ARTICLE pubs.acs.org/IECR
Minimum Energy Dissipation under Cocurrent Flow in Packed Beds 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, Czech Republic ABSTRACT: Functional analysis of mass balances on gas and liquid, and pressure drop equations describing the cocurrent flow in packed beds has yielded two criteria. Positive values of these criteria has been shown sufficient to achieve energy savings by synchronized periodic pulsations of inlet gas and liquid velocities or by liquid velocity only compared to the situation under the same mean steady inlet phase velocities. This, however, is not the case of pulsation by gas inlet velocity alone. The developed criteria have been evaluated from data obtained by experimental measurement of liquid holdup and gas pressure drop in a column 0.05 m in diameter packed to 1 m by 0.003 m glass spheres using water/air and toluene/air system. Our experiments showed these criteria to be fulfilled in the whole domain of gas and liquid velocities tested. Evaluation of the energy savings under pulsations relative to the steady state operation showed that the savings for water/air system may reach up to 4% with the maximum in region of high and intermediate liquid velocities and low gas velocities. For the toluene/air system the savings are somewhat lower and occur in the same velocity domain.
’ INTRODUCTION Functional analysis may become a powerful tool for the analysis of the model equations. In our previous papers we have used it to analyze the equations describing the countercurrent flow in a packed bed (Jiricny et.al.,1 Stanek et.al.,2 Akramov et. al.,3,5,6 Svoboda et.al.4). In the first paper (Jiricny et.al.1) we have observed and described for the first time the phenomenon of holdup and pressure drop overshoot following a sudden change of fluid velocity. In the three following papers (Stanek et.al.,2 Akramov et.al.,3 Svoboda et.al.4) the conditions leading to overshoots have been analyzed and the final paper (Akramov et.al.5) related these conditions to the properties of the liquid holdup as a function of fluid phase velocities using functional analysis. In a more recent paper (Akramov et.al.6) we have turned our attention to cocurrent flow beds and using again functional analysis have been able to formulate a criterion determining the conditions leading to natural pulsing flow. This criterion was expressed in terms of the properties of liquid holdup as a function of fluid phase velocities and the kinetic constant of the rate of liquid holdup formation. Literature on pulsing flow in fixed beds is extensive.722 Individual papers have been concerned with the transition from trickle to pulsing flow regime,713 induced pulsing,14,15 transient behavior of trickle bed reactors11,18,20,22 and the characteristics of the pulsing flow under various conditions.10,14,19,21 In the present paper we intend to analyze by functional analysis the response of a cocurrent flow column to periodic pulsations at the inlets. This analysis should provide useful insight into the transient behavior of the column/reactor when inlet velocity changes bring about changes of liquid holdup that further interact with pressure losses of the flowing gas. Second, the generally nonlinear dependence of the rate of energy dissipation on gas velocity still can be addressed by the functional analysis even though the analysis employs linearized formulas expressing the dependence of liquid holdup and gas pressure drop on the velocities of phases. This should allow us to find out if periodic pulsations of inlet phase velocities could lead to lower rate of gas energy dissipation in the column. Eventual r 2011 American Chemical Society
lower energy dissipation under inlet pulsations may be possible due to the nonlinearity of the expression for the rate of energy dissipation.
