Transport dimensionless

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...

The dimensionless numbers in tlris equation are the Reynolds, Schmidt and the Sherwood number, A/ sh. which is defined by this equation. Dy/g is the diffusion coefficient of the metal-transporting vapour species in the flowing gas. The Reynolds and Schmidt numbers are defined by tire equations... 

All criteria proposed here are constructed such that if absolutely no gradient of a particular type exists, then the value of the corresponding criterion is zero. For fast catalytic processes this is not reasonable to expect and therefore a value judgment must be made for how much deviation from zero can be ignored. For the dimensionless expressions the Damkdhler numbers are used as these are applied to each particular condition. The approach is that the Damkdhler numbers can be calculated from known system values, which are related to the unknown driving forces for the transport processes. 

Dimensionless groups for a proeess model ean be easily obtained by inspeetion from Table 13-2. Eaeh of the three transport balanees is shown (in veetor/tensor notation) term-by-term under the deseription of the physieal meanings of the respeetive terms. The table shows how various well-known dimensionless groups are derived and gives the physieal interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

Experiments on the scale of 1 to 1 are often used to study the local ventilation around an operator s workplace. Tracer gas is used to simulate the contaminant transport, and a high concentration level of the model tracer gas makes it possible to work with a convenient level of concentration for the measurements. Figure 12.31 shows an enclosure with an emission source S and a laboratory. setup with a model source 5,. The dimensionless concentration c/cg is... 

Using the mathematical technique of dimensionless group analysis, the rate of mass transport (/ m) in terms of moles per unit area per unit time can be shown to be a function of these variables, which when grouped together can be related to the rate by a power term. For many systems under laminar flow conditions it has been shown that the following relationship holds ... 

In the universal velocity profile a dimensionless velocity + is plotted against lnyf, where y+ is a dimensionless distance from the surface. For the region where eddy transport dominates (eddy kinematic viscosity kinematic viscosity), the ratio of the mixing length (Ag) to the distance (>>) from the surface may be taken as approximately constant and equal to 0.4. Obtain an expression for d +/dy+ in terms of y+. 

The ratio of the observed reaction rate to the rate in the absence of intraparticle mass and heat transfer resistance is defined as the elFectiveness factor. When the effectiveness factor is ignored, simulation results for catalytic reactors can be inaccurate. Since it is used extensively for simulation of large reaction systems, its fast computation is required to accelerate the simulation time and enhance the simulation accuracy. This problem is to solve the dimensionless equation describing the mass transport of the key component in a porous catalyst[l,2]... 

The effect of transport limitations can conveniently be evaluated by considering the spherical catalyst particle shown in Fig. 5.32. We will introduce a dimensionless quantity called the Thiele diffusion modulus (Og) [W. Thiele Ind. Eng. Chem. 31... 

Dimensionless numbers (Reynolds number = udip/jj., Nusselt number = hd/K, Schmidt number = c, oA, etc.) are the measures of similarity. Many correlations between them (known also as scale-up correlations) have been established. The correlations are used for calculations of effective (mass- and heat-) transport coefficients, interfacial areas, power consumption, etc. 

In order to characterize the erosion properties of specific systems, the Thiele modulus 0 has been used to describe those systems where chemical reaction and transport are both important. By making the above Equations (1 - 4) dimensionless, several Thiele moduli are noted. The one which describes the transport of water compared to its consumption by chemical reaction is... 

Two examples will now be given of solution of the convective diffusion problem, transport to a rotating disk as a stationary case and transport to a growing sphere as a transient case. Finally, an engineering approach will be mentioned in which the solution is expressed as a function of dimensionless quantities characterizing the properties of the system. 

The original seven variables in this problem can now be replaced by an equivalent set of four dimensionless groups of variables. For example, if it is desired to determine the driving force required to transport a given fluid at a given rate through a given pipe, the relation could be represented as... 

In this case, the flow rate is to be determined when a given fluid is transported in a given pipe with a known net driving force (e.g., pump head, pressure head, and/or hydrostatic head). The same total variables are involved, and hence the dimensionless variables are the same and are related in the same way as for the unknown driving force problems. The main difference is that now the unknown (Q) appears in two of the dimensionless variables (/ and 7VRe), which requires a different solution strategy. 

In this problem, it is desired to determine the size of the pipe (D) that will transport a given fluid (Newtonian or non-Newtonian) at a given flow rate (Q) over a given distance (L) with a given driving force (DF). Because the unknown (D) appears in each of the dimensionless variables, it is appropriate to regroup these variables in a more convenient form for this problem. 

Molerus (1993) developed a state diagram that shows a correlation between these dimensionless groups based on an extremely wide range of data covering 25 < D < 315 mm, 12 < d < 5200/am, and 1270 < ps < 5250 kg/m3 for both hydraulic and pneumatic transport. This state diagram is shown in Fig. 15-3 in the form... 

The dimensionless group Pep is essentially the ratio of the rate of convective transport to the rate of diffusive transport. Similarly, Nr describes the relative importance of radioactive decay to convective flow as a method of removing radon from the soil pores. In the case of Pep >>1/ diffusion can be neglected and the first term in equation (1) drops out. If in addition Nr >>1, then radioactive decay can be neglected as a removal term. If Pep 1, then diffusive radon migration dominates, and the second term in equation (1) can be neglected. 

On the submicron scale, the current distribution is determined by the diffusive transport of metal ion and additives under the influence of local conditions at the interface. Transport of additives in solution may be non-locally controlled if they are consumed at a mass-transfer limited rate at the deposit surface. The diffusion of additives in solution must then be solved simultaneously with the flux of reactive ion. Diffusive transport of inhibitors forms the basis for leveling [144-147] where a diffusion-limited inhibitor reduces the current density on protrusions. West has treated the theory of filling based on leveling alone [148], In his model, the controlling dimensionless groups are equivalent to and D divided by the trench aspect ratio. They determine the ranges of concentration within which filling can be achieved. 

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