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:
- the equation of state of an ideal gas describes the relationships for
each of the gas components
- the system is operated under isothermal conditions.
- the pressure drop inside the channels is neglected (inner
electronically controlled pressure regulator provide precise gas
control)
Analysis of the gas flow line (Fig. 1) leads to the following
conclusions:
- The valve dumps interactions of the zones 1 and 4 with zones 2a and
zone 2b, respectively,
- the zones 2a and 2b interacts;
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.