Project Report – Modelling the Porocell System (Sept – Dec 2016)

Research Associate - Simon Coleman

1 Introduction

1.1 Project Overview

A Porocell is being designed to recover toxic metals from industrial effluent. The Porocell is an electrochemical reactor which recovers metal by electrodeposition onto a carbon felt porous electrode. Professor Roy’s research group at Strathclyde University is in collaboration with Hanoi University of Science and Technology (Vietnam) to develop and scale-up this electrochemical technology which can be applied in developing countries

It has been proven that toxic metals such as copper and gold can be removed using this technique at lab scale using a 1L size cell. This project required the development of a model for the Porocell system to determine its operational parameters, keeping in view safety and energy efficiency for a large scale industrial system. The development of mathematical models was carried out using COMSOL Multiphysics software. The results will be used to formulate reactor specifications and operational parameters.

1.2 Porocell system

The diagram in Fig. 1a shows the reactor and flow system used in the porocell system. The industrial effluent (i.e. electrolyte) enters at the bottom, flow through the porous electrode and up through the rop of the reactor. The cathode is the carbon felt porous electrode arranged in a cylinder shape, the anode is a cylindrical titanium sheet with a coating of mixed tantalum/iridium oxides. Porous electrodes are used for this system due to the high surface area/volume ratio of this 3-D electrode. This allows for efficient extraction of metal ions even from relatively dilute solutions. The dimensions of the cell are shown in Fig. 1b.

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Fig. 1 – Diagram of Porocell system (a) and Porocell reactor with dimensions (b)

2 Modelling of simple copper plate system

2.1 Geometry

A 2D asymmetric geometry was used for the model, which is revolved around the radius to create a cylinder shape. Initially a simple model was set-up with the cathode modelled as thin copper metal plates in order to test a simple version of the system. The geometry of this is shown in Fig. 2. The blue lines represent the cathode boundary and the red line represents the anode boundary. The other boundaries were modelled as insulation.

2.2 Description of model for simple copper plate system

A secondary current distribution was developed in COMSOL using the Butler-Volmer expression to describe the charge transfer reactions. The input values used for the model were provided by Dr. Todd Green and are described in table 1 and 2. Two reactions were modelled for the cathode; the copper electrodeposition reaction and hydrogen reaction. An oxygen reaction was modelled for the anode. These reactions and parameters are shown in table 2.

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Fig. 2 – Geometry of simple copper plate system

Table 1 – Input values used for the model

Properties ValueElectrolyte Conductivity 20 S/m

Eeq, Cu 0.31 V

Eeq, H 0.00 V

Eeq, O 1.23 V

A

potential was applied to this system using the following method. The electric potential at the cathode was set to 0V which implies that the electric potential at the anode is equal to the cell voltage. The potential of the electrolyte floats and adapts to satisfy the balance of current. This means an equal amount of current that leaves at cathode also enters at the anode, this therefore determines the overpotential at the anode and cathode. The real system applies constant current but this method is suitable for the initial simple model.

2.3 Current distribution for simple copper plate system

Fig. 3 shows the potential and current distributions for the simple copper plate system in 3D. The shape of the distribution looks reasonable for this type of system. The distribution is easier to illustrate in 2D, the following current distributions will therefore be shown in 2D as shown in fig. 4.

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Fig. 3 - Potential distribution (left) and current distribution (right) in 3D for a simple copper plate systemFig. 4 – Current distribution with short anode (left) and long anode (right) in 2D section of

simple copper plate system

Table 2 – Kinetic parameters for the reactions in the Porocell system

Electrode αc αa i0 (A/m-2)

Cu2+

+ 2e- Cu cathode 0.5 1.5 10

2H+

+ 2e- H2 cathode 0.5 4 x 10-4

2H2O O2 + 4H+

+ 4e-

anode 0.5 1 x 10-2

The current density distribution is shown in the left image of fig. 4 along with the current lines. It illustrates that when using a height of anode which is shorter that the height of the cathode an uneven distribution of current occurs across the surface of the cathode. This is not desired for electrowinning processes as it means there will be a non-uniform electrodeposition of copper across the cathode. The image on the right of fig. 4 shows the current distribution with a longer anode, clearly showing a more uniform current distribution. The following models will continue to use the shorter height of anode, as this is what was used in previous practical experiments. However, the length of anode may be something to consider when building the large scale system.

