Coupled heat and mass flows

For a binary fluid at mechanical equilibrium and for diffusion based on the mass-average velocity, we can now establish a set of phenomenological equations (Eqs. 7.6 and 7.7) with nonvanishing cross coefficients, and hence represent the coupled heat and mass flows... 

Example 7.4 Modified Graetz problem with coupled heat and mass flows The Graetz problem originally addressed heat transfer to a pure fluid without the axial conduction with various boundary conditions. However, later the Graetz problem was transformed to describe various heat and mass transfer problems, where mostly heat and mass flows are uncoupled. In drying processes, however, some researchers have considered the thermal diffusion flow of moisture caused by a temperature gradient. 

Here, Kt are the mass transfer coefficients (permeabilities) for each wall, and CL, and CUi are the ambient concentrations of each component i outside the lower and upper walls, respectively. Sometimes, selective membranes may be used as the walls. These membranes may be permeable to selected components only. For example, in a purification process, the membrane would be permeable to one of the solutes only. In a concentration process, both walls can be impermeable to the selected solute. Equations (7.151) and (7.152) describe the thermodynamically and mathematically coupled heat and mass flows at stationary conditions and may be solved with boundary conditions and with some simplifications (Coelho and Telles, 2002). 

Chemical reaction with coupled heat and mass flows... 

CHEMICAL REACTION WITH COUPLED HEAT AND MASS FLOWS... 

Using Eqs. (9.62) and (9.63) with the Arrhenius equation k = k(le, IRT, we have the transient forms of the coupled heat and mass flows for a single component... 

Equations (9.69) and (9.70) represent the modeling of reaction-diffusion systems with the thermodynamically coupled heat and mass flows excluding the coupling effects due to reaction. After combining Eqs. (9.64), (9.69), and (9.70) steady-state balance equations with the coupled heat and mass transfer become... 

Equation above shows the three contributions to the rate of entropy production due to heat flow, mass flow, and the chemical reaction, respectively, and excludes the viscous and electrical effects. As the membrane is assumed to be an isotropic medium, there will be no coupling between the vectorial heat and mass flows and scalar chemical reaction, according to the Curie-Prigogine principle. Under these conditions, entropy production equation identifies the conjugate forces and flows, and linear relations for coupled heat and mass flows become... 

These equations display the spatial order with the thermodynamically coupled heat and mass flows. Here, the coupling between chemical reactions and transport processes of heat and mass is excluded. The analysis of reaction-diffusion systems would be more complete if we consider heat effects and coupling among fluxes of mass and heat. The nonequi-librium thermodynamics approach is useful for incorporating the coupling phenomena into reaction-diffusion systems. 

Example 9.4 Coupled heat and mass flows in oxidation of CH3OH to CH2O Estimate the dimensionless concentration and temperature surfaces in time and length. 

The above phenomenological equations represent coupled heat and mass flows with chemical reaction in an anisotropic medium. The phenomenological equations above in vector form are... 

Use the modeling Eqns (9.69) and (9.70) with the parameters in Table 9.1 to describe the coupled heat and mass flows by plotting coupled concentration and temperature profiles in NH3 synthesis. Assume the values e = 0.(X)01 and w = 0.0001. 

E -1- F rate constant. Using the coupled heat and mass flows relations... 

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