The Thermal Boundary Layer

There is a striking similarity between Eq. (5-25) and the momentum equation for constant pressure, 

The solution to the two equations will have exactly the same form when a = v. Thus we should expect that the relative magnitudes of the thermal diffusivity and kinematic viscosity would have an important influence on convection heat transfer since these magnitudes relate the velocity distribution to the temperature distribution. This is exactly the case, and we shall see the role which these parameters play in the subsequent discussion. 

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 

The conditions which the temperature distribution must satisfy are 

As a standard for the y coordinate the mean thickness of the thermal boundary layer is introduced. An approximation of the thermal boundary layer has already been given in (3.8) 

The thermal diffusivity is defined here by a = A/gcp. The mean thickness Tm of the thermal boundary layer 

Pr = vja is the Prandtl number. With Tm we form the dimensionless coordinate 

By this normalisation the temperature in the thermal boundary layer varies between the values 0 d+ 1. With these dimensionless quantities the energy equation (3.115) can be rearranged into 

3 Convective heat and mass transfer. Single phase flow 

---

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 ... 

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 ... 

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... 

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... 

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)... 

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... 

ITie Henry and Fauske model employs curves similar to Fig. 16. Immediately upon initial contact, they assume that there is rapid pressurization at the interface. Nucleation in this vicinity is then prevented [Po in Eq. (7) is large and so is Dq] until the pressure is acoustically relieved by the wave moving to a free surface and returning. During this period, the thermal boundary layer in the cold liquid continues to develop. At relief, there still may be no intersection of the t-Do curve (in Fig. 16), so until such a time... 

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

By comparison, all bonds other than Sn - C in the tin hydroxides are quite strong. In ClsSnOH the bond energies are 125 kcalmor 95 kcal mol and 87 kcal mol for the 0 - H, Sn - 0, and Sn - Cl bonds, respectively. Thus, it appears likely that the hydroxide ligand is quite stable and could survive transit through the thermal boundary layer in a CVD reactor and form tin oxide. 

Regarding the heat transfer coefficient, let us write that the thickness 8 of the thermal boundary layer is related, for sufficiently large Prandtl numbers, to that of the hydrodynamic boundary layer by... 

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. 

It is difficult to solve the system of Eqs. (39)—(41) for these boundary conditions. However, certain simplifying assumptions can be made, if the Prandtl number approaches large values. In this case, the thermal boundary layer becomes very thin and, therefore, only the fluid layer near the plate contributes significantly to the heat transfer resistance. The velocity components in Eq. (41) can then be approximated by the first term of their Taylor series expansions in terms of y. In addition, because the nonlinear inertial terms are negligible near the wall, one can further assume that the combined forced and free convection velocity is approximately equal to the sum of the velocities that would exist when these effects act independently. Therefore, for assisting flows at large Prandtl numbers (theoretically for Pr -> oo), Eq. (41) can be rewritten in the form ... 

As with other matters concerned with transport to electrodes, detailed treatments were set up very early. Various boundary layers at interfaces under flow were suggested by Prandtl as early as 1904. Three are shown in Fig. (7.93). The 8y is die well-known diffusion layer due to Nernst (Section 7.9.9). The 8 is the thermal boundary layer and the 8V signifies the thickness of the layer (Prandtl s layer) in a flowing liquid in which the velocity slows an approach peipendicular to the surface. 

Determine the effect of Prandtl number on the thermal boundary-layer thickness. Consider the range 1 < Pr < 100. 

The equivalent equation for the thermal boundary layer will be ... 

A special complication is formed by thermal inhomogeneities. The polymer follows a circular path in the cross-channel direction, which implies that only a certain amount of polymer passes the thermal boundary layer where it is cooled directly. Therefore, for deep cut channels the heat of reaction in the middle of the channel is difficult to remove. Janeschitz Kriegl [18] compared the thickness of the thermal boundary layer with the channel depth, and identified a criterion for thermal homogeneity in a single-screw extruder. This can be adapted to twin-screw extruders with m thread starts per screw. It may be concluded that thermal inhomogeneities become important if ... 

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. 

---