Dynamic Tray Model To Predict Start-Up Transients in Concentrated

The model accounts for gas holdup and pressure, solvent evaporation, weeping, entrainment, and downcomer holdup. The main assumptions in the model are...
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Ind. Eng. Chem. Res. 2000, 39, 2525-2533

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Dynamic Tray Model To Predict Start-Up Transients in Concentrated Absorbers Praveen Gunaseelan and Phillip C. Wankat* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283

In the presence of a sink, a gas stream with high solute concentration can undergo large decreases in pressure and flow rate because of the transfer of solute. This behavior has been observed during the loading of concentrated gas adsorbers. In this paper we introduce a dynamic tray model capable of predicting gas-phase transients during the start-up of concentrated absorbers. The model accounts for gas holdup and pressure, solvent evaporation, weeping, entrainment, and downcomer holdup. The main assumptions in the model are equilibrium trays, well-mixed gas and liquid phases, and ideal gas behavior. Simulation results for the absorption of concentrated HCl gas in water and concentrated NH3 gas in water are discussed. Predictions with the model reveal severe flow and pressure transients capable of inducing excessive weeping. An alternate start-up procedure to reduce transient weeping is discussed. Introduction When a gas stream comes into contact with a sink (adsorbent, solvent, etc.), there is a decrease in the moles of solute in the gas. For concentrated gas streams, this can result in severe decreases in pressure and gas flow rate. This behavior has been confirmed by theory and experiment1 for the loading of fixed-bed adsorption columns with concentrated feed gas. Oscillations in pressure and gas flow rate have been reported1,2 for adsorption columns connected in series. The start-up of counter-current absorbers is somewhat analogous to the loading of gas adsorption columns. From the modeling standpoint, dynamic simulation of tray absorbers is more complex because of the mobility of the separating agent and tray hydraulics. Despite these differences, we expect similar decreases in pressure and gas flow during concentrated absorber start-up. If the feed gas to a sieve-tray column is concentrated, rapid absorption at start-up will result in significant loss of pressure, which could lead to excessive weeping. Because this is undesirable, the start-up behavior of concentrated absorbers warrants investigation. The objective of this work is to gain insight into the startup behavior in absorbers using a dynamic tray model. Because most classical tray models ignore gas holdup and are therefore incapable of predicting transients in the gas, we developed a tray model that includes gas holdup. Transient studies of absorbers have been reported as early as 1950.3 Shortly thereafter, nonanalytical solutions for transient absorption4 (using analogue computers) were published. Holland and Liapis5 discuss dynamic equilibrium tray models that include gas holdup and provide simulation results for a dilute lean-oil absorber. Kooijman and Taylor6 provide a chronological review of the evolution of tray models for distillation and absorption. In the same paper, they introduce a dynamic nonequilibrium tray model that includes gas holdup. As an illustration, they simulate a 30-tray acetone absorber with an inlet gas mole fraction of * To whom correspondence should be addressed.

0.091. The examples in these papers are not concentrated enough to illustrate the behavior shown in this paper. We introduce a dynamic, equilibrium, nonisothermal, sieve-tray model that accounts for pressure effects, weeping, entrainment, variable liquid flow rates, and downcomer hydraulics. It has been successfully implemented to predict start-up transients in the gas phase in concentrated absorbers. Dynamic Sieve-Tray Model The important assumptions in the model are equilibrium trays, well-mixed liquid and gas phases, and ideal gas behavior. The HCl-water system, with nitrogen gas as the carrier, is used as the primary model system for simulation. We include a few simulations with the NH3-water-N2 system to illustrate the effect of a significantly lower heat of absorption. We chose the HCl-water-N2 system because of the availability of data7-9 and because HCl absorption in water to make hydrochloric acid is commercially important. The absorption is very rapid at high HCl gas concentrations,10 which supports the equilibrium tray assumption. Because of the large heat effect, the process is typically run in cooled wetted-wall absorbers.10 The literature also mentions HCl absorption in packed towers.11 Although HCl absorption is not normally done in tray columns, we consider the process suitable for the purposes of simulation and for making qualitative deductions. Material Balance Equations Liquid Phase. The balance equations are written for the absorption of a gaseous species i. The carrier gas is assumed insoluble in the solvent. As a result, there are two components in the liquid phase (e.g., HCl and water); thus, two material balance equations are required. Because of the well-mixed assumption, the concentration of liquid leaving the tray due to overflow, weeping, or entrainment is the same as that on the tray, that is, ) xentr,out ) xi. xi,out ) xweep,out i i

