Computer-based refrigerant thermodynamic properties. Part 1: Basic equations

C. Y. Chan and G. G. Haselden

Propri6t6s thermodynamiques des frigorig nes calcul6es par ordinateur, l e partie. Equations fondamentales On d#crit un ensemble de m#thodes uti/isant /'ordinateur pour calculer la densitY, la pression de

vapeur, /'entha/pie, /'#nergie interne et /'entropie des frigorig#nes, tout en effectuant des ca/cu/s simples du cycle frigorifique. La premiere partie pr#sente les #quations fondamentales et /es ddrivations pour le R 11, le R 12, le R 13,/e R 13B1,/e R 14, le R 22, /e R 113,/e R 114 et /e R 502 uti/is#s par/ ' / IF pour ses derniers diagrammes. On donne aussi /es #quations pour/'ammoniac d'apr~s des pub//cations de Dvorek et Petrak.

A set of computer-based methods for calculating densities, vapour pressures, enthalpies, internal energies and entropies of refrigerants, together with simple refrigeration cycle calculations are to be described. Part 1 presents the basic equations

derivations for refrigerants R 11, R 12, R 13, R 13B1, R 14, R 22, R 113, R 114 and R 502 as used by the IIR in its latest charts. Ammonia equations are also given based on Dvorek and Petrak's publications.

There is a need, both for designers and plant users, for a computer data-base covering the thermody- namic properties of common refrigerants. These data need to be of high accuracy, consistent with reasonable access time, so that the potential of computers for increasing design precision can be exploited. It is also desirable that the same data are used by suppliers and customers, so that performance can be checked without arguments about the basis of assessment.

For most common halocarbon refrigerants the IIR has recently published new charts and tables, in SI units, having the best precision available from current experimental measurements. The compilation of this material was undertaken by a working party under the leadership of Dr Andre Gac. The equations and constants used by the Working Party have been made available to the Authors, therefore the programs to be presented are consistent with the II R data.

For ammonia, which is not yet covered by the new IIR charts, the equations presented by Dvorak and Petrak 1, based on the earlier work of Rombusch and Giessen, are used. The form of the equations they used was different, so both sets will be presented in full.

The authors are from the Department of Chemical Engineering, University of Leeds, Leeds, UK. Paper received August 1980.

Ideally all thermodynamic properties can be generated from a single equation of state, covering the vapour and liquid phases, and a correlation of zero-pressure specific heats. In practice it is more accurate, and computationally faster, to use an equation of state only for the vapour region supplemented by vapour pressure and liquid density equations to embrace the liquid phase. This approach will be used.

Consideration was given to fitting polynomials to calculated sets of values of derived properties such as density, enthalpy and entropy. Whilst the speed of access to data is appreciably faster by this route, it necessarily involves a loss of accuracy which will vary rather unpredictably over the phase regions. Since with modern computers, even of relatively modest size, the property computations to be presented generally involve a run time of less than a second, this approach was not pursued. Instead, effort was concentrated on making the solution of the basic equations efficient and robust.

Halocarbons The following halocarbon refrigerants will be covered: R 11, R 12, R 13, R 13B1, R 14, R 22, R 113, R 114 and R 502. The equations used are in British units: pressure, psia; volume, ft 3 Ib -1 temperature, °R; liquid density, Ib ft-3; C °, Btu Ib -1 o R --1.

01 40-7007/81/01 0007-0652.00 Volume 4 Num#ro 1 Janvier 1981 @1981 IPC Business Press Ltd and IIR 7

However, the subsequent programming to be presented will yield results in Sl units.

Equation of State:

RT p - V - b

A (2) + B(2) T+ C(2) [#exp( - kT/T~) + v/T 3] + ( V - b ) 2

A (3) + B(3) T+ C(3)exp( - kT/T~) + ( V - b ) 3

A (4) + B(4) T+ C(4) [/~exp ( - kT/To) + v/T 3] + ( V - b ) 4

A (5) + B(5) T+ C(5)exp( - kT/Tc) + ( V - b ) 5

-I A (6) + B(6) T+ C(6)exp( - kT/Tc) (1) exp(aV) [1 + C'exp(aV)]

Saturated vapour pressure:

LogloP=AVP(1 ) +AVP(2)/T+AVP(3) LOgl0T

+A VP(4) T+ [/4 VP(5) (A VP(6) - T)

AVP(7) Log~o(IAVP(6)-TI) 4 T2

Liquid density:

+ AVP(8) T 2

(2)

PL = RL (1) + RL (2)(1 - T/Tc) 1/3

+ RL (3) (1 - T/Tc) 2/3

+ RL (4)(1 - T/Tc) + RL (5)(1 - T/To) 4/3

+ RL(6) (1 -T/Tc)I"2+RL(7)(1 -T/Tc) 2 (3)

Specific heat capacity of vapour at zero pressure:

C°=ACV(1 ) + ACV(2 ) T + ACV(3) T 2 + ACV( 4 ) T 3

+ACV(5)/T 2 (4)

The constants used in these equations will be presented in the program listings.

