## 3 Calculation Methods
### ADINA Software  
The ADINA software, version 9.2, was used to solve the stated problem. In addition to Equations (1) and (2) that express the conservation of energy principle in the fluid and foil domains, respectively, equations that describe the conservation of mass and momentum principles were solved [23]. As a consequence, velocity \(\mathbf{v}=\left[v_{x},v_{y}\right]\) and pressure \(p\) fields in fluid domain are independent variables, and it was not necessary to assume a fixed shape of the \(x\)-component of the velocity field \(\mathrm{v}_{x}\) as in Equation (10). Instead, additional boundary conditions had to be introduced as follows:  
\[\frac{\partial T_{F}}{\partial y}(x,\delta_{H}+\delta_{M},t)=0 \tag{11}\]  
\[\mathbf{v}(x,\delta_{H},t)=\left[0,\;0\right] \tag{12}\]  
\[\mathbf{v}(x,\delta_{H}+\delta_{M},t)=\left[0,0\right] \tag{13}\]  
\[\mathbf{v}(0,y,t)=\left[v_{ave}(t),0\right] \tag{14}\]  
\[p(L,y,t)=p_{out}(t) \tag{15}\]  
where \(0\leq x\leq L\), \(\delta_{H}\leq y\leq\delta_{H}+\delta_{M}\), \(p_{out}\) is the absolute pressure at the outlet (known from the experiment--see Table 1).  
Equation (11) represents the assumption that there are no heat losses to the environment at the boundary \(y=\delta_{H}+\delta_{M}\) of the minichannel. Equations (12) and (13) represent the standard no-slip condition at the boundaries perpendicular to the flow direction. Equations (14) and (15) set the velocity of the fluid at the inlet and the absolute pressure of the fluid at the outlet, respectively.  
The transient type of analysis was applied, and 2D 4-node, planar FCBI elements (flow-condition based interpolation) were used. The mesh used in the simulations was regular: elements were rectangles, and their sides along the length of the minichannel (\(\alpha\) direction, Figure 2) have equal lengths in the perpendicular direction (\(y\) direction, Figure 2), the lengths of sides gradually decreased with the smallest sides near the fluid-foil interface. The number of subdivisions of lines in the direction perpendicular to the minichannel length (in the fluid and foil domains) were proportional to the number of subdivisions in the direction along the minichannel length. The mesh density depended only on one parameter, the number of subdivisions in the direction along the length of the minichannel (let us denote this number by \(N\)). The typical FE mesh used in the calculation in ADINA software is illustrated in Figure 3.  
The unconditionally stable implicit time integration method was used with a time step equal to 1 s.  
The mesh density, as essential for the simulations, was taken into special consideration. Various values of the parameter \(N\) (i.e., mesh densities) were tested, and the relationship between \(N\) and the values of the heat transfer coefficient in time for fixed \(x\) was investigated. An \(N\) was selected, for which the maximum relative difference between the distributions of the heat transfer coefficients in time and the fixed coordinate \(x\) for subsequent values of \(N\) was less than 5%. This goal was achieved for \(N=500\). The resulting mesh contains 18,000 elements (2000 elements in the foil domain and 16,000 in the fluid domain) and 19,038 nodes.  
The procedure described above was carried out for the FEM simulation of the experiment at \(Q_{m}=30\)\([\mathrm{kg/h}]\). However, the same FE mesh was also used in other calculations.