Stagewise Absorption and Extraction Equipment Transient and Unsteady State Operation LEON LAPIDUS UNIVERSITY OF
NEAL R. AMUNDSON
AND
MINNESOTA, MINNEAPOLIS, MINN.
consideration is given t o t h e transient and unsteady state behavior of stagewise absorption and extraction equipment, both crosscurrent and countercurrent. T h e treatment is entirely mathematical and no experimental data are included. It is assumed t h a t a plate-type absorber Is fed with a f a t gas stream and a lean oil stream a t a constant inert rate and t h a t t h e compositions of these streams are functions of the time. It is assumed further that, initially, any arbitrary composition can exist on the pldtes. Formulas are then developed t o show how t h e fat oil and lean gas compositions will vary with time. It i s necessary t o assume t h a t a linear equilibrium relation
obtains between the two phases. W i t h these formulas it Is possible t o compute the composition on any plate a t any time. Two numerical examples are included t o show how feed and product compositions vary with t h e time and t o illustrate the t i m e necessary t o return t o near equilibrium conditions when t h e unit Is upset by variations in the feed streams. In t h e second part of the paper crosscurrent operation is considered and analogous formulas are developed. This problem is not as complex and probably n o t as pertinent. The methods of solution used in this paper can be used in other stagewise problems. T h e paper generalizes results of Marshall and Pigford (12).
U
Since the problems to be solved are complicated physically, the mathematical formulas were expected to be complex. This is borne out, but the formulas actually include only elementary mathematical functions. The use of modern calculating and computing machines would facilitate the application of these equations. As the problems discussed here are concerned with the time of operation, it is apparent that the compositions must always be a function of two variables-the stage number, n, and the time, t. It follows that the processes will be described by partial differential-difference equations, and the method which lends itself most readily to their solution is that of the Laplace transformation taken with respect to the time. This reduces the equations to difference equations whose solutions can be obtained by standard methods. Since there are several well-known books on the LaPlace transform and since this should now be a standard tool of all engineers, details of its manipulation will not be presented. For the uninitiated, however, the authors suggest the book of Churchill ( 8 ) to which some reference will be made later. Example 1 is a problem of countercurrent absorption in a plate column of N plates. It is clear, of course, that the same equations are applicable to countercurrent extraction. Example 2 is a problem of crosscurrent or cocurrent extraction. This problem is less important than the first one but is included for the sake of complete-
NTIL quite recently the calculations involved in the stagewise unit operations of chemical engineering were those of the steady state-that is, the operation of any piece of equipment was independent of the time. In distillation, the methods of calculation were based on the premise that feed and product compositions remain fixed and that constant sources for heating and s i n k for cooling were available. In absorption and extraction, it has been assumed that feed compositions remain invariant and similarly feed rates do not change. These stagewise operations can be described mathematically by finite difference equations. Tiller (18, 19) was the first to study systematically the application of difference equations to chemical engineering. In another paper Tiller ( 1 7 ) showed how a great variety of difference equations, both linear and nonlinear, could be solved by graphical means. Smoker (18) and Amundson (1) showed how such equations could be applied to binary rectification, and Murdock (14) and Underwood (20)showed how simultaneous difference equations could be used in multicomponent rectification. Brown and Souders ( 4 ) and Kremser (10) obtained formulas for absorption, whereas MacMullin and Weber (11) and later Kirillov (9) and Eldridge (6) considered systems of continuous, stirred, chemical tank reactors. In papers of very recent origin the question of the transient behavior has been considered by a few authors. Berg and James ( 8 ) discussed the rate of approach to equilibrium of a still operating a t total reflux assuming a linear equilibrium relationship of a special kind. Marshall and Pigford (12) in chapter X I of their book discussed the transient behavior of both distillation and absorption equipment, and Kirillov (9) and Mason (IS)have considered the transient state in continuous, stirred tank reactors. However, in none of these papers have the authors considered the problem of unsteady feed compositions. The purpose of this paper is to consider the operations of absorption and extraction and to solve the problem of the transient and unsteady state completely for a general linear equilibrium relation. The authors have consideredchanges in feed compositionsand previous history of the equipment, but have supposed that feed rates remain invariant. It is hoped that this latter problem can be considered in future work. The methods used in this paper can also be applied to distillation and the purging of chemical reaction systems; these problems also will be considered in future work. The use of a linear equilibrium relation restricts the application of the derived formulas, but it is felt that qualitative conclusions for the nonlinear cmes can be drawn from the results obtained.
