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Dimensional Analysis of the Mass Transfer Equation

DIMENSIONAL ANALYSIS OF THE EQUATIONS OF CHANGE FOR FLUID DYNAMICS WITHIN THE MASS TRANSFER BOUNDARY LAYER... [Pg.361]

Step 1. Use the integral form of a linear-least squares analysis to determine the best value of the pseudo-first-order kinetic rate constant, i, that will linearize the reaction term in the mass transfer equation. It is necessary to apply the Leibnitz rule for differentiating a one-dimensional integral with constant limits to the following expression ... [Pg.455]

The radial variable r is dimensionalized to isolate the Damkohler number in the mass balance. It is important to emphasize that dimensional analysis on the radial coordinate must be performed after implementing the canonical transformation from Ca to iJia- If the surface area factors of and 1/r are written in terms of as defined by equation (13-9), prior to introducing the canonical transformation given by equation (13-4), then the mass transfer problem external to the spherical interface retains variable coefficients. If diffusion and chemical reaction are considered inside the gas bubble, then the order in which the canonical transformation and dimensional analysis are performed is unimportant. Hence,... [Pg.371]

Equation 6.10 is a definition of the Reynolds number based on power dissipation using a velocity term derived from dimensional arguments. This prompted Middleman (1965) to comment that derivation of Equation 6.11 using Kolmogorov s theory is a sophisticated form of dimensional analysis. Even with this oversimplification. Equation 6.10 still needs to be modified for gas-liquid mass transfer for which has a wide variation, and hence, an average is difficult to define. In view of this, most investigators resorted to correlating the volumetric mass transfer coefficient,. The correlations proposed for stirred tank reactors were therefore of the form (Hickman 1988 Middleton 2000)... [Pg.112]

To understand the mechanism further, a microscopic investigation on the moisture content in clay is required. The osmotic suction potential was introduced as the driving force of moisture movement, described in Section 31.3, and was successively applied to the prediction of moisture movement in wet clay [15,16]. The theoretical analysis on the two-dimensional moisture transfer of cylindrical clay was performed taking into account the effects of both osmotic suction and strain-stress caused by the shrinkage [17,18]. However, only the transient mass-transfer equation was analyzed, assuming a constant drying rate on the external surface of the... [Pg.753]

The similarity theory and the dimensional analysis give the following types of equations for the partial mass transfer coefficients for gas phase controlled processes... [Pg.79]

Clearly, the maximum degree of simplification of the problem is achieved by using the greatest possible number of fundamentals since each yields a simultaneous equation of its own. In certain problems, force may be used as a fundamental in addition to mass, length, and time, provided that at no stage in the problem is force defined in terms of mass and acceleration. In heat transfer problems, temperature is usually an additional fundamental, and heat can also be used as a fundamental provided it is not defined in terms of mass and temperature and provided that the equivalence of mechanical and thermal energy is not utilised. Considerable experience is needed in the proper use of dimensional analysis, and its application in a number of areas of fluid flow and heat transfer is seen in the relevant chapters of this Volume. [Pg.14]

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine... [Pg.428]

In fact, the advantage of these combinations of numbers obtained by making differential equations dimensionless, over those combinations delivered by dimensional analysis, is that they characterize certain types of mass and heat transfer, respec-... [Pg.181]

We consider the following physical situation. A first-order exothermic reaction runs on a smooth, nonporous catalytic element having a length L (thread, fiber, tube). Assume that there is no temperature distribution at the element cross-section, thus reducing the analysis to a one-dimensional case. Assume also that the reaction runs under conditions of transversal flow, and the heat and mass transfer between the catalyst surface and the bulk-flow are described by the effective coefficients a and respectively. Under these assumptions, the model can be written in the form of two equations, one describing the heat balance in the solid catalyst phase, and another the reactant balance in the gaseous phase of a certain characteristic layer adjacent to the catalyst surface ... [Pg.554]

FLUIDS FLOWING NORMALLY TO A SINGLE TUBE. The variables affecting the coefficient of heat transfer to a fluid in forced convection outside a tube are D , the outside diameter of the tube Cp, /r, and k, the specific heat at constant pressure, the viscosity, and the thermal conductivity, respectively, of the fluid and G, the mass velocity of the fluid approaching the tube, Dimensional analysis gives, then, an equation of the type of Eq, (12,27) ... [Pg.360]

In view of the complexity of mass transfer in actual equipment, fundamental equations for mass transfer in actual equipment are rarely available, and empirical methods, guided by dimensional analysis and by semitheoretical analogies, are relied upon to give workable equations. The approach to the problem has been made in several steps in the following manner. [Pg.663]

In cases where hydrodynamic dispersion and the corresponding broadening of residence-time distributions deteriorate the performance of a process, the question arises as to which channel design minimizes dispersion. Already from the analysis of Taylor and Aris it becomes clear that an enhanced mass transfer perpendicular to the main flow direction reduces the broadening of concentration tracers. Such a mass-transfer enhancement can be achieved by the secondary fiow occurring in a curved channel. This aspect was investigated by Daskopoulos and Lenhoff [78] for ducts of circular cross section. They assumed the diameter of the duct to be small compared to the radius of curvature and solved the convection-diffusion equation for the concentration field numerically. More specifically, a two-dimensional problem defined on the cross-sectional plane of the duct was solved based on a combination of a Fourier series expansion and an expansion in Chebyshev polynomials. The solution is of the general form... [Pg.65]

Kinematic similarity is concerned with the motion of phases within a system and the forces inducing that motion. For example, in the formation of boundary layers during flow past flat plates and during forced convection in regularly shaped channels, there are usually three dominant forces pressure, inertia, and viscous forces. If corresponding points in two different-sized cells show at corresponding times identical ratios of fluid velocity, the two units are said to be kinematically similar and heat and mass transfer coefficients will bear a simple relation in the two cells. It can be shown by means of dimensional analysis that for a closed system under forced convection the equation of motion for a fluid reduces to a function of Re, the Reynolds number, which we have met in Chapter 2. To preserve kinematic similarity under those circumstances, Reynolds numbers in the two cells must be identical. [Pg.200]

Eq.(79) is the second Pick s low. Its structure is the same as that of the differential equation of the convective h t transfer (in case of a st y state process Eq. (56)). This gives the possibility, as shown later, to calculate the heat transfer processes by means of experimental data or equations for mass transfer. The basic methods for these calculations are the similarily theory and the dimensional analysis. That is why before considering the theory of mass transfor processes, we present these important methods largely used in chemical engineering and in particular In the area of packed columns. [Pg.25]


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