Heat Conduction with Flux Boundary Conditions

Example 4.3. Heat Conduction with Flux Boundary Conditions 

In the previous two examples, the temperature (dependent variable) at x = 0 was specified. The same technique can be applied for the case where the derivative of the dependent variable is known at the boundary x = 0 (flux boundary conditions). Consider the transient heat conduction problem in a slab. [4] The governing equation in dimensionless form is 

The flux boundary condition has to be considered while taking the Laplace transform. Equation (4.3) is solved in Maple below  

4 Partial Differential Equations in Semi-infinite Domains 

---

These authors numerically solved the system of equations with appropriate boundary conditions to derive the time-averaged radiant and conductive heat fluxes between the fluidized bed and the heat transfer surface. Using... 

Under these conditions, the variation of the temperature profile (during irradiation with constant flux) could be computed from the heat-conduction equation in terms of the thermal conductivity, k the absorbance, a the incident flux, H the specific heat capacity, c and the thermal diffusivity, a. With the boundary conditions ... 

In this work, heat and fluid flow in some common micro geometries is analyzed analytically. At first, forced convection is examined for three different geometries microtube, microchannel between two parallel plates and microannulus between two concentric cylinders. Constant wall heat flux boundary condition is assumed. Then mixed convection in a vertical parallel-plate microchannel with symmetric wall heat fluxes is investigated. Steady and laminar internal flow of a Newtonian is analyzed. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. In this study, the usual continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump. 

C. N. Sokmen and M. M. Razzaque, Finite Element Analysis of Conduction-Radiation Heat Transfer in an Absorbing-Emitting and Scattering Medium Contained in an Enclosure with Heat Flux Boundary Conditions, ASME HTD-vol. 81, pp. 17-23,1987. 

At low Re and when conjugate effects have to be considered, the temperature distribution along the microchannel is not linear. Under constant heat flux boundary conditions, Nu decreases with decreasing ratio of outer to inner channel diameter, approaching the constant temperature solution. A decrease in Nu is also seen with increasing wall conductivity. For constant temperature boundary conditions, Nu will increase approaching the constant heat flux solution with axial heat conduction in the wall. The values for local Nusselt number for the conjugated problem lie between the values for the two boundary conditions constant heat flux and constant temperature. 

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. 

The wall temperature maps shown in Fig. 28 are intended to show the qualitative trends and patterns of wall temperature when conduction is or is not included in the tube wall. The temperatures on the tube wall could be calculated using the wall functions, since the wall heat flux was specified as a boundary condition and the accuracy of the values obtained will depend on their validity, which is related to the y+ values for the various solid surfaces. For the range of conditions in these simulations, we get y+ x 13-14. This is somewhat low for the k- model. The values of Tw are in line with industrially observed temperatures, but should not be taken as precise. 

Prior to the tests, all the samples were dried in a vacuum oven at 80°C for at least 72 h to minimize the moisture effect and then transferred to a desiccator. Measurements were carried out on a cone calorimeter provided by the Dark Star Research Ltd., United Kingdom. To minimize the conduction heat losses to insulation and to provide well-defined boundary conditions for numerical analysis of these tests, a sample holder was constructed as reported in [14] with four layers (each layer is 3 mm thick) of Cotronic ceramic paper at the back of the sample and four layers at the sides. A schematic view of the sample holder is shown in Figure 19.12. Three external heat fluxes (40, 50, and 60kW/m2) were used with duplicated tests at each heat flux. 

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... 

Consider a spherical container of inner radius r, outer radius rj, and thermal conductivity k. Express the boundary condition on the inner surface of the container for steady one-dimensional conduction for the following cases (a) specified temperature of 50°C, (b) specified heat flux of 30 W/m toward the center, (c) convection to a medium at 7. with a heal transfer coefficient of/i. 

Consider a short cyUnder of radius r<, and height H in which heat is generated at a constant rate of Heat is lost from the cylindrical surface at r = r by convection to the surrounding medium at temperature with a heat transfer coefficient of /i. The bottom surface of the cylinder at z = 0 is insulated, while the top surface at z — is subjected to uniform heat flux Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not. solve. 