Cell Voltage (V)

Average Current Density from

model (A/m2) Current from model (A)Current from

experiment (A)1.6 88 1.9 32 340 7.2 6

2.5 740 15.7 94.7 943 20.0 18

Before continuing with modelling the system with a porous electrode, a verification of the simple copper plate model was required. This was carried out by applying various different cell voltages, calculated the current from the model and comparing with the values of current from experimental data. The comparison of these values is shown in table 3 and shows that the values of current are in reasonable agreement with each other.

3 Model of porous electrode system (1L reactor)

3.1 GeometryThe geometry is separated into three domains for the porous electrode model, as

shown in the left image of fig. 5. Domains 1 and 2 are electrolyte domains and domain 3 is the porous electrode domain. The mesh was a physics controlled mesh automatically built by the software and is shown the right image of fig. 5.

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Table 3 – Comparison of current from model and current from experimental data

3.2 Description of the porous electrode model

The porous electrode was modelled as a mixed material, one electrode phase and one electrolyte phase. It is therefore modelled by defining two separate current balances, as shown in equation 1, where ‘l’ is for electrolyte and ‘s’ is for electrode. The current balances in the pore electrode and electrode matrix contain sources and sinks according to the transfer reactions. The charge transfer reaction is a source for the current balance in the electrode, therefore the electrode receives current from the pore electrode. The charge transfer reaction is a sink for the current balance in the pore electrolyte, therefore current is transferred from the pore electrolyte to the electrode in a cathodic reaction. The charge transfer reactions for a porous electrode are therefore described by the equations 2 and 3.

(1)

(2)

(3)

The term Av is the Active Specific Surface Area (L2/L3) which specifies the area of the electrode-electrolyte interface that is active for the porous electrode reaction. It was calculated by dividing the 3D area by the volume of the porous electrode. The 3D area was estimated as 1.5 m2 from material data and the volume was calculated as 0.000178 m2. Therefore, the active specific area used for this porous electrode model was 8427 1/m. The other parameters used for the porous electrode model are shown in table 4. The electrode conductivity that was used for the model is the conductivity of copper. This assumes that the carbon felt is immediately covered in a layer of electrodeposited copper at the start of the process. The volume fraction values were taken from material property data sheets.

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Fig. 5 – Geometry for model of porous electrode system (left) and mesh (right)

The same method of applying potential for the simple copper plate model was used. However, the electrode potential of an entire domain cannot be set, therefore the electrode potential was set to zero at a point on the cathode near to where the electrical connection is made. This will change later on when creating a more realistic model of the current feeder at the back, but this method was suitable for an initial model for the porous electrode

3.3 Current distribution of the porous electrode model

Fig. 6 shows the current density distribution for the porous electrode system described above. It can be seen that the current lines are clearly penetrating through the electrode as you would expect. Interestingly, the current lines tend to go towards the top of the electrode towards the point that is set to 0 V. A more realistic model where current is being applied by a current feeder at the back of the porous electrode therefore needed to be developed.

4 Improvement of Porous Electrode model

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Fig. 6 – Current distribution of porous electrode model

Table 4 – Model input values for porous electrode model

Properties ValueElectrolyte Conductivity 20 S/m

Electrode Conductivity 5.96 x 107 S/m

Electrolyte volume fraction 0.95

Electrode volume fraction 0.05

4.1 Geometry of improved model

The Porocell system used in previous experiments applies a constant current via a current feeder at the back of the electrode. The current feeder is made of four copper rings, connected electrically together via a copper rod. The porous electrode is contacted by these four rings at the back of the electrode. This geometry was modelled by creating four sections on the back side of the porous electrode domain, shown in blue in fig. 7.

