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UDC 517.946.9:518
SOLUTION OF CERTAIN PROBLEMS OF ELECTROCHEMISTRY BY THE METHOD OF STRAIGHT LINES
V. T. IVANOV
The distribution of current in an electrolyzer plays an extremely important role in problems of electroplating, corrosion theory and electrochemical protection, electrolysis under the conditions of electrochemical production, etc.
It is shown in [1] that the problem of finding the current in an electrolytic cell, when the electrode process is limited by chemical polarization, reduces to integrating the Laplace equation in a domain whose boundary consists of electrodes and insulators. On the electrolyte–insulator boundary the normal derivative is equal to zero, while on the electrolyte–electrode boundary, in the general case, the potential is expressed as a nonlinear function of the current density.
The boundary conditions on the electrodes are, as a rule, determined from experiment.
An analytical solution of this problem has been found only for the simplest domains, taking into account linear polarization of the electrodes.
In the present work we consider the solution, by the method of straight lines, of problems of current distribution on the electrodes for cylindrical and rectangular electrolyzers in the case of linear and nonlinear polarization of the electrodes.
Calculation of the current distribution between coaxial cylinders leads to the integration of a boundary-value problem for the Laplace equation in cylindrical coordinates with axial symmetry.
Let it be required to find, in the rectangle \(\Pi(r_0<r<R,\;0<z<d)\), a solution of the equation
\[ \frac{\partial^2 \Phi}{\partial r^2} +\frac{1}{r}\frac{\partial \Phi}{\partial r} +\frac{\partial^2 \Phi}{\partial z^2}=0, \tag{1} \]
satisfying the boundary conditions:
\[ \Phi(r_0,z)-C_1\eta_1\bigl(\chi\Phi'_r(r_0,z)\bigr)=a, \qquad 0\le z\le l, \tag{2} \]
\[ \Phi'_r(r_0,z)=0, \qquad l<z\le d, \tag{3} \]
\[ \Phi'_z(r,0)=0,\qquad \Phi'_z(r,d)=0, \qquad r_0<r<R, \tag{4} \]
\[ \Phi(R,z)+C_2\eta_2\bigl(\chi\Phi'_r(R,z)\bigr)=b, \qquad 0\le z\le d, \tag{5} \]
where \(r_0\) is the radius of the cathode, \(l\) its height; \(R\) is the radius of the anode, and \(d\) its height; \(\chi\) is the coefficient of specific electrical conductivity of the electrolyte; the constants \(C_1\), \(C_2\), \(a\), and \(b\), as well as the nonlinear functions \(\eta_1\) and \(\eta_2\), are determined from experiment according to the form of the polarization curves.
If chemical polarization at the electrode is absent (such electrodes in electrochemistry are customarily called nonpolarizable), then the problem of primary current distribution arises. The primary current distribution
allows one to determine approximately the actual current distribution, if one uses the calculation procedure developed in [2].
Conditions (2) and (5) in this case take the form
\[ \Phi(r_0,z)=a_1,\quad 0\leq z\leq l;\qquad \Phi(R,z)=b_1,\quad 0\leq z\leq d. \tag{6} \]
If the current densities are sufficiently small, so that the polarization is expressed by a linear law (a weakly polarizable electrode), then instead of conditions (2) and (5) we have the following linear boundary conditions:
\[ \Phi(r_0,z)-A\Phi'_r(r_0,z)=a_2,\quad 0\leq z\leq l, \tag{7} \]
\[ \Phi(R,z)+B\Phi'_r(R,z)=b_2,\quad 0\leq z\leq d, \tag{8} \]
where \(A\) and \(B\) are positive constants.
In [1] the possibility is shown of simplifying the boundary conditions (2) and (5) for electrodes of finite dimensions; this simplification also reduces to the linear boundary conditions (7) and (8).
