An inlet concentration is given by (a rectangular signal pulse)
\(c_{0}=\left\{\par \begin{matrix}0\ \ \ \ \ \ \ \ \ \ \ \ \ \ t<0\\ c_{T}\ \ 0\leq t\leq\frac{V_{\text{imp}}}{F_{v}}\\ 0\ \ \ \ \ \ \ \ \ \ \ \ t>\frac{V_{\text{imp}}}{F_{v}}\\ \end{matrix}\right.\ \) (1)
where: \(c_{T}=\frac{P}{R_{g}\cdot T}\mathrm{\text{\ \ }}\)
Models are based on the following assumptions:
Analysis of the gas flow line (Fig. 1) leads to the following conclusions:
Interactions mentioned in the second bullet point can be treated as negligible, then the resulted mathematical model is much simpler for developing and easier for solution. They can also be treated as non-negligible, then the model derivation is a rather hard task. Since we could not safely accept a hypothesis on negligible interactions, we had to test two models. The first model, called further “model A”, neglects interactions between zones 2a and 2b while the second one, “model B”, takes them into account. The mass balance of nitrogen is presented by equation (2).
Mass balance equations.
\(\frac{\partial c\left(x,t\right)}{\partial t}=D_{L,i}\frac{\partial^{2}c\left(x,t\right)}{\partial x^{2}}-\frac{F_{v}}{\varepsilon_{e}A_{i}}\bullet\frac{\partial c\left(x,t\right)}{\partial x}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i=1,2a,2b,3\)(2)
IC (for all zones)
\(c\left(x,0\right)=0\) (3)
BC for Model A (interactions between zones 2a and 2b are not remarkable)
zone 1
\(F_{v}c\left(0^{-},t\right)=F_{v}c\left(0^{+},t\right)-A_{1}D_{L,1}\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x=0^{+}}\)(4)
\(\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x=L_{1}^{-}}=0\)(5)
zone 2a
\(F_{v}c\left(L_{1}^{-},t\right)=F_{v}c\left(L_{1}^{+},t\right)-A_{2a}D_{L,2a}\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x=L_{1}^{+}}\)(6)
\(\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x={L_{1}+L}_{2a}^{-}}=0\)(7)
zone 2b
\(F_{v}c\left(L_{1}+L_{2a}^{-},t\right)=F_{v}c\left(L_{1}+L_{2a}^{+},t\right)-A_{2b}D_{L,2b}\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x={L_{1}+L}_{2a}^{+}}\)(8)
\(\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x={L_{1}+L_{2a}+L}_{2b}^{-}}=0\)(9)
zone 3
\(F_{v}c\left({L_{1}+L_{2a}+L}_{2b}^{-},t\right)=F_{v}c\left({L_{1}+L_{2a}+L}_{2b}^{+},t\right)-A_{3}D_{L,3}\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x={L_{1}+L_{2a}+L}_{2b}^{+}}\)(10)
\(\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x=L_{1}+L_{2a}+L_{2b}+L_{3}}=0\)(11)
BC for Model B (interactions between zones 2a and 2b are remarkable)
Various boundary conditions on the border of the zone 2a and 2b is the difference between the models A and B. Presented here boundary conditions make allowance for interactions between the zones.
zone 2a + zone 2b
\(c\left({L_{1}+L}_{2a}^{-},t\right)=c\left({L_{1}+L}_{2a}^{+},t\right)\)(12)
\(F_{v}c\left(L_{1}+L_{2a}^{-},t\right)-A_{2a}D_{L,2a}\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x={L_{1}+L}_{2a}^{-}}=F_{v}c\left(L_{1}+L_{2a}^{+},t\right)-A_{2b}D_{L,2b}\left.\ \frac{\partial c\left(x,t\right)}{\partial x}\right|_{x={L_{1}+L}_{2a}^{+}}\)(13)
where
\(A_{i}=\frac{\pi d_{i}^{2}}{4}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i=1,2a,2b,3\)(14)
For empty zones \(\varepsilon_{e}=1\).
c (L 1+L 2a+L 2b+L3,t ), that is, the outlet concentration from the last zone corresponds to the signal recorded by the TCD-detector.
In both cases models consist of four partial differential equations with the initial and boundary conditions. Model A and B differ one to another in boundary conditions for zones 2a and 2b. This observation gave us an idea on how to improve and speed-up analysis of the problem. Numerical solution of the large system of equations, that may be stiff is rather tedious. We propose application of the transfer function concept. A transfer function is the ratio of an output variable to any input variable, in the Laplace domain. We choose outlet concentration from a zone as an outlet variable and, similarly, input concentration as an input variable. In that case each zone is described by its own transfer function and, next, we can multiply the individual transfer function of each subsystem (zone) to obtain the overall transfer function. Model reconstruction is an easy task and, in practice, reduces to replacing a transfer functions for a subsystem by another one.