Turbulent heat transfer

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Set. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Jnt. J. Heat Ma.s.s Tran.sfer, 6, 483, 495 (1963)]. Lee fnt. J. Heat Ma.s.s Tran.sfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Np < 30, lO < < 2 X 10, 1.01 <... 

There have been several analyses of turbulent heat transfer in annuh for example, Deissler and Taylor (NACA Tech. Note 3451, 1955), Kays and Leung [Int. J. Heat Ma.ss Transfer, 6, 537 (1963)], Lee [Int. J. Heat Transfer, 11, 509 (1968)], Sparrow, HaUman and Siegel [Appl. Sci. Res., 7A, 37 (1958)], and Johnson and Sparrow [Am. Soc. Mech. Eng. J. Heat Transfer, 88, 502 (1966)]. The reader is referred to these for details of the analyses. 

Metzner and Friend [Ind. Fng. Chem., 51, 879 (1959)] present relationships for turbulent heat transfer with nonnewtouiau fluids. Relationships for heat transfer by natural convection and through laminar boundaiy layers are available in Skelland s book (op. cit.). 

Kato H NtSHtWAKi, N. and Hirata, M. Im. Jl. Heat Mass Transfer 11 (1968) 1117. On the turbulent heat transfer by free convection from a vertical plate. 

Direct measurements of turbulent heat transfer in smooth pipes led to the correlation known as the Dittus-Boelter equation... 

Deissler, R. G., 1955, Analysis of Turbulent Heat Transfer, Mass Transfer, and Friction in Smooth Tubes at High Prandtl and Schmidt Numbers, NACA Rep. 1210, Lewis Res. Ctr., Cleveland, OH. (5)... 

ShaJSc113 as indicated by the thin solid line. This 0.67 power of Re agrees with the result of a turbulent heat transfer measurement on a rotating sphere [40], Since the flow induced by a rotating sphere is also characterized by an outflowing radial jet at the equator caused by the collosion of two opposing flow boundary layers on the sphere, the 0.67 power dependence on Re is clearly related to the radial flow stream away from the equator. 

In a system with both heat and mass transfer, an extra turbulent factor, kx, is included which is derived from an adapted energy equation, as were e and k. The turbulent heat transfer is dictated by turbulent viscosity, pt, and the turbulent Prandtl number, Prt. Other effects that can be included in the turbulent model are buoyancy and compressibility. 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

Figure 2.5 Temperature gradient in turbulent heat transfer from solid to fluid.

For a sufficient increase in the braking and heat transfer time we may imagine that the slowest reactions, e.g., combustion of dusts, will keep up with the detonation wave and can lead to detonation. Large losses in mixtures in which the detonation velocity is larger turn out to be related to intensification of the turbulent heat transfer and braking for an increase in the velocity proportional to D. 

Therefore, since (-kdT/dx) and (-k3T/dy) are the time-averaged heat conduction rates per unit area in the x- and y-directions respectively, it will be seen that the effects of the additional turbulence terms are the same as an increase in the heat transfer rate. For this reason, these extra terms pcpu T and pcpv V are often termed the turbulent heat transfer terms. Their presence can be demonstrated in a more physical maimer using the same line of reasoning as was adopted in the discussion of the turbulent stresses. For examole. considering rhe plpm nt chnum in... 

A consideration of the right-hand sides of these two equations indicates that the turbulence terms in these equations have, as discussed in Chapter 2, the form of additional shearing stress and heat transfer terms although they arise, of course, from the momentum transfer and enthalpy transfer produced by the mixing that arises from the turbulence. Because of their similarity to the molecular terms, the turbulence terms are usually called the turbulent shear stress and turbulent heat transfer rate respectively. Thus, the following are defined ... 

Tp being the turbulent shear stress and qj being the turbulent heat transfer rate. 

By analogy with the form of the molecular shearing stress and molecular heat transfer rate relations, as given in Eqs. (5.6) and (5.7), it is often convenient to express the turbulent shearing stress and turbulent heat transfer in terms of the velocity and temperature gradients in the following way ... 

It is also assumed that the turbulent heat transfer rate is is proportional to v 7, i.e., that ... 