10.1021/ie990587u CCC: $19.00 © 2000 American Chemical Society Published on Web 04/20/2000

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Solute balance:

trays, downcomers, and the column shell.

d entr,in (M x ) ) (Lin + Lweep,in)xin i + Lentr,inxi dt l i (Lout + Lweep,out + Lentr,out)xi + Ntr,i (1)

Energy balance:

Overall balance: d (M ) ) (Lin + Lweep,in + Lentr,in) dt l (Lout + Lweep,out + Lentr,out) + Ntr,i - Ntr,solvent (2) Gas Phase. The model accounts for solvent evaporation, which is significant in the HCl-water system because of the large heat of absorption. Consequently, the gas phase is ternary (e.g., HCl, water, and N2); thus, three balance equations are required. We assume that the gas phase is well-mixed.

Solute balance: d (M y ) ) Ginyin i - Goutyi - Ntr,i dt g i

(3)

Solvent balance: d in (M y ) ) Ginysolvent - Goutysolvent + Ntr,solvent (4) dt g solvent Overall balance: d (M ) ) Gin - Gout - Ntr,i + Ntr,solvent dt g

(5)

The sign convention for transfer rates Ntr,i and Ntr,solvent indicates that we expect net absorption of solute and net evaporation of solvent from a tray. However, these quantities can be negative, which translates to net desorption of solute and net condensation of solvent. The model has no explicit equations for calculating transient transfer rates; hence, Ntr,i and Ntr,solvent are treated as unknowns. However, compared to classical tray models,4 the model has two additional material balance equations per tray that compensate for the unknown transfer rates. For a simplified isothermal model with a nonvolatile solvent and an equilibrium expression of the form yi ) f(xi), we were able to derive the following expression for the transfer rate (see Appendix for derivation):

Ntr,i )

in MlGin(yin i - yi) + Mgf ′(xi)Lin(xi - xi )

Ml(1 - yi) + Mgf ′(xi)(1 - xi)

(6)

This expression requires the equilibrium relationship to be explicit in y and differentiable. The values of Ntr,i obtained with eq 6 were identical to those calculated by the dynamic process simulation software, SPEEDUP, for an isothermal system. Energy Balance Equation The absorption of HCl in water is very exothermic. Accordingly, the model includes an overall tray energy balance equation. The form of the energy balance (eq 7) was chosen because it requires only one initial condition per tray, the obvious choice being initial tray temperature. The equation assumes thermal equilibrium. This is a weak assumption because the absorption of HCl in water is heat-transfer-controlled.8 The equation neglects the heat capacity of metal associated with

d (M h + MgHg) ) GinHg,in - GoutHg + dt l l [(L + Lweep)hl]in + Lentr,inhentr,in [(L + Lweep + Lentr)hl]out + Q (7) The heat of absorption does not enter explicitly into the energy balance but is accounted for in the liquid enthalpy calculations. This has to do with the choice of aqueous HCl at 25 °C for the base liquid enthalpy, which is a function of liquid concentration and is calculated using the heat of absorption. Himmelblau12 lists data for the heat of absorption (in which the heat of condensation of HCl gas is already incorporated). Ullman’s encyclopedia7 provides specific heat data for aqueous HCl. Specific heat data for the gases was taken from Reid et al.13 Other Equations Because this model differs from classical models primarily in the material and energy balance equations, they have been explained in detail in preceding sections. The current section briefly discusses other equations used in the model. Vapor-liquid equilibrium data for the HCl-water system is available in the form of partial pressure data.9 The partial pressures of HCl and water in the gas phase were fitted to modified forms of the Antoine equation that accurately capture the liquid concentration and temperature dependence. The equations are

