Temperature thermal boundary layer

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. 

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 ... 

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 ... 

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 /... 

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 ... 

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. 

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. 

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... 

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... 

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... 

Next, consider the gradients of temperature. If the reaction is exothermic, the center of the particle tends to be hotter, and conversely for an endothermic reaction. Two sets of gradients are thus indicated in Figure 8.9. Heat transfer through the particle is primarily by conduction, and between exterior particle surface (Ts) and bulk gas (Tg) by combined convection-conduction across a thermal boundary layer, shown for convenience in Figure 8.9 to coincide with the gas film for mass transfer. (The quantities T0, ATp, A7y, and AT, are used in Section 8.5.5.)... 

Equation 9.1-15 equates the rate of heat transfer by conduction at the surface to the rate of heat transfer by conduction/convection across a thermal boundary layer exterior to the particle (corresponding to the gas film for mass transfer), expressed in terms of a film coefficient, h, and the difference in temperature between bulk gas at Tg and particle surface at Ts ... 

Figure 26.56 is the corresponding plot for 12% inlet H2 in air. In this case, there is an extinction at about 1000 K for both reactors. The qualitative features are similar to that of the PSR discussed above for 28% H2 in air. For such fuel-lean mixtures, the flame is attached to the surface. As a result, the thermal coupling between the surface and the gas phase is strong, and reduction in surface temperature affects the entire thermal boundary layer resulting in significant reduction of NOj,. These results indicate that the bifurcation behavior, in terms of extinction, determines the role of flame-wall thermal interactions in emissions. 

The velocity field is caused in free convection by the temperature field. Therefore, the thickness 8 of the thermal boundary layer can be used as the single length scale that characterizes both the temperature and velocity fields. Denoting the velocity scale in the x direction by u0, the continuity equation [Eq. (39)] shows that the velocity scale v0 in the y direction is of the order of u08/x. 

A hydrodynamic boundary layer begins to develop immediately as the unreacted flow enters the channel. A thermal boundary layer begins to grow at z = 5 mm as the wall temperature ramps linearly from the inlet temperature of 7in = 600 K to the final catalyst-wall temperature of Tw = 1290 K at z = 10 mm. At z = 10 mm the channel wall becomes... 

The wall cooling has a major effect when there are large changes in reactor throughput. When turning down a gasifier, the temperature of the bed will be lowered due to heat loss to the environment, and the thermal boundary layer will penetrate inwards to the central core. 

The effects of heat transfer are found only within the thermal boundary layer. The fluid outside the thermal boundary layer will be unaffected by the heat transfer, and have a uniform temperature Tin at the entrance where x = 0. 

Consider fully developed flow in a pipe. A thermal boundary condition is applied, starting at a distance x = x<,. For the four different cases listed below, sketch the temperature profiles in the pipe as the thermal boundary layer develops, and the temperature profiles after the thermal boundary layer has fully developed ... 

It can be seen that the expression for the average Nusselt number for Pr 1 is closer in form to the case where Pr — oo, than the case where Pr —> 0. The reason for this is that in natural convection, the driving force is caused by the temperature gradients, and thus defined by the thermal boundary layer. When Pr 1 and when Pr — co, the thermal boundary layer is thicker than the velocity boundary layer. Hence, the behavior of the Nusselt number would be similar in form for both cases. When Pr — 0, the behavior of the kinematic viscosity relative to the thermal diffusivity is going to be different from that of the other two cases. In addition, the right-hand side of the expression for Pr — 0 is independent of o, as one would expect for this case where the effects of the kinematic viscosity are very small or negligible. 

Consider convection with incompressible, laminar flow of a constant-temperature fluid over a flat plate maintained at a constant temperature. With the velocity distributions found in either Prob. 10.1 or Prob. 10.2, compute the dimensionless temperature distribution within the thermal boundary layer for the Peclet number equal to 0.1,1.0,10.0,100.0. Use the ADI method. 

Now, in general, the effects of viscosity and heat transfer do not extend to the same distance from the surface. For this reason, it is convenient to define both a velocity boundary layer thickness and a thermal or temperature boundary layer thickness as shown in Fig. 2.14. The velocity boundary layer thickness is a measure of the distance from the surface at which viscous effects cease to be important while the thermal boundary layer thickness is a measure of the distance from the wall at which heat transfer effects cease to be important. 

As with the velocity boundary layer, the thermal boundary layer is assumed to have a definite thickness, dr, and outside this boundary layer the temperature is assumed to be constant. 

The assumption that the temperature profiles are similar is equivalent to assuming that 0 depends only on the similarity variable, 17, because the thermal boundary layer thickness is also of order xl jRex. 

The coefficients in this equation, i.e., e,f, g, and h, are determined by applying the boundary conditions on temperature at the inner and outer edges of the thermal boundary layer. Three such boundary conditions, which are analogous to those given for the velocity in Eq. (3.123), are ... 

St is, of course, the thickness of the thermal boundary layer. The first of these conditions follows from the requirement that die fluid in contact with the wall must attain the same temperature as the wall. The other two conditions follow from the requirement that the boundary layer temperature profile must blend smoothly into the freestream temperature distribution at the outer edge of the boundary layer. 

Air at 300 K Ad 1 atm flows at a velocity of 2 m/s along a flat plate which has a length of 0.2 m. The plate is kept at a temperature of 330 K. Plot the variations of the velocity and thermal boundary layer thicknesses along the plate. 

If the Darcy assumptions are used then with forced convective flow over a surface in a porous medium, because the velocity is not assumed to be 0 at the surface, there is no velocity change induced by viscosity near the surface and there is therefore no velocity boundary layer in the flow over the surface. There will, however, be a region adjacent to the surface in which heat transfer is important and in which there are significant temperature changes in the direction normal to the surface. Under many circumstances, the normal distance over which such significant temperature changes occur is relatively small, i.e., a thermal boundary layer can be assumed to exist around the surface as shown in Fig. 10.9, the ratio of the boundary layer thickness, 67, to the size of the body as measured by some dimension, L, being small [15],[16]. 

Just as the hydrodynamic boundary layer was defined as that region of the flow where viscous forces are felt, a thermal boundary layer may be defined as that region where temperature gradients are present in the flow. These temperature gradients would result from a heat-exchange process between the fluid and the wall. 

Consider the system shown in Fig. 5-7. The temperature of the wall is T ., the temperature of the fluid outside the thermal boundary layer is T, and the thickness of the thermal boundary layer is designated as 8,. At the wall, the velocity is zero, and the heat transfer into the fluid takes place by conduction. Thus the local heat flux per unit area, q", is... 

Consider the control volume bounded by the planes 1, 2, A-A, and the wall as shown in Fig. 5-8. It is assumed that the thermal boundary layer is thinner than the hydrodynamic boundary layer, as shown. The wall temperature is 7 ,., the free-stream temperature is Tx, and the heat given up to the fluid over the length dx is dqw. We wish to make the energy balance... 

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