Thermal boundary layer heat balance

The procedure here is similar to that adopted previously. A heat balance, as opposed to a momentum balance, is taken over an element which extends beyond the limits of both the velocity and thermal boundary layers. In this way, any fluid entering or leaving the element through the face distant from the surface is at the stream velocity u and stream temperature 0S. A heat balance is made therefore on the element shown in Figure 11.10 in which the length l is greater than the velocity boundary layer thickness S and the thermal boundary layer thickness t. 

Consider the control volume bounded by the planes 1, 2, A-A, and the wall as shown in Fig. 5-8. It is assumed that the thermal boundary layer is thinner than the hydrodynamic boundary layer, as shown. The wall temperature is 7 ,., the free-stream temperature is Tx, and the heat given up to the fluid over the length dx is dqw. We wish to make the energy balance... 

Manipulate the multicomponent thermal energy balance in the gas-phase boundary layer that surrounds each catalytic pellet. Estimate the external resistance to heat transfer by evaluating all fluxes at the gas/porous-solid interface, invoking continuity of the normal component of intrapellet mass flux for each component at the interface, and introducing mass and heat transfer coefficients to calculate interfacial fluxes. 

An integral procedure similar to that adopted previously for momentum boundary layers will be used here to obtain the expression for the rate of heat exchange between the fluid and the plane surface. A heat balance will be made over a control voliune (Figure 7.4) which extends beyond the limits of both the momentum and thermal boimdaiy layers. 

The effective reaction rate tm eff (related to the mass of catalyst/solid) already considers all extra- and intraparticle mass and heat transfer effects (Sections 4.5-4.7). Thus the pseudo-homogeneous model does not distinguish between the conditions in the fluid and in the sohd phase, as more sophisticated heterogeneous models do, as discussed, for example, in Baerns ef al. (2006), Froment and Bischoff (1990), and Westerterp, van Swaaij, and Beenackers (1998). Thus, gradients of temperature and concentration within the particle and in the thermal and diffusive boundary layers are combined by the use of an overall effectiveness factor that enables the system of four equations (mass and heat balances for solid and fluid phase) to be replaced by just two equations, Eqs. (4.10.125) and (4.10.126). 

The appearance of a loose packing in the boundary layers si uJBes the transition to a less balanced equilibrium position, to less probable conformation. One should have expected that a lengthy thermal treatment of the boundary layers similar to the effect of such treatment on the approach of glass-like polymers to equilibrium, will also affect molecular mobility. However, the study of PMMA and of copolymer methylmethacrylate and styrene filled with aerosil irulicates that the heat treatment leads to a smaller shift of the low-temperature maximums, ie. to a mote dense packing, and yet does not affect the position of the hi -temperature maximum. 

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