Bh(x) T

(8)

Bw(x) - Cw(x) log(T) T

(9)

log10(pHCl) ) Ah(x) log(pwater) ) Aw(x) -

where partial pressures are in millimeters of mercury, T is in Kelvin, and

Ah(x) ) 11.9756 - 22.5606x + 96.7354x2 206.7324x3 Bh(x) ) 4839.2 - 19425 x + 53292x2 - 89476x3 Aw(x) ) 22.1197 - 52.7290x - 219.9083x2 + 2879.1x3 Bw(x) ) 2882.1 - 2103.2x - 9962.2x2 + 151350x3 Cw(x) ) 4.4784 - 18.2505x - 69.2924x2 + 954.6807x3 A similar fitting procedure was followed for specific heat and density data.7 The partial pressure tables could alternatively be used to calculate the separation factor (Ki), but it was difficult to fit the behavior of KHCl as a function of xHCl and T in a compact equation. This is due to the abnormal behavior of the HCl-water system, which has an azeotrope at around 20 wt % HCl. Below this concentration, KHCl is less than unity, which

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translates to high HCl solubility in water. Above the azeotropic concentration, KHCl is greater than unity, and hence solubility is lower. This VLE behavior influences the transients for HCl absorption and will be discussed later in more detail. The model uses the Francis weir formula14 for triangular weirs to calculate liquid overflow rate from a tray. An equation for sieve tray pressure drop14 is used to calculate gas flow rate. The rate of gas exiting the column is calculated using a simplified pressure drop equation that relates exit gas flow rate to pressure difference across the outlet pipe. Examples of equations used are the modified equation for laminar compressible flow through a circular pipe15

Goutflow ) Klaminar(Ptop2 - Pout2)

(10)

and the modified equation for turbulent incompressible flow, which is applicable for gases when density changes within the system are less than 10 percent.15

Goutflow ) KturbulentxPtop - Pout

(11)

The model discussed in this paper uses eq 10, but eq 11 is preferable because the flow is usually turbulent. The terms Klaminar and Kturbulent are obtained by lumping terms in the original equations to simplify the model, making it unnecessary to calculate gas properties in the exit pipe. With reasonable pipe dimensions (20-cm internal diameter and 10-m length) and use of physical properties of the exit gas stream, the value of Kturbulent was calculated as 2.2 × 10-3. The pipe dimensions were chosen to ensure an outflow rate large enough to prevent back-pressure in the column. When the constant in eq 10 was adjusted (Klaminar ) 3 × 10-10) to match the steady-state pressure drop using eq 11, the transient results for the two cases were almost identical. Weep rates are calculated using the procedure outlined by Hsieh and McNulty.16 The equations for downcomer and entrainment calculations were taken from Ludwig.14 For entrainment prediction, we use an average liquid surface tension of 55 dyn/cm2. We model the downcomer as a tank, and ignore temperature and concentration changes that can occur in it. For the gasphase equation of state, we use the ideal gas law, which is reasonable because the simulation pressures are low. Model Testing and Validation To assess model performance, the following test case was conceived using guidelines for column design17 and published data for HCl absorption.18 A counter-current sieve-tray absorber with six trays is brought to an initial steady state, using pure nitrogen at 0 °C as the feed gas and pure liquid water at 100 °C as the solvent. The column operates adiabatically at atmospheric pressure, and the molar feed rate of the solvent is 6 times the molar feed rate of gas. (Adiabatic operation allows for heat removal by evaporation of water as opposed to using heat-transfer media. A combination of the two methods is also used industrially19). The column is designed to have minimal weeping at steady state. Table 1 lists the important operating conditions and design parameters for the test case. After attainment of the initial steady state, solute concentration in the feed gas is increased by switching to feed containing predominantly HCl at 0 °C. In