The presence of the solid wall has a considerable influence on the turbulence structure near the wall. Because there can be no flow normal to the wall near the wall, v decreases as the wall is approached and as a result the turbulent stress and turbulent heat transfer rate are negligible in the region very near the wall. This region in which the effects of the turbulent stress and turbulent heat transfer rate can be neglected is termed the sublayer or, sometimes, the laminar sublayer [1],[2], [26],[27],[28],[29]. In this sublayer ... 

Kozlu, H.. Mikic, B.B., and Patera, A.T., Turbulent Heat Transfer Augmentation Using... 

Kasagi, N., Kuroda, A., and Hirata, M., Numerical Investigation of Near-Wall Turbulent Heat Transfer Taking Into Account the Unsteady Heat Conduction in the Solid Wall , J. of Heat Transfer, Vol. Ill, pp. 385-392, 1989. 

Azer, N.Z.,and Chao, B.T., Turbulent Heat Transfer in Liquid Metals—Fully Developed Pipe Row with Constant Wall Temperature , Int. J. Heat Mass Transfer, Vol. 3, p. 77, 1961. 

Sparrow, E.M., Hallman, T.M., and Siegel, R., Turbulent Heat Transfer in the Thermal Entrance Region of a Pipe with Uniform Heat Rux , Appl. Sci. Rest. Section A, Vol. 7, p. 37. 1957. 

Deissler, R.G., Turbulent Heat Transfer and Friction in the Entrance Regions of Smooth Passages , Trans. ASME, Vol. 77, pp. 1211-1234, 1955. 

Choi, J.M., and Anand, N.K, Turbulent Heat Transfer in a Serpentine Channel with a Series of Right-Angle Turns , Inter. J. of Heat and Mass Transfer, Vol. 38, No. 7, pp. 1225-1236, 1995. 

Garcia. A., and Sparrow, E.M., Turbulent Heat Transfer Downstream of a Contraction-Related, Forward-Facing Step in a Duct , J. Heat Trans., Vol. 109, pp. 621-626, 1987. 

Kim, S.H., and Anand, N.K., Turbulent Heat Transfer Between a Series of Parallel Plates with Surface-Mounted Discrete Heat Sources , J. Heat Transfer, Vol. 116, pp. 577-587, 1994. 

Lau, S.C., Kukreja, R.T., and McMillin. R.D., Effects of V-Shaped Rib Arrays on Turbulent Heat Transfer and Friction of Fully Developed Flow in a Square Channel , /nr. J. 

Liou, Tong Miin, Hwang, Jenn Jiang Chen, Shih Hui, Simulation and Measurement of Enhanced Turbulent Heat Transfer in a Channel with Periodic Ribs on One Principal Wall , Int. J. Heat and Mass Transfer. Vol. 36. No. 2, pp. 507-517, 1993. 

Next, consider (the turbulent heat transfer. The buoyancy forces have no effect on the temperature fluctuations so Eq. (9.81) still applies. Hence, again assuming ... 

Turbulent heat transfer is analogous to turbulent momentum transfer. The turbulent momentum flux postulated by Eq. (5-59) carries with it a turbulent... 

U Turbulent Heat Transfer Based on Fluid-Friction Analogy... 

Schlichting [1] has surveyed experimental measurements of friction coefficients for turbulent flow on flat plates. We present the results of that survey so that they may be employed in the calculation of turbulent heat transfer with the fluid-friction-heat-transfer analogy. The local skin-friction coefficient is given by... 

As we shall see in Chap. 6, Eq. (5-116 ) predicts heat-t rar.sfcr coefficients that are somewhat higher than those observed in experiments. The purpose of the discussion at this point has been to show that one may arrive at a relation for turbulent heat transfer in a fairly simple analytical fashion. As we have indicated earlier, a rigorous development of the Reynolds analogy between heat transfer and fluid friction involves considerations beyond the scope of our discussion and the simple path of reasoning chosen here is offered for the purpose of indicating the general nature of the physical processes. 

The agreement between Pr and the assumed value is sufficiently close. The other properties to be used in the turbulent heat-transfer analysis are... 

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