Peclet number, equation

From this discussion of parameter evaluation, it can be seen that more research must be done on the prediction of the flow patterns in liquid-liquid systems and on the development of methods for calculating the resulting holdups, pressure drop, interfacial area, and drop size. Future heat-transfer studies must be based on an understanding of the fluid mechanics so that more accurate correlations can be formulated for evaluating the interfacial and wall heat-transfer coefficients and the Peclet numbers. Equations (30) should provide a basis for analyzing the heat-transfer processes in Regime IV. 

As the fluid was air, Pr = 0.72 the bed voidage e was taken as 0.4. The ratios kj-g/kg and kp/kg were related using the formulas of Zehner and Schliinder (27). Preliminary sensitivity tests showed that the parameter Nufs had very little effect on the profiles in the presence of reaction this parameter would be more important. The correlations for the fluid-phase Peclet numbers, equations (18) and (19), were obtained by analogy from reported mass transfer work. 

Diffusion in liquids can be slow, and this makes the problems hard to solve, as shown below. The best way to learn how important diffusion is for any problem is to calculate the Peclet number. Equation (11.3) is made nondimensional using the techniques illustrated in Eq. (10.42). Let c (x, y, f ) be the nondimensional concentration, and c = Csc (x, y, f ). Equation (11.3) for a constant diffusivity becomes ... 

The dimensionless parameter in Equation 6.109 is the Peclet number (Equation 6.108). The remaining parameter in Equation 6.110 determines the ratio of effective mass transport to convection, vhich is defined by the modified (effective) Stanton number (Steff) ... 

Therefore, by calculating the variance, it can be seen the deviation from ideal behavior of the reactor using axial dispersion by equation that relates the variance with Peclet number (Equation 15.25) ... 

Equation 74 is shown graphically ia Figure 19a for a given set of conditions. Curves such as these cannot be directly used for design, however, because the Peclet number contains the tower height as a characteristic dimension. Therefore, new Peclet numbers are defined containing as the characteristic length. These relate to the conventional Pe as... 

Peclet number specifically defined for phase /, i = x or jy, see equation 77... 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

The value of n is the only parameter in this equation. Several procedures can be used to find its value when the RTD is known experiment or calculation from the variance, as in /i = 1/C (t ) = 1/ t C t), or from a suitable loglog plot or the peak of the curve as explained for the CSTR battery model. The Peclet number for dispersion is also related to n, and may be obtainable from correlations of operating variables. 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

The solution of this partial differential equation is recorded in the literature (Otake and Kunigata, Kngaku Kogaku, 22, 144 [1958]). The plots of E(t ) against t are bell-shaped, resembling the corresponding Erlang plots. A relation is cited later between the Peclet number,... 

Estimate the Peclet number from Equation 8-141. If the value of Npg is 10 or greater, accept it. Otherwise, use a refinement of the theory, which accounts for the boundary conditions at the outlet, or formulate another model. 

Equations 8-148 and 8-149 give the fraction unreacted C /C o for a first order reaction in a closed axial dispersion system. The solution contains the two dimensionless parameters, Np and kf. The Peclet number controls the level of mixing in the system. If Np —> 0 (either small u or large [), diffusion becomes so important that the system acts as a perfect mixer. Therefore,... 

N = L/d, or that each row of pellets amounts to a mixing cell. This is intuitively reasonable. However, the parallel dispersion for heat has a Peclet number equal to 0.5, which would argue that four rows of pellets should be considered to be a mixing cell. The heat balance equation for cell i in a cascade of N cells is... 

Chapter 11 consists of following Sect. 11.2 deals with the pattern of capillary flow in a heated micro-channel with phase change at the meniscus. The perturbed equations and conditions on the interface are presented in Sect. 11.3. Section 11.4 contains the results of the investigation on the stability of capillary flow at a very small Peclet number. The effect of capillary pressure and heat flux oscillations on the stability of the flow is considered in Sect. 11.5. Section 11.6 deals with the study of capillary flow at a moderate Peclet number. 

Perturbed Energy Equation for Small Peclet Number... 

The form of the solution of the dispersion equation (11.61) depends on the sign of the determinant D = q + Pl, i.e., on the values of the characteristic parameters g and P. The latter are determined by the physical properties of the liquid and its vapor, as well as the values of the Peclet number. This allows us to use g and P as some general characteristics of the problem considered here. 

The perturbed energy equation for moderate Peclet number has (at = 0) the following form ... 

Via a passive scalar method [6] where or, denotes the volume fraction of the i-th phase, while T, represents the diffusivity coefiBcient of the tracer in the i-th phase. The transient form of the scalar transport equation was utilized to track the pulse of tracer through the computational domain. The exit age distribution was evaluated from the normalized concentration curve obtained via measurements at the reactor outlet at 1 second intervals. This was subsequently used to determine the mean residence time, tm and Peclet number, Pe [7]. 

Parameter estimation 112 Partial differential equation 578 Partial differentials 154 Peclet number 243, 579 Penetration distance 654 Perfect mixing 159 Perfectly mixed 142... 

These substitutions replace the eight dimensional parameters in the original equations by the four nondimensional parameters above. The parameter Pe is the Peclet number (37) and shows the relative importance of convection compared to diffusion. The advantage of this formulation becomes obvious when typical parameter values are substituted into the equations. 

In practice, the Peclet number can always be ignored in the diffusion-convection equation. It can also be ignored in the root boundary condition unless C > X/Pc or A, < Pe. Inspection of the table of standard parameter values (Table 2) shows that this is never the case for realistic soil and root conditions. Inspection of Table 2 also reveals that the term relating to nutrient efflux, e, can also be ignored because e < Pe [Pg.343]

Because the term r(CA T) is exponentially dependent on T and can be nonlinear as well, a numerical solution or piecewise linearization must be used. To simplify the numerical manipulations, equations in Table IX are normalized by = z/L, r = ut/L, and jc = 1 - C,/(C,)0, where i is normally S02. y also is a normalized quantity. The Peclet numbers for mass and heat are written PeM = 2Rpu/D) and PeH = 2Rpcpul t for a spherical particle. They are also written in terms of bed length as Bodenstein numbers. It is... 

A Peclet number of zero indicates perfect mixing and a value of oo indicates plug flow. For bubble-cap and sieve plates the eddy diffusivity can be estimated from the equation ... 

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