Table 1. Test Case Specifications feed conditions feed stage pressure (bar) temperature (°C)

gas

liquid

6 ∼1.08 (variable) 0

1 1 100

component flows (kmol/s) before step (t < 200 s) HCl water N2 after step (t g 200 s): HCl water N2 initial conditions in column pressure drop per tray (Pa) 800 tray temp. (°C) 55 HCl mole fraction 0 in liquid transfer rates (kmol/s) 0 crest height (mm) 20 downcomer liquid height (cm)

35

gas

liquid

0 0 0.05

0 0.3 0

0.05 0 0

0 0.3 0

operating conditions molar (L/G)feed 6 feed temp. (°C) liquid 100 gas feed liquid pressure (bar) heat removal (J/s)

column and tray design specifications number of trays 6 weir length (m) tray diameter (m) 1.25 weir height (m) tray spacing (m) 0.5 hole area (% active) hole diameter (mm) 4.5 downcomer area (m2) active area (m2) 0.789 area under apron (m2)

0 1 0

0.875 0.05 0.1275 0.108 0.0219

response to this step, the column goes via a transient to a new steady state. Model predictions for the test case show good agreement with steady-state observations reported in the literature.10,18 For pure HCl feed after the step (yHCl,in ) 0-1.0 at t ) 200 s), the model predicts a steady-state temperature of 80.44 °C and concentration of 32.5 wt % HCl in the exiting liquid, which compares well with 80 °C and 32 wt % HCl reported by Oldershaw et al.18 who used a graphical method. The model also predicts decreases in pressure and gas flow rate at start-up which agree qualitatively with transients observed in concentrated gas adsorption.1,2 In addition, the model predicts a maximum in the column temperature profile after the step, which is a common feature of exothermic absorption with cold feed gas.10 Model Application and Results For start-up of dilute absorbers (yHCl,in < 0.1 after the step), the model predicts a small decrease in the gas flow rates and a modest increase in temperature, but no noticeable effect on column operation. This is because the gas phase is predominantly an insoluble carrier gas, which maintains sufficient pressure in the column, thus preventing weeping. This agrees with typical results and explains why the issue of start-up transients in dilute absorbers is not very important. Model predictions for concentrated absorbers are markedly different. Prior to the concentration step, the column is allowed to stabilize, during which a temperature gradient is established in the column due to the difference in feed temperatures. In addition, evaporation of water, due to dry feed gas and boiling water as the

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Figure 3. Transient solute transfer rates for HCl test case.

Figure 1. Transient gas flow rates for HCl test case. (a) Complete transient. (b) Details of transient. Feed gas switched from N2 to HCl at t ) 200 s.

Figure 4. Transient weep rates for HCl test case.

Figure 5. Transient tray temperatures for HCl test case. Figure 2. Transient tray pressures for HCl test case.

feed solvent, results in an increase in gas flow rates. When the column is subjected to a large concentration step (yHCl,in ) 0-1.0 at t ) 200 s), gas flow rates (Figure 1) and pressures (Figure 2) in the column drop sharply in response to rapid absorption on the bottom tray (Figure 3), resulting in severe weeping in the column (Figure 4). The bottom tray does not weep as much because of the constant supply of feed gas. Table 1 lists the process conditions for this simulation. Simultaneous with the initial flow decrease is a dramatic increase in tray temperatures (Figure 5), because of the heat of

absorption and decrease in moles of gas. This large heat effect leads to evaporation of solvent from the trays (Figure 6), which aids the recovery of gas flow rates (Figure 1b) and thus shortens the period when weeping is excessive (Figure 4). Thus, there are two time scales in the transient: a short initial response, during which solvent evaporation on the bottom tray leads to revival of gas flow rates, followed by a longer response caused by slow upward movement of solute in the column (Figure 7a,b). The short response is due to the large heat effect and the assumption of thermal equilibrium. It is discussed later in more detail. The long response is a consequence of

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Figure 6. Transient solvent transfer rates for HCl test case.

