Some Dimensionless Parameters

Having introduced some theoretical and empirical developments, it is also important to introduce several dimensionless parameters. These parameters could basically characterise the transport processes of gases. 

A fluid is usually considered as a continuous substance under normal conditions. In fact, it is composed of myriads of continuously moving molecules. For gases at 1 atm, 25°C, the spacing between molecules is on the order of 10 3 pm, and for the liquids it is on the order of 10 4 pm. The number of molecules per cubic micrometer is 2.69 x 107 for air and 3.21 x 1010 for water. Hence, it is reasonable to assume and treat the fluid as a continuum. 

The Knudsen number (Kn) is used to determine the different regimes of the gas flow. These regimes can be divided into continuous flow, transitional flow and free molecular flow. This division is based on the understanding that the flow behaviour differs within each of the flow regimes. The Knudsen number is defined as 

When X L or Kn 0.01, many collisions between molecules occur, hence the gas can be regarded as a continuum. Under this condition or regime, the flow of 

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  

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The main purpose in transforming the above set of dimensional equations to an equivalent dimensionless form is to enable us to identify any terms which may be particularly important in determining the behaviour of the system. Generally, such terms will be revealed by some dimensionless parameters being much larger than others. An additional bonus of the process, however, is often that we can also reduce the number of parameters in the equations. Thus it may turn out that the behaviour is not determined primarily by the absolute values of all the rate constants, but only by one or two ratios of them. 

Scaling is the art of choosing the dimensionless variables so that we know the ranges of magnitude in which they will lie. Clearly, this is an indispensable prelude to calculation. Usually there are some dimensionless parameters that... 

Leentvaar J., Ywema T.S.J. (1980), Some dimensionless parameters of impeller power in coagulation-flocculation processes. Water Research, 14, 135-140. 

How crude are the results we might get from such ideal models Are there some cases where they work well, and some where they fail There is a special trick that often helps us to decide. It involves finding some dimensionless parameters, either large or small, which describe the system. For example, a gas can be characterized by the fi action of volume which is taken up by the molecules. If this parameter is much less than one. 

TTirough a dimensional analysis it can be recognized that some dimensionless parameters determine the behaviour of the system. Of course the choice is not unique, but a convenient combination is K (the equiUbrium constant), 1 — and 5 —. ... 

The previous equations are some of the major dimensionless parameters. For the flow to remain dynamically similar, all the parameters must remain constant however, constancy is not possible in a practical sense, so one must make choices. 

There was some argument in the literature over the relative merits and demerits of the JKR and the DMT theories [23-26], but the controversy has now been satisfactorily resolved. A critical comparison of the JKR and DMT theories can be obtained from the literature [23-30]. According to Tabor [23], JKR theory is valid when the dimensionless parameter given by Eq. 25 exceeds a value of about five. 

Tables IV and V contain appropriate balance equations for nonisothermal free-radical polymerizations and copolymerizations, which are seen to conform to equation 2k. Following the procedure outlined above, we obtain the CT s for homopolymerizations listed in Table VI. Corresponding CT s for copolymerizations can be. obtained in a similar way, and indeed the first and fourth listed in Table VII were. The remaining ones, however, were derived via an alternate route based upon the definitions in Table VI labeled "equivalent" together with approximate forms for pj, which were necessitated by application of the Semenov-type runaway analysis to copolymerizations, and which will subsequently be described. Some useful dimensionless parameters defined in terms of these CT s appear in Tables VIII, IX and X.

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. 

In later sections, the use of the scaling relationships to design small scale models will be illustrated. For scaling to hold, all of the dimensionless parameters given in Eqs. (36), (37) or (39) must be identical in the scale model and the commercial bed under study. If the small scale model is fluidized with air at ambient conditions, then the fluid density and viscosity are fixed and it will be shown there is only one unique modeling condition which will allow complete similarity. In some cases this requires a model which is too large and unwieldy to simulate a large commercial bed. 

Most early experiments devoted to verifying the scaling relationships have dealt with the full set of scaling relationships. Several more recent experiments have dealt with a reduced set of dimensionless parameters. In some experiments, additional scaling parameters were unintentionally matched. 

Exercise 6. Show that the equilibrium point of the model defined by Eq.(34) and the simplified model R given by Eq.(35), i.e. when the dynamics of the jacket is considered negligible, are the same. Deduce the Jacobian of the system (35) at the corresponding equilibrium point. Write a computer program to determine the eigenvalues of the linearized model R at the equilibrium point as a function of the dimensionless inlet flow 4 50. Values of the dimensionless parameters of the PI controller can be fixed at Ktd = 1-52 T2d = 5. The set point dimensionless temperature and the inlet coolant flow rate temperature are Xg = 0.0398, X40 = 0.0351 respectively. An appropriate value of dimensionless reference concentration is C g = 0.245. Does it exist some value of 2 50 for which the eigenvalues of the linearized system R at the equilibrium point are complex with zero real part Note that it is necessary to vary 2 50 from small to great values. Check the possibility to obtain similar results for the R model. 

You are taking some experimental data on pipe friction and want the analyzed results to apply to other pipes and sizes. Using the dimensionless Navier-Stokes equations, determine the important dimensionless parameters in these experiments. 

Dimensionless correlations based on momentum-or energy-balances and using the Lockhart and Martinelli parameter, Xc, or some similar parameter. The practical relevance of such correlations is limited since their use requires the knowledge of the single-phase pressure drop of both gas and liquid furthermore, the influence of the geometry of the bed is not always well described by these single-phase pressure drops alone. 

