Turbulent thermal diffusivity

Cartesian coordinate vector (x, y, z) Molecular thermal diffusivity Turbulent thermal diffusivity Molecular kinematic viscosity Turbulent kinematic viscosity Karman constant Mass density See Eq. (26)... 

Turbulent Thermal Diffusivity Model Table 2.1 Model constants of Eq. (2.10) by different aufhcn ... 

Matrix of correction factor Dimensionless distance, m Turbulent thermal diffusivity, m"... 

The turbulent thermal diffusivity t is calculated by using the T — Zf two-equation model ... 

The turbulent thermal diffusivity at can also be calculated by using two-equation model as shown in Fig. 7.5, in which, similar to the turbulent mass diffusivity Dj, the t reaches almost steady condition after traveling a distance about 50-fold of the effective catalyst diameter from the entrance and decreases sharply afterward. 

If two vessels each containing completely mixed gas, one at temperature T, and the other at a temperature T2, are connected by a lagged non-conducting pipe in which there are no turbulent eddies (such as a capillary tube), then under steady state conditions, the rate of transfer of A by thermal diffusion and molecular diffusion must be equal and opposite, or. 

The large fluctuations in temperature and composition likely to be encountered in turbulence (B6) opens the possibility that the influence of these coupling effects may be even more pronounced than under the steady conditions rather close to equilibrium where Eq. (56) is strictly applicable. For this reason there exists the possibility that outside the laminar boundary layer the mutual interaction of material and thermal transfer upon the over-all transport behavior may be somewhat different from that indicated in Eq. (56). The foregoing thoughts are primarily suppositions but appear to be supported by some as yet unpublished experimental work on thermal diffusion in turbulent flow. Jeener and Thomaes (J3) have recently made some measurements on thermal diffusion in liquids. Drickamer and co-workers (G2, R4, R5, T2) studied such behavior in gases and in the critical region. 

Satterfield (S2, S3) carried out a number of interesting macroscopic studies of simultaneous thermal and material transfer. This work was done in connection with the thermal decomposition of hydrogen peroxide and yielded results indicating that for the relatively low level of turbulence experienced the thermal transport did not markedly influence the material transport. However, the results obtained deviated by 10 to 20 from the commonly accepted macroscopic methods of correlating heat and material transfer data. The final expression proposed by Satterfield (S3), neglecting the thermal diffusion effect (S19) in the boundary layer, was written as... 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

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. 

When the transport is considered without turbulence we have, in general, Dj- u is the cinematic viscosity for the momentum transport a = A,/(pCp) is the thermal diffusivity and D is the diffusion coefficient of species A. Whereas with turbulence we have, in general, Dj-, w, is the cinematic turbulence viscosity for the momentum transport a, =, /(pCp) is the thermal turbulence diffusivity and D t is the coefficient of turbulent diffusion of species A frequently = a = D t due to the hydrodynamic origin of the turbulence. 

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,... 

It is at = At/gc with the turbulent thermal conductivity or eddy diffusivity for heat transfer1 At (SI units W/K m). The total heat flux is... 

For KjpCpD — 1, the relation between concentration and temperature. (9.9), is independent of the nature of the flow, either laminar or turbulent. It applies to both the instantaneous and time-averaged concentration and temperature fields, but only in regions in which condensation has not yet occurred. When the equations of transport for the jet flow are reduced to the form used in turbulent flow, the molecular diffusivity and thermal diffusivity are usually neglected in comparison with the turbulent diffusivities. This is acceptable for studies of gross transport and the time-averaged composition and temperature. However, this frequently made assumption is not correct for molecular scale processes like nucleation and condensation, which depend locally on the molecular transport properties. 

When the same chemical compositions of the reactants are used for each type of flame, the chemical reaction rate is considered to be the same for each. However, the reaction surface area of the turbulent flame is increased because of the nature of eddies, and the overall reaction rate at the combustion wave appears to be much higher than that for the laminar flame. Furthermore, the heat transfer process from the unburned gas to the burned gas at the combustion wave is different because the thermophysical properties such as the thermal diffusivity are higher for the turbulent flame than for the laminar flame. Thus, the speed of a turbulent flame appears to be much higher than that of a laminar flame. 

In the above equations, H, fit, and fcv are eddy thermal diffusivity, eddy dif-fusivity, and eddy kinematic viscosity, respectively, all having the same dimensions (L2T ). It should be noted that these are not properties of fluid or system, because their values vary with the intensity of turbulence which depends on flow velocity, geometry of flow channel, and other factors. 

The condensed phase density p, specific heat C, thermal conductivity A c, and radiation absorption coefficient Ka are assumed to be constant. The species-A equation includes only advective transport and depletion of species-A (generation of species-B) by chemical reaction. The species-B balance equation is redundant in this binary system since the total mass equation, m = constant, has been included the mass fraction of B is 1-T. The energy equation includes advective transport, thermal diffusion, chemical reaction, and in-depth absorption of radiation. Species diffusion d Y/cbfl term) and mass/energy transport by turbulence or multi-phase advection (bubbling) which might potentially be important in a sufficiently thick liquid layer are neglected. The radiant flux term qr... 

Uniform Fluid Properties. Analyses of turbulent boundary layers experiencing surface transpiration employ a hierarchy of increasingly complex models of the turbulent transport mechanisms. Most of the analyses, supported by complementary experiments, have emphasized the transpiration of air into low-speed airstreams [110-112], Under these conditions, the fluid properties in the boundary layer are essentially constant, and the turbulent boundary layer can be described mathematically with Eqs. 6.170 and 6.179. In addition, when small quantities of a foreign species are introduced into the boundary layer for diagnostic purposes or by evaporation, the local foreign species concentration in the absence of thermal diffusion is given by... 

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