4.2 Modelling applying constant current

The current could be set at the boundaries displayed in blue in fig. 7 by setting the total current without imposing the current density distribution. The potential along the boundary is calculated in order to satisfy the total value of current. Additionally, one electrode needs to be grounded for this to work. The electric potential of the anode (shown in red in fig 7) was therefore set to zero.

Although this method of applying current to the system was closer to the real system, it created issues which needed to be resolved. When defining currents in COMSOL the current will be defined such that positive current values means positive current flow into the electrode. To drive the copper reaction in the correct direction the applied current needs to be set to a negative value. However, this can be difficult to get the model to converge as solving for galvanostatic control numerically is more complex.

COMSOL Multiphysics has an inbuilt tool for helping highly non-linear electrochemical simulations to converge by providing initial conditions better suited to the system. This is the ‘Current Distribution Initialization’ study step. This was added to the Study sequence before the Stationary Study Step that actually solves the physics. Using the ‘Current Distribution Initialization’ as a first step and solves for the potentials first and therefore improves convergence and stability. The model was then able to converge with -3A applied current with no issues.

Another improvement related to the current is the inclusion of the limiting current density into the model. The limiting current density option in COMSOL is used to provide a

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Fig. 7 – Geometry of Porous electrode model with current feeder

cut-off to the current density for a reaction due to a transport-limited process. Seeing as the exchange current density of the copper reaction is much higher than that for hydrogen evolution, the copper reaction will dominate for low overall applied current. When the applied current density exceeds the limiting current density for copper electrodeposition, hydrogen evolution will kick in, and the surface potential will also change accordingly.

4.3 Improvement of Mesh

The mesh required improvement to have a more accurate view of the current distribution within the porous electrode. A finer mesh was also required in the porous electrode domain in order to improve the quality of the potential and current profiles though the porous electrode, to be discussed later. A new mesh was created using a user-controlled mesh. A free triangular mesh was used with a maximum element size of 1 mm within the porous electrode domain. This gave the finer mesh shown in fig. 8. It should be noted that with this new geometry, applying constant current and improved mesh the time taken for the model to converge was 8 seconds and had a number of iterations of 7.

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Fig. 8 – Improved mesh diagram for the porous electrode model

4.4 Current distribution of improved porous electrode model

The current distribution of the improved model described above is shown in the left image in fig. 9. There is now a more detailed plot of the current distribution within the porous electrode. Also, the current lines fully penetrate the electrode and curve around the back of the electrode where the current feeder is positioned.

4.5 Effect of porous electrode thickness

One of the major parameters which effects the performance of a porous electrode is the porous electrode thickness. The thickness of the electrode was therefore reduced from 10 mm to 5 mm to see what effect this would have. The right image in fig. 9 shows the current distribution plot for a 5 mm thick porous electrode. The current distribution looks similar to that for the 10 mm, however there is actually a difference in the current distribution within the porous electrode domain. This is easier to see by analysis of the current and potential profiles through the porous electrode.

The profiles were taken from a distance of 0.05 m from the bottom of the cathode, this is indicated by the dotted line in fig. 7. Fig. 10 and 11 shows the potential and current profiles through porous electrodes. It should be noted that when these profiles were calculated using the physics controlled mesh with element size of 2.5 mm within the porous electrode domain, the profiles did not give an accurate representation. This was improved when using the free triangular mesh with a maximum element size of 1 mm as described above. However, the plots of the profiles still had a ‘blocky’ appearance even with the improved mesh. This is because it is assembled from numerical derivatives. The quality of the plots was improved by using a Polynomial-Preserving Derivative Recovery method which created smoother profiles.

