Modeling of a Two-Phase Electrochemical Reactor for the Fluorination

Ind. Eng. Chem. Res. 2001, 40, 3117-3126

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Modeling of a Two-Phase Electrochemical Reactor for the Fluorination of Organic Compounds. 2. Multiple Steady States Javit A. Drake,* Clayton J. Radke, and John Newman Department of Chemical Engineering, University of California, Berkeley, California 94720-1462

A three-cell, continuous-flow electrochemical reactor is numerically simulated, revealing multiple steady states in terms of production rate. The bipolar electrochemical reactor fluorinates organic compounds in liquid solution in parallel flow cells, accompanied by the generation of hydrogen gas. The mathematical model of two-phase flow, phase equilibrium, gas-phase molecular association, and thermal energy considerations by Drake et al. (Ind. Eng. Chem. Res. 2001, 40, 3109; J. Electrochem. Soc. 1998, 145, 1578) is extended to address interactive multiple cells. Necessary model extensions include intercell heat transfer through the bipolar electrodes, inlet flow-distributor connectivity, and serial current redistribution. Multiphase heat transfer, among these other cell-to-cell interactions, affects the occurrence of multiple states and profiles within differently operating parallel cells. A trifurcation point occurs at 17.3 V in the operating curve of total current versus applied potential. Here, this three-cell production rate curve branches into three possible steady states of one, two, or three high-potential, inefficient cells. These states are unfavorable in terms of power consumption because the same total current can be accomplished by an efficient state requiring less potential. The pressure profiles of an inefficient state also reveal that the system is vulnerable to physical failure with the collapse of the narrow flow channels. 1. Introduction Multiple steady states of reactors are well studied in chemical engineering. Multiplicity of states can arise when a phenomenon caused by the process limits the rate. For example, in a fixed-bed catalytic reactor, heat loss to an exchanger decreases catalytic activity and thus limits the reaction rate.3 This thermal energy transfer, caused by temperature elevation from an exothermic reaction, leads to multiple steady states. Similarly, in enzymatic chemical reactors, generation of a reaction-inhibiting intermediate can lead to multiple steady states.4,5 However, to our knowledge, the occurrence of multiple steady states in electrochemical flow reactors has not been addressed in the literature. In our latest work, (part 1 of this pair),1 we postulate the multiplicity of steady states for a gas-liquid electrochemical flow reactor. As in other reactors, the multiple states arise because of a local rate-inhibiting phenomenon: the development of non-liquid-phasecontinuous flow with high applied potential. The liquidfed electrochemical reactor performs the Simons process for the fluorination of organic compounds.6,7 Gas generation in the process leads to a maximum in the production rate. About this maximum, a high-potential state and a low-potential state exist for an individual cell at a given production rate. For multiple cells in parallel flow and serial electrical configuration, Nc + 1 steady states are anticipated, where Nc is the total number of cells.1 Investigation of multiple steady states is important for reactor performance. As described in part 1,1 the proposed high-potential states are inefficient in terms of power consumption. Thus, maintaining a given total * To whom correspondence should be addressed.

current does not guarantee a desired efficient operating state if the high-potential states exist. In addition to performance, the physical stability of the Simons process electrochemical reactor is an issue. In real systems of this type, the thin flow channels collapse occasionally, resulting in a shutdown of the process.8 Although the buildup of deposits is the apparent cause of this problem, the occurrence reveals that a change in the thin dimension of the cell can cause failure. The sensitivity of the narrow flow system raises the question of mechanical stability in multiple-cell systems. In practice, electrochemical fluorination reactors consist of adjacent parallel flow cells in a bipolar arrangement. This electrically serial configuration has the anode of one cell act as the cathode of the next cell and so on. Several features external to the reaction zone are necessary to achieve this multiple-cell arrangement. Jha et al.9 addressed the pressure changes associated with common inlet and exit manifolds in the Simons flow reactor. However, the cells are assumed to operate identically. In reality, one cell in a cell pack may fail. Jha et al.10 recently simulated thermal and flow interaction among multiple gas-liquid flow cells. An upper estimate of cell-to-cell heat transfer was used in a dynamic relaxation to a steady state. However, twophase effects on heat transfer were not included. Twophase flow, for instance, significantly amplifies thermal energy transfer.11 Last, no study to date demonstrates multiple steady states in electrochemical flow reactors. Here, we address the detailed interaction of cells in multiple steady states. Unique to this work are electrical interactions and the complex thermal effects between adjacent cells. As outlined below, these intercell interactions demand extensions beyond previous mathematical models of the Simons process in continuous-flow cells.1,2,9 Simulation of high- and low-potential modes of opera-

10.1021/ie0006108 CCC: $20.00 © 2001 American Chemical Society Published on Web 06/16/2001

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Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001

tion permits the exploration of multiple steady states for a collection of cells. 2. Mathematical Model 2.1. Primary Equations. The mathematical model is an extension from part 11 to address interacting electrochemical cells in a bipolar array. Nine of ten primary equations from part 1 are used in the present work. The last equation, the statement of uniform electrode potential, is replaced by a cell-interactive equation, addressing current redistribution as described below. All ten equations and variables apply to each respective cell. Thus, for the three-cell system, there are 30 primary, independent equations and variables. Also, boundary conditions for two equations become different under the consideration of flow and electrical interactions, respectively, as discussed below. Heat-transfer interaction adds auxiliary expressions for the thermal energy equation.1 2.2. Intercell Phenomena. The present model of multiple interactive cells features common inlet manifolds, multiphase cell-to-cell heat transfer, and serial electrical configuration. The derivations are also described elsewhere in thorough detail.12 2.2.1. Inlet Flow Distributors. As described in part 1,1 flow distributors partition the reactor inlet flow stream to each cell and merge the cell exit streams. The inlet flow distributors and manifold become important here for describing cell-to-cell flow interaction. Namely, the average (or total) flow rate to the cells is specified as a condition as in part 1. Then, rather than specify the flow rate to each cell, a constraint is introduced that the cells must have the same inlet manifold pressure, in in in pman,1 ) ... ) pman,Jc ) ... ) pman,Nc

