Pipe-flow

Pressure drop calculations for simple geometries like pipe flows are straightforward. The Fanning friction factor is given by  

This correlation is good for a Reynolds number range of between 2,100 and 106. The Fanning equation for pressure drop is  

Combining both expressions gives a simple relation for estimating the frictional pressure drop in straight run pipe. 

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  

The above expression can be solved by a trial and error calculation, whereby the following key assumptions are applied (1) the flow is turbulent, i.e., Re i 2,100, and (2) motive power is supplied by a prime mover such as a pump or a compressor. To make this expression easier to use, the terms are rearranged in the following manner  

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The pressure drop accompanying pipe flow of such fluids can be described in terms of a generalized Reynolds number, which for pseudoplastic or dilatant fluids takes the form ... 

The phenomena are quite complex even for pipe flow. Efforts to predict the onset of instabiHty have been made using linear stabiHty theory. The analysis predicts that laminar flow in pipes is stable at all values of the Reynolds number. In practice, the laminar—turbulent transition is found to occur at a Reynolds number of about 2000, although by careful design of the pipe inlet it can be postponed to as high as 40,000. It appears that linear stabiHty analysis is not appHcable in this situation. 

The velocity head JT in a pipe flow is related to Hquid velocity hy H = I Qc The Hquid velocity in a mixing tank is proportional to impeller tip speed 7zND. Therefore, JTin a mixing tank is proportional to The power consumed by a mixer can be obtained by multiplying and H and is given... 

Here, h is the enthalpy per unit mass, h = u + p/. The shaft work per unit of mass flowing through the control volume is 6W5 = W, /m. Similarly, is the heat input rate per unit of mass. The fac tor Ot is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, Ot = 1. In turbulent flow, Ot is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circiilar pipe with a parabohc velocity profile, Ot = 2. 

In laminar flow,/is independent of /D. In turbulent flow, the friction factor for rough pipe follows the smooth tube curve for a range of Reynolds numbers (hydrauhcaUy smooth flow). For greater Reynolds numbers,/deviates from the smooth pipe cui ve, eventually becoming independent of Re. This region, often called complete turbulence, is frequently encountered in commercial pipe flows. The Reynolds number above which / becomes essentially independent of Re is (Davies, Turbulence Phenomena, Academic, New York, 1972, p. 37) 20[3.2-2.46ln( /D) ... 

Residence Time Distribution For laminar Newtonian pipe flow, the cumulative residence time distribution F(0) is given by... 

The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = X/D. Molecular flow is characterized by Kn > 1.0 continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. 

Slip Flow In the transition region between molecular flow and continuum viscous flow, the conductance for fully developed pipe flow is most easily obtained by the method of Brown, et al. (J. Appl. Phys., 17, 802-813 [1946]), which uses the parameter... 

Curved Pipes and Coils For flow through curved pipe or coil, a secondary circiilation perpendicular to the main flow called the Dean effect occurs. This circulation increases the friction relative to straight pipe flow and stabilizes laminar flow, delaying the transition Reynolds number to about... 

FricHonal pipe flow is not isentropic. Strictly speaking, the flashes must be carried out at constant h + V /2 + gz, where h is the enthalpy... 

Feed or withdraw from both ends, reducing the pipe flow velocity head and required hole pressure drop by a factor of 4. 

Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant. 

For isothermal compressible flow of a gas with constant compressibility factor Z through a packed bed of granular solids, an equation similar to Eq. (6-114) for pipe flow may be derived ... 

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

For permanent pressure loss with segmental and eccentric orifices with laminar pipe flow see Lakshmana Rao and Sridharan, Proc. Am. Soc. Civ. Eng., ]. Hydraul. Div., 98 (HY 11), 2015-2034 (1972). 

Hanratty (pipe flow) Hinze (turbulence theory) -1.2 0.6 -0.6 ... 

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 ... 

Since pipe flow is more nearly isenthalpic, the flash fraction x is found from an enthalpy balance between the stagnation point and a point z downstream. Accounting for changes in potential energy, kinetic energy, and heat added or removed from the pipe Q, x is given by ... 

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-case solution for compressible, inclined pipe flow is next stated, then the solution is developed for the special case of horizontal compressible flow... 

HEM for Horizontal Pipe Discharge For horizontal pipe flow, Fi = q = 0, and ... 

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.)... 

Skin friction loss. Skin friction loss is the loss from the shear forces on the impeller wall caused by turbulent friction. This loss is determined by considering the flow as an equivalent circular cross section with a hydraulic diameter. The loss is then computed based on well-known pipe flow pressure loss equations. 

Carbon residue, pour point, and viseosity are important properties in relation to deposition and fouling. Carbon residue is found by burning a fuel sample and weighing the amount of earbon left. The earbon residue property shows the tendeney of a fuel to deposit earbon on the fuel nozzles and eombustion liner. Pour point is the lowest temperature at whieh a fuel ean be poured by gravitational aetion. Viseosity is related to the pressure loss in pipe flow. Both pour point and viseosity measure the tendeney of a fuel to foul the fuel system. Sometimes, heating of the fuel system and piping is neeessary to assure a proper flow. 

FIQURE 5-3.1.1 Predicted charging current for long pipe flow [Eq. (5-3.1.3)]. 

The charging current is therefore roughly proportional to the square of (ucO, the velocity-diameter product. An important outcome is that the velocity-diameter product can be used to characterize charging current in pipe flow and as a basis for setting flow limits when filling tanks (5-4). 

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