7424
Ind. Eng. Chem. Res. 2008, 47, 7424–7432
GENERAL RESEARCH Analysis of the Conditions for the Inception of Natural Pulsing Flow in Cocurrent Packed Columns Talgat A. Akramov,† Petr Stavarek, Vladimir Jiricny, and Vladimir Stanek* Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, RozVojoVa 135, Prague 6, 16502 Czech Republic
Functional analysis of mass balances in a cocurrent packed-bed column has been carried in conjunction with analysis of the locally linear relationship between liquid holdup and phase velocities and first-order kinetics of the rate of holdup formation. The analysis indicates that the stability of the hydrodynamic regime in the bed is limited by the condition k(λ - γ)L < 2. Furthermore, for k(λ - γ)L > 0.5, the amplitude of the pulsations appearing in the bed in response to small flow perturbations was found to amplify. For k(λ - γ)L approaching 2, the flow disturbances are amplified very strongly. Experimental measurement of the liquid holdup, analysis of the kinetics of liquid holdup formation, and determination of the limits of the appearance of natural pulsations from pressure drop fluctuations show that the limit of inception of natural pulsations closely adheres to the line k(λ - γ)L ) 2 where our analysis predicts very strong amplification of flow disturbances. Introduction In our recent articles,1-3 we formulated a mathematical model of the hydrodynamic transients of countercurrent columns. This model was successfully used to explain and describe the existence of the overshoot phenomenon observed in the transient profiles of liquid holdup and pressure drop,4 as well as the behavior of the column near the flooding point.5 The aim of the current work is to solve the transient mass balances of gas and liquid together with the linearized liquid holdup function in a cocurrent column and analyze its properties. The work examines the behavior of the column when it is exposed to periodic variations of liquid and/or gas velocity and the stability of the hydrodynamic regime. The focus is on revealing the conditions, in terms of the model parameters, that lead to the appearance of the natural pulsing regime. The pulsing regime has been observed in real columns and described in the literature.6-9 A one-dimensional two-fluid model of the transition to periodic flow was formulated by Attou et al.10 and analyzed by Kostoglou et al.11 The transition to pulsing has been studied in trickle beds by Tsochatzidis et al.,12 in reactors by Boelhouwer et al.,13 under elevated temperature and with non-Newtonian liquids by Aydin et al.,14 and at elevated pressure by Urseanu et al.15 and Wammes et al.16 The transition to pulsing and pulse attenuation were studied by Giakoumakis et al.17 Cocurrent Flow with Infinite Rate of Holdup Formation With acceptable assumptions of constant void fraction of the bed and constant densities of the flowing phases, the transient one-dimensional model has the following form of differential balance equations on the mass of flowing liquid and gas ∂h ∂VL )0 (1a) ∂t ∂z * To whom correspondence should be addressed. Tel.: +420 220 390 233. Fax: +420 220 920 661. E-mail:
[email protected]. † On leave from the Russian State Commercial and Economic University, Ufa, Russia.
∂h ∂VG + )0 ∂t ∂z
(1b)
These equations implicitly entail the assumption that the local liquid holdup, h(z,t), forms in the bed instantly according to the locally prevailing liquid and gas velocities. For the sake of enabling functional analysis of the model, we assume that the liquid holdup is a linear function of the superficial velocities of liquid and gas h(VG,VL) ) λVL + γVG + κ
(1c)
