Prandtl number dependence

The Sherwood number can be correlated with the Reynolds number just as the Nusselt number is. The correlations depend on the Schmidt number, which is comparable to the Prandtl-number dependence of the Nusselt number. 

TABLE 4.1 Prandtl Number Dependence of Various Coefficients... 

It can be generalized somewhat more by accounting for the Prandtl number dependence power. The correlation is based on a pseudo-single-phase model. It follows from the correlation that the gas approaches the temperature of the solid in the very first centimeters of the bed. This is confirmed by a recent correlation of Balakrishnan and Pei [20]... 

The Prandtl number depends on the thermophysical properties of the fluid only. Typical values of the Prandtl number are 0.001-0.03 for liquid metals, 0.2-1 for gases, 1-10 for water. 

The Prandtl number depends on the thermophysical properties of the fluid only. Typical values of the Prandtl number are 0.001-0.03 for liquid metals, 0.2-1 for gases, 1-10 for water, 5-50 for organic liquids and 50 - 2000 for oils. The Prandlt number depends on the bulk temperature of the fluid since the viscosity is a strong function of temperature for this reason, especially in very narrow microchannels in which the viscous heating effects are not negligible (see viscous heating and viscous dissipation), the Prandlt number cannot be considered as a constant along the channel. 

Sahoo, G., Perlekax, R, Panditn, R. Systematics of the magnetic-Prandtl-number dependence of homogeneous, isotropic magnetohydrodynamic turbulence. New J. Phys. 13, 1367-2630 (2011)... 

However, the heat transfer coefficient was averaged over the cel Vs height and the Prandtl Number dependence was doubled to reflect this averaging process. Radiation was Included using the optically thick approximation. 

The average Nusselt number is not very sensitive to changes in gas velocity and Reynolds number, certainly no more than (Re)I/3. The Sherwood number can be calculated with the same formula as the Nusselt number, with the substitution of the Schmidt number for the Prandtl number. While the Prandtl number of nearly all gases at all temperatures is 0.7 the Schmidt number for various molecules in air does depend on temperature and molecular type, having the value of 0.23 for H2, 0.81 for CO, and 1.60 for benzene. 

For an incompressible fluid, the density variation with temperature is negligible compared to the viscosity variation. Hence, the viscosity variation is a function of temperature only and can be a cause of radical transformation of flow and transition from stable flow to the oscillatory regime. The critical Reynolds number also depends significantly on the specific heat, Prandtl number and micro-channel radius. For flow of high-viscosity fluids in micro-channels of tq < 10 m the critical Reynolds number is less than 2,300. In this case the oscillatory regime occurs at values of Re < 2,300. 

Thus at small Pol the growth rate of the oscillations is negative and the capillary flow is stable. The absolute value of sharply increases with a decrease of the capillary tube diameter. It also depends on the thermal diffusivity of the liquid and the vapor, as well as on the value of the Prandtl number. 

Convective heat transfer to fluid inside circular tubes depends on three dimensionless groups the Reynolds number. Re = pdtu/ii, the Prandtl number, Pr = Cpiilk where k is the thermal conductivity, and the length-to-diameter ratio, L/D. These groups can be combined into the Graetz number, Gz = RePr4/L. The most commonly used correlations for the inside heat transfer coefficient are... 

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 a supersonic gas flow, the convective heat transfer coefficient is not only a function of the Reynolds and Prandtl numbers, but also depends on the droplet surface temperature and the Mach number (compressibility of gas). 154 156 However, the effects of the surface temperature and the Mach number may be substantially eliminated if all properties are evaluated at a film temperature defined in Ref. 623. Thus, the convective heat transfer coefficient may still be estimated using the experimental correlation proposed by Ranz and Marshall 505 with appropriate modifications to account for various effects such as turbulence,[587] droplet oscillation and distortion,[5851 and droplet vaporization and mass transfer. 555 It has been demonstrated 1561 that using the modified Newton s law of cooling and evaluating the heat transfer coefficient at the film temperature allow numerical calculations of droplet cooling and solidification histories in both subsonic and supersonic gas flows in the spray. 

Pr (Cp/Lt/X) is constant for most gases over wide ranges of temperature and pressure and this fact may be used to estimate the thermal conductivity at high temperatures. The Prandtl number is between 0.65 and 1.0, depending on the molecular complexity of the gas. 

