Illustrative example of development of transfer function
Development of transfer function will be presented for the zone 1. The
mathematical model in real domain is presented by equation (2). In the
first step, the governing equation with the boundary condition should be
converted to the Laplace domain:
\(sC+\frac{F_{v}}{A_{1}}\frac{\text{dC}}{\text{dx}}-D_{L,1}\frac{d^{2}C}{\text{dx}^{2}}=0\)(15)
BC for x=0
\(F_{v}X=F_{v}\bullet C-A_{1}D_{L,1}\frac{\text{dC}}{\text{dx}}\)(16)
for x=L1
\(\frac{\text{dC}}{\text{dx}}=0\) (17)
where:
capital letters describe the complex functions corresponding to the real
functions,
X is the complex function that describe inlet concentration for the zone
1, i.e. \(\mathcal{L}\left(c_{0}\right)\)
and \(w_{2}=\frac{4F_{v}}{\pi d_{w,2}^{2}}\).
Integration to find a general solution and substitution of initial
condition leads to
\(C=K_{1}e_{1}^{x}+K_{2}e_{2}^{x}\) (18)
where
\(r_{1}=\sqrt{4A_{1}^{2}D_{L,1}s+F_{v}^{2}}\) (19)
\(e_{1}=exp\left(\frac{\left(\text{Fv}+r_{1}\right)}{2A_{1}D_{L,1}}\right)\)(20)
\(e_{2}=exp\left(\frac{\left(\text{Fv}-r_{1}\right)}{2A_{1}D_{L,1}}\right)\)(21)
Constants of integration can be obtained from boundary conditions.
\(K_{1}=\frac{F_{v}Xe_{2}^{L_{1}}\left(2A_{1}^{2}D_{L,1}s+F_{v}^{2}-F_{v}r_{1}\right)}{\left[\left(A_{1}^{2}D_{L,1}s+\frac{F_{v}^{2}}{2}-\frac{F_{v}r_{1}}{2}\right)e_{2}^{L_{1}}-\left(A_{1}^{2}D_{L,1}s+\frac{F_{v}^{2}}{2}+\frac{F_{v}r_{1}}{2}\right)e_{1}^{L_{1}}\right]\left(F_{v}-r_{1}\right)}\)(22)
\(K_{2}=\frac{F_{v}\left(F_{v}+r_{1}\right)Xe_{1}^{L_{1}}}{\left(2A_{1}^{2}D_{L,1}s+F_{v}^{2}-F_{v}r_{1}\right)e_{2}^{L_{1}}-\left(2A_{1}^{2}D_{L,1}s+F_{v}^{2}+F_{v}r_{1}\right)e_{1}^{L_{1}}}\)(23)
Substitution of integration constants and x=L1 (zone
outlet) to the general solution leads to outlet concentration. According
to the definition, outlet concentration divided by inlet concentration
(X) gives the transform function. After simplifications we obtain
\(G_{1}\left(s\right)=\frac{C\left(L_{1}\right)}{X}=\frac{2F_{v}\left(r_{1}^{2}-F_{v}r_{1}\right)e^{\frac{F_{v}L_{1}}{A_{1}D_{L,1}}}}{\left(F_{v}-r_{1}\right)\left[\left({2A}_{1}^{2}D_{L,1}s+F_{v}^{2}+F_{v}r_{1}\right)e_{1}^{L_{1}}-\left({2A}_{1}^{2}D_{L,1}s+F_{v}^{2}-F_{v}r_{1}\right)e_{2}^{L_{1}}\right]}\)(24)
Transfer functions for the other zones can be obtained in the same way.
The overall transfer function of a system can be easily obtained by
multiplication of transfer functions of the zones 1..3:
\(G\left(s\right)=G_{1}\left(s\right){\bullet G}_{2a}\left(s\right)\bullet G_{2b}\left(s\right)\bullet G_{3}\left(s\right)=\frac{\left.\ C\right|_{x=L_{1}}}{C_{0}\left(s\right)}\frac{\left.\ C\right|_{x=L_{1}+L_{2a}}}{\left.\ C\right|_{x=L_{1}}}\frac{\left.\ C\right|_{x=L_{1}+L_{2a}+L_{2b}}}{\left.\ C\right|_{x=L_{1}+L_{2a}}}\frac{\left.\ C\right|_{x=L_{1}+L_{2a}+L_{2b}+L_{3}}}{\left.\ C\right|_{x=L_{1}+L_{2a}+L_{2b}}}=\frac{\left.\ C\right|_{x=L_{1}+L_{2a}+L_{2b}+L_{3}}}{C_{0}\left(s\right)}\)(25)
Multiplication by C0 and application of inverse Laplace
transform gives outlet concentration from the system that is
concentration recorded by TCD
\(\left.\ c\right|_{x=L_{1}+L_{2a}+L_{2b}+L_{3}}=\mathcal{L}^{-1}\left(G\left(s\right)\bullet c_{0}\left(s\right)\right)\)(26)
All calculations were carried out using Maple®2017.
den Iseger algorithm was used for finding inverse Laplace transform of
right-hand side of above equation. Precision of the algorithm is high
(see Appendix 1). Since numerical inverse Laplace transform is not
supported by Maple, the implementation of den Iseger algorithm was coded
by the authors. Determination of DL-value requires
application of optimization methods (the inverse boundary value
problem). The build-in global optimization procedure of Maple
(GlobalSolve) was used. The aim function was based on areas ratio of
experimental and calculated signal curves (factor f presented in the
Tables below is a reflection of the areas ratio; f=1 corresponds to the
perfect fit). From an optimal fit of experimental and calculated signal
curves we get DL-value.