The pressure equation

The pressure equation is computed in this section for a system of spherical particles. This choice is made only because of notational convenience. We shall quote the corresponding equation for nonspherical particles at the end of this section. 

The pressure is obtained from the Helmholtz free energy by 

Note that the dependence of Q on the volume comes only through the configurational partition function, hence 

In order to perform this derivative, we first express Zj explicitly as a function of V, For macroscopic systems, we assume that the pressure is independent of the geometric form of the vessel. Hence, for convenience, we choose a cube of edge so that the configurational partition function is written as 

With the new set of variables, the total potential becomes a function of the volume  

The relation between the distances expressed by the two sets of variables is 

Combining (5.8.34) and (5.8.35) and transforming back to the original variables, we obtain 

The pressure in the T, V, N ensemble is obtained from the Helmholtz energy by 

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It the pressure is doubled, the driving force is doubled, Cr/Cgm is essentially unaltered, and the diffiisivity, being inversely proportional to the pressure (equation 10.43) is halved. The mass transfer rate therefore remains the same. 

The solution of Equations (5.23) or (5.24) is more straightforward when temperature and the component concentrations can be used directly as the dependent variables rather than enthalpy and the component fluxes. In any case, however, the initial values, Ti , Pi , Ui , bj ,... must be known at z = 0. Reaction rates and physical properties can then be calculated at = 0 so that the right-hand side of Equations (5.23) or (5.24) can be evaluated. This gives AT, and thus T z + Az), directly in the case of Equation (5.24) and imphcitly via the enthalpy in the case of Equation (5.23). The component equations are evaluated similarly to give a(z + Az), b(z + Az),... either directly or via the concentration fluxes as described in Section 3.1. The pressure equation is evaluated to give P(z + Az). The various auxiliary equations are used as necessary to determine quantities such as u and Ac at the new axial location. Thus, T,a,b,. .. and other necessary variables are determined at the next axial position along the tubular reactor. The axial position variable z can then be incremented and the entire procedure repeated to give temperatures and compositions at yet the next point. Thus, we march down the tube. 

Another equation from General Relativity (expressing the First Law of Thermodynamics) is the pressure equation ... 

In the preceding chapters we considered Raoult s law and Henry s law, which are laws that describe the thermodynamic behavior of dilute solutions of nonelectrolytes these laws are strictly valid only in the limit of infinite dilution. They led to a simple linear dependence of the chemical potential on the logarithm of the mole fraction of solvent and solute, as in Equations (14.6) (Raoult s law) and (15.5) (Heiuy s law) or on the logarithm of the molality of the solute, as in Equation (15.11) (Hemy s law). These equations are of the same form as the equation derived for the dependence of the chemical potential of an ideal gas on the pressure [Equation (10.15)]. 

Finally, the subject of bubble dynamics, in which the pressure equation is coupled to the convective diffusion equation, offers a number of unsolved problems which will be considered in a succeeding volume of this series. 

Note that the equations for estimating the pressure dependencies of 7 and aw (Eqs. 2.87 and 2.90) depend on the Pitzer equations (Eqs. 2.76, 2.80, and 2.81) but this is not the case for the pressure dependence of the equilibrium constants (Eq. 2.29) the latter equation is based entirely on partial molar volumes at infinite dilution, which are independent of concentration. Also, compared to the pressure-dependent equation for the equilibrium constant (Eq. 2.29), the pressure equations for activity coefficients (Eq. 2.87) and the activity of water (Eq. 2.90) do not contain compressibilities (K) because the database for these terms and the associated Pitzer parameters are lacking at present (Krumgalz et al. 1999). The consequences of truncating Eqs. 2.80 and 2.81 for ternary terms and Eqs. 2.87 and 2.90 for compressibilities will be discussed in Sect. 3.6 under limitations. 

If the pressure is 1.00 atm, how high will a column of mercury be To answer this question we need to rearrange the pressure equation above by solving for h to get ... 

Note in the last step we simply rearranged the symbols in order to write the expression in a manner typically used in textbooks. This expression P = pgh allows us to calculate the pressure at any depth in any fluid, as long as we know the density of the fluid. We are now ready to substitute the appropriate values into the pressure equation to get ... 

How much pressure will the water exert at a depth of only 5 cm Using the pressure equation we calculate ... 

