Dimensionless number equation

Physical modeling is not as accurate as mathematical modeling. This should be attributed to the fact that in dimensionless equations, the dependent number is expressed as a monomial product of the determining numbers, whereas the corresponding phenomena are described by polynomial differential equations. Moreover, errors in the experimental determination of the several constants and powers of the dimensionless equations can also lead to inaccuracies. We should also keep in mind that the dimensionless-number equations are only valid for the limits within which the determining parameters are varied in the investigations of the physical models. 

Where no complete mathematical description of the process and no dimensionless-numbers equations are available, modeling based on individual ratios can be employed. This is the most characteristic case for a number of industrial processes, especially in the field of organic-chemicals technology. This method is referred to as scale-up modeling (Mukhyonov et al., 1979). In such cases, individual ratios for the model and the object, which should have a constant value, are employed. For instance, there should be a constant ratio between the space velocity of the reacting mixture in the model and the industrial object. Some of the dimensionless numbers mentioned in physical modeling are also employed in this case. 

Considering the above-defined dimensionless numbers. Equation 7.11 would modify to... 

A dimensionless number equation is a substitution of the Il-theorem for the case at hand. Here is an example, for mass transfer to or from a fluid moving aroimd a spherical object ... 

Dimensionless numbers are not the exclusive property of fluid mechanics but arise out of any situation describable by a mathematical equation. Some of the other important dimensionless groups used in engineering are Hsted in Table 2. 

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... 

There are tluee dimensionless numbers used in these equations, and their dehnitions are ... 

Damkdhler (1936) studied the above subjects with the help of dimensional analysis. He concluded from the differential equations, describing chemical reactions in a flow system, that four dimensionless numbers can be derived as criteria for similarity. These four and the Reynolds number are needed to characterize reacting flow systems. He realized that scale-up on this basis can only be achieved by giving up complete similarity. The recognition that these basic dimensionless numbers have general and wider applicability came only in the 1960s. The Damkdhler numbers will be used for the basis of discussion of the subject presented here as follows ... 

Equation 46 is a general expression that may be applied to the treatment of experimental data to evaluate exponent a. This, however, is a cumbersome approach that can be avoided by rewriting the equation in dimensionless form. Equation 42 shows that there are n = 5 dimensional values, and the number of values with independent measures is m = 3 (m, kg, sec.). Hence, the number of dimensionless groups according to the ir-theorem is tc = 5 - 3 = 2. As the particle moves through the fluid, one of the dimensionless complexes is obviously the Reynolds number Re = w Upl/i. Thus, we may write ... 

As one of two possible dimensionless numbers is now known, the second one can be obtained by dividing both sides of the equation through by the remaining values ... 

And introducing the ratio of accelerations, = ag/g, where indicates the relative strength of acceleration, ag, with respect to the gravitational acceleration g. This is known as the separation number. The LHS of equation 60 contains a Reynolds number group raised to the second power and the drag coefficient. Hence, the equation may be written entirely in terms of dimensionless numbers ... 

Substituting these dimensionless numbers into Equation 7-12 yields,... 

The number of tanks in series from the dimensionless varianee (Equation 8-45) from the traeer experiment is... 

TT-theorem) or from the governing equations of the flow. The latter is to be preferred because this method will give a sufficient amount of dimensionless numbers. Furthermore, it will connect the numbers to the physical process via the equations and give important information in cases where it is necessary to make approximations. 

It is observed that the following dimensionless numbers appear in the equations ... 

The dimensionless numbers are important elements in the performance of model experiments, and they are determined by the normalizing procedure ot the independent variables. If, for example, free convection is considered in a room without ventilation, it is not possible to normalize the velocities by a supply velocity Uq. The normalized velocity can be defined by m u f po //ao where f, is the height of a cold or a hot surface. The Grashof number, Gr, will then appear in the buoyancy term in the Navier-Stokes equation (AT is the temperature difference between the hot and the cold surface) ... 

It is only po.ssible to obtain similar solutions in situations where the governing equations (Eqs. (12.40) to (12.44)) are identical in the full scale and in the model. This tequirement will be met in situations where the same dimensionless numbers are used in the full scale and in the model and when the constants P(i, p, fj.Q,.. . have only a small variation within the applied temperature and velocity level. A practical problem when water is used as fluid in the model is the variation of p, which is very different in air and in water see Fig. 12.27. Therefore, it is necessary to restrict the temperature differences used in model experiments based on water. 

Before discussing the on.set, and nature, of fluid turbulence, it is convenient to first recast the Navier-Stokes equations into a dimensionless form, a trick first used by Reynolds in his pioneering experimental work in the 1880 s. In this form, the Navier-Stokes equations depend on a single dimensionless number called Reynolds number, and fluid behavior from smooth, or laminar, flow to chaos, or turbulence,... 

In 1953, Rushton proposed a dimensionless number that is used for scale-up calculation. The dimensionless group is proportional to NRe as shown by the following equation 2,3... 

The mean molality values m (moles per kilogram), mole fractions x (dimensionless number) and concentrations c (moles per cubic decimetre) are related by equations similar to those for non-electrolytes (see Appendix A). 

Integration of the above equation can be represented by three dimensionless numbers... 

Integration of this equation can be expressed in terms of the dimensionless numbers used already... 

The dimensionless distance y+ has the form of a Reynolds number. Equation 2.58 fits the experimental data in the range 0 y+ 5. In the viscous sublayer, the velocity increases linearly with distance from the wall. 

The values for % and 0 are as defined above and used as dimensionless numbers. Property as in Hammett relations should be the logarithm of a rate constant, equilibrium constant, etc. An important refinement for the steric contribution was made after the recognition that below a certain threshold of cone angles the property no longer changed and one should use the value 0 for c [29], Above the threshold a linear relation was found. Thus, for each property a steric threshold 0 was defined and the equation now reads ... 

Literature values of SPMD-water partition coefficients (iifswS) should be used with caution. Different units of Ks are generated when the concentrations in SPMD and water are expressed on a mass basis (e.g., ng g ) or a volume basis (e.g., ng mL ). Depending on the choice of concentration units, values will have units of g g mL mL or mL g or Ksv, values are given as dimensionless numbers in the former two cases. Care should be taken to distinguish between the different versions of Ksv,. Preferably, mL mL units should be used, because most of the equations for SPMD uptake kinetics use the volume of an SPMD rather than its mass. The two most frequently used versions of Ks are interrelated by... 

In Eq. (14), Ck is another dimensionless number. Up to an overall multiplicative constant the coefficients Cd and Ck are derived now by imposing that the energy is conserved. The relevant condition is written in Eq. (11). Inserting into this equation the expression of j given in Eq. (12) and using the expressions for D and K given in (13, 14), one obtains the condition equivalent to the conservation of energy ... 

We have found now an equation for the evolution of the density inside the cluster without any uncontrolled parameter, except for the dimensionless number C. Below we shall do two things. First, in Section V, we shall find the steady solutions for the density, that turns out to transform into a quite simple problem, mathematically equivalent to the equilibrium of self-gravitating atmosphere. Then, in Section VI we shall look at the possible existence of finite time singularities in the dynamical problem. 

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