Abstract
This paper presents a one dimensional model for the gas dynamics in a chimney. This is a prototype example of a small Mach number flow with strong heat sources. Due to the small Mach number of the gas flow an asymptotic model is derived from a fully compressible model, which then is compared to the original model and to the often used Boussinesq approximation. The Boussinesq approximation is shown to be inappropriate for this application, whereas the small Mach number asymptotic model we propose is shown to be a very good approximation. In particular it allows very fast numerical simulations. Finally, all this is underlined by numerical simulations where we validate the various models.MSC:35C20, 35Q35, 76M45, 76N15, 80A20.
Highlights
Chimneys are typical representatives of the beginning of the industrial era in the th century
Just to give some examples, we can think to modern heating systems, whit different kinds of fuel, and to Controlled Ventilation Systems, where the basic principle of chimney is used, often with the help of a fan, to keep an healthy level of oxygen in modern buildings [ ]
On the other hand one can find sophisticated CFD gasdynamics simulations based on time expensive numerical simulations
Summary
Chimneys are typical representatives of the beginning of the industrial era in the th century. We describe the gas dynamics by a one dimensional model, where the single dimension x starts at the base of the structure and follows the air flow towards the top of the chimney. To determine reasonable values to these coefficients we recall that burning wood produces ∼ J/kg of thermal energy ([ , ] and [ ]), while chimneys used for heating in common handholds have usually a power of kW [ , ]. Considering this and taking into account the reference values for density and velocity, we will use a heat production coefficient cq ∼ – kg m/s.
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