Conduction equation effectiveness, heat-transfer

The fin surface area will not be as effective as the bare tube surface, as the heat has to be conducted along the fin. This is allowed for in design by the use of a fin effectiveness, or fin efficiency, factor. The basic equations describing heat transfer from a fin are derived in Volume 1, Chapter 9 see also Kern (1950). The fin effectiveness is a function of the fin dimensions and the thermal conductivity of the fin material. Fins are therefore usually made from metals with a high thermal conductivity for copper and aluminium the effectiveness will typically be between 0.9 to 0.95. 

Suppose the bottom temperature of the liquid is maintained at 25 °C for a thin pool. Let us consider this case where the bottom of the pool is maintained at 25 °C. For the pool case, the temperature is higher in the liquid methanol as depth increases. This is likely to create a recirculating flow due to buoyancy. This flow was ignored in developing Equation (6.33) only pure conduction was considered. For a finite thickness pool with its back face maintained at a higher temperature than the surface, recirculation is likely. Let us treat this as an effective heat transfer coefficient, between the pool bottom and surface temperatures. For purely convective heating, conservation of energy at the liquid surface is... 

When the effects of heats of adsorption cannot be ignored—the situation in most industrial adsorbers—equations representing heat transfer have to be solved simultaneously with those for mass transfer. All the resistances to mass transfer will also affect heat transfer although their relative importance will be different. Normally, the greatest resistance to mass transfer is found within the pellet and the smallest in the external boundary film. For heat transfer, the thermal conductivity of the pellet is normally greater than that of the boundary film so that temperatures through a pellet are fairly uniform. The temperature... 

The coefficient of hydride bed effective heat transfer aef is proportional in general to layer effective thermal conductivity and inversely proportional to layer thickness. The analysis of this equation shows that the heat transfer coefficient is less than a smaller coefficient of heat emission and, consequently, it is meaningless to increase strongly one of them without changing the other. The experimental results show that for the soldered and diffusion welded connections of sorber case and heat-conducting insertion R = (0.5-1.5)-10 5 (m2 K)/W. If contact between insertion and a case is tight fit, then R increases in 10-100 times and influence of contact resistance becomes comparable with influence of reduced heat emission of a hydride bed. 

The time variation of enthalpy per m (left hand side of the equation) is caused by forced convection (first term on the right hand side), the effective heat transfer (second term on the right hand side) and the heat release caused by the chemical reaction (third term on the right hand side) )i is the effective coefficient of thermal conductivity of the mixture of substances in W/(m K), p is its density in kg/m, Cp the corresponding heat capacity at constant pressure in J/(kg K) and AHrj the enthalpy of reaction of reaction j in J/kg (with a negative sign for exothermic reactions). 

The specific heat not only describes the capacity to store heat, but also influences the dynamics of heat transfer within aerogels according to the equation of heat transfer (23.1) and (23.4). The higher the specific heat, the slower the heat propagation within a material, if the density and the effective thermal conductivity remain constant. In the late eighties and early nineties several authors investigated the specific heat of silica aerogels in detail [9,51, 52, 54, 64-67]. The focus of their research work was the study of the density of vibrational states g(a>) correlated to the specific heat and the solid thermal conductivity by phonons ... 

For the determination of effective thermal conductivity and wall heat transfer coefficient, the following heat balance equation is solved ... 

Laminar Flow Normally, laminar flow occurs in closed ducts when Nrc < 2100 (based on equivalent diameter = 4 X free area -i-perimeter). Laminar-flow heat transfer has been subjected to extensive theoretical study. The energy equation has been solved for a variety of boundaiy conditions and geometrical configurations. However, true laminar-flow heat transfer veiy rarely occurs. Natural-convecdion effects are almost always present, so that the assumption that molecular conduction alone occurs is not vahd. Therefore, empirically derived equations are most rehable. 

The basic equations for filmwise condensation were derived by Nusselt (1916), and his equations form the basis for practical condenser design. The basic Nusselt equations are derived in Volume 1, Chapter 9. In the Nusselt model of condensation laminar flow is assumed in the film, and heat transfer is assumed to take place entirely by conduction through the film. In practical condensers the Nusselt model will strictly only apply at low liquid and vapour rates, and where the flowing condensate film is undisturbed. Turbulence can be induced in the liquid film at high liquid rates, and by shear at high vapour rates. This will generally increase the rate of heat transfer over that predicted using the Nusselt model. The effect of vapour shear and film turbulence are discussed in Volume 1, Chapter 9, see also Butterworth (1978) and Taborek (1974). 

Yoder showed that radiation heat transfer and axial conduction heat transfer in the tube wall have a negligible effect on predicting wall temperatures. The following equations were used by Yoder and Rohsenow (1980) as well as previous investigators such as Bennett et al. (1967b), Hynek (1969), and Groeneveld (1972). 

Equations (8) are based on the assumption of plug flow in each phase but one may take account of any axial mixing in each liquid phase by replacing the molecular thermal conductivities fc, and ku with the effective thermal conductivities /c, eff and kn eff in the definition of the Peclet numbers. The evaluation of these conductivity terms is discussed in Section II,B,1. The wall heat-transfer terms may be defined as... 

To study the effects due to droplet heating, one must determine the temperature distribution T(r, t) within the droplet. In the absence of any internal motion, the unsteady heat transfer process within the droplet is simply described by the heat conduction equation and its boundary conditions... 

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)]... 

For small particles, subject to noncontinuum effects but not to compressibility, Re is very low see Eq. (10-52). In this case, nonradiative heat transfer occurs purely by conduction. This situation has been examined theoretically in the near-free-molecule limit (SI4) and in the near-continuum limit (T8). The following equation interpolates between these limits for a sphere in a motionless gas ... 

Here we consider a spherical catalyst pellet with negligible intraparticle mass- and negligible heat-transfer resistances. Such a pellet is nonporous with a high thermal conductivity and with external mass and heat transfer resistances only between the surface of the pellet and the bulk fluid. Thus only the external heat- and mass-transfer resistances are considered in developing the pellet equations that calculate the effectiveness factor rj at every point along the length of the reactor. 

Here, k is the effective thermal conductivity, A is the effective contact area between the adjacent cells, l is the characteristic conduction length scale, hconv is the convection heat transfer coefficient, Aext external surface area of the cell exposed to the ambient air, 7 x is the ambient temperature and P is the cell power. The characteristic conduction length is calculated as the volume of the bipolar plate divided by the cell normal area. Factor /3 is an empirical constant which is the ratio of the heat generated to the power produced by the cell, i.e. (1 - rj), rj being the efficiency. When radiation is considered, should be included in Equation (5.64). The heat transfer relationships between the gas channels and the solid regions are given by ... 

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