Laminar flow in pipes

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. 

Pulse Response Experiment and the E Curve for Laminar Flow in Pipes... 

Fig. 10. Axial dispersion in laminar flow in pipes, dispersed plug flow model. Adapted from (B13).

Note roughness has no effect on laminar flow in pipes. 

Viscous Transport. Low velocity viscous laminar tiow in gas pipes is commonplace. Practical gas flow can be based on pressure drops of <50% for low velocity laminar flow in pipes whose length-to-diameter ratio may be as high as several thousand. Under laminar flow, bends and fittings add to the frictional loss, as do abrupt transitions. 

The friction factor for laminar flow in pipes Re < 2300) is given by fo = 4/i = For turbulent flow in rough pipes the friction factors depends on both the Reynolds number and the surface roughness of the tube. Colebrook [35] devised an implicit relation for the Darcy friction factor which reproduce the well known Moody diagram quite well. 

Humphrey, J. A. C., Numerical Calculation of Developing Laminar Flow in Pipes of Arbitrary Curvature Radius, Can J. Chem. Engg., 56, 151-164 (1978). 

Fig. 9. Relationship between pipe friction coeilicient X and Re3molds number for various pulp water suspensions. I depicts the theoretical equation of the flow resistance in laminar flow in pipe and II depicts the Blasius resistance formula of turbulent flow. Cp, % , 0 (Water) , 0.30 , 0.55 v, 0.72----. Adapted from Figure 9 of Ref. 166.

For laminar flow in pipes, one finds for the mass transfer at the pipe wall relation (4.23) with the following approximate values for the constants ... 

F(t) is plotted versus reduced time, t/t, in Figure 8.29, where we also compare F(t) against values for plug flow, laminar flow in a pipe, and a continuous stirred tank reactor. F(t) is seen to lie between plug and laminar flow in pipes. F(t) for the extruder is rather narrow with no long tails. 

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. 

FIG. 6-10 Parabolic velocity profile for laminar flow in a pipe, with average velocity V. 

For developed laminar flow in smooth channels of t/h > 1 mm, the product ARe = const. Its value depends on the geometry of the channel. For a circular pipe ARe = 64, where Re = Gdh/v is the Reynolds number, and v is the kinematic viscosity. 

Below a Reynolds number of about 2000 the flow in pipes will be laminar. Providing the natural convection effects are small, which will normally be so in forced convection, the following equation can be used to estimate the film heat-transfer coefficient ... 

Careful study of various fluids in tubes of different sizes has indicated that laminar flow in a tube persists up to a point where the value of the Reynolds number (NRt = DVp/n) is about 2000, and turbulent flow occurs when NRe is greater than about 4000, with a transition region in between. Actually, unstable flow (turbulence) occurs when disturbances to the flow are amplified, whereas laminar flow occurs when these disturbances are damped out. Because turbulent flow cannot occur unless there are disturbances, studies have been conducted on systems in which extreme care has been taken to eliminate any disturbances due to irregularities in the boundary surfaces, sudden changes in direction, vibrations, etc. Under these conditions, it has been possible to sustain laminar flow in a tube to a Reynolds number of the order of 100,000 or more. However, under all but the most unusual conditions there are sufficient natural disturbances in all practical systems that turbulence begins in a pipe at a Reynolds number of about 2000. 

Under normal circumstances, the laminar-turbulent transition occurs at a Reynolds number of about 2100 for Newtonian fluids flowing in pipes. 

When analysing simple flow problems such as laminar flow in a pipe, where the form of the velocity profile and the directions in which the shear stresses act are already known, no formal sign convention for the stress components is required. In these cases, force balances can be written with the shear forces incorporated according to the directions in which the shear stresses physically act, as was done in Examples 1.7 and 1.8. However, in order to derive general equations for an arbitrary flow field it is necessary to adopt a formal sign convention for the stress components. 

Laminar flow in a pipe showing a typical fluid element and the velocity profile. The negative sign convention for stress components is shown... 

In the case of laminar flow in a pipe, work is done by the shear stress component rTX and the rate of doing work is the viscous dissipation rate, that is the conversion of kinetic energy into internal energy. The rate of viscous dissipation per unit volume at a point, is given by... 

The velocity profile for steady, fully developed, laminar flow in a pipe can be determined easily by the same method as that used in Example 1.9 but using the equation of a power law fluid instead of Newton s law of viscosity. The shear stress distribution is given by... 

The occurrence of slip invalidates all normal analyses because they assume that the velocity is zero at the wall. Returning to the Rabinowitsch-Mooney analysis, the total volumetric flow rate for laminar flow in a pipe is given by... 

Ryan, N.W. and Johnson, M.M., Transition from laminar to turbulent flow in pipes, AIChE Journal, 5, pp. 433-5 (1959). 

Chapters 13 and 14 deal primarily with small deviations from plug flow. There are two models for this the dispersion model and the tanks-in-series model. Use the one that is comfortable for you. They are roughly equivalent. These models apply to turbulent flow in pipes, laminar flow in very long tubes, flow in packed beds, shaft kilns, long channels, screw conveyers, etc. 

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