Solution of the equation of state for halocarbons

Since the equation of state is explicit in pressure it has to be solved iteratively if the volume is unknown. Thus the specific volume values required in enthaipy and entropy calculation can only be obtained by numerical methods. Two approaches

were considered, one is the binary search and the other is the Newton-Raphson method. Both methods iterate in pressure as shown in the flowchart in Fig. 1,

In the Newton-Raphson method, the new estimate is given by

/dP VEST = VEST-}- ( P - i f ) /

dV (new)(old) /

where the pressure differential term

dP RT dV ( V - b ) 2

_ 2[A (2) + B(2)T + C(2){;~exp (-kT/To)+ ~/T3}] L ( V - b ) 3 J

31-A (3) + B(3) T+ C(3)exp( - kT/Tc)]

L (¢-%Y J _ 4[A (4)+B(4)T+C(4){#exp(-kT/Tc)~5. + v/T3}]

_ 5 [ A ( 5 ) +B(5)T+C(5)exp(-kT/T~)](V_b) 6

- a [ I(A (6) + B(6) T+ C(6) exp( - kT/To))

L exp(aV){1 + C'exp(aV)} 2

{1 + 2C'exp(aV)} ] exp(aV){1 + C'exp(aV)} 2 (5)

In the binary search method, the new estimate is obtained by adding a positive or negative increment, depending on the approach to convergence, to the old value. When the desired value is within the range of the increments, the increment is harved successively until the tolerance is met.

Both iterative methods have been tested for speed of convergence. Data for R 22 were used under saturation, superheated and critical conditions. With a maximum of 50 iterations allowed, the Newton- Raphson method does not converge at temperatures within about 30°C of the critical value. This is because the differential term (dP/dV) reaches a magnitude of the order of 105 and the correction term is then too small to allow convergence in a reasonable time. At temperatures below -50°C, both methods have comparable speed. At temperatures between -50°C to 20°C, the Newton- Raphson method is faster. The difference in processing time is generally less than 0.1 s. As the temperature increases, it requires a greater number of iterations and its performance deteriorates.

The binary search method is stable and has proved robust under all conditions, This method was adopted therefore for all regions, though both numerical subroutines were programmed.

8 International Journal of Refrigeration

From the four basic thermodynamic equations, and Maxwel rs relationships of a one component system, the differential enthalpy can be expressed as:

dH = C°d T+ d ( PV) - [ P - T(aP/ST)v] d V (6)

w

P ISpecified P and TI

I Fram Ideal Gas EquationJ I st estimate Ves t = RT/P I

--IK=K+i|

Calculate P' from Equation of State (I)

Derivation of enthalpy

Improve the I/=tby either -Binary Search method -Newton-Raphson method

J lndicate iteration exceeded limitJ

j -

Fig. 1 Flowchart of numerical solution of equation of state

Fig. 1 Diagramme d'emploi de la r~solution num~rique de I'#quation d'#tat

For a given reference temperature and pressure (Tref, Pref), the enthalpy at T and P,

T P,V

H(T,P)= fC°dT+ f d(PV) Tref Pref. Vref

V

- ~ [ P - T(aP/aT)v]d

Vref

V+ H(Tref,Pref) (7)

Each of these integral terms is treated separately.

The specific heat integral term using (4) gives

T

f C°dT=ACV(1 ) T+ 2ACV(2) T 2

Tref

+ 3ACV(3)T3+ 1ACV(4)T 4

-ACV(5)/T-CH1 (8)

CH1 iS a constant for the same expression with Tref replacing T. To evaluate the two pressure-volume integral terms, the partial differential term (c3P/aT)v has to be evaluated first.

