fl = order of reaction with respect to oxygen A = nonintrinsic parameter
Literature Cited
Rus'yanova, N. D., Kostromin, A. S., et a/., Chem. Absfr., 64, 19354e (1966). Sampson, R . J., Shooter, D., Oxid. Combust. Rev., 1, 225 (1965). Subramanian, P.,Murthy, M . S., Ind. Eng. Chem., Process Des. Develop. 11, 242 (1972). Subramanian P., Murthy, M. S., Chem. Eng. Sci.. in press, 1974.
Andraikov. E. I., Rus'yanova. N. D., Koks, Smola, Gaz, 12 ( q l ) , ' 289 (1967); Chem. Abstr. 6 9 , 6 6 3 8 ~(1968). Mezaki, R., Kittrell, J. R., Can. J. Chem. Eng., 44, 285 (1966).
Received f o r review March 5, 1973 Accepted November 6, 1973
Gas Absorption with Heat Effects. 1. A New Computational Method John R. Bourne,* Urs von Stockar, Technisch-chemisches Laboratorium ETH, CH-8006 Zurich, Switzerland
and George C. Coggan Gamma Associates Ltd., Nottingham, Great Britain
A dynamic simulation method has been applied to determining the state of a stagewise gas absorption column. The unsteady state heat and mass balances for each real stage have been integrated up to the steady state, taking into consideration all possible heat effects for physical gas absorption. The new method resulted in stable convergence of the two point boundary value problem and was superior to the method employed earlier. After a description of this improved method, the earlier studies of the influences of typical operating and system variables upon the capacity of an absorber have been extended. The transport capacity of the gas phase for heat, solvent evaporation, the L/G ratio, the solvent feed temperature, and the feed gas humidity were investigated. The limitation of the absorber capacity through the formation of a temperature plateau has been confirmed and extended to the case of a column, where both absorption and desorption occur. The proper description of these phenomena has proved possible only by correctly formulating the heat and mass transport phenomena and using dynamic simulation for their solution.
Introduction Because of the generally strong dependence of the solubility of the solute gas on temperature, the temperature distribution in an absorption tower should be considered when determining its capacity. The heat effects, influencing this temperature profile, are (1) heat of absorption of the solute, leading to a rise in the temperature of the liquid phase; (2) partial evaporation of the solvent, tending to lower the liquid temperature, provided that the solvent is volatile, e.g., water; under certain conditions, the reverse processes (condensation and heating) occur; (3) transfer of sensible heat by direct contact between gas and liquid phases; (4) transfer of sensible heat between the fluids in the column and cooling coils and/or the column wall. Temperature in turn influences the degrees of heat and mass transfer thermodynamically, through the phase equilibrium relationships, e.g., the solubilities, and kinetically, through variations of the transfer coefficients, e.g., plate efficiencies in the case of stagewise gas-liquid contact. The overall effect of temperature variations on a gas absorption process can be very significant. Typical conditions for substantial heat effects are the following. (a) Normal and high values for the enthalpy changes, when solute and solvent are transferred between phases. (b) The equilibrium partial pressures of solute and solvent are highly dependent upon the temperature. Exceptions to this generalization are, however, systems where a fast, quantitative chemical reaction occurs. The following mixtures, for example, have equilibrium partial pressures for the solute which are so small that their temperature
dependence is of little significance: HCl/H20, C02/MEA, COa/DEA, HCHO/H20 (Stockar, 1972). Such absorptions are often conducted without cooling. (c) Absorption towers operate predominantly adiabatically, especially when they process large volumes of gas, i.e., at industrial scale. The following two types of simplification were introduced many years ago to aid the theoretical treatment of gas absorption. (1) Isothermal process, whereby the temperature of the liquid phase was assumed to be everywhere the same, e.g., equal to its inlet value. This method implicitly ignored all heat effects and produced the simplest calculations, e.g., on a McCabe-Thiele diagram. (2) Simple adiabatic model, whereby the heat of solution was assumed to manifest itself only in the liquid phase. Its temperature could then be determined, as a function of solute concentration, from a simple adiabatic energy balance. The real situation, resulting from the factors described earlier, exhibits however a high degree of interaction between all factors and does not allow simplifying approximations (Coggan and Bourne, 1969). Rather a digital computer should be applied to a comprehensive generalized model of the absorption process. Our earlier method, however, resulted sometimes in difficult convergence and therefore a degree of uncertainty. It is therefore the objective of this paper to show a better, more certain computational method, to test its capacities and convergence properties, and to draw some further generalizations about the nature of heat effects in gas absorption. In the second part (Bourne, e t al., 1974), the accuracy and reliability of the method developed here will be demonstrated Ind. Eng. Chem.,
Process
Des. Develop., Vol.
13, No. 2, 1974
115
by comparison with the results of a lengthy experimental study of absorption and desorption. General Considerations of Absorber Calculations Although there exists a large amount of literature on the calculation of stagewise equilibrium processes, the majority is devoted to fractional distillation. The status of gas absorption with heat effects is, however, special because of the difficulties which arise when otherwise trouble-free methods are applied to this processes. Characteristics of Gas Absorption. A gas absorption process operating with C components, exhibits 2C 4 degrees of freedom. The degrees of freedom are the variables which the designer or operator of the process is free to specify, e.g., flow rate, composition and temperature of an entering stream, the operating pressure, the number of theoretical plates. If one of the feed gases in insoluble (inert), 2C 3 degrees of freedom remain, e.g., nine for the simplest case of a solvent, an inert gas, and a solute. Usually seven of these are taken up by specifying the states of the gas and liquid streams entering the absorber and after definition of the operating pressure, only one remains. In a design calculation, one specifies further the desired value of one quantity in a product stream, e.g., through the purity specification. In a performance calculation ( i e . , the analysis of the operation of an existing column), one specifies the number of stages. In both cases, it is clear that isothermal operation may not be specified. The derivation of the number of degrees of freedom and a discussion of the differences between absorption and distillation have recently been published (Stockar, 1972). It follows that the states of the product streams may not be fully specified, rather they are determined by the separation process itself. Although for certain absorptions, simplifying assumptions can have pragmatic value, it is impossible in principle to select arbitrary states for the product streams. With countercurrent flow, the states of both streams at either end of the column cannot be specified and the two-point boundary value problem for gas absorption is quite complex. A second difficulty is presented by the fact that the states of the streams a t one end of a .gas absorber are not coupled, e.g., uia a condenser, as in the case of distillation. Large deviations from thermal and diffusional equilibrium usually exist. They create irregular concentration and temperature profiles along the absorber, about which no general predictions can be made. When a volatile solvent is used, a temperature maximum is common. The gas and liquid flow rates are not necessarily either constant or monotonically changing. A third difficulty arises from the nonideality of most solutions of dissolved gases, which increases the nonlinearity and the degree of coupling of the systems of balance equations. One seeks, therefore, a robust, reliable method of computing stagewise gas absorption, with proper consideration of the degrees of freedom available for this process. The method should be sufficiently versatile to cover stages with coolers, pressure variations, and unequal plate efficiencies, for example. Computational Methods. The analytical method of multicomponent absorption calculations, using absorption factors (Horton and Franklin, 1940; Edmister, 1943), demands a priori knowledge of the distributions of A i , j . These depend upon the distributions of T, L, and G, which are, in fact, dependent variables and therefore unknown. This method allows only very rough estimates of absorber performance, when temperature variations occur. The classical McCabe-Thiele construction also fails, principally because the location of the equilibrium line
+
+
116
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
also depends on the temperature, which is itself dependent and unknown. Early work (Coggan and Bourne, 1969) dealt with the implicit, nonlinear plate equations of the steady-state model by a n iterative procedure that was nested inside another iterative loop, producing the overall component and heat balances. Due to negative curvature of the objective function and virtual discontinuities in slope near the solution, in some cases (e.g., ammonia absorption) the method became sensitive to round-off errors and would not converge from a randomly chosen starting point. Nagel and Thielen (1972) proposed the separation of the a t least 5N equations into three classes, followed by the simultaneous solution of all the equations in one class by Newton’s method. The above-mentioned difficulties suggest that only a calculation beginning with a good estimate of the final state of an absorption column would converge. Nagel and Thielen communicated nothing about the convergence properties of the Newton method, but the adapted variable metric method used earlier (Coggan and Bourne, 1969) is thought to be more stable. The state-space or dynamic model, in which the process is described by simultaneous differential equations expressing the rates of change of composition and enthalpy on each stage, has several advantages. The equations are explicit and need not be linearized or simplified and the accuracy of the initial guesses for the state variables is not critical. The unsteady state can be simulated and, because the modeling of the physical state of a column is realistic, convergence is certain. Furthermore such refinements as coolers and variable plate efficiencies are readily incorporated into the model. (This method (Hanson, et al., 1962) has however been rejected for distillation columns (Sargent and Murtagh, 1969) because of slow convergence, which may have been due to the spread of system time constants produced by the variations in liquid flow and the positive feedback through the reflux stream.) The dynamic response of the column may be a useful by-product of the computation, but as it is not essential here, it is better for the stage hold-ups to be adjusted to bring these time constants closer together. If the integration subinterval is one tenth of the smallest time constant and convergence occurs after six times the sum of the time constants, clearly computing time is a function of the ratio 6027/rmln. This is a minimum when all time constants are equal. Shortly after our earlier work on absorption (Coggan and Bourne, 1969), we switched to state-space models and have been using them ever since. The more recent experience of Stichlmair (1972) with a similar program is also good. State-Space Model of a Gas Absorption Column General Principles. As only a steady-state solution is required here, any unnecessary dynamic characteristics can be omitted and the differential mass and energy balances integrated up to the steady state. With an inert, insoluble carrier gas, a stage is described by X I , ] the mole fraction of solute in the liquid and the mean liquid molar enthalpy. Referring to Figure 1, heat and solute balances in a mixed stage give
In some cases, eq 2 was extended to nonadiabatic absorption, with results for optimum cooler positions, etc., simi-
Read m physical data
Figure 1. Quantities entering the mass and energy balances for stage j .
