Mass dimensionless

Dimensions moles moles mass dimensionless mass... 

Yp/s Product mass produced per substrate mass Dimensionless 12.14... 

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

Vibrational anharmonicity constant Vibrational coordinates Internal coordinates Normal coordinates, dimensionless Mass adjusted Vibrational force constants "eAe A,s get Ri, r 0J, etc. Qr m-i ... 

Values of the mass-transfer coefficient k have been obtained for single drops rising (or falling) through a continuous immiscible Hquid phase. Extensive Hterature data have been summarized (40,42). The mass-transfer coefficient is often expressed in dimensionless form as the Sherwood number ... 

In this equation, represents the rate of energy dissipation per unit mass of fluid. In pulsed and reciprocating plate columns the dimensionless proportionahty constant K in equation 38 is on the order of 0.3. In stirred tanks, the proportionaUty constant has been reported as 0.024(1 + 2.5 h) in the holdup range 0 to 0.35 (67). The increase of drop si2e with holdup is attributed to the increasing tendency for coalescence between drops as the concentration of drops increases. A detailed survey of drop si2e correlations is given by the Hterature (65). 

In a quiescent fluid, the dimensionless mass-transfer coefficient, or the Nusselt number, djkj for a sphere is two. In fluidized beds the Nusselt... 

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

Dimensionless Quantities. Certain quantities, eg, refractive index and relative density (formerly specific gravity), are expressed by pure numbers. In these cases, the corresponding SI unit is the ratio of the same two SI units, which cancel each other, leaving a dimensionless unit. The SI unit of dimensionless quantities may be expressed as 1. Units for dimensionless quantities such as percent and parts per million (ppm) may also be used with SI in the latter case, it is important to indicate whether the parts per million are by volume or by mass. 

Density and Relative Density. Density is mass per unit volume and in SI is normally expressed as kilograms per cubic meter (density of water = 1000 kg/m or 1 g/cm ). The term specific gravity was formerly the accepted dimensionless value describing the ratio of the density of sohds and Hquids to the density of water at 4°C or for gases to the density of ak at standard conditions. The term specific gravity is being replaced by relative mass density, a more descriptive term. 

The result is shown in Figure 10, which is a plot of the dimensionless effectiveness factor as a function of the dimensionless Thiele modulus ( ), which is R.(k/Dwhere R is the radius of the catalyst particle and k is the reaction rate constant. The effectiveness factor is defined as the ratio of the rate of the reaction divided by the rate that would be observed in the absence of a mass transport influence. The effectiveness factor would be unity if the catalyst were nonporous. Therefore, the reaction rate is... 

Fig. 10. The Thiele plot accounting for the influence of intraparticle mass transport on rates of catalytic reaction. The dimensionless terms Tj and ( ) are the...

Lr = dimensionless refractoiy-wall loss. m = mass flow rate. n = refractive index. 

Chilton-Colbum factor for mass transfer, Eq. (5-291) Dimensionless Dimensionless... 

A/og Number of overall gas-pbase mass-transfer units Dimensionless Dimensionless... 

Sohd Catalysts Processes with solid catalysts are affected by diffusion of heat and mass (1) within the pores of the pellet, (2) between the fluid and the particle, and (3) axially and radially within the packed bed. Criteria in terms of various dimensionless groups have been developed to tell when these effects are appreciable. They are discussed by Mears (Ind. Eng. Chem. Proc. Des. Devel., 10, 541-547 [1971] Jnd. Eng. Chem. Fund., 15, 20-23 [1976]) and Satterfield (Heterogeneous Cataly.sls in Practice, McGraw-Hill, 1991, p. 491). 

Mass-transfer theory indicates that for trays of a given design the factors most hkely to inflnence E in absorption and stripping towers are the physical properties of the flnids and the dimensionless ratio Systems in which the mass transfer is gas-film-controlled may be expected to have plate efficiencies as high as 50 to 100 percent, whereas plate efficiencies as low as 1 percent have been reported for the absorption of gases of low sohibility (large m) into solvents of relatively high viscosity. 

