N e w Thermodynamic Equation for Testing Consistency of liquid-Vapor Phase Equilibria Data The following new thermodynamic equation i s derived
where u ii s the mole fraction of the ith component in the charge fed to an equilibrium cell, VTis the total volume of the cell, n is the total number of moles in the cell, and f c is the fugacity of the ith component, with liquid and vapor in equilibrium. This equation does not contain any of the thermodynamic properties of the liquid phase. It can b e used to test an assumed equation of state for the ability to calculate consistent fugacities for the vapor phase. For a two-component system, the derivatives, (d In ft/dP)T, along the coexistence curve can b e determined from phase equilibrium data and used to calculate the molal volumes of the liquid and vapor phases.
IN
MAKING thermodynamic consistency tests on liquidvapor phase equilibria data, it has always been necessary to make some assumptions, as experimental values for all the thermodynamic quantities entering into the equation have not been available. Assumptions are usually made about the properties of the equilibrium vapor phase. These can take the form of assuming the Lewis and Randall rule or assuming that some equation of state, such as the Redlich-Kwong or the B-W-R equation, can be used to calculate the necessary properties of the vapor. I n addition, in the usual consistency test, it is necessary to know the molal volume of the liquid phase. I have discovered an equation that does not require any information at all about the liquid phase, except the certainty that it exists. We start with the general Gibbs-Duhem equation relating changes in pressure, temperature, 8nd fugacity:
(z,d 1nfJ
=
1
V L dP - __ dT RT RT2 -
(1)
where z c is the mole fraction of the ith component, f i is the fugacity of the ith component, V is the molal volume of the phase, and L = H - H o is the molal relative enthalpy (Van Ness, 1964). (Instead of accepting Lewis and Randall's nomenclature for the relative enthalpy, Van Kess changes the sign and calls AH = H o - H the enthalpy deviation.) Equation 1 is true for any change of a phase, whether or not it remains in two-phase equilibrium during the change. We may therefore write an equation similar t o Equation 1 for each phase, with the understanding that during the change the phases remain in equilibrium. We then have
C y t d In f c = z
VG LO E dP - -dT RT2
+ nLzdd lnft
C(nGyt i
=
+
+
(nGVG nLVL)d p - (n'La nLLL) RT RT2
where nt is the total number of moles of the ith component in the container. Also
n"VG
+ nLVL = VT
(6)
where V T is the total inside volume of the container, and
+
nGLG nLLL = LT = H r - HT'
(7)
where L T is the total relative enthalpy of the contents of the container. Substituting Equations 5, 6, and 7 into Equation 4 we have
Enid lnfc i
=
VT LT RT - dP - R T2 dT
(8)
Dividing Equation 8 by n, the total number of moles in the container, we have (9)
where n f / n = ut represents the mole fraction of the ith component in the feed composition. Now consider a change restricted to constant temperature. We then have
and (3) Equation 2 applies to the vapor phase and Equation 3 applies to the liquid phase. Consider that these two phases are in equilibrium in a closed container. Let no be the total number of moles in the vapor phase and nLbe the total number of moles in the liquid phase. Riultiplying Equation 2 by nGand Equation 3 by n L and adding the resulting equations, we have 298
Ind. Eng. Chem. Fundam., Vol.
