void IonFlow::frozenIonMethod(const double* x, size_t j0, size_t j1)
{
for (size_t j = j0; j < j1; j++) {
double wtm = m_wtm[j];
double rho = density(j);
double dz = z(j+1) - z(j);
double sum = 0.0;
for (size_t k : m_kNeutral) {
m_flux(k,j) = m_wt[k]*(rho*m_diff[k+m_nsp*j]/wtm);
m_flux(k,j) *= (X(x,k,j) - X(x,k,j+1))/dz;
sum -= m_flux(k,j);
}
// correction flux to insure that \sum_k Y_k V_k = 0.
for (size_t k : m_kNeutral) {
m_flux(k,j) += sum*Y(x,k,j);
}
// flux for ions
// Set flux to zero to prevent some fast charged species (e.g. electron)
// to run away
for (size_t k : m_kCharge) {
m_flux(k,j) = 0;
}
}
}
void LiquidTransport::getSpeciesFluxesExt(size_t ldf, doublereal* fluxes)
{
stefan_maxwell_solve();
for (size_t n = 0; n < m_nDim; n++) {
for (size_t k = 0; k < m_nsp; k++) {
fluxes[n*ldf + k] = m_flux(k,n);
}
}
}
void IonFlow::electricFieldMethod(const double* x, size_t j0, size_t j1)
{
for (size_t j = j0; j < j1; j++) {
double wtm = m_wtm[j];
double rho = density(j);
double dz = z(j+1) - z(j);
// mixture-average diffusion
double sum = 0.0;
for (size_t k = 0; k < m_nsp; k++) {
m_flux(k,j) = m_wt[k]*(rho*m_diff[k+m_nsp*j]/wtm);
m_flux(k,j) *= (X(x,k,j) - X(x,k,j+1))/dz;
sum -= m_flux(k,j);
}
// ambipolar diffusion
double E_ambi = E(x,j);
for (size_t k : m_kCharge) {
double Yav = 0.5 * (Y(x,k,j) + Y(x,k,j+1));
double drift = rho * Yav * E_ambi
* m_speciesCharge[k] * m_mobility[k+m_nsp*j];
m_flux(k,j) += drift;
}
// correction flux
double sum_flux = 0.0;
for (size_t k = 0; k < m_nsp; k++) {
sum_flux -= m_flux(k,j); // total net flux
}
double sum_ion = 0.0;
for (size_t k : m_kCharge) {
sum_ion += Y(x,k,j);
}
// The portion of correction for ions is taken off
for (size_t k : m_kNeutral) {
m_flux(k,j) += Y(x,k,j) / (1-sum_ion) * sum_flux;
}
}
}
/*
*
* Solve for the diffusional velocities in the Stefan-Maxwell equations
*
*/
void LiquidTransport::stefan_maxwell_solve() {
doublereal tmp;
size_t VIM = m_nDim;
m_B.resize(m_nsp, VIM);
//! grab a local copy of the molecular weights
const vector_fp& M = m_thermo->molecularWeights();
/*
* Update the concentrations in the mixture.
*/
update_conc();
double T = m_thermo->temperature();
m_thermo->getStandardVolumes(DATA_PTR(volume_specPM_));
m_thermo->getActivityCoefficients(DATA_PTR(actCoeffMolar_));
/*
* Calculate the electrochemical potential gradient. This is the
* driving force for relative diffusional transport.
*
* Here we calculate
*
* c_i * (grad (mu_i) + S_i grad T - M_i / dens * grad P
*
* This is Eqn. 13-1 p. 318 Newman. The original equation is from
* Hershfeld, Curtis, and Bird.
*
* S_i is the partial molar entropy of species i. This term will cancel
* out a lot of the grad T terms in grad (mu_i), therefore simplifying
* the expression.
*
* Ok I think there may be many ways to do this. One way is to do it via basis
* functions, at the nodes, as a function of the variables in the problem.
*
* For calculation of molality based thermo systems, we current get
* the molar based values. This may change.
