2. Reference solution

2.1. Calculation method used for the reference solution

The reference result is of the semi-analytical type. The 1D equation to be solved is as follows:

\(\{\begin{array}{}V\beta {(T)}_{,x}-K{T}_{,\mathrm{xx}}=0\\ \text{avec}{T}_{(x=0)}={T}_{0}\text{et}{T}_{(x=L)}={T}_{L}\end{array}\) eq 2.1-1

by integrating the equation [éq 2.1-1] we get:

\(\frac{V}{K}\beta (T)-\frac{\mathrm{dT}}{\mathrm{dx}}=A\) eq 2.1-2

where \(A\) is a constant depending on the boundary conditions, the \(\frac{V}{K}\) ratio, and the enthalpy function \(\beta (T)\).

This constant will be determined analytically.

Equation [éq 2.1-2] leads to:

\(x={\int }_{{T}_{0}}^{T(x)}\frac{\mathrm{dT}}{A+\frac{V}{K}\beta (T)}\) eq 2.1-3

who should check:

\(L={\int }_{{T}_{0}}^{{T}_{L}}\frac{\mathrm{dT}}{A+\frac{V}{K}\beta (T)}\) eq 2.1-4

Knowing \({T}_{\mathrm{0,}}{T}_{L},L,V,t\) and \(\beta (T)\), equation [éq 2.1-4] should give the value of the integration constant \(A\).

However, it is difficult (if not impossible) to determine this constant analytically, hence the use of numerical resolution of equation [éq 2.1-4] to determine \(A\).

With the data for problem \(({T}_{\mathrm{0,}}{T}_{L},{T}_{\mathrm{1,}}{T}_{\mathrm{2,}}{C}_{S}={C}_{l},{C}_{\mathrm{Sl}}\mathrm{.}\mathrm{.}\mathrm{.})\), we got the (physical) solution of \(A\) which takes the value \(A=-294.9117\).

Based on this constant, the analytical solution of the problem [éq 2.1-1] is analytical.

2.2. Benchmark results

Abscissa	Temperature
0.6	387.98514
0.7	451.51001
0.725	469.72232
0.750	488.97505
0.775	509.32766
0.80	530.84296
0.825	553.58738
0.85	577.63114
0.9	683.71269
0.9125	719.51615
0.925	756.32221
0.9375	794.16795
0.95	833.07971
0.9625	873.08751
0.9750	914.22222
0.9875	956.51557