’ THEORY Assuming constant void fraction of the bed and constant densities (incompressibility) of both liquid and gas a transient one-dimensional model of the cocurrent flow of gas and liquid trough a packed bed can be formulated in the following form. ∂h ∂vL ¼0 ∂t ∂z
ð1Þ
∂h ∂vG þ ¼0 ∂t ∂z
ð2Þ
For the purpose of our functional analysis we express the holdup and pressure gradient as linear functions of liquid and gas velocities that may be viewed as truncated Taylor expansions of the more complex nonlinear functions. h ¼ λvL þ γvG þ h0
∂P ¼ KL v L þ KG v G þ K0 ∂z
ð3aÞ ð3bÞ
Let us consider the situation that the bed is exposed to periodic perturbations of inlet velocities that pose the boundary conditions for our problem v0L ðtÞ ¼ vL0 ð1 þ ε1 pðtÞÞ, jε1 j e 1
ð4aÞ
v0G ðtÞ ¼ vG0 ð1 þ ε2 qðtÞÞ, jε2 j e 1
ð4bÞ
Received: January 3, 2011 Accepted: August 19, 2011 Revised: June 20, 2011 Published: August 19, 2011 10824
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where p(t) and q(t) are time-periodic functions with the period τ such that their means over the period τ equal zero. These definitions ensure that the means of the superficial velocities v0L(t), v0G(t) over the period τ = 2π/ω equal vL0 and vG0 respectively. We note that vL0 and vG0 are positive numbers as they represent liquid and gas superficial velocities. As an example we may take pðtÞ ¼ qðtÞ ¼ sinðωtÞ
J2 ¼
ð5Þ J11
1 v2L0 ¼ Lτ ðλ γÞ2
ð6aÞ
0
0
ðKL vL vG þ KG v2G þ K0 vG Þdzdt
J12 ¼
where J0 ¼
1 Lτ
Z τZ 0
L 0
ðKL vL0 vG0 þ KG v2G0 þ K0 vG0 Þdzdt
1 vL0 Lτ ðλ γÞ
Z τZ 0
L
Z τZ 0
L
½KL ðqðξÞ qðtÞÞγðλqðtÞ γqðξÞÞ
0
1 vL0 vG0 Lτ ðλ γÞ2
Z τZ 0
L
½KL fðλpðξÞ γpðtÞÞðλqðtÞ γqðξÞÞ
0
þλγðqðξÞ qðtÞÞðpðtÞ pðξÞÞg þ 2KG fλðpðtÞ pðξÞÞðλqðtÞ γqðξÞÞgdzdt
ð7Þ
For the simplicity of subsequent analysis we will assume that p(t) = q(t) = sin(ωt) and evaluate the integrals in eqs 11. For the sake of brevity of notation let us define the following parameters
ð9Þ
ð10Þ
k ¼ KL =KG , k0 ¼ K0 =KG , n ¼ vL0 =vG0 , l ¼ λ=γ
ð12Þ
Based on their physical meaning the first three of the above parameters are positive. However, liquid holdup in cocurrent beds decreases with increasing gas velocity and, therefore, γ is expected to be negative. Consequently, also the parameter l is negative. With the newly defined parameters and evaluated integrals in eqs 11, we can now rewrite expressions for J’s to the following form where η= ω (λγ)L Z Z 1 τ L ðKL vL0 vG0 þ KG v2G0 þ K0 vG0 Þdzdt J0 ¼ Lτ 0 0 ¼ KG v2G0 ðkn þ 1 þ k0 n=vG0 Þ ¼ KG v2G0 I0 where I0 =(kn+1+k0n/vG0)>0. Further J1 = J2 = 0 as follows from the above evaluated integrals. Next we get KG v2G0 sin η J11 ¼ 1 lð2l kl kÞ n2 ð13Þ η 2ðl 1Þ2 KG v2G0 sin η sin η þ 1 1 k l η η 2ðl 1Þ2 sin η þ ðl 1Þ2 þ 2l 1 η KG v2G0 sin η 2 ¼ ðl 1Þ þ ð2l lk kÞ 1 η 2ðl 1Þ2 ð14Þ