Divide the rectangle \(\Pi\) into \(n+1\) equal parts by the straight lines \(z=z_k\) \((k=1,2,\ldots,n)\), replace \(\partial^2\Phi/\partial z^2\) and the boundary conditions (4) by difference quotients; then we obtain a system of second-order differential equations
\[ \Phi''_k+\frac{1}{r}\Phi'_k+\frac{1}{h^2}\left[\Phi_{k+1}-2\Phi_k+\Phi_{k-1}\right]=0 \tag{9} \]
\[ (k=1,2,\ldots,n), \]
\[ \Phi_0=\Phi_1,\quad \Phi_{n+1}=\Phi_n, \tag{10} \]
which must satisfy the boundary conditions
\[ \Phi_k(r_0)-C_1\eta_1(\chi\Phi'_k(r_0))=a \quad (k=1,2,\ldots,m),\quad mh<l<(m+1)h, \tag{11} \]
\[ \Phi'_k(r_0)=0\quad (k=m+1,\ldots,n), \tag{12} \]
\[ \Phi_k(R)+C_2\eta_2(\chi\Phi'_k(R))=b\quad (k=1,2,\ldots,n). \tag{13} \]
Here \(h=\dfrac{d}{n+1}\), and \(\Phi_k=\Phi_k(r)\) is the approximate solution on the straight line \(z=z_k\). If the polarization is expressed by a linear law, then conditions (11) and (13) take the following form:
\[ \Phi_k(r_0)-A\Phi'_k(r_0)=a_2 \quad (k=1,2,\ldots,m),\quad mh<l<(m+1)h, \tag{14} \]
\[ \Phi_k(R)+B\Phi'_k(R)=b_2 \quad (k=1,2,\ldots,n). \tag{15} \]
We shall seek particular solutions of this system in the form
\[ \Phi_k(r)=\gamma(k)u(r). \tag{16} \]
After substituting (16) into (9) and (10) and separating the variables, we obtain a differential equation for determining the function \(u(r)\) and a difference boundary-value problem of Sturm–Liouville type. It is required to find the values \(\delta^2\), called eigenvalues, for which nontrivial solutions of the problem exist,
\[ \gamma(k+1)-[2-h^2\delta^2]\gamma(k)+\gamma(k-1)=0, \tag{17} \]
\[ \gamma(0)=\gamma(1),\quad \gamma(n)=\gamma(n+1), \tag{18} \]
called eigenvectors, and also to find these vectors.
We seek the solution of problem (17), (18) in the form
\[ \gamma(k)=c_1\lambda_1^k+c_2\lambda_2^k, \tag{19} \]
where \(\lambda_1\) and \(\lambda_2\) are the roots of the characteristic equation
\[ \lambda^2-[2-h^2\delta^2]\lambda+1=0. \tag{20} \]
Using conditions (18)—(20), we find the eigenvalues and eigenvectors of the boundary-value problem, which have the form
\[ \delta_s^2=\frac{4}{h^2}\sin^2\frac{\pi s}{2n},\qquad \gamma_s(k)=c\, \frac{\cos \dfrac{\pi s(k-1/2)}{n}} {\cos \dfrac{\pi s}{2n}} \tag{21} \]
\[ (s=0,1,\ldots,n-1). \]
To determine the function \(u(r)\), we obtain the modified Bessel equation of order zero
\[ u''+\frac{1}{r}u'-\delta_s^2u=0, \]
whose general solution is
\[ u_0=A_0+B_0\ln r,\qquad u_s=A_s I_0(r\delta_s)+B_sK_0(r\delta_s) \tag{22} \]
\[ (s=1,2,\ldots,n-1), \]
where \(I_0\) and \(K_0\) are Bessel functions of order zero of imaginary argument. It is not difficult to compose the general solution of the system (9), (10)
\[ \Phi_k(r)=A_0+B_0\ln r+\sum_{s=1}^{n-1}\bigl[A_s I_0(r\delta_s)+ \]
\[ +\,B_sK_0(r\delta_s)\bigr]\, \frac{\cos \dfrac{\pi s(k-1/2)}{n}} {\cos \dfrac{\pi s}{2n}}. \tag{23} \]
The coefficients \(A_s\) and \(B_s\) are determined from conditions (11)—(13); to determine them we have \(2n\) nonlinear equations with \(2n\) unknowns. In the case of boundary conditions (12), (14), (15), the determination of the unknown constants reduces to the solution of a linear algebraic system of equations.
Below we consider the correctness of the method of straight lines for the case of linear polarization of the electrodes.