From (1)

R

v = V - b

B(2) - C(2){#k/Tcexp( - kT/T~) + 3v/T 4} 4 ( V - b ) 2

B(3) - C(3)k/Tcexp ( - kT/Tc) ( V - b ) 3

B(4) - C(4){#k/Tcexp( - kT/Tc) + 3v/T 4} ( V - b ) 4

B ( 5 ) - C ( 5 ) k / T c e x p ( - kT/Tc) -+ ( V - b ) ~

+B (6 ) - C(6)k/Tcexp( - kT/Tc) exp(aV){1 + C'exp(aV)}

Substitut ing (9) and (1) into [P-T(aP/aT)v ] and integrating the two terms,

P,V V

Pref, Vref Vref

A(2) + C(2){#(1 +kT/Tc)exp(-kT/Tc) +4v /T 3}

(9)

( V - b )

A(3) +C(3 ) (1 +kT/Tc)exp(-kT/Tc) 4 2 ( V - b ) 2

Volume 4 Number 1 January 1981 9

A (4) + C(4){/~(1 + kT/To)exp( - kT,/T~) + 4v/T 3} Jr- . . . . . . . . .

3 ( V - b ) s

A(5) + C(5) (1 +kT/To)exp(-kT/Tc) + . . . . . . 4-(V--- b) q . . . . .

_ [ ,4 (6 ) +C(6) (1 +kT/To)eXP(a -kT/Tc)l x

[ (C' /n {C" +exp(-aV)}-exp(-aV) ]-CH2 (lO)

CH2 is the same expression described by P,ef, Vref and Tre f. The enthalpy expression is then obtained from the combination of (8) and (10).

Der iva t ion of en t ropy

Using the same basic thermodynamic equations, the differential entropy is given by

0 aP

The expression for entropy, at given specified reference conditions, is

(11)

T and P, with

T V aP

S{7.P)= f C~dT + f t~tvdV+S(Tr~f, Pr~f) Tref Vref

(12)

Integrating the specific heat integral term,

T

fE~dT=ACV(1 +IAcv(3)T2 ) I n T+ACV(2)T Tref

1 +~ACV(4) T 3 -ACV(5)/2 T 2 - Cs, ,~ - - (13)

the pressure-volume integral term becomes,

V

dV:R ,n, v - b,-- Vref

B(2) - C(2){I~k/Tc exp( - kT/To) + 3v/T 4} ( V - b )

B(3) -C(3)k/Tcexp( - kT/Tc) 2 ( V - b ) 2

B (4) - C(4){/~k/Toexp ( - kT/Tc) + 3v/T 4} 3 ( V - b) 3

B(5) - C(5)k/Toexp( - kT/Tc) 4 ( V - b ) 4

+ IB (6) -C(6)k /Tcexp(-kT/Tc)~a

• i IC' tn {C'+exp(--aV)}--exp(-aV) C,: (14~

Csl and Cs2 are constants for a given Tre, and P,,f.

The entropy expression is obtained by substituting (1 3) and (14 ) in to (12).

Der iva t ion of internal energy

The differential internal energy expression is

T aP (15)

It is similar to that of enthalpy wi thout the d(PV) term. Hence,

T V

= P T aP

Tref Vref (1 6 )

The

7- aP

term can be obtained from (10) by deleting the PV term and replacing the C.2 by CHs. The specific heat integral term is solved in (8).

Choice of re ference condi t ions

The values of the enthalpy and entropy calculated from (7) and (12) respectively depend on their datum values, H(Tr~f, Pref) and S(T,of, Pref)- A fixed reference enthalpy and entropy are chosen so that all calculated enthalpies and entropies are relative. It is appreciated that values of internal energy fo l low directly from the reference value of enthalpy.

There are two different reference condit ions used in published tables and charts. ASHRAE defines as zero both the enthalpy and entropy of saturated liquid at -40°C. The IIR used 0°C as the reference temperature, and stipulated that saturated liquid enthalpy and entropy equal 200 kJ kg -1 and 1 kJ kg -~ K -1, respectively. The latter reference does not have zero datum points for both enthalpy and entropy. The zero entropy is determined to be at approximately lOOK, whi le zero enthalpy occurs at a lower temperature.

In view of this, a more flexible calculation program is required to embrace the available options. The reference condit ions are chosen such that cross- reference can be made to charts or tables.

A common point of these two reference condit ions is that they both refer to saturated liquid. The

10 Revue Internationale du Froid

reference state is finalised to be the saturated liquid state at - 4 0 ° C (provision for adopting a different reference temperature is included). Equations (7) and (1 2) can only go down. to the saturated vapour condition. An additional term is needed to allow for the latent heat of phase change. For enthalpy, the latent heat of vaporization H ET ,should be

• ~ reff subtracted from the constant C,~ in (8). The change in entropy in phase change, HL(Tref)/Tref s h o u l d be subtracted from the constant Csl in (1 3).