Even though the liquid holdup is clearly a complex nonlinear function of phase velocities, this linear relationship can be viewed as a truncated Taylor series expansion of the real dependence valid over a limited domain of phase velocities. The two parameters, λ and γ, in eq 1c thus represent partial derivatives of the holdup with respect to the superficial liquid and gas velocities, respectively. Already at this stage, we note that existing experimental studies (e.g., Moreira et al.18) indicate that the liquid holdup under cocurrent flow in packed beds increases with increasing liquid velocity but decreases with increasing gas velocity. In our subsequent analyses of cocurrent beds, we therefore confine ourselves to positive values of λ and negative values of γ. The boundary conditions stipulate the velocities of gas and liquid at the inlet end of the bed VG|z)0 ) ψ(t)
VL|z)0 ) φ(t)
(2)
and the initial conditions within the bed are generally given by VG|t)0 ) VG0(z)
VL|t)0 ) V0L(z)
(3)
From the subtraction of eqs 1a and 1b, it follows that VG(z,t) + VL(z,t) ) S(t)
(4)
where S(t) is a function of time only. For the inlet end (z ) 0), using the inlet conditions in eq 2, we can write ψ(t) + φ(t) ) S(t)
10.1021/ie800501q CCC: $40.75 2008 American Chemical Society Published on Web 08/30/2008
(5)
Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7425
With the linear dependence of the liquid holdup on the phase velocities, the problem stated in eqs 1a-1c, 2, and 3 can be solved explicitly with the result VL(z,t) )
γ 1 f [t + z(λ - γ)] S(t) λ-γ (λ - γ)
(6a)
VG(z,t) )
λ -1 f [t + z(λ - γ)] + S(t) λ-γ (λ - γ)
(6b)
The thus-far-unknown function f(ξ) can be easily specified in terms of the boundary and initial conditions φ(t), Ψ(t), VG0(z), and VL0(z) λφ(t) + γω(t) ) f(t) (7) For our purpose of analyzing the conditions leading eventually to natural pulsing flow, the above form of the solution is sufficient. We now define the characteristic time, τ0, of the system as τ0 ) (λ - γ)L, where L is the length of the packed bed. (For the derivation of the characteristic time, τ0, see Akramov et al.2) Based on the above-mentioned λ and γ values relevant for cocurrent flow we note that, in cocurrent beds, τ0 is greater than 0. On defining the mean liquid holdup in the column, h(t), as a linear functional given by 1 L 1 ) L
h(t) )
∫ ∫
L
0 L
0
∫ ∫
1 L 1 ) τ0
)
L
0
h[V(z,t)] dz (λVL + γVG + k) dz
Finite Kinetics of Holdup Formation f [t + (λ - γ)z] dz + κ
t+τ0
t
f (ξ) dξ + κ
(8)
it is seen that, as long as the boundary condition function λφ(t) + γψ(t) is τ-periodic, irrespective of whether either φ(t) or Ψ(t) or both vary, the mean holdup is also τ-periodic for t g τ0. Indeed, for t g τ0, we obtain 1 h(t + τ) ) τ0 1 ) τ0
∫ ∫
f(τ + η) dη + κ
1 τ0
∫
[λφ(τ + η) + γψ(τ + η)] dη + κ
1 τ0
∫
[λφ(η) + γψ(η)] dη + κ
1 τ0
∫
f(η) dη + κ
)
)
)
t+τ+τ0
t+τ t+τ0
t
t+τ0
t
t+τ0
t
) h(t)
t+τ0
t
conditions originally prevailing in the packed bed. (3) Because the function f(t) depends on the sum λφ(t) + γψ(t) only, one can maintain constant mean holdup in the bed by keeping λφ(t) + γψ (t) constant, even though the functions Ψ(t) and φ(t) might change individually. The mean holdup can never remain constant if either φ(t) or Ψ(t) (not both) are periodic. This conclusion might be useful for control purposes if keeping the mean holdup in the bed constant is important. We have also analyzed the hypothetical cases of λ ) γ > 0 and λ < γ. However, because of their practical irrelevance to cocurrent beds, we only briefly present the conclusions: For λ ) γ > 0, the analysis indicates that the local holdup would not depend on the variable z and therefore would equal the mean holdup h(t). If either φ(t) or Ψ(t) were periodic, then the amplitude of the fluctuations of the velocities would increase with the distance from the inlet, but the mean holdup would remain the same. For λ < γ, the only way to operate the bed would be to control the inlet conditions so as to keep the mean liquid holdup constant and equal to that determined by the initial conditions. In all other cases, the hydrodynamic regime of the system would break down. The analysis of the solution of the model in eqs 1a-1c, 2, and 3 revealed no conditions that could cause a transition of the system to the natural pulsing flow under realistic values of the parameters characterizing liquid holdup (λ > 0, γ < 0) in cocurrent trickle beds. In the following section, we therefore concentrate on a finite rate of holdup formation.
