Boundary layers temperature distribution

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. 

Flow type Hydrodynamic boundary layer Velocity distribution in the flow direction Friction factor Thermal boundary layer Dimensionless temperature distribution in the flow Nusselt number... 

Nonuniform Surface Temperature. Nonuniform surface temperatures affect the convective heat transfer in a turbulent boundary layer similarly as in a laminar case except that the turbulent boundary layer responds in shorter downstream distances The heat transfer to surfaces with arbitrary temperature variations is obtained by superposition of solutions for convective heating to a uniform-temperature surface preceded by a surface at the recovery temperature of the fluid (Eq. 6.65). For the superposition to be valid, it is necessary that the energy equation be linear in T or i, which imposes restrictions on the types of fluid property variations that are permitted. In the turbulent boundary layer, it is generally required that the fluid properties remain constant however, under the assumption that boundary layer velocity distributions are expressible in terms of the local stream function rather than y for ideal gases, the energy equation is also linear in T [%]. 

FIGURE 4.24 Laminar and turbulent boundary layers and temperature distribution inside the boundary layer. 

In the stratification strategy with a replacing air distribution in the lower zone, the height of the boundary layer between the lower and upper zones can be determined with the criteria of the contaminant interfacial level.This level, where the air mass flow in the plumes is equal to the air mass flow of the supply air, IS presented in Fig. 8,4. In this ideal case the wait and air temperatures are equal on the interfacial level. In practical cases they are not usually equal and the buoyancy flows on the walls will raise the level and decrease the gradient. 

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 /... 

Figure 3.26 Velocity and temperature distribution in a subcooled boiling flow (bubble boundary-layer concept). (From Larson and Tong, 1969. Copyright 1969 by American Society of Mechanical Engineers, New York. Reprinted with permission.)...

The heat transfer across the vapor layer and the temperature distribution in the solid, liquid, and vapor phases are shown in Fig. 13. In the subcooled impact, especially for a droplet of water, which has a larger latent heat, it has been reported that the thickness of the vapor layer can be very small and in some cases, the transient direct contact of the liquid and the solid surface may occur (Chen and Hsu, 1995). When the length scale of the vapor gap is comparable with the free path of the gas molecules, the kinetic slip treatment of the boundary condition needs to be undertaken to modify the continuum system. Consider the Knudsen number defined as the ratio of the average mean free path of the vapor to the thickness of the vapor layer ... 

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... 

The catalyst activity depends not only on the chemical composition but also on the diffusion properties of the catalyst material and on the size and shape of the catalyst pellets because transport limitations through the gas boundary layer around the pellets and through the porous material reduce the overall reaction rate. The influence of gas film restrictions, which depends on the pellet size and gas velocity, is usually low in sulphuric acid converters. The effective diffusivity in the catalyst depends on the porosity, the pore size distribution, and the tortuosity of the pore system. It may be improved in the design of the carrier by e.g. increasing the porosity or the pore size, but usually such improvements will also lead to a reduction of mechanical strength. The effect of transport restrictions is normally expressed as an effectiveness factor q defined as the ratio between observed reaction rate for a catalyst pellet and the intrinsic reaction rate, i.e. the hypothetical reaction rate if bulk or surface conditions (temperature, pressure, concentrations) prevailed throughout the pellet [11], For particles with the same intrinsic reaction rate and the same pore system, the surface effectiveness factor only depends on an equivalent particle diameter given by... 

Accurate modeling of the temperature distribution in a PEFC requires accurate information in four areas heat source, thermal properties of various components, thermal boundary conditions, and experimental temperature-distribution data for model validation. The primary mechanism of heat removal from the catalyst layers is through lateral heat conduction along the in-plane direction to the current collecting land (like a heat sink). Heat removed by gas convection inside the gas channel accounts for less than 5% under typical PEFC operating conditions. 

Further the pressure and temperature dependences of all the transport coefficients involved have to be specified. The solution of the equations of change consistent with this additional information then gives the pressure, velocity, and temperature distributions in the system. A number of solutions of idealized problems of interest to chemical engineers may be found in the work of Schlichting (SI) there viscous-flow problems, nonisothermal-flow problems, and boundary-layer problems are discussed. 

Such expressions can be extended to permit the evaluation of the distribution of concentration throughout laminar flows. Variations in concentration at constant temperature often result in significant variation in viscosity as a function of position in the stream. Thus it is necessary to solve the basic expressions for viscous flow (LI) and to determine the velocity as a function of the spatial coordinates of the system. In the case of small variation in concentration throughout the system it is often convenient and satisfactory to neglect the effect of material transport upon the molecular properties of the phase. Under these circumstances the analysis of boundary layer as reviewed by Schlichting (S4) can be used to evaluate the velocity as a function of position in nonuniform boundary flows. Such analyses permit the determination of material transport from spheres, cylinders, and other objects where the local flow is nonuniform. In such situations it is not practical at the present state of knowledge to take into account the influence of variation in the level of turbulence in the main stream. 

A principal assumption for similarity is that there exists a viscous boundary layer in which the temperature and species composition depend on only one independent variable. The velocity distribution, however, may be two- or even three-dimensional, although in a very special way that requires some scaled velocities to have only one-dimensional content. The fact that there is only one independent variable implies an infinite domain in directions orthogonal to the remaining independent variable. Of course, no real problems have infinite extent. Therefore to be of practical value, it is important that there be real situations for which the assumptions are sufficiently valid. Essentially the assumptions are valid in situations where the viscous boundary-layer thickness is small relative to the lateral extent of the problem. There will always be regions where edge effects interrupt the similarity. The following section provides some physical evidence that supports the notion that there are situations in which the stagnation-flow assumptions are valid. 