(1)

where pman,Jc refers to manifold pressure computed using quantities from cell Jc. This generates Nc - 1 entrance conditions where Nc is the total number of cells. The application of this constraint requires a relation of cell inlet variables to the inlet manifold pressure. Namely, an expression for the pressure drop across the inlet flow distributors is employed. The pressure difference across the inlet flow distributor contains head losses due to gravity, friction, contraction, and expansion (cf. Figure 2 of part 11). In the inlet manifold, the velocity is considered negligible relative to the velocities in the distributors or cells. Thus, for the all-liquid feed, the following relation holds between the cell entrance pressure, pz)0, and manifold pressure, in pman , 1 in 2 pin man ) ∆pf + FlgHdist + /2FlVsl,z)0 + pz)0

fHdist 1 2 ( /2FlVdist )+ RH 1 2 ein exp( /2FlVsl,z)0)

ficients and relates the inlet cell velocity to the distributor velocity, Vdist ()WbVsl,z)0/Adist). Thus, inlet manifold in , in eq 1 is calculable from the respective pressure, pman entrance velocity, Vsl,z)0, and pressure, pz)0, of each cell. Thus, one boundary condition per cell results from setting the average flow rate and requiring a common inlet manifold pressure. 2.2.2. Serial Electrical Connectivity. A unique feature of the present model is the account for serial electrical connectivity among cells operating differently. This consideration generates the final independent equation in the mathematical model. In practice, the potential is applied across an entire array of cells7 (cf. Figure 2 of part 11). Thus, the total potential may be monitored, although the potential drop (and thus local current density) from cell to cell may differ. Over the entire height of the reactor, the unequal potential drop leads to a redistribution of the total current through the metal separating two adjacent cells. Nevertheless, the total current (or average current density) across the cells must be identical, neglecting any current inefficiency, as validated elsewhere.12 The local-current-density mismatch between adjacent cells necessitates a compensating current flow in the vertical direction inside the bipolar electrode. Figure 1 shows a vertical section of the electrode wall between two cells. The horizontal current flux passes into the wall from cell Jc + 1 and from the wall to cell Jc. Continuity for the electrical current density holds,

diz,wall iJc+1 - iJc ) dz bw

(4)

(2)

where Hdist is the height of the narrow distributor, F is the phase density, and g is gravitational acceleration. The frictional pressure drop, ∆pin f , contains contraction, length, and expansion losses, in 1 2 ∆pin f ) econ( /2FlVdist) +

Figure 1. Current density and heat-transfer coefficients from cell to cell. The difference in current densities leads to a vertical current inside the wall and thus electrical potential variation in this dimension. Driven by a temperature difference, an intercell heat flux depends on two-phase flow, wall and film heat-transfer coefficients, and gas evolution rate.

where iJc denotes the local current density of cell Jc, bw is the wall thickness, z refers to the height above the cell entrance, and iz,wall is the vertical current density inside the electrode. Ohm’s law expresses iz,wall in terms of the vertical potential variation in the wall,

iz,wall ) -κw

dVJc dz

(5)

(3)

in where ein con and eexp represent the head-loss coefficients from contraction and expansion, respectively.13 The change in cross-sectional area determines these coef-

where κw is the conductivity of the electrode metal and VJc refers to the local potential of the anode of cell Jc. Combining eq 4 and eq 5, we relate this potential and local cell-to-cell current density:

Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3119

d2VJc dz2

iJc - iJc+1 κwbw

)

fully developed flow,

(6)

Next, the local potential drop across a cell, ∆VJc, comes from the difference between adjacent electrodes:

∆VJc ) VJc - VJc-1

(7)

Note that ∆VJc is not set. Rather, the local, variable potential drop is calculated self-consistently as the difference of the variable electrode potentials. For the end monopolar electrodes (shown in Figure 2 of part 11), the potential is considered uniform throughout. We use the first electrode as our reference, with a potential of zero. This cathodic potential does not appear as a variable. For instance, for the first (leftmost) cell, the potential drop is simply V1. The specification of uniform potential for the monopolar anode or eq 6 for the bipolar electrodes creates the final independent equation per cell. Then, eq 6 requires two boundary conditions per cell. The conditions are that the vertical electrode current density, iz,wall, vanishes at the ends of the reactor, z ) 0 and z ) H. That is, this current must terminate where the electrodes meet the electrically insulating, flow distributors. From eq 5, the potential gradient must also vanish at these locations. This condition is written at each cell entrance (z ) 0):

dVJc dz

|

z)0

)0

(8)

The corresponding requirement at the exits (iz,wall ) 0 at z ) H) is replaced with an equivalent condition, namely, the constraint of equal total current for all cells. [Upon integrating eq 4 to z ) H, the cumulative cellto-cell difference in horizontal current density, iz,wall, returns to zero. This means that the integrated production rates, Iz)H, for all cells are equal, eq 9.] That is, redistribution of current from cell to cell is subject to the stipulation that extents of reaction at the cell exits are all equal,

I1,z)H ) ... ) IJc,z)H ) ... ) INc,z)H

(9)

as calculated by integrating the local current density. Equation 9 generates Nc - 1 boundary conditions. The remaining end condition is the specification of the total applied potential across all cells (VNc ) Vtot) or the total production rate, Itot, for the reactor. 2.2.3. Intercell Heat Transfer. Cell-to-cell heat transfer is another multiple-cell consideration. Thermal energy is transferred between adjacent cells as determined by conduction, two-phase flow convection, and nucleate gas evolution. The latter two effects are not previously addressed in multiple-cell interaction. Figure 1 shows the transfer coefficients and locality of the processes. Gas-liquid forced convection occurs on both sides of each bipolar electrode. Conduction occurs through the anodic film and wall. Last, on the cathodic side, heat transfer is due to flow convection coupled with nucleate gas evolution. First, forced convection in the cells is addressed using a heat-transfer model for gas-liquid flow by Steiner and Taborek.11 The treatment requires the respective heattransfer coefficients of the gas and liquid, hl and hg. We employ either a laminar14 or turbulent15 correlation for

hl or g )

( )( ) ( )

thermal 2RePr 1 κl,g 4 2b 3z/b Γ( /3)

hl or g ) 0.02

1/3

for laminar

(10)

κthermal l,g Re0.8Pr1/3 for turbulent 2b

(11)

where κthermal is the phase thermal conductivity, Pr ˆ p refers ()µC ˆ p/F) denotes the phase Prandtl number, C to the phase heat capacity per unit mass, µ is the viscosity, and the Reynolds number, Re, contains the total mass flow rate (FlVsl + FgVsg). [This original formula is for annular regions, including a factor (Douter/ Dinner); we apply this relation treating the thin channel as an annular region with Douter ) Dinner]. The larger of the two values, h(laminar) or h(turbulent), is used. The flow convection transfer coefficient, hflow, comes from the liquid coefficient multiplied by an enhancement factor, Ftp, due to two-phase flow,

hflow ) hlFtp

(12)

where Ftp depends on the liquid volume fraction, φ:

