Variables dimensionless

Once the vessel type has been selected and the rate expression established for the cleaning process, the design engineer must decide which variables are significant for the scaling process. Mathematical tools in the form of dimensionless variables are useful for determining these variables. Every chemical or physical process can be defined by a set of dimensionless ratios or variables intrinsic to the process. An example is the Reynolds number, which is defined for a process with internal flow through a tube by solvent properties, flow rate of the solvent, and the diameter of the tube. l 

V = velocity or average velocity of the solvent flow through the tube 

The Reynolds number is useful for characterizing flow of a Newtonian fluid through a tube. The variables in dimensional analysis can be arranged to suggest a valid ratio when none exists. Therefore, established dimensionless variables should be used where applicable. A number of these dimensionless variables have been proven to work in scaleup applications. Examples include the Reynolds, Nusselt, Grashof, and Sherwood numbers, all of which are completely described in Perry s Chemical Engineers Handbook. 

The impeller Reynolds number is proportional to the standard Reynolds number and is arrived at by assuming that the velocity of the solvent flow through the chamber is proportional to the speed of the impeller and the area through which it acts. The impeller Froude number is arrived at in a similar manner. 

using these variables, assume a prototype to be designed with an internal volume 125 times that of the model. The solvent density and viscosity are considered fixed. Then for a geometrically similar vessel  

Here aX is the entropy of vaporization and AVv is the volume change on vaporization. Using now 

1 Frederick Thomas Trouton, bom Nov. 24,1863, in Dublin, Ireland, died Sep. 21,1922, inDowne, Kent. 

Tc is the critical temperature and f is the reduced temperature. The same notation is used for the pressure. 

only reduced, dimensionless variables appears, and this equation should no be longer dependent on the individual properties of the materials. Note that T AS/pAV is some ratio of thermal energy to volmne energy. 

In fact, a close similarity for the various substances is monitored however, it can be seen that the monoatomic gases are separated from the polyatomic gases. The difference is not very much pronounced, however, and we neglect this difference here. Further, we deal with the plot as a straight line as a first approximation. Under these conditions, immediately 

Substituting this value into the equation for dynamic similarity, as based on the power number, gives  

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The upper limit of the dimensionless variable Vp, is typically written in tenns of the Debye temperature 9, as... 

For deep quenches, where the post-quench T is far below T, the equations are conveniently written in temis of scaled (dimensionless) variables t) = mid x... 

Finally we must consider the effectiveness factor. Introducing the dimensionless variables Into equation (11.27), it reduces to... 

Detailed Modeling Results. The results of a series of detailed calculations for an ideal isothermal plug-flow Langmuir system are summarized in Figure 15. The soHd lines show the form of the theoretical breakthrough curves for adsorption and desorption, calculated from the following set of model equations and expressed in terms of the dimensionless variables T, and P ... 

Theorem 5. The transpose of is a complete B-matrrx of equation 13. It is advantageous if the dependent variables or the variables that can be regulated each occur in only one dimensionless product, so that a functional relationship among these dimensionless products may be most easily determined (8). For example, if a velocity is easily varied experimentally, then the velocity should occur in only one of the independent dimensionless variables (products). In other words, it is sometimes desirable to have certain specified variables, each of which occurs in one and only one of the B-vectors. The following theorem gives a necessary and sufficient condition for the existence of such a complete B-matrix. This result can be used to enumerate such a B-matrix without the necessity of exhausting all possibilities by linear combinations. 

The film thickness 6g depends primarily on the hydrodynamics of the system and hence on the Reynolds number and the Schmidt number. Thus, various correlations have been developed for different geometries in terms of the following dimensionless variables ... 

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. 

Dimensionless variables can be defined for time, the axial coordinate, and velocity ... 

Develop (e.g., write) the hyperbolic equation in terms of the dimensionless variables. This breaks the interdependence of exponential and pre-exponential terms. 

It is often useful to write a model equation sueh as Equation 8-121 in terms of dimensionless variables. This introduees the Peelet number Npg = uL/Dg [, whieh represents the ratio of eharaeteristie dispersion time to eharaeteristie eonveetion time (residenee time), and the Damkdhler number,... 

Since there can be an infinite number of combinations of creep and recovery periods it has been found convenient to express this behaviour in terms of two dimensionless variables. The first is called the Fractional Recovery, defined as... 

The second dimensionless variable is called the Reduced Time, tg, defined as... 

The factor depends on the Reynolds number, which is a dimensionless variable that denotes the nature of flowr... 

The energy s and the distance r are both real physical quantities, with a measure md a unit. If we define the variables r,ed = r/rio and fired = / h, then both id fired are dimensionless. The idea is to rewrite the electronic Schrodinger equation in terms of the dimensionless variables, giving a much simpler dimensionless equation. 

It is instructive to look at the form of the Is, 2s and 3s orbitals (Table 9.1). By convention, we use the dimensionless variable p = Zrjaa rather than r. Here 2 is the nuclear charge number and oq the first Bohr radius (approximately 52.9 pm). The quantity Z/n is usually called the orbital exponent, written These exponents have an increasing number of radial nodes, and they are orthonormal. 

Introducing the dimensionless variables x = and 6 = t/t, where t is the switch time interval, and is the length of one SMB column, the model equations become ... 

Next, we substitute these dimensionless variables into the incompressible Navier-Stokes equations (equation 9.16). In Cartesian coordinates, the T component of the first equation reads... 

What this transcription into dimensionless variables means physically is very interesting. It means that, if expressed in terms of the dimensionless variables v, x and t, any two fluid problems will have essentially the same flow solutions whenever their Reynolds numbers are equal. This is of considerable practical importance. of course, since it implies that the air flow past an airplane wing, for example,... 

Flow field calculations are conveniently performed in dimensionless variables defined in terms of the orifice parameters ... 

Assuming steady state in Eqs. (10.8-10.10) and (10.18-10.20), we obtain the system of equations, which determines steady regimes of the flow in the heated miero-channel. We introduce values of density p = pp.o, velocity , length = L, temperature r = Ti 0, pressure AP = Pl,o - Pg,oo and enthalpy /Jlg as characteristic scales. The dimensionless variables are defined as follows ... 

This messy result apparently requires knowledge of five parameters k, (A )o> Poo, and po- However, conversion to dimensionless variables usually reduces the number of parameters. In this case, set Y = Na/(Na)o (the fraction unreacted) and r = t/thatch (fractional batch time). Then algebra gives... 

The Lotka-Volterra reaction described in Section 2.5.4 has three initial conditions—one each for grass, rabbits, and lynx—all of which must be positive. There are three rate constants assuming the supply of grass is not depleted. Use dimensionless variables to reduce the number of independent parameters to four. Pick values for these that lead to a sustained oscillation. Then, vary the parameter governing the grass supply and determine how this affects the period and amplitude of the solution. 

The use of dimensionless variables will be illustrated using Equation (8.12) but with an added term for axial diffusion ... 

The velocity profile is scaled by the mean velocity, m, giving the dimensionless profile z(- ) = Ez(r)/ . To complete the conversion to dimensionless variables, the dependent variable, a, is divided by its nonzero inlet concentration. The dimensionless version of Equation (8.12) is... 

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