Fractional time

The cloud inventory is equal to the flash fraction times the amount of fuel released. To allow for spray and aerosol formation, the cloud inventory should be multiplied by 2. This number may not, of course, exceed the total amount of fuel released. 

Wilkinson" has generalized the fractional time method in the following way. For rate equation dcldt = —kc", the integrated equation, for n 1, is... 

Using the same operating reflux (same fraction times the minimum) as was used in Example 8-10, calculate the theoretical plates required for feed of the following thermal conditions Use Figure 8-27. 

The time required to convert a given fraction of the limiting reagent is a characteristic of the rate equation. A comparison of successive half-times, or any other convenient fractional time, reveals whether a reaction follows any simple-order rate law. Thus, the ratio of the time to reach 75 percent completion to that for 50 percent is characteristic of the reaction order. Values of this ratio for different orders are as follows ... 

The partial pressure of a component in a gas mixture is its mole fraction times the total pressure. Example illustrates calculations with gas mixtures. 

Process Unit Building No. Side-on Overpressure (psi) Incident Frequency Building Type Occupant Vulner- ability Fractional Time of Attendance Individual 1 Risk I... 

One approach that allows increased chromatographic flow rates without loss of resolution entails the use of microparticulate stationary-phase media of very narrow diameter. This effectively reduces the time required for molecules to diffuse in and out of the porous particles. Any reduction in particle diameter dramatically increases the pressure required to maintain a given flow rate. Such high flow rates may be achieved by utilizing high-pressure liquid chromatographic systems. By employing such methods, sample fractionation times may be reduced from hours to minutes. 

In the first fractional time step, the micromixing term is solved separately for each notional particle 153... 

The overall fractional-time-stepping process can be represented by... 

Fractional time stepping is widely used in reacting-flow simulations (Boris and Oran 2000) in order to isolate terms in the transport equations so that they can be treated with the most efficient numerical methods. For non-premixed reactions, the fractional-time-stepping approach will yield acceptable accuracy if A t r . Note that since the exact solution to the mixing step is known (see (6.248)), the stiff ODE solver is only needed for (6.249), which, because it can be solved independently for each notional particle, is uncoupled. This fact can be exploited to treat the chemical source term efficiently using chemical lookup tables. 

The intra-cell processes are common to all PDF codes, and are treated the same in both Eulerian and Lagrangian PDF codes.8 On the other hand, inter-cell processes are treated differently in Eulerian PDF codes due to the discrete representation of space in terms of x . In PDF codes, fractional time stepping is employed to account for each process separately. Methods for treating chemical reactions and mixing are described in Section 6.9. Thus we will focus here on the treatment of inter-cell processes in Eulerian PDF codes. 

In the MC simulation, these equations are treated numerically using fractional time stepping. For unsteady flow, the simulation time step At is determined from the FV code by... 

The crux of the argument is that the average molecule is representative of a large number of identical molecules of the solute. Hence R, defined previously as the fractional time the average molecule spends in the mobile phase, may also be viewed as the fraction... 

The local mole fractions cannot be measured easily, but must be related to the overall composition. A simple correlation can be obtained from statistical thermod5mamics for which the quotient of the local mole fractions equals the quotient of the total mole fractions times a Boltzmann s factor according to ... 

The radius of unreacted core in terms of fractional time for complete conversion is obtained by combining Eqs. 8 and 9, or... 

Specie Bliaa-Calculated LCtso, Concentration, mg>min/m3 mg/m) Duration of Expoaure min, Mortality Fraction Time to Death, hb... 

Note that in die leapfrog method, position depends on the velocities as computed one-half time step out of phase, dins, scaling of the velocities can be accomplished to control temperature. Note also that no force-deld calculations actually take place for the fractional time steps. Eorces (and thus accelerations) in Eq. (3.24) are computed at integral time steps, halftime-step-forward velocities are computed therefrom, and these are then used in Eq. (3.23) to update the particle positions. The drawbacks of the leapfrog algorithm include ignoring third-order terms in the Taylor expansions and the half-time-step displacements of the position and velocity vectors - both of these features can contribute to decreased stability in numerical integration of the trajectoiy. 

The partial pressures are given by the mole fractions times the total pressure, and are substituted into the Kv expression, to give... 

Therefore, the bubble-point pressure of an ideal liquid solution at a given temperature is simply the summation of the products of mole fraction times vapor pressure for each component. 

WC2 lb ethane/lb mole mix (mole fraction times molecular weight of component)... 

The product x m is equal to W /M/ where W/ is the weight rate of fuel flow and Mj is the molecular weight. The concentrations may be expressed as the mole fraction times the molar density or... 

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