Ftp ) ([φ1.5 + 1.9(1 - φ)0.6φ0.01(Fl/Fg)0.35]-2.2 + {(hl/hg)(1 - φ)0.01[1 + 8φ0.7(Fl/Fg)0.67]}-2)-0.5 (13) At the anode side of a cell, convection occurs because of two-phase flow only. At the cathode, gas generation also affects heat transfer. At the current densities in our simulations, generation of hydrogen is a nucleate gas-evolution process.16 For hydrogen evolution in anhydrous HF, a semitheoretical relation expresses the heat-transfer coefficient, hgas, in terms of current density,16

hgas )

2a thermal FlC ˆ p,l)1/2i1/3 (κ π l

(14)

where the parameter a is a constant. We use a combining rule by Steiner and Taborek11 for the heat-transfer coefficient, hgas-flow, at a surface with simultaneous nucleate boiling and two-phase flow:

hgas-flow ) [(hgas)3 + (hflow)3]1/3

(15)

The anodic film and the metal electrode impose some heat-transfer resistances as well. The film is mainly composed of high molecular weight organic compounds.17 Hence, the film heat-transfer coefficient, hfilm, is estimated from thermal conductivity values18 for polymeric layers. The heat-transfer coefficient of the nickel bipolar electrode is denoted hwall and is the thermal conductivity19 divided by the wall thickness, bw, in the Nomenclature section. The overall heat-transfer coefficient across an intermediate electrode thus depends on coefficients from the flow on the anodic side, a film at the anode, the nickel electrode, and the combined nucleate gas evolution and two-phase flow at the cathode,

hJc,Jc+1 )

[

1 1 1 1 + + + hflow,Jc hfilm hwall hgas-flow,Jc+1

]

-1

(16)

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where hJc,Jc+1 refers to the overall transfer coefficient between cells Jc and Jc + 1. From the overall heat-transfer coefficient, the net thermal-energy flux to a cell, qht,Jc is calculated:

qht,Jc ) hJc,Jc+1(TJc+1 - TJc) + hJc-1,Jc(TJc-1 - TJc) (17) This net heat transfer appears in the thermal energy equation.1 The end monopolar electrodes of the first and last cells are considered adiabatic. For example, qht,Nc has the first term of eq 17 omitted because there is no cell to the right in Figure 2 of part 1. In summary, detailed considerations of parallel flow connectivity, bipolar electrode potential variation, and intercell heat transfer are added to the model. The flow cells interact through a common inlet manifold. The potential of a bipolar electrode varies in the vertical direction when there is a local horizontal current density imbalance. Last, the heat-transfer model, original to electrochemical flow reactors, includes two-phase flow, solid resistances, and gas evolution at the cathode. With these interactions, the multiple-cell model seeks to examine the proposed multiple steady states. The mathematical model of ten primary, independent equations and variables per cell with auxiliary algebraic relations and boundary conditions is solved numerically. Fifty mesh spaces are used in the finite-difference discretization. An iterative, multivariable NewtonRaphson method involving the BAND subroutine by Newman calculates solutions to the system.20 3. Results and Discussion Simulation results for a three-cell Simons electrofluorination reactor are presented. The reactor specifications and inlet stream conditions are the same as those in part 1. The average inlet velocity, Vsl,z)0, to the cells is 0.15 m/s. Each cell has a 3.66-m height, H, a 1.2-m breadth, W, and a cell thickness, b, of 3.2 mm. The 50 °C inlet stream has a molar composition of 79.5% HF, 20% RH (organic reactant), and 0.5% CE. We specify an exit manifold pressure of 3 atm. 3.1. Three-Cell Multiple Steady States. Using a three-cell model with interactions, we calculate the production rates versus average applied potential, Vtot/ Nc. Figure 2 shows that multiple steady states exist. We identify up to four states for a given total current. The solid curve represents the rates for three cells operating identically. This curve is the same as the single-cell production curve (cf. Figure 3 in part 1), demonstrating three low-potential cells, state A, and three high-potential cells, state D. In these two states, there are no driving forces for the mentioned cell-tocell interactions. For instance, identical temperatures result in no heat transfer. A maximum production rate of 6950 A occurs at a 14.5-V potential per cell. We refer to this potential as ∆Vmax. As described, the solid curve also shows that each cell can have a high-potential (inefficient) and a low-potential (efficient) state for a given total current. The drop in Itot in state D comes only from the gradual encroachment of non-liquidphase-continuous flow. At an average potential of 17.3 V, two curves in Figure 2 branch from the solid curve. Because three total states split from the point where Vtot/Nc ) 17.3 V, this potential is a trifurcation point. Steady-states B and C emerging from the solid curve at 17.3 V in Figure 2 involve differently operating cells.

Figure 2. Three-cell production-rate curve. Different multiple steady states exist in terms of total current and applied potential. The production-rate curve branches into three possible states of one (state B), two (state C), or three (state D) high-potential cells at 17.3 V applied potential per cell yielding 6500 A. Cell-to-cell heat transfer, inlet and exit flow distributors, serial connectivity of cells, and bipolar-wall-potential variation are incorporated into the calculations.

The long-dashed line contains the production rates of state B, where one cell operates inefficiently and two run efficiently. The short-dashed curve traces states C, having two high-potential cells and one lowpotential cell. For example, a calculated state C yielding 6400 A involves cells with the following potential drops: 17.8|17.1|17.8. This format indicates ∆VJc values ordered left to right in Figure 2 in part 1. Throughout the present study, we refer to the cells numbered in this left-to-right order. For all states B and C presented, the two outer cells (in positions 1 and 3) have essentially the same potential drop, ∆VJc, while the middle potential drop differs from the others. As discussed later, additional states with different orderings of low- and high-potential cells probably exist but are not simulated. Efficient, low-potential operation refers to a cell or a group of cells functioning in the lower of two possible potentials. Yet efficient operation does not imply an individual cell potential drop below ∆Vmax. In the state-C example (17.8|17.1|17.8), the middle cell is the low-potential cell, and the others are high-potential cells. Each of the outer cells consumes 114 kW ()17.8 V × 6400 A); cell 2 requires only 109 kW. Hence, states B, C, and D are undesirable and inefficient because they involve one or more high-potential cells. That is, at least one cell operates at a potential higher than necessary to achieve the same Itot. In Figure 2, we see that the production rates of states B and C decline more sharply with increased Vtot/Nc than those of states D. The reason is seen in referring back to the identical- or single-cell curve, the solid curve in Figure 2. State B entails two cells operating on lowpotential curves (similar to curve A) and one on a highpotential curve (similar to curve D). Thus, when the production rate is lowered for a system in state B in Figure 2, two (low-∆V) cells decrease in potential and one (high-∆V) cell increases. A noteworthy feature is that the steady-state-B curve undergoes a minimum in potential. Consider the successive states as curve B branches from the solid curve at the trifurcation point. Initially, the potential drop of the two cells running on low-potential curves is larger than the rise in potential of the other cell. Over a short segment of curve B in Figure 2, this causes a decrease in Vtot/Nc with lowered production rate. Soon, however, the high-potential cell (as with state D) dramatically


