Modelling the dynamics of a heating tank (1st order, perfect mixing, incompressible flow)

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 12:52

*Enlace a Spanish version

Summary:

This video derives a first-order model for a heating tank (perfectly stirred so everything is assumed to mix instantly, yielding an homogeneous temperature in all of the tank’s volume). If it had non-perfect mixing (or, say, if it were a long pipe) we would need Partial Differential Equations (PDE) as done in most heat exchanger theory; the most complex cases would need full computational fluid-dynamics code; for simplicity, this will not be the case here and we’ll just consider perfect mixing, 1st order dynamics.

A heating resistor providing thermal power $Q$ is present, in order to heat a flow $F$ of incompressible liquid, density $\rho$, specific heat $c$.

The final equation is $V\rho cṪ=F\rho c{T}_{in}-T)+Q-\kappa T$, and the video justifies it in terms of a power balance. Formally, as there are no pressure/volume changes, no work is done and increments of “energy” and “enthalpy” coincide.

There are other alternative approaches to build a model for a tubular heat exchanger; we have, of course, the PDE one (video [termedpEN]) and first-order approximations with nonconstant longitudinal temperature profile (video [term1expEN]).

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