Axial heat flow

Mullite has the advantage that its lower thermal conductivity diminishes axial heat flow along the central heater more effectively. 

The primary concern for accurate thermal conductivity measurements using this technique is to eliminate axial heat flow. As long as the central heater is long, its temperature near the central portion is uniform. Devoid of an axial temperature gradient, heat will strictly flow radially outward from the central... 

Two 3.0-cm-diameter 304 stainless-steel bars, 10 cm long, have ground surfaces and are exposed to air with a surface roughness of about I fim. If the surfaces are pressed together with a pressure of 50 atm and the two-bar combination is exposed to an overall temperature difference of I00°C. calculate the axial heat flow and temperature drop across the contact surface. 

Two 1-in-diameter bars of stainless steel [k = 17 W/m °C] are brought into end-to-end contact so that only 0.1 percent of the cross-sectional area is in contact at the joint. The bars are 7.5 cm long and subjected to an axial temperature difference of 300°C. The roughness depth in each bar LJ2) is estimated to be 1.3 fun. The surrounding fluid is air, whose thermal conductivity may be taken as 0.035 W/m - °C for this problem. Estimate the value of the contact resistance and the axial heat flow. What would the heat flow be for a continuous 15-cm stainless-steel bar ... 

The effect of geometric parameters on the adiabaticity of a test reactor can be deduced from Table V. It can be seen that for improperly designed laboratory reactors the axial and radial heat flows can be quite appreciable even when the net heat loss is zero. From this table it follows that the radial heat flow is reduced as the bed diameter is increased, whereas the axial heat flow diminishes as the reactor length is increased. Hence, long pilot plant reactors of wide diameter will perform best as adiabatic reactors even with suboptimal design of compensation heaters. 

The thermal botmdary conditions of approximately constant axial heat flow rate per unit length of the microchannel are referred to as H boundary conditions. 

Fig. 2. Unidirectional heat flow patterns (a) parallel faces with axial heat flow (b) cii> cular cylinder with radial heat flow.

Flat-plate Methods. The principle of the axial heat-flow pattern and flat-plate design is illustrated in Figure 2a. In this category, a variety of apparatuses are in general use the three most widely used are descrihed helow. 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

Analysis of Heat Transfer. In the vertical Bridgman-Stockbarger system shown in Figure la, the axial temperature gradient needed to induce solidification is created by separating hot and cold zones with a diabatic zone in which radial heat flow from the ampoule to the furnace is suppressed. Analyses of conductive heat transfer have focused on this geometry. 

It is assumed that e i ec and es ec. With these conditions, the equivalent thermal resistance is approximatively equal to the thermal resistance of the activated carbon. Therefore, the equivalent thermal conductivity along the radial direction is considered as equal to the activated carbon conductivity (Xr Xj. Along the axial direction, the thermal conductivity, Xy, is assumed to be the same as the aluminum conductivity. This condition is deduced from the electrical analog used to represent the heat flow inside the DLC by the parallel thermal resistances as follows ... 

In case of application of high heat-conducting materials (Am Xh) takes place cffr m(l-s), effa h/s- That is application of the radial copper ribs established, for example, with porosity s=0.9 results in increase in effective heat conductivity in a radial direction (conterminous with a direction of a heat flow) up to values 40 W/(m-K) and during too time heat conductivity in an axial direction remains small ( effa l-lXh). This restriction and also not adaptability to manufacture of a design result in limited use of plate reactors. 

We now wish to examine the applications of Fourier s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification. 

Imagine two solid bars brought into contact as indicated in Fig. 2-14, with the sides of the bars insulated so that heat flows only in the axial direction. The materials may have different thermal conductivities, but if the sides are insulated, the heat flux must be the same through both materials under steady-state conditions. Experience shows that the actual temperature profile through the two materials varies approximately as shown in Fig. 2-14b. The temperature... 

Here, ae is the effective thermal diffusivity of the bed and Th the bulk fluid temperature. We assume that the plug flow conditions (v = vav) and essentially radially flat superficial velocity profiles prevail through the cross-section of the packed flow passage, and the axial thermal conduction is negligible. The uniform heat fluxes at each of the two surfaces provide the necessary boundary conditions with positive heat fluxes when the heat flows into the fluid... 

In separation processes and chemical reactors, flow through cylindrical ducts filled with granular materials is important. In such systems conduction, convection, and radiation all contribute to the heat flow, and thermal conduction in axial ke x and radial ke r directions may be quite different, leading to highly anisotropic thermal conductivity. For a bed of uniform spheres, the axial and radial elements are approximated by... 

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