increases in potential. That is, each incremental decrease in the Itot corresponds to a greater increase in ∆V for this cell. Hence, the three-cell, average potential for state B begins to increase as Itot lowers. This turning point, occurring at Itot ) 6400 A, is denoted state Bmin because it has the lowest average potential (16.6 V) of any state B. For production rates lower than 6400 A, greater applied potentials lead to a steadily smaller Itot. On the other hand, state C always shows an increase in average potential with total current decrease. The combined potential increase of two high-potential cells of this state is greater than the corresponding decrease of one low-potential cell. In states B and C, at least one inefficient cell demands an increasing potential drop while each low-potential cell requires lower ∆V as the production rate falls. In state D, there is no efficient cell, decreasing in potential with lower total current. Hence, curve D displays the most gradual decrease in Itot with average potential. Also worth discussion is the position of the trifurcation point; namely, the three-cell production-rate curve has the trifurcation point at a potential higher than expected from the single-cell results. Referring to the prediction of multiple steady states in part 1, we find that a hypothetical state b entails a cell on the highpotential side functioning with two corresponding lowpotential cells. On the solid curve of Figure 2, any noninteractive cell of potential greater than ∆Vmax ()14.5 V) has a production rate matching a lowpotential state. Such a scenario is seen in a 14-, 15.3-, 14-V case inferred. From this finding, for a multicell system, one expects an identical-cell curve to split at a potential of 14.5 V into curves representing differently operating cells. This reasoning suggests ∆Vmax as the trifurcation potential for a three-cell reactor. However, the complete model with full interactions shows otherwise. Heat transfer between the cells predominantly determines the potential of the trifurcation point, as discussed later. Despite the disparity, a similarity exists between the simple expected and actual trifurcation point for a threecell system. On the basis of the single-cell curve, the state b case, mentioned above, has an average potential of 14.4 V versus ∆Vmax ) 14.5 V. Thus, the minimum potential for an undesired steady state is below the expected trifurcation point. From the full model, this is shown to be the case as well. The state Bmin, mentioned earlier, has cell potentials, 14.0|21.8|14.0. This minimum average potential (of 16.6 V) for an inefficient state is less than the trifurcation potential of 17.3 V. This similarity to the result expected from the single-cell curve suggests that a trend in part 1 applies for fully interactive cells. Namely, the safe potential limit to avoid undesirable states decreases (relative to the furcation potential) with increasing number of cells. However, there is no indication of how the furcation potential changes with larger-cell systems. This information requires full simulations of the various manycell cases. A related diagnostic of the importance of the cell interactions is a greater-than-expected potential in states B and C. For instance, in part 1, a state b is envisioned from the single-cell curve (cf. solid curve in Figure 2) involving two 12-V cells and a 19.6-V cell, because they independently demonstrate Itot ) 5930 A. However, for three interactive cells, we find that a state B with this production rate occurs with 12.6|25.5|12.6

V. The high-potential cell runs at nearly 26 V rather than 19.6 V. The effects of the cell interactions can be explored by examining the reactor profiles for this steady-state B. Finally, in reference to Figure 2, we elaborate that permutations of high- and low-potential cells in the states B and C are expected to exist. The states simulated involve two almost identically functioning cells at the ends in positions 1 and 3. Cell 2 runs rather differently in the cases mentioned, such as in the scenario 12.6|25.5|12.6 V. However, states probably exist with a different ordering of the high- and lowpotential cells. For instance, a state B can have the highpotential cell in position 1 or 3 rather than in the middle. Therefore, within each state designated B or C, there may be three permutations because the differently running cell can be in any of the three positions. Permutations are irrelevant for an identically functioning array or for a system with no interactions. However, one high-potential cell in position 2 versus at one of the ends makes a difference. In the former case, the highpotential cell transfers excess heat to two neighbors; in the latter case, the cell can pass thermal energy to only one adjacent channel. The different permutations are not explored here. Instead, one state-B scenario is discussed in detail here to capture the essence of differences and interactions between high- and lowpotential cells operating simultaneously in an array. 3.2. Reactor Profiles. We present one-dimensional profiles for the state-B scenario, 12.6|25.5|12.6 V, and an identical-cell state A (12 V) both at the same production rate, 5930 A. State B involves cells 1 and 3 as low-potential cells. Again, positions 1 and 3 represent the outer cells in a three-cell version of the reactor in Figure 2 of part 1.1 The variables these two state-B, lowpotential cells differ but indiscernibly. As discussed later, a very small difference results from heat transfer. For now, the differences are ignored as we report the changes in liquid volume fraction, reaction rate, temperature, and pressure with height (z) inside the reactor for the two steady states. These profiles demonstrate the various flow regimes, the reactor efficiency, and the interactions of cells functioning differently at the same Itot. As described (cf. Figure 1 of part 1), liquid-phase- and gas-phase-continuous (mist) flow regimes are possible within the reactor. In order of increasing gas flow rate, bubble flow, heading flow, slug flow, and transition-tomist flow can be encountered. Seen in Figure 3, the liquid-volume-fraction (φ) profile inside the cells of states A and B reflects the flow-regime changes. The initial fraction is unity because the reactor is liquid-fed. The identical cells of state A show the most gradual decrease in liquid fraction, interrupted by the change from bubble flow to slug flow through the heading flow regime around z ) 1.2 m. The stream remains liquid-phase continuous over the entire height of the reactor. For the 12.6-V cells of state B, the regime transitions occur closer to the inlet, and the exit volume fraction (0.13) is lower than for the state-A case (φz)H ) 0.22). Although not noticeable in this graph, the transition-to-mist flow begins in the top 0.1 m of the state-B, low-potential cells. Finally, φ in the 25.5-V cell falls the fastest. Bubble flow ends 0.22 m into the channel, shortly after which slug flow begins at roughly z ) 0.29 m. The transition-to-mist flow starts at a height of 1.0 m. The liquid volume fraction at this point is about

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Figure 3. Liquid-volume-fraction profiles for cells within different steady states mentioned in Figure 2. The low-potential cells of steady-state B progress through the liquid-phase-continuous flow regimes at lower heights than in the steady-state-A case. These cells all maintain above 15% liquid. The high-potential cell of steady-state B has the most rapid decrease in liquid volume fraction. After the transition to mist flow begins, liquid occupies less than 5% of the channel.

