Boundary layer thickness temperature

When the two liquid phases are in relative motion, the mass transfer coefficients in eidrer phase must be related to die dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive Uansfer to the Schmidt number. Another complication is that such a boundaty cannot in many circumstances be regarded as a simple planar interface, but eddies of material are U ansported to the interface from the bulk of each liquid which change the concenuation profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most indusuial chcumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass U ansfer model must therefore be replaced by an eddy mass U ansfer which takes account of this surface replenishment. 

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. 

For a Prandtl number, Pr. less than unity, the ratio of the temperature to the velocity boundary layer thickness is equal to Pr 1Work out the thermal thickness in terms of the thickness of the velocity boundary layer... 

The value of E is insensitive to small changes in ocean temperature but is quite sensitive to wind speed over the sea surface (boundary layer thickness, wave action, and bubble formation are functions of wind speed). Therefore changes in surface wind speed accompanying a climate change could affect rates of air-sea CO2 exchange. 

The speed of agitation governs the rate. The boundary layer thickness decreases at high speeds. A physical process such as the dissolution of an ionic salt in plain water is associated with an activation energy of the order of less than about 5 kcal mol-1. The influence of temperature on physical processes is less pronounced than that from agitation. 

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 (Ta - Ts)/8 term is the temperature gradient, which correlates (dTIdy through the boundary layer thickness. The fact that 8 can be correlated with the Reynolds number and that the Colburn analogy can be applied leads to the correlation of the form... 

Although the correlations given by Eq. (6.48) are useful for practical evaluation of heat transfer to a wall, one must not lose sight of the fact that the temperature gradient at the wall actually determines the heat flux there. In transpiration cooling problems, it is not so much that the injection of the transpiring fluid increases the boundary layer thickness, thereby decreasing the... 

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. 

Once the system of equations has been solved, the nondimensional temperature gradient can be easily evaluated at the surface, providing the Nusselt number. It should be expected that the heat transfer depends on the boundary-layer thickness, which in turn depends on the flow field, which is principally governed by the Reynolds number. Figure 6.9 shows a correlation between the Nusselt number and the Reynolds number that was obtained by solving the nondimensional system for several Reynolds numbers. 

As in the case of normal supported catalysts, we tried with this inverse supported catalyst system to switch over from the thin-layer catalyst structure to the more conventional powder mixture with a grain size smaller than the boundary layer thickness. The reactant in these studies (27) was methanol and the reaction its decomposition or oxidation the catalyst was zinc oxide and the support silver. The particle size of the catalyst was 3 x 10-3 cm hence, not the entire particle in contact with silver can be considered as part of the boundary layer. However, a part of the catalyst particle surface will be close to the zone of contact with the metal. Table VI gives the activation energies and the start temperatures for both methanol reactions, irrespective of the exact composition of the products. 

In this situation, a film is grown on the hot surface (Tw), and its thickness will increase without limit as long as fresh reactants are provided and products can be removed. The gas state will be in quasiequilibrium far from the hot surface and in a strongly nonequilibrium condition close to it. The change from one to the other will occur across a boundary layer where temperature, velocity, and species concentration vary rapidly. The behavior of this boundary layer will be determined by gas transport properties such as viscosity, thermal conductivity, as well as gas-phase kinetics and diffusion coefficients. So, even if the kinetics at the surface are very fast, we must deal with quasiequilibrium phenomena where gas conditions vary rapidly over short distances. 

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. 

No distinction is made at this stage between the velocity and temperature boundary layer thicknesses since both are assumed to be of the same order of magnitude. 8 is, therefore, a measure of the order of magnitude of both boundary layer thicknesses. 

Consider next the application of the conservation of energy principle to the control volume that was used above in the derivation of the momentum integral equation. The height, , of this control volume is taken to be greater than both the velocity and temperature boundary layer thicknesses as shown in Fig. 2.21. 

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. 

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. 

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. 

As with the program for laminar boundary layer flow discussed in Chapter 3, the turbulent boundary flow program calculates the velocity and temperature boundary layer thicknesses using, as discussed above, the assumption that A is the value of Y at which U = 0.99U and At is the value of Y at which 0 = 0.010. If either A or At is greater than the number of grid points is increased by ten. 

Here, L is a reference length that characterizes the size of the surface, e.g., its length. Twr is again some reference wall temperature. If some measure of the boundary layer thickness, 8, is also introduced, then the governing equations can be written in terms of the following dimensionless variables ... 

Plot the free-convection boundary-layer thickness along a 0.3-m high vertical plate which is maintained at a uniform surface temperature of 50CC and exposed to stagnant air at ambient pressure and a temperature of 10°C. Assume the flow remains laminar. 

A vertical flat plate is maintained at a uniform surface temperature and is exposed to air at standard ambient pressure. At a distance of 10 cm from the leading edge of the plate the boundary layer thickness is 2 cm. Estimate the thickness of the boundary layer at a distance of 25 cm from the leading edge. Assume a laminar boundary layer flow. 

Air at a temperature of 10°C flows upward over a 0.25 m high vertical plate which is kept at a uniform surface temperature of 40°C. Plot the variation of the velocity boundary layer thickness and local heat transfer rate along the plate for air velocities of between 0.2 and 1.5 m/s. Assume two-dimensional flow. 

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]. 

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