The gas film coefficient is dependent on turbulence in the boundary layer over the water body. Table 4.1 provides Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1,000 and 10, respectively, results in the entire concentration boundary layer being inside of the laminar sublayer of the momentum boundary layer. In air, both the Schmidt and Prandtl numbers are on the order of 1. This means that the analogy between momentum, heat, and mass transport is more precise for air than for water, and the techniques apphed to determine momentum transport away from an interface may be more applicable to heat and mass transport in air than they are to the liquid side of the interface. 

This Prandtl-number expression is independent of temperature, since both the viscosity and conductivity expressions have the same temperature dependence. For monatomic gases, y as 5/3, so the expression shows Pr 0.67, which is close to that observed experimentally. For diatomic gases with y = 1.4, the expression yields Pr = 0.74, which is a bit high. 

In fact both the Prandtl number and the heat capacity are temperature-dependent. For gases, however, the dependency is relatively weak, especially for the Prandtl number. The heat capacity cp of air increases by about 30% between 300 K and 2000 K. Because of these temperature dependencies, it may be anticipated (e.g., from Eq. 3.144) that the viscosity and the thermal conductivity generally show slightly different temperature dependencies. 

The equation is nearly analogous to the thermal energy equation with the Schmidt number replacing the Prandtl number, although the density dependence is different. The Schmidt number and Reynolds numbers are defined as... 

There is a natural draw rate for a rotating disk that depends on the rotation rate. Both the radial velocity and the circumferential velocity vanish outside the viscous boundary layer. The only parameter in the equations is the Prandtl number in the energy equation. Clearly, there is a very large effect of Prandtl number on the temperature profile and heat transfer at the surface. For constant properties, however, the energy-equation solution does not affect the velocity distributions. For problems including chemistry and complex transport, there is still a natural draw rate for a given rotation rate. However, the actual inlet velocity depends on the particular flow circumstances—there is no universal correlation. 

Discuss how the surface heat-transfer coefficient h depends on the strain rate and the Prandtl number. 

The diffusion coefficient as defined by Fick s law, Eqn. (3.4-3), is a molecular parameter and is usually reported as an infinite-dilution, binary-diffusion coefficient. In mass-transfer work, it appears in the Schmidt- and in the Sherwood numbers. These two quantities, Sc and Sh, are strongly affected by pressure and whether the conditions are near the critical state of the solvent or not. As we saw before, the Schmidt and Prandtl numbers theoretically take large values as the critical point of the solvent is approached. Mass-transfer in high-pressure operations is done by extraction or leaching with a dense gas, neat or modified with an entrainer. In dense-gas extraction, the fluid of choice is carbon dioxide, hence many diffusional data relate to carbon dioxide at conditions above its critical point (73.8 bar, 31°C) In general, the order of magnitude of the diffusivity depends on the type of solvent in which diffusion occurs. Middleman [18] reports some of the following data for diffusion. 

The functions f5-f8 and ip5-ip8 may be determined from a system of two ordinary differential equations. In addition, the form of the functions /7, /8, Prandtl number. It, is interesting that along a laminar ascending flow, in both the plane and radially-symmetric cases, the Reynolds number (defined as ub/v) increases as x3/5 and x1/2, respectively.1 Consequently, at a sufficient height disruption of the laminar flow and transition to turbulent flow should take place. 

These six quantities contain four base dimensions [L, T, , H], wherein H means the amount of heat with calorie as measuring unit. According to the pi-theorem, a dependence between two pi-numbers will result. Rayleigh obtained the following two pi-numbers which are today named The Nusselt number Nu and the Peclet number Pe, the latter being the product of Reynolds and Prandtl numbers, Pe = RePr ... 

Thus, the vorticity and temperature fields are governed by equations having the same basic form. When Pr is equal to 1, the equations have exactly the same form. Even in this case, however, the vorticity and temperature fields will not be identical because the boundary conditions on the two fields at the surface will not in general be identical. However, there will obviously be similarities between the two fields. Vorticity is generated in the flow by the action of viscosity due to the presence of the surface. The temperature differences arise in the flow because the surface is at a temperature which is different from the flowing fluid. Thus, Eq. (2.76) essentially describes the rate at which viscous effects spread into the fluid while Eq. (2.77) describes the rate at which the effects of the temperature changes at the surface spread into the fluid. It will be seen that the relative rates of spread depend on the value of the Prandtl number. 

The flows over a series of bodies of the same geometrical shape will be similar, i.e., will differ from each other only in scale, if the Reynolds and Prandtl numbers are the same in all the flows. From this it follows that the Nusselt number in forced convection will depend only on the Reynolds and Prandtl numbers. 

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