// Given the pressure, P, and the mole fraction of THIS sorbate in the
// gas phase, YA, returns the selectivity of THIS sorbate vs sorbate B.
public double selectivity(Isotherm B, double P, double YA)
{
double XA = x(B, P, YA);
double XB = 1 - XA;
double YB = 1 - YA;
return(((XA) / (YA)) / ((XB) / (YB)));
}
public void getN_forEachComponent(Isotherm B, double P, double YA, out double NA, out double NB)
{
double XA, XB, YB;
get_SorbedAndBulkMoleRatios_fromYA(B, P, YA, out XA, out YB, out XB);
double NA_stp = n(P * YA / XA);
double NB_stp = B.n(P * YB / XB);
double Ntot_stp = (NA_stp * NB_stp) / (NB_stp * XA + NA_stp * XB);
NA = XA * Ntot_stp;
NB = XB * Ntot_stp;
}
// Given a pressure, P, the mole fraction of sorbate A (THIS sorbate) in the
// gas phase, X_A the mole fraction of sorbate A (THIS sorbate) in the sorbed
// phase, Y_A, and isotherm model for species B, spreading_pressure_diff()
// calculates the difference in spreading pressure between species A (the
// species for THIS isotherm) & B.
double spreading_pressure_diff(
Isotherm B, // isotherm for species B
double P, // pressure
double X_A, // mole fraction of sorbate A in the gas phase
double Y_A // mole fraction of sorbate A in the sorbed phase
)
{
double spreadingPressure_A = pi(P * Y_A / X_A);
double spreadingPressure_B = B.pi(P * (1.0 - Y_A) / (1.0 - X_A));
return(spreadingPressure_B - spreadingPressure_A);
}
// Qst( A_T1, A_T2, n ) returns the heat of adsorption associated with a molar uptake, n, given two isotherms
// for a single species, recorded at temperatures T1 and T2
public static double Qst(Isotherm Isotherm_T1, Isotherm Isotherm_T2, double n)
{
const double R = .008314462175; // kJ/(mol*K)
double P1 = Isotherm_T1.P(n);
double T1 = Isotherm_T1.temp;
double P2 = Isotherm_T2.P(n);
double T2 = Isotherm_T2.temp;
double numerator = -R *Math.Log(P2 / P1);
double denominator = (T1 - T2) / (T1 * T2);
return(numerator / denominator);
}
public void get_SorbedAndBulkMoleRatios_fromYA(Isotherm B, double P, double YA, out double XA, out double YB, out double XB)
{
XA = x(B, P, YA);
XB = 1 - XA;
YB = 1 - YA;
}
// Given a pressure, P, the mole fraction of sorbate A in the sorbed phase, Y_A, and
// isotherm model for species B, x() narrows in on a gas phase mole fraction for sorbate
// A (THIS sorbate), X_A, accurate to the limits of machine precision.
double x(
Isotherm B, // isotherm for species B
double P, // pressure
double Y_A // mole fraction of sorbate A in the sorbed phase
)
{
// Bracket the root
double x = 0;
double lo_x = 1E-14; // 0 + dx
double hi_x = 1.0 - 1E-14; // 1 - dx
const double STEPS = 100;
// factor will invert the value of the spreading pressure difference
// function in the event that the values start out negative and go
// positive.
double factor = ((spreading_pressure_diff(B, P, lo_x, Y_A) >= 0)?(1.0):(-1.0));
if (factor * spreading_pressure_diff(B, P, hi_x, Y_A) > 0)
{
bool foundInterval = false;
for (int dx = 1; dx < 100; dx++)
{
double diff = factor * spreading_pressure_diff(B, P, dx * 0.01 * (hi_x - lo_x), Y_A);
if (diff <= 0)
{
foundInterval = true;
hi_x = dx * 0.01;
}
}
if (!foundInterval)
{
throw(new System.Data.ConstraintException("No equilibrium condition found for pressure " + P));
}
}
// Test values of x in range (0,1 (or hi_x)) until the sign changes.
//
// (We are looking for a root and are trying to find the X_A
// value where the difference in the spreading pressure for A & B
// is 0, i.e. the equilibrium point.)
//
// Each time the sign changes, test values of x in the range of x
// just before the sign of delta_of_spreading_pressure (spDiff)
// changed, and the x that caused the spDiff sign to change.
// We will narrow this testing region by a factor of 'steps' a
// total of MAX_ITER times.
const int MAX_ITER = 10;
for (int j = 0; j < MAX_ITER; j++)
{
int i = 0;
double spDiff = factor * spreading_pressure_diff(B, P, lo_x, Y_A);
// Go until we've reached "steps" iterations or until spDiff
// changes sign.
while (i <= STEPS && spDiff > 0)
{
x = lo_x + (hi_x - lo_x) * (double)(i) / (double)(STEPS);
spDiff = factor * spreading_pressure_diff(B, P, x, Y_A);
i++;
}
lo_x = lo_x + (hi_x - lo_x) * (double)(i - 2) / (double)(STEPS);
hi_x = x;
}
double mid_x = (lo_x + hi_x) / 2.0;
double lo = Math.Abs(spreading_pressure_diff(B, P, lo_x, Y_A));
double mid = Math.Abs(spreading_pressure_diff(B, P, mid_x, Y_A));
double hi = Math.Abs(spreading_pressure_diff(B, P, hi_x, Y_A));
if (lo < mid)
{
if (lo < hi)
{
return(lo_x);
}
}
else
{
if (mid < hi)
{
return(mid_x);
}
}
return(hi_x);
}