*
* Note, we have broken the symmetry of the matrix here, due to
* consideratins involving species concentrations going to zero.
*
*/
for (size_t i = 0; i < m_nsp; i++) {
double xi_denom = m_molefracs_tran[i];
for (size_t a = 0; a < VIM; a++) {
m_ck_Grad_mu[a*m_nsp + i] =
m_chargeSpecies[i] * concTot_ * Faraday * m_Grad_V[a]
+ concTot_ * (volume_specPM_[i] - M[i]/dens_) * m_Grad_P[a]
+ concTot_ * GasConstant * T * m_Grad_lnAC[a*m_nsp+i] / actCoeffMolar_[i]
+ concTot_ * GasConstant * T * m_Grad_X[a*m_nsp+i] / xi_denom;
}
}
if (m_thermo->activityConvention() == cAC_CONVENTION_MOLALITY) {
int iSolvent = 0;
double mwSolvent = m_thermo->molecularWeight(iSolvent);
double mnaught = mwSolvent/ 1000.;
double lnmnaught = log(mnaught);
for (size_t i = 1; i < m_nsp; i++) {
for (size_t a = 0; a < VIM; a++) {
m_ck_Grad_mu[a*m_nsp + i] -=
m_concentrations[i] * GasConstant * m_Grad_T[a] * lnmnaught;
}
}
}
/*
* Just for Note, m_A(i,j) refers to the ith row and jth column.
* They are still fortran ordered, so that i varies fastest.
*/
switch (VIM) {
case 1: /* 1-D approximation */
m_B(0,0) = 0.0;
for (size_t j = 0; j < m_nsp; j++) {
m_A(0,j) = M[j] * m_concentrations[j];
}
for (size_t i = 1; i < m_nsp; i++){
m_B(i,0) = m_ck_Grad_mu[i] / (GasConstant * T);
m_A(i,i) = 0.0;
for (size_t j = 0; j < m_nsp; j++){
if (j != i) {
tmp = m_concentrations[j] / m_DiffCoeff_StefMax(i,j);
m_A(i,i) += tmp;
m_A(i,j) = - tmp;
}
}
}
//! invert and solve the system Ax = b. Answer is in m_B
solve(m_A, m_B);
break;
case 2: /* 2-D approximation */
m_B(0,0) = 0.0;
m_B(0,1) = 0.0;
for (size_t j = 0; j < m_nsp; j++) {
m_A(0,j) = M[j] * m_concentrations[j];
}
for (size_t i = 1; i < m_nsp; i++){
m_B(i,0) = m_ck_Grad_mu[i] / (GasConstant * T);
m_B(i,1) = m_ck_Grad_mu[m_nsp + i] / (GasConstant * T);
m_A(i,i) = 0.0;
for (size_t j = 0; j < m_nsp; j++) {
if (j != i) {
tmp = m_concentrations[j] / m_DiffCoeff_StefMax(i,j);
m_A(i,i) += tmp;
m_A(i,j) = - tmp;
}
}
}
//! invert and solve the system Ax = b. Answer is in m_B
solve(m_A, m_B);
break;
case 3: /* 3-D approximation */
m_B(0,0) = 0.0;
m_B(0,1) = 0.0;
m_B(0,2) = 0.0;
for (size_t j = 0; j < m_nsp; j++) {
m_A(0,j) = M[j] * m_concentrations[j];
}
for (size_t i = 1; i < m_nsp; i++){
m_B(i,0) = m_ck_Grad_mu[i] / (GasConstant * T);
m_B(i,1) = m_ck_Grad_mu[m_nsp + i] / (GasConstant * T);
m_B(i,2) = m_ck_Grad_mu[2*m_nsp + i] / (GasConstant * T);
m_A(i,i) = 0.0;
for (size_t j = 0; j < m_nsp; j++) {
if (j != i) {
tmp = m_concentrations[j] / m_DiffCoeff_StefMax(i,j);
m_A(i,i) += tmp;
m_A(i,j) = - tmp;
}
}
}
//! invert and solve the system Ax = b. Answer is in m_B
solve(m_A, m_B);
break;
default:
printf("uninmplemetnd\n");
throw CanteraError("routine", "not done");
break;
}
for (size_t a = 0; a < VIM; a++) {
for (size_t j = 0; j < m_nsp; j++) {
m_flux(j,a) = M[j] * m_concentrations[j] * m_B(j,a);
}
}
}
void LiquidTransport::stefan_maxwell_solve()
{
doublereal tmp;
m_B.resize(m_nsp, m_nDim, 0.0);
m_A.resize(m_nsp, m_nsp, 0.0);
//! grab a local copy of the molecular weights
const vector_fp& M = m_thermo->molecularWeights();
//! grad a local copy of the ion molar volume (inverse total ion concentration)
const doublereal vol = m_thermo->molarVolume();
/*
* Update the temperature, concentrations and diffusion coefficients in the mixture.