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Fig. 9 – Current distribution of improved porous electrode with constant current for a 10 mm thick electrode (left) and a 5 mm thick porous electrode (right)

Fig. 10 shows the potential and current profiles through the 10 mm thick porous electrode. The left hand side of the plot is next to the current feeder and the right hand side is the nearest to the anode. The potential becomes less negative as you move towards the area of the electrode nearer to the anode, as would be expected. Interestingly, the current density only penetrates to about half of the porous electrode. This suggests that no deposit would be formed in this part of the electrode. This would be inefficient as not only half the area of the electrode is being used, but also when recovering the metal by incineration of the cathode more energy will be used to incinerate the unused half of the carbon felt.

The potential and current profiles through a 5 mm thick porous electrode is shown in fig. 11. The profiles show that more of the electrode is operating above 0 current compared to the 10 mm thick electrodes. This suggests that a larger percentage of the cathode area would be used with 5 mm electrodes. The following models will still use the 10 mm thick electrodes as this is what was used in the practical experiments. However, the results shown in the current profiles indicates that the porous electrode thickness is a factor which needs to be taken into account as the model is further improved and also when building further porocell reactors.

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Fig. 10 – Potential and Current profiles through porous electrode with 10 mm thickness. Electrolyte Potential Profile (left) and Electrolyte Current Density (right)

Fig. 11 – Potential and Current profiles through porous electrode with 5 mm thickness. Electrolyte Potential Profile (left) and Electrolyte Current Density (right)

4.6 Current distribution with lower electrode conductivity

The conductivity for the porous electrode material that has been used so far is the conductivity of copper. This assumes that the carbon felt is immediately covered in a layer of electrodeposited copper, as explained above. It was also important to see what happens in the initial stages of the reaction in the porocell when copper deposit hasn’t yet formed on the surface of the porous electrode. The conductivity of the porous electrode material was therefore lowered to 14 S/m; the conductivity of the carbon felt.

Fig. 12 shows the current distribution when using the lower conductivity of the porous electrode material. The current lines move towards the areas where the current feeder is placed, and also pass directly through the porous electrode and around the back. This suggests that in the first few moments of deposition there may be some deposit formed on the current feeder itself. This indicates that it could be important to carry out regular checks on the current feeder in-between operation.

4.7 Calculation of hydrogen current

It was important to obtain the hydrogen current from the model in order to work out the hydrogen gas evolution rate due to safety considerations for the process. To measure the hydrogen current, first the appropriate expression for the current density due to hydrogen evolution was extracted from the model. This expression was then integrated over the porous

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Fig. 12 – Current distribution of porous electrode when using a lower conductivity for the porous electrode material

electrode domain. The multiple of of ‘2*pi*r’ was introduced to the integrand so it is computed in the revolved geometry. A user-defined hydrogen current variable was defined which integrates the current source (per volume of porous material) across the cathode volume.

When applying a current of 3A a hydrogen current of 0.3A was calculated. The mass evolution rate of hydrogen gas was calculated from Faraday’s laws. The hydrogen evolution rate for this 1L size system under these conditions was 3.15 x 10-9 kg/s.

5 Large-scale porous electrode reactor (50L)

The improvements to the model of the small-scale 1L size reactor explained above has made a more realistic model of the porocell reactor. A larger-scale 50L model of the porocell was therefore developed using the same model parameters as the small-scale model described in sections 4.1, 4.2 and 4.3.

5.1 Geometry of large-scale porous electrode reactor

The 50L size porocell was modelled using the same ratio of dimensions as the 1L size porocell. The height of the reactor is approximately 1 m. The thickness of the porous electrode remained the same at 10 mm. The geometry for the 50L porocell reactor is shown in fig. 13. The current feeders at the back of the electrode was again modelled in the geometry by creating four sections on the back side of the porous electrode domain, marked in blue in fig. 13.

The current was applied to the system using the same method described in section 4.2, and again the anode was grounded by setting the electric potential of the anode boundary to 0V. The area of the porous electrode in the large-scale porocell model is 70 times larger than that of the small-scale model, therefore the large scale system required 70 times the amount of current in order to obtain the same current density. A current of -210A was therefore applied to the boundaries marked in blue in fig. 13.

