Heat transfer

Heat Transfer is part of a free web series, ChemPlugin Modeling with Python, by Aqueous Solutions LLC.

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Introduction

A ChemPlugin instance can trace how temperature varies over the course of a simulation by keeping track of the net accumulation or loss of heat energy during the time marching procedure.

If Q, as before, is the flow rate in m³ s⁻¹ across a link, the rate of advective heat transfer is given by

in units of J s⁻¹. Here, ρ_w is the fluid density in kg m⁻³, C_w is fluid heat capacity in J kg⁻¹ K⁻¹, and T is temperature in K.

A client uses the “FlowRate()” member function to set the flow rate Q across a link, as described earlier. Once Q is specified, the instance computes the effects of advective heat transport whenever the client program calls the “AdvanceHeatTransport()”member function.

Fourier's law gives the rate of conductive heat transfer in J s⁻¹ by

In this equation, A is the link's cross-sectional area in m², K_T is the thermal conductivity in W m⁻¹ K⁻¹, and dT/dx is the temperature gradient across the plane, in K m⁻¹.

ChemPlugin calculates the conductive heat flux according to an approximate equation

where τ_T is the thermal transmissivity in W K⁻¹ , and T^j and T^linked are temperature of the originating and linked instances, respectively, in K. Calculation of the thermal transmissivity τ_T closely parallels determination of the transmissivities τ for mass transport, as described earlier.

The thermal transmissivity is set using the “HeatTrans()” member function. The default units are W K⁻¹. In the time marching loop, use “AdvanceHeatTransport()” to transfer heat energy. For more information, see the Heat Transfer chapter in the ChemPlugin User's GuIde.

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Authors

Craig M. Bethke and Brian Farrell. © Copyright 2016–2024 Aqueous Solutions LLC. This lesson may be reproduced and modified freely to support any licensed use of The Geochemist's Workbench® software, provided that any derived materials acknowledge original authorship.

References

Bethke, C.M., 2022, Geochemical and Biogeochemical Reaction Modeling, 3^rd ed. Cambridge University Press, New York, 520 pp.

Bethke, C.M., 2024, The Geochemist's Workbench®, Release 17: ChemPlugin™ User's Guide. Aqueous Solutions LLC, Champaign, IL, 303 pp.

Carslaw, H.S. and Jaeger, J.C., 1959, Conduction of Heat in Solids. Clarendon Press, Oxford, 510 pp.

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