The IPM is a simple application of the slow-flow approximation to the pressure equation (2)... 

Since no terms are omitted when deriving the pressure equation used by SIMPLER, the resulting pressure field corresponds with the velocity field. Therefore, unlike SIMPLE, the correct velocity field results in the correct pressure field. Consequently, SIMPLER does not require under-relaxation of pressure and performs significantly better than SIMPLE. 

For SIMPLER, the pressure equation is solved based on the updated velocity field. 

To determine the dependence of the activity coefficient yp of a gas in a mixture on the pressure, equation (30.15) is written in the logarithmic form. 

In the above equation, L is the length over which the pressure drop p — p2 is to be calculated pi is the absolute pressure of the flow at an upstream point 1, and p2 is the absolute pressure of the flow at a downstream point 2 / is the Darcy-Weisbach friction factor that can be determined from the Moody Diagram y is the adiabatic exponent (equal to 1.4 for air) and Mi is the Mach number of the flow at the upstream point 1. In addition to the pressure equations, the following equation of state of ideal gas is also needed ... 

The Laplacian operator on the LHS of the pressure equation is the product of the divergence operator originating from the continuity equation and the gradient operator that comes from the momentum equations. The RHS of the pressure equation consists of a sum of derivatives of the convective terms in the three components of the momentum equation. In all these terms, the outer derivative stems from the continuity equation while the inner derivative arises from the momentum equation. In a numerical approximation, it is essential that the consistency of these operators is maintained. The approximations of the terms in the Poisson equations must be defined as the product of the divergence and gradient approximations used in the basic equations. Violation of this constraint may lead to convergence problems as the continuity equation is not appropriately satisfied. 

For SIMPLER, solve the pressure equation for p after is obtained above. 

Thus, another route to estimating A is by integration of the pressure equation (equation (2.8.11)) with respect to volume. This calculation requires that the dependence of g r) on volume or density be known. Since this is usually not known, this method of estimating A is also not convenient in most cases. 

For low densities, this equation gives the same value of PV/(Nk T) as that estimated from the pressure equation (equation (2.9.9)). However, at high densities, the estimate from the compressibility equation is much higher (see fig. 2.10). The disagreement between the equations of state obtained by the two different methods clearly is a result of the approximation made in deriving the PY equation. 

Values of PV/ Nk T) have been ealeulated for the hard-sphere system by eomputer simulation. At high densities, the results obtained in this way fall between those estimated by the pressure and eompressibility equations. This led Carnahan and Starling [24] to propose an equation of state obtained by eombin-ing one-third of the result from the pressure equation (equation (2.9.9)) together with two-thirds of the compressibility result (equation (2.9.10)). The result is... 

The data designated ( ) were obtained using the pressure equation (2.9.9), those designated ( ) using the compressibility equation (2.9.10), and the smooth curve, using the Carnahan-Starling equation (2.9.11). 

This is the pressure equation for a one-component system of spherical particles obeying the pairwise additivity for total potential energy. Note that the first term is the ideal gas pressure. The second is due to the effect of the inter-molecular forces on the pressure. Note that in general, g(R) is a function of the density hence, the second term in (3.37) is not the second-order term in the density expansion of the pressure. 

The pressure equation is very useful in computing the equation of state of a system based on the knowledge of the form of the function g(R). Indeed, such computations have been performed to test theoretical methods of evaluating g(R). 

For a system of rigid, nonspherical molecules, the derivation of the pressure equation is essentially the same as that for spherical molecules. The result is... 

Clearly, in order to express a in terms of g(R), we must know the explicit dependence of g(R) on the density. Thus, if we used the pressure equation in the integrand of (3.43) we need a second integration over the density to get the Helmholtz energy per particle. 

Substituting this into the pressure equation (25) gives... 

Note that at this point no recourse need be made to any theory of energy transport, since the only equation necessary to describe the motion of the fluid is the pressure equation. The calculated pressure difference as a function of subcooling is shown in Fig. 20 from the data of Gunther and Kreith (GIO) at a heat flux of 2.75 Btu/(sq in.)(sec) and a superficial liquid velocity of 10 ft/sec past the heating strip. One interpretation of these results is that the liquid surrounding the bubble has been given an initial supply of kinetic... 

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