This section describes three performance measures that are useful for industrial electrolytic systems. We begin with the faradaic efficiency, which is the ratio of the product mass to the amount that could be obtained based on the current and Faraday’s law, as was introduced previously in Chapter 1. It can be written as

(14.1)

where m_i is the mass of the product, Q is the total charge passed during electrolysis,  is the mass flow rate of the desired product, and I is the total current (assumed constant). In cases where multiple products are formed, the faradaic efficiency for each product may be different. Note that the faradaic efficiency is dimensionless and has the same value if based on mass, moles, or amperes since it is calculated for a single chemical species. In Illustration 14.2, we rework the problem from Illustration 14.1 while accounting for the faradaic efficiency of the chlorine reaction, which for a diaphragm cell is ∼0.96 or 96%. It turns out that the faradaic efficiency for the Cl₂ reaction in a chlor-alkali cell is quite high. However, that is not the case for reactions in general, and faradaic efficiencies much less than 1 are frequently encountered.

ILLUSTRATION 14.2

Calculate the electrical power needed to produce the global supply of chlorine using diaphragm cells. Use an annual production of 44 million metric tons Cl₂ per year, and a faradaic efficiency of 96%.

1. As before, we convert the rate of production to kg·s⁻¹ of chlorine:
2. With a known mass flow rate and a known efficiency, we can use Equation 14.1 to determine the theoretical mass flow rate and the total required current.
3. To calculate the power, we simply multiply the total current by the operating voltage:

Another way to approach this problem is as follows:

1. Determine I_reaction, which is the current associated with just the desired product.
2. Determine The power can then be determined as above.

There are a number of reasons why the faradaic efficiency may be less than 1. One of the most important reasons is the presence of side reactions, which are reactions driven by the current flow that do not produce the desired product. We saw previously in Chapter 3 how a current efficiency can be used to account for such reactions. In a chlor-alkali cell, oxygen evolution at the positive electrode is an example of a parasitic (current consuming) side reaction.

The faradaic efficiency is generally different for the anode and the cathode. For example, the evolution of oxygen, an anodic reaction, reduces the amount of chlorine evolved, and thus, lowers the faradaic efficiency of the anode. However, oxygen evolution does not affect the efficiency of the cathode, since it does not influence the amount of caustic or hydrogen produced at that electrode.

Note that the faradaic efficiency as defined in Equation 14.1 is slightly different from the current efficiency introduced in Chapter 3. You should carefully compare the definitions and note the difference. The faradaic efficiency is focused on the final rate of product formation and not just on the electron transfer reaction(s) at the electrode. Therefore, while the current efficiency constitutes an important part of the faradaic efficiency, there are processes that affect the faradaic efficiency, but do not affect the current efficiency. One such process that causes a diminished faradaic efficiency involves the transport of material across the cell. For instance, current is consumed to produce chlorine gas at the positive electrode of a diaphragm cell. This chlorine has a small but significant solubility in the anolyte. The dissolved chlorine can be transported across the diaphragm to the catholyte. In the catholyte, chlorine reacts with NaOH and is not recovered. The chlorine lost to reaction in the catholyte represents a reduction in the faradaic efficiency, even though the current efficiency has not changed. Similarly, hydrogen and caustic can diffuse from the catholyte to the anolyte. This diffusion process lowers the faradaic efficiency and may also contaminate the products.

Contaminants that react with the desired products represent another possible source of lower faradaic efficiency. For example, sodium carbonate in the brine feed reacts with chlorine to reduce the amount of chlorine produced in the process per coulomb passed in the cell,

Finally, product recovery can affect the faradaic efficiency. For example, because of its solubility, a small fraction of the chlorine evolved in a diaphragm cell will be removed with the flow of the anolyte. This chlorine is lost and not recovered as product.

Even though it is not the only contributing factor, the current efficiency remains a critical component of the faradaic efficiency. The current efficiency for multiple electrochemical reactions that take place on a single electrode can be calculated as a function of potential and concentration if the kinetics of the reactions are known as a function of those variables. Such a calculation, useful for process optimization, is given in Illustration 14.2.

The next performance measure of interest is the space–time yield, which is the rate of production per volume of reactor. It is essentially a measure of reactor efficiency and is defined as

(14.2)

where a_r, the specific area for the reactor, is the area of the electrode at which the production takes place divided by the volume of the reactor. It is similar to the specific area, a, that was defined in Chapter 5 for porous electrodes. The difference is that the volume used for a_r is the total reactor volume rather than just the electrode volume used previously for a. M_i is the molecular weight, and  is the reactor volume. Y has units of kg s⁻¹·m⁻³. The current, I, and the current density, i, both correspond only to the portion of the current associated with the product of interest. In a situation where there are multiple products, for example, one product at the anode and another at the cathode, a space–time yield can be specified for each product with use of the area of the corresponding electrode. The quantity ia_r represents the current per unit volume of reactor. Economic analysis is at the heart of industrial processes—both capital and operating costs must be considered. The space–time yield is a parameter that includes the reactor volume, which will directly impact the capital cost. Our main tool to minimize the volume of the reactor is to raise the operating current density. The interplay of the current and volume is apparent from the space–time yield. For a fixed rate of production, , the volume is inversely proportional to the current. Thus, high currents or high current densities lead to smaller volumes and lower capital cost. On the other hand, high current densities result in high cell potentials, and high operating costs. Operating efficiency is addressed below with the third performance measure.

ILLUSTRATION 14.3

In a chlor-alkali cell, it is possible to have oxygen evolution in addition to chlorine evolution at the anode. The evolution of unwanted oxygen is, of course, a side reaction that reduces the faradaic efficiency. Determine the reaction rates for each of the two reactions for a chlorine overpotential of 0.08 V. Also calculate the faradaic efficiency at the anode, considering just the relative rate of these two reactions. Both reactions can be represented reasonably well with a Tafel expression. The pH = 4 on the anodic side of the cell, and the temperature is 60 °C. The equilibrium potentials versus SHE are given below at the conditions in the reactor.

Let’s first determine the overpotentials for the two reactions.

Now, we can calculate the current corresponding to each reaction. Please review Chapter 3 for the definition of the Tafel slope if necessary.

Based on the relative rates of these reactions alone (ignoring other contributions to the faradaic efficiency),

The magnitude of the current density also influences the type of reactor that can be used economically. When the current density is sufficiently high (>100 A·m⁻²), then simple two-dimensional electrodes can be used. However, high current densities are not possible for some systems due, for example, to mass-transfer limitations or side reactions. Processes with very low current densities (≤10 A·m⁻²) may require three-dimensional electrodes, such as the porous electrodes considered in Chapter 5, in order to be economically feasible; such electrodes can have high specific areas of ∼5000 m²·m⁻³. The reactor configurations considered in this chapter are quite simple. In contrast, a large variety of configurations of varying complexity are used in practice (see Further Reading section for more details).

The third performance measure we consider in this section is the energy efficiency defined as

(14.3)

This parameter is the product of the faradaic efficiency and the ratio of the equilibrium potential to the cell potential. The last term on the right side is the voltage efficiency defined for an electrolytic cell. Physically, the energy efficiency is the theoretical power required to complete the chemical conversion (I_RxU) divided by the actual power used (IV_cell). A key element of this efficiency is calculation of the cell potential, V_cell, which is discussed in the next section.

ILLUSTRATION 14.4

If the equilibrium voltage of the diaphragm cell considered above is 2.25 V, determine the energy efficiency of the cell if the operating voltage is 3.45 V.

Thus, only two-thirds of the energy added goes into the reaction itself. The balance of the energy ends up as heat or drives unwanted reactions.