Theorem 1. The system of differential equations (9), (10), satisfying the boundary conditions (12), (14), (15), has a unique solution.
Let \(\Phi_{1k}\) and \(\Phi_{2k}\) be different systems of solutions of the boundary-value problem (9), (10), (12), (14), (15). Then the system of functions \(\Phi_k=\Phi_{1k}-\Phi_{2k}\) satisfies the homogeneous system of differential equations (9), (10) and the following homogeneous boundary conditions:
\[ \Phi_k(r_0)-A\Phi'_k(r_0)=0 \quad (k=1,2,\ldots,m),\qquad mh<l<(m+1)h, \tag{24} \]
\[ \Phi'_k(r_0)=0 \quad (k=m+1,\ldots,n). \tag{25} \]
\[ \Phi_k(R)+B\Phi'_k(R)=0 \quad (k=1,2,\ldots,n). \tag{26} \]
Multiplying each equation of system (9) by \(-r\Phi_k\), summing the resulting expressions from 1 to \(n\), and integrating from \(r_0\) to \(R\), we obtain the following equality:
\[ -\int_{r_0}^{R}\sum_{k=1}^{n}\frac{d}{dr}\left(r\frac{d\Phi_k}{dr}\right)\Phi_k\,dr -\int_{r_0}^{R}\frac{r}{h^2}\sum_{k=1}^{n}(\Phi_{k+1}-2\Phi_k+\Phi_{k-1})\Phi_k\,dr=0. \tag{27} \]
Integrating the first term by parts, and taking into account conditions (24)—(26), we obtain
\[ -\int_{r_0}^{R}\sum_{k=1}^{n}\frac{d}{dr}\left(r\frac{d\Phi_k}{dr}\right)\Phi_k\,dr = \sum_{k=1}^{n}B^{-1}R(\Phi_k(R))^2 + \sum_{k=1}^{m}A^{-1}r_0(\Phi_k(r_0))^2 + \int_{r_0}^{R}r\sum_{k=1}^{n}\left(\frac{d\Phi_k}{dr}\right)^2\,dr. \]
By regrouping the terms, we transform the sum in the integrand of the second integral, taking account of conditions (10). Then we obtain
\[ -\int_{r_0}^{R}\frac{r}{h^2}\sum_{k=1}^{n}(\Phi_{k+1}-2\Phi_k+\Phi_{k-1})\Phi_k\,dr = \int_{r_0}^{R}\frac{r}{h^2}\sum_{k=1}^{n-1}(\Phi_{k+1}-\Phi_k)^2\,dr. \]
Thus, equality (27) takes the form
\[ \sum_{k=1}^{n}B^{-1}R(\Phi_k(R))^2 + \sum_{k=1}^{m}A^{-1}r_0(\Phi_k(r_0))^2 + \int_{r_0}^{R}r\sum_{k=1}^{n}\left(\frac{d\Phi_k}{dr}\right)^2\,dr + \]
\[ + \int_{r_0}^{R}\frac{r}{h^2}\sum_{k=1}^{n-1}(\Phi_{k+1}-\Phi_k)^2\,dr=0. \]
Since \(A>0\), \(B>0\), and \(r>r_0>0\), we conclude that \(\Phi_k\equiv0\).
From the uniqueness of the solution of the boundary-value problem it follows that the determinant of the system for finding the coefficients \(A_s\) and \(B_s\) is nonzero. Thus the existence of a solution of problem (9), (10), (12), (14), (15) has been proved.
