2.1.4. Coupling of the Flow Distribution and Heat Transfer Submodels

Each major iteration of the FEA-based solver consists of two steps. The first step is a fixed temperature field analogy of the adiabatic model (as described in [26]; robustness of the model can be improved by carrying out this first step repeatedly until the respective residual falls below a predefined threshold). New estimates of the temperature fields for both the tube and the shell sides are then obtained in the second step. Here, the necessary values of *CU* are updated using edge mass flow rates from the first step and the corresponding new estimates of cumulative overall heat transfer coefficients. To solve the respective combined linear system for the tube-side and the shell-side temperatures, one boundary temperature must be provided for each stream (for instance, at the inlet of each tube in the bundle and for each inlet cell in the discretized shell side). The resulting temperature matrix could look, for example, like the one in Figure 6. Even though linear systems represented by such matrices can be solved quite easily, it is obvious from the figure that implementation of a reordering algorithm would be necessary should one want to improve performance via preconditioning in case of much larger linear systems.

**Figure 6.** Non-zero values in the sparse temperature field matrix used in the second step of the FEA solver. Please note that, for the sake of clarity, only a small matrix with the rank slightly below 800 is shown here, which originates from a flow distribution system with two bundles consisting of just four tubes each.

As the convergence criterion, the fluid flow submodel used the scaled norm of the difference between the solutions from the predictor step and the corrector step. The corresponding scaled residual limit was 10<sup>−</sup>5. In the case of the heat transfer submodel, if we denote the original linear system **Ax** = **b**, then the scaled residual norm is computed from **b** − **Ax** just before the heat transfer submodel is solved. (If it were done after the respective solution process, the norm would be equal to zero.) The same residual limit, that is, 10<sup>−</sup>5, was used here.

All physical property data (mean specific heat capacity, dynamic viscosity, etc.) are always taken for the current conditions from the IAPWS [34] or the CoolProp [35] physical property libraries, or, in special cases (e.g., flue gas), are computed using various interpolation functions or tabulated data depending on the actual compositions. Thermal properties of the tube and fin materials are always obtained using tabulated data from literature (for example, from the technical standard [36]).
