External Resistance to Heat Transfer

This appendix considers the appropriate wall boundary condition for temperature when the external resistances to heat transfer are significant. We suppose the tube is jacketed with a fluid at temperature Text that transfers heat to the outer wall of the 

This same amount of heat must be transferred into the reacting fluid by conduction  

The interested reader should explore the values of p as either or both ho and /Cwaii approach infinity. 

Equation 8.56 is the appropriate wall boundary condition associated with Equation 8.24 when there is external resistance to heat transfer. To implement it as part of the method of lines, an estimate for dTjdr at the wall is needed. A first-order approximation is just 

A more accurate estimate of the first derivative is obtained from a third-order approximation. Fit a cubic through the points T(R), T(R — Ar), T R —2 Ar), and T(R — 3 Ar) and differentiate to estimate the slope at point R. The derivative approximation is 

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The simple boundary condition for temperature is 7/ = T waii(z), where T waii(z) is the external temperature, but see Appendix 8.2 for the case where there is external resistance to heat transfer. The simple boundary condition for adiabatic reactors is... 

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. 

The external resistance to heat transfer is incorporated in reactor design simulations by expressing the normal component of interfacial flux in terms of a transfer coefficient and a driving force, the latter of which is sensitive to the direction of the unit normal vector n and the fact that Fourier s law and Pick s law require a negative sign to calculate the flux in a particular coordinate direction. These considerations produce the following expressions for the conductive energy flux ... 

Dnring each iteration of the numerical algorithm described on pages 846-850, it is instructive to use the current values of Ca. bulk gas and Ca, surface, estimate temperature differences inside and outside of the pellet, and determine the importance of the external resistance to heat transfer in packed catalytic tubular reactors. For example, equations (30-31) and (30-41) suggest that if the chemical reaction is exothermic and the following ratio is large ... 

A —B C based on pellet surface temperature and bulk fluid concentrations. In the absence of external resistance to heat transfer, the surface temperature is equal to the bulk fluid temperature. Therefore, (Rg)bA )X be point yield when only the diffusional resistance is present. For the rate expression re = kf (composition), the point yield when the reactions are affected by both diffusion and external heat transport, (/ c) /(jRg)a s simply ... 

The Biot number is essentially the ratio of the resistance to heat transfer within the particle to that within the external fluid. At first sight, it appears to be similar in form to the Nusselt Number Nu where ... 

Catalyst pellets often operate with internal temperatures that are substantially different from the bulk gas temperature. Large heats of reaction and the low thermal conductivities typical of catalyst supports make temperature gradients likely in all but the hnely ground powders used for intrinsic kinetic studies. There may also be a him resistance to heat transfer at the external surface of the catalyst. 

Although resistance to heat transfer goes up as the thickness of pipe insulation is increased, the external surface also increases a thickness may be reached at which the heat transfer becomes a minimum and then becomes larger. In accordance with this kind of behavior, heat pickup by insulated refrigerated lines of small diameters can be greater than that of bare lines. In another instance, electrical transmission lines often are lagged to increase the rate of heat loss. An example worked out by Kreith (Principles of Heat Transfer, Intext, New York, 1973, p. 44) reveals that an insulated 0.5 in. OD cable has a 45% greater heat loss than a bare one. 

External transfer transfer of energy from the surface of the particle into the fluid stream. The properties of flowing fluids are such that the resistance to heat transfer can be larger than that for mass transfer, so that a negligible concentration difference may exist between bulk fluid and particle surface and yet the corresponding temperature difference will be significant. 

Measurement of radial temperature gradient inside the adsorbent particle during the sorption test showed that the internal resistance to heat transfer may not be negligible in comparison with the external film [13-16]. 

The first term depends entirely on the physical properties of the reactor contents and degree of agitation. It represents resistance to heat transfer of the internal film and of eventual deposits at the wall, which may determine the overall heat transfer [3], Therefore, the reactor should be regularly cleaned with a high pressure cleaner. Both last terms depend on the reactor itself and on the heat exchange system, that is, reactor wall, fouling in the jacket, and external liquid film. They are often grouped under one term the equipment heat transfer coefficient (cp) [4, 5],... 

