Mathematical Modeling of the Formation of Calcareous Deposits on Cathodically Protected Steel in Seawater #977

tjczec01 · 2023-11-30T13:33:54Z

tjczec01
Nov 30, 2023

I have been trying to write Python code to solve the set of PDEs listed in the paper here: https://www.researchgate.net/publication/238140924_Mathematical_Modeling_of_the_Formation_of_Calcareous_Deposits_on_Cathodically_Protected_Steel_in_Seawater

The simplest equations are as follows:

$$ \frac{dc_{i}}{dt} = -\nabla \cdot N_{i}$$

with

$$N_{i} = -\frac{z_{i}D_{i, e} F c_{i}}{RT}\frac{d\phi}{dy} - D_{i, e}\frac{dc_{i}}{dy}$$

How would I generally use fipy to represent these equations? I figured a transient term for the time derivative but not sure for the other parts.

Also if possible, how would I continue to use this to represent equations 15 & 16?

guyer · 2023-11-30T17:27:46Z

guyer
Nov 30, 2023
Maintainer

It's a little weird to state one as a vector equation and the other as a 1D ODE. Further, all derivatives should be partials.

First, write the second equation in vector form:

$$N_i = -\frac{z_i D_{i,e} F c_i}{R T}\nabla \phi - D_{i,e} \nabla c_i$$

then substitute that equation for $N_i$ into the equation for $\partial c_i/\partial t$:

$$\begin{aligned} \frac{\partial c_i}{\partial t} &= -\nabla\cdot\left\{ -\frac{z_i D_{i,e} F c_i}{R T}\nabla \phi - D_{i,e} \nabla c_i \right\} \\ \frac{\partial c_i}{\partial t} &= \nabla\cdot\left\{ \frac{z_i D_{i,e} F c_i}{R T}\nabla \phi \right\} + \nabla\cdot\left\{ D_{i,e} \nabla c_i \right\} \end{aligned}$$

These terms are now in canonical form for FiPy.

The term on the left is a TransientTerm in $c_i$.
The second term on the right is a DiffusionTerm in $c_i$.
The first term on the right presents a choice; it can either be written as a ConvectionTerm in $c_i$ or as a DiffusionTerm in $\phi$. If the equations are solved separately and sequentially, then a ConvectionTerm is appropriate. If the equations are to be solved coupled, then a DiffusionTerm is better. Nine equations is a lot to couple, but they might converge better than solving them sequentially.

So, either

eq1 = (fp.TransientTerm(var=C1)
       == fp.ConvectionTerm(coeff=(z1 * D1e * F / (R * T)) * phi.faceGrad, var=C1)
       + fp.DiffusionTerm(coeff=D1e, var=C1))
eq2 = (fp.TransientTerm(var=C2)
       == fp.ConvectionTerm(coeff=(z2 * D2e * F / (R * T)) * phi.faceGrad, var=C2)
       + fp.DiffusionTerm(coeff=D2e, var=C2))
  :
  :

and solve them one by one, or

eq1 = (fp.TransientTerm(var=C1)
       == fp.DiffusionTerm(coeff=(z1 * D1e * F * C1.faceValue/ (R * T)), var=phi)
       + fp.DiffusionTerm(coeff=D1e, var=C1))
eq2 = (fp.TransientTerm(var=C2)
       == fp.DiffusionTerm(coeff=(z2 * D2e * F * C2.faceValue/ (R * T)), var=phi)
       + fp.DiffusionTerm(coeff=D2e, var=C2))
  :
  :
eq = eq1 & eq2 & ... & eq9

and solve the coupled equation eq.

As to [15] & [16], they would need to be coupled. I would try

eq15 = (fp.TransientTerm(var=C_CO3m2) + fp.TransientTerm(var=C_HCO3m1)
        == fp.DiffusionTerm(coeff=(z_CO3m2 * D_CO3m2_e * F * C_CO3m2.faceValue/ (R * T)), var=phi)
        + fp.DiffusionTerm(coeff=D_CO3m2_e, var=C_CO3m2)
        + fp.DiffusionTerm(coeff=(z_HCO3m1 * D_HCO3m1_e * F * C_HCO3m1.faceValue/ (R * T)), var=phi)
        + fp.DiffusionTerm(coeff=D_HCO3m1_e, var=C_HCO3m1))

and similarly for $\mathrm{CO_3^{2-}}$ and $\mathrm{OH^-}$.

