Momentum pipe flow

Pipe Flow For steady-state flow through a constant diameter duct, the mass flux G is constant and the governing steady-state momentum balance is ... 

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. 

Derive the Taylor-Prandtl modification of the Reynolds analogy between heat and momentum transfer and express it in a form in which it is applicable to pipe flow. 

Viscometric flow theories describe how to extract material properties from macroscopic measurements, which are integrated quantities such as the torque or volume flow rate. For example, in pipe flow, the standard measurements are the volume flow rate and the pressure drop. The fundamental difference with spatially resolved measurements is that the local characteristics of the flows are exploited. Here we focus on one such example, steady, pressure driven flow through a tube of circular cross section. The standard assumptions are made, namely, that the flow is uni-directional and axisymmetric, with the axial component of velocity depending on the radius only. The conservation of mass is satisfied exactly and the z component of the conservation of linear momentum reduces to... 

Fanning (Darcy) friction factor f(f or fD) e, D 2 V2L fo = 4f TW yv2 e, = friction loss (energy/mass) rw = wall stress (Energy dissipated)/ (KE of flow x 4L/D) or (Wall stress)/ (momentum flux) Flow in pipes, channels, fittings, etc. 

Reynolds number flows /vRe N -°Vp /vRe — pV2 pV/D AQp izDp PV2 Tw/8 Pipe flow rw =wall stress (inertial momentum flux)/ (viscous momentum flux) Pipe/internal flows (Equivalent forms for external flows)... 

Momentum Balance in Dimensionless Variables For pipe flow, it is necessary to solve the momentum balance. The momentum balance is simplified by using the following dimensionless variables ... 

For pipe flow, HEM requires solution of the equations of conservation of mass, energy, and momentum. The momentum equation is in differential form, which requires partitioning the pipe into segments and carrying out numerical integration. For constant-diameter pipe, these conservation equations are as follows ... 

The role of the pressure gradient may be shown in the momentum equation of a gas-solid mixture. Consider a steady pipe flow without mass transfer and with negligible interparticle collisions. From Eq. (5.170), the momentum equations for the gas and particle phases can be given by... 

Consider a one-dimensional gas-solid pipe flow where the electrostatic effect is negligible. The momentum equation of the particle phase can be expressed as [Klinzing, 1981]... 

Velocity measurements made in a trapezoidal canal, reported by O Brien, yield the distribution contours, with the accompanying values of the correction factors for kinetic energy and momentum. The filament of maximum velocity is seen to lie beneath the surface, and the correction factors for kinetic energy and momentum are greater than in the corresponding case of pipe flow. Despite the added importance of these factors, however, the treatment in this section will follow the earlier procedure of assuming the values of a and p to be unity, unless stated otherwise. Any thoroughgoing analysis would, of course, have to take account of their true values. 

As mentioned in the discussion of pipe flow, because an explicit finite-difference procedure is being used to solve the momentum and energy equations, the solution can become unstable, i.e., as the solution proceeds it can diverge increasingly from the actual solution as indicated in Fig. 4.29. To determine the conditions under... 

In turbulent pipe flow it is again also often convenient to write the turbulence quantities in terms of the eddy viscosity and diffusivity and when this is done the momentum and energy equations become ... 

If the puncture occurs on the vessel or on a line shorter than 0.5 m, the discharge is likely to be nonhomogeneous, meaning the gas and liquid velocities are not equal and the phases are not likely to be in equilibrium. For this case, various models have been developed, including some of considerable complexity, accounting for interphase heat, mass, and momentum transfer. These are generally used in the nuclear power industry. For most engineering applications, simpler models suffice. A reasonably simple nonequilibrium model (NEM) is developed here. We also provide an HEM for orifice flow, since it helps to develop the HEM for pipe flow, and its inaccuracies may at times be tolerable. 

For integrating Eq. (4-9), vji= ei Er) should be known as a function of and operating variables. However, the momentum diffusivity is the only term we know, with essentially no systematic data for In the case of free turbulence of a homogeneous fluid, the diffusivity of a scalar quantity like heat and mass is estimated to be about two times that of momentum (S4) and the two diffusivities are not far apart for turbulent pipe flow (S8). However, such a relation is not available yet for gas-liquid bubble flow in bubble columns. Generally the local radial mass diffusivity may be expressed by a, with a being a numerical coefficient of order unity. 

To assess the physical deviation between the average of products and the product of averages a momentum velocity correction factor can be defined by Cm = vz) / v1)a- By use of the Hagen-Poiseuille law (1.353) and the power law velocity profile (1.354) it follows that at steady state Cm has a value of about 0.95 for turbulent flow and 0.75 for laminar flow [55]. In practice a value of 1 is used in turbulent flow so that v1)a is simply replaced by the averaged bulk velocity vz) - On the other hand, for laminar flows a correction factor is needed. For more precise calculations a simplified (not averaged ) 2D model is often considered for ideal axisymmetric pipe flows [52, 69]. 

Unfortunately, it is not possible to derive an analogue velocity profile for turbulent flow in an anal dical manner based on the generalized momentum equations. However, a number of entirely empirical relations of similar simplicity exist for the velocity profile in turbulent pipe flow. One such relation often found in introductory textbooks on engineering fluid flow is the power law velocity profile. ... 

Erst consider steady, fully developed pipe flow. Hie conservation of momentum for the control volume shown in Fig. 6.1, noting that momentum flow pAV2 does not change in fully developed flow, gives... 

So fax, we have studied the friction factor and the drag coefficient associated with a number of common cases. We may now utilize the analogy between heat and momentum transfer, obtaining the heat transfer indirectly from the friction associated with these cases. Combining Eq. (6.15) with Eq. (5.63) yields a relation for the heat transfer in fully developed turbulent pipe flow,... 

The analogy between heat and momentum does not account for the pressure drop in pipes and is not, strictly speaking, valid for pipe flow. However, the effect of pressure drop on this analogy appears to remain within the uncertainty of the available experimental data and is usually ignored. Next, introducing Eq. (6.16) into Eq. (5.63) gives the heat transfer in fully developed turbulent flow over a flat plate,... 

The prevalence of pipe flows in engineering (heating, cooling, power plants, water transport, etc.) makes pipe flow the most important application of internal flows. Because of this importance, there exist a number of correlations of experimental data on pipe flow. Before listing these correlations, however, let us recall Eq. (6.20), obtained from the analogy between heat and momentum transfer. All of the physical properties associated with the dimensionless numbers of this equation depend on the fluid temperature. Therefore a reference temperature is needed for the evaluation of the properties. A commonly used temperature for this purpose is the bulk temperature 7j, associated with the enthalpy flow in the first law (recall of Eq. (1.10)),... 

For a fully developed nonnewtonian laminar pipe flow, the governing momentum equation can be written as... 

It is well known that for newtonian fluids in turbulent pipe flow, an analogy between momentum and heat transfer can be drawn and expressed in the following form ... 

Y. I. Cho and J. P. Hartnett, Analogy for Viscoelastic Fluids—Momentum, Heat and Mass Transfer in TUrbulent Pipe Flow, Letters in Heat and Mass Transfer (7/5) 339-346,1980. 

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