Intermediate Heat Exchanger

This section is essentially a copy from an article that has appeared much earlier in the old sci.math and sci.physics groups.
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 primary or shell-side flow. The tubes are filled with a medium which is called the secondary or tube-side flow. The primary flow is "hot" while the secondary flow is "cold". The primary flow streams from top to bottom while the secondary flow streams from bottom to top. In this way, the secondary flow is heated up by the primary flow. Which explains the device's name.
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 thermal stresses caused by temperature gradients.

Fluid-Tube Continuum

In order to describe the shell-side flow and temperature distribution, investigators have attempted the so-called fluid-tube continuum approach. The idea of the method is: setting up partial differential equations for a kind of porous medium. Tube bundles are treated this way because, from a practical point of view, it is hopeless to apply the basic laws of flow (Navier Stokes) and of heat transfer directly, without approximations. (Requiring for example that velocities should be zero at all parts of the solid structure.)
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 true continuous media. In my (not so) humble opinion, the Fluid-Tube Continuum is in no way different from other continuous media, like rock, water, air or even space-time itself. Physics allows continuous media only to exist by approximation. Which is due to the fact that "real" numbers, covering observations, are essentially inaccurate. As far as the fluid-tube continuum in a heat exchanger is concerned, that inaccuracy has as an order of magnitude the distance (pitch) between two adjacent tubes.

Governing Equations

It can be argued that, as a first approximation, the shell-side flow in the tube bundle is not only incompressible, but also irrotational. The latter can be understood intuitively. The size of a fluid particle in the fluid-tube continuum model is, "by definition", greater than (say) the pitch between two neighboring tubes. It is reasonable to assume that a fluid particle of this size will experience almost equal friction at all of its sides, and therefore will not rotate. Consequently, the (Partial Differential !) equations for flow in a tube bundle may be assumed to be those for ideal flow, in a cylindrically symmetric geometry: $$ \frac{\partial ru}{\partial r} + \frac{\partial rv}{\partial z} = 0 \qquad ; \qquad \frac{\partial v}{\partial r} - \frac{\partial u}{\partial z} = 0 $$ Here: $u$ = horizontal velocity component, $v$ = vertical velocity component, $r$ = horizontal radius, $z$ = vertical distance.
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 heat transfer coefficient; $P=$ primary; $S=$ secondary. The boundary conditions should not be forgotten: $$ T_P = T_{PL} \qquad \mbox{at the primary inlet (upper perforation)} \\ T_S = T_{S0} \qquad \mbox{at the secondary inlet (tube plate below)} $$ Remember. These equations are ONLY simple and elegant because very CRUDE approximations are involved. Replacing the Discrete (tubes in fluid) by an unstructured amorphous mixture of fluid and tubes means that a lot of (useful?) information has been simply thrown away, for the purpose of the end-result being applicable in engineering.
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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