Figure 4. Current distribution for steady-states A and B, with the same total production rate, 5930 A. The current density at the reactor inlet, with no gas present, is proportional to the cell potential. For the steady-state A, the decrease is slightly more gradual than for the 12-V cells in steady-state B, due to additional heat exchange from the adjacent high-potential cell. A dramatic drop in current density occurs for the high-potential cell where non-liquid-phase-continuous flow begins.

0.18. Beyond this height, the channel is less than 5% liquid, with φz)H ) 0.019. Despite the small liquid fraction, full gas-phase continuous mist flow does not occur. Intermittent slugs of liquid still pass throughout the upper portion of the reactor. Corresponding current distributions, shown in Figure 4, follow the liquid-volume-fraction changes. The local reaction rate, i, is plotted versus the reactor height for the state-A and -B cells. Because all cells have the same total production rate, 5930 A, the integrated areas under the curves are equal. However, the reaction-rate distributions differ. The initial current density originates from the potential drop reflecting the anodic-film and liquid resistances only (φz)0 ) 1). The state-A rate is initially lower than the other cases because ∆V is the smallest, and all resistances are identical. The state-A cells show a slight decrease in current density with height. The decline is small because the dominant electrical resistance comes from the anodic film.2 A small dip in the current density occurs at the transition from bubble flow at 1.2 m, as displayed in the liquidvolume-fraction profile of Figure 3. State-A cells generate the least gas because of relatively low ohmic heating and adiabatic operation. The gas-phase resistance is smallest, and the decline in reaction rate is the least

Figure 5. Temperature profile for cells of different steady states. From the inlet temperature of 50 °C, the rise is sharpest for the high-potential cell of steady-state B because of ohmic heating. The low-potential cells of this state increase in temperature more than those of the steady-state-A case because of heat transfer from the 25.5-V cell in state B.

but still 19%. The current distributions show less uniformity in the state-B cells. The 12.6-V-cell profile closely follows that of state A. The 12.6-V initial reaction rate is only slightly higher because the potentials are close. Somewhat more gas is produced, leading to the noticeable drop in current density upon entering nonliquid-phase-continuous flow very near the cell exit. The reaction rate diminishes 35% from entrance to exit. Still, the majority of the resistance is due to the anodic film. The high-potential cell, however, demonstrates a dramatic decrease in the local current density due to gas-phase resistance. The initial reaction rate starts the highest of the three cases, 362 mA/cm2. The currentdensity profile shows the sharpest initial decline because this cell produces gas at the fastest rate. As with the other cases, the profile shows a minor dip in the current density as bubble flow ends at z ) 0.22 m. Unlike the other cells, however, the transition-to-mist flow, occurring over the majority of the channel, dramatically reduces the local reaction rate. At the end of liquid-phase-continuous flow at z ) 1.0 m, the current density is 79% of its initial value or 287 mA/cm2. Over the next 0.5 m, the reaction rate drops by a factor of 4. Over the remainder of the cell, the current density steadily declines, reaching 22 mA/cm2 at the exit. The high-potential cell sustains a 94% or 16-fold drop in reaction rate overall. More than 88% of the reaction occurs in the lower half of the channel height. In Figure 5, the temperature profile starting at the inlet temperature of 50 °C is plotted versus height. From the thermal energy equation in part 1,1 the main causes for temperature change are electrical or joule heating (∆ViW) and evaporative cooling (Σ∆Hvap,iRvap,i). Also, the influence of cell-to-cell heat transfer can be seen in state B. Electrical heating leads to the initial increase in temperature for all three cases and results in a steady, linear climb in the 12-V, state-A cells over the first 1.0 m. Then, evaporative cooling causes the curve to level between 70 and 75 °C. The reactant mixture approaches the boiling point of HF. Thus, evaporation prevents significant temperature rise for z > 1.5 m in the 12-V case. Although electrochemical hydrogen gas evolution initiates the gas phase, most of the gas is due to evaporation of liquid components, as quantitatively described in detail previously2 and briefly below. The 25.5-V cell of state B increases in temperature the most. Joule heating causes a rapid rise to 73 °C at z ) 0.29 m. Here, appreciable evaporative cooling begins, ac-


companied by a decline in the joule heating rate. Slug flow occurs (0.29 < z < 1.1 m), marked by a drop in liquid volume fraction (cf. Figure 3). The current density falls during slug flow (cf. Figure 4), reducing joule heating. The significant drop in i in the transition-tomist flow (z > 1.1 m) further decreases joule heating. This current-density decline also means little H2 generation. As a result, evaporative cooling reduces as well. Thus, the primary causes of temperature change, joule heating, and evaporation, essentially cease once nonliquid-phase-continuous flow begins. Consequently, the temperature changes by less than 0.5 °C in the transition regime (z > 1.1 m). Also, along this height, heat transfer to the adjacent, low-potential cells is small because their temperatures are elevated as well. Over the full channel height, the temperatures in the stateB, low-potential cells lie between the state-A and highpotential-cell values. The 12.6-V cells begin with the same rise as the state-A case over the first 0.1 m. However, heat transfer from the high-potential cell soon elevates the temperature relative to the state-A cells. Upon reaching 72 °C at z ) 0.8 m, the streams in the 12.6-V cells undergo significant evaporative cooling, as with the other cases. Extensive evaporation occurs with electrochemical H2 generation for two reasons. The volatility of liquid components, RH, HF, and RF (the fluorinated product), results in high local gas mole fractions for these species, and the temperature increase further increases this volatility. For instance, at z ) 1.5 m inside the state-A cell, the temperature-dependent, gas-phase fractions are 0.77, 0.12, 0.07, and 0.03 for HF, H6F6, RH, and H2, respectively, at the local liquid composition. (Less than 1 mol % is RF.) This composition stipulates that the non-H2 species are 32 times more abundant in the gas. That is, 32 mol of evaporated species accompany every mole of hydrogen generated by the reaction. Moreover, the increase in temperature further skews this factor, necessitating an even greater evaporation-rateto-current-density ratio. As mentioned, all variables inside the two state-B, low-potential cells 1 and 3 do differ, but indiscernibly. For instance, the exit temperatures differ by 3 × 10-4 °C. This miniscule difference comes from heat transfer. Thermal transfer between cells 1 and 2 involves the current density of the cell 2, the cathodic (gas-evolving) side of the wall (cf. eqs 14, 15, and 16). However, the heat flux from cell 2 to cell 3 depends on the cathodic gas-evolution rate of cell 3. The middle channel has a nonuniform current distribution versus the relatively level current density of cell 3, leading to a difference in the heat-transfer coefficient, eq 14. A greater disparity is expected when the combination of high- and lowpotential cells is not symmetric. For instance, an inefficient, end-monopolar cell in state B loses heat to only one neighbor. The heat received in the middle cell probably causes more gas generation to occur in this cell than in the end, low-potential cell. The final graph, Figure 6, shows the pressure variation with height for the three cases. Referring to the momentum equation in part 1,1 we find that the gravitational head, 〈F〉g, and to some extent the wall stress, τw/b, determine the pressure profiles. The stateB, 12.6-V cells and the state-A cells have similar liquid fractions, making the gas-liquid densities, 〈F〉, close; thus, the two cases have essentially parallel pressure profiles. In the high-potential cell of state B, the liquid