*/
update_T();
update_C();
if (!m_diff_temp_ok) {
updateDiff_T();
}
double T = m_thermo->temperature();
update_Grad_lnAC();
m_thermo->getActivityCoefficients(DATA_PTR(m_actCoeff));
/*
* Calculate the electrochemical potential gradient. This is the
* driving force for relative diffusional transport.
*
* Here we calculate
*
* X_i * (grad (mu_i) + S_i grad T - M_i / dens * grad P
*
* This is Eqn. 13-1 p. 318 Newman. The original equation is from
* Hershfeld, Curtis, and Bird.
*
* S_i is the partial molar entropy of species i. This term will cancel
* out a lot of the grad T terms in grad (mu_i), therefore simplifying
* the expression.
*
* Ok I think there may be many ways to do this. One way is to do it via basis
* functions, at the nodes, as a function of the variables in the problem.
*
* For calculation of molality based thermo systems, we current get
* the molar based values. This may change.
*
* Note, we have broken the symmetry of the matrix here, due to
* considerations involving species concentrations going to zero.
*/
for (size_t a = 0; a < m_nDim; a++) {
for (size_t i = 0; i < m_nsp; i++) {
m_Grad_mu[a*m_nsp + i] =
m_chargeSpecies[i] * Faraday * m_Grad_V[a]
+ GasConstant * T * m_Grad_lnAC[a*m_nsp+i];
}
}
if (m_thermo->activityConvention() == cAC_CONVENTION_MOLALITY) {
int iSolvent = 0;
double mwSolvent = m_thermo->molecularWeight(iSolvent);
double mnaught = mwSolvent/ 1000.;
double lnmnaught = log(mnaught);
for (size_t a = 0; a < m_nDim; a++) {
for (size_t i = 1; i < m_nsp; i++) {
m_Grad_mu[a*m_nsp + i] -=
m_molefracs[i] * GasConstant * m_Grad_T[a] * lnmnaught;
}
}
}
/*
* Just for Note, m_A(i,j) refers to the ith row and jth column.
* They are still fortran ordered, so that i varies fastest.