There is a natural draw rate for a rotating disk that depends on the rotation rate. Both the radial velocity and the circumferential velocity vanish outside the viscous boundary layer. The only parameter in the equations is the Prandtl number in the energy equation. Clearly, there is a very large effect of Prandtl number on the temperature profile and heat transfer at the surface. For constant properties, however, the energy-equation solution does not affect the velocity distributions. For problems including chemistry and complex transport, there is still a natural draw rate for a given rotation rate. However, the actual inlet velocity depends on the particular flow circumstances—there is no universal correlation. 

The details of the transitions and the vortex behavior depend on the actual channel dimensions and wall-temperature distributions. In general, however, for an application like a horizontal-channel chemical-vapor-deposition reactor, the system is designed to avoid these complex flows. Thus the ideal boundary-layer analysis discussed here is applicable. Nevertheless, one must exercise caution to be sure that the underlying assumptions of one s model are valid. 

We shall find the temperature distribution in the gas in those layers adjacent to x — 0 in which the chemical reaction has not yet started. The evaporation heat or the heat of endothermic reaction of gas-formation, L, is equal to the jump in the thermal energy of the original substance. Thus, at x = 0 at the phase-boundary the magnitude of the thermal flux experiences a jump. Using one prime for the c-phase and two primes for the gas, we construct the equation... 

Thermal stress calculations in the five cell stack for the temperature distribution presented above were performed by Vallum (2005) using the solid modeling software ANSYS . The stack is modeled to be consisting of five cells with one air channel and gas channel in each cell. Two dimensional stress modeling was performed at six different cross-sections of the cell. The temperature in each layer obtained from the above model of Burt et al. (2005) is used as the nodal value at a single point in the corresponding layer of the model developed in ANSYS and steady state thermal analysis is done in ANSYS to re-construct a two-dimensional temperature distribution in each of the cross-sections. The reconstructed two dimensional temperature is then used for thermal stress analysis. The boundary conditions applied for calculations presented here are the bottom of the cell is fixed in v-dircction (stack direction), the node on the bottom left is fixed in x-direction (cross flow direction) and y-direction and the top part is left free to... 

Although the assumption of a quasistationary distribution of the concentration of component A within the diffusion boundary layer seems to be very rough, nevertheless under conditions of sufficiently intensive convection the dissolution kinetics of solids in liquids is well described by equations (5.1) and (5.8)-(5.10) (see Refs 301, 303, 304, 306-308). Clearly, these equations are generally applicable at a low solubility of the solid in the liquid phase (about 10-100 kg m or up to 5 mass %). Note that they may also describe fairly well the dissolution process in systems of much higher solubility. An example is the Al-Ni binary system in which the solubility of nickel in aluminium amounts to 10 mass % even at a relatively low temperature of 700°C (in comparison with the melting point of aluminium, 660°C).308... 

For the temperature distribution within the boundary layer, the following polynomial is selected ... 

Pohlhausen, 1911 [1] solved these equations first, whereas Schmidt and Beckmann, 1930 [2] solved them for Pr = 0.733 in 1930. Ostrach, 1953 [3], solved the same equations for the range 0.01 to 1000. For free convection laminar boundary layer on a heated vertical plate in that range of Pr, the velocity and the temperature distributions are shown in Figs. 9.2 and 9.3, respectively. 

Use the linear Burgers equation for heat convection in a channel where the water is flowing with uniform velocity of 0.1 m/s across the cross section of the channel (boundary layers are neglected). The water is initially at 25°C throughout. At time t = 0 sec, waste heat is continuously rejected at x = 0 m, and the channel is long such that dT/dx = 0 for x > 1 m. The amount of heat rejected is 6.23 W/m2 for t > 0. Using the MacCormack explicit scheme, calculate the first 9 time steps to show the transient temperature distributions. 

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. 

When functions 0n(r ) are known, the problem of determining the temperature distribution T(boundary layer reduces to evaluating unknown expansion coefficients an in Eq. (14.31). The boundary condition Eq. (14.34a) is now utilized to determine these coefficients. From Eq. (14.31),... 

Consider two-dimensional laminar boundary layer flow over a flat isothermal surface. Very close to the surface, the velocity components are very small. If the pressure changes are assumed to be negligible in the flow being considered, derive an expression for the temperature distribution near the wall. Viscous dissipation effects should be included in the analysis. 

A comparison of Eqs. (3.52) and (3.53) and also of their boundary conditions as given in Eqs. (3.24) and (3.54) respectively, shows that these equations are identical in all respects. Therefore, for the particular case of Pr equal to one, the distribution of 9 through the boundary layer is identical to the distribution of uJu ). In this par-ticular case, therefore, Fig. 3.4 also gives the temperature distribution and the two boundary layer thicknesses are identical in this case. Now many gases have Prandtl numbers which are not very different from 1 and this relation between the velocity and temperature fields and the results deduced from it will be approximately correct for them. 

Temperature distributions in a laminar boundary layer with and without viscous dissipation. 

Air flows at a velocity of 25 m/s ova a wide flat plate that is aligned with the flow. The mean air temperature in the boundary layer is 30°C. Plot the mean velocity distribution near the wall in the boundary layer assuming that flow is turbulent and that the wall shear stress is given by ... 

---