Pressure terms

The Washburn equation has most recently been confirmed for water and cyclohexane in glass capillaries ranging from 0.3 to 400 fim in radii [46]. The contact angle formed by a moving meniscus may differ, however, from the static one [46, 47]. Good and Lin [48] found a difference in penetration rate between an outgassed capillary and one with a vapor adsorbed film, and they propose that the driving force be modified by a film pressure term. 

Elimination of the pressure term from the equation of motion does not automatically yield a robust scheme for incompressible flow and it is still necessary to satisfy the BB stability condition by a suitable technique in both forms of the penalty method. 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

In the continuous tenalty method prior to the discretization of the flow equations the pressure term in the equation of motion is substituted by the penalty relationship, given as Equation (3.6). Therefore using Equations (4.4) and (4.1), we have... 

As described in the discrete penalty technique subsection in Chapter 3 in the discrete penalty method components of the equation of motion and the penalty relationship (i.e. the modified equation of continuity) are discretized separately and then used to eliminate the pressure term from the equation of motion. In order to illustrate this procedure we consider the following penalty relationship... 

Figure 1-3 presented a general form performance curve for each of the compressors. The centrifugal compressor exhibited a relatively flat curve compared to the other machines. Flat is defined as a relatively low head rise for a volume change. Translated to pressure terms, it means a relatively low pressure change for a given volume change. It is important to understand some of the basics that contribute to the curve shape. 

Fig. 5.10 shows the exergy losses as a fraction of the fuel exergy (including the partial pressure terms referred to in Section 2.4) for the General Electric LM 2500 [CBT]ic plant. 

Film coefficients calculated by this equation, while useful, are quite strongly influenced by the effects of pressure of the system. In order to somewhat compensate for this, variable exponents for the pressure term of the equation are used as follows ... 

Incompressible Limit In order to obtain the more familiar form of the Navier-Stokes equations (9.16), we take the low-velocity (i,e. low Mach number M = u I /cs) limit of equation 9,104, We also take a cue from the continuous case, where, if the incompressible Navier-Stokes equations are derived via a Mach-number expansion of the full compressible equations, density variations become negligible everywhere except in the pressure term [frisch87]. Thus setting p = peq + p and allowing density fluctuations only in the pressure term, the low-velocity limit of equation 9,104 becomes... 

There is a method that can be used for this analysis. It is extremely complex so it requires using a computer. In general, equations are generated to determine the moment and thrust created in the invert area of the deflected pipe, where a pressure term is superimposed. This analysis must examine the strains in the outer and innermost fibers of the pipe to verify that its wall structure is adequate and not overstrained. During this analysis the pipe must be examined under conditions of no pressure, minimum pressure, and maximum pressure. 

Pred = reduced pressure termed maximum internal... 

For this reason, a square root of pressure term was introduced into the equation for Kw. Further experimental work would be desirable if pressures greatly different from 25 atm were to be used. 

Characteristic of most equations for surface-controlled kinetics, as opposed to diffusion-controlled kinetics, are a number of partial pressure terms, often to high powers. When large changes in partial pressures are made, differences between the observed and the calculated reaction can easily equal a factor of 1000 or more. When diffusion-type kinetics are used, one seldom finds differences exceeding a factor of two or three. While this may not seem very accurate, comparison of the two methods is rather startling. 

Pressures, Gun. Pressures within a gun tube or barrel, a used in design practices. Because of the wide variations in size, wall ratios, heat dissipation, required safety actors, etc, design practices, vary for the different types of weapons. Some pressure terms have significance in the design of (a) all tubes and barrels, (b) cannon tubes, (c) recoilless rifle tubes, and (d) small arms, barrels, viz ... 

The wearless friction consists of two components, namely the adhesion term Cj and the pressure term CiP xt-Similar expressions for the shear stress, e.g., Tc=TQ + iJiP, were also derived by other investigators [26,27], where tq represents the contribution from adhesion and yu, is referred to as the friction coefficient. 

