Non-dimensional form

Equation (77) is converted first to a non-dimensional form and the Fourier Transform Series is used to solve the differential equation for specified initial and boundary conditions. The final solution of Eq. (77) in a non-dimensional form is given by ... 

The transient diffusion of hydrogen in non-dimensional form can be expressed as9... 

In bringing the models to a non-dimensional form, the presence of dominant Peclet and Damkohler numbers in reactive flows is observed. The problems of interest arise in complex geometries-like porous media or systems of capillary tubes. 

Let us write non-dimensional forms for both non-linear adsorption laws. 

Treact — superficial chemical reaction time scale = H/fciR Then after a short calculation we get the non-dimensional form of Equation (24) ... 

The conservation equations representing the intraparticle de-activatiorf in non-dimensional form for spherical geometry are given as follows ... 

Here the equilibrium flow is considered to be in the x — z) - plane with y indicating the wall-normal direction. Above equations are written in non-dimensional form with the shear layer edge velocity, Ug (or an appropriate velocity for other flows), as the velocity scale and L as the length scale so that Re = is the Reynolds number. 

If we introduce a length scale (L), velocity scale U ), temperature scale (AT = T — Too) and pressure scale pU ), then the above equations can be represented in non-dimensional form by,... 

The following substitutions are made to non-dimensionalize the equation by letting u = u/Um, p-y/L and Kn = A7L where L is the characteristic length of the channel. In the non-dimensional form, after substitution of (4.4) into (4.3) the following can be obtained after rearrangement of the terms (Note that the superscript has been dropped from the equation) ... 

To model such an electroosmotic flow and mixing process, we need the following equations. The flow field is described by the Navier-Stokes Equations and the continuity Equation (given below in non-dimensional form) ... 

Application to sintering data has shown that m = 1 or 2, depending on the sample under analysis [6,7]. In non-dimensional form, the integrated equation for either order becomes ... 

The mathematical problem outlined above can be expressed in a variety of ways. Non-dimensional form allows the problem to be expressed in terms of the minimum number of parameters and independent variables. 

The non-dimensional form of the mechanical energy balance is then ... 

Redox parameters analogous to those for acid-base chemistry can be defined for all aqueous systems. The redox intensity factor pE is an energy parameter in non-dimensional form that describes the ratio of electron acceptors (oxidants) and donors (reductants) in a redox couple. The redox potential (Ej ) of the system is an alternative and equivalent intensity factor. Table I summarizes the complete thermodynamic analogy between pH and pE. An analogy between acid-base and redox systems can also be made for capacity factors. 

The mass balance equations given above can be represented in non-dimensional form by employing the following dimensionless variables and parameters ... 

The coordinates X and 2 define the orientation of the plane perpendicular to the thickness of the lubricant film. Transformations may be employed to accommodate the geometry of the boundary surfaces or to throw the treatment into non-dimensional form. 

Very often when we solve the mass balance equation under a transient condition, equations are more conveniently cast into nondimensional form for subsequent numerical analysis. What we will show in this example is the non-dimensional form of the diffusion matrix B. To do this, we need a... 

The mass balance equations presented above are non-linear and must be solved numerically. We have maintained our practice that before the equations are solved numerically, they must be cast into a non-dimensional form, at least the spatial variable must be normalised as it is convenient to discretise a spatial domain having a range of (0, 1). We define the following nondimensional variables... 

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