In this section, we examine the different polarizations or voltage losses present in typical electrolytic cells. The analysis is largely the same as for any electrochemical cell. Again, the objective is to establish the relationship between the potential of the cell and the current density. Recall from Chapter 4 for an electrolytic cell,
(4.58b)
We again use a diaphragm cell as an example. Let’s start by examining ohmic losses, which are particularly important for industrial electrolysis cells. The potential drop across a gap of distance h due to current flow between two parallel electrodes is
(14.4)
For the diaphragm cell, however, the situation is a bit more complex as shown in Figure 14.3. Separating the two electrodes is a diaphragm, which is needed to prevent the product gases from mixing. The diaphragm of thickness hd is a porous sheet with an effective conductivity. We can estimate the effective conductivity by modifying the conductivity of the electrolyte to account for the porosity and tortuosity of the diaphragm, similar to what was done in Chapter 5. Additionally, there is a gap between the diaphragm and each of the two electrodes, ha and hc. Finally, there is resistance associated with current flow through and connections to the electrical leads. Thus, the total ohmic resistance of the diaphragm cell is
(14.5)
The first three terms on the right side of Equation 14.5 can be readily evaluated. Rleads depends on the specific connections and bus bars used in the system. For our purposes, you will be given the value for this resistance. Some systems may also have an additional term in Equation 14.5 to account for ohmic losses across imperfect interfaces, frequently referred to as contact resistance.
Figure 14.3 Cell resistances.
Gas evolution occurs in many industrial electrolysis cells and can be associated with a product (e.g., Cl2 in the diaphragm cell) or with side reactions. In Chapter 4, we saw how gas evolution increases the rate of mass transfer in a cell. Gas evolution can also have a negative effect by increasing ohmic losses. As shown in Figure 14.4, gas that evolves at an electrode produces bubbles that rise along the length of the electrode due to buoyancy forces. Gas bubbles displace the electrolyte and cannot carry current. These bubbles reduce the effective conductivity of the solution in a fashion similar to the porous membrane discussed in Chapter 5. One simple expression for this effect, useful for gas fractions up to about 40%, is
(14.6)
where εg is the volume fraction of gas in the gap. As expected, the conductivity decreases as the volume fraction of bubbles increases, providing greater resistance to current flow. A simple way to use this expression is to assume that the distribution of bubbles in the electrolyte is uniform and that the bubbles occupy a certain fraction of the volume, which can be estimated from the height change that occurs in the level of the electrolyte as a result of the gas evolution.
Figure 14.4 Evolution of gas on an electrode.
To more accurately account for the volume fraction of bubbles as a function of height and to connect the local volume fraction explicitly to the local current density, we consider some early work in this area by Charles Tobias that applies at low current densities where the bubble formation is not sufficient to cause circulation of the electrolyte. Specifically, the analysis assumes a stagnant electrolyte, no interaction between bubbles, and a single, constant bubble velocity for all bubbles. The ideal gas law is also assumed to apply. We make the additional assumption that kinetic overpotentials are not significant and that the current is controlled by ohmic losses at these low current densities. With these assumptions, the following expressions result:
(14.7)
(14.8)
where K is the gas effect parameter,
(14.9)
In these equations, ix is the local current density beginning at the bottom of the electrode, where x = 0, and iavg is obtained by integrating the local current density over the electrode surface. The local current density decreases with increasing height due to the presence of more bubbles (see Figure 14.4). Also, as before, h is the gap between electrodes and L is the vertical height of the electrode.
With the assumptions described above, we can also write
(14.10)
Combining Equations 14.8 and 14.10 yields
(14.11)
and
(14.12)
Our goal is to find the potential of the cell that corresponds to a particular current density, which of course is related to the production rate. To do this, we substitute the known current density into iavg in Equation 14.8 and solve for . Note that K is a function of . K also includes the bubble velocity, which depends on the size of the bubble, the difference in density, and viscosity of the liquid. For single, small (less than 0.7 mm in diameter), spherical bubbles at low Re, Stokes flow gives
(14.13)
Bubble diameters (db) of 0.05−0.1 mm are typical. Use of these equations to calculate is shown in Illustration 14.5.
DIMENSIONALLY STABLE ANODE (DSA)
One of the most significant advancements in industrial electrochemistry is the development of stable metal oxide electrodes or DSA. The first anodes for chlor-alkali were carbon and then graphite, both of which were consumed over time, increasing the electrode gap and requiring costly maintenance. Titanium is stable in the harsh environment of the chlor-alkali cell, but forms a nonconducting oxide. The key innovation was the application of conductive oxides on Ti to produce a “nonconsumable,” dimensionally stable electrode. Also, with titanium as a support, the electrode could be constructed in the form of meshes and expanded metals. The open structure allowed gases to be removed more easily, thus allowing the gap to be reduced.