The formulation of the model in eqs 1a and 1b tacitly assumed that the local holdup adjusts instantaneously to the local velocities of liquid and gas according to eq 1c, i.e., with infinitely fast kinetics. We now relax this assumption by adopting first-order kinetics for the rate of holdup formation as ∂h ) -k[h - h∞(VL,VG)] ∂t
(10)
where the limiting value of the holdup, h∞(VL,VG), is a function of the current local gas and liquid phase velocities and, as before, is related by a simple linear formula, eq 1c, where the parameters λ, γ, and κ are again constants and, without loss of generality, one can set κ ) 0. The boundary and the initial conditions are given again by eqs 2 and 3. From the initial conditions, we obtain specifically
f(ξ) dξ + κ
∂h )0 (11) ∂t t)0 From eqs 1a and 1b, we can now obtain a new system of equations in the form [h(z,t) - h∞(VL,VG)]|t)0 ) 0 or
∫
z
0
(9)
which proves that h(t + τ) ) h(t). The obtained solution of the model in eqs 1a-1c, 2, and 3 for the values of the parameters relevant to cocurrent packed beds indicates the following: (1) If the conditions at the inlet are bounded, then the solutions VL(z,t) and VG(z,t) and the local holdup h(z,t) and mean holdup h(t) are also bounded. (2) If the inlet function is τ-periodic, then the functions VL(z,t) and VG(z,t) and the local and mean holdups are also τ-periodic for times greater than the characteristic time, τ0. This means that, after passage of the characteristic time, the system “forgets” the initial
∂h dz ) VL(z,t) - φ(t) ∂t
(12)
∫
∂h dz ) -VG(z,t) + ψ(t) (13) ∂t Multiplying eq 12 by λ and eq 13 by -γ and summing the result, we obtain z
0
∂h ) -k[h - F(t)] + m ∂t
∫
z
0
∂h dz ∂t
(14)
where F(t) and m are defined as F(t) ) λφ(t) + γψ(t)
(15)
m ) k(λ - γ)
(16)
7426 Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008
From the initial conditions, it further follows that [h(z,t) - F(t)]|t)0 ) 0
(17)
Stability of the Steady-State Solution of the Model We first investigate the stability of the steady-state solution of the system of eqs 12 and 13. Under the steady-state conditions, the inlet rates of gas are constant, which means ψ(t) ≡ constant, φ(t) ≡ constant, and then F(t) ) λφ(t) + γψ(t) ) F(0) ≡ constant. On designating u(z,t) ) h(z,t) - F(0) (18) one can obtain from eq 14 the following equation defining the stability of the trivial steady-state solution
∫
z ∂u ∂u ) -ku + m 0 dz ∂t ∂t to small perturbations of the initial conditions as
∫ [u(z,t) - m∫ u(x,t) dx] L
2
z
0
(21)
Differentiating the functional A(u) with respect to time gives ∂ A(u) ) 2 ∂t )2
∫ [u(z,t) - m∫ L
z
0
0
∂u u(x,t) dx (z,t) - m ∂t
][
∫
z
0
∂u (x,t) dx dz ∂t
]
L
) -2k
z
0
∫ u (z,t) dz + 2km∫ u(z,t)[∫ u(x,t) dx] dz L
L
2
0
) -2k
L
L
2
2
z
0
0
L
2
z
2
0
L
2
L
2
0
0
0
L
2
-2k + k|m|L < 0 then the trivial solution of eq 19 is stable in the norms L
2
(z,t)
J [√4k (λ - γ)t(z - ς)] dς + ∫e ∫ kF(τ)e J [√4k (λ - γ)z(t - τ)] dτ}(33) z -k(λ-γ)ς
kτ
2
0
The first two terms in the braces in the above solution represent the transition of the hydrodynamics of the trickle bed from the profile of the holdup initially prevailing in the bed (constant F0). Because of the presence of the factor e-kt, however, the initial profile is rapidly “forgotten” by the system after passage of the characteristic time of the bed, τ0 ) (λ - γ)L. The asymptotic behavior of the system for large times is controlled by the third term in the braces in eq 33. Thus h(z,t f ∞) ) e-k[t-(λ-γ)z]
{∫ kF(τ)e t
kτ
0
}
J0[√4kmz(t - τ)] dτ
(34)
[∫
√4kmzt
(
F t-
)
]
2 λ2 J (λ) d(e-λ /4mz) 4kmz 0
(36) 19
and using the integral formula (Tikhonov et al., ∞
(23)
0
Jn(λF)e-θλ λn+1 dλ ) 2
p 669)
1 F n -F2/4θ e 2θ 2θ
( )
(37)
and Euler’s formula eiξ ) cos ξ + i sin ξ On now defining an angle R as
dz on 0 e z e L
From eq 23, it is seen that, for real cocurrent trickle beds where m > 0, the regime is stable to small periodic perturbations if k(λ - γ)L < 2
2
0
0
t
0
from eq 22, it follows that, if
0
F0k(λ - γ)
∫
0
√∫ u
{
h(z,t) ) e-k[t-(λ-γ)z] F0J0[√4k2(λ - γ)zt] -
(22)
[∫ u(z,t) dz] e L∫ u (x,t) dx 2
which states that the bed initially has the same holdup throughout its length. Derivation of the solution of eq 25 is given in the Appendix. Substituting for m in eq 32 in the Appendix, we finally obtain the solution of the profiles of liquid holdup in the bed for general inlet conditions in terms of the parameters k, λ, and γ in the form
