Calculation of Pressure

The weight of an object that is subject to Earth s gravitational pull is equal to its mass times the gravitational constant, 9.80665 m/s. Thus, the force exerted by this column of air is 

Pi essure, though, is force per unit area. Specifically, pressure in pascals is equal to force in i tons per square meter. We must first convert area from cm to m.  

This pressure is roughly equal to 1 atm ( 1 X 10 Pa), which we would expect at sea level. 

We can calculate the pressure exerted by a column of any fluid (gas or liquid) in the same way. In fact, this is how atmospheric pressure is commonly measured—by determining the height of a column of mercury it can support. 

In this section, several strategies for determining the pressure in the incompressible flow limit is outlined for pressure-based methods. The extension of the pressure-correction approach to arbitrary Mach numbers is examined. 

For pressure-based techniques, the lack of an independent equation for the pressure complicates the solution of the momentum equation. Furthermore, the continuity equation does not have a transient term in incompressible flows because the fluid transport properties are constant. The continuity reduces to a kinematic constraint on the velocity held. One possible approach is to construct the pressure field so as to guarantee satisfaction of the continuity equation. In this case, the momentum equation still determines the respective velocity components. A frequently used method to obtain an equation for the pressure is based on combining the two equations. This means that the continuity equation, which does not contain the pressure, is employed to determine the pressure. If we take the divergence of the momentum equation, the continuity equation can be used to simplify the resulting equation. 

For the case of constant fluid properties (e.g., density and viscosity), the equation reduces to  

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. 

Even though the original derivation of this method was based on the governing equations for incompressible flows, in which the fluid properties are constant, this concept can be adapted to many flows with variable fluid properties and weak compressibility. 

In this section, an explicit time advance scheme for unsteady flow problems is outlined [35]. The momentum equation is discretized by an explicit scheme, and a Poisson equation is solved for the pressure to enforce continuity. The continuity is discretized in an implicit manner. In the original formulation, the spatial derivatives were approximated by finite difference schemes. 

To illustrate the basic principles of the projection method, the semi-discretized (discrete in space but not in time) momentum equations are written symbolically as  

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The measurement of a crude oil s viscosity at different temperatures is particularly important for the calculation of pressure drop in pipelines and refinery piping systems, as well as for the specification of pumps and exchangers. 

The well-known inaccuracy of numerical differentiation precludes the direct calculation of pressure by the insertion of the computed velocity field into Equation (3.6). This problem is, however, very effectively resolved using the following variational recovery method Consider the discretized form of Equation (3.6) given as... 

Find radial component of velocity (v ) and (r) at the reduced integration points and calculate v,./r. Include this temi in the calculation of pressure via the penalty relation. 

In order to select the pipe size, the pressure loss is calculated and velocity limitations are estabHshed. The most important equations for calculation of pressure drop for single-phase (Hquid or vapor) Newtonian fluids (viscosity independent of the rate of shear) are those for the deterrnination of the Reynolds number, and the head loss, (16—18). 

The viscous or frictional loss term in the mechanical energy balance for most cases is obtained experimentally. For many common fittings found in piping systems, such as expansions, contrac tions, elbows and valves, data are available to estimate the losses. Substitution into the energy balance then allows calculation of pressure drop. A common error is to assume that pressure drop and frictional losses are equivalent. Equation (6-16) shows that in addition to fric tional losses, other factors such as shaft work and velocity or elevation change influence pressure drop. 

Calculation of pressure drops in steam lines is a time-consuming task and requires the use of a number of somewhat arbitrary factors for such functions as pipe wall roughness and the resistance of fittings. To simplify the choice of pipe for given loads and steam pressures. Figure 22.4 will be found sufficiently accurate for most practical purposes. 

The correlation studies of heat and mass transfer in pellet beds have been investigated by many, usually in terms of the. /-factors (113-115). According to Chilton and Colburn the two. /-factors are equal in value to one half of the Fannings friction factor / used in the calculation of pressure drop. The. /-factors depend on the Reynolds number raised to a factor varying from —0.36 to —0.68, so that the Nusselt number depends on the Reynolds number raised to a factor varying from 0.64 to 0.32. In the range of the Reynolds number from 10 to 170 in the pellet bed, jd should vary from 0.5 to 0.1, which yields a Nusselt number from 4.4 to 16.1. The heat and mass transfer to wire meshes has received much less attention (110,116). The correlation available shows that the /-factor varies as (Re)-0-41, so that the Nusselt number varies as (Re)0-69. In the range of the Reynolds number from 20 to 420, the j-factor varies from 0.2 to 0.05, so that the Nusselt number varies from 3.6 to 18.6. The Sherwood number for CO is equal to 1.05 Nu, but the Sherwood number for benzene is 1.31 Nu. 

Calculation of pressure drop for liquid flowing in a pipe... 

Consideration will now be given to the various flow regimes which may exist and how they may be represented on a Flow Pattern Map to the calculation and prediction of hold-up of the two phases during flow and to the calculation of pressure gradients for gas-liquid flow in pipes. In addition, when gas-liquid mixtures flow at high velocities serious erosion problems can arise and it is necessary for the designer to restrict flow velocities to avoid serious damage to equipment. 

Considerably more work has been carried out on horizontal as opposed to vertical pneumatic conveying. A useful review of relevant work and of correlations for the calculation of pressure drops has been given by Klinzing et a/.(68). Some consideration will now be given to horizontal conveying, with particular reference to dilute phase flow, and this is followed by a brief analysis of vertical flow. 

