Partial Differential Equations tubular heat exchanger: PDE solution via Laplace transform (transfer function)

Antonio Sala, UPV

Difficulty: ***** ,       Relevance: ,      Duration: 19:57

*Enlace a Spanish version

Summary:

This video details how to obtain the solution to the partial differential equations describing the dynamics ofa tubular heat exchanger with a resistor inside. Solution is obtained via Laplace transform techniques on the ”temporal” side. Further detail on how to obtain the PDE model can be found in the video [termedpEN]. Given the tubular geometry, PDE is one-dimensional in space, for simplicity. The PDE is linear if flow traversing the heat exchanger is constant, so it will be assumed as such.

Really, we don’t obtain the actual “solution” understood as $T(x,t)$, but a transfer function representation of the outlet temperature as a function of the heating power increments or those from variations of inlet temperature. Actual solutions and simulations will be the objective of future materials.

The resulting expression contains terms arising from the transport delay phenomena that arise in the underlying physics. It has the form:

$T_{out}(s)=\frac{b\cdot (1-e^{-\varphi s}{e}^{-\varphi a})}{s+a}Q(s)+e^{-\varphi s}{e}^{-\varphi a}{T}_{in}(s)$

The step response of these transfer functions is simulated in the video [termedpstepEN], continuation of this one.

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