A finite correlation may replace the slip equations (6-8). With assumptions (1) and (2) and use of Eqs. (6-8)--(6-13), instead of two Eqs. (6-5), the pressure term is present only in Eq. (6-13), which may be solved separately. With assumptions (1) and (3),the phase energy equation (6-9) becomes equivalent to the phase mass conservation equation (6-3), thus reducing the order of the set. 

The fluid-particle interaction force, omitting the virtual mass term and combining the pressure terms in the equation of motion becomes... 

These convective transport equations for heat and species have a similar structure as the NS equations and therefore can easily be solved by the same solver simultaneously with the velocity field. As a matter of fact, they are much simpler to solve than the NS equations since they are linear and do not involve the solution of a pressure term via the continuity equation. In addition, the usual assumption is that spatial or temporal variations in species concentration and temperature do not affect the turbulent-flow field (another example of oneway coupling). 

The first term on the right-hand side of Eqs. (145) and (146) is a pressure term shared by both phases. The purpose of this term (when ps and pg are constant) is to ensure that the volume-average velocity, defined by... 

Thus, just as for incompressible single-phase flow, the pressure p constrains the velocity fields to ensure (in the case of multiphase flows) that the sum of the phase volume fractions equals unity. In the presence of mass transfer, the right-hand side of Eq. (148) is nonzero nevertheless, the role of the pressure is still the same. Finally, we should note that in gas-solid flows the maximum volume fraction of the solid phase is less than unity due to physical constraints (i.e., when particles are close packed there is still room for the gas phase so that 0solid-pressure term ps that becomes extremely large when ag approaches its minimum value (e.g., oc — 0.4). 

Equal and constant specific heats, cp = constant. From the Reynolds transport theorem, Equation (3.9), the pressure terms can be combined as... 

This simple calculation shows why it is more difficult to breathe when up a mountain than at ground level the pressure term in Equation (8.18) decreases, so the rate at which oxygen enters the blood decreases in proportion to the decrease in the oxygen partial pressure. And the partial pressure is smaller at high altitudes than at sea level. 

It is appropriate here to define some pressure terms. Consider Bernoulli s equation for frictionless flow with no pump in the section ... 

The pressure term in (2.121) can be rewritten using the continuity equation as... 

The pressure term is thus responsible for spatial transport due to the fluctuating pressure field. 

DNS of homogeneous turbulence thus involves the solution of a large system of ordinary differential equations (ODEs see (4.3)) that are coupled through the convective and pressure terms (i.e., the terms involving T). 

Turbulent mixing (i.e., the scalar flux) transports fluid elements in real space, but leaves the scalars unchanged in composition space. This implies that in the absence of molecular diffusion and chemistry the one-point composition PDF in homogeneous turbulence will remain unchanged for all time. Contrast this to the velocity field which quickly approaches a multi-variate Gaussian PDF due, mainly, to the fluctuating pressure term in (6.47). 

For regular solutions, the influence of the solvent is determined by molar volumes and internal pressure terms. Since the molar volumes do not vary greatly, the internal pressure factor is more important. If the internal pressures of solvent, reactants and activated complex are similar, the solvent will have little effect on the rate of reaction as compared to a solvent in which reaction behaves ideally. If the internal pressure of the solvent is close to that of reactants but appreciably different from that of the activated complex, the rate of reaction in this solvent will be low. On the other hand, if solvent has an internal pressure similar to that of activated complex, but different from one or both the reactants, rate of reaction in this solvent will be high. Since the activated complex has properties which approach the properties of the product, it may be concluded, in general, that the reaction in which the products are of higher internal pressure than the reactants, it is accelerated by solvent of high internal pressure. 

Values based on Eq. (39) are given in Fig. 6 in terms of S ps and in Fig. 7 in terms of equivalent charge-to-mass ratio for various values of surface tension. Actually the maximum value of Sps before breakup, as given by Eq. (39), is independent of pressure. The pressure term has been incorporated in Figs. 6 and 7 in order to permit direct comparison with later developments of instability due to corona. For present purposes, Figs. 6 and 7 will give a correct representation of Eq. (39) if all pressure terms (in the coordinates or in the parameter) are ignored or assumed to be unity. 

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