Ind. Eng. Chem. Res. 2004, 43, 5899-5903
5899
Mathematical Analysis of the Conditions Necessary for the Appearance of the “Overshoot” Phenomenon in Countercurrent Packed Columns Talgat A. Akramov,† Petr Svoboda, Vladimir Jiricny, and Vladimir Stanek* Institute of Chemical Process Fundamentals, Academy of Science of the Czech Republic, 135 Rozvojova, 165 02 Prague, Czech Republic
This paper analyzes the model describing the transient behavior of a countercurrent packedbed column exposed to a change of the inlet fluid flow rate. Functional mathematical analysis of the model equations is employed to establish the conditions that would lead to the overshoot phenomenon of liquid holdup in the column undergoing a sudden change of the inlet fluid velocity. It has been proven that a mere nonlinear relationship between the liquid holdup and liquid velocity is not a sufficient condition for the overshoot. The liquid holdup overshoot can appear if the liquid holdup is also a function of the gas velocity, if the implemented step change of either the gas or liquid velocity is to a higher velocity, and if the higher value of the velocity is maintained for a sufficiently long time. Introduction In several earlier papers1-3 concerning the hydrodynamics of countercurrent packed-bed columns, the authors observed and reported overshoots on the transient profiles of pressure drops and liquid holdup. By overshoot, it is meant that the gas pressure and liquid holdup, in response to a step increase of the inlet velocity of either gas or liquid, temporarily exceeds the value corresponding to their final steady-state values. These overshoots were observed experimentally when the bed was exposed to a sudden increase of either the gas or liquid inlet velocity. Another condition for the appearance of the overshoots was that the step change in the inlet velocity brings the system into the proximity of the flooding point. In a recent paper,4 a mathematical model of the hydrodynamic transients of the countercurrent column was formulated. This model proved to be capable of explaining the cause and nature of these overshoots. Solutions of the model further indicated that an essential condition for the appearance of the overshoots is that the step change of the inlet fluid velocity must bring the system into the proximity of the flooding point. The aim of this paper is to define exactly the conditions for the appearance of the overshoot of liquid holdup by means of functional mathematical analysis of the model equations. Posing a Problem The transient one-dimensional model presented earlier4 has the form of differential balance equations on the mass of liquid and gas:
∂(hFL) ∂(vLFL) )0 ∂t ∂z
(1)
∂[( - h)FG] ∂(vGFG) + )0 ∂t ∂z
(2)
The boundary conditions stipulate the velocities of gas and liquid at their respective inlet ends. For gas at z ) 0 and for liquid at z ) L such as
vG|z)0 ) ψ(t); vL|z)L ) φ(t)
and the initial conditions for gas and liquid velocities are as follows:
vG|t)0 ) v0G(z); vL|t)0 ) v0L(z)
(4)
Here h ) h(vG,vL) designates the local holdup of liquid expressed as a volume fraction of the empty bed. The holdup is considered to be generally a function of both liquid and gas velocities. Further, we assume that both liquid and gas are incompressible fluids so that we have FG ) constant and FL ) constant. The pressure drop for a 1-m-high packed bed where “overshoots” have been observed1,4 never exceeded 0.2 m water head, which corresponds to about a 2% change of the gas density. This makes the assumption that a constant gas density is acceptable. Further, we also assume that void fraction is constant. By subtracting eqs 1 and 2 and using the assumptions of constant fluid densities and constant void fraction, we obtain that the derivative of the difference of phase velocities with respect to coordinate z is zero. For the difference of phase velocities, we thus may write the following:
vG(z,t) - vL(z,t) ) S(t) * To whom correspondence should be addressed. E-mail:
[email protected]. † On leave from Baskhir State University, Ufa, Russian Federation.