¼ KL vL0 vG0 þ KG v2G0 þ K0 vG0 J1 ¼
½KL ðλpðξÞ γpðtÞÞλðpðtÞ pðξÞÞ
0
ð6bÞ
The result of substitution for vL and vG from eqs 6a,6b and from eqs 4a,4b into eq 9 for E may be formally written as E̅ ¼ J0 þ J1 ε1 þ J2 ε2 þ J11 ε21 þ J12 ε1 ε2 þ J22 ε22
0
L
þ KG ðλqðtÞ γqðξÞÞ2 dzdt
Because p(t) and q(t) are τ-periodic functions and the parameters λ, γ, h0, KL, KG, K0, vL0 and vG0 are taken to be constants, the local liquid holdup, h(z,t), and pressure, P(z,t), are also τ-periodic functions. The mean liquid holdup, h, and the mean energy of gas, E, dissipated in a bed of length L per unit volume of the bed and time averaged over the period τ are defined as Z Z 1 τ L ðλvL þ γvG þ h0 Þdzdt ð8Þ h̅ ¼ Lτ 0 0 L
1 v2G0 Lτ ðλ γÞ2
J22 ¼
where ξ, having the dimension of time, designates
Z τZ
Z τZ
þ KG ðλpðtÞ λpðξÞÞ2 dzdt
ε1 vL0 ðλðpðtÞ pðξÞÞÞ þ ε2 vG0 ðλqðtÞ γqðξÞÞ þ λγ
for 0 e z e L
½KL vL0 ðλqðtÞ γqðξÞÞ
0
ð11Þ
ε1 vL0 ðλpðξÞ γpðtÞÞ þ ε2 vG0 γðqðξÞ qðtÞÞ λγ
ξ ¼ t ðλ γÞz
L
þ2KG vG0 ðλqðtÞ γqðξÞÞ þ K0 ðλqðtÞ γqðξÞÞdzdt
ðλðv0L ðtÞ v0L ðξÞÞÞ þ ðλv0G ðtÞ γv0G ðξÞÞ ¼ vG ¼ vG ðz, tÞ ¼ λγ
1 E̅ ¼ Lτ
0
λv0L ðξÞ γv0L ðtÞ þ γðv0G ðξÞ v0G ðtÞÞ ¼ λγ
vL ¼ vL ðz, tÞ ¼
¼ vG0
Z τZ
þ KL vG0 ðγqðξÞ γqðtÞÞ
A solution of the transients of liquid and gas velocities (Akramov et.al.6) is following
¼ vL0 þ
1 vG0 Lτ ðλ γÞ
J22 ¼
½KL vL0 ðλpðtÞ λpðξÞÞ
0
þ KL vG0 ðλpðξÞ γpðtÞÞ þ2KG vG0 ðλpðtÞ λpðξÞÞ þ K0 ðλpðtÞ λpðξÞÞdzdt 10825
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KG v2G0 sin η sin η 2 sin η þ 2l þ l k 4l þ η η η 2ðl 1Þ2 sin η n¼ þ 2 l2 þ l ðl2 þ lÞ η KG v2G0 sin η 2 ¼ ð1 þ lÞ k 4l þ η 2ðl 1Þ2 sin η þ 2ðl2 þ lÞ 1 n ð15Þ η
J12 ¼
From here on we will continue working in terms of the dimensionless I’s defined as follows
This form shows us an important conclusion that λ1 e λ2 since the eigenvalues λ1,λ2 are the roots of the characteristic equation 1 λ21, 2 ðI11 þ I22 Þλ1, 2 þ I11 I22 ðI12 Þ2 ¼ 0 4
whose discriminant d2 = (I11 I22)2 + I212 is positive. The mathematics shows us that the quadratic form [Iε,ε] may be expressed as ½Iε, ε ¼ λ1 η21 þ λ2 η22 1 η1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðε1 θε2 Þ 1 þ θ2
ð16aÞ I22
¼ ðl 1Þ2 þ A11 I12
1 η2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðε1 θ þ ε2 Þ 1 þ θ2
2ðl 1Þ sin η 2 ¼ J22 ¼ ðl 1Þ þ ð2l lk kÞ 1 η KG v2G0 2
where
A11 ¼
1
sin η ð2l kl kÞ η
ð16cÞ
ð17Þ
Now we can investigate the possibility of reducing the mean rate of dissipation of the energy of gas in the form defined in the following equation E̅ ¼ KG v2G0 ðI0 þ I1 ε1 þ I2 ε2 ¼ 1 2 2 ðI ε þ I ε ε þ I ε Þ þ 11 1 12 1 2 22 2 2ðl 1Þ2 ! 1 ¼ KG v2G0 I0 þ ½Iε, ε 2ðl 1Þ2
ð18Þ
where [Iε,ε] designates the scalar product of the vector ε defined as ! ε1 ε¼ ε2 and the vector Iε, being the product of the vector ε with the symmetric matrix I given by 0 1 1 I I B 11 2 12 C C ð19Þ I ¼B @1 A I12 I22 2 Let λ1,λ2 designate the eigenvalues of the matrix I which are real numbers given in the following form λ1 ¼