Figure 6. Pressure profile for cells of different steady states. The steady-state-B low-potential cells trace a profile parallel to that of steady-state A. The curves do not overlap because of a redistribution of flow in steady-state B. The high-potential cell of state B has a more gradual decrease in pressure because of a smaller gravitational head, due to a lower liquid volume fraction. The cell-to-cell pressure difference in steady-state B may lead to mechanical failure.

volume fraction is the smallest of the cases, resulting in the least gravitational head. Hence, this cell displays a more gradual pressure decrease with height. The different cell exit pressures are due to the flow distributors. Here, the exit manifold pressure is set to 3 atm. Thus, the head loss inside the distributor determines the cell pressure at z ) H. The pressure curves of the state-B efficient cells and the state-A cells are parallel but offset because of the different exitdistributor pressure drops. Shear stress dominates this pressure drop because of high velocities inside the small cross-section distributors. The gas velocities are particularly high. Each 12.6-V cell of state B produces 50% more gas (by volume) than a state-A, 12-V cell. The difference is largely due to evaporation caused by heat transferred from the high-potential cell in state B. Because the shear stress depends on the square of velocity, the frictional pressure drop in the low-potential cells of state B is more than twice that in cells of state A. Thus, the exit pressure (3.35 atm) of the 12.6-V cell is greater than pz)H ()3.24 atm) for the state-A cells. The 25.5-V cell of state B generates 20% more gas than each state-B, low-potential cell. Consequently, the 25.5-V exit pressure drop is the greatest at 0.40 atm. Because of common inlet and exit manifolds, different behavior inside the state-B cells leads to a nonuniform inlet flow distribution. As mentioned in the mathematical model section, the average inlet flow rate from a common manifold is specified. For identically functioning cells, the inlet flow rates are all equal to this average. The state-A cells have an inlet velocity of 0.152 m/s. For cells running differently, the rate to each channel is different. For state B, a 0.157 m/s stream enters the 12.6-V cells, and the flow in the inefficient cell starts at 0.143 m/s, for an average of 0.152 m/s. The flow distributes itself so that the pressure drop from the inlet manifold to exit manifold is the same for all channels. Hence, the high-potential cell (with more gas and thus more flow resistance) must have a lower inlet flow rate than the low-potential cells. The pressure profiles in Figure 6 show that a force imbalance exists for the adjacent cells in state B. In light of the known failure of the system due to blockage of the thin cells,8 we examine the mechanical stability of the narrow-channel reactor. The maximum deflection, δ, of a bipolar electrode wall sustaining a pressure

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difference comes from mechanical beam theory,21

δ)

5 |pJc - pJc+1| W4 32 E b 3

(18)

w

where E is the modulus of elasticity. For the maximum cell-to-cell pressure difference (0.049 atm), the deflection is 0.83 m, much larger than the channel thickness, 3.2 mm. Thus, elastic deformation theory finds that the bipolar electrode bends, closing the low-potential cells. For this state B to exist without mechanical failure of the system, additional structural support is needed. Using eq 18, we determine that 6 times the present bipolar wall thickness, bw ) 2.26 × 10-3 m, is required to keep the deflection smaller than the channel thickness. Alternatively, vertical supports in the breadth dimension spaced 0.30 m apart can prevent closure; the current breadth dimension, W, of the modeled reactor is 1.22 m. The beam eq 18 provides only an estimate because the pressure difference varies with height. Also, the requirements to prevent mechanical failure may be more stringent because an acceptable deflection is only a fraction of the channel thickness, b. In summary, we demonstrate the likely failure of the 12.6|25.5|12.6 V case with the dimensions specified. This plate-deflection calculation reveals another unfavorable result of nonidentical cell operation. Vertical potential variation inside the bipolar electrodes is another multiple-cell consideration. From the simulations, we find that this potential change over the reactor height is small. For the state B, 12.6|25.5|12.6, the intermediate electrode potentials change by about 0.13 V over the entire reactor height. For the calculated state B with the greatest cell-to-cell current density difference, 9.1|43.5|9.1 V, the variation is also small, 0.10 V. The vertical current density, iz,wall, in the electrode walls does become large, on the order of 100 A/cm2. However, the high electrical conductivity of electrode material, nickel, results in very little resistance, making the vertical potential differences small. Of the cell-to-cell interactions addressed, heat transfer most significantly affects the three-cell production rates; in particular, the trifurcation point (cf. Figure 2) is largely determined by thermal-energy transfer. As mentioned, the expected trifurcation potential from the single-cell (solid) curve is 14.5 V rather than the threecell model result of 17.3 V. The disparity is explained by considering the total current of each individual cell with and without heat transfer. The single-cell, production-rate curve applies for adiabatic operation. However, net heat input lowers the total current. This occurs because thermal energy evaporates excess gas, increasing the electrical resistance. The maximum in Itot is lower and is reached at a potential less than ∆Vmax. Conversely, thermal-energy loss increases the cell production rate. Thus, for a (cooled) high-potential cell at the production rates just beyond ∆Vmax, no (heated) lowpotential cells in the same array can reach these rates. Heat transfer to the low-∆V cells prevents them from achieving the elevated Itot of a high-potential cell. Hence, combinations of low- and high-potential cells do not exist for production rates immediately after the maximum in the three-cell system. The high-potential-cell production rate falls enough at an applied potential of 17.3 V for a lower-potential, heat-receiving neighbor to exist with the same Itot. The respective elevated and lowered rate for heated and cooled cells can be seen in comparing the state-B

