Dimensionless terms

The flowrate of oil into the wellbore is also influenced by the reservoir properties of permeability (k) and reservoir thickness (h), by the oil properties viscosity (p) and formation volume factor (BJ and by any change in the resistance to flow near the wellbore which is represented by the dimensionless term called skin (S). For semisteady state f/owbehaviour (when the effect of the producing well is seen at all boundaries of the reservoir) the radial inflow for oil into a vertical wellbore is represented by the equation ... 

Fig. 10. The Thiele plot accounting for the influence of intraparticle mass transport on rates of catalytic reaction. The dimensionless terms Tj and ( ) are the...

Application of the Baker-Strehlow method for evaluating blast effects from a vapor cloud explosion involves defining the energy of the explosion, calculating the scaled distance (/ ), then graphically reading the dimensionless peak pressure (Ps) and dimensionless specific impulse (i ). Equations (4.41) and (4.42) provide the means to calculate incident pressure and impulse based on the dimensionless terms. 

Reduce the number of variables by combining the variables into a few dimensionless terms. These dimensionless terms can be used to assist in the design of a physical model that, in turn, can be used for experimentation (on an economic scale) that will yield insight into the prototype device or system. 

Develop descriptive governing relationships (in equation and/or graphical form) utilizing the dimensionless terms (with the variables) and the model exp>erimental results that describe the performance of the physical model. 

Develop device or system designs utilizing the descriptive governing relationships between the dimensionless terms (and subsequently between the variables) that are similar to those of the tested physical model, but are the dimensional scale of the desired device or system. 

Applications of the Buckingham FI theorem results in the formulation of dimensionless terms called FI ratios. These FI ratios have no relation to the number 3.1416. 

Assuming that the initial sodium concentration of glass particle with a radius R is C, and its surface sodium concentration Cj is zero at t > 0, dimensionless terms can be written as... 

Further examples of the use of dimensionless terms in dynamic modelling applications are given in Sec. 1.2.5.1, Sec. 4.3.6.1 and 4.3.7 and in the simulation examples KLADYN, DISRET, DISRE, TANKD and TUBED. 

For a first-order reaction n=l, and the dimensionless time is given by k t. Make a series of runs with different initial concentrations and compare the results, plotting the variables in both dimensional and dimensionless terms. 

The above model equations are also presented in dimensionless terms by Ramirez (1989). 

In dimensionless terms, there is a critical value for S (Damkohler number) that makes ignition possible. From Equation (4.23), this qualitatively means that the reaction time must be smaller than the time needed for the diffusion of heat. The pulse of the spark energy must at least be longer than the reaction time. Also, the time for autoignition at a given temperature T is directly related to the reaction time according to Semenov (as reported in Reference [5]) by... 

It is very useful to minimize significant variables by expressing the equations in dimensionless terms. To do this we need a characteristic length to normalize z. If we had a finite source diameter, D, it would be a natural selection however, it does not exist in the point source problem. We can determine this natural length scale, zc, by exploring the equations dimensionally. We equate dimensions. From Equation (10.22),... 

This set of dimensionless groups and variables represents a fairly complete set of dependent variables and the independent coordinates, time, property and geometric parameters for fire problems. The presentation of dimensionless terms reduces the number of variables to its minimum. In some cases, restrictions (e.g. steady state, two-dimensional conditions, etc.) will lead to a further simplification of the set. However, in general, we should consider fire phenomena, with water droplet interactions, that have the functional dependence as follows ... 

Defining Z = z/L, t=T/T0, C=CA/CA0> =Dt/L2, the model equations can be recast into dimensionless terms to give the following dimensionless model equations ... 

There are three levels of increasing difficulty in computing the mathematical expressions defining, in dimensionless terms, the current responses in cyclic voltammetry or with any other analytical techniques. The simplest case is that of an analytical expression. This is found, for example, for a Nernstian... 

Coming back to the original space, we obtain integral relationships linking, in dimensionless terms, the concentrations at the electrode surface to the current ... 

The pure kinetic conditions still apply if electron transfer is not unconditionally fast and Nemst s law has to be replaced by the law that governs the electron transfer kinetics as boundary condition, that is, in dimensionless terms,... 

Investigation of the competition between the ECE and DISP pathways requires considering the full partial derivative equation system involving all three species A, B, and C. In dimensionless terms,... 

The potential distance between the first and second waves depends, in dimensionless terms, on the parameter... 

The concentration profile of B is squeezed within the reaction layer. It may be analyzed in dimensionless term so as to obtain the expression of the yields with introduction of a minimal number of parameters. This is arrived at by normalizing the space variable versus the reaction layer thickness as y = xy/k /D(y = 1 corresponds to x = fi) and the concentrations as... 

AU is a dimensionless term known as the absorbance unit. The term AU was originally introduced because some analysts thought that all parameters should have units ... 

Either v, or Ea can be calculated from Eq. (9.20), if there is one more relationship between the two terms. In Fig. 9.12, v,. related to the singledrop velocity Vp according to Eq. (9.15) is plotted as a function of the drop holdup for droplet swarms with the Archimedes number as a dimensionless term for the drop diameter for the measured values. It can be seen that the relative velocity constantly decreases, as the holdup of the drops, e, increases and the size of the drops in the swarm decreases. 

Reduction of the number of parameters required to define the problem. The n theorem states that a physical problem can always be described in dimensionless terms. This has the advantage that the number of dimensionless groups, which fully describe it, is much smaller than the number of dimensional physical quantities. It is generally equal to the number of physical quantities minus the number of basic units contained in them. 

The previous chapter has provided some indication of the behaviour which can be exhibited by the simple cubic autocatalysis model. In order to make a full analysis, it is convenient both for algebraic manipulation and as an aid to clarity to recast the rate equations in dimensionless terms. This is meant to be a painless procedure (and beloved of chemical engineers even though traditionally mistrusted by chemists). We aim wherever possible to make use of symbols which can be quickly identified with their most important constituents thus for the dimensionless concentration of A we have a, with / for the dimensionless concentration of B. Once this transformation has been achieved, we can embark on a quite detailed and comprehensive analysis of the behaviour of this prototype chemical oscillator. 

We may now allow for reactant consumption explicitly by restoring the exponential decay, p = p0e- °. In dimensionless terms this means recognizing that p is a time-dependent parameter, p = /i0e . The governing rate equations are... 

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