I11 þ I22 d , 2
θ¼
ð16bÞ
2ðl 1Þ2 ¼ J12 ¼ ½nðkðl 1Þ2 þ A11 ðl þ 1ÞÞ KG v2G0
λ2 ¼
I11 þ I22 þ d 2
ð20Þ
ð22Þ
where
2ðl 1Þ sin η ð2l kl kÞln2 ¼ A11 ln2 ¼ J11 ¼ 1 2 η KG vG0 2
I11
ð21Þ
ðI11 I22 Þ þ d , I12
ð23Þ
I12 6¼ 0
Let us now explore various situations affecting the mean rate of energy dissipation ( ) 1 2 E̅ ¼ KG vG0 I0 þ ½Iε, ε 2ðl 1Þ2 ( ) 1 ¼ KG v2G0 I0 þ ðλ1 η21 þ λ2 η22 Þ ð24Þ 2ðl 1Þ2 where we can distinguish three cases: Case 1. Both eigenvalues λ1, λ2 defined in eq 20 are nonnegative (positive). In this case there is no way of decreasing E by inlet pulsations and E g E0, where E0 = KGv2G0I0, for any values from the range defined in eqs 4a and 4b. The energy losses under pulsating input velocities will therefore be always higher compared to those under the equivalent mean steady inlet rates. Case 2. The first eigenvalue is negative, λ1 < 0, while the second eigenvalue is nonnegative, λ2 g 0. In this case we should choose such values of ε1,ε2 that make the number η2 = (ε1θ + ε2)/(1 + θ2)1/2 vanish. This will take place when ε2 = ε1θ and then (
E̅ ¼
KG v2G0
)
1 I0 þ λ1 ð1 þ θ2 Þε21 2ðl 1Þ2
< E̅ 0 , for ε1 6¼ 0:
It is clear that the minimum of the mean rate of energy dissipation is determined by the properties of the system, namely by ( ) 1 2 2 E̅ min ¼ KG vG0 I0 þ λ1 ð1 þ θ Þ < E̅ 0 ð25Þ 2ðl 1Þ2 provided that the system satisfies the following inequality jθj e 1
ð26Þ
Then from eqs 25 and 26 it follows that the limiting minimum rate of energy dissipation per unit volume of bed for the extreme 10826
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ð27Þ
Case 3. Both eigenvalues are negative while λ1 < λ2 < 0. We will not examine this case because for real cocurrent flow systems where l = λ/γ < 0 this situation cannot occur. The General Case of Pulsations Both at Gas and Liquid Inlet. Let us now investigate the magnitude of energy dissipation under periodic pulsations both at the gas and liquid inlets while the mean liquid holdup averaged over the length of the bed and over the period of pulsations remains constant. Considering eq 23 indicating that generally η21 + η22 = ε21 + 2 ε2 e 2 we can evaluate the mean liquid holdup from the integral in eq 8 as Z Z 1 τ L ðλvL þ γvG þ h0 Þdzdt h̅ ¼ Lτ 0 0 ¼ γvG0 ðnl þ 1 þ h0 =ðvG0 γÞÞ ¼ const
ð28Þ
The constancy of the mean holdup h dictates that nl = (vL0)/ (vG0)(λ)/(γ) is constant too. Therefore we have that ( ) 1 2 2 λ1 ð1 þ θ Þ < E̅ min < E̅ 0 ð29Þ KG vG0 I0 þ ðl 1Þ2 which is now generally valid for all θ, not only for |θ| e 1. Consequently, we can put 2 in the denominator. In the general case of synchronized (p(t) = q(t) = sin(ωt) pulsations at both inlets it is true that for negative λ1 we can write ( ) 1 KG v2G0 I0 þ λ1 ð1 þ θ2 Þ < KG v2G0 ðl 1Þ2 ( 1 λ1 ð1 þ θ2 Þ 2ðl 1Þ2