cells to their adiabatic counterparts. In the state B of 5930 A, heat is transferred from the middle 25.5-V cell to the adjacent 12.6-V cells. For an isolated 25.5-V cell, Itot is only 4840 A. Yet, heat loss in state B increases this rate to 5930 A. An adiabatic 12.6-V cell runs at about 6200 A; heat input in state B drops this Itot to 5930 A. Therefore, heat transfer increases the highpotential cell rate and lowers the 12.6-V cell rate, making the total currents match. The matching can occur only for rates lower than that of the trifurcation point of 6500 A. It is unclear how the furcation potential changes with larger-cell systems. On one hand, a high-potential cell can have many low-potential neighbors in large-cell systems to disperse the excess heat. This effective cooling reservoir makes a high-potential cell run with a greater-than-adiabatic total current. The low-potential cells then may not be able to achieve the elevated highpotential rates, thereby increasing the furcation potential. On the other hand, each low-potential cell accepts less heat if there are more cells. Thus, the total current in the efficient cells is reduced less severely, driving the furcation potential lower. The position of the furcation point is important in estimating the minimum potential to avoid undesired states. The full model with interactions can be used to map precisely the production-rate-versus-potential curves for many-cell systems. However, application of the full model to large-cell systems requires long computation time. Thus, knowledge of the furcation point along with the trend in Figure 4 of part 1 can be used to estimate the lowest potential of the unfavorable states. This potential is the upper limit to guarantee all efficient cells. The graph shows that this limit decreases with an increased number of cells. The nonuniformity of the current density in state B and the excess power consideration reveal the inefficiency of the nonidentical cell states. In the state-B high-potential cell, more than 88% of the reaction occurs in the lower half of the channel height. Reactor space and electrode height are poorly utilized. Also, each highpotential state has a low-potential counterpart, achieving the same production rate with less power consumption. Hence, this work demonstrates that operating states with a high-potential cell are a waste of power and capital costs through poor utilization. However, to conduct a full economic optimization, one needs additional information. First, one needs to weigh the manufacturing and finance costs of the system. The expense of reactor materials, especially the nickel electrodes, encourages operation yielding a relatively uniform reaction rate. Uniformity of the rate is accomplished by avoiding non-liquid-phase-continuous flow. This objective, in turn, can be accomplished by running below a safe average cell potential. However, because of the value of the product, one seeks the highest, safe production rate. Reaction selectivity and frequency of shutdowns under various conditions are additional considerations addressed by others.8,17 Still, given the financial parameters, the production-rate curve and variable profiles allow for economic optimization of the Simons process flow reactor. The model presented here includes the important aspects of a continuously operating system. The work presented provides directions for future studies. Experimental verification of the flow-regime and production-rate curves is important. A gas-evolving


reaction in a safer solvent than HF may be used in a model system, for example. In addition, a study of dynamics considering multiple flow regimes may determine how to maintain efficient states and avert undesired states over time. 4. Conclusions Cell-to-cell interactions are considered in the proposed mathematical model and finite-difference simulation of a three-cell electrochemical flow reactor. The liquid-fed process for fluorination of organic compounds generates H2 gas as a byproduct, creating a two-phase system. Account for gas-liquid flow regimes, chemical association, two-phase electrical resistance, and phase equilibrium from previous work is used.1,2 Yet, the cell interactive considerations, distinct from previous efforts, allow for the simulation of multiple steady states where parallel flow cells in a serial electrical array operate differently. Bipolar electrodes, modeled to vary vertically in potential, separate adjacent cells. Convection from two-phase flow and nucleate gas evolution account for cell-to-cell heat transfer. Other reactor features include inlet and exit manifolds connected to the cells through flow-constrictive distributors. For the three-cell system, we find multiple steady states for a given production rate, as predicted by in part 1 of this work.1 States with one to three inefficient cells branch from the production-rate-curve trifurcation point: Itot ) 6500 A, (Vtot/Nc) ) 17.3 V. The minimum potential of an undesired state (16.6 V per cell) is lower than the trifurcation potential, agreeing with a singlecell extrapolation.1 Height profiles of liquid volume fraction, current density, temperature, and pressure further reveal undesired aspects of a high-potential steady state. A threecell scenario consisting of a middle 25.5-V cell with adjacent 12.6-V cells is compared to a 12.0-V identicalcell case, all with Itot ) 5930 A. The large amount of gas generated in the 25.5-V cell, seen in the volumefraction profile, leads to non-liquid-phase-continuous flow beyond 1.0 m of the 3.66-m height reactor. The corresponding drastic reduction in current density over most of the cell height demonstrates poor utilization of reactor volume and wasted power in the state of differently operating cells. This inefficient case requires more downstream cooling because its cells also have higher exit-stream temperatures than the low-potential, identical cells. Finally, the pressure profiles for differently operating cells in an array can lead to a physical instability of the reactor. For the three-cell, 5930-A scenario with a middle high-potential cell, the separating bipolar walls bend, collapsing the end channels for the dimensions specified. Additional structural support and the avoidance of differently operating cells are recommended. Acknowledgment We thank Alcoa for support of this work through an unrestricted research grant. Nomenclature a ) constant for heat-transfer coefficient in eq 14, 0.0576 m2/3 s-1/2 A-1/3 Adist ) cross-sectional area per flow distributor, 6.5 × 10-4 m2

b ) cell channel thickness (excluding anodic film), 0.0032 m bw ) thickness of bipolar nickel electrode, 2.16 × 10-3 m Cp ) molar heat capacity, J/(mol‚K) C ˆ p ) average mass heat capacity of a phase, J/(kg‚K) CE ) conductivity enhancer ein con ) contraction head-loss coefficient, inlet manifold to distributor, 0.44 ein exp ) expansion head-loss coefficient inlet distributor to cell, 25.0 E ) modulus of elasticity for nickel,19 2.07 × 1011 N/m2 f ) friction factor F ) Faraday’s constant, 96487 C/equiv Ftp ) two-phase enhancement factor for heat transfer g ) gravitational acceleration, 9.8 kg‚m/s2 hJc,Jc+1 ) overall heat-transfer coefficient between cells Jc and Jc+1, W/m2‚K hfilm ) heat-transfer coefficient of anodic film18 (Value uses a film thickness of 0.1 mm and a temperature-interpolated polymer conductivity18), 1590 W/m2‚K hflow ) heat-transfer coefficient due to gas-liquid flow, W/m2‚K hgas ) heat-transfer coefficient due to H2 gas evolution, W/m2‚K hgas-flow ) heat-transfer coefficient due to combined gasliquid flow and gas evolution, W/m2‚K hwall ) heat-transfer coefficient of nickel electrode,19 40 800 W/m2‚K Hdist ) flow distributor height, 0.15 m ∆Hrxn ) enthalpy of overall reaction, 26 700 J/mol ∆Hvap ) enthalpy of vaporization (31 900 for RH, 32 040 for RF) J/mol i ) local current density, A/m2 iz,wall ) vertical current density inside bipolar electrode, A/m2 I(z) ) cumulative current up to local position z, A Jc ) cell reference number Khex ) equilibrium parameter for HF hexamerization, (N/ m2)-5 Kvap ) equilibrium parameter for HF evaporationsliquid to monomer, N/m2 n ) molar flow rate, mol/s Nc ) total number of cells in reactor p ) pressure, N/m2 exit pman ) exit manifold pressure, 304 000 N/m2 (3 atm) in pman ) inlet manifold pressure, N/m2 pvap ) vapor pressure of pure species, N/m2 ∆p ) contraction or expansion pressure drop, N/m2 Pr ) Prandtl number qht ) net heat-transfer flux into cell, W/m2 R ) ideal-gas constant, 8.314 J/(mol‚K) Rfilm ) specific area resistivity of anodic film, 0.006 Ω‚m2 Rvap ) rate of evaporation, mol/(m‚s) Re ) Reynolds number RH ) hydraulic radius of distributor, 7.58 × 10-4 m RH ) organic reactant RF ) fluorinated organic product S ) slip velocity, m/s T ) absolute temperature, K U ) open-circuit potential from linear approximation,22 3.0 V v ) partial molar volume, m3/mol VJc ) potential of cell Jc anode, V Vsg ) gas superficial velocity, m/s Vsl ) liquid superficial velocity, m/s Vtot ) total applied potential, V W ) channel breadth, 1.22 m x ) liquid mole fraction y ) gas mole fraction z ) height above reactor entrance, m