4 replies

tjczec01 Dec 6, 2023
Author

Here is what I have so far:

# -*- coding: utf-8 -*-

import math
from fipy import CellVariable, Grid1D, Viewer, DiffusionTerm, TransientTerm, ConvectionTerm

def SurfaceCoverage(A_s, A_0):
	sc = A_s / A_0
	return sc

def depositPorosity(V_s, V_d):
	dp_1 = V_s - V_d
	dp = dp_1 / V_s
	return dp

def eD(A_s, A_d):
	ed_1 = A_s - A_d
	ed = ed_1 / A_s
	return ed

def eS(A_0, A_d):
	es_1 = A_0 - A_d
	es = es_1 / A_0
	return es

def constant1(z_i, D_i, Fc_i, R, T):
	top = z_i * D_i * Fc_i
	bottom = R * T
	C_1 = top / bottom
	return C_1

def v_y(a_p, omega, v, y):
	p1 = a_p * omega
	p2 = omega / v
	p3 = math.sqrt(p2)
	p4 = p1 * p3
	return p4

def MacMullin(tau, sig_d):
	mac = tau / sig_d
	return mac

def ediff(D_i, theta, mac):
	D_1 = D_i * (1 - theta)
	D_2 = (D_i / mac) * theta
	D_ie = D_1 + D_2
	return D_ie

def i_j(sig_s, J_j):
	I_J = sig_s * J_j
	return I_J

def i_oj_ref(i_oj_data, y_ij, c_ref, c_data):
	cl = c_ref / c_data
	cy = cl ** y_ij
	i_ref = cl * cy
	return i_ref

def U_ref(U_theta, R, T, F, n_j, p_0, c_1_ref, U_RE, S_ij):
	sl = []
	for s in range(0, S_ij, 1):
		st = c_1_ref / p_0
		sv = math.log(st, math.e)
		sl.append(sv)
	sf = sum(sl)
	c_1 = (R * T) / (n_j * F)
	c_2 = c_1 * sf
	c_3 = U_theta - c_2
	U_ref = c_3 - U_RE
	return U_ref

def R_if(s_i, I_j, n_j, F, J):
	R_l = []
	for j in range(0, J, 1):
		top = s_i * I_j
		bottom = n_j * F
		Rv = top / bottom
		R_l.append(Rv)
	Rs = sum(R_l)
	r_if = -1 * Rs
	return r_if

def Y_ij(p_ij, alpha_cj, s_ij, n_j):
	y_ij = p_ij - ((alpha_cj * s_ij)/(n_j))
	return y_ij

def Y_ijb(q_ij, alpha_cj, s_ij, n_j):
	y_ijb = q_ij + ((alpha_cj * s_ij)/(n_j))
	return y_ijb

def getPQ(s_ij):
	if s_ij > 0:
		p_ij = s_ij
		q_ij = 0
	elif s_ij < 0:
		p_ij = 0
		q_ij = - s_ij

	return p_ij, q_ij

def J_j(R, T, F, N_j, alpha_a, alpha_c, p_ij, q_ij, ioj_ref, c_i_ref, c_i_o):
	tl_1 = []
	tl_2 = []

	exp_term1 = math.exp((alpha_a * F * N_j) / (R * T))
	exp_term2 = math.exp((-1 * alpha_c * F * N_j) / (R * T))
	for c_i in range(0, len(c_i_o), 1):
		C_term = c_i / c_i_ref
		C_f = (C_term ** p_ij) * exp_term1
		C_b = (C_term ** q_ij) * exp_term2
		tl_1.append(C_f)
		tl_2.append(C_b)

	term_left = math.prod(tl_1)
	term_right = math.prod(tl_2)
	final_term = term_left - term_right
	j_j = ioj_ref * final_term
	return j_j

def J_jb(R, T, F, N_j, alpha_a, alpha_c, p_ij, q_ij, ioj_ref, c_i_ref, c_i_o, c_j_o):
	tl_1 = []
	tl_2 = []

