Dimensionless form boundary conditions

Boundary condition Dimensionless form Boundary condition ... 

Simila.rityAna.Iysis, Similarity analysis starts from the equation describing a system and proceeds by expressing all of the dimensional variables and boundary conditions in the equation in reduced or normalized form. Velocities, for example, are expressed in terms of some reference velocity in the system, eg, the average velocity. When the equation is rewritten in this manner certain dimensionless groupings of the reference variables appear as coefficients, and the dimensional variables are replaced by their normalized relatives. If another physical system can be described by the same equation with the same numerical values of the coefficients, then the solutions to the two equations (normalized variables) are identical and either system is an accurate model of the other. 

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

Bunimovich et al. (1995) lumped the melt and solid phases of the catalyst but still distinguished between this lumped solid phase and the gas. Accumulation of mass and heat in the gas were neglected as were dispersion and conduction in the catalyst bed. This results in the model given in Table V with the radial heat transfer, conduction, and gas phase heat accumulation terms removed. The boundary conditions are different and become identical to those given in Table IX, expanded to provide for inversion of the melt concentrations when the flow direction switches. A dimensionless form of the model is given in Table XI. Parameters used in the model will be found in Bunimovich s paper. 

Since the dimensionless equations and boundary conditions governing heat transfer and dilute-solution mass transfer are identical, the solutions to these equations in dimensionless form are also identical. Profiles of dimensionless concentration and temperature are therefore the same, while the dimensionless transfer rates, the Sherwood number (Sh = kL/ ) for mass transfer, and the Nusselt number (Nu = hL/K ) for heat transfer, are identical functions of Re, Sc or Pr, and dimensionless time. Most results in this book are given in terms of Sh and Sc the equivalent results for heat transfer may be found by simply replacing Sh by Nu and Sc by Pr. 

We now consider the dependence of the stationary-state solution on the parameter d. To represent a given stationary-state solution we can take the dimensionless temperature excess at the middle of the slab, 0ss(p = 0) or 60,ss-With the above boundary conditions, two different qualitative forms for the stationary-state locus 0O,SS — <5 are possible. If y and a are sufficiently small (generally both significantly less than i), multiplicity is a feature of the system, with ignition on increasing <5 and extinction at low <5. For larger values of a or y, corresponding to weakly exothermic processes or those with low temperature sensitivity, the hysteresis loop becomes unfolded to provide... 

The Dimensionless Parameter is a mathematical method to solve linear differential equations. It has been used in Electrochemistry in the resolution of Fick s second law differential equation. This method is based on the use of functional series in dimensionless variables—which are related both to the form of the differential equation and to its boundary conditions—to transform a partial differential equation into a series of total differential equations in terms of only one independent dimensionless variable. This method was extensively used by Koutecky and later by other authors [1-9], and has proven to be the most powerful to obtain explicit analytical solutions. In this appendix, this method will be applied to the study of a charge transfer reaction at spherical electrodes when the diffusion coefficients of both species are not equal. In this situation, the use of this procedure will lead us to a series of homogeneous total differential equations depending on the variable, v given in Eq. (A.l). In other more complex cases, this method leads to nonhomogeneous total differential equations (for example, the case of a reversible process in Normal Pulse Polarography at the DME or the solutions of several electrochemical processes in double pulse techniques). In these last situations, explicit analytical solutions have also been obtained, although they will not be treated here for the sake of simplicity. 

The analytical solution to Equation 2 for a semi-infinite medium with first-type boundary conditions may be written in dimensionless form as (Powers et cil, 1991) ... 

To generalize the analysis, various equations and boundary conditions are expressed in dimensionless forms. Consider the following dimensionless quantities... 

The dimensionless models keep the same form as with the original although there are differences at. V = 1 on the membrane side only. The two boundary conditions, used to find the roots, Qo or T o, are simplified. 

This boundary condition is easily incorporated into the solution procedure that was outlined above, the set of equations governing the dimensionless temperature in this case having the form ... 

Although it is not a necessary step, the governing equations and the boundary conditions will be written in dimensionless form before obtaining the numerical... 

This is written out in order to make the next point. The first and last equation in the set are superfluous, because the boundary concentrations Co and C/y I are not subject to diffusion changes, but to other conditions. Also, where the boundary values appear in the other equations, they must be replaced with what we can substitute for them. The outer boundary value, C/y I, is (almost always) equal to the initial bulk concentration C, usually equal to unity in its dimensionless form. This means that the last term in each equation separates out as a constant term and makes for a constant vector [Hgw+iC II 2,n+iC. .. H jv+iC ]7, which will be called Z here. The concentration at the electrode Co is handled according to the boundary condition. For Cottrell, for example, it is set to zero throughout and thus simply drops out of the set. For other conditions, for example constant current or an irreversible reaction, a gradient C is involved, as described in Chap. 6. In that chapter, the gradient was expressed as a possibly multipoint approximation,... 