Latent heat of vaporisation

From Ctapeyron-Clausius equation for a single component system,

~t- T(V- VL) (1 7)

The saturated vapour specific volume, V, is obtained by the numerical method in Fig. 1. The saturated liquid specific volume, V L is determined by (3). The differential term, (dP/dT) has to be derived from the vapour pressure equation (2).

() dP =2.3026 + t-AVP(4) dT s~t T 2 2.3026T

AVP(5) ,~ -2 .3026Tt~ +AVP(6) In (IAVP(6)-7-1)}

2AVP(7) ] T3 {- 2AVP(8)T (1 8)

Equations for ammonia

The equations published by Rombusch and Giessen, and used by Dvorek and Petrak for ammonia, are:

Saturated vapour pressure

( , t -1 in P=AVP(1) +AVP(2) ~

(19)

where P is measures in bars and T in degrees Kelvin.

Liquid density

3

P' = Pc + ~ RL (i) ( Tc - T) j/3 (20) j = l

where p' is measured in kg m -3. Pc is the critical density of vapour, kg m -3, and T cis the critical temperature, K.

Constant pressure specific heat at zero pressure

C°= ~ ACP(I) j = l

where C O is measured in kJ kg -~ K -1

Equation of state (reduced parameters)

(21)

The equation is different from the one given in the paper, and it is the correct form.

I A (2 ) (A (3 ) _~)~2] ~ = 1 - ( 1 - 5 ~) 1+A(1)5-1 ] ~ 7 _ ~ 7 J

6 +(O-1)~B(/ ' )~s+ (O-1) ;~C( i )SJ (22)

0 j = l / = 2

where ~ is P/Pc being reduced pressure, & is Vo/V or P/Pc being reduced density and 19 is T/To being reduced temperature.

Latent heat of vaporisation

L = 100PT(V- VL) {ALH(1 ) T 5

+ ALH(2) T -1 + ALH(3) T -2} (23)

where L is measured in kJ kg -1, V is vapour specific volume, m 3 kg -1 and V L is liquid specific volume• m 3 kg -1.

The derivation of enthalpy uses (7), the differential term being given by:

OP\ PoF6 ( t 1 4 •

The corresponding integral terms are:

T

Ic e Tref

dT= (ACP(1) - R) T + 100ACP(2)

T 6 IOOACP(i) T -u--2) CH1

(25) P.V V

f d ( P V ) - f [ P-T(OP'~]dV\OT,Iv]

Pref.Vref Vref

=PV+PoVo[(g-A(1 ))In5 + 5(A(1 ) - 2 ) 5

+ 5(1-A(1))~2 +5(2A(1)-1)a3+ ( 1 - 5 A ( 1 ) ) a 4

1 ) ( A ( 3 ) - 1 ) / ' 1 2 ) + 5 A(1)~6+A(1 128 t~-m + l n m - 2 m

Volume 4 Num~ro 1 Janvier 1981 11

A(2)//1 5' 1 3 , 1 ~ A(2) 1 - - 4 = - k ~ u t l ~ U t l ~ U ) + ~2~tan (2u)

- B ( 1 ) l n S - ~ ~,~:-.,~) + 2 ( 1 - 1) /=2

C~I) ~ J - C H 2 /=2

where u = l - & and m = l +4uL

Entropies are calculated from (1 2). where: T

fCTdT=(ACP(1)-R)InT Tref

__ ~ A C P ( j ) ( T ~ (/-1, /:2 ~ I I 1 ) \100J -- CS'

and

V

f(~T)v dV= Vref

(26)

(27)

_ c B(1 )In&+ ~,Bo)0.Z~ ) /=2

+(1 1 ~A ~_1 I (28)

CH1, CH2, Cs1 and Cs2 are constants defined by the reference temperature and pressure.

The numerical constants for the ammonia equations will also be included in the listings.

C o m p u t e r l i s t ings

Part 2 of this paper (to be included in the March issue) will give the listings of all the above equations and their solutions using standard Fortran IV coding. Part 3 (May issue) will illustrate the use of data in the computation of standard refrigeration cycles.

This work was sponsored by HalI-Thermotank Ltd and the Authors appreciate the willingness of the Company to allow publication.

R e f e r e n c e

1 Dvorak, Petrak K/ima-Kalte-/ngenieur 10 (1975) 319

12 International Journal of Refrigeration