We therefore advise that the reader should consult a recent series of papers published by Galvez et al. [171, 172] encompassing all the mechanisms mentioned in Sect. 7.1, elaborated for both d.c. and pulse polarography. The principles of the Galvez method are clearly outlined in the first part of the series [171]. It is similar to the dimensionless parameter method of Koutecky [161], which enables the series solutions for the auxiliary concentration functions cP and cQ exp (kt) and

combined directly with the partial differential equations of the type of eqn. (203). In some of the treatments, the sphericity of the DME is also accounted for. The results are usually visualized by means of predicted polarograms, some examples of which are reproduced in Fig. 38. Naturally, the numerical description of the surface concentrations at fixed potential are also immediately available, in terms of the postulated power series, and the recurrent relationships obtained for the coefficients of these series. 

B. If a parametric study of the effect of some quantity is to be done, that quantity should appear in the numerator of one and only one of the dimensionless parameters. 

When the electrochemical properties of some materials are analyzed, the timescale of the phenomena involved requires the use of ultrafast voltammetry. Microelectrodes play an essential role for recording voltammograms at scan rates of megavolts-per-seconds, reaching nanoseconds timescales for which the perturbation is short enough, so it propagates only over a very small zone close to the electrode and the diffusion field can be considered almost planar. In these conditions, the current and the interfacial capacitance are proportional to the electrode area, whereas the ohmic drop and the cell time constant decrease linearly with the electrode characteristic dimension. For Cyclic Voltammetry, these can be written in terms of the dimensionless parameters yu and 6 given by... 

Let us have a closer look at the differences between the minimal and the extended set of equations and follow these differences along some paths in the parameter space. As mentioned in Sect. 2.3, we can omit some of the physical parameters by using dimensionless parameters. In Figs. 5-9 we show the dependence of the critical values of the tilt angle and wave vector on the dimensionless parameters (as defined... 

Solutions of the diffusion equation inevitably involve the dimensionless parameter Dtla2 in such a way that diffusive redistribution becomes significant as this parameter approaches unity. Here a is some characteristic dimension of the diffusive region. In the case where the medium is homogeneous, without a microscopic substructure (e.g., a glass or a liquid), a is the macroscopic dimension. In rocks that consist of numerous mineral grains, the relevant a is usually the individual grain size rather than the macroscopic dimension. 

Typical cascade impactors consist of a series of nozzle plates, each followed by an impaction plate each set of nozzle plate plus impaction plate is termed a stage. The sizing characteristics of an inertial impactor stage are determined by the efficiency with which the stage collects particles of various sizes. Collection efficiency is a function of three dimensionless parameters the inertial parameter (Stokes number, Stk), the ratio of the jet-to-plate spacing to the jet width, and the jet Reynolds number. The most important of these is the inertial parameter, which is defined by Equation 2) as the ratio of the stopping distance to some characteristic dimension of the impaction stage (10), typically the radius of the nozzle or jet (Dj). 

Besides, r ° is a dimensionless parameter running between 0 and 1, susceptible to reflect some break of the IR selection rule probably following from some lack of symmetry of the dimer or perhaps from dynamic vibronic structure of the H-bond system [77]. 

First we must normalise some quantities, to make them compatible with the other dimensionless parameters already used. We refer to the normalisation formulae on p.26. Recall that we have normalised voltage by the factor and that the time unit r for LSV is equal to (v being the sweep rate), or the time the sweep takes to traverse one normalised potential unit V-... 

The quality factor, or Q-factor, is a general dimensionless parameter, used in mechanical, electrical, electromagnetic, and optical contexts. Given some signal intensity S(co) as a function of frequency m, the Q-factor is defined as the resonance frequency divided by the bandwidth A (see Fig. 9.4) ... 

It is also often useful, given some quantity m(x) and some weighting function h x), to define an overalO quantity U as the weighted integral of (x) over the label range U = and, should u(x) also depend on some other variable T,u = u(x, t), one, of course, has U(t) = . Often we will restrict our attention to the special case where h(x) is identically equal to unity.Finally, for purposes of analysis it is often useful to choose a specific mathematical form for M(x). A powerful form is that of a gamma distribution, which contains only one dimensionless parameter, a ... 

Reduced (dimensionless) parameters for some of the IE systems subjected to numerical calculations are shown in Table 1. Variants I through III (high selectivity, Kp /Kps > 1, of invading B ion or favorable isotherm) and variants I.e through Ill.e (low selectivity, Kp /Kpa < 1, of B ion or unfavorable isotherm) are listed in this Table. 

The dimensionless form of the equation contains one dimensionless parameter as a multiplier of the first term of the right-hand side and maybe some additional dimensionless parameters, which may appear within the dimensionless source term, S. Depending on the general variable, 0, the effective diffusion coefficient, F, appearing in this dimensionless number will be different, leading to different dimensionless numbers. For the species mass fraction, momentum and enthalpy transport equations, the effective diffusion coefficient will be molecular diffusion coefficient, the kinematic viscosity of the fluid and the thermal diffusivity of the fluid respectively. The corresponding dimensionless numbers are, therefore, defined as follows. 

In the interpretation of the spectral data it is usually the constants A and B in Eq. (3.27) or some other set of reduced potential constants that are evaluated. The barrier height, AB2/4 for B < 0, is thus determined directly. However, if one wants to relate the value of the dimensionless parameter Z I at the minimum of a double-minimum potential function to the absolute geometry of the puckered ring, the reduced mass must be known in order to find x from IZI. One is then required to introduce assumptions about the dynamical model of the ring-puckering motion. 

Some of the early work is in German and the stability parameters , Af, and N are related to Damkohler numbers see, for example, p. 196 of the translation of Brotz text cited above in the references to Chap. 1. Some dimensionless graphs of the first order plot are given in ... 

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