Infinite heat transfer

The drag coefficients for disks (flat side perpendicular to the direction of motion) and for cylinders (infinite length with axis perpendicular to the direclion of motion) are given in Fig. 6-57 as a Function of Reynolds number. The effect of length-to-diameter ratio for cylinders in the Newton s law region is reported by Knudsen and Katz Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, 1958). 

The process is indicated on the chart in Figure 24.9, taking point B as the tube temperature. Since this would be the ultimate dew point temperature of the air for an infinitely sized coil, the point B is termed the apparatus dew point (ADP). In practice, the cooling element will be made of tubes, probably with extended outer surface in the form of fins (see Figure 7.3). Heat transfer from the air to the coolant will vary with the fin height from the tube wall, the materials, and any changes in the coolant temperature which may not be constant. The average coolant temperature will be at some lower point D, and the temperature difference B — D will be a function of the conductivity of the coil. As air at condition A enters the coil, a thin layer will come into contact with the fin surface and will be cooled to B. It will then mix with the remainder of the air between the fins, so that the line AB is a mix line. 

Analytical solutions of equation 9.44 in the form of infinite series are available for some simple regular shapes of particles, such as rectangular slabs, long cylinders and spheres, for conditions where there is heat transfer by conduction or convection to or from the surrounding fluid. These solutions tend to be quite complex, even for simple shapes. The heat transfer process may be characterised by the value of the Biot number Bi where ... 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

It is seen from equation 11.66 that the heat transfer coefficient theoretically has an infinite value at the leading edge, where the thickness of the thermal boundary layer is zero, and that it decreases progressively as the boundary layer thickens. Equation 11.66 gives the point value of the heat transfer coefficient at a distance x from the leading edge. The mean value between. v = 0 and x = x is given by ... 

A large block of material of thermal diffusivity Du — 0.0042 cm2/s is initially at a uniform temperature of 290 K and one face is raised suddenly to 875 K and maintained at that temperature. Calculate the time taken for the material at a depth of 0.45 m to reach a temperature of 475 K on the assumption of unidirectional heat transfer and that the material can be considered to be infinite in extent in the direction of transfer. 

To identify the governing processing and material parameters, a one dimensional case was analyzed. The heat transfer problem renders an exact solution, [10], which can be presented as an infinite series... 

Consecutive reactions, isothermal reactor cmi < cw2, otai = asi = 0. The course of reaction is shown in Fig. 5.4-71. Regardless the mode of operation, the final product after infinite time is always the undesired product S. Maximum yields of the desired product exist for non-complete conversion. A batch reactor or a plug-flow reactor performs better than a CSTR Ysbr.wux = 0.63, Ycstriiuix = 0.445 for kt/ki = 4). If continuous operation and intense mixing are needed (e.g. because a large inteifacial surface area or a high rate of heat transfer are required) a cascade of CSTRs is recommended. 

The first and easiest example to look at when discussing heat transfer in PCM is a one-dimensional semi-infinite layer as shown in Figure 125. 

Analytical Model for a Simple Heat Storage The solution for the semi-infinite layer can give quite some insight into heat transfer within PCM. However, it is also clear that for a real heat storage as shown in Figure 127, the one-dimensional approach is insufficient. 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

The interfacial heat transfer coefficient can be evaluated by using the correlations described by Sideman (S5), and then the dimensionless parameter Ni can be calculated. If the Peclet numbers are assumed to be infinite, Eqs. (30) can be applied to the design of adiabatic cocurrent systems. For nonadiabatic systems, the wall heat flux must also be evaluated. The wall heat flux is described by Eqs. (32) and the wall heat-transfer coefficient can be estimated by Eq. (33) with... 

Classical heat transfer provides expressions for quantities such as view factors, radiation and temperature fields in semi—infinite bodies. The lining materials studied here were treated as semi-infinite bodies since the test duration is relatively short. 

Considering the extreme case when there are an infinite number of beds (Fig. 13) and ideal heat transfer, the maximum amount of recovered heat is calculated by reflecting the lower curve in the upper half of the diagram, as is shown by the dashed line. The input heat that must be obtained from an external source is shown by the shaded area. The COP is the maximum that may be obtained from a single effect adsorption machine. 

Schellenz et al. [ 1.133] confirmed that the assumption of an infinite plate in Eq. (12) is a reasonable approximation, even for drying of products in vials. They show by the measurement of temperature profiles and by X-ray photos during drying of a 5 % mannitol solution, 23 mm filling height, that the sublimation front retreats mostly from the top parallel to the bottom. The heat transfer from glass vials deforms the flat surface only to some extent close to the wall. 

Let us consider the semi-infinite (thermally thick) conduction problem for a constant temperature at the surface. The governing partial differential equation comes from the conservation of energy, and is described in standard heat transfer texts (e.g. Reference [13]) ... 

You can see, using Equation (d) only, that Cop is minimized by setting TH— Ts (infinitesimal boiler AT). However, this outcome increases the required boiler heat transfer area to an infinite area, as can be noted from the calculation for the area... 

Heat-transfer surfaces, 76 717 Heat-transfer surface area, 73 249, 256 infinite, 73 253... 

Inferred petroleum reserves, 18 595 Infinite dilution coefficients, 8 743 Infinite heat-transfer surface area, 13 253 Infinity point, 14 611... 

There is apparently an inherent anomaly in the heat and mass transfer results in that, at low Reynolds numbers, the Nusselt and Sherwood numbers (Figures. 6.30 and 6.27) are very low, and substantially below the theoretical minimum value of 2 for transfer by thermal conduction or molecular diffusion to a spherical particle when the driving force is spread over an infinite distance (Volume 1, Chapter 9). The most probable explanation is that at low Reynolds numbers there is appreciable back-mixing of gas associated with the circulation of the solids. If this is represented as a diffusional type of process with a longitudinal diffusivity of DL, the basic equation for the heat transfer process is ... 

The first part of the analysis is focused on the energy transfer to the solid bed and what assumptions might be reasonable regarding the temperature profile in the solid bed. For this analysis the barrel and solid interface will be addressed. It is desired that an infinite bed assumption can be justified. Once this assumption is justified, the heat transfer analysis for the melting is quite straightforward. 

Assuming one-dimensional heat transfer is the mode of the solid bed heating due to the heating of the film by conduction and dissipation, the temperature will only change in the y direction. The same assumption that was made by Tadmor and Klein will be made here that the heat transfer model is a semi-infinite slab moving at a velocity Vsy c (melting velocity) with the boundary conditions T(0) = and j(-oo) = 7 , This assumption is not strictly correct because it will also be proposed that the other four surfaces are melting. The major error will occur at the corners of the solid bed. is the velocity of the solid bed surface adjacent to Film C as it moves toward the center of the solid bed in the y direction. 

As pointed out by Tadmor and Klein [1], the solid bed temperature decreases very rapidly from the melting surface, and thus, a reasonable first assumption is that it can be considered to be infinitely deep for heat transfer and melting. Given this assumption the heat flux away from melt film into the solid will be ... 

The ideal finite-time Rankine cycle and its T-s diagram are shown in Figs. 7.14 and 7.15, respectively. The cycle is an endoreversible cycle that consists of two isentropic processes and two isobaric heat-transfer processes. The cycle exchanges heats with its surroundings in the two isobaric external irreversible heat-transfer processes. The heat source and heat sink are infinitely large. Therefore, the temperature of the heat source and heat sink are unchanged during the heat-transfer processes. 

The heat source and heat sink are not infinitely large. Therefore, the temperature of the heat source and heat sink change during the heat-transfer processes. 

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