Conduction heat transfer with other boundary conditions

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. 

Consider a long rectangular bar of length a in the. r-direction and width b in the y-direction that is initially al a uniform temperature of T). The surfaces of the bar at x = 0 and y = 0 are insulated, while heat is lost from the other two surfaces by convection to the surrounding medium at temperature with a heat transfer coefficient of h. Assuming constant thermal conductivity and transient two-dimensional heat transfer with no heat generation, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve. 

If heat is transferred by conduction between two bodies (1) and (2) which are in contact with each other (boundary condition of the 3rd kind), then not only the heat fluxes, but also the temperatures will be equal. In contrast... 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

Consider a large plate of thickness L and thermal conductivity k in which heat is generated uniformly at a rate of One side of the plate is insulated while the other side is exposed to an enviromnent at with a heat transfer coefficient of h. (a) Express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the plate, (fc) determine the variation of temperature in the plate. 

This time is reached after 40 steps with M = 1, 20 steps for M = 2, 8 steps for M = 5 and finally 4 steps for M = 10. The temperatures for M = 1 and M = 2 agree very well with each other and with the analytical solution. The values for M = 5 yield somewhat larger deviations, while the result for M = 10 is useless. This large step produces temperature oscillations which are physically impossible. In [2.57], p. 122, a condition for the restriction of the step size, so that oscillations can be avoided, is given for a transient heat conduction problem with boundary conditions different from our example. The transfer of this condition to the present task delivers the limit... 

Mashayek and Ashgriz [98] considered effects of the heat transfer from the liquid to the surrounding ambient, the liquid thermal conductivity, and the temperature-dependent surface tension coefficient on the jet instability and the formation of satellite drops. Two different disturbances were imposed on the jet. In the first case, the jet is exposed to a spatially periodic ambient temperature. In addition to the thermal boundary condition, an initial surface disturbance with the same wave number as the thermal disturbance is also imposed on the jet. Both in-phase and out-of-phase thermal disturbances with respect to surface disturbances are considered. For the in-phase thermal disturbances, a parameter set is obtained at which capillary and thermocapiUary effects can cancel each other and the jet attains a stable configuration. No such parameter set can be obtained when the thermocapillary flows are in the same direction as the capillary flows, as in the out-of-phase thermal disturbances. In the second case, only an initial thermal disturbance is imposed on the surface of the liquid while the ambient temperature is kept spatially and temporally uniform (Fig. 1.20). 

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