Pipe, laminar flow fully-developed

The character of the fully developed region is determined by the character of the flow in the boundary layer at the cusp (Fig. 2.4). If it is laminar, the fully developed flow will be laminar (Fig. 2.4a). Or the flow may be transitional (Fig. 2.4b) or completely turbulent (Fig. 2.4c). The latter, in a pipe, channel, or annulus, has three zones of flow behavior which reflect those of the turbulent boundary layer. The velocity distribution depicted in Fig. 2.1 corresponds to the latter regime. 

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... 

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... 

Calculate the thickness of the laminar sub-layer when benzene flows through a pipe 50 mm in diameter at 2 1/s. What is the velocity of the benzene at the edge of the laminar sub-layer Assume that fully developed flow exists within the pipe and that for benzene, p — 870 kg/m3 and p = 0.7 mN s/m2. 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

For steady, uniform, fully developed flow in a pipe (or any conduit), the conservation of mass, energy, and momentum equations can be arranged in specific forms that are most useful for the analysis of such problems. These general expressions are valid for both Newtonian and non-Newtonian fluids in either laminar or turbulent flow. 

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 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... 

Steady-state, fully 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. 

For a fixed spherical particle in a fully developed laminar pipe flow, determine the Saffinan force on the particle at various radial positions. Identify the location of the maximum Saffman force. Discuss the case if the flow is turbulent (using the 1/7 power law for the velocity profile). 

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

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. 

This is, of course, the well-known parabolic velocity profile for fully developed laminar pipe flow. 

This shows, incidentally, that the mean velocity in fully developed laminar pipe flow is half of the center line velocity, a well-known result. 

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 ... 

The above results show that the flow is laminar and that the flow will not be fully developed at the exit of the pipe. The program discussed above gives the variation of dimensionless temperature with dimensionless radius at the pipe exit. The program has therefore been run up to a maximum Z value of 0.00004163. This gives the dimensionless temperature variation with dimensionless radius at the exit listed in Table E4.4. The actual radius and temperature are then obtained by recalling that ... 

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. 

Because the velocity field is fully developed, the variations of U and E with R are known. The solution to Eq. (7.93) can therefore be obtained using a similar procedure to that used in Chapter 4 to solve for thermally developing laminar pipe flow, i.e., using separation of variables. Here, however, a numerical finite-difference solution procedure will be used because it is more easily adapted to the situation where the wall temperature is varying with Z. 

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... 

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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