	exp_term1 = math.exp((alpha_a * F * N_j) / (R * T))
	exp_term2 = math.exp((-1 * alpha_c * F * N_j) / (R * T))
	for c_i in range(0, len(c_i_o), 1):
		C_term = c_i / c_i_ref
		C_f = (C_term ** p_ij) * exp_term1
		C_b = (C_term ** q_ij) * exp_term2
		tl_1.append(C_f)
		tl_2.append(C_b)

	term_left = math.prod(tl_1)
	term_right = math.prod(tl_2)
	final_term = term_left - term_right
	j_j = ioj_ref * final_term
	return j_j

def NJ(V, phi_o, U_ref):
	n_j = V - phi_o - U_ref
	return n_j

def D_ie(Di, theta, tau, Ed):
	nm = MacMullin(tau, Ed)
	t1 = Di * (1- theta)
	t2 = (Di * theta) / nm
	d_ie = t1 + t2
	return d_ie


species_index = {"1": "O2",
				 "2": "H2",
				 "3": "OH",
				 "4": "Mg",
				 "5": "Ca",
				 "6": "CO3",
				 "7": "HCO3",
				 "8": "Na",
				 "9": "Cl"}

var_index = {"1": "O2",
			 "2": "H2",
			 "3": "OH",
			 "4": "Mg",
			 "5": "Ca",
			 "6": "CO3",
			 "7": "HCO3",
			 "8": "Na",
			 "9": "Cl",
			 "10": "theta",
			 "11": "phi"}

Di_l = [2.90, 6.28, 5.27, 0.705, 0.793, 0.955, 1.19, 1.34, 2.03]

Cref_l = [2.11E-8, 6.7E-11, 1.6E-9, 5.45E-5, 1.05E-5, 2.07E-7, 1.54E-6, 4.3E-4, 5.58E-4]

K_sp = [6.96E-13, 4.50E-19]

k = [1.13E-12, 3.7E7]

z = [0, 0, -1, 2, 2, -2, -1, 1, -1]
T = 298.15 # K
omega = 50 # RPM
rho_0 = 1.0234E-3 # Kg/cm**3
v = 9.33E-2 # cm**2 / s
tau = 1.0
sig_d = 0.25
y_re = 1.5E-2 # cm
delta = 1.0E-3 # cm
V = 0.0 # V
phi_re = 0.9 # V
U_re = 0.242 # V
m = 1.7
a_a1 = 1
a_a2 = 3
a_c1 = 1
a_c2 = 1
n_j1 = 2
n_j2 = 4
ioj_data1 = 1.24E-24 # A / cm**2
ioj_data2 = 2.0E-11 # A / cm**2
R = 8.314 # J / mol·K
F = 96487 # C/mol
U_theta1 = 0.401 # V
U_theta2 = -0.828 # V

CO2_data = 1.0E-7 # mol / cm **3
COH_data1 = 1.0E-10 # mol / cm **3

CH2_data = 6.7E-10 # mol / cm **3
COH_data2 = 1.6E-9 # mol / cm **3

MWCaCo3 = 100.0869 # g/mol
rho_CaCo3 = 2.711 # g/cm3

MWMgOH2 = 58.3197 # g/mol
rho_MgOH2 = 2.3446 # g/cm3

Keq = 8.4E7 # cm ** 3 / mol

CO2_bulk = 2.1E10-7 # mol/cm 3
COH_bulk = 1.6E10-9 # mol/cm 3
CCa_bulk = 1.1E10-5 # mol/cm 3
CMg_bulk = 5.5E10-5 # mol/cm 3
CCO3_bulk = 1.8E10-7 # mol/cm 3

U_O2_ref = U_ref(U_theta1, R, T, F, 2, rho_0, Cref_l[0], U_re, 1)
N_j_O2 = NJ(V, phi_re, U_O2_ref)
P_O2, Q_O2 = getPQ(1)
Gamma_O2 = Y_ij(P_O2, 1, 1, 2)
I_oj_ref_O2 = i_oj_ref(ioj_data1, Gamma_O2, Cref_l[0], CO2_data)
# J_j_O2 = J_j(R, T, F, N_j_O2, 1, 1, P_O2, Q_O2, I_oj_ref_O2, Cref_l[0], c_i_o)