h(z,t f ∞) ) -emz
Because, from the Cauchi-Bunykovskii inequality, we have L
h(z,0) ) F0 ) constant(25)
0
∫ u (z,t) dz + km[∫ u(x,t) dx] L
with
F(t) ) λφ(t) + γψ(t) ) c + A sin(ωt) (35) The form of the asymptote in eq 34 specific for the sinusoidal inlet disturbance, eq 35, can be obtained by applying the following transformation of variables
dz
∫ u (z,t) dz + km[∫ u(x,t) dx] | 0
) -2k
0
∫ u (z,t) dz + km∫ dzd [∫ u(x,t) dx] 0
) -2k
z
0
z
0
and we now examine its properties for a specific form of the function F(t). Let us assume that the hydrodynamics of the bed is perturbed at the inlet by periodic variation of the inlet liquid or gas velocity, or both, of the following type
∫ [u(z,t) - m∫ u(x,t) dx][-ku(z,t)] dz 0
∫ h(x,t) dx] ) -kh + kF(t)
0
dz
0
[
(19)
u(z,0) ≡ constant (20) Solutions of eqs 19 and 20 clearly exist for all t g 0 for an arbitrary perturbation. On this solution, we now construct a Lyapunov’s functional of the form A(u) )
∂ h-m ∂t
(24)
Solution for General Inlet Conditions Using the Riemann Function When the conditions at the inlet vary in time, the above function F(t) ) λφ(t) + γψ(t), eq 15, is no longer a constant. The problem in eqs 14 and 17 can then be recast in the form of an integral-differential equation
cos R )
k
√ω
2
and sin R )
+k
2
(38) ω
√ω
2
(39)
+ k2
we finally obtain the asymptotic behavior of the liquid holdup in the bed responding to an external sinusoidal perturbation in the form of eq 35 as
[
h(z,t f ∞) ) c + A cos R exp[k(λ - γ)z sin2 R] sin ωt + k(λ - γ)z sin(2R) - R (40) 2 For z ) 0, we obtain from eq 40 a periodic behavior of the
]
Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7427
Figure 1. Angle of lag and amplification factor of the amplitude of the flow disturbance as functions of the dimensionless parameter k(λ - γ)L.
liquid holdup at the inlet end with the same order of magnitude of the error in the form h(0,t f ∞) ) c + A cos R sin(ωt - R) indicating that the angle R determines the phase shift of the holdup at the inlet relative to the inlet velocity perturbation. From eq 40, one can see that the amplitude of the perturbation, A, at some distance z from the inlet will change according to to A cos R exp[k(λ - γ)z sin2 R], which reaches its maximum when 1 for k(λ - γ)z > 0.5 (41) √2k(λ - γ)z Through the presence of the angle R, the amplitude depends also on the frequency of the sinusoidal perturbation. By substituting for the angle R, we can obtain an expression for the frequency that maximizes the amplification of the amplitude cos R )
ωmax ) k√2k(λ - γ)z - 1
Figure 2. Experimental apparatus: (1) column, (2) tank, (3) gear pumps, (4) solenoid valves, (5) rotameters, (6) gas flow meter, (7) strain gauge sensor, (8) pressure transducers, (9) gas outlet, (10) data acquisition system.
(42)
It is seen that, although the angle of lag R depends solely on the dimensionless criterion k(λ - γ)z, the frequency of the perturbation causing maximum amplification of its amplitude depends on k(λ - γ)z as well as k alone. Figure 1 shows a plot of the lag angle R and the coefficient of gain of the amplitude as functions of the criterion k(λ - γ)L. The figure shows that the amplification of the amplitude of perturbation begins for k(λ - γ)L ) 0.5 and then increases progressively. Experimental Section The aim of the experimental part of the work was to obtain data enabling us to evaluate the parameters λ, γ, and k and to determine the limits of the transition of the regime to natural pulsing flow in the VL-VG domain. These data were needed for testing of the results of the above analysis and its ability to predict the inception of natural pulsing flow. For this purpose, the experiments were designed to obtain the values of liquid holdup and pressure drop and the kinetics of liquid holdup formation in a cocurrent packed-bed column. A sketch of the experimental apparatus is shown in Figure 2. The principal part (1) of the apparatus was a glass laboratory column 0.05 m in inner diameter packed to a height of 1 m with 3-mm glass spheres. Air was used as the gas, and toluene was used as the liquid. Toluene was selected for its properties similar to petrochemicals often processed in trickle-bed reactors. The column was equipped with sensitive membrane pressure
Figure 3. Comparison of experimental and correlated values of liquid holdup.