Note that the group on the left side of Eq. (14-182) is dimensionless. When turbulence promoters are used at the inlet-gas seclion, an improvement in gas mass-transfer coefficient for absorption of water vapor by sulfuric acid was obsei ved by Greenewalt [Ind. Eng. Chem., 18, 1291 (1926)]. A falhug off of the rate of mass transfer below that indicated in Eq. (14-182) was obsei ved by Cogan and Cogan (thesis, Massachusetts Institute of Technology, 1932) when a cauTiiug zone preceded the gas inlet in ammonia absorption (Fig. 14-76). 

Two dimensionless variables play key roles in the analysis of single transition systems (and some multiple transition systems). These are the throughput parameter [see Eq. (16-129)] and the number of transfer units (see Table 16-13). The former is time made dimensionless so that it is equal to unity at the stoichiometric center of a breakthrough cui ve. The latter is, as in packed tower calculations, a measure of mass-transfer resistance. 

Separation Factor By analogy with the mass-action case and appropriate for both adsorption and ion exchange, a separation factor / can be defined based on dimensionless system variables [Eq. (16-10)] by... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

For other mechanisms, the particle-scale equation must be integrated. Equation (16-140) is used to advantage. For example, for external mass transfer acting alone, the dimensionless rate equation in Table 16-13 would be transformed into the ( — Ti, Ti) coordinate system and derivatives with respect to Ti discarded. Equation (16-138) is then used to replace cfwith /ifin the transformed equation. Furthermore, for this case there are assumed to be no gradients within the particles, so we have nf=nf. After making this substitution, the transformed equation can be rearranged to... 

Isocratic Elution In the simplest case, feed with concentration cf is apphed to the column for a time tp followed by the pure carrier fluid. Under trace conditions, for a hnear isotherm with external mass-transfer control, the linear driving force approximation or reaction kinetics (see Table 16-12), solution of Eq. (16-146) gives the following expression for the dimensionless solute concentration at the column outlet ... 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

The maximum value for G,p is G,, evaluated with zero pipe length. Denoting this value as G,, Fig. 26-68 plots the dimensionless mass... 

In this section we shall expand upon the problem of one-dimensional motion in a potential V x). Although it is a textbook example, we use here the less traditional Feynman path-integral formalism, the advantage of which is a possibility of straightforward extension to many dimensions. In the following portion of the theory we shall use dimensionless units, in which h = i,k = 1 and the particle has unit mass. 

Hh and Hg are the height of heat and mass transfer units, respectively, in multiples of the cell height, 1. H and M are the same in dimensionless units. 

Dimensional analysis leads to various dimensionless parameters, wliieli are based on the dimension s mass (M), length (L), and time T). Based on these elements, one ean obtain various independent parameters sueh as density (p), viseosity (/i), speed (A ), diameter ( )), and veloeity (V). The independent parameters lead to forming various dimensionless groups, whieh are used in fluid meehanies of turbomaehines. Reynolds number is the ratio of the inertia forees to the viseous forees... 

Thus, the larger the value of (K), the more the solute will be distributed in the stationary phase. (K) is a dimensionless constant and, in gas/liquid and liquidAiquid systems, (Xs) and (Xm) can be measured as mass of solute per unit volume of phase. In gas/solid and liquid/solid systems, (Xs) and (Xm) can be measured as mass of solute per unit mass of phase. 

In vertical downward flow as well as in upward and downward inclined flows, the flow patterns that can be observed are essentially similar to those described above, and the definitions used can be applied. Experimental data on flow patterns and the transition boundaries are usually mapped on a two dimensional plot. Two basic types of coordinates are generally used for this mapping - one that uses dimensional coordinates such as superficial velocities, mass superficial velocities, or momentum flux and another that uses dimensionless coordinates in which some kind of dimensionless groups are used as coordinates. The dimensional coordinates maps are inherently limited to the range of data and flow conditions under which the experiments were conducted. In spite of this limitation, it is widely used because of its simplicity and ease of use. Figure 24 provides an example of such a map. 

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