Let \(\alpha_k(r)\) be the error of the differential-difference method; then the system of functions \(\alpha_k(r)\) satisfies the system of differential equations
\[ L_n(\alpha_k)=\alpha''_k+\frac{1}{r}\alpha'_k+\frac{1}{h^2}[\alpha_{k+1}-2\alpha_k+\alpha_{k-1}] = R_k(r)\quad (k=1,2,\ldots,n), \tag{28} \]
\[ \frac{\alpha_1-\alpha_0}{h}=R_0(r), \qquad \frac{\alpha_{n+1}-\alpha_n}{h}=R_{n+1}(r) \tag{29} \]
with the accompanying boundary conditions
\[ \alpha_k(r_0)-A\alpha'_k(r_0)=0 \quad (k=1,2,\ldots,m), \qquad mh<l<(m+1)h, \tag{30} \]
\[ \alpha_k'(r_0)=0 \qquad (k=m+1,\ldots,n), \tag{31} \]
\[ \alpha_k(R)+B\alpha_k'(R)=0 \qquad (k=1,2,\ldots,n). \tag{32} \]
If there exists a sufficiently smooth solution of problem (1), (3), (4), (7), (8), which has in \(\overline{\Pi}\) (the closure of the domain \(\Pi\)) bounded derivatives up to and including the fourth order, then, putting
\[ M_4=\max_{\overline{\Pi}}\left|\frac{\partial^4\Phi}{\partial z^4}\right| \quad \text{and} \quad M_2=\max_{\overline{\Pi}}\left|\frac{\partial^2\Phi}{\partial z^2}\right| \]
and using Taylor’s formula, it is not difficult to estimate the functions \(R_k(r)\) \((k=1,2,\ldots,n)\) and \(R_j(r)\) \((j=0,n+1)\):
\[ |R_k(r)|\leq M_4\frac{h^2}{12}, \qquad |R_j(r)|\leq \frac{M_2h}{2}. \]
Lemma 1. If \(R_k(r)>0\) \((R_k(r)<0)\) \((k=1,2,\ldots,n)\), then for the system of functions \(\alpha_k(r)\) \((k=1,2,\ldots,n)\), twice differentiable in \((r_0,R)\) and continuous in \([r_0,R]\), a positive maximum (negative minimum) is attained only on the boundary of the domain \(\Pi_h\).
The boundary of the domain \(\Pi_h\) includes the endpoints \((r_0,z_k)\) and \((R,z_k)\) \((k=1,2,\ldots,n)\), as well as the extreme straight lines \(z_0\) and \(z_{n+1}\).
The proof of this lemma is carried out by the usual method of contradiction.
Lemma 2. Suppose the conditions of Lemma 1 are satisfied, and suppose that a positive maximum (negative minimum) is attained at some endpoint of the segment \(z_{k_0}\); then the derivative of the function \(\alpha_{k_0}(r)\) at this point in the direction of the outward normal to \(\Pi_h\) is positive (negative).
If a positive maximum (negative minimum) is attained at the point \((r,0)\) or \((r,d)\), then for this value of \(r\) one has
\[ \frac{\alpha_1-\alpha_0}{h}<0 \quad \text{or} \quad \frac{\alpha_{n+1}-\alpha_n}{h}>0 \left( \frac{\alpha_1-\alpha_0}{h}>0 \quad \text{or} \quad \frac{\alpha_{n+1}-\alpha_n}{h}<0 \right). \]
Lemma 2 is the differential-difference analogue of Zaremba’s lemma [2].
On the basis of Lemmas 1 and 2, the following lemma is proved.
Lemma 3. If on the mesh there are given two systems of functions \(\alpha_k(r)\) and \(\beta_k(r)\), continuous in \([r_0,R]\) and twice differentiable in \((r_0,R)\), such that inside the domain
\[ L_h(\beta_k)<-|L_h(\alpha_k)|, \]
and on the boundary of \(\Pi_h\) the following conditions are satisfied:
\[ \text{a)}\quad \beta_k'(r_0)\leq -|\alpha_k'(r_0)| \qquad (k=m+1,\ldots,n), \]
\[ \text{b)}\quad \frac{\beta_1-\beta_0}{h}\leq -\left|\frac{\alpha_1-\alpha_0}{h}\right|, \qquad \frac{\beta_{n+1}-\beta_n}{h}\geq \left|\frac{\alpha_{n+1}-\alpha_n}{h}\right|, \]
\[ \text{c)}\quad \beta_k(r_0)-A\beta_k'(r_0)\geq |\alpha_k(r_0)-A\alpha_k'(r_0)| \qquad (k=1,2,\ldots,m), \]
\[ \text{d)}\quad \beta_k(R)+B\beta_k'(R)\geq |\alpha_k(R)+B\alpha_k'(R)| \qquad (k=1,2,\ldots,n), \]
then everywhere in the domain \(\Pi_h\)
\[ |\alpha_k(r)|\leq \beta_k(r). \]
Theorem 2. Let \(a_k(r)\) be the error of the differential-difference method; then for the system of functions \(a_k(r)\) the following estimate holds:
\[ |\alpha_k(r)|\leq \frac{1}{2}\bigl[(R-r_0)^2+2(R-r_0)B-(r-r_0)^2\bigr]\times \]
\[ \times\left(\frac{M_4h^2}{12}+\frac{M_2h}{d-h}\right) +\frac{M_2h}{2(d-h)} \left(z_k-\frac{d}{2}\right)^2 . \]
By verification we see that the majorant
\[ \beta_k(r)= \frac{1}{2}\bigl[(R-r_0)^2+2(R-r_0)B-(r-r_0)^2\bigr]\times \]
\[ \times\left(\frac{M_4h^2}{12}+\frac{M_2h}{d-h}\right) +\frac{M_2h}{2(d-h)} \left(z_k-\frac{d}{2}\right)^2 \]
satisfies all the conditions of Lemma 3.