*/
double condSum1;
const doublereal invRT = 1.0 / (GasConstant * T);
switch (m_nDim) {
case 1: /* 1-D approximation */
m_B(0,0) = 0.0;
//equation for the reference velocity
for (size_t j = 0; j < m_nsp; j++) {
if (m_velocityBasis == VB_MOLEAVG) {
m_A(0,j) = m_molefracs_tran[j];
} else if (m_velocityBasis == VB_MASSAVG) {
m_A(0,j) = m_massfracs_tran[j];
} else if ((m_velocityBasis >= 0)
&& (m_velocityBasis < static_cast<int>(m_nsp))) {
// use species number m_velocityBasis as reference velocity
if (m_velocityBasis == static_cast<int>(j)) {
m_A(0,j) = 1.0;
} else {
m_A(0,j) = 0.0;
}
} else {
throw CanteraError("LiquidTransport::stefan_maxwell_solve",
"Unknown reference velocity provided.");
}
}
for (size_t i = 1; i < m_nsp; i++) {
m_B(i,0) = m_Grad_mu[i] * invRT;
m_A(i,i) = 0.0;
for (size_t j = 0; j < m_nsp; j++) {
if (j != i) {
tmp = m_molefracs_tran[j] * m_bdiff(i,j);
m_A(i,i) -= tmp;
m_A(i,j) = tmp;
}
}
}
//! invert and solve the system Ax = b. Answer is in m_B
solve(m_A, m_B);
condSum1 = 0;
for (size_t i = 0; i < m_nsp; i++) {
condSum1 -= Faraday*m_chargeSpecies[i]*m_B(i,0)*m_molefracs_tran[i]/vol;
}
break;
case 2: /* 2-D approximation */
m_B(0,0) = 0.0;
m_B(0,1) = 0.0;
//equation for the reference velocity
for (size_t j = 0; j < m_nsp; j++) {
if (m_velocityBasis == VB_MOLEAVG) {
m_A(0,j) = m_molefracs_tran[j];
} else if (m_velocityBasis == VB_MASSAVG) {
m_A(0,j) = m_massfracs_tran[j];
} else if ((m_velocityBasis >= 0)
&& (m_velocityBasis < static_cast<int>(m_nsp))) {
// use species number m_velocityBasis as reference velocity
if (m_velocityBasis == static_cast<int>(j)) {
m_A(0,j) = 1.0;
} else {
m_A(0,j) = 0.0;
}
} else {
throw CanteraError("LiquidTransport::stefan_maxwell_solve",
"Unknown reference velocity provided.");
}
}
for (size_t i = 1; i < m_nsp; i++) {
m_B(i,0) = m_Grad_mu[i] * invRT;
m_B(i,1) = m_Grad_mu[m_nsp + i] * invRT;
m_A(i,i) = 0.0;
for (size_t j = 0; j < m_nsp; j++) {
if (j != i) {
tmp = m_molefracs_tran[j] * m_bdiff(i,j);
m_A(i,i) -= tmp;
m_A(i,j) = tmp;
}
}
}
//! invert and solve the system Ax = b. Answer is in m_B
solve(m_A, m_B);
break;
case 3: /* 3-D approximation */
m_B(0,0) = 0.0;
m_B(0,1) = 0.0;
m_B(0,2) = 0.0;
//equation for the reference velocity
for (size_t j = 0; j < m_nsp; j++) {
if (m_velocityBasis == VB_MOLEAVG) {
m_A(0,j) = m_molefracs_tran[j];
} else if (m_velocityBasis == VB_MASSAVG) {
m_A(0,j) = m_massfracs_tran[j];
} else if ((m_velocityBasis >= 0)
&& (m_velocityBasis < static_cast<int>(m_nsp))) {
// use species number m_velocityBasis as reference velocity
if (m_velocityBasis == static_cast<int>(j)) {
m_A(0,j) = 1.0;
} else {
m_A(0,j) = 0.0;
}
} else {
throw CanteraError("LiquidTransport::stefan_maxwell_solve",
"Unknown reference velocity provided.");
}
}
for (size_t i = 1; i < m_nsp; i++) {
m_B(i,0) = m_Grad_mu[i] * invRT;
m_B(i,1) = m_Grad_mu[m_nsp + i] * invRT;
m_B(i,2) = m_Grad_mu[2*m_nsp + i] * invRT;
m_A(i,i) = 0.0;
for (size_t j = 0; j < m_nsp; j++) {
if (j != i) {
tmp = m_molefracs_tran[j] * m_bdiff(i,j);
m_A(i,i) -= tmp;
m_A(i,j) = tmp;
}
}
}
//! invert and solve the system Ax = b. Answer is in m_B
solve(m_A, m_B);
break;
default:
printf("unimplemented\n");
throw CanteraError("routine", "not done");
break;
}
for (size_t a = 0; a < m_nDim; a++) {
for (size_t j = 0; j < m_nsp; j++) {
m_Vdiff(j,a) = m_B(j,a);
m_flux(j,a) = concTot_ * M[j] * m_molefracs_tran[j] * m_B(j,a);
}
}
}
