Turbulent-Boundary-Layer Heat Transfer

The average shear stress at the wall is computed frdm Eq. (5-52)  

The drag force is the product of this shear stress and the area, 

The physical mechanism of heat transfer in turbulent flow is quite similar to that in laminar flow the primary difference is that one must deal with the eddy properties instead of the ordinary thermal conductivity and viscosity. The main difficulty in an analytical treatment is that these eddy properties vary across the boundary layer, and the specific variation can be determined only from experimental data. This is an important point. All analyses of turbulent 

If one observes the instantaneous macroscopic velocity in a turbulent-flow system, as measured with a laser anemometer or other sensitive device, significant fluctuations about the mean flow velocity are observed as indicated in Fig. 5-11, where u is designated as the mean velocity and is the fluctuation from the mean. The instantaneous velocity is therefore 

The mean value of the fluctuation u must be zero over an extended period for steady flow conditions. There are also fluctuations in the y component of velocity, so we would write 

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Various analyses, similar to the one for the universal velocity profile above, have been performed to predict turbulent-boundary-layer heat transfer. The analyses have met with good success, but for our purposes the Colburn analogy between fluid friction and heat transfer is easier to apply and yields results which are in agreement with experiment and of simpler form. 

Some more comprehensive methods of correlating turbulent-boundary-layer heat transfer are given by Churchill [11]. 

Thus we conclude that both laminar and turbulent boundary-layer heat transfer must be considered. We first determine the reference temperatures for the two regimes and then evaluate properties at these temperatures. 

R. E. Mayle, M. F. Blair, and F. C. Kopper, Turbulent Boundary Layer Heat Transfer on Curved Surfaces, J. Heat Transfer (101) 521-525, August 1979. 

Another concept sometimes used as a basis for comparison and correlation of mass transfer data in columns is the Clulton-Colbum analogy (35). This semi-empirical relationship was developed for correlating mass- and heat-transfer data in pipes and is based on the turbulent boundary layer model... 

Eckert, E.R.G., Engineering Relations for Heat Transfer and Friction in High-Velocity Laminar and Turbulent Boundary Layer Flow over Surfaces with Constant Pressure and Temperature , Trans. ASME, Vol. 78, pp. 1273-1284, 1956. 

The effects of fluid property variations on heat transfer in turbulent boundary layer flow over a flat plate have also been numerically evaluated. This evaluation indicates that if the properties are as with If minar boundary layers evaluated at ... 

The roof of a building is flat and is 20 m wide and long. If the wind speed over the roof is 10 m/s, determine the convective heat transfer rate to the roof (i) on a clear night when the roof temperature is 2°C and the air temperature is 12°C and (ii) on a hot, sunny day when the roof temperature is 46°C and the air temperature is 28°C. Assume two-dimensional turbulent boundary layer flow. 

A numerical procedure for calculating the heat transfer rate with turbulent boundary layer flow was discussed in Chapter 5. This procedure used a mixing length-based turbulence model. Discuss the modifications that must be made to this procedure to apply it to mixed convective flow over a vertical plate. 

Consider laminar film condensation on a vertical plate when the vapor is flow ing parallel to the surface in a downward direction at velocity, V. Assume that a turbulent boundary layer is formed in the vapor along the outer surface of the laminar liquid film. Determine a criterion that will indicate when the effect of the shear stress at the outer edge of the condensed liquid film on the heat transfer rate is less than 59c. Assume that pv [Pg.602]

The average heat transfer over the entire laminar-turbulent boundary layer is... 

Leontev, A. I. Heat and Mass Transfer in Turbulent Boundary Layers, Adv. Heat Transfer, vol. 3, 1966. 

Equations (45) and (46) are only two of many formulas that have been used to describe erosive burning [8]. Most of the formulas that have been suggested are based on physical concepts of influences of crossflow on propellant burning. Among these concepts is the idea that high external velocities produce a turbulent boundary layer (see Chapter 12) on the propellant surface and thereby effectively increase the thermal diffusivity of the gas, which in turn increases the rate of heat transfer to the propellant and hence the burning rate [99]. The idea that turbulent convective heat transfer from the hot combustion products outside the boundary layer provides an additive contribution to the heat flux reaching the propellant surface and,... 

The random eddy motion of groups of particles resembles the random motion of molecules in a gas—colliding with each other after traveling a certain distance and exchanging momentum and licat in the process. Therefore, momentum and beat transport by eddies in turbulent boundary layers is analogous to the molecular momentum and heat diffusion. Then turbulent wall shear stress and turbulent heat transfer can be expressed in an analogous manner as... 

In turbulent flow, momentum is constantly fed into the layer adjacent to the wall because of the momentum transfer between layers at different velocities. The kinetic energy of the fluid elements close to the wall does not decrease as rapidly as in laminar flow. This means that turbulent boundary layers do not become detached as quickly as laminar boundary layers. Heat and mass transfer close to the wall is not only promoted by turbulence, the fluid also flows over a larger surface area without detachment. At the same time the pressure resistance is lower because the fluid flow does not separate from the surface for a longer flow path. 

Eckert, E.R.G. Engineering relations for heat transfer and friction in high-velocity laminar and turbulent boundary-layer flow over surfaces with constant pressure and temperature. Trans. Amer. Soc. Mech. Eng., J. Heat Transfer 78 (1956) 1273-1283... 

The modeling procedure can be sketched as follows. First an approximate description of the velocity distribution in the turbulent boundary layer is required. The universal velocity profile called the Law of the wall is normally used. The local shear stress in the boundary layer is expressed in terms of the shear stress at the wall. From this relation a dimensionless velocity profile is derived. Secondly, a similar strategy can be used for heat and species mass relating the local boundary layer fluxes to the corresponding wall fluxes. From these relations dimensionless profiles for temperature and species concentration are derived. At this point the concentration and temperature distributions are not known. Therefore, based on the similarity hypothesis we assume that the functional form of the dimensionless fluxes are similar, so the heat and species concentration fluxes can be expressed in terms of the momentum transport coefficients and velocity scales. Finally, a comparison of the resulting boundary layer fluxes with the definitions of the heat and mass transfer coefficients, indiates that parameterizations for the engineering transfer coefficients can be put up in terms of the appropriate dimensionless groups. 

It has been shown that there exists a continuous change in the physical behavior of the turbulent momentum boundary layer with the distance from the wall. The turbulent boundary layer is normally divided into several regions and sub-layers. It is noted that the most important region for heat and mass transfer is the inner region of the boundary layer, since it constitutes the major part of the resistance to the transfer rates. This inner region determines approximately 10 — 20% of the total boundary layer thickness, and the velocity distribution in this region follows simple relationships expressed in the inner variables as defined in sect 1.3.4. 

To proceed we need to put up dimensionless relations for the heat and mass transfer fluxes in the turbulent boundary layer using a procedure analoguous to the one applied for the momentum flux (5.249) in which the Boussinesq s turbulent viscosity hypothesis is involved. 

Troshko AA, Hassan YA (2001) Law of the wall for two-phase turbulent boundary layers. Int J Heat and Mass Transfer 44(4) 871-875... 

Kader, B. A., Temperature and concentration profiles in fully turbulent boundary layers, Int. J. Heat Mass Transfer, V. 24, No. 9, pp. 1541-1544, 1981. 

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