If one considers the boundary-value problem with a nonzero right-hand side in (1), (3), and (4), then, by a corresponding choice of a majorizing function, one can find an a priori estimate of the solutions for the system of differential equations of the method of straight lines with boundary conditions on the electrodes (7) and (8). From this estimate there will follow the continuous dependence of the solution on the right-hand side of the equation and on the boundary conditions, and at the same time the stability of the differential-difference method.
The error of the differential-difference method can be reduced by means of a better approximation of the boundary conditions (4). Let us divide the domain by the straight lines \(z=z_{k+1/2}\) \((k=0,1,\ldots,n-1)\).
To determine the approximate solution \(\Phi_{k+1/2}\), we shall solve the system of differential equations
\[ \Phi''_{k+1/2} +\frac{1}{r}\Phi'_{k+1/2} +\frac{1}{h^2}\left[\Phi_{k+3/2}-2\Phi_{k+1/2}+\Phi_{k-1/2}\right]=0 \]
\[ (k=0,1,\ldots,n-1), \]
\[ \Phi_{-1/2}=\Phi_{1/2},\qquad \Phi_{n-1/2}=\Phi_{n+1/2} \]
with the accompanying boundary conditions
\[ \Phi_{k+1/2}(r_0)-C_1\eta_1\bigl(\chi\Phi'_{k+1/2}(r_0)\bigr)=a \qquad (k=0,1,\ldots,m), \]
\[ (m+1/2)h<l<(m+3/2)h, \]
\[ \Phi'_{k+1/2}(r_0)=0 \qquad (k=m+1,\ldots,n-1), \]
\[ \Phi_{k+1/2}(R)+C_2\eta_2\bigl(\chi\Phi'_{k+1/2}(R)\bigr)=b \qquad (k=0,1,\ldots,n-1). \]
The approximation of the boundary conditions here has order \(h^2\). The general solution of the system of differential equations has the form
\[ \Phi_{k+1/2}(r)= A_0+B_0\ln r+ \sum_{s=1}^{n-1} \left[A_sJ_0(r\delta_s)+B_sK_0(r\delta_s)\right] \cos\frac{\pi s(k+1/2)}{n}. \]
The method of straight lines can also be used to compute the current distribution on electrodes in a rectangular electrolytic cell [3].
We shall assume that the relation between the polarization and the current density at the cathode is expressed by the Tafel equation, and that the anode is nonpolarizable. In this case the calculation of the current distribution reduces to integration in the domain \(\Pi(0<x<a,\ 0<y<b)\) of the equation
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} =0, \]
under the boundary conditions
\[ u(0,y)=\frac{RT}{\alpha F}\ln\bigl(\chi u_x'(0,y)\bigr),\quad 0\leq y\leq l;\qquad u_x'(0,y)=0,\quad l<y\leq c; \]
\[ u(d,y)=1,\quad 0\leq y\leq c; \]
\[ u_y'(x,0)=0,\quad u_y'(x,c)=0,\quad 0\leq x\leq d, \]
where \(R\) is the universal gas constant; \(T\) is the temperature of the electrolyte; \(F\) is the Faraday number.
