SDM-LKM Equivalence - Molass Library Essence

Relation among DSM, LKM and EDM¶

With regard to the relation to the stochastic models, there is a notable paper Felinger et al. (2004) which suggests:

SDM \sim LKM \supseteq EDM

(1)

Notational Correspondence in Relevant Papers¶

Here are summarized notational correspondence among the relevant papers:

Table 1:Summary of Notational Correspondence between SDM and { LKM $\supseteq$ EDM }

Dondi-2002 SDM	Felinger-2004 LKM	Rehman-2021 EDM	Parameter Despription
$N_0$	$N_d = \frac{Lu}{2D}$	$\frac{Lu}{2D_z}$	Number of Plates in Mobile Zone; $N_0 = N_d = (\frac{t_0}{\sigma_0})^2$
$n_i$	$N_m$		Average Number of Stays in Stagnant Zone; $N_m=F k_a t_0 = k' k_d t_0$ ; $k' = \frac{F k_a}{k_d}$
$\tau_i$	$\frac{1}{k_d}$		Average Time of Stays in Stagnant Zone; $\frac{1}{k_d} = \frac{k' t_0}{N_m}$
$K_{SEC}$	$e$ (porosity)	$e$ (porosity)	$\rho = \min(1, \frac{R_g}{R_p})$ ; $K_{SEC} = (1 - \rho)^m$ ; $(K_{SEC})^{\frac{1}{m}} = 1 - \rho$ ; $R_g \sim R_p(1 - porosity^{\frac{1}{m}})$

SymPy Examples to Understand the Papers¶

To follow the papers, you can use SymPy. Here are some examples.

from sympy import symbols, Eq, Function, dsolve

f, g = symbols("f g", cls=Function)
x = symbols("x")
eqs = [Eq(f(x).diff(x), g(x)), Eq(g(x).diff(x), f(x))]
dsolve(eqs, [f(x), g(x)])

[Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), C1*exp(-x) + C2*exp(x))]

dsolve(eqs, [f(x), g(x)], ics={f(0): 1, g(2): 3})

[Eq(f(x), (1 + 3*exp(2))*exp(x)/(1 + exp(4)) - (-exp(4) + 3*exp(2))*exp(-x)/(1 + exp(4))),
 Eq(g(x), (1 + 3*exp(2))*exp(x)/(1 + exp(4)) + (-exp(4) + 3*exp(2))*exp(-x)/(1 + exp(4)))]

eqn = Eq(f(x).diff(x), f(x))
dsolve(eqn, f(x), ics={f(x).diff(x).subs(x, 1): 2})

from sympy import symbols, Function, dsolve
t = symbols('t')
y = Function('y')(t)
y

yp = y.diff(t)
ypp = yp.diff(t)
eq = ypp + 2*yp + y
eq

dsolve(eq, y)

dsolve(eq, y, ics={y.subs(t, 0): 0})

References¶

Felinger, A., Cavazzini, A., & Dondi, F. (2004). Equivalence of the microscopic and macroscopic models of chromatography: stochastic–dispersive versus lumped kinetic model. Journal of Chromatography A, 1043(2), 149–157. https://doi.org/10.1016/j.chroma.2004.05.081
Dondi, F., Cavazzini, A., Remelli, M., Felinger, A., & Martin, M. (2002). Stochastic theory of size exclusion chromatography by the characteristic function approach. Journal of Chromatography A, 943(2), 185–207. https://doi.org/10.1016/S0021-9673(01)01443-1
Ur Rehman, J., Muneer, A., & Qamar, S. (2021). Analysis of equilibrium dispersive model of liquid chromatography considering a quadratic-type adsorption isotherm. Thermal Science, 26(3), 2069–2080. 10.2298/TSCI201229179U