Using Dimensionless Criteria

When a liquid warms up, its density decreases, which results in buoyancy and an ascendant flow is induced. Thus, a reactive liquid will flow upwards in the center of a container and flow downwards at the walls, where it cools this flow is called natural convection. Thus, at the wall, heat exchange may occur to a certain degree. This situation may correspond to a stirred tank reactor after loss of agitation. The exact mathematical description requires the simultaneous solution of heat and impulse transfer equations. Nevertheless, it is possible to use a simplified approach based on physical similitude. The mode of heat transfer within a fluid can be characterized by a dimensionless criterion, the Rayleigh number (Ra). As the Reynolds number does for forced convection, the Rayleigh number characterizes the flow regime in natural convection ... 

A dimensionless criterion, the Biot number, is often used in transient heat transfer problems by comparing the heat transfer resistance within the body with... 

To link the dimensionless model temperature T to that of the experimental study, we use the criterion that the critical solution temperature of the experimental system, is to be matched by that of our model mixture, T. ... 

The nature of tbe rate-determining step can be predicted by use of the simple dimensionless criterion given by Helfferich IS... 

The dimensionless boundary conditions are listed in Table 2. The dimensionless mass transfer coefficient K is defined as kMdjl, and the parameter ij(see Eq. 7) relating the concentration of the oxidized and reduced species, Cq and Cr, respectively, is determined using the criterion for equal diffusivity using the expression = c o Icr. There are ten dimensionless groups, each shown within a box in Eqs. 5, 6, and 7 and in Table 2 since we have defined only five dimensionless variables in Eq. 4, we need five dimensionless parameters that characterize the problem. To determine these parameters, we define the characteristic quantities as shown in Eq. 8 ... 

These data points represent extensive damage and effective failure the loss in stiffness can be used to determine a suitable failure criterion. The normal failure strain provides a useful dimensionless parameter for characterizing fatigue. 

To link the dimensionless model temperature T to that of the experimental study, we use the criterion that the critical solution temperature of the experimental system, 7, is to be matched by that of our model mixture, 7. Moreover, we introduce a temperature offeet, T r, because this lattice fluid model is less suitable at low temperatures. Because this model is based on a mean-field treatment and does not account for the difference in molecular size... 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

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. 

The two current transients are shown in Fig. 10.6. The curve for progressive nucleation rises faster at the beginning because not only the perimeter of the clusters increases but also their number it drops off faster after the maximum. Such dimensionless plots are particularly useful as a diagnostic criterion to determine the growth mechanism. Real current transients may fit neither of these curves for a number of reasons, for example, if the growth starts from steps rather than from circular clusters. 

To ensure the system is probing reactions in a kinetically controlled regime, the reaction conditions must be calculated to determine the value of the Wiesz-Prater criterion. This criterion uses measured values of the rate of reaction to determine if internal dififusion has an influence. Internal mass transfer effects can be neglected for values of the dimensionless number lower than 0.1. For example, taking a measured CPOX rate of 5.9 x 10 molcH4 s g results... 

For the pseudohomogeneous analysis criterion, a predictive expression for the temperature difference between the gas- and solid-phase temperatures for a fixed bed reactor is given by Wei et al. (1984). In this expression, p is the dimensionless heat transfer coefficient, AA is the total heat generation in the reactor, and y is the fraction of the heat effect going to the solid phase. For the example packed bed reactor used throughout this chapter, the temperature difference between the gas and solid phases is in many cases up to 10 K. 

The following criterion could also be used to find the expected fluidization regime for a specified system. Experiments on particulate fluidization show that particle and fluid densities and fluid viscosity are the most significant factors affecting fluidization behavior. On the basis of this, a dimensionless discrimination number /)n has been suggested to... 

Secondary atomization, the breakup of the drops first formed, has been studied by Littaye (11C), who assumes a necessary criterion that the drag forces exceed the inertia forces. Ohnesorge (17C) makes use of the principles of mechanical similarity by introducing dimensionless coefficients to help explain jet breakup. Above certain well defined numbers, the jet completely atomizes at the nozzle. Lower values indicate the formation of a jet which disintegrates, owing to helical vibrations which later change into Rayleigh vibrations. 

If a large amount of (bio)catalyst is added to a substrate solution better results are achieved than if just a small amount of a highly productive catalyst is used. To even call an agent a catalyst, one condition is that the agent be added, as a minimum, in substoichiometric amounts. The smaller the amount of catalyst that has to be added for the same result, the better its performance. The relevant criterion is the dimensionless turnover number, TON [Eq. (2.23)]. 

