## 3 Calculation Methods  
Unsteady state heat transfer in a group of minichannels was modelled in two-dimensional domains, Figure 2. Variations in temperature over the width of the minichannels were not considered. The \(x\)-axis represents the flow of the fluid direction, and the \(y\) coordinate denotes the depth of the minichannel. The heat transfer problem was formulated for the laminar flow of an incompressible fluid (Fluorinert FC-72) in minichannels. The independence of material properties from temperature was also assumed. In mathematical modelling, the central minichannel was chosen as the representative channel. The heat equation [19] with an internal heat source was used to describe the temperature distribution in the heated wall of the minichannel:  
\[\kappa_{H}\,\
abla^{2}T_{H}-\frac{\partial T_{H}}{\partial t}=q_{V}(t)\text{ for }(x,y)\in\Omega_{H},\;t>0, \tag{1}\]  
where: \(T_{H}\)--the heated wall temperature, \(\Omega_{H}=\big{\{}(x,y)\in R^{2}:\;0\leq x\leq L,\;0\leq y\leq\delta_{H}\big{\}}\), \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Experimental Parameter** & & & & & & & & & & & \\
**[Unit]** & & & & & & & & & & & & \\ \hline Mass flow rate, \(Q_{m}\left[\frac{kg}{h}\right]\) & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 & 55 \\ \hline Average fluid velocity, \(v_{F,\text{\tiny{env}}}\left[\frac{\text{m}}{\text{s}}\right]\) & 0.03 & 0.06 & 0.09 & 0.12 & 0.15 & 0.18 & 0.21 & 0.24 & 0.27 & 0.3 & 0.33 \\ \hline Average inlet pressure at the inlet, \(p_{F,\text{\tiny{fin}}}\left[\text{kPa}\right]\) & 98.4 & 104.9 & 101.7 & 129.6 & 107 & 101.8 & 99.4 & 127.8 & 100.8 & 104.1 & 103.3 \\ \hline Heat flux \(q\left[\frac{\text{kW}}{\text{m}^{2}}\right]\) the entire experiment & 4–22 & 4–32 & 4–40 & 4–47 & 4–50 & 4–50 & 4–59 & 5–57 & 4–58 & 6–57 & 5–57 \\ \hline Heat flux \(q\left[\frac{\text{kW}}{\text{m}^{2}}\right]\) part of the experiment & 8–11 & 10–14 & 10–14 & 10–14 & 13–19 & 13–19 & 13–19 & 14–20 & 17–31 & 17–23 & 17–25 \\ \hline Average fluid temperature & 289.7 & 289.4 & 288.1 & 287.9 & 287.1 & 286.5 & 285.8 & 286.8 & 286.4 & 286.3 & 287.1 \\ \hline Average fluid temperature & 303.9 & 302.2 & 298 & 295.3 & 296.1 & 293.9 & 291.8 & 291.4 & 293.1 & 292.4 & 293.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The basic data and parameters of the experimental series. Figure 2: The scheme of the main domains of the test section (fluid and foil) considered in the calculations. \(L\)--the minichannel length, \(\delta_{H}\) --the heated wall thickness, \(\
abla^{2}\)--Laplacian, \(\kappa_{H}\)--the heated wall thermal diffusivity coefficient, \(\kappa_{H}=\frac{\lambda_{H}}{\rho_{H}\,c_{pH}}\),  
\(\lambda_{H}\)--the heated wall thermal conductivity, \(\rho_{H}\) --the heated wall density,  
\(c_{pH,H}\)--the heated wall specific heat,  
\(q_{V}\)--the internal heat source, \(q_{V}(t)=-\frac{q(t)}{\delta_{H}\cdot\lambda_{H}}\), \(q\)--the heat flux, \(q(t)=\frac{I(t)\Delta U(t)}{A}\),  
\(A\)--cross-sectional area of the heated wall, \(I\)--the current supplied to the foil, \(\Delta U\)--the voltage drop across the heated wall. The Fourier-Kirchhoff equation [19] defines the non-stationary temperature field of the flowing fluid as follows:  
\[\kappa_{F}\,\
abla^{2}T_{F}-v_{x}(y,t)\,\frac{\partial T_{F}}{\partial x}- \frac{\partial T_{F}}{\partial t}=0\text{ for }(x,y)\in\Omega_{M},\ t>0 \tag{2}\]  
where \(\Omega_{M}=\big{\{}(x,y)\in R^{2}:0\leq x\leq L,\ \delta_{H}\leq x\leq \delta_{H}+\delta_{M}\big{\}}\), \(T_{F}\)--the fluid temperature, \(\delta_{M}\) --the minichannel depth, \(\kappa_{F}\)--fluid thermal diffusivity coefficient, \(\kappa_{F}=\frac{\lambda_{F}}{\rho_{F}\,c_{pH}}\),  
\(\lambda_{F}\)--fluid thermal conductivity, \(\rho_{F}\) --fluid density, \(c_{pH,F}\) --fluid specific heat, \(v_{x}(y,t)\)--component of the fluid velocity vector.