Figure 7. Transient solute concentrations for HCl test case: (a) liquid phase; (b) gas phase.

the well-mixed assumption, and the large ratio of liquid to gas molar holdup (∼100) on the trays, because of which the tray liquid concentration increases sluggishly. As a result of being tied to the liquid concentration through the VLE relationship, the tray gas concentration also increases slowly. The initial weep rates (Figure 4) are reasonably high; they are in the order of magnitude of liquid flow rates in the column. It was pointed out to us that this may explain the failure of absorbers used for emergency absorption of vent gases.20 The model predicts very little entrainment during start-up. However, the model predicts significant entrainment in processes with large

Figure 8. Transient weep rates for modified HCl test case. Feed liquid at 0 °C.

rates of solute desorption, such as shutdown of concentrated absorbers and start-up of concentrated stripping columns. The chain of events in the short response resulting from reduction of gas flow is intricate. At start-up, the liquid solute concentration is low and KHCl is very small, which translates to high solubility of HCl and high volatility of water. Thus, initial absorption of HCl on the bottom tray is rapid and the large heat of absorption results in a sharp rise in tray temperature (Figure 5). The temperature rise continues until the boiling point of the mixture is reached, culminating in a high rate of water evaporation (Figure 6). However, when HCl liquid concentration on the bottom tray becomes greater than the azeotropic concentration of around 20 wt % (Figure 7a), water becomes less volatile and the evaporation rate decreases (Figure 6), coincident with a sharp fall in the rate of HCl absorption (Figure 3). Hot water vapor leaving the bottom tray condenses on the colder trays above, and the heat of condensation brings these trays successively to their boiling point (Figure 5). This is responsible for the dramatic recovery in gas flow rates and shortening of the period where weeping is excessive. The fact that gas flow rates and pressures after recovery are greater than their values at the initial steady state is testimony to the large rates of water evaporation in the column. Because a large part of the condensed water vapor on a tray flows back to the tray it originated from, there is, in a sense, solvent recycling within the column. Because the large heat effect is instrumental in the recovery of gas flow rates, and thus in the suppression of weeping, colder columns are expected to have longer and more intense weep transients. Simulations of columns identical to the column in Table 1, except for the cold feed liquid (Figure 8; compare with Figure 4) or heat removal (Figure 9), show this to be true. For simulations with a cold feed liquid (T ) 0 °C), although transient weeping lasts longer than that in the test case (Figure 4), there is enough solvent evaporation to prevent steady-state weeping. For simulations of columns with cooling, if heat removal is optimized to prevent excessive solvent evaporation, we observe the onset of severe steady-state weeping (Figure 9). This is an important result because prevention of this weeping would require modification of column design. (We used the simplified equation Q ) UA(Tbottom - Tcoolant) to model heat removal from the bottom tray, with UA ) 5

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Figure 9. Transient weep rates for modified HCl test case. Bottom tray is cooled.

Figure 11. Transient weep rates for modified HCl test case. Feed solvent flow rate increased to 0.5 kmol/s at 0 °C.

Figure 10. Transient KG/L for HCl test case.

Figure 12. Transient weep rates for modified HCl test case. Two smaller step increases used instead of one large step.

× 104 J/(s °C) and Tcoolant ) 12 °C. The value of UA was chosen to restrain solvent evaporation.) Apart from the large heat effect and hot feed liquid, the test case is unique because of another reason. The parameter KG/L indicates the direction of solute movement in the column. Values greater than unity indicate upward movement of solute, while values less than unity indicate the opposite. Values close to unity result in a sluggish transient because of slow solute movement. The latter is applicable for the test case (Figure 10). This occurs because we use a molar (L/G)feed ratio of 6 for the test case (based on recommendations in the literature18). Simulations for columns with lower steady-state values of KG/L were contrived by increasing the L/G ratio (by increasing Lfeed while keeping Gfeed constant; Figure 11) and decreasing K (by decreasing column temperature; Figures 8 and 9). As expected, these simulations showed more intense weeping (as compared to that in the test case; Figure 4). It should be noted that KG/L is not a reliable indicator of weeping behavior in general, but it works well for concentrated systems. For the HCl-water system, it predicts weeping behavior for molar feed gas concentrations of 40% or more. In dilute HCl absorption, however, despite low KG/L (because K is very low), transient weeping is insignificant because of adequate flow of the carrier gas. It is evident that transient behavior is closely related to the magnitude of the concentration step. This fact was utilized to develop a scheme to reduce transient weeping. The large step in solute concentration was replaced by two steps of equal magnitude (step 1: yHCl,in

