Turbulent Prandtl Number Model

Turbulent Thermal Dififusivity Model 2.2.1 Turbulent Prandtl Number Model... 

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). 

In applying the turbulence kinetic energy model it is common to assume that the turbulent Prandtl number, Prr, is constant. 

Cebeci, T., A model for Eddy-conductivity and Turbulent Prandtl Number, Rep. MDC-j0747/01. McDonnell Douglas Co., 1970. 

The modeled transport equations for z differ mainly in the diffusion and secondary source term. Launder and Spalding (1972) and Chambers and Wilcox (1977) discuss the differences and similarities in more detail. The variable, z = e is generally preferred since it does not require a secondary source, and a simple gradient diffusion hypothesis is fairly good for the diffusion (Launder and Spalding, 1974 Rodi, 1984). The turbulent Prandtl number for s has a reasonable value of 1.3, which fits the experimental data for the spread of various quantities at locations far from the walls, without modification of any constants. Because of these factors, the k-s model of turbulence has been the most extensively studied and used and is recommended as a baseline model for typical internal flows encountered by reactor engineers. 

The simplest way to close equations (3.1.37) is to use the hypothesis that the turbulent Prandtl number for the examined process is a constant quantity. Then it readily follows from Eq. (3.1.39) that the turbulent diffusion coefficient is proportional to the turbulent viscosity >t = i /Pr,. By using the expression for vx borrowed from the corresponding hydrodynamic model, one can obtain the desired value of Dt. In particular, following Prandtl s or von Karman s model, one can use formula (1.1.21) or (1.1.22) for vx. 

Eddy Diffusivity Models. The mean velocity data described in the previous section provide the bases for evaluating the eddy diffusivity for momentum (eddy viscosity) in heat transfer analyses of turbulent boundary layers. These analyses also require values of the turbulent Prandtl number for use with the eddy viscosity to define the eddy diffusivity of heat. The turbulent Prandtl number is usually treated as a constant that is determined from comparisons of predicted results with experimental heat transfer data. 

The equations solved formulate fhe principles of conservation of mass, momentum in the X, y, and z directions, thermal energy, and mass of secondary medium (cyclohexane). Turbulence modeling is performed by using a standard two-equation k-e) turbulence model with near-wall grid node wall functions used to evaluate surface friction and surface heat transfer. The effects of turbulence are represented via a turbulent viscosity and thermal dif-fusivity where these quantities are linked by the turbulent Prandtl number, which takes a constant value of 0.9. 

Prandtl numbers. Thus, the coefficient near the pseudocriticai temperature, where the Prandtl number becomes large, may be smaller. The ideal coefficient calculated by the Jones-Launder k-e model at the pseudocriticai temperature is plotted in Fig. 2.4. It is calculated by fixing the thermophysical properties at the pseudocriticai temperature. This value is higher than that shown by the curve of 2.33 x 10 W m . When the Jones-Launder k-e model is used, it is known that the wall shear stress is relatively large and the heat transfer coefficient is also large with a constant turbulent Prandtl number. As indicated by Jackson and Hall [4], the heat transfer coefficient is the maximum when the heat flux is zero and it monotonically decreases as the heat flux increases. The calculation supports their assertion. 

The turbulent viscosity i/j is determined using the WALE model [330], similar to the Smagorinski model, but with an improved behavior near solid boundaries. Similarly, a subgrid-scale diffusive flux vector Jfor species Jk = p (uYfc — uYfc) and a subgrid-scale heat flux vector if = p(uE — uE) appear and are modeled following the same expressions as in section 10.1, using filtered quantities and introducing a turbulent diffusivity = Pt/Sc], and a thermal diffusivity Aj = ptCp/Pr. The turbulent Schmidt and Prandtl numbers are fixed to 1 and 0.9 respectively. 

The influence of a wall on the turbulent transport of scalar (species or enthalpy) at the wall can also be modeled using the wall function approach, similar to that described earlier for modeling momentum transport at the wall. It must be noted that the thermal or mass transfer boundary layer will, in general, be of different thickness than the momentum boundary layer and may change from fluid to fluid. For example, the thermal boundary layer of a high Prandtl number fluid (e.g. oil) is much less than its momentum boundary layer. The wall functions for the enthalpy equations in the form of temperature T can be written as ... 