Agitated jacketed vessel the main resistance to heat transfer is located at the wall, where there is practically no resistance to heat transfer inside the reaction mass. Due to agitation, there is no temperature gradient in the reactor contents. Only the film near the wall presents a resistance. The same happens outside the reactor in its jacket, where the external film presents a resistance. The wall itself also presents a resistance. In summary, the resistance against heat transfer is located at the wall. 

Storage silo containing a solid the main resistance to heat transfer is located in the bulk of the product contained in the silo. The resistance of the wall and the external film are low compared to the conductive resistance of the solid. 

If a hot steel ball were immersed in a cool pan of water, the lumped-heat-capacity method of analysis might be used if we could justify an assumption of uniform ball temperature during the cooling process. Clearly, the temperature distribution in the ball would depend on the thermal conductivity of the ball material and the heat-transfer conditions from the surface of the ball to the surrounding fluid, i.e., the surface-convection heat-transfer coefficient. We should obtain a reasonably uniform temperature distribution in the ball if the resistance to heat transfer by conduction were small compared with the convection resistance at the surface, so that the major temperature gradient would occur through the fluid layer at the surface. The lumped-heat-capacity analysis, then, is one which assumes that the internal resistance of the body is negligible in comparison with the external resistance. 

Although resistance to heat transfer goes up as the thickness of pipe insulation is increased, the external surface also increases a thickness may be reached at which the heat transfer becomes a minimum and then becomes larger. In accordance with this... 

As we can see from the equations given above, large heat transfer coefficients are achieved when the temperature difference 1 S — 1 0 and the height of the wall are small. In both cases the condensate film is thin and so the resistances to heat transfer are low. The results from above are also valid for condensation of vapours on the internal and external walls of vertical tubes, if the tube diameter is large in comparison to the film thickness. The width b = nd has to be inserted into (4.13a). 

Analogously, resistances to heat transfer can occur between bulk and catalyst as well as within the catalyst pellet. The first of these, external heat transfer limitation is observed occasionally, particularly in stagnant or slow flowing liquids. Significant resistance to heat transfer within a pellet internal resistance to heat transfer is only observed for highly endo- or exothermic reactions. This is because the heat is more easily distributed across the pellet by means of conduction through the solid, than by convection in the pores. 

Consider a thermocouple inside a thermowell as shown in Figure PHI.3b. Assume that the resistance to heat transfer does not come only from the external film between the surrounding fluid and the thermowell wall but also from the internal film between the thermowell wall and the thermocouple (Figure PIII.5). Let hcxt and hint be the heat transfer coefficients for these two films. 

III.41 Thermocouples are commonly used to measure the temperature of a process fluid. Figure PIII.3b shows a thermocouple within its thermowell. Assume that all resistance to heat transfer resides with the external film around the thermowell wall (see Figure PIII.3c). In other... 

Important results from earlier sections are summarized here to develop reactor design strategies when external resistances to heat and mass transfer cannot be neglected. Intrapellet resistances require information about... 

ANALYSIS OF FIRST-ORDER IRREVERSIBLE CHEMICAL KINETICS IN IDEAL PACKED CATALYTIC TUBULAR REACTORS WHEN THE EXTERNAL RESISTANCES TO HEAT AND MASS TRANSFER CANNOT BE NEGLECTED... 

An experimental setup for gaseous systems is shown in Fig. 7. The actual ZLC column consists of a thin layer of adsorbent material placed between two porous sinter discs. The individual particles (or crystals) are dispersed approximately as a monolayer across the area of the sinter. This minimizes the external resistances to heat and mass transfer, so that the adsorption cell can be considered as a perfectly mixed isothermal, continuous-flow cell. The validity of this assumption has been examined in detail [52]. The isothermal approximation is generally valid for studies of diffusion in zeoHte crystals, but it can break down for strongly adsorbed species in large composite particles under conditions of macropore diffusion control. 

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