Laminar velocity distributions

Comment by J. M. Geist, Air Products, Inc. We would like to add that because of the laminar velocity distribution, back-diffusion may occur in an "annular ring," where the velocity is less than the critical, and that back-diffusion does not occur in the center core, where the critical velocity is exceeded. 

The author is grateful to R. L. Turner for discussions of his work on laminar velocity distributions and for pointing out the method of L. H. Thomas. 

Weilbacher confirmed that the parabolic laminar velocity distribution forms near the upper end of the Gonell elutriator. He noted that the arrangement at the top of the tube for the removal of fineparticles fractionated in the chamber moved them through the boundary region of low velocities at the side of the tube, thus causing resedimentation into the boundary region and down the elutriation column thus prolonging the separation process. 

A. Laminar, vertical wetted wall column Ws/, 3.41 — D 5fa (first term of infinite series) [T] Low rates M.T Use with log mean concentration difference. Parabolic velocity distribution in films. 

Few experimental data exist on laminar jets (see Gutfinger and Shinnar, AJChE J., 10, 631-639 [1964]). Theoretical analysis for velocity distributions and entrainment ratios are available in Schhcht-ing and in Morton Phy.s. Fluids, 10, 2120-2127 [1967]). 

Laminar or power law velocity distribution in which the linear velocity varies with radial position in a cylindrical vessel. Plug flow exists along any streamline and the mean concentration is found by integration over the cross section. 

The relation between the mean velocity and the velocity at the axis is derived using this expression in Chapter 3. There, the mean velocity u is shown to be 0.82 times the velocity us at the axis, although in this calculation the thickness of the laminar sub-layer was neglected and the Prandtl velocity distribution assumed to apply over the whole cross-section. The result therefore is strictly applicable only at very high Reynolds numbers where the thickness of the laminar sub-layer is vety small. At lower Reynolds numbers the mean velocity will be rather less than 0.82 times the velocity at the axis. 

Maynes and Webb (2002) presented pressure drop, velocity and rms profile data for water flowing in a tube 0.705 mm in diameter, in the range of Re = 500-5,000. The velocity distribution in the cross-section of the tube was obtained using the molecular tagging velocimetry technique. The profiles for Re = 550,700,1,240, and 1,600 showed excellent agreement with laminar flow theory, as presented in Fig. 3.2. The profiles showed transitional behavior at Re > 2,100. In the range Re = 550-2,100 the Poiseuille number was Po = 64. 

Assume laminar flow and a parabolic velocity distribution. Calculate the temperature and composition profiles in the reactor. Start with 7=4 and double until your computer cries for mercy. Consider two cases (a) 7 = 0.01 m (b) 7 = 0.20 m. 

In equation 1.14, z, P/(pg), and u2/(2ga) are the static, pressure and velocity heads respectively and hf is the head loss due to friction. The dimensionless velocity distribution factor a is for laminar flow and approximately 1 for turbulent flow. 

Laminar flow In circular tubes with parabolic velocity distribution Is known as Poiseuille flow. This special case is found frequently in vacuum technology. Viscous flow will generally be found where the molecules mean free path is considerably shorter than the diameter of the pipe X d. 

Clearly the assumption of a flat velocity profile is not correct. For a film in steady, laminar motion one may obtain an expression for the velocity distribution from the Navier-Stokes equations of motion [Eq. (9)]. For this case the Navier-Stokes equations simplify to... 

Eigure 2.4a shows the velocity distribution in a steady isothermal laminar flow of an incompressible Newtonian fluid through a straight, round tube. The velocity distribution in laminar flow is parabolic and can be represented by... 

Figure 2.4b shows, conceptually, the velocity distribution in steady turbulent flow through a straight round tube. The velocity at the tube wall is zero, and the fluid near the wall moves in laminar flow, even though the flow of the main body of fluid is turbulent. The thin layer near the wall in which the flow is laminar is called the laminar sublayer or laminar film, while the main body of fluid where turbulence always prevails is called the turbulent core. The intermediate zone between the laminar sublayer and the turbulent core is called the buffer layer, where the motion of fluid may be either laminar or turbulent at a given instant. With a relatively long tube, the above statement holds for most of the tube length, except for... 

Velocity distributions in turbulent flowthrough a straight, round tube vary with the flow rate or the Reynolds number. With increasing flow rates the velocity distribution becomes flatter and the laminar sublayer thinner. Dimensionless empirical equations involving viscosity and density are available that correlate the local fluid velocities in the turbulent core, buffer layer, and the laminar sublayer as functions of the distance from the tube axis. The ratio of the average velocity over the entire tube cross section to the maximum local velocity at the tube axis is approximately 0.7-0.85, and increases with the Reynolds number. 

The usual velocity distributions in a steady flow of liquid through a tube are shown in Figure 2.4. In either laminar or turbulent flow, the velocity at the tube wall is zero but is maximum at the tube axis. The ratio of the average velocity to the maximum velocity is 0.5 for laminar flow and approximately 0.8... 

From the force balance acting on an imaginary coaxial fluid cylinder of radius r and length L, the velocity distribution in laminar flow through a straight round tube is given as follows,... 

Fig. 2.11. Actual velocity distribution in a pipe, (a) Laminar flow (b) Turbulent flow...

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

In laminar flow the velocity distribution, and hence the frictional energy loss, is governed entirely by the rheological constitutive relation of the fluid. In some cases it is possible to derive theoretical expressions for the friction factor. Where this is possible, a three-step procedure must be followed. 

Heat Transfer and Velocity Distribution in Hydrodynamically and Thermally Developed Laminar Flow in Conduits... 

Obtain the steady-state, fully developed velocity distribution for laminar flow between two parallel plates, in the absence of body forces. 

Figure 9.2 Dimensionless velocity distribution for laminar free convection on a vertical flat plate. Ostrach, 1953 [3].

Consider convection with incompressible, laminar flow of a constant-temperature fluid over a flat plate maintained at a constant temperature. With the velocity distributions found in either Prob. 10.1 or Prob. 10.2, compute the dimensionless temperature distribution within the thermal boundary layer for the Peclet number equal to 0.1,1.0,10.0,100.0. Use the ADI method. 

Consider the tube-flow system in Fig. 5-13. We wish to calculate the heat transfer under developed flow conditions when the flow remains laminar. The wall temperature is Tw, the radius of the tube is rc, and the velocity at the center of the tube is u0. It is assumed that the pressure is uniform at any cross section. The velocity distribution may be derived by considering the fluid element shown in Fig. 5-14. The pressure forces are balanced by the viscous-... 

Using the velocity distribution for developed laminar flow in a tube, derive an expression for the friction factor as defined by Eq. 5-112. 

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