Turbulent heat transfer terms

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... 

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). 

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 ... 

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 ... 

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 ... 

So far, we have studied a number of illustrative examples for two-phase laminar heat transfer following the analytical approach we used in Chapter 5. For two-phase turbulent heat transfer we use an approach based on two-length scale dimensional analysis and the correlation of experimental data in terms of dimensionless numbers resulting from this analysis. 

The term L dvfldy by Eq. (3.10-29) is the momentum eddy diffusivity e,. When this term is in the turbulent heat-transfer equation (5.7-24), it is called a, eddy thermal diffusivity. Then Eq. (5.7-24) becomes... 

Figures 2.5 and 2.6 reveal that deterioration is caused by a different mechanism at low flow rates. The calculation results at G = 39 kg m s and 7 = T, which gives the Reynolds number 10,000, are rearranged in terms of the Grashof number and the Nusselt number in Fig. 2.8. Nu has a minimum value at Gr = 2 x 10. Nu is constant when Gr is lower than it, which means forced convection is dominant. On the other hand, Nu increases linearly when Gr is larger than the minimum point, which implies that natural convection is dominant. The minimum point emerges at the boundary between the two convection modes. Flow velocity and turbulence energy profiles are depicted in Fig. 2.9. When the heat flux is enhanced, the flow velocity increases near the wall and the profile becomes flat. Since turbulence energy is produced by the derivative of flow velocity, it is reduced. Hence, heat transfer is deteriorated. When the heat flux is enhanced above the minimum point, the flow velocity profile is more distorted and turbulent heat transfer is then enhanced. This type of heat transfer deterioration is attributed to acceleration as well as buoyancy. In the present analysis, buoyancy force is dominant. The computational results without the buoyancy force term in the Navier-Stokes equations are...

In fully developed flow, equations 12.102 and 12.117 can be used, but it is preferable to work in terms of the mean velocity of flow and the ordinary pipe Reynolds number Re. Furthermore, the heat transfer coefficient is generally expressed in terms of a driving force equal to the difference between the bulk fluid temperature and the wall temperature. If the fluid is highly turbulent, however, the bulk temperature will be quite close to the temperature 6S at the axis. 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

Now let us consider the mixing time, t. This will be estimated by an order of magnitude estimate for diffusion to occur across the boundary layer thickness, <5Bl- If we have turbulent natural conditions, it is common to represent the heat transfer in terms of the Nusselt number for a vertical plate of height, , as... 

The study of fire in a compartment primarily involves three elements (a) fluid dynamics, (b) heat transfer and (c) combustion. All can theoretically be resolved in finite difference solutions of the fundamental conservation equations, but issues of turbulence, reaction chemistry and sufficient grid elements preclude perfect solutions. However, flow features of compartment fires allow for approximate portrayals of these three elements through global approaches for prediction. The ability to visualize the dynamics of compartment fires in global terms of discrete, but coupled, phenomena follow from the flow features. 

Alternatively, one might satisfy convection near a boundary by invoking Il6 and Ilg where the heat transfer coefficient is taken from an appropriate correlation involving Re (e.g. Equation (12.38)). Radiation can still be a problem because re-radiation, n7, and flame (or smoke) radiation, II3, are not preserved. Thus, we have the art of scaling. Terms can be neglected when their effect is small. The proof is in the scaled resultant verification. An advantage of scale modeling is that it will still follow nature, and mathematical attempts to simulate turbulence or soot radiation are unnecessary. 

Other terms which can be defined quantitatively are introduced in the following sections. Some other terms, such as turbulence, viscosity, and diffusivity are used without definition. For a full explanation of these terms, we refer the reader to standard texts in fluid meehanies, heat transfer, and mass transfer. 

In process operations, simultaneous transfer of momentum, heat, and mass occur within the walls of the equipment vessels and exchangers. Transfer processes usually take place with turbulent flow, under forced convection, with or without radiation heat transfer. One of the purposes of engineering science is to provide measurements, interpretations and theories which are useful in the design of equipment and processes, in terms of the residence time required in a given process apparatus. This is why we are concerned here with the coefficients of the governing rate laws that permit such design calculations. 

In the derivation of these unsteady flow equations, no assumptions regarding the nature of the unsteadiness are made. In turbulent flow, therefore, the instantaneous values of the variables will satisfy these equations. Numerical solutions to the above equations are very difficult to obtain and for most purposes it is only the mean values of these variables and the mean heat transfer rate that are required. An attempt is, therefore, usually made to express them in terms of the mean values of the variables. 

Some simple methods of determining heat transfer rates to turbulent flows in a duct have been considered in this chapter. Fully developed flow in a pipe was first considered. Analogy solutions for this situation were discussed. In such solutions, the heat transfer rate is predicted from a knowledge of the wall shear stress. In fully developed pipe flow, the wall shear stress is conventionally expressed in terms of the friction factor and methods of finding the friction factor were discussed. The Reynolds analogy was first discussed. This solution really only applies to fluids with a Prandtl number of 1. A three-layer analogy solution which applies for all Prandtl numbers was then discussed. 

This chapter has been concerned with flows in wb ch the buoyancy forces that arise due to the temperature difference have an influence on the flow and heat transfer values despite the presence of a forced velocity. In extemai flows it was shown that the deviation of the heat transfer rate from that which would exist in purely forced convection was dependent on the ratio of the Grashof number to the square of the Reynolds number. It was also shown that in such flows the Nusselt number can often be expressed in terms of the Nusselt numbers that would exist under the same conditions in purely forced and purely free convective flows. It was also shown that in turbulent flows, the buoyancy forces can affect the turbulence structure as well as the momentum balance and that in turbulent flows the heat transfer rate can be decreased by the buoyancy forces in assisting flows whereas in laminar flows the buoyancy forces essentially always increase the heat transfer rate in assisting flow. Some consideration was also given to the effect of buoyancy forces on internal flows. 

The linear velocity (LV) or superficial velocity is an important engineering term because it relates to pressure drop and turbulence. This parameter is often increased in fixed bed reactors to enhance bulk mass transfer and heat transfer. 

Select the appropriate heat-transfer coefficient equation. Heat-transfer coefficients for fluids flowing inside helical coils can be calculated with modifications of the equations for straight tubes. The equations presented in Example 7.18 should be multiplied by the factor 1 + 3.5/1, /D,. where Di is the inside diameter and Dc is the diameter of the helix or coil. In addition, for laminar flow, the term (Dc/Dj)1/6 should be substituted for the term (L/Zl)1 3. The Reynolds number required for turbulent flow is 2100[1 + I2(/1,//1C)I/2. ... 

In general, the initiation of the precipitation process may result from the presence of particulate matter in the bulk water that seeds the crystallization. The process is usually termed heterogeneous nucleation. It is possible for homogeneous nucleation to occur when the nucleation is spontaneous. Once nucleation has occurred, crystals can grow, provided that the solution is supersaturated. Suitable nucleation points on the heat transfer surface facilitate deposit formation on the surface. In turbulent flow, it is possible that crystallites that are formed in the bulk fluid may be carried into regions, where they can redissolve. 

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