Dimensionless criterion

In preparative selective chromatography, the formation of broad zones of the substances is determined by the formation of sharp boundaries of each zone. The formation of these sharp boundaries of substance zones in column sorption processes for systems in which the interphase transfer is limited by substance diffusion in sorbent grains [104, 122, 123] is determined by the dimensionless criterion X ... 

The dimensionless yield depends on the mobility of the reaction but we must decide on a measure of this mobility. We turn first to the rate coefficient of the reaction at the maximum temperature k(Tm), for brevity denoted km. The dimensionless yield can, however, depend only on a dimensionless criterion. The rate coefficient of a bimolecular reaction has the dimensions (time)-1 (concentration)-1. Hence we obtain a dimensionless criterion of the mobility if we multiply the rate coefficient by the reaction time r (which depends on the cooling rate) and the concentration. As concentration unit we have already chosen above the equilibrium concentration at the theoretical temperature [NO]. The dimensionless criterion of the mobility thus has the form... 

The dimensionless criterion factor f = qv/a is of great importance in estimating liquid flow parameters the discussion of its values is beyond the frames of the present work (as outlined in the references cited). It may be noted, however, that the same expression appears in the equation evaluating the ability of polymers for thread and fiber spinning 15>. To the first approximation, the criterion f = qv/o may be treated as a measure of shape-keeping ability of liquid in flow. 

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... 

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

The substitution of 5d into Eq. (7.52) allows to derive a dimensionless criterion. 

In the theory of transport in electrolysers, it is convenient to introduce a dimensionless criterion called the Sherwood number and defined as... 

The condition (6.14) suggests an additional dimensionless criterion. Indeed, let us introduce dimensionless parameters... 

Efficiency of shock-inertial sedimentation ri is function of following dimensionless criterion ... 

For a vertical flow the amount of potassium bichromate dust of various fractions settling (Fig. VI.23) on spherical and cylindrical surfaces covered with a layer of vaseline was determined experimentally in [117] and the dimensionless criterion C2 was calculated ... 

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. 

Similarly it may be shown that, at the critical conditions, the flowrate is a maximum for a given value of the specific energy J. At the critical velocity, (ir/gD) is equal to unity. This dimensionless group is known as the Froude number Fr. For velocities greater than the critical velocity Fr is greater than unity, and vice versa. It may be shown that the velocity with which a small disturbance is transmitted through a liquid in an open channel is equal to the critical velocity, and hence the Froude number is the criterion by which the type of flow, tranquil or rapid, is determined. Tranquil flow occurs when Fr is less than unity and rapid flow when Fr is greater than unity. 

The stability criterion. Equation (8.29), can be converted to dimensionless form. The result is... 

Flow of the liquid past the electrode is found in electrochemical cells where a liquid electrolyte is agitated with a stirrer or by pumping. The character of liquid flow near a solid wall depends on the flow velocity v, on the characteristic length L of the solid, and on the kinematic viscosity (which is the ratio of the usual rheological viscosity q and the liquid s density p). A convenient criterion is the dimensionless parameter Re = vLN, called the Reynolds number. The flow is laminar when this number is smaller than some critical value (which is about 10 for rough surfaces and about 10 for smooth surfaces) in this case the liquid moves in the form of layers parallel to the surface. At high Reynolds numbers (high flow velocities) the motion becomes turbulent and eddies develop at random in the flow. We shall only be concerned with laminar flow of the liquid. 

Agglomerates in a sheared fluid rupture when the hydrodynamic stress exceeds a critical value in dimensionless form the criterion for rupture is Fa > Facrjt. Rupture occurs within a short time of application of the critical stress, and thus can be distinguished from erosion, which occurs over much longer time scales. 

To treat current distribution in through-holes, Hazlebeck and Talbot [140] formulated a model and established dimensionless criteria for ohmic and convective regimes. The most uniform distribution is obtained in the ohmically limited regime bounded by a numerical criterion based on a Peclet number and a Thiele modulus yT. In the notation of Takahashi and Gross,... 

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. 

Combining the defining parameters of equation (3) and modifying this equation into a dimensionless form, it is possible to demonstrate that criterion (p a, is the function of only two parameters r a and xi,2. 

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... 

The dimensionless parameter, DAtjAt is known as the diffusion number, Di. For stability of an explicit computational solution (i.e., for the solution to make any sense at all), this number must be less than 1/2. The logic behind this stability criterion can be seen in equation (E7.1.4). If Di is greater than 1 /2, then the relation between Q, +i and is negative, creating a computational solution that can have serious errors. 

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... 

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