Simultaneous Solution

When only the total system composition, pressure, and temperature (or enthalpy) are specified, the problem becomes a flash calculation. This type of problem requires simultaneous solution of the material balance as well as the phase-equilibrium relations. 

This appears to be a complex problem requiring simultaneous solution of the sequence together with heat integration. 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

Equations of continuity and motion in a flow model are intrinsically connected and their solution should be described simultaneously. Solution of the energy and viscoelastic constitutive equations can be considered independently. 

For an isothermal system the simultaneous solution of equations 30 and 31, subject to the boundary conditions imposed on the column, provides the expressions for the concentration profiles in both phases. If the system is nonisotherm a1, an energy balance is also required and since, in... 

Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the flmd. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-HiU, New York, 1958 and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949 vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation hL I L p gp At cjl... 

The variables that are combined hnearly are In / 17T, and In C, Multilinear regression software can be used to find the constants, or only three sets of the data smtably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of / C, and Ci,. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. 

Simultaneous solution by the Newton-Raphson method yields x = 0.9121, y = 0.6328. Accordingly, the fractional compositions are ... 

The time-dependent nature of the emergency pressure relieving event is obtained by the simultaneous solution of Eqs. (26-27) and (26-28). Generally, the only unknown parameters in these two equations are the venting rate W and the vent stream quahty (Xo). The vent rate W at any instant is a func tion of the upstream conditions and the relief system geometry. 

The couphng equation is a vapor mass balance written at the vent system entrance and provides a relationship between the vent rate W and the vent system inlet quahty Xq. The relief system flow models described in the following section provide a second relationship between W and Xo to be solved simultaneously with the coupling equation. Once W andXo are known, the simultaneous solution of the material and energy balances can be accomplished. For all the preceding vessel flow models and the coupling equations, the reader is referred to the DIERS Project Manual for a more complete and detailed review. 

Other cases, neglecting heat effects would cause serious errors. In such cases the mathematical treatment requires the simultaneous solution of the diffusion and heat conductivity equations for the catalyst pores. 

Industrial design problems often oeeur in tubular reaetors that involve the simultaneous solution of AP, energy, and mass balanees. 

EquatitHis (8.4)-(8.8) represent a complete mathematical description of the chemical equilibrium between a rich phase and the y th MSA. The simultaneous solution... 

The experiments were performed on two sets of beams with the beam axis at 0 and 90°, respectively, to the fiber direction of the odd-numbered layers. The beams were 1-in (25.4-mm) wide,. 12-in (3-mm) thick, and of 6-in (152-mm) span. Strain rosettes were located on the upper and lower beam surfaces so that the middle-surface strains and curvatures can be calculated from simultaneous solution of... 

To see the type of differences that arises between an iterative solution and a simultaneous solution of the coefficient equations, we may proceed as follows. Bor the thirteen moment approximation, we shall allow the distribution function to have only thirteen nonzero moments, namely n, v, T, p, q [p has only five independent moments, since it is symmetric, and obeys Eq. (1-56)]. For the coefficients, we therefore keep o, a, a 1, k2), o 11 the first five of these... 

In the theoretical treatment, the heat- and mass-transfer processes shown in Fig. 6 were considered. Simultaneous solution of the equations describing the behavior of the unsteady-state reaction system permits the temperature history of the propellant surface to be calculated from the instant of oxidizer propellant contact to the runaway reaction stage. 

Simultaneous solution of these equilibrium relations (coupled with the conservation equations x+ x-f = 1 and x/ + x/ = 1) gives the coexistence curve for the two-phase system as a function of pressure. 

When the three coefficients oc12, oc13, and a23 are known, the coexistence curve can be found by simultaneous solution of Eqs. (119) and (120). A numerical iterative technique given by Hennico and Vermeulen (HI) was used by Balder for performing these calculations with a digital electronic computer. 

In Eq. (128), the superscript V stands for the vapor phase v2 is the partial molar volume of component 2 in the liquid phase y is the (unsym-metric) activity coefficient and Hffl is Henry s constant for solute 2 in solvent 1 at the (arbitrary) reference pressure Pr, all at the system temperature T. Simultaneous solution of Eqs. (126) and (128) gives the solubility (x2) of the gaseous component as a function of pressure P and solvent composition... 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

The simultaneous solution of Eqs. (1) and (2) is shown in Fig. 2, in which the two curves represent the two relations between the variables. Their intersection corresponds to 0 = 0.935a and /S = 0.87a, which are thus minimum values for these exchange integrals. 

Again, a negative root was rejected. The simultaneous solution also produces the stoichiometric relationship... 

Note that Step 4 in this procedure uses the old value for u since the new value is not yet known. The new value could be used in Equation (3.29) if Umw is found by simultaneous solution with Equation (3.30). However, complications of this sort are not necessary. Taking the numerical limit as Az 0 removes the errors. As a general rule, the exact sequence of calculations is unimportant in marching schemes. What is necessary is that each variable be updated once during each Az step. 

Since a stable steady state is sought, the method of false transients could be used for the simultaneous solution of Equations (5.29) and (5.31). However, the ease of solving Equation (5.29) for makes the algebraic approach simpler. Whichever method is used, a value for UAext pQCp is assumed and then a value for Text is found that gives 413 K as the single steady state. Some results are... 

Equation (8.9) can be applied to any reaction, even a complex reaction where ctbatch(t) must be determined by the simultaneous solution of many ODEs. The restrictions on Equation (8.9) are isothermal laminar flow in a circular tube with a parabolic velocity profile and negligible diffusion. 

The numerical techniques of Chapter 8 can be used for the simultaneous solution of Equation (9.3) and as many versions of Equation (9.1) as are necessary. The methods are unchanged except for the discretization stability criterion and the wall boundary condition. When the velocity profile is flat, the stability criterion is most demanding when at the centerline ... 

Solution The axial dispersion model requires the simultaneous solution of Equations (9.14) and (9.24). Piston flow is governed by the same equations except that D = E = Q. The following parameter values give rise to a near runaway ... 

The point contact thin him lubrication problem with micropolar fluids requires the simultaneous solution of several governing equations as described below. 

The determination of the steam density, pj, therefore requires the simultaneous solution of two algebraic equations. This represents an IMPLICIT algebraic loop and cannot be solved within a simulation program without the incorporation of a trial and error convergence procedure. 

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