Partial Differential Equation (PDE) modelling of a one-dimensional tubular heater for liquid fluids

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 16:16

*Enlace a Spanish version

Summary:

This video pursues modelling a tubular heat exchanger with at heating resistor alongside it, as a connection of multiple (infinitely many of them) elements where each of the elements is a mini-heating-tank modelled as a first-order dynamical systems; modelling is briefly outlined in the first two minutes of this video, detailed explanations appear at video [term1eEN].

Once the inputs and outputs of each element are suitably named, the separation between elements will be set to “$dx$”. Conceptually, if one element’s temperature is $T(x)$, that of the next element will be $T(x+dx)$. As we are analysing the time evolution of temperature, we actually pursue findin the equations governing the evolution of $T(x,t)$ at different positions and time instants.

Taking limits when $dx\to 0$, the following PDE (partial differential equation) is obtained:

The final part of the video discusses two frequent particular cases, well studied:

$\bullet$ The first one is transport delay (no heating, perfect insulation), i.e., the transport PDE: $\frac{\partial T}{\partial t}=-v\frac{\partial T}{\partial x}$ being $v=F∕S$ the linear speed of the fluid.

$\bullet$ The second particular case is the steady-state heat exchanger (enforcing equilibrium condition $\frac{\partial T}{\partial t}=0$), considered for simplicity under no heating, i.e., $\overline{Q}=0$. In that case, we obtain an ordinary differential equation (ODE) in the longitudinal position variable $\frac{\partial T_{eq}}{\partial x}=-\frac{\overline{\kappa }}{F\rho C_e}\cdot T_{eq}$ which obviously has an exponential solution which is widely used and explained in depth in heat-exchanger textbooks.

Numerical simulation of PDE requires, in a general nonlinear case, the approximation (discretization) to a finite number of non-infinitesimal elements, as considered in the starting phases of the modelling procedure in this material. Nevertheless, in this particular case, for constant flow, there is a Laplace-transform solution (with delay) that can be simulated with the control system toolbox in Matlab; details on how to solve the PDE are in video [termedpsolEN], the step response of such solution is detailed in video [termedpstepEN].

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.