In laminar flow

In configurations more complex than pipes, eg, flow around bodies or through nozzles, additional shearing stresses and velocity gradients must be accounted for. More general equations for some simple fluids in laminar flow are described in Reference 1. 

TABLE 5-4 Values of Limiting Nusselt Number in Laminar Flow in Closed Ducts... 

N tubes in a vertical row where the total condensate flows smoothly from one tube to the one beneath it, without splashing, and still in laminar flow on the tube, the mean condensing coefficient h i for the entire row of N tubes is related to the condensing coefficient for the top tube hi by... 

The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Jn.st. Chem. Eng., 31,365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theoiy. 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

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) ... 

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-PoiseuiUe equation, gives the velocity i as a Innction of radial position / in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity t ce the average velocity, is shown in Fig. 6-10. 

Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fuUy developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara and Riza, Can. J. Chem. Engr, 71, 472 76 [1993])... 

In laminar flow, the friction factor for curved pipe/ may be expressed in terms of the straight pipe friction factor/= 16/Re as (Hart, Chem. Eng. ScL, 43, 775-783 [1988])... 

For friction loss in laminar flow through semicircular ducts, see Masliyah and Nandakumar AlChE J., 25, 478-487 [1979]) for curved channels of square cross section, see Cheng, Lin, and On ]. Fluids Eng., 98, 41-48 [1976]). 

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. 

Laminar Flow With highly viscous fluids the linear velocity along a streamline varies with the radial position. Laminar flow is charac teristic of some polymeric systems. Figure 23-21 shows how the conversion is poorer in laminar flow than with uniform flow over the... 

Permanent pressure loss across quadrant-edge orifices for turbulent flow is somewhat lower than given by Eq. (10-30). See Alvi, Sridharan, and Lakshmana Rao, loc. cit., for values of discharge coefficient and permanent pressure loss in laminar flow. 

Shear stresses are developed in a fluid when a layer of fluid moves faster or slower than a nearby layer of fluid or a solid surface. In laminar flow, the shear stress is equal to the product of fluid viscosity and velocity gradient or rate of shear. Under laminar-flow conditions, shear forces are larger than inertial forces in the fluid. 

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

Suspensions of fine sohds may have pseudoplastic or plastic-flow properties. When they are in laminar flow in a stirred vessel, motion in remote parts of the vessel where shear rates are low may become negligible or cease completely. To compensate for this behavior of slurries, large-diameter impellers or paddles are used, with (D /Df) > 0.6, where Df is the tank diameter. In some cases, for example, with some anchors, > 0.95 Df. Two or more paddles may be used in deep tanks to avoid stagnant regions in slurries. 

In laminar flow < 10), 1/A Re nd P c< [LN D. Since shear stress is proportional to rotational speed, shear stress can be increased at the same power consumption by increasing N proportionally to as impeller diameter is decreased. 

Fluid circulation probably can be increased at the same power consumption and viscosity in laminar flow by increasing impeller diameter and decreasing rotational speed, but the relationship between Q, N, and for laminar flow from turbines has not been determined. 

If the system is badly fouled, m - 0, and increasing or decreasing flow at constant pressure has httle effect on flux. However, raising the pressure may raise flux. For an unfouled system in laminar flow 0.33 [Pg.2041]

For a gas in laminar flow over a condensed phase sample of length L, the mass transport across the boundary layer, in terms of the flux of molecules from the sample to die gas phase, is therefore... 

Substituting the expression for C, we again obtain the settling velocity of an isolated particle in laminar flow ... 

In applying this equation it is possible to determine the maximum size particle in laminar flow, taking into account the given conditions of sedimentation (p, Pp, n and a ). However, this equation does not determine what the flow regime is when d > d . 

Statie mixers, as reviewed in Chapter 7, eontain mixing elements enelosed in a tubular housing through whieh radial mixing is aehieved. They redistribute fluid aeross the flow ehannel and eonsequently rearrange temperature and eomposition distributions. They are often used to promote mixing in laminar flow systems thus having a pro-nouneed effeet on the RTD. 

In laminar flow there are no disturbances, and therefore all flow particles move in the same direction. Transitional flow is the flow regime that takes place during the change from streamline to turbulent flow. In the case of turbulent flow the particles move in a given flow direction, but the flow is erratic and random. 

In laminar flow, a processed material divides at the leading edge of each element and follows the channels created by the element shape. At each succeeding element the two channels are further divided, resulting in an exponential increase in stratification. The number of striations produced is 2 where n is the number of elements. 

The flow changes from laminar to turbulent in the range of Reynolds numbers from 2,100 to 4,000 [60]. In laminar flow, the friction pressure losses are proportional to the average flow velocity. In turbulent flow, the losses are proportional to the velocity to a power ranging from 1.7 to 2.0. 

The plate heat exchanger, for example, can be used in laminar flow duties, for the evaporation of fluids with relatively high viscosities, for cooling various gases, and for condensing applications where pressure-drop parameters are not excessively restrictive. 

Correlations for heat transfer and pressure drop in laminar flow are ... 

The overhung design of the rotor (i.e., no outboard bearing) increases the potential for radical shaft deflection. Any variation in laminar flow, volume, or load of the... 

While designers of fluid power equipment do what they can to minimize turbulence, it cannot be avoided. For example, in a 4-inch pipe at 68°F, flow becomes turbulent at velocities over approximately 6 inches per second (ips) or about 3 ips in a 6-inch pipe. These velocities are far below those commonly encountered in fluid power systems, where velocities of 5 feet per second (fps) and above are common. In laminar flow, losses due to friction increase directly with velocity. With turbulent flow, these losses increase much more rapidly. 

General laws for the flow of fluids were determined by Reynolds, who recognized two flow patterns, laminar and turbulent. In laminar flow the fluid can be considered as a series of parallel strata, each moving at its own speed, and not mixing. Strata adjacent to walls of the duct will be slowed by friction and will move slowest, while those remote from the walls will move fastest. In turbulent flow there is a general forward movement together with irregular transfer between strata. 

In the present discussion only the problem of steady flow will be considered in which the time average velocity in the main stream direction X is constant and equal to ux. in laminar flow, the instantaneous velocity at any point then has a steady value of ux and does not fluctuate. In turbulent flow the instantaneous velocity at a point will vary about the mean value of ux. It is convenient to consider the components of the eddy velocities in two directions—one along the main stream direction X and the other at right angles to the stream flow Y. Since the net flow in the X-direction is steady, the instantaneous velocity w, may be imagined as being made up of a steady velocity ux and a fluctuating velocity ut, . so that ... 

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