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EXAMPLE 1. COUNTERCURRENT ABSORPTION
Fi ure 1. Diagrammatic b e w of a Plate Absorber
Consider a plate absorber (Figure 1) which contains N theoretical plates, and suppose that the amount of inert c o m p o n e n t in either the gas phase holdup or liquid phase holdup per plate is independent of the platenumber, nthat is, holdup on each
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INDUSTRIAL AND ENGINEERING CHEMISTRY
plate is the same. This is really not an essential assumption for the problem could still be solved but it does simplify the analysis appreciably. Most absorbers approximate to this assumption The following nomenclature is used in Example 1:
x,
composition of liquid leaving nth plate, Ib. solute/lb. inert absorbent I-n = composition of vapor leaving nth plate, Ib. solute/lb. inert gas liquid holdup on any plate, Ib. of absorbent (soluteh = free) H = vapor holdup on any plate, lb. of inert gas (solufefree) absorbent rate, lb./hr. (solute-free) L = inert gas rate, Ib./hr. (solute-free) G = composition of liquid on lzth plate at time, t = 0, A n = Ib. solute/lb. inert absorbent X d t ) = f ( t ) =. composition of lean oil to top plate, Ib. solute/ lb. inert absorbent Y N + ~= ( ~g )( t ) = composition of fat gas to bottom plate, lb. solute/lb. inert absorbent a, p = constants in equilibrium relationship t = time, hours s = Laplace transform parameter corresponding to t L Ga ( a H h)s a = Ga =
+
+
+
=
-
fj
=
-
WL
=
aH
xn = F ( s ) , when n = 0
and
(7)
+1
x7L = G(s), when n = N
(8)
Equation 6 is a nonhomogenepus difference equation of the second order whose solution will be obtained by the operational method (3). If E is an operator such that Exn = xnt1 , then Equation 6 can be written in the form (E
- ai)(E - a n ) ~ n - l =
-dA,
(9)
where a1 and a, are the two roots of the quadratic equation Z2
+c = 0
- aZ
(10)
The method of solution for Equation 9 is the following: Obtain a particular solution of each of ( E - al)z,-l = -dA,
(11)
( E - a2)xnm1= -dA,
(12)
calling the two solutions x,:and z: respectively. Then find the general solution of
- a l ) ( E - a2)x:':
(E
=
0
which is
L Ga
c
Vol. 42, No. 6
12.
+ Club
= Clay
x:" T
The general solution of Equation 9 is then
Gff kfh root of equation, Z"+' - 1 = 0
+
r, = PIX:
Referring to Figure 1, if inflow minus outflow to the accumulation in the section bounded by the n and n - 1 plates, is equated
PZX:
+ Cia; + Cza;
where P I and ~2 are two numbers determined from the partial fraction expansion of 1
ZS-aZ+c
=-
PI
Z-a1
+L Z-an
The two first-order equations, 11and 12, can be solved by a simple iteration procedure to obtain
Making use of the equilibrium relation
+B
z: = -d[Ala;-' x: = -d[Ala",-'
Y d t ) = LYX7d1) results in
+ A&-2 + . . . . . . . . + + A,] + A?a';-2 + . . . . + A ~ - I +~ zAn1 ,,
,
It can be shown that Pl
The conditions a t the t n o ends of the column are described by
X,(t) = f(t), when n = 0 (upper end)
+ 1 (lower end)
Y v + , ( t )= g ( t ) , when n = N and the condition of the column a t t
=
(3)
1
m = - - al - a2
and hence the general solution of Equation 9 is
0 is given by
X,(O) = A ,
(5)
X4S)
To determine the values of C1and C2,Equations 7 and 8 are used, and it follows that F(s) =
c 1
+
c 2
and
ZIf(t)l = F ( s )
G(s) = B
+ Cla':+' + C2a;+l
where
= G(s)
Z[g%]
1 - a2
a1
(4)
The four equations 1 or 2, 3, 4, and 5 make up a complete mathematical description of the problem. It is the solution of this system which must be found, and it is a special case of this system which was solved by Marshall and Pigford ( I d ) . Use 2 to denote the Laplace transform operator, so that b:[X,(t)l =
=
On applying the transform to Equation 2
Lx,-~- ( L
+ C ~ ) X+, Gc~x,+~= ( H a + ~)(sz, - A , )
where x,,(s) is abbreviated as xn. This equation can be written, along Xvith the column terminal conditions, as x,,+I
+
- ax,,
CXA
=
-dA,
(6)
These two equations can be solved for Cl and C2 and these values substituted back into Equation 13 to obtain, after a little algebraic juggling,
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
June 1950
1013 to the factor a? - n l , so
Thevaluesof 2 4 a n d -2