) 0-0.5 at t ) 200 s; step 2: yHCl,in ) 0.5-1.0 at t ) 250 s). The second step was imposed after the column had stabilized from the first. The result is a significant decrease in transient weep rates (Figure 12; compare with Figure 4). Finally, we present results for the NH3-water-N2 systems to illustrate the effect of a significantly lower heat of absorption. Tables of partial pressure data for the NH3-water system8 were fitted in a manner described earlier for the HCl-water system. The simulations use feed solvent at 0 °C, and there is a provision for heat removal from the bottom tray. With no heat removal, the simulation predicts high NH3 concentration in the vent gas (Figure 13a). To model heat removal, we use a value of UA () 5 × 103 J/(s °C)) that is optimized for low vent gas concentration (Figure 13b) and results in a steady-state heat removal rate of 2.7 × 105 J/s. Simulation results with the above heat removal rate show expected decreases in gas flow rates (Figure 14) and pressures (Figure 15) because of rapid absorption at start-up. As a result of a smaller heat of absorption and heat removal, the temperature increase (Figure 16) is not as severe as that for HCl absorption, and rates of water evaporation are accordingly smaller. As a consequence, there is intense weeping during the transient and at steady state (Figure 17). For comparison, consider an identical simulation for the HCl-water-N2 system with heat removal. We use the same value of UA () 5 × 103 J/(s °C)), which corresponds to a steady-state heat removal rate of 4.6

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Figure 15. Transient tray pressures for NH3 absorption in water. Feed solvent at 0 °C; heat removal from bottom tray (UA ) 5 × 103).

Figure 13. Transient solute concentrations in gas phase for NH3 absorption in water: (a) No heat removal; (b) heat removal from bottom tray (UA ) 5 × 103, Tcoolant ) 12 °C). Feed solvent at 0 °C. Feed gas switched from N2 to NH3 at t ) 200 s.

Figure 14. Transient gas flow rates for NH3 absorption in water. Feed solvent at 0 °C; heat removal from bottom tray (UA ) 5 × 103).

× 105 J/s, which is larger because steady-state ∆T is larger for the HCl simulation. However, this heat removal is not enough to prevent solvent evaporation at steady state, and as a result we observe only transient weeping, but no steady-state weeping (Figure 18). It must be noted that the simulation for the NH3water system uses operating parameters (L/G, number of trays, etc.) that were optimized for the HCl case (Table 1), and the different VLE behavior also has a bearing on the transient. As a result, comparisons between the two systems are very qualitative.

Figure 16. Transient tray temperatures for NH3 absorption in water. Feed solvent at 0 °C; heat removal from bottom tray (UA ) 5 × 103).

Figure 17. Transient weep rates for NH3 absorption in water. Feed solvent at 0 °C; heat removal from bottom tray (UA ) 5 × 103).

Summary and Conclusions A dynamic equilibrium sieve-tray model with separate material balances for gas and liquid phases has been developed, which is capable of predicting transients in the gas during start-up of concentrated absorbers. The model predicts sharp decreases in gas flow rates and pressures, which lead to severe transient weeping during start-up. To alleviate this problem, a two step start-up procedure is recommended. In systems with

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Liquid-phase solute balance: xi

d d (M ) + Ml (xi) ) Linxin i - Loutxi + Ntr,i dt l dt

(I)

Liquid-phase overall balance: d (M ) ) Lin - Lout + Ntr,i dt l

(II)

