Equation dimensionless

Simplified Dimensional Equations Equation (5-32) is a dimensionless equation, and any consistent set of units may be used. Simphfied dimensional equations have been derived for air, water, and organic hquids by rearranging Eq. (5-32) into the following form by collecting the fluid properties into a single factor ... 

Norris and Spofford [Tron.s. Am. Soc. Mech. Eng., 64, 489 (1942)] correlate their results for air by the dimensionless equation of Pohlhausen ... 

Operating holdup may be estimated by the dimensionless equation of Buchanan [Jnd. Eng. Chem. Eundam., 6,400 (1967)] ... 

This expression represents the first form of the general dimensionless equation of sedimentation theory. As the desired value is the velocity of the particle, equation 64 is solved for the Reynolds number ... 

This is the second form of the dimensionless equation for sedimentation. The Reynolds number also may be calculated from this equation ... 

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. 

P0 = Np = Power number, dimensionless, Equation 5-19 Ppew = Plate coil width, one plate, ft Ap = Pressure drop, psi AP0 = Pressure drop for open pipe, psi AP, = Static mixer pressure drop in turbulent flow, psi Q = Flow rate or pumping capacity from impeller, cu l t/sec, or Ls/1... 

Significant simplification of the governing equations may be achieved by using a quasi-one-dimensional model for the flow. Assume that (1) the ratio of meniscus depth to its radius is sufficiently small, (2) the velocity, temperature and pressure distributions in the cross-section are close to uniform, and (3) all parameters depend on the longitudinal coordinate. Differentiating Eqs. (8.32-8.35) and (8.37) we reduce the problem to the following dimensionless equations ... 

The ratio of the observed reaction rate to the rate in the absence of intraparticle mass and heat transfer resistance is defined as the elFectiveness factor. When the effectiveness factor is ignored, simulation results for catalytic reactors can be inaccurate. Since it is used extensively for simulation of large reaction systems, its fast computation is required to accelerate the simulation time and enhance the simulation accuracy. This problem is to solve the dimensionless equation describing the mass transport of the key component in a porous catalyst[l,2]... 

The significance of this dimensionless equation form is now that only the parameter (k x) is important and this alone determines the system dynamics and the resultant steady state. Thus, experiments to prove the validity of the model need only consider different values of the combined parameter (k x). 

The model is written in both dimensional and dimensionless forms. This example provides experience in the use of dimensionless equations. 

For simplicity the prime ( ) is dropped in the dimensionless equations. Example KLADYN DYNAMIC KLA-MEASUREMENT... 

The shaft power required to drive an agitator can be estimated using the following generalised dimensionless equation, the derivation of which is given in Volume 2, Chapter 13. 

For viscous dominated flows, it can be assumed that the gas inertia and the gas gravitational forces are negligible. By dropping the gas inertia and gravity time from the gas momentum equation and simplifying the dimensionless drag coefficient to the linear viscous term, the set of dimensionless equations does not include gas-to-solid density ratio as a parameter. 

Verify the form of the dimensionless equations. Rewrite the program in dimensional form. 

FIGURE 2.1. EC reaction scheme in cyclic voltammetry. Kinetic zone diagram showing the competition between diffusion and follow-up reaction as a function of the equilibrium constant, K, and the dimensionless kinetic parameter, X. The boundaries between the zones are based on an uncertainty of 3 mV at 25°C on the peak potential. The dimensionless equations of the cyclic voltammetric responses in each zone are given in Table 6.4. 

Section 2.2.6) situation defined by the following dimensionless equation ... 

We may thus translate the current and potential relationships of Section 1.3.1 [equations (1.11)] into the following dimensionless equations ... 

Homogeneous Catalytic EC Mechanism The system is governed by the following dimensionless equations (we need not consider equations involving p, since as in all preceding cases, p = 1 — q), where two additional normalized rate parameters are introduced ... 

By substituting Eq.(33) into Eq.(27) the dimensionless equations of the reactor are simplified as follows ... 

By appropriate rescaling of the variables x t and h to X, T and H, the parameters in the equation of motion can be scaled out, leading to a dimensionless equation. Considering the present form (4) of the current, without the ) term, the equation of motion of the surface can be written in form of a functional derivative ... 

X by the terrace width a/s, y by the distance / over which the saturated value of a/ reaches the terrace width squared, a/sy, and t by the corresponding saturation time 4 we should arrive at an isotropic, dimensionless equation, i.e.. 

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. 

The third technique uses the fundamental equations, and converts them into dimensionless equations with length and velocity scales important to the application. From these dimensionless equations, dimensionless numbers will evolve. It is this technique that will be described herein. 

This equation may be solved for C if k -A and the kinetic constants are known (Mosbach, 1976) or C j may be obtained graphically (Mosbach, 1976) using the following dimensionless equation ... 

Note that in the following dimensionless equations the asterisk is dropped from p, c g and M and that... 

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