= heat transfer coefficient, Btu/hr ft2, O F K = thermal conductivity, Btu/hr ft, O F k = mass transfer coefficient, lb H2O/hr ft2mm Hg
h
= Stefan-Boltamann constant, 1.73 X l o W 9Btu/hr ft2(O R) 8 = time, hr w = 2 ~ / 2 4hr-* u
m = ho/hKo, mm Hg/OF p* = saturation vapor pressure of brine, mm Hg p , = partial pressure of water in air, mm Hg q = abbreviation defined for Equation 15 Q = intensity of direct radiation on horizontal surface, Btu/hr ft2 Q’ = intensity of indirect (“scattered”) radiation, Btu/hr ft2 Qr = re-radiation from the pond, Btu/hr ft2 r = ( k 2 / h ~d)u / 2 p ~ dimensionless , R = linearization constant for Qr, Equation 5, Btu/hr ft2 s = abbreviation defined for Equation 15 S = linearization constant for Qr, Equation 5, Btu/hr ft2,O F T = temperature, O F V = wind speed, miles/hr W = rate of evaporation, lb HnO/hr ft2 z = vertical coordinate measured below soil surface, ft
SUBSCRIPTS a = air G = gas film (brine-air) L = liquid film (brine-soil) n = summing index s = soil surface 1 = brine 2 = ground ’ = indirect radiation over bar = integrated average for 24 hr literature Cited
Cawlaw, H. S., Jaeger, J. C., “Conduction of Heat in Solids,” 2nd ed., pp 62-68, Oxford Univ. Press, 1949. Ferguson, J., Auslr. J. 9ci. Res., 5, 315-30 (1952). Pancharatnam, S., Ind. Eng. Chem. Process Des. Develop., 11, 287 (1972).
Penman, H . L., Proc. Roy. BOC.(London),A193, 120-45 (1948).
GREEKLETTERS LY = fractional absorption of radiation bo, p1 = linearization constants for p*, Equation 6 e = spectral average emissivity of brine = heat of vaporization of water from brine, Btu/lb p = thermal diffusivity, (KIpC,), ft2/hr Y = a wave number defined for Equation 12, ft-’ p = density, lb/ft*
SUBRAMANIAN PANCHARATNAM Department of Chemical Engineering Stanford University Stanford, Calif. 94906 RECEIVED for review February 7, 1972 ACCEPTEDMay 19, 1972
Dynamic Models for pH and Other Fast Equilibrium Systems A general principle is presented to be used to formulate dynamic models for systems in which fast equilibrium reactions occur.
I n a recent article (McAvoy et al., 1972), the derivation of dynamic equations for pH-controlled stirred tank reactors (CSTR’s) was examined. The use of electroneutrality and water equilibrium in deriving the equations was discussed in detail. However, one additional principle was used but not specifically pointed out. This principle concerns what to do about making dynamic mass balances on species involved in fast equilibrium reactions. To illustrate the principle and the problem which generates it, it is convenient first to consider a trivial case. Consider a stirred vessel in which the B occur. The forward and simple equilibrium reactions A reverse reactions are assumed to be instantaneous. The feed to the tank is lOOyo A which changes to lOOyo B a t t = 0. The overall mass in the system remains constant with time. Two formulations of the governing equations can be given. Correct Formulation.