A solar heat flux q, is incident on a sidewalk whose thermal conductivity is k, solar absorptivity is a and convective heat transfer coefficient is h. TaWng the positive x direction to be towards the sky and disregarding radiation exchange with the surroundings surfaces, the correct boundary condition for this sidewalk surface is... 

If the thermally conductive body is in contact with another medium, several different boundary conditions can apply, each depending on whether the other medium is a solid or fluid, and its respective material properties. When the other medium is another solid, the heat flux at the interface of body 1 to body 2 is the same for both bodies. According to (2.18), at the interface, index I, it is valid that... 

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... 

In this lecture, a variety of results for convective heat transfer in microtubes and microchannels in the slip flow regime under different conditions have been presented. Both constant wall temperature and constant wall heat flux cases have been analyzed in microtubes, including the effects of rarefaction, axial conduction, and viscous dissipation. In rough microchannels the abovementioned effects have also been investigated for the constant wall temperature boundary condition. Then, temperature-variable dynamic viscosity and thermal conductivity of the fluid were considered, and the results were compared with constant property results for microchannels, with the effects of rarefaction and viscous dissipation. 

The effect of single and multiple isotropic layers or coatings on the end of a circular flux tube has been determined by Antonetti [2] and Sridhar et al. [107]. The heat enters the end of the circular flux tube of radius b and thermal conductivity k3 through a coaxial, circular contact area that is in perfect thermal contact with an isotropic layer of thermal conductivity k, and thickness This layer is in perfect contact with a second layer of thermal conductivity k2 and thickness t2 that is in perfect contact with the flux tube having thermal conductivity k3 (Fig. 3.22). The lateral boundary of the flux tube is adiabatic and the contact plane outside the contact area is also adiabatic. The boundary condition over the contact area may be (1) isoflux or (2) isothermal. The dimensionless constriction resistance p2 layers = 4k3aRc is defined with respect to the thermal conductivity of the flux... 

Thermally Developing Flow. Wibulswas [160] and Aparecido and Cotta [161] have solved the thermal entrance problem for rectangular ducts with the thermal boundary condition of uniform temperature and uniform heat flux at four walls. However, the effects of viscous dissipation, fluid axial conduction, and thermal energy sources in the fluid are neglected in their analyses. The local and mean Nusselt numbers Nu j, Num T, and Nu hi and Num Hi obtained by Wibulswas [160] are presented in Tables 5.32 and 5.33. 

This problem is analogous to the problem of heat flow in a slab of thermal conductivity k, with constant temperature at x = 0 and constant heat flux, F0, at x = 8. The solution of this problem has been published, but seems to have received less attention (6,7) than the solution for the boundary conditions ... 

A surface heat transfer coefficient h can be defined as the quantity of heat flowing per unit time normal to the surface across unit area of the interface with unit temperature difference across the interface. When there is no resistance to heat flow across the interface, h is infinite. The heat transfer coefficient can be compared with the conductivity the conductivity relates the heat flux to the temperature gradient the surface heat transfer coefficient relates the heat flux to a temperature difference across an unknowm distance. Some theoretical work has been done on this subject [8], but since it is rarely possible to achieve in practice the boundary conditions assumed in the mathematical formulation, it is better to regard it as an empirical factor to be determined experimentally. Some typical values are given in Table 2. Cuthbert [9] has suggested that values greater than about 6000 W/m K can be regarded as infinite. The spread of values in the Table is caused by mold pressure and by different fluid velocities. Heat loss by natural convection also depends on whether the sample is vertical or horizontal. Hall et al. [10] have discussed the effect of a finite heat transfer coefficient on thermal conductivity measurement. 

In the pure convection problem, heat transfer through the wall is characterized by an appropriate thermal boundary condition directly or indirectly specified at the wall-fluid interface. In a pure convection problem, the solution of the temperature problem for the solid wall is not needed the velocity and temperature are determined only in the fluid region. However, the heat transfer through the sohd walls of the microchannel by conduction may have significant normal and/or peripheral as well as axial components, or the wall may be of ncmuniform thickness. In these cases the temperature problem for the solid wall needs to be analyzed simultaneously with that for the fluid in order to calculate the real wall-fluid interface heat flux distribution. In this case the wall-fluid heat transfer is referred to as conjugate heat tranter. 

---