9,No. 2, 1970
Equation 10 is a new thermodynamic equation that does not involve any of the thermodynamic properties of the liquid phase. It could be used in the following manner: n moles of a feed composition ut would be added to a container of total volume V T , at temperature T , where two phases exist. At equilibrium, 5, y, and P values would be determined. The experiment would be repeated a number of times, varying the number of moles, n , or the feed composition, ut.Using the experimental y, P, and T values, fugacities would be calculated from an assumed equation of state or possibly obtained by
some other means. One would then develop In fl as a function of pressure. The derivatives (d In f l / d P ) ~would then be evaluated a t the experimental conditions where ut and VT/n values would be known. Substitution of the various quantities into Equation 10 would then be a test of the ability of the assumed equation of state to generate fugacities consistent with the vapor, two-phase equilibria data. For a two-component system Equation 10 could also be used in the following manner: The equilibrium conditions 2, y, and P could be developed as a function of n, with fixed composition. The feed composition would then be changed and again equilibrium conditions determined as a function of n. Experimental conditions with the same pressure, in these two series of experiments, would have the same equilibrium conditions. At the same pressure they would therefore have the same values of (d In f i / d P ) ~ We . would then have two equations like Equation 10 with different n and ut values, but with the same values of (d In f,/dP) T. The two values of (d In f i / d P ) T would then be determined by the two equations. These values of the derivatives could then be substituted, along with x and y, into the isothermal form of Equations 2 and 3 and the molal volume of the liquid and vapor calculated. With the derivatives (d In f i / d P ) ~known as a function of pressure, In fi values could be calculated by integration
except for a constant of integration. Such In f l values would of course be consistent with Equation 10. If the (d In fi/dP)* values were developed from an assumed equation of state and found to be consistent with the data entering into Equation 10, Equations 2 and 3 could be used isothermally and isobarically to generate molal volumes and molal relative enthalpies for each phase. With molal relative enthalpies and ideal enthalpies, total enthalpies can be calculated and constant pressure, H - 2: (Ponchon) diagrams constructed. The accuracy with which this can be done will, of course, depend upon the accuracy of the data available. The quantities uiand VT/n have not customarily been determined in conjunction with phase equilibria data. It is hoped this paper will encourage their determination. literature Cited-
Van Ness, H. C., “Classical Thermodynamics of Nonelectrolyte Solution,” p. 36, Eq. 2-75, Pergamon, New York, 1964. ROBERT E. BARIEAU
Helium Research Center Bureau of Mines Amarillo, Tex. 79106
RECEIVED for review July 16, 1969 ACCEPTED February 4, 1970
CORRESPONDENCE Oscillatory Operation of Jacketed Tubular Reactors SIR:Recently, Chang and Bankoff (1968),in referring to one of our papers (Bailey et al., 1968) gave an incorrect summary of the concept of relaxed steady-state control. Also, statements regarding slow cycling appearing in Chang and Bankoff’s paper (1968) and the work of Laurence and Vasudevan (1968) are somewhat misleading. We feel that the relaxed steady-state approach is an extremely important tool for the analysis of periodic processes, while the investigation of slow oscillations may be facilitaced by employing the attainable set concept in conjunction with calculation of system steady states. The latter technique may be summarized as follows: The economics of a cyclically operated system-e.g., a chemical reactor-often depend directly on the values of several objective variables gi; i = 1,2,. . ., m-e.g., average compositions in the reactor outlet stream-which may be computed by time-averaging functions of the state vector x and control vector u over one period I.
The values of the objective variables for a given periodic process may be represented by a point 7 = (gl, g2,. . . ,g”) in m-dimensional Euclidean space. The union of all such points corresponding to possible steady-state operations where u is restricted to some set U is called the steady-state attainable set for the control set U and will be denoted by S ( U ) , Now suppose that the system is operated in a periodic
state where the period T is much larger than the characteristic time for the system to reach steady state. In many such cases the process dynamics can be described by applying the “steady-state approximationJ’-that is, if for a time-invariant control usseU the corresponding steady state x,, is given by
xss = g(u,s)
(2)
then, for very slow changes in the control, x(t) and u(t) are related by
x(t) = g [ W I
(3)
For a general class of problems it can be shown that when the steady-state approximation applies, cycling can lead only to objective vectors 7, which are elements of the closed convex hull of S ( U*)-Le., y€cl[co S ( U * ) ]
(4)
where U* denotes the range of the control for the operation in question. An outline of the proof of Relationship 4 is as follows: Suppose that for a given control u(t) the period I can be divided into a finite number of subintervals T ~ such , , ] ; i = 1,2,. . . , m, are that all of the functions y * { g [ u ( t ) ]u(t) continuous in t over each subinterval. This will be so for most models of physical systems. Then it is possible to rewrite the integrals in Equation 1 as a sum of several (Riemann) integrals, each of which exists. Each integral may be approximated arbitrarily closely by a Riemann sum. By employing the same partitions of the subintervals I, for all Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
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