ILLUSTRATION 14.5
Estimation of Ohmic Resistance in a Cell Gap with Gas Evolution
A two-electron reaction with a gas-phase product takes place in an undivided cell (no separator) at an average current of 20 A. The pressure is 250 kPa and the temperature is 300 K. The gap between the electrodes is 4 mm, the length (vertical) of the electrode is 0.5 m, and its width is 0.5 m. The conductivity of the solution without bubbles is 5 S m−1. The density and viscosity of the liquid are, respectively, 1100 kg·m−3 and 0.00105 Pa·s. The gas density is approximately 3.2 kg·m−3 at the stated pressure.
SOLUTION:
All the quantities needed for the gas-effect parameter, K, are known except the bubble velocity and ΔV. Therefore, we will first determine the bubble velocity. Once that is known, we can express K as a function of ΔV and then substitute that function into Equation 14.9 in order to solve for ΔV.
We can now calculate K as function of ΔV. Substituting the appropriate values into Equation 14.9 yields K = 1.419 ΔV.
Next we substitute this expression for K into Equation 14.10 for the average current density. The average current density is
Therefore,
Solving this expression for ΔV gives ΔV = 0.0687 V.
The voltage drop is not large due to the low current density, which is a requirement for use of the simplified analysis for the effect of bubbles. The impact of the bubbles on the effective conductivity is
This value is 93% of the original value without bubbles.
The procedure just illustrated only applies at very low currents where the electrolyte is stagnant. This condition is satisfied, at least approximately, when the Reynolds number defined earlier in Chapter 4,
(4.48)
is less than about 3; even so, the procedure is frequently used as a first approximation at significantly higher values of Re. Note that the velocity used in Equation 4.48 is the superficial gas velocity defined as
(14.14)
Also, the viscosity and density in Equation 4.48 are based on the liquid properties. Although this method only applies at low current densities, it provides a useful illustration of the effect of bubbles on the ohmic drop. In general, the flow is more complex as bubbles induce electrolyte flow and interact with each other. Some of the factors that are important include coalescence of bubbles, turbulence, the nonspherical shape of the bubbles, and the effect of the walls on the flow. Regardless of the complexity, there is a relationship between the gas evolution rate and the void volume, and that the void volume impacts the ohmic losses in solution.
The impact of gas evolution can be reduced by lowering the current density, increasing the pressure, and increasing the gap distance. At the same time, there are strong incentives to avoid these remedies in order to keep the overall size and cost low, and the energy efficiency high. There is clearly a need for system optimization. One notable engineering solution that has significantly reduced the ohmic resistance associated with bubbles is the use of perforated electrodes in some types of cells that allow gas to be removed from the backside of the electrode out of the current path, specifically the DSA used in the chlor-alkali industry. Another important strategy is the use of convective flow through the gap between electrodes to limit gas buildup by sweeping the gas out of the cell.
In addition to the ohmic losses described previously, kinetic losses or surface overpotentials can also be important. Industrial processes can most often be described with Tafel kinetics, and the corresponding overpotential can be calculated with use of that expression. This is true for a chlor-alkali cell where both electrode reactions are a bit sluggish. For chlorine evolution at the anode,
(14.15)
With equal to 2, the Tafel slope is roughly 30 mV per decade. Using i0 = 10 A·m−2 at 60 °C, the anode overpotential at a current density of 1940 A·m−2 is about 75 mV. Similarly, for the cathodic hydrogen reduction process,
(14.16)
With αc = 1 and i0 = 0.07 A·m−2, the overpotential at 1940 A·m−2 is about 0.29 V. The various polarizations or voltage losses at this current density are shown in Figure 14.5. Now that the voltage losses are known, we can estimate the cell voltage with Equation 4.58b to be 3.45 V. Concentration overpotential, which is not likely to be large, has been ignored. For a diaphragm cell operating at 60 °C, the equilibrium potential is 2.25 V. This is the same cell voltage used in Illustration 14.4 to determine the energy efficiency.
Figure 14.5 Polarization in an operating diaphragm chlor-alkali cell.
This methodology can be extended to develop a full polarization curve that relates the cell potential to the operating current density. Just as in other systems, this relationship is essential in designing electrolytic systems.
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