transducers (8) (Omega Engineering, Inc., Stamford, CT) located 0.2 m apart along the packed section measuring the pressure difference from the atmosphere. Toluene was introduced by two gear pumps (3) from tank (2), via flow meters (5) and liquid distributor into the column. Air flow was controlled by a mass flow meter (6) (Bronkhorst, Vordingborg, Denmark). Only one gear pump was operating under steady-state experiments. The gas contaminated with toluene vapors leaving the column (9) was fed to plant off gas pipe. The whole column, including the liquid distributor, was suspended on a strain gauge sensor (7; Single Point - D5075, Eilersen Electric, Kokkedal, Denmark), measuring the weight
7428 Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008
h ) 0.5069VL0.3089VG-0.1292
Figure 4. Comparison of experimental and correlated values of the kinetic constant, k, of liquid holdup formation.
of the column under operating conditions. The total weight of the setup was about 18 kg, which is just below the 20-kg allowable maximum load of the strain gauge. The total liquid holdup in the column was on the order of 0.5 kg, which compares favorably with the 1-g sensitivity of the load cell. The total liquid holdup in the column was evaluated by subtracting the weight of the dry column from the column weight under operating conditions. To obtain the values of the kinetic constant of liquid holdup formation, k, the following experiments were designed: The column was first brought to the steady state under preselected inlet velocities of liquid and gas. Subsequently, the inlet velocity of the liquid was increased in a stepwise manner by about 10% by opening a magnetic valve in the liquid supply of the second gear pump (3) providing the extra liquid. Simultaneously, the weight of the column (liquid holdup) was continuously recorded on a hard disk of a personal computer (PC) at a sampling rate of 100 Hz. Triggering of the magnetic valves (4) of the gear pumps and data logging on the hard disk were both controlled by a PC (10 in Figure 2) using LabVIEW software (National Instruments, Austin, TX). Additional experiments were designed to provide data for evaluation of the onset of natural pulsation. For this purpose, the pressure fluctuations experienced by the column under constant liquid and gas inlet velocities were continuously monitored and recorded on a PC at a sampling rate of 100 Hz. These time series of pressure were always obtained for a selected liquid flow rate and a number of gas flow rates. Then, a new liquid flow rate was selected, and the procedure was repeated until the whole domain of liquid and gas flow rates was covered. All experiments were carried out at a laboratory temperature of 25 °C. The measurements covered the range between 0.002 and 0.018 m/s for the liquid velocity and between 0.014 and 1.0 m/s for the gas superficial velocity. Experimental Data Processing The literature on liquid holdup is extensive. Some authors use the Lockhart-Martinelli parameter to correlate liquid holdup (e.g., Rao et al.20), whereas others have developed models of the hydrodynamics8,16,21-24 using criteria involving characteristics of the bed and properties of the liquid and gas to describe the regime in the bed. The CFD technique was used by Gunjal et al.25 In light of subsequent use of the liquid holdup function for evaluation of the parameters λ and γ, the experimentally obtained liquid holdups were correlated by the following type of the function (superficial velocities in m/s units) employed, for example, by Moreira et al.18 with the result
(43)
The corresponding correlation coefficient was 0.9409, and the quality of the fit is shown in Figure 3. The Moreira et al.18 correlation has exponents for the Reynolds numbers for liquid and gas equal to 0.31 and -0.10, respectively. Similarly, Satterfield et al.26 found the exponent for the superficial velocity of the liquid to equal 1/3. Charpentier and Favier9 list the exponents of the liquid velocity for four types of particles and eight liquids in the range between 0.30 and 0.65 with the exponent of gas velocity being 0. These values compare reasonably well with the results in eq 43. The weighting method employed to measure liquid holdup yields the weight of all of the liquid in the bed under operating conditions and therefore provides values of the total holdup in contrast to the dynamic holdup often presented in the literature. The total holdup is clearly the relevant quantity in eqs 1a and 1b balancing the total mass of liquid and gas in the column. The obtained correlation now yields the needed expression for the parameter (λ - γ) in the form
(
λ-γ)h
)
0.3089 0.1292 + ) 0.5069VL0.3089VG-0.1292 × VL VG 0.3089 0.1292 + (44) VL VG
(
)
The transients of the liquid holdup in the bed exposed to a step change in the inlet liquid velocity were processed using the least-squares routine in Excel software fitting the data to the formula h - h∞ ) (h0 - h∞)e-kt
(45)
