On the role of applied PDE analysts in non-academic R&D workplaces

Question

What is the role covered by PDE analysts in research and development in non-academic R&D workplaces?

For instance in medical or pharmaceutical companies, tech companies that produce hardware (CPUs, GPUs, and so on), engineering firms, etc.

• I'm looking for answers that are backed up by references and/or direct experience.
• I'm interested in the role of researchers focusing on both theoretical and numerical aspects of PDEs.

Answer 1

Backed up by direct experience and the following reference:

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).

Focusing on both theoretical and numerical aspects of PDEs:

Intermediate Heat Exchanger

Summary

Problem description: 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.
In order to describe the shell-side flow and temperature distribution, investigators have attempted the so-called fluid-tube continuum approach. The basic idea of the method is: setting up partial differential equations for a kind of porous medium.
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: $$ T_P = T_{PL} \qquad \mbox{at the primary inlet (upper perforation)} \\ T_S = T_{S0} \qquad \mbox{at the secondary inlet (tube plate below)} $$

One would think that what engineers really want is a realistic flow. Unexpectedly perhaps, that's NOT what they really want. With the apparatus I've been working on, what the engineers want is the flow field, calculated in such a way that temperature stresses cannot be worse in reality than they are in the calculations. We call such calculations conservative. An insightful moment of thinking has revealed that not a realistic flow simulation but rather an Ideal Flow simulation has the desired properties:

Ideal Internal Flow

Consequently, the (Partial Differential) equations for flow in a tube bundle may be assumed to be those for incompressible and irrotational 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. With proper boundary conditions. This completes the analytical description.

The numerical treatment of such systems of first order PDE's is quite another story; there exist visualizations of some results. And there are related references at Mathematics Stack Exchange:

Any employment for the Varignon parallelogram?