(3)
(5)
where S(t) is a function of time, t, only. Let us now investigate the possibility of predicting the holdup overshoot under various types of dependen-
10.1021/ie030850y CCC: $27.50 © 2004 American Chemical Society Published on Web 07/15/2004
5900 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
cies of the holdup of liquid, hL(vG,vL), on the velocities of liquid, vL, and gas, vG. Quadratic Form of the Liquid Holdup versus Liquid Velocity Function First let us examine the simple case of a nonlinear relationship when liquid holdup is a function of the liquid velocity only. For brevity, we shall now drop the subscript L from the symbol of the liquid velocity. Thus, v ) vL(z,t) ) v(z,t). In this case we may rewrite eq 1 in the following form:
2. On the lines of discontinuity, z ) z(t), there exist the left-hand side, v(z-0,t), and the right-hand side, v(z+0,t), limiting values. At the discontinuity, we adopt the equality v(z,t) ) v(z+0,t). As the weak solution of eq 8 in Q ) [0,L] × [0,∞), we designate the function v(z,t) ∈ K satisfying the following integral equality:
∫Ch[v(z,t)] dt + v(z,t) dt ) 0
(12)
Because our intention is to find out whether a nonlinear relationship of liquid holdup with the liquid velocity may be the cause of the “overshoot”, we now define, with sufficient generality for our purposes, the liquid holdup as the following function of the liquid velocity:
for an arbitrary piecewise smooth and closed contour C lying fully in Q. Let v(z,t) ∈ K, and z ) z(t) is an equation of one of the lines of discontinuity of the function v(z,t). D ) z′(t), v-(t) ) v[z(t)-0,t], and v+(t) ) v[z(t)+0,t], and [v] ) v+(t) v-(t) is the jump of the function v(z,t) on the line of discontinuity, while [h(v)] ) h[v+(t)] - h[v-(t)] is the jump of the function h(v) ) Av2 on the line of discontinuity. From the integral form of the equation of continuity (12), we get the following equality:
h(v) ) Av2
[h(v)]D + [v] ) 0
∂h(v) ∂v )0 ∂t ∂z
(6)
(7)
where A is a positive constant. On adopting the quadratic form of the dependence of the liquid holdup on the liquid velocity, we are dealing with the type of so-called Hopf equation. In our notation, this equation has the form
∂ ∂ (Av2) - (v) ) 0 ∂t ∂z
(8)
with the following initial and boundary conditions:
v(z,0) ) v0
(9)
v(L,t) ) φ(t)
(10)
where v0 is a constant initial liquid inlet velocity. φ(t) is a piecewise smooth function with discontinuities of the first type and with a finite number of discontinuities in the interval [0, T] and equal to a constant for t > T. Under these conditions, it is known that the hyperbolic equation (8) does not possess a differentiable solution.5 We shall therefore construct a weak solution of the problem defined by the set of equations (8) and (10), assuming that φ(t) is a piecewise constant function of the form
{
φ- ) v0 φ(t) ) φ+ φ-
for 0 e t < tfor t- e t < t+ for t+ e t < ∞
(11)
where φ- and φ+ are constants related by
0 < φ- < φ+ and t+ and t- designate the points of discontinuity of the function φ(t). Upon designation by K, the class of functions v(z,t) ∈ K satisfy the two following conditions: 1. In an arbitrary part of a cylinder Q ) [0,L] × [0,∞), there is a finite number of lines, z ) z(t), and a finite number of points of discontinuity. Outside of these lines and points, the function v(z,t) is continuous and has continuous derivatives.
(13)
From this, we get for [v] ) v+(t) - v-(t) * 0 that
D)-
1 ) z′(t) A(v+ + v-)
(14)
Equation 13 is known in gas dynamics as the Hugoniot condition at discontinuity. Below we shall show that if the jump [v] ) v+(t) - v-(t) * 0, then v((t) does not depend on time. From eq 14, it therefore follows that the line of discontinuity is a straight line with slope D. We will construct the weak solution by the method of characteristics. Because φ(t) is a piecewise constant function, this method allows us to write the solution in an explicit form. The system of characteristic equations for eq 8 may be written as
dv ) 0, v|t)t0 ) vR dt dz 1 ), z|t)t0 ) z0 dt 2Av
(15)
Solution of the Cauchy problem in eq 15 may be written explicitly as
v ≡ vR z ) z0 -
t - t0 2AvR
(16)
The solution v(z,t) of eq 8 along the characteristic line, which is now a straight line in Q, is a constant that can be determined from
v(z,t) ) vR )
t - t0 2A(z0 - z)
(17)
when the point (z, t) lies on the characteristic line passing through the point (z0, t0). We will construct the solution of our boundary value problem (8)-(10) in the neighborhood of the point of discontinuity (L, t-), where t- is the point of discontinu-
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5901
ity of the function φ(t). From the point (L, t-), we shall draw two characteristics (straight lines):
z0(t): z ) L -
t - t2Aφ-
and
z1(t): z ) L -
t - t2Aφ+
which represent solutions of the Cauchy problem of eq 15 for z0 ) L, t0 ) t-, and vR equal correspondingly to φ- and φ+. The given straight lines divide Q into regions G1 and G2 defined as
Because the solution of the Cauchy problem (18) continuously depends on the parameter R and the smooth solution of eq 8 is constant along the characteristic line, the necessary condition of stability of the weak solution for the given initial and boundary conditions dictates [ref 5, pp 480 and 481] that the weak solution (8) on G2 is determined by the following rules: If the point (z, t) ∈ G2 lies on the straight line z ) zR(t): z ) L - (t - t-)/2AvR, then the value of the solution is v ) v(z,t) ≡ vR. This follows from eq 18. Now we express vR in the last expression in eq 18. We already have vR ) (t - t-)/(L - z)2A from which, with the aid of the first expression in eq 18, we obtain for (z, t) ∈ G2 that
v(z,t) ≡ vR )
G1 ) {(z,t): 0 e z < L; 0 e t < (L - z)2Aφ- + t-} G2 ) {(z,t): 0 e z < L; (L - z)2Aφ- + t- e t < (L - z)2Aφ+ + t-} Analogously, we will draw from the point (L, t+) the characteristic (straight line)
z2(t): z ) L -
t - t+ A(φ- + φ+)
that is the solution of the Cauchy problem of eq 15 for z0 ) L, t0 ) t+, and vR ) (φ+ + φ- )/2. This straight line is the line of discontinuity of the weak solution v(z,t), which for AL(φ+ - φ-) < t+ - tdivides the region Q into subsets
G3 ) {(z,t): 0 e z < L; (L - z)2Aφ+ + t- e t < (L - z)A(φ+ + φ-) + t+} G4 ) {(z,t): 0 e z < L; (L - z)A(φ+ + φ-) + t+ e t < ∞} In regions G1 and G4, the solution v(z,t) is a constant and equal to φ-. In the region G3, the solution v(z,t) is also a constant and equal to φ+. On the straight line z ) z2(t), the solution v(z,t) is discontinuous and exhibits a jump [v] ) [φ+ - φ-] for 0 e z < L. In the common region G1 ∪ G2 ∪ G3, the solution is continuous, but the derivatives are discontinuous on the contours z0(t), z1(t), and z2(t). In the point (L, t-), there is a singular solution of the type (t - t-)/2A(L - z) appearing in the gas dynamic as the centered shock wave [ref 5, p 481]. In the region G2, the weak solution v(z,t) is not a constant function, but it continuously varies. In order that we can find the solution v(z,t) for (z,t) ∈ G2, we will use the following construction. We solve a singleparameter Cauchy problem (15) for t0 ) t- and z0 ) L, where vR ) φ+R + (1 - R)φ- and R is a parameter satisfying 0 e R e 1. As a solution, we obtain
z ) zR(t): z ) L -
t - t2AvR
(18)
which for R ) 0 and 1 gives the straight lines z0(t) and z1(t).
