton))

Property

CH2F2(R32)/CH3Cl(R40)

0.344/0.656 (pred)

CHClF2(R22)/CCl3F(R11)

0.7044/0.2956 (pred)

CHClF2(R22)/CCl3F(R11)

0.7044/0.2956 (expt)

Comp Disp

CD 0.0517 0.1013 0.0984

Oz Dep Pot

ODP 0.0013 0.4448 -

Evap Press

Pvp(Te) 5.45 2.56 2.70

Cond Press

Pvp(Tcd) 14.0 7.42 7.45

TABLE 5: Refrigerant mixtures vs Single component refrigerants.

Wt1 = 1.0, Wt2 = 1.0. Te = 30oF, Ts = 30oF, Tcd = 110oF, Pvp(Te) ≥ 1.4 bar, Pvp(Tcd) ≤ 14.0 bar.

(Units: CD [=] m3/(min)(ton), Pvp [=] bar)

Ref

CD

pred

CD

expt

ODP

pred

ODP

expt

Ob Fn

pred

Ob Fn

expt

Pvp

evap

pred

Pvp

evap

expt

Pvp

cond

pred

Pvp

cond

expt

Mix ** 0.0718 - 0.0015 - -.0733 - 3.92 - 14.0 -

CH3Cl 0.1078 0.1077 0.002 0.002 -.1098 -.1097 2.47 2.46 9.36 9.27

CH2F2 0.0438 0.0423 0.0 0.0 -.0438 -.0423 7.85 7.84 26.73* 26.86*

CH3CHF2 0.1187 0.1130 0.0 0.0 -.1187 -.1130 2.61 2.57 9.94 9.96

* Upper bound on the condenser pressure is violated. ** CH2F2(0.227)/CH3Cl(0.773)