The mesh for the 50L porocell model used a user-controlled mesh with a free triangular mesh with maximum element size of 1 mm within the porous electrode domain. The mesh is shown in fig. 14. For this model of the 50L porocell, the time for the model to converge was 16 seconds and the number of iterations was 8.

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Fig. 13 – Geometry of Porous electrode model with current feeder

Fig. 14 –Mesh diagram for the large-scale 50L porocell model

5.2 Current distribution of large-scale porous electrode reactor

Fig. 15 shows the potential and current distributions for the model of the 50L porocell reactor. The shapes of the distributions look similar to the distributions for the 1L porocell model and the values of current and potentials are within a reasonable range. Notice how the current lines penetrate the porous electrode indicating that the current flows through the entire thickness of the electrode. It should be noted that the top third of the porous electrode is at a much lower current density due to the non-uniformity of the current distribution caused by the anode height and its position.

6 Hydrogen current comparison

An important factor to compare for the scale-up of this process in terms if safety is the hydrogen evolution rate. The hydrogen current was extracted from the model for the 50L porocell system using the same method as described in section 4.7 and the evolution rate was calculated using Faraday’s laws. The comparison of the hydrogen current and evolution rate for the two systems is shown in table 5.

Reactor Volume (L) Current Applied (A) Hydrogen Current (A) Hydrogen Evolution

Rate (kg/s)1 L -3 0.3 3.15 x 10-9

50 L -210 12.9 1.34 x 10-7

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Fig. 15 – Potential (left) and current (right) distribution of large-scale 50L porous electrode model with constant current for a 10 mm thick electrode

Table 5 – Comparison of hydrogen current and hydrogen gas evolution rate for 1L and 50L Porocell systems

The hydrogen evolution rate in the 50L porocell system when applying a current of -210 A is 1.34 x 10-7 kg/s. This is over 40 times the rate of hydrogen evolution for the 1L when applying the same current density, but it is still a small amount of hydrogen production which is suitable for industrial use at this scale.

7 Summary and conclusions

A model has successfully been developed for the 1L size porocell reactor in a 2D asymmetric geometry. A porous electrode system was modelled with 2 cathodic reactions and 1 anodic reaction using the Butler-Volmer kinetic expressions. The limiting current of copper electrodeposition was also included in the model.

Model improvements were made by developing a method of applying constant current from the back of the electrode to simulate the current feeder. Models with galvanostatic control and non-linear electrochemical simulations can have problems in converging, therefore a current distribution initialization step was added as an initial study step improve convergence and stability. Mesh improvements were also made with inclusion of a free triangular mesh with a maximum element size of 1 mm within the porous electrode domain. The model convergence time was 8 seconds and had a number of iterations of 7.

Potential and current distribution plots were calculated from the model. These showed that the length of the anode needs to be taken into consideration in further designs to provide a more uniform current distribution across the cathode. Current profiles through the porous electrode demonstrated the importance of the thickness of the porous electrode. It was suggested that 5 mm carbon felt electrodes should be investigated when carrying out further experiments and designing larger scale reactors in order to increase the efficiency of the process.

The porous electrode model that was designed for the 1L size Porocell was used to build a model for a large-scale 50L Porocell reactor and potential and current distributions were calculated for the large-scale reactor. A method of calculating the hydrogen gas evolution rate was also developed. This was achieved by integrating the expression for the hydrogen current across the porous electrode domain and a multiple of 2*pi*r was introduced to compute the integral in the revolved geometry. Faraday’s law was then used to calculate the hydrogen gas evolution rate for each system.

According to this model, the amount of hydrogen evolution from the 50L system was relatively small and is suitable for industrial use at this scale under the operating conditions described. As the model is further improved and other sizes of reactors are tested, the value of hydrogen evolution can be checked using this method which has been developed and values can be checked with industrial standards to decide what size system would be most suitable for this process.

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