(L - z)2A
(19)
It is easy to verify that on G2 the function v(z,t) represents in the above form (19) the solution of eq 8. The formula (19) is even differentiable on G2, which although continuous on G1 ∪ G2 ∪ G3 has a singularity in the form of a centered shock wave at the point (L, t-). Let us inspect now the behavior of the weak solution in the neighborhood of the point (L, t+), where φ(t) exhibits a jump [φ] ) φ- - φ+ < 0. In this case, the method of characteristics shows that the weak solution v(z,t) is discontinuous. On the contours of discontinuity passing through the point (L, t+), the Hugoniot conditions (13) and (14) are fulfilled. Therefore, the contour of discontinuity originating from the point (L, t+) is the straight line
z2(t): z ) L -
t - t+ A(φ- + φ+)
{
By setting up the solution v(z,t) on Q in the form
φt - t-
for (z, t) ∈ G1
for (z, t) ∈ G2 v(z,t) ) (L - z)2A φ+ for (z, t) ∈ G3 φfor (z, t) ∈ G4
(20)
we obtain the weak solution for which, on the contour of discontinuity, the condition of stability and the Hugoniot condition (14) are fulfilled. It can be shown that the weak solution constructed in this way is unique on class K. Analysis of the Mean Holdup of Liquid for the Weak Solution The mean holdup of liquid in the packed column is defined as
h(t) )
v(z,t) ≡ vR
t - t-
∫0Lh[v(z,t)] dz
1 L
(21)
where in the above-analyzed case h(v) ) Av2(z,t) and L designates the depth of the packed section. We will now evaluate the mean holdup, h(t), using the above definition of the weak solution (12). Let the contour C ) C(t) be the boundary of a rectangle
5902 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004
delimited as
C ) {(z,τ): 0 e z e L, 0 e τ e t} Then from the obtained weak solution (12) we have
∫0Lh[v(z,0)] dz + ∫L0h[v(z,t)] dz + ∫0tv(L,τ) dτ + ∫t0v(0,τ) dτ ) 0,
(22)
it thus turns out that, in the model case of the liquid holdup being a quadratic function of the liquid velocity, the overshoot of the mean liquid holdup cannot take place. We thus may conclude that when the mean liquid holdup, h(t), obeys eq 8, then for an arbitrary type of dependence of h(v) on v (satisfying the condition h′′v(v) * 0) the overshoot phenomenon cannot, in principle, take place. This is so because of the validity of the inequality h(t) e h[max φ(t)].
from which we obtain
h(t) ) h(0) +
∫0tφ(τ) dτ - L1 ∫0tv(0,τ) dτ ) h(0) + I(t)
1 L
(23) where
I(t) )
1 L
∫0tφ(τ) dτ - L1 ∫0tv(0,τ) dτ
(24)
On designating by τ-, τ0, and τ+ the parameters
τ- ) 2ALφ-; τ0 ) AL(φ- + φ+); τ+) 2ALφ+ (25)
Linear Form of the Liquid Holdup versus Gas and Liquid Velocities Function Let us now analyze the possibility of the existence of the liquid holdup overshoot for the case when the liquid holdup is simultaneously a function of the liquid and gas velocities. Let us assume in particular that h(vL,vG) is a linear function of vL(z,t) and vG(z,t) so that we may write h(vL,vG) ) AvL + BvG, where A and B are positive constants. Using eq 5, the model equations (6)-(10) with h(v) ) (A + B)v + BS(t) then take the form
