For the pipe flow

Finally, we consider the type (ii) homogeneous boundary condition in physical terms. For the pipe flow problem, if we had stipulated that the tube wall was well insulated, then the heat flux at the wall is nil, so... 

These dimensionless groups have been used ever since as the basis for the pipe flow analogy in almost all packed bed correlations. 

Looking at the friction factor chart (Fig. 48.6), we see how the experimental data further confirms Reynolds earlier work and helps us even today with our own field work. Once we have determined the friction factor (j) for the pipe flow we are studying, all that remains is to evaluate the frictional pressure loss, AP from... 

Using the weighing system example of dead time, a 10 kg increase in material on the conveyor belt would result in a 10 kg increase at the weight sensor or a static gain of one. Similarly, it can also be shown that the above analysis holds for the pipe flow situation. In each of these cases a pure dead time exists since Kss = However, consider the scenario shown in Figure 3.22. 

The equations for nozzle flow, Eqs. (6-114) through (6-118), remain valid for the nozzle section even in the presence of the discharge pipe. Equations (6-116) and (6-120), for the temperature variation, may also be used for the pipe, with Mo, po replacing Mi, pi since they are valid for adiabatic flow, with or without friction. 

For turbulent pipe flow, the friction velocity u, = used earlier... 

HEM for Two-Phase Pipe Discharge With a pipe present, the backpressure experienced by the orifice is no longer qg, but rather an intermediate pressure ratio qi. Thus qi replaces T o iri ihe orifice solution for mass flux G. ri Eq. (26-95). Correspondingly, the momentum balance is integrated between qi and T o lo give the pipe flow solution for G,p. The solutions for orifice and pipe now must be solved simultaneously to make G. ri = G,p and to find qi and T o- This can be done explicitly for the simple case of incompressible single-phase (hquid) inclined or horizontal pipe flow The solution is implicit for compressible regimes. 

The general compressible flow solution simplifies for horizontal pipe flow to ... 

FIG. 26-68 Ratio of mass flux for horizoutal pipe flow to that for orifice discharge for flashing liquids hy the homogeueoiis eqiiilihriiim model, (Leung and Gmlmes, AIChE J, 33 (3), pp. 524-527, 1987 reproduced by permission of AIChE. copy-right 1987. All rights reseroed.)... 

FIG. 26-69 Ratio of mass flux for inclined pipe flow to that for orifice discharge for flashing liquids by the homogeneous equilibrium model. Leung, J. of Loss Prev. Process Ind. 3 pp. 27-32, with kind peimission of Elsevier Science, Ltd, The Boulevard, Langford Lane, Kidlington, 0X5 IGB U.K., 1990.)... 

The following analysis can be used to determine economic pipe diameters for the turbulent flow of Newtonian fluids. The working expression that can be used is ... 

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final velocity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully developed flow. 

Equation (14.126) is our final result it can be used for calculating the pressure loss in the pipe flows. 

Process flowsheets do not normally show companion flanges for valves unless these serve as blinds or for orifice plates. This detail is sometimes shown on the piping flow-... 

Determine Cj and C2 from Figure 2-31 and Table 2-11 for the steam flow rate and assumed pipe size respectively. Use Table 2-4 or Table 2-8 to select steam t eloc-ily for line size estimate. 

Discharge piping must be sized for low pressure drop at maximum flow not only from any one valve, but for the combined flow possibilities in the discharge collection manifold all the w ay to the vent release point, whether it be a flare, incinerator, absorber or other arrangement [13]. See Figures 7-20 illustrations. 

Since in the energy balance equation, the kinetic energy per unit mass is expressed as a2/2a, hence a = 0.5 for the streamline flow of a fluid in a round pipe. 

For the fluicj flowing as shown in Figure 3.15, from section 1 (the pressure just inside the enlargement is found to be equal to that at the end of the smaller pipe) to section 2, the net force = the rate of change of momentum, or ... 

This expression enables v2, the specific volume at the downstream end of the pipe, to be calculated for the fluid flowing at a mass rate G from an upstream pressure P. ... 

In some cases, particularly for the radial flow of heat through a thick pipe wall or cylinder, the area for heat transfer is a function of position. Thus the area for transfer applicable to each of the three media could differ and may be A, A2 and A3. Equation 9.3 then becomes ... 

The right-hand side of equation 10.224 gives numerical values which are very close to those obtained from the Blasius equation for the friction factor (j> for the turbulent flow of a fluid through a smooth pipe at Reynolds numbers up to about 106. 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

A simple approximate form of the relation between u+ and y+ for the turbulent flow of a fluid in a pipe of circular cross-section may be obtained using the Prandtl one-seventh power law and the Blasius equation. These two equations have been shown (Section 11.4) to be mutually consistent. 

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