Incorrect Formulation : Balance on A
It
V -dCA at
+ Cg = constant
Equilibrium
CB 630
K ~ ~ C A
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
(1)
=
- FCA
- RI + R,
or
Balance on B 0
/I
dCB V - at
Overall balance
CA
0
or
=
FCej
- FCB + R , - R,
(4)
I n Equations 3 and 4 the difference between the reverse and forward rates of reaction have been canceled because of the assumption of instantaneous equilibrium. The resulting Equations 3a and 4a are both incorrect. Equations 1 and 2 correctly predict that the CSTR has no dynamics while Equations 3a and 4a predict an exponential decay. I n physical systems it is impossible to achieve instantaneous equilibrium. If the actual rates of reaction, R f and R , were known and used, then there would be a short period of time where R R , was nonzero. This in turn would couple Equations 3 and 4 so that they would be correct. Suppose, however, that for a system under study, the assumption of instantaneous equilibrium is an excellent assumption, and that the exact R f and R, are unknown. All that is known is that R and R , are very fast. The question then arises as to how correct mass balances can be formulated assuming instantaneous equilibrium. The answer can be stated in the following principle. First, mass balances are written down for all components which are not involved in any equilibrium reactions. Next, all equilibrium equations are written down. N o attempt is made to balance any component in equilibrium. If more equations are required, then they can be obtained by balancing groupings of atoms for which there is no net loss or gain via reactions. These last equations will be of the type: Accumulation = flow in flow out. A good choice for these last equations would be elemental balances. I n the previous study (McAvoy et al., 1972), the balance on the variable 6, which was the sum of acetate ion and acetic acid concentration, was of this last category. Effectively the 6 balance was an elemental carbon balance. To illustrate the above principle, two examples will be considered. Example 1, NaOH Neutralizing HzC03 in a CSTR. The different chemical species whose concentrations are to be found are: Na+, OH-, H+, H2C03, HC03-, and C032-. The equilibrium reactions are
+ H+ HC03- FI COS2- + H S €I20 8 H f + OH-
HZCO, iS HCO3-
The following approach will yield a well-defined set of equations. Balance NB+, write down three equilibrium equations and an electroneutrality equation. Last, balance carbon
by balancing the sum of the concentrations of H2COa,HCOsand c03'-, to give six equations in six unknowns. Example 2, a Tubular Reactor in Which the Following Reactions Occur :
2 NzOs + 2 NzO4
+ 02
Nz04 F! 2 NO2 The concentration of the following species must be found as functions of distance and time: NzOs, Nz04, Oz, and NO*. Direct balances can be made on NzOs and 02. (The balance would consider only 02 and not the oxygen in the other species. It would not be an elemental balance in the sense previously discussed.) To finish the formulation, an equilibrium equation and a balance on nitrogen can be made. A direct balance on Nz04 should not be made. Dynamic energy balances on fast equilibrium systems would have to be achieved by the fundamental approach of writing internal energy as a function of such items as composition and teniperature. It would not be possible a priori to write down an energy equation with rate of reaction times heat of reaction terms. Nomenclature = concentration of A , mol/l. Cg = concentration of B , mol/l. F = flow K,, = equilibrium constant R f = forward reaction rate R, = reverse reaction rate t = time V = volumeof CSTR
CA
SUBSCRIPT i
=
inlet
literature Cited
McAvoy, T. J., Hsu, E., Lowenthal, S., Ind. Eng. Chem. Process Des. Develop., 11 (I), 68-70 (1972).
Department of Chemical Engineering University of Massachusetts Amherst, Mass.
THOMAS J. h4CAVOY
RECEIVED for review March 20, 1972 ACCEPTED May 8, 1972
CORRESPONDENCE Parameters in Wilson Equation SIR:I would like to add a comment to a recent paper by Schreiber and Eckert (1971), in which they deal with the use of infinite-dilution activity coefficients with a single-parameter Wilson equation. Wilson's parameters are given by
v, A j + = - exp [-(A,, Vt
- A,,)/RTl
(1)
However, it is possible to assume as
4, = exp
[-(Xi,
- A,d/RTl
(2)
If we assume that like-pair potentials, Ai,, are estimated from the energy of vaporization as A t t = - ( 2 / z ) u f , and A21 = X,Z = A, a ratio of the two parameters is given by AZI Vz2exp (2 u Z / z R T ) AH VI2exp (2 u l / z R T )
(3)
from Equation 1, where 2/2 is a term which accounts for the fraction of ut which contributes to a pair potential. According to the Flory-Huggins equation, it is expected to obtain
provided that A t t is related to the interaction energy per molecule, not per molecular segment.
(4) Ind. Eng. Chem. Process Der. Develop., Vol. 1 1, No. 4, 1972
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