Methods for the calculation of pressure drop through pipes and fittings are given in Section 5.4.2 and Volume 1, Chapter 3. It is important that a proper analysis is made of the system and the use of a calculation form (work sheet) to standardize pump-head calculations is recommended. A standard calculation form ensures that a systematic method of calculation is used, and provides a check list to ensure that all the usual factors have been considered. It is also a permanent record of the calculation. Example 5.8 has been set out to illustrate the use of a typical calculation form. The calculation should include a check on the net positive suction head (NPSH) available see section 5.4.3. 

Venkateswararao et al. (1982), in evaluating the flow pattern transition for two-phase flow in a vertical rod bundle, suggested the calculation of pressure gradient for annular flow by... 

Keil F, Dahnke S (1996) Numerical calculation of pressure field in sonochemical reactor. Chem Ing Tech 68 419 -22... 

A number of equations have been proposed for use in the calculation of pressure drop in coils of constant curvature [Srinivasan et al (1968)]. The latter are known as helices. For laminar flow, Kubair and Kuloor (1965) gave an equation for the Reynolds number range 170 to the critical value. In terms of the Fanning friction factor, their equation can be written as... 

Calculation of pressure drop for turbulent flow in a pipe 75... 

Calculation of pressure drop in a boiler tube using the 260... 

Govier and Omer, in a recent article in the Canadian Journal of Chemical Engineering, have summarized the present (G4) state of knowledge concerning the calculation of pressure drop and other quantities in two-phase flow The principal flow patterns are understood in a qualitative way and the effect on flow pattern of the major variables, mainly the lineal or maffi velocities of the phases, is recognized. In nearly all cases the real significance of the viscosities of the phases, the possible separate roles of the densities of the phases, and the influence of the diameter of the pipe is not known. 

In summary, the calculation of pressure drops by the Lockhart-Marti-nelli method appears to be reasonably useful only for the turbulent-turbulent regions. Although it can be applied to all flow patterns, accuracy of prediction will be poor for other cases. Perhaps it is best considered as a partial correlation which requires modification in individual cases to achieve good accuracy. Certainly there seems to be no clear reason why there should be a simple general relationship between the two-phase frictional pressure-drop and fictitious single-phase drops. As already pointed out, at the same value of X in the same system, it is possible to have two different flow patterns with two-phase pressure-drops which differ by over 100%. The Loekhart-Martinelli correlation is a rather gross smoothing of the actual relationships. 

The calculation of pressure drop in vertical slug flow requires an understanding of the distribution of shear stress and pressure around a... 

Calculations of pressures developed on detonation were made by Brinkley Wilson for several explosives as described in OSRD Repts 1231 1510 (Refs 32b 33a)... 

Other prior-art procedures for calculation of pressure drop in the... 

TABLE 6.10. Equations for the Calculation of Pressure Drop in Gas-Solid Transport... 

Corner s Equation of State. At the high temps pressures encountered in explosives technology, the perfect gas law is not applicable. The calculation of pressures developed by expls therefore requires the adoption of a suitable equation of state. Furthermore, in calculating the explosion products it is necessary to correct the ideal thermodynamic equilibria of the relevant chem reactions for the effect of the gas imperfection. This correction also depends on the equation of state adopted. Most equations of state, whether empirical or theoretical, are not suitable for application at the high temps pressures developed by expls. Comer has recently discussed a theoretical equation of state applicable to propellent expls (Ref 3)... 

Two common pipe flow problems are calculation of pressure drop given the flow rate (or velocity) and calculation of flow rate (or velocity) given the pressure drop. When flow rate is given, the Reynolds number may be calculated directly to determine the flow regime, so that the appropriate relations between/and Re (or pressure drop and flow rate or velocity) can be selected. When flow rate is specified and the flow is turbulent, Eq. (6-39) or (6-40), being explicit in/ may be preferable to Eq. (6-38), which is implicit in/and pressure drop. 

However, these are not adequate for the strong non-Newtonian polymer melts under discussion here. The latest flow modeling tools are used for this task. They allow highly accurate calculation of pressure build-up and energy consumption. 

Lahti, G. P., Calculation of Pressure Drops and Outlets, SPE Journal, Jul. 1963. 

In this paper, a model for predicting trickling-pulsing transition, as proposed by Ng (6), is extended to include large-size columns. Preliminary calculations of pressure drop and holdup as a function of bed height indicate several interesting features associated with large-scale reactors. 

Recommendations For large-size packings, use of Eq. (7-2) is recommended for the calculation of pressure drop in the bubble- and pulsed-flow regimes. For small-size packings, wherever possible, PERC data22 or the data of Sato et al.27 for the pressure drop could be used. More experimental as well as theoretical studies on the pressure drop for hydrocarbon systems are needed. 

Meirovitch developed the scanning method to study a system of many chains with excluded volume contained in a box on a square lattice.With this method, an initially empty box is filled with the chain monomers step by step, with help of transition probabilities. The probability of construction of the whole system is the product of the transition probabilities selected, and therefore, the entropy of the system is known. Consequently standard thermodynamic relations can be used to make highly accurate calculations of pressure and chemical potential, directly from the entropy. In principle, all these quantities can be obtained from a single sample without the need to carry out any thermodynamic integration. 

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