Analysis shows that the sign of I11 can be positive as well as negative. When I11 < 0 the mean dissipation of energy, E, can be decreased. Pulsations of Inlet Gas Velocity Only. If now only the inlet gas stream is subject to periodic pulsations, i.e. when ε1 = 0, then we get ( ) 1 2 2 2 E̅ ¼ J0 þ J2 ε2 þ J22 ε2 ¼ KG vG0 I0 þ I22 ε2 2ðl 1Þ2 Analysis shows that the sign of I22 can be positive as well as negative. When I22 < 0 the mean dissipation of energy, E, can be decreased. Inspection of the expressions for I11 and I22 in eqs 16a and 16b gives us that if I11 < 0 (meaning that A11 > 0) then it is possible to decrease E by pulsations of liquid velocity only and impossible to decrease E by pulsations of gas only. If I11 > 0 (that is A11 < 0) then, in contrast, it is not possible to decrease E by pulsations only of liquid and but possible to decrease E by pulsations only of gas velocity. Sufficient Conditions. Above we have seen that specific synchronized periodic pulsations at both inlets (p(t) = q(t) = sin(ωt), maintaining the same mean liquid holdup, can cause lower energy losses only for case 2, that is, when λ1 < 0 and λ2 > 0. In this case the necessary and sufficient condition for lowered energy losses by pulsations is 1 I11 I22 ðI12 Þ2 < 0 4 Now a sufficient condition for this is
) I0 þ
Pulsations of Inlet Liquid Velocity Only. If only the inlet liquid stream is subject to periodic pulsations, that is, when ε2 = 0, then we get ( ) 1 2 2 2 I11 ε1 E̅ ¼ J0 þ J1 ε1 þ J11 ε1 ¼ KG vG0 I0 þ 2ðl 1Þ2
I11 I22 ¼ A11 ln2 ððl 1Þ2 þ A11 Þ < 0
¼ E̅ min < E̅ 0
The algorithm for selecting the optimum pulsations around constant velocities vG0, vL0 with the amplitudes |ε2| e 1,|ε1| e 1 that lead to minimum energy losses may be as follows Step 1. For the given parameters λ, γ, h0, KL, KG, K0 calculate the elements I11, I22 and I12 of the matrix I from eqs 16a, 16b, and 16c. Step 2. Calculate the discriminant d2 and the eigenvalues λ1,λ2 from eqs 20 and 21. Step 3. Compare the signs of the eigenvalues λ1, λ2. Then: If the eigenvalues λ1,λ2 are positive no pulsations of the inlet gas, liquid or both velocities can reduce the energy losses of gas. Else if the eigenvalues are such that λ1 < 0, λ2 > 0 then simultaneous pulsations of liquid inlet velocity with the amplitude ε1 and gas with the amplitude ε2=ε1θ while |θ| e 1 will minimize the mean rate of energy dissipation of gas as may be seen from formula in eq 25. Further if |θ| > 1 then |ε1| = (2/(1 + θ2))1/2 and ε2 = ε1θ then the mean rate of energy dissipation can also be reduced as may be seen again from eq 25. Analysis of the elements I11, I22, I12 of the matrix in eq 19 has shown that all these cases can occur.
