Heat flux boundary conditions

Heat flux is an important variable in fire growth and its determination is necessary for many problems. In general, it depends on scale (laminar or turbulent, beam length ), material (soot, combustion products) and flow features (geometric, natural or forced). We 

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NUh2 is the Nusselt number for uniform heat flux boundary condition along the flow direction and periphery. 

Yet another boundary condition encountered in polymer processing is prescribed heat flux. Surface-heat generation via solid-solid friction, as in frictional welding and conveying of solids in screw extmders, is an example. Moreover, certain types of intensive radiation or convective heating that are weak functions of surface temperature can also be treated as a prescribed surface heat-flux boundary condition. Finally, we occasionally encounter the highly nonlinear boundary condition of prescribed surface radiation. The exposure of the surface of an opaque substance to a radiation source at temperature 7 ,-leads to the following heat flux ... 

For a uniform heat flux boundary condition, we define r , u and z as before, and the dimensionless temperature as... 

The critical Reynolds number Reor is typically taken as 5 x 105, lie, < Re, < 3 x 107, and 0.7 < Pr < 400. The fluid properties are evaluated at the film temperature (7, + I )/2 where 7, is the free-stream temperature and 7 is the surface temperature. Equation (5-60) also apphes to the uniform heat flux boundary condition provided h is based on the average temperature difference between 7 and 7, ... 

For a plate of thickness L subjected to heat flux of 50 W/m into the medium from both sides, for example, the specified heat flux boundary conditions can be expressed as... 

FIGURE 2-29 Specified heat flux boundary conditions on both surfaces of a plane wall. 

Discussion This example demonstrates how steady one-dimensional heat conduction problems in composite media can be solved. We could also solve this problem by determining the heat flux at the interface by dividing the total heat generated in the wire by the surface area of the wire, and then using this value as the specifed heat flux boundary condition for both the wire and the ceramic tayer. This way the two problems are decoupled and can be solved separately. 

Combined Convection, Radiation, and Heat Flux Boundary Condition... 

Ethylene glycol-distilled water mixture with a ihass fraction of 0.72 and a flow rale of 2.05 X 10 mVs flows inside a tube with an inside diameter of 0.0158 m with a uniform wail heat flux boundary condition. For this flow, determine the Nusseli number at the location.t/D = 10 for the inlet lube configuration of (a) belt-mouth and (b) re-entrant. Compare the re suits for parts (a) and (b). Assume the Grashof number is Or = 60,000. The physical properties of ethylene glycol-distilled water mixture ate Pr - 33.46,1 = 3,45 X 10 mVsand 2.0. 

Table 2 demonstrates the effects of the Knudsen and the Brinkman numbers on heat transfer in a tube flow. As it can be seen, the Nusselt number decreases with the increases in both the Brinkman number and the Knudsen number, since the increasing temperature jump decreases heat transfer. Also, under the constant wall temperature boundary conditions, the Nusselt numbers are greater than under constant heat flux boundary conditions when the Brinkman number is nonzero [51, 521. 

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

Kavehpour et al. [20] solved the compressible two-dimensional fluid flow and heat transfer characteristics of a gas flowing between two parallel plates under both uniform temperature and uniform heat flux boundary conditions. They compared their results with the experimental results of Arkilic [3] for Helium in a 52.25x1.33x7500 mm channel. They observed an increase in the entrance length and a decrease in the Nusselt number... 

This behaviour can be explained if we consider the wall and the fluid temperatures as a function of the channel length (Figure 23). The longitudinal profiles are presented for two Reynolds numbers. For the first, with a Reynolds number much higher than 500 (fie = 4004), it is seen that the two temperature profiles are parallel as expected for uniform heat flux boundary conditions. For the second Reynolds number, smaller than 500 (Re = 381), the two profiles are no longer parallel. 