Replacing the derivatives with respect to \(y\) by difference quotients, we arrive at the system of differential equations
\[ u_k''+\frac{1}{h^2}\,[u_{k+1}-2u_k+u_{k-1}]=0, \tag{33} \]
\[ u_0=u_1,\quad u_n=u_{n+1} \tag{34} \]
with boundary conditions
\[ u_k(0)=\frac{RT}{\alpha F}\ln\bigl(\chi u_k'(0)\bigr) \quad (k=1,2,\ldots,m),\qquad mh<l<(m+1)h, \tag{35} \]
\[ u_k'(0)=0\quad (k=m+1,\ldots,n), \]
\[ u_k(c)=1\quad (k=1,2,\ldots,n). \tag{36} \]
Solving the system of equations (33), (34), taking into account the conditions (36), we find the general solution, which has the form
\[ u_k(x)=1+B_0(x-a)+\sum_{s=1}^{n-1}B_s \frac{\operatorname{sh}\delta_s(x-a)}{\operatorname{ch}\delta_s a}\, \frac{\cos \dfrac{\pi s(k-1/2)}{n}}{\cos \dfrac{\pi s}{2n}}, \]
where
\[ \delta_s^2=\frac{4}{h^2}\sin^2\frac{\pi s}{2n}. \]
For linear boundary conditions the stability is proved analogously to Theorem 2.
To determine the arbitrary constants \(B_s\ (s=0,1,\ldots,n-1)\) from conditions (35), we have a system of \(n\)-transcendental equations with \(n\) unknowns, which is conveniently written in the following matrix form:
\[ \mathbf{A}\mathbf{B}=f(\mathbf{B}), \tag{37} \]
where \(\mathbf{A}=[a_{ij}]\) is the coefficient matrix;
\[ \mathbf{B}= \begin{bmatrix} B_0\\ \vdots\\ B_{n-1} \end{bmatrix} \]
is the vector sought, and the transcendental right-hand side of this system has the form
\[ f(\mathbf{B})= \begin{bmatrix} \dfrac{RT}{\alpha F}\ln(C_{00}B_0+\ldots+C_{0\,n-1}B_{n-1})-1\\ \hline \dfrac{RT}{\alpha F}\ln(C_{m0}B_0+\ldots+C_{m\,n-1}B_{n-1})-1\\ 0\\ \hline 0 \end{bmatrix}. \]
Let \(B_0^0, \ldots, B_{n-1}^0\) be numbers close to the solutions of system (37); refinement of the solutions can be carried out by the method of iteration according to the following scheme:
\[ \mathbf{B}^1=\mathbf{A}^{-1} f(\mathbf{B}^0),\quad \mathbf{B}^2=\mathbf{A}^{-1} f(\mathbf{B}^1),\ldots,\quad \mathbf{B}^{k+1}=\mathbf{A}^{-1} f(\mathbf{B}^k), \]
where \(\mathbf{B}^k=(B_0^k,\ldots,B_{n-1}^k)\) is the \(k\)-th approximation; \(\mathbf{A}^{-1}\) is the inverse matrix.
As the initial approximation \((B_0^0,\ldots,B_{n-1}^0)\), it is convenient to take the solution of the system of equations (37) in whose right-hand sides the logarithms are assigned zero values.
Such a choice of the zeroth approximation is dictated by the physical formulation of the problem. It is not difficult to see that the numbers taken as the zeroth approximation correspond to the values of arbitrary constants in the general solution of the problem of the primary current distribution.
Calculations performed on an electronic computer (\(d=82\ \mathrm{mm}\), \(c=70\ \mathrm{mm}\), \(l=54\ \mathrm{mm}\), \(n=12\), \(RT/\alpha F=1/20\)) showed satisfactory convergence of the iteration method.
Using the methods developed in works [4–6], one can obtain error estimates for the differential-difference method for certain boundary-value problems for equations of elliptic and parabolic types.
I express my sincere gratitude to Candidate of Physical and Mathematical Sciences O. A. Liskovets for valuable comments on this work.
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Received by the editors
February 22, 1965
Bashkir State University
named after the 40th Anniversary of October