we obtain from the weak solution in regions Gi, i ) 1-4 for t+ - t- g τ+ that
{
φ(t - t-)φv(0,t) ) τφ+ φ-
{
for 0 e t < t- + τfor t- + τ- e t < t- + τ+
(26)
for t- + τ+ e t < t+ + τ0 for t+ + τ0 e t < ∞
Using φ(t) and v(0,t) from eq 26, we can evaluate the integral I(t) with the result I(t) ) 0 1 (φ - φ-)(t - t-) L + φ-τ- φ- (t - t-)2 1 φ+(t - t-) L 2 τ2
[ [
]
2
φ-τ- φ- τ+ 1 φ τ ) I(t+) L + + 2 τ- 2 1 I(t+) + (φ- - φ+)(t - t+) L 0
(30)
v|z)L ) φ(t)
(31)
v|t)0 ) v0L(z) ) v0 > 0
(32)
Here v0 is also a constant and the function φ(t), just as above, is a piecewise constant of the type (11). We shall employ the method of characteristics to obtain the classic solution of eq 30. The set of characteristic equations for eq 30 has the following form:
dz 1 ); z|t)t0 ) z0 dt A+B
for 0 e t < t-
]
∂[(A + B)v(z,t) + BS(t)] ∂v(z,t) )0 ∂t ∂z
for t- e t < t- + τ-
dv B )S′(t); v|t)t0 ) v0 dt A+B
for t- + τ-e t < t- + τ+ for t- + τ+ e t < t+
(33)
Solution of the Cauchy problem gives the next two equalities
for t+ e t < t+ + τ0
z(A + B) + t ≡ z0(A + B) + t0 ) constant
for t+ + τ0 e t < ∞ (27)
It is easy to prove that the maximum value of I(t), designated as Imax, is reached on the interval t ∈ [t- + τ+, t+] and equals
Imax )
[
]
2
φ-τ- φ-τ+ 1 φ τ L + + 2 2τ-
v+
B B S(t) ≡ v0 + S(0) ) constant A+B A+B
From this, the general solution of eq 30 takes the form5
) 2
2
A(φ+ - φ- ) ) [h(v)] (28)
Upon evaluation of the inequalities
h(t) ) h(0) + I(t) e Aφ-2 + Imax ) Aφ+2 ) A[max φ(t)]2 (29)
v(z,t) ) f[z(A+B)+t] -
B S(t) A+B
(34)
where f(ξ) is an arbitrary function of its argument. We shall now express the unknown function f(ξ) from the initial and boundary conditions (31) and (32) utilizing the function φ(t). Let τ0 ) (A + B)L; then from the initial and boundary conditions (31) and (32), we obtain
Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5903
{
f(ξ) )
B for 0 e ξ < τ0 φ- + S(0) A+B (35) B S(ξ - τ0) for τ0 e ξ φ(ξ - τ0) + A+B
Analogously, for the mean liquid holdup, we obtain from the definition of the weak solution the following formula:
h(t) )
∫0Lh[v(z,t)] dz ) 1 t 1 t h(0) + ∫0 φ(τ) dτ - ∫0 v(0,τ) dτ L L
1 L
(36)
Substituting in the above formula the expression for v(0,t) obtained from eq 34 for z ) 0, we get
h(t) ) h(0) + I1(t) + I2(t)
(37)
Here the symbols on the right-hand side designate
I1(t) ) I2(t) )
1 L
∫0tφ(τ) dτ - L1 ∫0tφ˜ (τ) dτ ∫0tS(τ) dτ - L1 A +B B∫0tS˜ (τ) dτ
1 B LA+B
(38)
(39)
Upon evaluation of the integrals in eqs 37-39 for 0 e θ+ < t < θ+ + τ0 < t+, the inequality
h(t) ) (A + B)φ+ +
1 BS (θ +τ -t) g (A + B)φ+ + τ0 + + 0 BS(t) ) h[φ(t)] (44)
proves the existence of overshoot. Conclusion Functional analysis of a one-dimensional transient model of countercurrent gas-liquid flow in packed beds has proven that, for the existence of the liquid holdup overshoot phenomenon under the linear relationship between the liquid holdup, on the one hand, and gas and liquid velocities, on the other hand, it is necessary that B > 0 and it suffices that the function S(t) makes the integral I2(t) positive over the interval [θ+, θ+ + τ0]. In practice, this means that the liquid holdup overshoot can appear if (i) liquid holdup is also a function of the gas velocity (not only liquid velocity), (ii) the implemented step change of either gas or liquid velocity takes place to higher velocities, and (iii) the higher velocity after the step change is maintained for a sufficiently long time. A mere nonlinear relationship between the liquid holdup and liquid velocity has been proven to be insufficient for the appearance of the liquid holdup overshoot. Acknowledgment
where
{ {
φfor 0 e τ < τ0 φ ˜ (τ) ) φ(τ - τ0) for τ0 e τ
(40)
for 0 e τ < τ0 S(0) S(τ - τ0) for τ0 e τ
(41)
S ˜ (τ) )
The structures of the integrals I1(t) and I2(t) are the same. Using the function φ(t) in eq 11, the integral I1(t) may be evaluated explicitly. For instance, for 0 e τ0 < t+ - t-, we get
{
I1(t) ) 0 (φ+ - φ-)(t - t-) 1 (φ - φ )τ + - 0 L (φ+ - φ- )τ0 - (φ+ - φ-)(t - t+) 0
for 0 e t < tfor t- e t < t- + τ0 for t- + τ0 e t < t+ for t+ e t < t+ + τ0 for t+ + τ0 e t < ∞ (42)
Analysis of the integrals in eq 37 shows that for B ) 0 the overshoot cannot appear. However, for B * 0 it is possible. For the overshoot to appear, it is necessary that the integral I2(t) becomes positive in the interval [t- + τ0, t+], which gives a positive contribution to the mean liquid holdup h(t). Let us formulate the properties of the function S(t) for the case of nonzero B. Let us define a new parameter θ+ satisfying the inequalities 0 e τ0 < t- + τ0 < θ+ < θ+ + τ0 < t+ and determine the function S(t) as follows:
{
for 0 e t < t0 S > 0 for t- e t < θ+ S(t) ) + for θ+ e t < ∞ 0
(43)
The authors gratefully acknowledge financial support of the project by the Grant Agency of the Czech Republic under Grant 104/03/1558. Nomenclature A, B ) parameters of the liquid holdup function h ) local liquid holdup L ) depth of the packed bed [m] vG ) superficial gas velocity [m‚s-1] v, vL ) superficial liquid velocity [m‚s-1] t ) time [s] z ) axial coordinate [m] FG ) gas density [kg‚m-3] FL ) liquid density [kg‚m-3] ) void fraction of bed
Literature Cited (1) Stanek, V.; Jiricny, V. Experimental observation of pressure drop overshoot following an onset of gas flow in counter-current beds. Chem. Eng. J. 1997, 68, 207-210. (2) Stanek, V.; Jakes, B.; Ondracek, J.; Jiricny, V. Characteristics of pressure and liquid holdup overshoot following a sudden increase of gas flow. Chem. Biochem. Eng. Q. 1999, 13 (2), 65-71. (3) Stanek, V.; Svoboda, P.; Jiricny, V. Experimental Observation of Pressure and Holdup Overshoot Following a Sudden Increase of Liquid Flow. Ind. Eng. Chem. Res. 2001, 40, 3230. (4) Svoboda, P.; Stanek, V. Theoretical explanation of pressure and holdup overshoots in countercurrent packed columns. Ind. Eng. Chem. Res. 2003, in press. (5) Rozhdestvenskii, V. L.; Yanenko, N. N. Sistemy kvazilinearnykh uravneinii i ikh prilozhenia v gazovoi dinamike; Nauka: Moscow, 1968; p 592 (in Russian).
Received for review November 19, 2003 Revised manuscript received May 6, 2004 Accepted June 9, 2004 IE030850Y
