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biot_numbers - Biot Numbers by David Adrian This document...

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Biot Numbers by David Adrian This document is a review * of some of the concepts of heat and mass transfer, particularly focusing on the dynamics at the interface between two disparate materials, such as the boundary of a solid particle submerged in a fluid. In the cases considered, the interface is stationary and there is no phase change or chemical reaction at the interface. The Biot number is a dimensionless group that compares the relative transport resistances, external and internal. It arises when formulating and non-dimensionalizing the boundary conditions for the typical conservation of species/energy equation for heat/mass transfer problems. If your problem consists of an object suspended in a well mixed fluid, commonly you only need to calculate the dynamics of the object (such as the temperature as a function of position and time). If we focus on the fluid/object interface, the convective flux from the bulk fluid to the object must equal the diffusive flux from the surface to the interior of the object. This is typically formulated as a Robin boundary condition at the interface. For example, consider the unsteady heat transfer in a solid sphere at initial temperature T 0 submerged in a fluid of temperature T (this is also the “bulk” temperature, and could be given the symbol T b ). At the fluid-solid interface, the flux of heat into the sphere from the fluid must equal the flux of heat from the surface of the sphere to the interior. r r q exterior = q interior q r exterior = h ( T s T ) n r q r interior = − k T T surface The variables are defined as follows: q is the heat flux, h is the heat transfer coefficient in the fluid, T is the temperature, n r is the outwardly pointing normal from the solid, and k T is the thermal conductivity of the solid. (Also recall that the heat transfer coefficient can be obtained from correlations, and is basically just k T , fluid / δ T in systems without any interfacial reactions or phase changes. The variable δ T is the thickness of the thermal boundary layer.) Since our system is spherically symmetric: q r exterior = h ( T s T ) e r r r e q r
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