The laminar gaseous flow heat convection problem was solved in a cylindrical microchannel with uniform heat flux boundary conditions in [20]. The fluid was assumed to be incompressible with constant properties, the flow was assumed to be steady and two-dimensional, and viscous heating was neglected. They used the results from a previous study, [21], of the same problem with uniform... 

Convective heat transfer analysis for a gaseous flow in microchannels was performed in [24]. A Knudsen range of 0.06-1.1 was considered. In this range, flow is called transition flow. Since the eontinuum assumption is not valid, DSMC technique was applied. Reference [24] considered the uniform heat flux boundary condition for two-dimensional flow, where the channel height varied between 0.03125 and 1 micrometer. It was concluded that the slip flow approximation is valid for Knudsen numbers less than 0.1. The results showed a reduction in Nusselt number with increasing rarefaetion in both slip and transition regimes. 

Since the definition of the Brinkman number is different for the case of the uniform heat flux boundary condition, a positive Br means that the heat is transferred to the fluid from the wall as opposed to the uniform temperature case. Therefore, we see in figure 6 that Nu decreases as Br increases when Br > 0. 

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. 

In this lecture, the effects of the abovementioned dimensionless parameters, namely, Knudsen, Peclet, and Brinkman numbers representing rarefaction, axial conduction, and viscous dissipation, respectively, will be analyzed on forced convection heat transfer in microchannel gaseous slip flow under constant wall temperature and constant wall heat flux boundary conditions. Nusselt number will be used as the dimensionless convection heat transfer coefficient. A majority of the results will be presented as the variation of Nusselt number along the channel for various Kn, Pe, and Br values. The lecture is divided into three major sections for convective heat transfer in microscale slip flow. First, the principal results for microtubes will be presented. Then, the effect of roughness on the microchannel wall on heat transfer will be explained. Finally, the variation of the thermophysical properties of the fluid will be considered. 

Nonisotfaermal reactor with isothermal cooling jacket. A coolant at constant temperature cooling jacket is added to the previous example to examine the perfomiancc of a nonisothermal reactor. In thi.s model, the boundary condition for the energy balance at the radial boundary is changed from the thermal insulation boundary condition to a heat flux boundary condition. 

Comparison With Data. Measurements for isothermal plates in air are compared to Eq. 4.40 in Fig. 4.13 for rectangular plates. Data lie within about 20 percent of the correlation. Measurements have also been done using water, but only with a uniform heat flux boundary condition. For water, the data of Fujii and Imura [103] for a simulated 2D strip lie about 30 percent below Eq. 4.40, but the data of Birkebak and Abdulkadir [20] lie about 3 percent... 

Heat Transfer on Walls With Uniform Heat Flux. For circular ducts with symmetrical heating, the same heat transfer results for fully developed flow and developing flow are obtained for boundary conditions through . Therefore, the uniform wall heat flux boundary conditions are simply designated as the boundary condition. Shah and Bhatti [2] have derived the temperature distribution and Nusselt number by recasting the results reported by Tyagi [6] for heat transfer in circular ducts. These follow ... 

The thermal entrance length for thermally developing flow under the uniform wall heat flux boundary condition is equal to the following ... 

The effects of viscous dissipation on the thermal entrance problem with the uniform wall heat flux boundary condition can be found in Brinkman [27], Tyagi [6], Ou and Cheng [28], and Basu and Roy [29]. Other effects, such as inlet temperature, internal heat source, and wall heat flux variation, are reviewed by Shah and London [1] in detail. 

Thermally Developing Flow. Numerous investigators [80, 89-94] have carried out the investigation of turbulent thermally developing flow in a smooth circular duct with uniform wall temperature and uniform wall heat flux boundary conditions. It has been found that the dimensionless temperature and the Nusselt number for thermally developing turbulent flow have the same formats as those for laminar thermally developing flow (i.e., Eqs. 5.34-5.37 and Eqs. 5.50-5.53). The only differences are the eigenvalues and constants in the equations. 

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