Balance in Dimensionless Variables

Momentum Balance in Dimensionless Variables For pipe flow, it is necessary to solve the momentum balance. The momentum balance is simplified by using the following dimensionless variables ... 

Stability and dynamics of chemical reactions within batch systems date back to the pioneering work of Russian scientist Nikolay Semenov on chain reactions and combustion, for which he shared the 1956 Nobel Prize with Sir Cyril Hinshelwood. There have since been considerable literature contributions, the most widely studied system being the irreversible first order reaction A B with Arrhenius kinetics k T) = kot p —E/RT). The mass balance (in dimensionless variables) within the reactor yields ... 

FIGURE 3 (Left) Incorrect and correct ways of making the balance when one phase is uniform. The latter is in dimensionless variables. (Right) Incorrect and correct solutions. (Top) How U(x) can become negative if 8 > I. 

Returning to the mass-balance equation, we write it in terms of fractional conversion (another dimensionless variable) as... 

In a more sophisticated analysis these functions can be found as the solutions of the dynamic and energy balance equations for filling a mold. 0m is the dimensionless temperature of the mold To is the initial temperature of the reactive mixture co = (H2kr,T)/a is the dimensionless factor characterizing the ratio of time scales for heat transfer and the chemical reaction. Other dimensionless variables are as follows ... 

The variables in the balance eqns. (6.279) to (6.283) can be scaled into dimensionless variables as... 

For a safe operation, the runaway boundaries of the phenol-formaldehyde reaction must be determined. This is done here with reference to an isoperibolic batch reactor (while the temperature-controlled case is addressed in Sect. 5.8). As shown in Sect. 2.4, the complex kinetics of this system is described by 89 reactions involving 13 different chemical species. The model of the system consists of the already introduced mass (2.27) and energy (2.30) balances in the reactor. Given the system complexity, dimensionless variables are not introduced. 

We denote by 07 = Hi/HijS the dimensionless variables corresponding to the energy flow rates Hiy i = 1,..., N (the subscript s denotes steady-state values). Appending a generic representation of the overall and component material-balance equations, with xtfc IRm being the material-balance variables, the overall mathematical model of the process in Figure 6.1 becomes... 

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... 

Because the quantity in curly braces is expressed entirely in terms of dimensionless variables, it is assumed to be 0(1). To preserve the balance of terms in (5-13), and thus ensure that mass conservation is respected, it follows that... 

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

Equations describing yioo and Tq can be obtained from the equations of mass and energy balance for a unit volume of the gas-liquid mixture, which, when written in the dimensionless variables introduced above, take the form ... 

Apply the method of successive approximations to solve the aggregation problem for the constant frequency. Assume that the population balance equation is given by Eq. (4.3.4) in terms of the dimensionless variables of Section 4.3 with a(x, x ) = 1 and that the initial distribution is given by (5(x — 1). 

The dimensionless energy balance in terms of the scaled variables becomes... 

According to Eqs. (5.1-5.5) and the theory of small disturbances, the transient flow is decomposed and determined by quantities of the time steady flow and disturbance variables. The disturbance variables describe the temporal and spatial evolution of superimposed small waveUke disturbances, which move alrnig the jet and finally lead to its breakup. Starting point of our analysis of the stability behavior is the pressure balance in radial direction, which we already used by the derivation of Eq. (5.8). Considering the normal stresses in radial direction we receive in dimensionless notation... 

Rewriting the governing equations in the dimensionless form simphfies the estimation of the behavior of processes on different scales. Since all terms in the equations have the same dimensions, we can obtain a dimensionless eqnation by dividing the terms in the equations by any of the other terms. However, dividing by the term that we assume is the most important will simplify the analysis. In many cases, the convective term is the dominating term. The tables in Appendix B show how the dimensionless variables are formed from momentum, heat, and mass balances. 

Using dimensionless variables allows for the development of scale-independent design equations. Note that the species balance equation is almost identical to the heat balance, and that, by replacing the Pr number in Equation (4.25) with the Schmidt number Sc, we obtain a similar correlation for mass transfer as for heat transfer. In principle they describe similar phenomena, and, if we replace heat conduction k with heat diffusion a = k/pcp containing the same dimension as diffusivity (m s ), we obtain the same expression. 

Measurements of Viscosity and Elasticity in Shear (Simple Shear) Shear viscosity J] and shear elasticity G are determined by evaluating the coefficients of the variables x and x, respectively, which result when the geometry of the system has been taken into account. The resulting equation of state balances stress against shear rate y (reciprocal seconds) and shear y (dimensionless) as the kinematic variables. For a purely elastic, or Hookean, response ... 

The energy balance is much closer to being satisfied but other functions show greater discrepancies than before. This is, in fact, no more or less than we should have expected. Since the equations are not dimensionless, we will find that the energy balance is always the last equation to be converged. The next set of values of the independent variables is found by solving Eq. C.2.5 again... 

The design equations and the species concentration relations contain another dependent variable, 6, the dimensionless temperature, whose variation during the reactor operation is expressed by the energy balance equation. For ideal batch reactors with negligible mechanical shaft work, the energy balance equation, derived in Section 5.2, is... 

Here u(f) is the inhibitor and a x, t) is the activator variable. In the semiconductor context u t) denotes the voltage drop across the device and a(x, t) is the electron density in the quantum well. The nonlinear, nonmonotonic function /(a, u) describes the balance of the incoming and outgoing current densities of the quantum well, and D(a) is an effective, electron density dependent transverse diffusion coefficient. The local current density in the device is j a, u) = (/(a, u) + 2a), and J = j jdx is associated with the global current. Eq. (5.22) represents Kirchhoff s law of the circuit (5.3) in which the device is operated. The external bias voltage Uq, the dimensionless load resistance r R, and the time-scale ratio e = RhC/ra (where C is the capacitance of the circuit and Ta is the tunneling time) act as control parameters. The one-dimensional spatial coordinate x corresponds to the direction transverse to the current flow. We consider a system of... 

The balance of Marangoni and viscous stresses (8.153), reformulated in terms of T, is integrated to obtain the surfactant distribution and yields T as a function of ( ) and the dimensionless Marangoni number Ma. The surfactant distribution can be integrated over the cap region to obtain the total amount on the surface, M. The variable M is also computed independently from the surfactant conservation equations and equating the two expressions yields Once ( ) is specified, the drag coefficient and terminal velocity can be calculated. 

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