Problem: how to describe the complex transport phenomena in the flow around the tubes of a shell-and-tube heat exchanger, such as the one depicted below.

A heat exchanger is a container with many (straight) tubes mounted in it. The container is filled with a medium which is called the

The flow-distribution on the shell-side of a heat exchanger has a significant influence on the temperature distribution across the tube-bundle and consequently on the

Classical theory of porous media describes flow and transport through soils, consisting of sand, clay, peat. Typical applications are in the field of petroleum reservoir engineering and groundwater hydrology. But looking around, we can see many non-classical examples of transport phenomena where porous media are involved. Such as potatoes, stored in a vessel, which form a porous medium for the cooling air flow. Other examples are filtration, chemical reactions using solid catalists, adsorption, and mass transfer in packed columns. Finally, the flow in the core structures of nuclear reactors, rod arrays, and also heat exchangers can be considered as a flow in a porous medium.

Before proceeding further, it is important to realize that there is nothing fake in considering tube bundles as

By assuming Ideal Internal Flow, the flow field is invariant for scaling with a factor $G_P$ (independent of the flow magnitude) and therefore will be normed in such a way that the absolute vaule $\left|(u,v)\right|$ is unity $= 1$ in the middle of the bundle. Furthermore, the Neratoom heat exchanger has been designed in such a way that the flow velocity in the middle of the bundle is the same as the flow velocity at the inlet perforation. So Ideal Flow calculations, with $u = -1$ at the inlet opening, can immediately be used and only have to be scaled to describe the real thing. So far so good for the flow, which is ideal in more than one respect.

Now wrap a control-volume around a couple of tubes. Set up the energy balances for this volume. Let the volume become "infinitesimally small", though still remaining larger than the distance between the tubes. Or throw away the integral signs after applying Gauss theorems. Or whatever. Then the following set of Partial Differential Equations may be inferred for Heat Exchange in a tube bundle: $$ c.G_P \left[ u.\frac{\partial T_P}{\partial r} + v.\frac{\partial T_P}{\partial z} \right] + a.(T_P - T_S) = 0 \qquad \mbox{: shell side} \\ c.G_S.\frac{\partial T_S}{\partial z} +a.(T_S - T_P) = 0 \qquad \mbox{: tube side} $$ Here: $c=$ heat capacity; $G=$ mass flow; $T=$ temperature; $(r,z)=$ cylinder coordinates; $(u,v)=$ normed velocities; $a=$ total

The system of Partial Differential Equations has served as an analytical framework whereupon Numerical Methods can be based. Like making powder from potatoes, and then preparing potatoes from the powder again (-: S.V. Patankar).

**Reference** (somewhat outdated):

H. de Bruijn and W. Zijl; Numerical Simulation of the
Shell-Side Flow and Temperature Distribution in Heat Exchangers;
Handbook of Heat and Mass Transfer; chapter 27; Volume 1: Heat Transfer Operations;
Nicholas P. Cheremisinoff, Editor; Gulf Publishing Company (1986).

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