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Greek δ ) deflection of electrode wall, m κ ) electrical conductivity of the gas-liquid mixture, (Ω‚m)-1 κl ) electrical conductivity of the liquid phase,23 (Ω‚m)-1 κw ) nickel electrical conductivity,19 1.28 × 107 (Ω‚m)-1 κthermal ) thermal conductivity of liquid (HF),24 0.4 W/m‚K l κthermal ) thermal conductivity of gas (HF),24 0.023 W/m‚K g µl ) liquid viscosity (HF),24 1.79 × 10-4 Pa‚s µg ) gas viscosity (HF),24 1.35 × 10-5 Pa‚s ν ) stoichiometric coefficient in overall reaction F ) mass density, kg/m3 τw ) wall shear stress, N/m2 φ ) liquid-phase volume fraction Subscripts con ) flow contraction dist ) flow distributor quantity exp ) flow expansion g ) gas phase l ) liquid phase man ) manifold quantity

Literature Cited (1) Drake, J. A.; Radke, C. J.; Newman, J. Modeling of a TwoPhase Electrochemical Reactor for the Fluorination of Organic Compounds: I. Maximum Production Rate. Ind. Eng. Chem. Res. 2001, 40, 3109. (2) Drake, J. A.; Radke, C. J.; Newman, J. Profiles and Performance Curves in a Parallel-Plate Reactor for the Electrochemical Fluorination of Hydrocarbons. J. Electrochem. Soc. 1998, 145, 1578. (3) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; John Wiley & Sons: New York, 1990; pp 425-429. (4) Fogler, H. S. Elements of Chemical Reaction Engineering; Prentice-Hall: Englewood Cliffs, NJ, 1992; pp 372-374. (5) Sheintuch, M.; Tartakovsky, B.; Narkis, N.; Rebhun, M. Substrate-Inhibition and Multiple States in a Continuous Nitrification Process.Water Res. 1995, 29, 953. (6) Simons, J. Fluorine Chemistry; Academic Press: New York, 1950; p 414. (7) Childs, W. V.; Klink, F. W.; Smeltzer, J. C.; Spangler, J. C. Bipolar Flow Cell and Process for Electrochemical Fluorination. U.S. Patent 5,322,597, 1994. (8) Schulte, D. C. et al. Electrochemical Fluorination Using Interrupted Current. International Patent WO 98/50603, 1998. [Additional authors in dispute.] (9) Jha, K.; Bauer, G. L.; Weidner, J. W. Steady-State Modeling of a Parallel-Plate Electrochemical Fluorination Reactor. J. Electrochem. Soc. 1998, 145, 3521.

(10) Jha, K.; Bauer G. L.; Weidner J. W. Dynamic Simulation of a Parallel-Plate Electrochemical Fluorination Reactor. J. Appl. Electrochem. 1999, 30, 85. (11) Steiner, D.; Taborek, J. Flow Boiling Heat Transfer in Vertical Tubes Correlated by an Asymptotic Model. Heat Transfer Eng. 1992, 13, 43. (12) Drake, J. Numerical Simulation of a Simons Gas-Liquid Electrochemical Flow Reactor: Cell Profiles, Multiple Steady States, and Transient Linear Stability. Ph.D. Thesis. University of California, Berkeley, CA, 2000. (13) Denn, M. M. Process Fluid Mechanics; Prentice Hall: Englewood Cliffs, NJ, 1980; p 93. (14) Pickett, D. J. Electrochemical Reactor Design; Elsevier Scientific Publishing Co.: New York, 1977; pp 125-135. (15) Monrad, C. C.; Pelton, J. F. Heat Transfer by Convection in Annular Spaces. Trans. Am. Inst. Chem. Eng. 1942, 38, 593. (16) Rousar, I.; Cezner, V. Transfer of Mass or Heat to an Electrode in the Region of Hydrogen EvolutionsI. Theory and II. Experimental Verification of Mass and Heat Transfer Equations. Electrochim. Acta 1975, 20, 289. (17) Childs, W. V.; Christensen, L.; Klink, F. W.; Kolpin, C. F. In Organic Electrochemistry: An Introduction and Guide; Lund, H., Baizer, M., Eds.; Marcel Dekker: New York, 1991; pp 11031112. (18) Ho, C. Y.; Desai, P. D.; Wu, K. Y.; Havill, T. N.; Lee T. Y. Thermophysical Properties of Polystyrene and Poly(vinyl chloride). Proc. Symp. Thermophys. Prop. 1977, 7, 198. (19) Weast, R. C., Ed. CRC Handbook of Chemistry and Physics; CRC Press Inc.: West Palm Beach, FL, 1978; pp. D-224, E-14, F-168. (20) Newman, J. Electrochemical Systems; Prentice Hall: Englewood Cliffs, NJ, 1991; pp 539-555. (21) Popov, E. P. Mechanics of Materials; Prentice Hall: Englewood Cliffs, NJ, 1976; pp 364-365. (22) Drakesmith, F. G.; Hughes, D. A. Electrochemical Fluorination Using Porous Nickel and Foam Anodes. J. Fluorine Chem. 1986, 32, 103. (23) Clark, R. P.; Moser, J. R. Electrical Conductivity Measurements in Anhydrous Hydrogen Fluoride. J. Electrochem. Soc. 1971, 118, 666. (24) Horvath, A. L. Physical Properties of Inorganic Compounds; Crane, Russak & Company, Inc.: New York, 1975; pp 87101.

Received for review June 26, 2000 Revised manuscript received February 16, 2001 Accepted April 23, 2001 IE0006108

Modeling of a Two-Phase Electrochemical Reactor for the Fluorination

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