U_H2_ref = U_ref(U_theta2, R, T, F, 4, rho_0, Cref_l[1], U_re, 1)
N_j_H2 = NJ(V, phi_re, U_H2_ref)
P_H2, Q_H2 = getPQ(1)
Gamma_H2 = Y_ij(P_H2, 1, 1, 4)
I_oj_ref_H2 = i_oj_ref(ioj_data2, Gamma_H2, Cref_l[1], CH2_data)
# J_j_O2 = J_j(R, T, F, N_j_H2, 3, 1, P_H2, Q_H2, I_oj_ref_H2, Cref_l[1], C2)

U_OH_ref_1 = U_ref(U_theta1, R, T, F, 2, rho_0, Cref_l[2], U_re, 4)
N_j_OH_1 = NJ(V, phi_re, U_OH_ref_1)
P_OH_1, Q_OH_1 = getPQ(4)
Gamma_OH_1 = Y_ij(P_OH_1, 1, 4, 2)
I_oj_ref_OH_1 = i_oj_ref(ioj_data1, Gamma_OH_1, Cref_l[2], COH_data1)
# J_j_OH_1 = J_j(R, T, F, N_j_OH_1, 1, 1, P_OH_1, Q_OH_1, I_oj_ref_OH_1, Cref_l[2], C3)

U_OH_ref_2 = U_ref(U_theta2, R, T, F, 4, rho_0, Cref_l[2], U_re, 2)
N_j_OH_2 = NJ(V, phi_re, U_OH_ref_2)
P_OH_2, Q_OH_2 = getPQ(2)
Gamma_OH_2 = Y_ij(P_OH_2, 1, 2, 4)
I_oj_ref_OH_2 = i_oj_ref(ioj_data2, Gamma_OH_2, Cref_l[2], COH_data2)
#J_j_OH_1 = J_j(R, T, F, N_j_OH_2, 3, 1, P_OH_2, Q_OH_2, I_oj_ref_OH_2, Cref_l[2], C3)
ny = 200
dy = 0.01


yre = 1.5E-2 # cm
L = ny * yre
mesh = Grid1D(dx = L /ny, nx = ny)

dt = 0.00025

y = CellVariable(mesh=mesh, name='y', value=0.0)

valueLeft = 0.0

valueRight = yre

valuethetaLeft = 0.

valuethetaRight = 1.

y.constrain(valueRight, mesh.facesRight)

y.constrain(valueLeft, mesh.facesLeft)

C1 = CellVariable(mesh=mesh, name='O2', value=CO2_bulk)
C1_charge = z[0] * C1
C2 = CellVariable(mesh=mesh, name='H2', value=0.)
C2_charge = z[1] * C2
C3 = CellVariable(mesh=mesh, name='OH', value=COH_bulk)
C3_charge = z[2] * C3
C4 = CellVariable(mesh=mesh, name='Mg', value=CMg_bulk)
C4_charge = z[3] * C4
C5 = CellVariable(mesh=mesh, name='Ca', value=CCa_bulk)
C5_charge = z[4] * C5
C6 = CellVariable(mesh=mesh, name='CO3', value=CCO3_bulk)
C6_charge = z[5] * C6
C7 = CellVariable(mesh=mesh, name='HCO3', value=0.)
C7_charge = z[6] * C7
C8 = CellVariable(mesh=mesh, name='Na', value=0.)
C8_charge = z[7] * C8
C9 = CellVariable(mesh=mesh, name='Cl', value=0.)
C9_charge = z[8] * C9

theta = CellVariable(mesh=mesh, name='theta', value=0.0)

theta.constrain(valuethetaLeft, mesh.facesLeft)
theta.constrain(valuethetaRight, mesh.facesRight)

phi = CellVariable(mesh=mesh, name='phi', value=0.0)