where h0 and h∞ designate the initial holdup in the bed prior to the step change in the liquid flow rate and the holdup after a new steady state has been reached. The obtained values of the kinetic constant were correlated with the liquid and gas velocities and yielded the following dependence (superficial velocities in m/s and k in s-1 units) k ) 162.153VL1.1065VG0.4847
(46)
with the correlation coefficient 0.8040. The quality of the fit is shown in Figure 4. It is acknowledged that the formula in eq 45 employed to evaluate the kinetic constant, k, does not yield values exactly corresponding to its meaning in the above presented model, eq 10. Analysis, however, showed that the differences are acceptable. The obtained time series of n experimental pressures, x, was processed to obtain the standard deviation of pressure, σ, from the formula
σ)
∑
(
∑x xn
)
2
(47) n Plots of the standard deviation as a function of the velocity of gas, for a given velocity of liquid, exhibit a characteristic S-shaped form whose inflection point corresponds to the transition from the film to natural pulsing flow.15 The obtained standard deviations of pressure for a given liquid velocity were therefore fitted to a four-parameter (a, b, c, d) sigmoid function of gas velocity of the type σ)
a +d 1 + exp(bVG - c)
(48)
The curve then was used to evaluate the inflection point from
Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7429
Figure 5. Evaluation of points of transition to natural pulsing flow from the development of the standard deviation of pressure.
Figure 7. Effect of bed length on the development of pulsing in a sinusoidally perturbed bed for k ) 0.3 s-1 and k(λ - γ) ) 2 m-1.
Figure 6. Comparison of experimental and correlated values of gas velocity leading to natural pulsing flow for a given velocity of liquid.
Figure 8. Effect of the parameter (λ - γ) (s/m) on the development of pulsing in a sinusoidally perturbed bed for k ) 0.3 s-1 and z ) 1 m.
its derivatives with respect to gas velocity. The fitting of the experimental standard deviations is demonstrated in Figure 5. The obtained inflection points, determining the inception of the natural pulsing flow as pairs of VG and VL values in the gas-liquid velocity domain, yielded the following correlation (velocities in m/s units) VG )
0.002972 - 0.21791 VL
(49)
with the correlation coefficient 0.9786. The quality of the fit is shown in Figure 6. Our experimental transition curve agrees fairly well with the experimental results of Boelhouwer et al.13 obtained for the same packing but for the air-water system. Results We now demonstrate the properties of the solution in eq 33 for the case of a cocurrent bed exposed to periodic variations of inlet conditions. The periodic disturbance was taken in the form of a sinusoid as F(t) ) 0.2 + 0.03 sin(ωt) for ω ) 2π/10 and F0 ) 0.2 (50) The effect of the length of the packed section of the column (or the distance from the inlet of the bed) on the character of the pulsations of the liquid holdup is shown in Figure 7 for several values of the axial coordinate (or several depths of the bed), for k ) 0.3 s-1 and k(λ - γ) ) 2 m-1. From this figure, it is seen that the amplification of the amplitude of the inlet disturbance increases dramatically with the depth of the bed or the distance from the top of the packed section. For bed lengths greater than 2 m, the maxima of the peaks of liquid holdup would exceed the physically realistic
values of the holdup (for spheres, about 0.4), and the minima would drop below zero. This situation would in reality probably represent the situation when plungers rich in liquid move down the bed interspersed with the zones nearly free of liquid. The figure also shows that, after the passage of the first pulse, the system is essentially in a pseudosteady state, pulsating with a constant amplitude. This can be explained by referring to eq 33 showing that, after the passage of the characteristic time τ0 ) (λ - γ)z, the terms related to the initial state of the bed are exponentially damped. The figure further indicates that the inception of natural pulsing flow begins from the bottom of the bed, which is in accord with the visual observations on our experimental setup. Figure 8 illustrates the pulsations in a bed exposed to the same sinusoidal disturbation at the inlet, eq 50, for several values of the parameter (λ - γ) in s/m units characterizing the sensitivity of the liquid holdup to the gas and liquid velocities for k ) 0.3 s-1 and z ) 1 m. From the figure, it is seen that the characteristics of the liquid holdup affect the appearance of the natural pulsing flow very strongly. Distinct amplifications of the disturbances and natural pulsations grow stronger with increasing value of (λ - γ), i.e., with increasing magnitude of the effect of the gas and liquid velocities on the holdup. Figure 9 again shows the development of pulsations in a bed induced by a sinusoidal perturbation in the dependence on the parameter k in s-1 units characterizing the kinetics of the holdup formation for constant values of (λ - γ) ) 6 s/m and z ) 1 m. The figure shows that the amplification of the amplitude of the disturbance grows stronger in the region with faster kinetics of the holdup formation, k. The effect of the parameter k, however, is somewhat weaker than that of (λ - γ).