Since l in real systems is negative the more stringent sufficient condition is A11 > 0 which gives sin η 0< 1 ð2l kl kÞ η Since the expression in the first parentheses on the RHS is always positive we get as the sufficient condition that ð2l kl kÞ > 0 In order to satisfy this inequality we have to satisfy the two following conditions λ > γ KG 1 1γ < þ 2λ KL 2 which give us sufficient conditions for lower gas energy dissipation by specific periodic pulsations of the inlet rates. The above sufficient conditions, if fulfilled, provide that A11 is positive which makes I11 negative and I22 positive. Consequently, the sufficient conditions provide for reduced energy losses in the 10827
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Table 1 parameters for liquid holdup and water/air system; velocities in m/s
’ EXPERIMENTAL SECTION The aim of the experimental part was to obtain data enabling us to evaluate the parameters λ, γ, KG, and KL. These parameters appear in the linearized expressions for liquid holdup and pressure drop. As such they represent derivatives of liquid holdup with respect to liquid velocity (λ) and gas velocity (γ). Analogously, KL and KG represent derivatives of pressure drop with respect to liquid and gas velocity respectively. There were essentially two ways how to evaluate these parameters. One would be to differentiate correlations for pressure drop8,13,17 and liquid holdup8,9,16,17 existing in the literature, or, create and differentiate a purely empirical correlation based on our experimental measurements. The advantage of the former method would be that the correlations would show us the dependence of the sought parameters on bed porosity, particle size, and particle shape factor, as well as fluid properties (density, viscosity). Nevertheless, we chose to use the latter method based on our own experimental data. The reason was that in order to get precise values of the derivatives we need generally extraordinarily accurate functions (correlations). The typical errors of published engineering correlations do not satisfy our needs. By employing our own data we get accurate correlations but the penalty we pay is that the validity of our empirical correlations is strictly limited only to the bed porosity, particle size, and particle shape factor, as well as fluid properties (density, viscosity) prevailing in our experiments. For the purpose of developing our empirical correlations we needed to evaluate liquid holdup and gas pressure drop in a cocurrent flow packed bed column in a wide range of gas and liquid velocities. A sketch of the experimental apparatus used was shown in our earlier paper (Akramov et.al.6). The essential part of the setup was a glass column 0.05 m in inner diameter packed to 1 m height by 0.003 m glass spheres. The gas and liquid phase were flowing concurrently from the top downward. The bed was irrigated in one set of experiments by water and in another set by toluene. Air was used as gas phase in both sets of experiments. The column was equipped with sensitive membrane pressure transducers (Omega Co., U.S.) measuring the pressure drop as a difference from the atmospheric pressure. Liquid (water or toluene) was fed by a gear pump from a tank via rotameters. Air flow was metered by a Bronkhorst (Denmark) mass flow meter. The whole column, including liquid distributor, was suspended on a strain gauge (Eilersen Electric, Denmark) measuring the weight of the column under operating conditions. This enabled us to evaluate liquid holdup under operating conditions. 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 of liquid velocity and between 0.014 and 1.0 m/s gas superficial velocity. ’ EXPERIMENTAL DATA PROCESSING The obtained data on liquid holdup were correlated by an empirical function of the following type h ¼ A þ BvL þ CvG þ Dv2L þ Ev2G þ FvL vG þ Gv2L vG þ HvL v2G
A
B
C
D
E
F
G
H
0.158813 7.31982 0.07573 9.238285 0.00971 16.9908 5.712237 22.21354
Table 2 parameters for liquid holdup and toluene/air system; velocities in m/s A
B
C
D
E
F
G
H
0.134217 7.892201 0.002252 5.99521 0.1217 36.2124 2.17637 47.89611
Figure 1. Correlated versus experimental holdup for water/air system.
