Pipe, laminar flow developing

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])... 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

Steady state, fuUy developed laminar flows of viscoelastic fluids in straight, constant-diameter pipes show no effects of viscoelasticity. The viscous component of the constitutive equation may be used to develop the flow rate-pressure drop relations, which apply downstream of the entrance region after viscoelastic effects have disappeared. A similar situation exists for time-dependent fluids. 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

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. 

Equations for the optimum pipe diameter with laminar flow can be developed by using a suitable equation for pressure drop in the equation for pumping costs. 

Determine the shear stress distribution and velocity profile for steady, fully developed, laminar flow of an incompressible Newtonian fluid in a horizontal pipe. Use a cylindrical shell element and consider both sign conventions. How should the analysis be modified for flow in an annulus ... 

The theory for the turbulent flow of fluids through pipes is far less developed than that for laminar flow. 

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... 

Some of the simplifications that may be possible are illustrated by the case of steady, fully-developed, laminar, incompressible flow of a Newtonian fluid in a horizontal pipe. The flow is assumed to be axisymmetric with no swirl component of velocity so that derivatives wrt 6 vanish and vg = 0. For fully-developed flow, derivatives wrt z are zero. With these simplifications and noting that the flow is incompressible, the continuity equation (equation A. 11) reduces to... 

Example 2-6 Consider the situation where the reactants at constant density are fed continuously into a pipe of length L instead of a tank of volume V as in the batch reactor. The reactants react as they flow down the tube with a speed u, and we assume that they flow as a plug without mixing or developing the laminar flow profile. Show that the conversion of the reactants is exactly the same in these very different reactor configurations. 

If there is laminar flow in a pipe or tube, equation (E4.3.15) is used to compute the friction factor accurately after some velocity profile development length. 

A somewhat more general approach which has been shown to be at least partially useful for the latter two problems as well as for correlation of laminar-flow data is that of Metzner and Reed (Mil). Their work is based upon an equation developed by Rabinowitsch (Rl) and Mooney (M15) for the calculation of shear rates at the wall of a tube or pipe ... 

More complex equations have been developed for the flow of power-law fluids under turbulent flow in pipes [85,86,90], The foregoing applies to smooth pipes. Surface roughness has little effect on the friction factor for laminar flow, but can have a significant effect when there is turbulent flow [85],... 

Example 7.8 Residence Time Distribution Functions in Fully Developed Laminar Flow of a Newtonian Fluid in a Pipe The velocity distribution... 

The mathematical analysis of flow in ducts of noncircular cross section is vastly more complex in laminar flow than for circular pipes and is impossible for turbulent flow. As a result, relatively little theoretical base has been developed for the flow of fluids in noncircular ducts. In order to deal with such flows practically, empirical methods have been developed. 

Consider a fully-developed steady-state laminar flow of a constant-property fluid through a circular pipe with a constant heat flux condition imposed at the duct wall. Neglect axial conduction, but include the effect of viscous dissipation. Obtain an expression for the Nusselt number. 

If the solution procedure is carried through as outlined above, the following is obtained for fully developed laminar flow through a pipe with constant wall temperature ... 

Next consider fully developed laminar flow through a plane duct whose wall temperature is kept constant As with pipe flow, for this boundary condition, Eq. (4.69) gives ... 

Consider fully developed laminar flow fluid through a circular pipe with a uniform wall heat flux. If heat is generated uniformly in the fluid, perhaps as the result of a chemical reaction, at a rate of q per unit volume, determine the value of the Nusselt number based on the difference between the wall temperature and the mean fluid temperature in the pipe. 

The kinetic-energy terms of the various energy balances developed h include the velocity u, which is the bulk-mean velocity as defined by the equati u = m/pA Fluids flowing in pipes exhibit a velocity profile, as shown in Fi 7.1, which rises from zero at the wall (the no-slip condition) to a maximum the center of the pipe. The kinetic energy of a fluid in a pipe depends on actual velocity profile. For the case of laminar flow, the velocity profile parabolic, and integration across the pipe shows that the kinetic-ertergy should properly be u2. In fully developed turbulent flow, the more common in practice, the velocity across the major portion of the pipe is not far fro... 

Water flows in a 2.5-cm-diameter pipe so that the Reynolds number based on diameter is 1500 (laminar flow is assumed). The average bulk temperature is 35°C. Calculate the maximum water velocity in the tube. (Recall that u, = 0.5wo.) What would the heat-transfer coefficient be for such a system if the tube wall was subjected to a constant heat flux and the velocity and temperature profiles were completely developed Evaluate properties at bulk temperature. 

Analysis of Steady Laminar Fully Developed Flow in a Pipe... 

The shell balance method will be used to examine steady laminar flow of a fluid in a pipe. For the geometrical system illustrated in Figure 3B-1 and for steady laminar fully developed flow of a fluid, a shell momentum balance can be conducted (Bird et al., 1960 Geankoplis, 1983) using the cylindrical coordinates, r, 6, andz. The momentum balance is conducted on a control volume shell at a radius r with dimensions Ar and Az. 

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