Eliminating the term d(Ml)/dt from eqs I and II and rearranging, we obtain

dxi Ntr,i(1 - xi) - Lin(xi - xin i ) ) dt Ml Figure 18. Transient weep rates for modified HCl test case. HCl absorption in water, feed solvent at 0 °C, and heat removal from bottom tray (UA ) 5 × 103).

large heats of absorption, evaporation of solvent can reduce the intensity of weeping. Simulations of colder columns show more transient and steady-state weeping than adiabatic columns. The parameter KG/L is a reliable indicator of weeping behavior in concentrated systems, but not in dilute systems. Concentrated columns with values of KG/L much smaller than unity are prone to severe weeping. The model is limited by its assumptions. The equilibrium tray assumption overestimates transfer rates. We would expect nonequilibrium models (which include mass and heat transfer) to predict slower and less severe transients. The assumption of thermal equilibrium is not appropriate for HCl absorption.8 This assumption is responsible for the dramatic temperature rise at startup. The well-mixed assumption for the liquid phase on the tray is questionable; a more detailed flow model is in order. The ideal gas assumption can be relaxed by choosing a more accurate equation of state for the gas phase. This would be necessary for systems with nonideal gas-phase behavior. Although the weep transient for the test case is shortlived and steady-state weeping is insignificant, these results are specific to the test case. Columns with conventional operating conditions (colder feed solvent, heat removal, etc.) experience more intense weeping. The limitations notwithstanding, the qualitative predictions with the model strongly indicate that start-up of sieve tray absorbers with concentrated feed gas is prone to severe transients that can be detrimental to column operation. Acknowledgment This research was partially supported by the Purdue Research Foundation and NSF Grants CTS-9401935 and CTS-9710553. Appendix In the following text, we derive an expression for the transient transfer rate of a solute for an isothermal tray. The derivation assumes nonvolatile solvent, ignores weeping and entrainment, and assumes an equilibrium expression of the form y ) f(x), where f is differentiable in x. With these assumptions, the solute and overall material balances become simpler.

(III)

A similar treatment for the gas phase gives

dyi Gin(yin i - yi) - Ntr,i(1 - yi) ) dt Mg

(IV)

Because yi ) f(xi), eq IV can be rewritten as

f ′(xi)

dxi Gin(yin i - yi) - Ntr,i(1 - yi) ) dt Mg

(V)

Eliminating the term dxi/dt from eqs III and V and solving for Ntr,i gives eq 6, which is the desired expression for the transient transfer rate of a solute. Nomenclature Ah, Bh ) concentration-dependent fitting parameters for the partial pressure of HCl Aw, Bw, Cw ) concentration-dependent fitting parameters for the partial pressure of water f(x) ) equilibrium relationship f ′(x) ) first derivative of f(x) G ) flow rate of gas, kmol/s hl ) enthalpy of a liquid on a tray, J/kmol hentr,in ) enthalpy of a liquid entering a tray because of entrainment, J/kmol Hg ) enthalpy of a gas on a tray, J/kmol Ki ) yi/xi ) separation factor Klaminar ) lumped constant in a modified laminar compressible flow equation, eq 10 Kturbulent ) lumped constant in a modified turbulent incompressible flow equation, eq 11 L ) flow rate of a liquid, kmol/s Lweep ) weep rate, kmol/s Lentr ) entrainment rate, kmol/s Ml ) molar liquid holdup on tray, kmol Mg ) molar gas holdup on tray, kmol Ntr,i ) transfer rate of component i (solute) across phases, kmol/s Ntr,solvent ) transfer rate of solvent across phases, kmol/s pi ) partial pressure of component i, mmHg Q ) heat input/removal from tray, J/s T ) temperature, °C UA ) overall heat-transfer coefficient multiplied by area, J/(s °C) xi ) mole fraction of component i (solute) in a liquid

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2533 xentr,in ) mole fraction of component i (solute) in a liquid i entering a tray because of entrainment y ) mole fraction in gas

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Received for review August 6, 1999 Revised manuscript received March 13, 2000 Accepted March 16, 2000 IE990587U