PHI = C1_charge + C2_charge + C3_charge + C4_charge + C5_charge + C6_charge + C7_charge + C8_charge + C9_charge


eq1 = (TransientTerm(var=C1)
       == ConvectionTerm(coeff=((z[0] * D_ie(Di_l[0], theta, tau, sig_d) * F )/ (R * T)) * phi.faceGrad, var=C1)
       + DiffusionTerm(coeff=D_ie(Di_l[0], theta, tau, sig_d), var=C1) + (-1 * ((1 - theta) + (sig_d * theta) * J_j(R, T, F, N_j_O2, 1, 1, P_O2, Q_O2, I_oj_ref_O2, Cref_l[0], C1))/(2 * F)))

from fipy import input

if __name__ == "__main__":
	eq1.solve(C1, dt=dt)
	viewer = Viewer(vars=(C1, eq1))

	viewer.plot()

	input("press key to continue")

eq2 = (TransientTerm(var=C2)
       == ConvectionTerm(coeff=(z[1] * D_ie(Di_l[1], theta, tau, sig_d) * F / (R * T)) * phi.faceGrad, var=C2)
       + DiffusionTerm(coeff=D_ie(Di_l[1], theta, tau, sig_d), var=C2) + (-1 * ((1 - theta) + (sig_d * theta) * J_j(R, T, F, N_j_H2, 3, 1, P_H2, Q_H2, I_oj_ref_H2, Cref_l[1], C2))/(4 * F) ))

eq3 = (TransientTerm(var=C6) + TransientTerm(var=C3)
        == DiffusionTerm(coeff=(z[5] * D_ie(Di_l[5], theta, tau, sig_d) * F * C6.faceValue/ (R * T)), var=phi)
        + DiffusionTerm(coeff=D_ie(Di_l[5], theta, tau, sig_d), var=C6)
        + DiffusionTerm(coeff=(z[2] * D_ie(Di_l[2], theta, tau, sig_d) * F * C3.faceValue/ (R * T)), var=phi)
        + DiffusionTerm(coeff=D_ie(Di_l[2], theta, tau, sig_d), var=C3) + (-1 * ((4 * (1 - theta) + (sig_d * theta) * J_j(R, T, F, N_j_OH_1, 1, 1, P_OH_1, Q_OH_1, I_oj_ref_OH_1, Cref_l[2], C3)) / (2 * F) + (2 * (1 - theta) + (sig_d * theta) * J_j(R, T, F, N_j_OH_2, 3, 1, P_OH_2, Q_OH_2, I_oj_ref_OH_2, Cref_l[2], C3)) / (4 * F))) + (-2 * ((1 - theta) * (k[1] * C4 * (C3 ** 2) - K_sp[1]))))

eq4 = (TransientTerm(var=C4)
       == ConvectionTerm(coeff=(z[3] * D_ie(Di_l[3], theta, tau, sig_d) * F / (R * T)) * phi.faceGrad, var=C4)
       + DiffusionTerm(coeff=D_ie(Di_l[3], theta, tau, sig_d), var=C4) + (-1 * ((1 - theta) * (k[1] * C4 * (C3 ** 2) - K_sp[1]))))

eq5 = (TransientTerm(var=C5)
       == ConvectionTerm(coeff=(z[4] * D_ie(Di_l[4], theta, tau, sig_d) * F / (R * T)) * phi.faceGrad, var=C5)
       + DiffusionTerm(coeff=D_ie(Di_l[4], theta, tau, sig_d), var=C5) + (-1 * ((1 - theta) * (k[0] * ((((C5 * C6)/ K_sp[0]) - 1) ** 1.7)))))

eq6 = (TransientTerm(var=C6) + TransientTerm(var=C7)
        == DiffusionTerm(coeff=(z[5] * D_ie(Di_l[5], theta, tau, sig_d) * F * C6.faceValue/ (R * T)), var=phi)
        + DiffusionTerm(coeff=D_ie(Di_l[5], theta, tau, sig_d), var=C6)
        + DiffusionTerm(coeff=(z[6] * D_ie(Di_l[6], theta, tau, sig_d) * F * C7.faceValue/ (R * T)), var=phi)
        + DiffusionTerm(coeff=D_ie(Di_l[6], theta, tau, sig_d), var=C7) + (-1 * ((1 - theta) * (k[0] * ((((C5 * C6)/ K_sp[0]) - 1) ** 1.7)))))

