Thermal boundary layers

Thus, a velocity boundary layer and a thermal boundary layer may develop simultaneously. If the physical properties of the fluid do not change significantly over the temperature range to which the fluid is subjected, the velocity boundary layer will not be affected by die heat transfer process. If physical properties are altered, there will be an interactive effect between the momentum and heat transfer processes, leading to a comparatively complex situation in which numerical methods of solution will be necessary. 

In general, the thermal boundary layer will not correspond with the velocity boundary layer. In the following treatment, the simplest non-interacting case is considered with physical properties assumed to be constant. The stream temperature is taken as constant In the first case, the wall temperature is also taken as a constant, and then by choosing the temperature scale so that the wall temperature is zero, the boundary conditions are similar to those for momentum transfer. 

It will be shown that the momentum and thermal boundary layers coincide only if the Prandtl number is unity, implying equal values for the kinematic viscosity (p./p) and the thermal diffusivity (DH = k/Cpp). 

The procedure here is similar to that adopted previously. A heat balance, as opposed to a momentum balance, is taken over an element which extends beyond the limits of both the velocity and thermal boundary layers. In this way, any fluid entering or leaving the element through the face distant from the surface is at the stream velocity u and stream temperature 0S. A heat balance is made therefore on the element shown in Figure 11.10 in which the length l is greater than the velocity boundary layer thickness S and the thermal boundary layer thickness t. 

The flow of fluid over a plane surface, heated at distances greater than. to from the leading edge, is now considered. As shown in Figure 11.11 the velocity boundary layer starts at the leading edge and the thermal boundary layer at a distance o from it. If the temperature of the heated portion of the plate remains constant, this may be taken as the datum temperature. It is assumed that the temperature at a distance y from the surface may be represented by a polynomial of the form ... 

Figure 11,11, Thermal boundary layer — streamline flow...

At the outer edge of the thermal boundary layer, the temperature is 9S and the temperature gradient (36/dy) = 0 if there is to be no discontinuity in the temperature profile. 

Thus the conditions for the thermal boundary layer, with respect to temperature, are the same as those for the velocity boundary layer with respect to velocity. Then, if the thickness of the thermal boundary layer is 5 the temperature distribution is given by ... 

It is assumed that the velocity boundary layer is everywhere thicker than the thermal boundary layer, so that 8 > 8, (Figure 11.11). Thus the velocity distribution everywhere within the thermal boundary layer is given by equation 11.12. The implications of this assumption are discussed later. 

The integral in equation 11.55 clearly has a finite value within the thermal boundary layer, although it is zero outside it. When the expression for the temperature distribution in the boundary layer is inserted, the upper limit of integration must be altered from /... 

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 ... 

Another important case is where the heat flux, as opposed to the temperature at the surface, is constant this may occur where the surface is electrically heated. Then, the temperature difference 9S — o will increase in the direction of flow (x-direction) as the value of the heat transfer coefficient decreases due to the thickening of the thermal boundary layer. The equation for the temperature profile in the boundary layer becomes ... 

If a concentration gradient exists within a fluid flowing over a surface, mass transfer will take place, and the whole of the resistance to transfer can be regarded as lying within a diffusion boundary layer in the vicinity of the surface. If the concentration gradients, and hence the mass transfer rates, are small, variations in physical properties may be neglected and it can be shown that the velocity and thermal boundary layers are unaffected 55. For low concentrations of the diffusing component, the effects of bulk flow will be small and the mass balance equation for component A is ... 

Again, the form of the concentration profile in the diffusion boundary layer depends on the conditions which are assumed to exist at the surface and in the fluid stream. For the conditions corresponding to those used in consideration of the thermal boundary layer, that is constant concentrations both in the stream outside the boundary layer and at the surface, the concentration profile is of similar form to that given by equation 11.70 ... 

Explain the concepts of momentum thickness" and displacement thickness for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this thermal thickness in terms of the thicknesses of the velocity and temperature boundary layers. 

Hsu YY, Graham RW (1961) An analytical and experimental study of the thermal boundary layer and ebullition cycle in nucleate boiling. NASA TN D-594 Hwan YW, Kim MS (2006) The pressure drop in micro-tubes and correlation development. Int J Heat Mass Transfer 49 1804-1812... 

Gap size, liquid thickness, thermal boundary layer... 

The temperature profiles along the x-axis at various times are shown in Figure 4. These values should be compared with the theoretical solution T - erfc [ (l-x)/(2jc t) ]. Some numerical oscillations are noted at the heated boundary at short times due to the inability of the rather coarse mesh and time Increment to capture the thermal boundary layer which forms there. However, this can easily be avoided if desired by using a finer mesh in that region, and also by stepping with shorter time increments initially. 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

As stated in Section 2.1, there is a waiting period between the time of release of one bubble and the time of nucleation of the next at a given nucleation site. This is the period when the thermal boundary layer is reestablished and when the surface temperature of the heater is reheated to that required for nucleation of the next bubble. To predict the waiting period, Hsu and Graham (1961) proposed a model using an active nucleus cavity of radius rc which has just produced a bubble that eventually departs from the surface and has trapped some residual vapor or gas that serves as a nucleus for a new bubble. When heating the liquid, the temperature of the gas in the nucleus also increases. Thus the bubble embryo is not activated until the surrounding liquid is hotter than the bubble interior, which is at... 

The thermal boundary-layer thicknesses in the liquid before bubble nucleation are much greater. 

Hsu and Graham (1961) took into consideration the bubble shape and incorporated the thermal boundary-layer thickness, 8, into their equation, thus making the bubble growth rate a function of 8. Han and Griffith (1965b) took an approach similar to that of Hsu and Graham with more elaboration, and dealt with the constant-wall-temperature case. Their equation is... 

Hsu, Y. Y., and R. W. Graham, 1961, An Analytical and Experimental Study of the Thermal Boundary Layer Ebullition Cycle in Nucleate Boiling, NASA TND-594, Lewis Res. Ctr., Cleveland, OH. (1)... 

Hull, L. M., and W. M. Rohsenow, 1982, Thermal Boundary Layer Development in Dispersed Flow Film Boiling, MIT Heat Transfer Lab. Rep. 85694-104, Massachusetts Institute of Technology, Cambridge, MA. (4)... 

The impact process of a 3.8 mm water droplet under the conditions experimentally studied by Chen and Hsu (1995) is simulated and the simulation results are shown in Figs. 16 and 17. Their experiments involve water-droplet impact on a heated Inconel plate with Ni coating. The surface temperature in this simulation is set as 400 °C with the initial temperature of the droplet given as 20 °C. The impact velocity is lOOcm/s, which gives a Weber number of 54. Fig. 16 shows the calculated temperature distributions within the droplet and within the solid surface. The isotherm corresponding to 21 °C is plotted inside the droplet to represent the extent of the thermal boundary layer of the droplet that is affected by the heating of the solid surface. It can be seen that, in the droplet spreading process (0-7.0 ms), the bulk of the liquid droplet remains at its initial temperature and the thermal boundary layer is very thin. As the liquid film spreads on the solid surface, the heat-transfer rate on the liquid side of the droplet-vapor interface can be evaluated by... 

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