Dimensionless ways of expressing an experimental result. This category forming dependent variables, but not model laws, can be derived from the boundary conditions of the model laws. 

This form is actually used by Koros and Paul (1978). Eq. (18.38) can be solved for the boundary conditions of transient permeation. The set of equations then obtained is given in Table 18.12. These Eqs. (18.39)-(18.42) enable us to derive SCfj and b from experimental sorption isotherms, as presented by Eq. (18.36), and the values of Dd and DH from transient permeation experiments. The equations contain a number of dimensionless groups ... 

We now compare the solution of the hyperbolic model with that of the parabolic model used widely in the literature to describe dispersion in tubular reactors. The parabolic model with Danckwerts boundary conditions (in dimensionless form) is given by... 

Total Energy for Heavy Neutral Atoms.—The fact that the neutral atom solution has the form I in Figure 1 implies that (x) - 0 as x - oo, in fact as 144/x which is readily verified to be an exact solution of the dimensionless TF equation (10), not however satisfying the atomic boundary condition (11). Since V(r) - 0 at infinity, it follows from equation (7) that, for this neutral case with N=Z, we must have p=0. The condition that, in the simplest density description of neutral atoms, the chemical potential is zero is important for the arguments which follow. We shall see below that one of the objectives of more sophisticated density descriptions must be to find p. Equations (25) and (26) can be rewritten in the form, using E= — T and p=0,... 

A mathematical model has been proposed to account for the mutual synergistic action of either particle component on the other in increasing the value of the dimensionless time 0 as shown in Fig. 9b. Thus, the dimensionless time 0X for the coarse particles could be assumed to exert a mass-fraction-based influence on the fine particles, proportional to 0 1 — x2), which is affected by certain interaction by the fines, inclusive of their ability to adhere to the surface of the coarse and form clusters among themselves, lumped in certain appropriate form, for instance, [1 + /(x2)], where the function /(x2) may again be assumed to possess certain appropriate form, for instance, exponential, x2, where n1 may be called the interactive exponent. This results in an overall contribution by the coarse particles, suitably corrected for the interaction of the fine particles, 0 1 — x2Xl + x2l). This function has the property of accommodating the following boundary conditions ... 

In practice, it is easy to introduce a pulse input. Under this condition, the boundary condition at Z — 0 in the dimensionless form would become... 

The Kossovich (Ko) and Posnov (Pn and PUp) numbers [(6.23), (6.28)] defined by Eqs. (6.188)-(6.190) are obtained from the drying model (Eqs. (6.178)-(6.183) completed with specific initial and boundary conditions). Moreover, they are converted into a dimensionless form by applying the pi theo-... 

The above governing equations and boundary conditions are in dimensionless form. The superscript indicates the dimensionless variables. Appropriate scalings for the non-dimensionalization are the wafer radius and the gap thickness h for the r- and z-coordinate, respectively. Obviously, the 0-directional component of the velocity should be scaled by R n . Since the two terms in the continuity equation (Equation (3)) should be balanced, the scale for u, should be also R 0,v. The four dimensionless parameters are appearing in these equations are... 

At the inlet to the finite element domain, the flow is parallel so the equations of the lubrication approximation are used to specify the inlet velocity profile. These equations are integrated from -oo to the inlet, generating an equation relating the flow rate to the inlet pressure. The remaining boundary conditions are as shown in Figure 3. The only complexity here is that the fluid traction, n T, at the free surface has to be specified as a boundary condition on the momentum equation. A force balance there gives, in dimensionless form,... 

The advantage of introducing dimensionless variables has already been shown in section 1.1.4. The dimensionless numbers obtained in that section provide a clear and concise representation of the physical relationships, due to the significant reduction in the influencing variables. The dimensionless variables for thermal conduction are easy to find because the differential equations and boundary conditions are given in an explicit form. 

The temperature held is dependent on this number when heat transfer takes place into a fluid. The Biot number has the same form as the Nusselt number defined by (1.36). There is however one very significant difference, A in the Biot number is the thermal conductivity of the solid whilst in the Nusselt number A is the thermal conductivity of the fluid. The Nusselt number serves as a dimensionless representation of the heat transfer coefficient a useful for its evaluation, whereas the Biot number describes the boundary condition for thermal conduction in a solid body. It is the ratio of L0 to the subtangent to the temperature curve within the solid body, cf. Fig. 2.4, whilst the Nusselt number is the ratio of a (possibly different choice of) characteristic length L0 to the subtangent to the temperature profile in the boundary layer of the fluid. 

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