The first model suggested for these dimensionless groups is named the Reynolds analogy. Reyuolds suggested that in fully developed turbulent flow heat, mass and momentum are transported as a result of the same eddy motion mechanisms, thus both the turbulent Prandtl and Schmidt numbers are assumed equal to unity ... 

Model Description. The chief characteristic of turbulent heat transfer is that (for a given Prandtl number and Rayleigh number) the heat transfer at a point on the surface depends only on the local surface angle (Fig. 4.5), and is independent of how far the point is from the leading edge. It follows that... 

Equation for C,. A number of experiments at different Prandtl numbers (mostly on tilted plates) have been carried out that permit the function C,(Q to be modeled. Observation has also revealed that there are two patterns of turbulent flow detached and attached. Attached flow, where the flow sticks to the body contour, is best exemplified by the flow on a vertical plate, and the C,(Q, = C, (90°) applying in this situation is denoted Cv,. Detached flow, where turbulent eddies rise away from the heated surface, is best exemplified by the flow on a horizontal upward-facing (heated) plate, and the C,(Q = C,(0°) applying in this situation is denoted Cf. The first step in establishing the C,(Q function has been to model how Cf and CJ depend on the Prandtl number the equations for these quantities (justified later) that have been found to best fit currently available data are... 

The first analytical study to predict the performance of tubes with straight inner fins for turbulent airflow was conducted by Patankar et al. [118]. The mixing length in the turbulence model was set up so that just one constant was required from experimental data. Expansion of analytical efforts to fluids of higher Prandtl number, tubes with practical contours, and tubes with spiraling fins is still desirable. It would be particularly significant if the analysis could predict with a reasonable expenditure of computer time the optimum fin parameters for a specified fluid, flow rate, etc. 

Under the above mentioned assumptions, the global mass conservation equation does not modify its structure in the regime of turbulent flows. However the chemical species conservation equations and the energy equations, in the framework of k-e models, make use of the "turbulent" Schmidt and Prandtl numbers. 

Generally speaking, the conventional numerical analysis with a k-e turbulence model and accurate treatment of thermophysical properties can successfully explain the unusual heat transfer phenomena of supercritical water. Heat transfer deterioration occurs due to two mechanisms depending on the flow rate. When the flow rate is large, viscosity increases locally near the wall by heating. This makes the viscous sublayer thicker and the Prandtl number smaller. Both effects reduce the heat transfer. When the flow rate is small, buoyancy force accelerates the flow velocity near the wall. This makes the flow velocity distribution flat and generation of turbulence energy is reduced. This type of heat transfer deterioration appears at the boundary between forced and natural convection. As the heat flux increases above the deterioration heat flux, a violent oscillation of wall temperature is observed. It is explained by the unstable characteristics of the steep boundary layer of temperature. 

These models are usually categorized according to the number of supplementary partial differential transport equations which must be solved to supply the modeling parameters. The so-called zero-equation models do not use any differential equation to describe the turbulent quantities. The best known example is the Prandtl (19) mixing length hypothesis ... 

Despite the fact that equation (3.37) is applicable to all kinds of time-independent fluids, numerous workers have presented expressions for turbulent flow friction factors for specific fluid models. For instance, Tomita [1959] applied the concept of the Prandtl mixing length and put forward modified definitions of the friction factor and Reynolds number for the turbulent flow of Bingham Plastic fluids in smooth pipes so that the Nikuradse equation, i.e. equation (3.37) with n = 1, could be used. Though he tested the applicability of his method using his own data in the range 2000 < Reg(l — 4>f 3 — )< 10, the validity of this approach has not been established using independent experimental data. 

The Prandtl mixing length model, as well as the k-e model, coupled with (8) and (5), have given quite good results predicting turbulent diffusion fluxes is a number of cases however quite clear discrepancies of (5) have been emphasized in other important cases, and other models can now be proposed. For more details concerning turbulence models, a good basic book is the one of Tennekes and Lumley [1]. 

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