Thermal Conductivity and Heat Transfer

Fig. 2.3 Effect of thickness on heat spreading for different heat source areas, material thermal conductivities, and heat transfer coefficients (A in cm, in W/mK, hinW/m K). Reprinted from Lasance and Simons (2005) with permission...

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Effective thermal conductivities and heat transfer coefficients are given by De Wasch and Froment (1971) for the solid and gas phases in a heterogeneous packed bed model. Representative values for Peclet numbers in a packed bed reactor are given by Carberry (1976) and Mears (1976). Values for Peclet numbers from 0.5 to 200 were used throughout the simulations. 

Further advancements in the theory of fixed bed reactor design have been made(56,57) but it is unusual for experimental data to be of sufficient precision and extent to justify the application of sophisticated methods of calculation. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on the reaction rate. 

W.H. Yu, D M. France, J.L. Routbort and S.U.S. Choi, Review and comparison of nanofluid thermal conductivity and heat transfer enhancements. Heat... 

The axial temperature rise in the coolant, Eq. (2.183), the radial temperature drop and the axial temperature distribution in the fuel, the gap, the clad, and the coolant, Eq. (2.188), are sketched in Fig. 2.52. Some typical values encountered in practice for the radial temperature drop are ATpuei 1500 °C, AToap 150 — 300 °C, ATaad 50 °C, and ATbooiam 5 °C (for water). Also, some values for the geometry, thermal conductivity and heat transfer coefficient are ... 

It might be possible to neglect the external gas phase resistance to mass transfer relative to intrapellet diffusional resistance through a tortuous pathway. An increase in the gas stream flow rate reduces the external mass transfer resistance further. Remember that diffusion coefficients and mass transfer coefficients increase as one progresses from solids to liquids to gases. Hence, gas-phase mass transfer resistances are small, but the intrapellet gas-phase diffusional resistances should be significant, particularly when the intrapellet Damkohler number is quite large. In contrast, thermal conductivities and heat transfer coefficients increase... 

Quasi-continuum models Of these, the quasi-continuum model is the most common. Here, the solid-fluid system is considered as a single pseudo-homogeneous phase with properties of its own. These properties, for example, diffusivity, thermal conductivity, and heat transfer coefficient, are not true thermodynamic properties but are termed as effective properties that depend on the properties of the gas and solid components of the pseudo-phase. Unlike in simple homogeneous systems, these properties are anisotropic, that is, they have different values in the radial and axial directions. KuUcami and Doraiswamy (1980) have compiled all the equations for predicting these effective properties. Both radial and axial gradients can be accounted for in this model, as well as the fact that the system is really heterogeneous and hence involves transport effects both within the particles and between the particles and the flowing fluid. 

Through virtually all of the filament length, the temperature is constant. But at the ends the temperature must drop almost to the cell body temperature. Heat is transferred to the body through the ends. Like thermal conductivity, this loss is proportional to the difference in temperature between filament and body, but unlike thermal conductivity, the heat transfer does not depend on conductivity of the gas. This amounts to 45 mW for the case cited. For the small-diameter wire and high-conductivity carrier gases used today, this heat loss mechanism can usually be neglected. 

The data needed are the rate equation, energy of activation, heat of reaction, densities, heat capacities, thermal conductivity, diffusivity, heat transfer coefficients, and usually the stoichiometry of the process. Simplified numerical examples are given for some of these cases. Item 4 requires the solution of a system of partial differential equations that cannot be made understandable in concise form, but some suggestions as to the procedure are made. 

Packed Bed Thermal Conductivity 587 Heat Transfer Coefficient at Walls, to Particles, and Overall 587 Fluidized Beds 589... 

Thus, unlike k (which is a thermal property), A is merely a definition and depends on flow (conditions). That is, unlike thermal conductivity, the heat transfer coefficient cannot be tabulated and needs to be determined for each flow condition. Accordingly, Chapters 5 and 6 are devoted to elaboration of Eqs. (1.60) and (1.61) and the solution of convection problems in teams of a heat transfer coefficient. Here, for some appreciation, an order-of-magnitude range of each heat transfer coefficient corresponding to natural or forced convection in different fluids is given in Table 1.2. The order-of-magnitude difference between the A values for natural convection and forced convection resulting from flow of the same fluid should be noted. 

I. Catton and J. H. lienhard V, Thermal Stability of Two Fluid Layers Separated by a Solid Interlayer of Finite Thickness and Thermal Conductivity, J. Heat Transfer (106) 605-612,1984. 

V. M. Zhukov, G. M. Kazakov, S. A. Kovalev, and Y. A. Kuzmakichta, Heat Transfer in Boiling of Liquids on Surfaces Coated With Low Thermal Conductivity Films, Heat Transfer Sov. Res. (7/3) 16-26,1975. 

B. Wright, D. Thomas, H. Hong, L. Groven, J. Puszynski, E. Duke, X. Ye, and S. Jin, Magnetic field enhanced thermal conductivity in heat transfer nanofluids containing Ni coated single wall carbon nanotubes, Appl. Phy.s Lett., 91,173116 (2007). 

All three transport properties, viscosity, diffusivity, and thermal conductivity, are important in reactor design. Viscosity is a measure of momentum transfer, diffusivity of mass transfer, and thermal conductivity of heat transfer. 

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