The pressure drop data were correlated by an analogous empirical function ΔP ¼ A þ BvL þ CvG þ Dv2L þ Ev2G þ F vL vG þ Gv2L vG þ H vL v2G Parameters of the above correlations were evaluated using the EXCEL 2010 software with the following results for liquid holdup and water/air system given in Table 1, for toluene/air system in Table 2. The best straight line fitted to the computed versus experimental holdup points yielded the following result: hcomputed = 0.9741hexperimental + 0.0046 with the correlation coefficient R = 0.9869. Fitting the same data but forcing zero holdup at zero phase velocities yielded: hcomputed = 0.9989hexperimental with R = 0.9865. The fit by the empirical formula is thus very good as may be also seen from Figure 1. The best straight lines fitted to the computed versus experimental holdup points for toluene/air system yielded the following result: hcomputed = 0.9890hexperimental + 0.0017 with the correlation coefficient R = 0.9942. Fitting the same data but forcing zero holdup at zero velocities yielded: hcomputed = 0.9995hexperimental with R = 0.9942. The fit by the empirical formula is thus very good as may be also seen from Figure 2. The best straight lines fitted to the computed versus experimental pressure drop points for the water/air system yields the following result: ΔPcomputed = 0.9859ΔPexperimental + 0.2035 with the correlation coefficient R = 0.9930. Fitting the same data but 10828
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The best straight lines fitted to the computed versus experimental pressure drop yielded for the toluene/air system the following result: ΔPcomputed = 0.9970ΔPexperimental + 0.0.0309 with the correlation coefficient R = 0.9985. Fitting the same data but forcing zero pressure drop at zero velocities yielded: ΔPcomputed = 0.9993ΔPexperimental with R = 0.9985. The fit by the empirical formula is thus very good as may be also seen from Figure 4. Also see Tables 3 and 4. The obtained coefficients were further processed to yield the holdup parameters from λ ¼ B þ 2DvL þ FvG þ 2GvL vG þ Hv2G γ ¼ C þ 2EvG þ FvL þ Gv2L þ 2HvL vG Figure 2. Correlated versus experimental holdup for toluene/air system.
and the pressure drop parameters analogously from KL ¼ B þ 2DvL þ F vG þ 2GvL vG þ H v2G KG ¼ C þ 2EvG þ F vL þ Gv2L þ 2H vL vG
Figure 3. Correlated versus experimental pressure drop for water/air system.
’ RESULTS From the experimentally obtained liquid holdup and gas pressure drop parameters we have evaluated 0.5 + 0.5γ/ λKG/KL and λ+γ criteria whose positive values, as derived in the theoretical part, provide energy savings under periodic inlet flow rate of liquid or synchronized inlet rates of gas and liquid compared to the steady state flow at the same mean velocities. These criteria are plotted in the four following Figures 5,8 as contours of constant value of the given criterion in the gas velocity versus liquid velocity domain for the water/air and toluene/air system. From these figures it is seen that both 0.5 + 0.5γ/λ KG/KL and λ + γ criteria remain positive in the whole studied domain of phase velocities for both liquid/air systems. Thus in the whole vL vG domain there exist synchronized periodic pulsations of inlet flow rates of gas and liquid, or pulsations of liquid alone that would result in lower gas energy dissipation. The following formula expresses the minimum rate of energy dissipation averaged over the period of pulsation expressed as a fraction of the rate of energy dissipation under steady state inlet velocity equal the mean velocity of pulsation. sin η λ KL λ KL λ vL0 2 1 2 E̅ η γ KG γ KG γ vG0 ! ¼1 þ 2 E̅ 0 λ KL vL0 K0 vL0 2 1 þ 1 þ γ KG vG0 KG ðvG0 Þ2
Figure 4. Correlated versus experimental pressure drop for toluene/air system.
forcing zero pressure drop at zero velocities yields: ΔPcomputed = 0.9963ΔPexperimental with R = 0.9930. The fit by the empirical formula is thus very good as may be also seen from Figure 3.
Computed values of relative energy dissipation under pulsation by liquid were plotted in Figure 9 for the water/air system. The empty white surface covers the region where the dissipation is reduced by less than 1%. Lower rates of relative energy dissipation occur in Figure 9 at high liquid velocities and low gas velocities. In the investigated domain of fluid velocities the maximum reduction can reach 3%. Similar situation under pulsations by liquid is seen in Figure 10 evaluated from the data for toluene/air system when the maximum savings amount to only 2%. 10829
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Table 3 parameters for gas pressure drop and water/air system; velocities in m/s, pressure in kPa A*
B*
C*
D*
E*
F*
G*
H*
4.41734
985.4558
29.8651
16.4116
3.36342
1725.431
13.395
986.173
Table 4 parameters for gas pressure drop and toluene/air system; velocities in m/s, pressure in kPa A*
B*
C*
D*
E*
F*
G*
H*
2.0098
256.7599
14.25876
591.4553
17.55137
3741.716
233.6228
4185.26
Figure 5. Criterion 0.5 + 0.5γ/λ KG/KL in the vL vG domain for water/air system.
