For laminar flow

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

Fig. 5. Entrance flows in a tube or duct (a) separation at sharp edge (b) growth of a boundary layer (illustrated for laminar flow).

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

For laminar flow a = 0.5. For fully developed turbulent flowO.88 < a < 0.98, but can be taken to be unity with Httle error. 

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

In the forced convection heat transfer, the heat-transfer coefficient, mainly depends on the fluid velocity because the contribution from natural convection is negligibly small. The dependence of the heat-transfer coefficient, on fluid velocity, which has been observed empirically (1—3), for laminar flow inside tubes, is h for turbulent flow inside tubes, h and for flow outside tubes, h. Flow may be classified as laminar or... 

The convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

For laminar flow (Re < 2000), generally found only in circuits handling heavy oils or other viscous fluids, / = 16/Re. For turbulent flow, the friction factor is dependent on the relative roughness of the pipe and on the Reynolds number. An approximation of the Fanning friction factor for turbulent flow in smooth pipes, reasonably good up to Re = 150,000, is given by / = (0.079)/(4i e ). 

In the simple Bunsen flame on a tube of circular cross-section, the stabilization depends on the velocity variation in the flow emerging from the tube. For laminar flow (paraboHc velocity profile) in a tube, the velocity at a radius r is given by equation 20 ... 

For laminar flow in vertical tubes a series of charts developed by Pigford [Chem. Eng. Prog. Symp. Sen 17, 51, 79 (1955)] maybe used to predict values of... 

Annuli Approximate heat-transfer coefficients for laminar flow in annuh may be predicted by the equation of Chen, Hawkins, and Sol-berg [Tron.s. Am. Soc. Mech. Eng., 68, 99 (1946)] ... 

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Set. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Jnt. J. Heat Ma.s.s Tran.sfer, 6, 483, 495 (1963)]. Lee fnt. J. Heat Ma.s.s Tran.sfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Np < 30, lO < < 2 X 10, 1.01 <... 

Equation (5-62) predicts the point of maximiim velocity for laminar flow in annuli and is only an approximate equation for turbulent flow. Brighton and Jones [Am. Soc. Mech. Eng. Basic Eng., 86, 835 (1964)] and Macagno and McDoiigall [Am. Inst. Chem. Eng. J., 12, 437 (1966)] give more accurate equations for predicting the point of maximum velocity for turbulent flow. 

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. 

Laminar and Turbulent Flow Below a critical Reynolds number of about 2,100, the flow is laminar over the range 2,100 < Re < 5,000 there is a transition to turbulent flow. For laminar flow, the Hagen-Poiseuille equation... 

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

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter AIChE J., 7, 445 48 [1961]), Wheeler and Wissler (AJChE J., 11, 207-212 [1965]), Bird, Armstrong, and Hassager Dynamics of Polymeric Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977), and Skelland Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). 

Economic Pipe Diameter, Laminar Flow Pipehnes for the transport of high-viscosity liquids are seldom designed purely on the basis of economics. More often, the size is dictated oy operability considerations such as available pressure drop, shear rate, or residence time distribution. Peters and Timmerhaus (ibid.. Chap. 10) provide an economic pipe diameter chart for laminar flow. For non-Newtouiau fluids, see SkeUand Non-Newtonian Flow and Heat Transfer, Chap. 7, Wiley, New York, 1967). 

For laminar flow the losses in sudden contraction may be estimated for area ratios Ao A < 0.2 by an equivalent additional pipe length Lc given by... 

For laminar flow, data for the frictional loss of valves and fittings are meager. (Beck and Miller,y. Am. Soc. Nav. Eng., 56, 62-83 [194fl Beck, ibid., 56, 235-271, 366-388, 389-395 [1944] De Craene, Heat. Piping Air Cond., 27[10], 90-95 [1955] Karr and Schutz, j. Am. Soc. Nav. Eng., 52, 239-256 [1940] and Kittredge and Rowley, Trans. ASME, 79, 1759-1766 [1957]). The data of Kittredge and Rowley indicate that K is constant for Reynolds numbers above 500 to 2,000, but increases rapidly as Re decreases below 500. Typical values for K for laminar flow Reynolds numbers are shown in Table 6-5. 

TABLE 6-5 Additional Frictional Loss for Laminar Flow through Fittings and Valves... 

For laminar flow of non-Newtouiau fluids across tube banks, see Adams and Bell Chem. Eng. Prog., 64, Symp. Ser, 82, 133-145 [1968]). 

See Benedict, loc. cit., for a general equation for pressure loss for nozzles installed in pipes or with plenum inlets. Nozzles show higher loss than venturis. Permanent pressure loss for laminar flow depends on the Reynolds number in addition to p. For details, see Alvi, Sri-dharan, and Lakshamana Rao, J. Fluids Eng., 100, 299-307 (1978). 

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. 

For laminar flow in a circular tube, the Leveque relationship is ... 

For laminar flow, the velocity at the centerline is Uq = 2u. For a power law rate equation = kO, the differential material balance on a streamline is... 

AP for laminar flow in a eireular tube is given by the Hagen-Poiseuille equation ... 

Figure 8-22 shows the F(6) eurves for laminar flow in a tubular reaetor and for other idealized flow patterns. 

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