7430 Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008
Figure 9. Effect of the parameter k (s-1) on the development of pulsing in a sinusoidally perturbed bed for (λ - γ) ) 6 s/m and z ) 1 m.
Figure 12. Comparison of the experimentally determined line of transition to natural pulsing flow and the contour of constant k(λ - γ)L ) 2 in the coordinates of the correlation of Charpentier and Favier.9
(
k(λ - γ)L ) 82.195VL1.4154VG0.3555
Figure 10. Effect of frequency ω (s-1) on the development of pulsing in a sinusoidally perturbed bed for k(λ - γ) ) 2 m-1, k ) 0.3 s-1, and z ) 1 m.
Figure 11. Comparison of the experimentally determined line of transition to natural pulsing flow (solid line with circles) with the computed contours of constant value of the dimensionless parameter k(λ - γ)L (solid lines with values shown by labels) in the VL-VG domain.
Finally, Figure 10 shows the effect of the frequency of the periodic disturbance, ω, on the amplitude of the pulsations for k ) 0.3 s-1 and k(λ - γ)z ) 2. The figure shows that, in accord with eq 42, the maximum amplification of the amplitude of pulsations is experienced for the frequency ω ) 2π/12.092 s-1. For frequencies lower than this, the amplification remains strong. Higher-frequency disturbances tend to be dampened. Nevertheless, the role of the frequency is clearly only secondary, as the key parameter is k(λ - γ), which controls the amplification of the disturbance. Figure 11 was constructed in the following manner: First, eq 44 was multiplied by eq 46 and by the length of the packed section L ) 1 m to obtain k(λ - γ)L as a function of phase velocities with the result
)
0.3089 0.1292 + (51) VL VG
This equation was subsequently solved to obtain VL for a set of preselected values of VG and k(λ - γ)L using Excel software. By repeated solution of eq 51, a plot of VG versus VL curves (solid lines) with k(λ - γ)L as a parameter was obtained (Figure 11). Added to the same figure is the curve (solid lines with full circles) computed from eq 49 characterizing the transition to pulsing flow based on our experimental results. Figure 11 shows that, for intermediate and high gas velocities, the experimental curve delimiting the transition to natural pulsing flow nearly coincides with the theoretical curve computed from eq 51 for k(λ - γ)L ) 2. We recall that k(λ - γ)L ) 2 was found in the theoretical analysis in eq 24 to be the limit of stability of the column to small perturbations. Even at low gas velocities, the experimental curve remains to the right of the theoretical curve for k(λ - γ)L ) 1.6 for which the above analysis showed very strong amplification of the amplitude of liquid holdup pulsations caused by flow disturbances. For comparison, the results presented in Figure 11 are plotted once again in Figure 12 in terms of the coordinates of the Charpentier-Favier diagram.9 The trends of the experimental curve delimiting the inception of natural pulsing flow and the contour k(λ - γ)L ) 2 agree with the published correlation,9 but our results are somewhat shifted upward, suggesting later inception of pulsing. Conclusions Functional analysis of the liquid and gas material balances in a cocurrent trickle-bed column together with the adopted firstorder kinetics of liquid holdup formation yielded limits of stability of the hydrodynamic regime to small flow perturbations. At the same time, explicit formulas have been obtained for the amplitude of pulsations appearing in the bed by amplification of small flow disturbances. These amplifications of flow disturbances become strong in region of the VL-VG domain that was identified experimentally as the inception of natural pulsations. The stability of the regime in the bed and the amplification of the disturbances were found to be determined by the characteristics of the liquid holdup function, λ ) ∂h/∂VL and γ ) ∂h/∂VG, and by the kinetic parameter of holdup formation, k. The analysis shows that, for the dimensionless parameter k(λ - γ)L > 2, the regime is unstable to small perturbations
Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7431
whereas, for k(λ - γ)L > 0.5, the amplitude of the disturbance will be strongly amplified in the bed. Theoretical predictions of the appearance of natural pulsations were confirmed by experiments determining the limits of inception of natural pulsations in the VL-VG domain. The experimentally determined limit of transition to natural pulsations approaches the theoretically found limits of stability of the bed at intermediate and high gas rates and passes through the region of strong amplification of the holdup pulsations caused, for instance, by minor perturbations of the inlet flow conditions at lower gas rates. The analysis further shows that longer beds have a greater tendency to natural pulsing under otherwise the same conditions. The agreement between the analysis and the experiments is good despite the fact that the theory assumed a linear dependence of the liquid holdup on the gas and liquid velocities that, in reality, can be justified only locally in the vicinity of a point in the gas-liquid velocity domain. Acknowledgment The authors gratefully acknowledge financial support of the project by the Ministry of Industry and Trade of the Czech Republic under Grant FT-TA039. Symbols F0 ) parameter defined in eq 25 GG, GL ) superficial mass velocity of gas and liquid, kg/(m2 s) h, h(VG,VL), h(z,t) ) liquid holdup h(t) ) mean liquid holdup in the bed k ) time constant of rate of holdup formation, s-1 L ) depth of packed section, m m ) parameter defined in eq 16, m-1 p(z,t) ) function defined in eq 28 t ) time coordinate, s u(z,t) ) function defined in eq 18 VG ) superficial velocity of the gas, m/s VL ) superficial velocity of the liquid, m/s z ) axial coordinate, m R ) angle defined in eq 39, rad γ, λ ) parameters of holdup function, eq 1c, s/m Λ ) [(Fgas/Fwater)(Fliquid/Fair)]0.5 ) Charpentier-Favier9 flow parameter µ ) viscosity [kg/m s] φ(t), ψ(t) ) functions defining inlet conditions, eq 2, m/s Ψ ) (σwater/σliquid)[(µliquid/µwater)(Fwater/Fliquid)2]0.33 ) CharpentierFavier9 flow parameter F ) density, kg/m3 σ ) surface tension, N/m τ ) time parameter, s τ0 ) (λ - γ)L ) characteristic time of the bed, s ω ) circular frequency, rad/s
Appendix For z ) 0, a solution of eq 25 gives h(0,t) ) e-kt F0 +
[
∫ kF(τ)e t
kτ
0
]
dτ
(26)
Differentiation of eq 25 with respect to z yields ∂2h ∂h ∂h ) -k +m ∂z ∂ t ∂z ∂t
(27)
with h(0,t) ) e-kt[F0 + ∫0t kF(τ)ekτ dτ] and h(z,0) ) F0. We now define the new function p(z,t) related to h(z,t) by
(28) h(z,t) ) p(z,t)e-kt+mz Substituting this expression into eq 27 gives the new problem ∂2p(z,t) ) -kmp(z,t) ∂z ∂ t with the boundary and initial conditions
[
p(0,t) ) F0 +
∫ kF(τ)e t
]
kτ
dτ
0
(29)
and p(z,0) ) F0e-mz
Equation 29 is the so-called “telegraph” equation (Tikhonov et al.,19 p 735). Next, we construct for our problem the Riemann function, R(z,t,ς,τ), (Tikhonov et al.,19 p 132), which, in the case of the telegraph equation, has the form R(z,t,ς,τ) ) J0[√4km(z - ς)(t - τ)]
(30)
where J0(ξ) is the (zero-order, first-kind) Bessel function satisfying the condition J0(0) ) 1. It is now easy to prove that, if p(z,t) is the solution of the problem in eq 29, then h(z,t) ) p(z,t)e-kt+mz is also a solution of eq 25. Now, it suffices to find the solution of the problem in eq 29 in an explicit form. Using the Riemann function in the integral representation, the solution of eq 29 gives
∫
∂p(ς,0) R(z,t,ς,0) dς ∂ς t ∂R(z,t,0,τ) dτ p(0,τ) 0 ∂τ z ∂p(ς,0) ) F0R(z,t,0,0) + 0 R(z,t,ς,0) dς + ∂ς
p(z,t) ) p(0,t) +
z
0
∫
∫
∫ kF(τ)e R(z,t,0,τ) dτ ∂p(ς,0) [ J √4kmt(z - ς)] dς + ) F J (√4kmzt) + ∫ ∂ς ∫ kF(τ)e J [√4kmz(t - τ)] dτ(31) t
kτ
0
z
0 0
0
0
t
kτ
0
0
Finally, we obtain the solution in terms of the parameters k and m in the form
{
h(z,t) ) e-kt+mz F0J0(√4kmzt) - mF0 J0(√4kmt(z - ς) ) dς+
∫e
z -mς
0
∫ kF(τ)e t
0
×
}
J0(√4kmz(t - τ) ) dτ (32)
kτ
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ReceiVed for reView March 31, 2008 ReVised manuscript receiVed June 30, 2008 Accepted July 15, 2008 IE800501Q