Figure 7. Criterion 0.5 + 0.5γ/λ KG/KL in the vL vG domain for toluene/air system.
Figure 6. Criterion λ+γ in the vL vG domain for water/air system. Figure 8. Criterion λ + γ in the vL vG domain for toluene/air system.
The expression for the minimum relative rate of energy dissipation under synchronized simultaneous pulsations by liquid and gas is following. E̅ λ1 ð1 þ θ2 Þ ! ¼1 þ 2 E̅ 0 λ KL vL0 K0 vL0 þ 1 þ 2 1 γ KG vG0 KG ðvG0 Þ2
The relative values of energy savings were computed for water/ air and toluene/air systems and the results are plotted in Figures 11 and 12. From these figures it is seen that for both fluids maximum energy savings occur at high liquid and low gas velocities while maximum energy savings are only slightly higher than those corresponding to pulsations by liquid alone. The last four figures confirm that if the two criteria for lower energy dissipation formulated above are met the energy 10830
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Figure 9. Contours of constant relative rate of energy dissipation under pulsations of liquid inlet velocity for water/air system.
Figure 10. Contours of constant relative rate of energy dissipation under pulsations of liquid inlet velocity for toluene/air system.
ARTICLE
Figure 12. Contours of constant relative rate of energy dissipation under synchronized pulsations of liquid and gas inlet velocities for toluene/air system.
’ CONCLUSION Functional analysis of the material balances on gas and liquid and an expression for the rate of energy dissipation in gas using linearized expressions for liquid holdup and gas pressure drop yielded two criteria: 0.5 + 0.5γ/λ KG/KL and λ + γ. Positive values of these criteria were shown to be sufficient for the energy savings by synchronized periodic pulsations of inlet gas and liquid velocities or by liquid velocity only, compared to the situation under the same mean steady inlet phase velocities. This, however, is not the case of pulsation by gas inlet velocity alone. Carried out experiments showed these criteria to be fulfilled in the whole domain of tested gas and liquid velocities. Evaluation of the energy savings under pulsations relative to the steady state operation showed the savings for water/air and toluene/air systems may reach up to 4% with the maximum in region of high and intermediate liquid velocities and low gas velocities. ’ AUTHOR INFORMATION Corresponding Author
*Phone: +420 220390 233; fax: +420 220 920 661; e-mail:
[email protected]. Notes §
On leave from Russian State Commercial and Economic University, Ufa, Russia.
’ ACKNOWLEDGMENT We gratefully acknowledge financial support of the project by the Grant Agency of the Czech Republic under Grant Number 104/09/0880.
Figure 11. Contours of constant relative rate of energy dissipation under synchronized pulsations of liquid and gas inlet velocities for water/air system.
reduction does occur. However, it is stressed that the presented conclusions are valid only for our experimental conditions (properties of the fluids, column, and packing characteristics).
’ SYMBOLS E mean energy of gas dissipated in bed over period τ per unit volume of bed and unit time, Pa/s h liquid holdup h mean liquid holdup in bed over period τ K0, KG, KL parameters defined in eq 3b, Pa/m; Pa s/m2; Pa s/m2 k, k0, l, n dimensionless parameters defined in eq 14 L length of bed, m P gas pressure, Pa 10831
dx.doi.org/10.1021/ie2000067 |Ind. Eng. Chem. Res. 2011, 50, 10824–10832
Industrial & Engineering Chemistry Research vG vL γ, λ λ1,λ2 τ ξ ω
superficial velocity of gas, m/s superficial velocity of liquid, m/s parameters defined in eq 3a, s/m eigenvalues time period, s variable defined in eq 7, s circular frequency [rad/s]
ARTICLE
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