eq7 = (TransientTerm(var=C7) == (C6 / (C3 - C7)))

eq8 = (TransientTerm(var=C8)
       == ConvectionTerm(coeff=(z[7] * D_ie(Di_l[7], theta, tau, sig_d) * F / (R * T)) * phi.faceGrad, var=C8)
       + DiffusionTerm(coeff=D_ie(Di_l[7], theta, tau, sig_d), var=C8))

eq9 = (TransientTerm(var=C9)
       == ConvectionTerm(coeff=(z[8] * D_ie(Di_l[8], theta, tau, sig_d) * F / (R * T)) * phi.faceGrad, var=C9)
       + DiffusionTerm(coeff=D_ie(Di_l[8], theta, tau, sig_d), var=C9))

#(-1 * ((1 - theta) * (k[0] * ((((C5 * C6)/ K_sp[0]) - 1) ** 1.7))))
#(-1 * ((1 - theta) * (k[1] * C4 * (C3 ** 2) - K_sp[1])))
eq10 = (TransientTerm(var=theta) == (1/((1 - sig_d) * delta)) * ((((-1 * ((1 - theta)*(k[0] * ((((C5 * C6)/ K_sp[0]) - 1) ** 1.7)))) * MWCaCo3)/rho_CaCo3) + (((-1 * ((1 - theta) * (k[1] * C4 * (C3 ** 2) - K_sp[1]))) * MWMgOH2)/rho_MgOH2)))


EQ = eq3 & eq4 & eq4 & eq5 & eq6 & eq7 & eq10

I am trying to solve the easiest equation first (eq1). When I try to solve it however I get this error:

== ConvectionTerm(coeff=((z[0] * D_ie(Di_l[0], theta, tau, sig_d) * F )/ (R * T)) * phi.faceGrad, var=C1)

TypeError: unsupported operand type(s) for *: 'binOp' and '_FaceGradVariable'

What should I do to fix this?

guyer Dec 6, 2023
Maintainer

D_ie is a function of theta, which is a rank-0 CellVariable. phi.faceGrad is a rank-1 FaceVariable. You cannot multiple those together. The different ranks are not a problem, but you cannot multiply values that are not defined at the same locations. D_ie(Di_l[0], theta, tau, sig_d).faceValue will resolve that error.

tjczec01 Dec 6, 2023
Author

eq1 = (TransientTerm(var=C1)
       == ConvectionTerm(coeff=((z[0] * D_ie(Di_l[0], theta, tau, sig_d).faceValue * F )/ (R * T)) * phi.faceGrad, var=C1)
       + DiffusionTerm(coeff=D_ie(Di_l[0], theta, tau, sig_d), var=C1) + (-1 * ((1 - theta) + (sig_d * theta) * J_j(R, T, F, N_j_O2, 1, 1, P_O2, Q_O2, I_oj_ref_O2, Cref_l[0], C1))/(2 * F)))

from fipy import input

if __name__ == "__main__":
	eq1.solve(C1, dt=dt)
	viewer = Viewer(vars=(C1, eq1))

	viewer.plot()

	input("press key to continue")

Now gives the error

    return _superlu.gstrf(N, A.nnz, A.data, indices, indptr,

RuntimeError: Factor is exactly singular

guyer Dec 7, 2023
Maintainer

Factor is exactly singular means that the matrix that results from your equations is not solvable, possibly because it has zeros on the diagonal, it admits infinite solutions, or there's some other pathology.
In your case, your explicit source is invalid:

>>> print(-1 * ((1 - theta) + (sig_d * theta) * J_j(R, T, F, N_j_O2, 1, 1, P_O2, Q_O2, I_oj_ref_O2, Cref_l[0], C1))/(2 * F))
[nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan nan
 nan nan]

which happens because

>>> print(J_j(R, T, F, N_j_O2, 1, 1, P_O2, Q_O2, I_oj_ref_O2, Cref_l[0], C1))
-inf

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Mathematical Modeling of the Formation of Calcareous Deposits on Cathodically Protected Steel in Seawater #977

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Mathematical Modeling of the Formation of Calcareous Deposits on Cathodically Protected Steel in Seawater #977

tjczec01 Nov 30, 2023

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