i
'2.0 x1.0 TL.0
Figure 3. Flow diagram for the calculation of absorption columns using the dynamic approach method.
TG,N+I
Figure 2. The absorption process.
lar to those published earlier (Coggan and Bourne, 1969). The following additional relationships were applied. overall balance
L J - l- L , +GI+] - G , = 0
( W , = c o n s t a n t ) (3)
concentration of solute in the gas phase Y1
J
=
Elyl
,* + (l -
E1)yl ) + I
(4)
e q u i l i b r i u m for t h e s o l u t e
(5) Equations similar to (4) and (5) were applicable to the solvent. liquid temperature
T L .= ~
fz(x1
,,hJ, liquid h e a t c a p a c i t i e s , h e a t of m i x i n g )
(6)
gas t e m p e r a t u r e
TG
j
=
TG.J-I + ET(TL1 - TGj + l )
G, = f 3 ( v a p o r c o m p o s i t i o n ) In the program, W is set initially
(7)
(8)
W = 8/(1OL, + GN+J (9 ) so that the mixing time constant T on any stage is approximately unity. For an N-stage column, the 2N differential equations are put together in one grand state equation where
i = f(2,u)
(10)
and u represents the inputs to the column (feed temperatures and compositions). The problem is closed by the specification of the nine degrees of freedom, conveniently chosen as follows (see Figure 2): feed gas: flow of inert gas, g 3 ; concentrations of ; T G , ~ +fresh l ; solsolute and solvent, y i , ~ + 1temperature, vent: flow of solvent, / 2 , 0 ; concentration of solute, ~1.0; temperature, TL,o; number of actual plates, N; pressure a t top of absorber, P. This format is convenient for performance calculations and it served to evaluate our experimental runs (see part 11). For design purposes, the number of actual plates may be found by interpolating between runs using the above format, as explained in the next section. Structure of the Main Program. The main program controls the input and output, initiates the integration, controls the convergence, and guides the repetition of the calculation, when working iteratively (Figure 3). The outer loop of Figure 4 distinguishes between design and performance calculations, whereby design requires iteration, and supervises systematic iteration. When determining, for example, the number of actual plates needed to reduce the concentration of solute in the gas phase to y ~ , this ~ , loop changes automatically the estimated value of N in proportion to the difference between calculated and desired y1.1 values, until convergence is obtained. Other options exist, e.g., a minimization routine whereby a suitable criterion is minimized in order to determine the best value of some parameters, such as plate efficiencies, from sets of experimental absorption data. The initial state of the column is defined by setting the compositions and mean molar enthalpies on all plates equal to the corresponding values in the entering solvent stream. The computation starts with the composition and temperature of the gas entering the first stage equal to the known values for the feed, and continues up the column calculating all gas states from the known states of the liquid phases. After setting the initial value of the state vector z, a routine for the numerical integration of ordinary, nonlinear differential equations is called. Contrary to all previous publications, where a simple linearization was used, a fourth order Runge-Kutta process was employed here. Together with physically realistic differential equations, Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2 , 1974
117
X1,l
hole-%I
15 Looate 0001er
Specify 7 , P t and ST
I n i t i a l s t a t e veotor
Call
P>JYsLCal
data
i
i
Figure 5. Change of ammonia concentration on the top plate during the simulation of the start-up: residence time in units which correspond to 12 times the length of an integration step; curve a, accelerated.
I
Examine d e r i v a t i v e
,
I I
I
