sci.math.num-analysis

SUNA38: Fluid-Tube Continuum
============================

This article has appeared earlier in the "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)
o o o o o o o \\           tubes mounted in it. The container is filled up with
o o o o o o o o \\         a medium which is called the _primary_ or shell-side
o o o o o o o o o \\       flow. The tubes are filled up with a medium which is
==  o o o o o o o o \\      called the secondary or tube-side flow. The primary
 \\   o o o o o o o o \\     flow is "hot" while the secondary flow is "cold".
   \\  o o o o o o o o ||    The primary flow streams from top to bottom while
    || o o o o o o o o ||    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.
    .. secondary outlet
-------|-|-|-|-|-|-|-|--           The flow-distribution on the shell-side of a
///////|/|/|/|/|/|/|/|/ |    p     heat exchanger has a significant influence
-----B-|-|-|-|-|-|-|-|-A=====r     on the temperature distribution across the
.   || | | | | | | | |  <--  i i   tube-bundle and consequently on the thermal
.   || | | | | | | | |  <--  m n   stresses caused by temperature gradients.
.   || | | | | | | | |  <--  a l
.   || | | | | | | | |  <--  r e   In order to describe the shell-side flow and
.   || | | | | | | | | |==== y t   temperature distribution, investigators have
.   || | | | | | | | | ||          attempted the so-called fluid-tube-continuum
.   || | | | | | | | | ||  H       approach. The idea of the method is: setting
.   || | | | | | | | | ||  E       up partial differential equations for a kind
.   || | | | | | | | | ||  A       of *porous medium*. Tube bundles are treated
.   || | | | | | | | | ||  T       this way because, from a practical point of
c   || | | T U B E | | ||          view, it is hopeless to apply the basic laws
e   || | B U N D L E | ||  E       of flow (Navier Stokes) and of heat transfer
n   || | | | | | | | | ||  X       directly, without approximations. (Requiring
t   || | | | | | | | | ||  C       for example that velocities are zero at all
r   || | | | | | | | | ||  H       parts of the solid structure.)
a   || | | | | | | | | ||  A
l   || | | | | | | | | ||  N       Classical theory of porous media describes
.   || | | | | | | | | ||  G       flow and tranport through soils, consisting
.   || | | | | | | | | ||  E       of sand, clay, peat. Typical applications of
t   || | | | | | | | | ||  R       this are in the field of petroleum reservoir
u   || | | | | | | | | ||     p    engineering and groundwater hydrology. But
b   || | | | | | | | | |===== r o  looking around, we can see many nonclassical
e   || | | | | | | | |   -->  i u  examples of transport phenomena where porous
.   || | | | | | | | |   -->  m t  media are involved. Such as potatoes, stored
.   || | | | | | | | |   -->  a l  in a vessel, which form a porous medium for
.   || | | | | | | | |   -->  r e  the cooling air flow. Other examples are
-----C-|-|-|-|-|-|-|-|-D===== y t  filtration, chemical reactions using solid
///////|/|/|/|/|/|/|/|//|          catalists, adsorption, and mass transfer in
-------|-|-|-|-|-|-|-|--           packed columns. Finally, a flow in the core
       secondary inlet             structures of nuclear reactors, rod arrays,
                                   and also heat exchangers can be considered
- Reference:                       as a flow in a porous medium.
H. de Bruijn and W. Zijl;
Numerical Simulation of the Shell-Side Flow and       Note: forget _all_ about
Temperature Distribution in Heat Exchangers;          the Least Squares Finite
Handbook of Heat and Mass Transfer; chapter 27;       Element Method, employed
Volume 1: Heat Transfer Operations;                   in this reference as our
Nicholas P. Cheremisinoff, Editor;                    numerical method. It did
Gulf Publishing Company (1986).                       NOT properly do the job.

| 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 Fuid Tube Continuum is concerned, the above mentioned inaccuracy
has as an order of magnitude the distance (pitch) between two adjacent tubes.

It is argued in our Reference that, as a first approximation, the shell-side
flow in the tube bundle is incompressible, and 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
neighbouring tubes. It is reasonable to assume that a fluid particle of this
size will experience allmost equal friction at all of his sides, and therefore
will not rotate. Consequently, the (Partial Differential!) equations for flow
in a tube bundle are those for potential flow. Especially in the corners A-B-C
and B-C-D this picture may even be simplified further by assuming that the flow
is approximately equivalent with PLANE flow (due to a rather fat central tube).
A consequence is that the velocities in the corners A-B-C and B-C-D (: figure)
vary approximately _linear_ with distances to the corner. (A fact that may be
used in rough analytical models, when discussing (temperature) singularities.)

Wrap a control-volume around a couple of tubes. Set up the energy balances for
this volume. Let the volume become "infinitesimally small", though remaining
bigger 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:

      c.Gp.[ (u/V).dTp/dr + (v/V).dTp/dz ] + a.(Tp - Ts ) = 0   : primary
                              c.Gs.dTs/dz  + a.(Ts - Tp ) = 0   : secondary

Where: c = heat capacity; G = mass-flow; T = temperature; (r,z) = coordinates;
       (u,v) = velocities; a = total heat transfer coefficient; p = primary;
       s = secondary; V = primary velocity in the middle of the bundle.

Remember: these equations are ONLY simple and elegant _because_ such very CRUDE
          approximations are involved.

BTW, I would'nt be surprised if the above statement can be generalized to _all_
kinds of "simple and elegant" physical laws.

Now I allmost forgot to mention the boundary conditions:

      Tp = Tp0  at the primary inlet  ;   Ts = Ts0  at the secondary inlet

The PDE system serves 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).
-
* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354   | would be NO idleness in  * Oil
* 2600 AJ  Delft; The Netherlands.     | Mathematics" (HdB).      * for
* E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood