Full flux models for optimization and control of heat exchangers

If convection is the dominate mechanism for heat transfer in a heat exchangers, then the devices are often modeled by hyperbolic partial differential equations. One of the difficulties with this approach is that for low (or zero) pipe flows, some of the imperial functions used to model friction can become singular. One way to address low flows is to include the full flux in the model so that the equation becomes a convection-diffusion equation with a “small” diffusion term. We show that solutions of the hyperbolic equation are recovered as limiting (viscosity) solutions of the convection-diffusion model. We employ a composite finite element - finite volume scheme to produce finite dimensional systems for control design. This scheme is known to be unconditionally L²-stable, uniformly with respect to the diffusion term. We present numerical examples to illustrate how the inclusion of a small diffusion term can impact controller design.