Convective diffusion of heat

Heat diffuses much like mass and is governed by similar equations. The temperature analog of Equation (8.12) is 

The inlet and centerline boundary conditions associated with Equation (8.52) are similar to those used for mass transfer  

Equation (8.52) has the same form as Equation (8.12), and the solution techniques are essentially identical. Replace a with T, 3)a with aj-, and 01 a with — AHr0X/ pCp), and proceed as in Section 8.3. 

Equation (8.52) has the same form as Equation (8.12), and the solution tech-  

Analogous to Equation 8.20 for mass, following is a convective diffusion equation for heat. 

More complicated forms can obviously be written, but Equation 8.24 is a good starting point. The boundary conditions associated with Equation 8.24 are 

Known inlet temperatures Known wall temperatures  

As written above, the wall boundary condition suggests that Twau(z) can be specified directly. This is a reasonable assumption for the common case of a metal wall in contact with an organic solution. A more complex situation arises when there is a significant thermal resistance in the wall (e.g., a plastic tube) or when the outside heat transfer coefficient is small. See Appendix 8.2 for the appropriate equations. 

The situation at r = 0 is directly analogous to Equation 8.23. In the absence of 6 dependence, the system must have radial symmetry, and the governing PDE takes a special form  

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Thermal runaway. Temperature control in a tubular polymerizer depends on convective diffusion of heat. This becomes difficult in a large-diameter tube, and temperatures may rise to a point where a thermal runaway becomes inevitable. 

The associated boundary conditions are zero slope at the walls and at any planes of symmetry. The counterpart of Equation 16.3 for the convective diffusion of heat is... 

Flow Rate Measurements, Methods, Fig. 4 Principle of a time-of-flight flow sensor using CFD simulation, (a) Convective diffusion of heat or of a molecular tracer... 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

Similar to the species transport, diffusion of heat compared to the convective transport is negligible. Thus, the energy balance is given by... 

Homemade models are often mass and energy balance spreadsheets, simplified kinetic models, or the simultaneous solution of the convection diffusion and heat equations together with nonlinear isotherms. All levels of models have their place. 

The dimensionless equations for the convective diffusion of mass and heat are... 

This section applies the method of lines to the equations for convective diffusion of mass and heat. It is convenient to replace r hy r = i Ar, i = 0,..., I, for coding... 

There is a close similarity between heat and mass transfer in terms of transport rate equation and transport conservation equation. The diffusion and convective mass transfer modes are similar to the conduction and convection modes of heat transfer. Both diffusion and convection mass transfer play a significant role in the transport of reactant gas species through the gas flow channels and gas diffusion layers/electrodes. 

A further complication arises in forced convection in fabric, which is a porous medium. There may be significant thermal dispersion, i.e., heat transfer due to hydrodynamic mixing of the fluid at the pore scale. In addition to the molecular diffusion of heat, there is mixing due to the nature of the fabric. 

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