Tubular heat exchanger: Comparison between exact EDP solution and 1st, 3rd order approximations

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 17:56

*Enlace a Spanish version

Summary:

This video compares the exact solution of the partial differential equation of a tubular heater (details of the PDE modelling in video [termedpEN]; solutions were obtained in transfer function form in the video [termedpsolEN]) with finite (1st and 3rd) order approximations of them.

Erratum: there is an error in the video at line 5 of code where a power unit conversion factor from W to kW is wrongly propagated. The MLX and PDF have been corrected, changing the heat transfer constant $\overline{\kappa }$ so that the numerical results of the video remain correct.

Once the typo has been clarified, let’s continue with the description.

Actually, an approximation of the exponential in the Laplace transfer functions via a certain parameter $\beta$, first order models are derived, which, luckily, are identical to those obtained based on physical insight with a non-uniform longitudinal temperature profile in video [term1expEN]. This is important, because models from first-principle insights can be simulated when input flow changes, whereas approximation in the Laplace domain cannot (transfer functions were obtained under a constant flow assumption).

A particular case $\beta =1$ renders the perfectly-stirred first-order approximation in video [term1eEN]; $\beta =0.5$ renders a linear temperature profile, see [term1expEN] for details.

Additionally, Pade approximations of the delay are compared to the above models (first and third order Pade approximations, via Matlab’s pade).

Comparisons are made in both steady state (DC gain) and transient response to steps (in inlet temperature and in resistor heating power).

A few of these low-order approximations may be cascade-connected to yield finite-element setups, discussed in other materials, such as the video [tubulFE1EN].

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