The turnover number is not used frequently in biocatalysis, possibly as the molar mass of the biocatalyst has to be known and taken into account to obtain a dimensionless number, but it is the decisive criterion, besides turnover frequency and selectivity, for evaluation of a catalyst in homogeneous (chemical) catalysis and is thus quoted in almost every pertinent article. Another reason for the low popularity of the turnover number in biocatalysis, apart from the challenge of dimensionality, is the focus on reusability of biocatalysts and the corresponding greater emphasis on performance over the catalyst lifetime instead of in one batch reaction as is common in homogeneous catalysis (Blaser, 2001). For biocatalyst lifetime evaluation, see Section 2.3.2.3. 

A more comprehensive approach consists of studying the variation of the Semenov criterion as a function of the reaction energy. Such an approach is presented in [12], where the reciprocal Semenov criterion is studied as a function of the dimensionless adiabatic temperature rise. This leads to a stability diagram similar to those presented in Figure 5.2 [11, 13]. The lines separating the area of parametric sensitivity, where runaway may occur, from the area of stability is not a sharp border line it depends on the models used by the different authors. For safe behavior, the ratio of cooling rate over heat release rate must be higher than the potential of the reaction, evaluated as the dimensionless adiabatic temperature rise. 

Thus, the equations describing the thermal stability of batch reactors are written, and the relevant dimensionless groups are singled out. These equations have been used in different forms to discuss different stability criteria proposed in the literature for adiabatic and isoperibolic reactors. The Semenov criterion is valid for zero-order kinetics, i.e., under the simplifying assumption that the explosion occurs with a negligible consumption of reactants. Other classical approaches remove this simplifying assumption and are based on some geometric features of the temperature-time or temperature-concentration curves, such as the existence of points of inflection and/or of maximum, or on the parametric sensitivity of these curves. 

Only the first factor is influenced by the physico-chemical separation process (the selectivity), while the other two factors are determined by the column and the operating conditions, respectively. If C is a continuous criterion (see table 4.7), then both C and C, can be transferred from one column to another. Both column dimensions and flow rate have trivial effects on the analysis time tm. However, if the final analysis is to be run on a different (optimized) column, then it is more logical to use the dimensionless, column-independent factor (1 + km) in eqn.(4.31) instead of tm ... 

Note that K, A, and B are dimensionless. The only criterion is that A and B be in the same measurement units. Some sample calculations using Eq. (1.1) can be seen in Table 1.3. 

The Grashof number may be interpreted physically as a dimensionless group representing the ratio of the buoyancy forces to the viscous forces in the free-convection flow system. It has a role similar to that played by the Reynolds number in forced-convection systems and is the primary variable used as a criterion for transition from laminar to turbulent boundary-layer flow. For air in free convection on a vertical flat plate, the critical Grashof number has been observed by Eckert and Soehngen [1] to be approximately 4 x 10". Values ranging between 10" and 109 may be observed for different fluids and environment turbulence levels. ... 

Using the velocity and temperature gradients, we obtain the dimensionless entropy production for the empty bed (Figure 4.10). Comparison of Figures 4.9 and 4.10 indicates that outside the wall region, the distribution of the rate of entropy production is uniform in the packed bed, which is the thermodynamic optimality criterion. The profile of entropy production shows a typical S shape in the empty bed. 

The Reynolds number is undoubtedly the most famous dimensionless parameter in fluid mechanics. It is named in honour of Osborne Reynolds, a British engineer who first demonstrated in 1883 that a dimensionless variable can be used as a criterion to distinguish the flow patterns of a fluid either being laminar or turbulent. Typically, a Reynolds number is given as follows ... 

The problem of many responses of approximately equal interest is often met in industrial synthesis. Cost, quality specifications, overall yield as well as pollution due to waste are necessary to control, and it is sufficient that one of the responses fails to meet the requirements for an overall result to be poor. In such cases it is possible to use a technique by which all responses are weighed together into one criterion which is then optimized. First, it is necessary to clearly define, what the desired result is for each response. The measured value of the response is then scaled to a dimensionless measure of the desirability, dj, of the response jj. The scaling is done so that dj is in... 

No details were given over the determination of the mixing time. Since the scale-up criterion must be formulated dimensionlessly and in the above-mentioned case Pe, n9, Ov/D or Q must be obviously excluded, the dimensionless numbers P(e/v) and 6(slDy, resp., came next into consideration, otherwise a combination of several relevant dimensionless numbers must be used. 

A criterion that can be used to predict which of these two steps is rate controlling is to equate the half-times calculated for ideal intraparticle- and ideal film-controlled exchange. This leads to a dimensionless Helfferich number (He) defined as [Helfferich, 1962a]... 

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