Velocity parabolic

This result describes a parabolic velocity profile, as sketched in Fig. 9.5b. 

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. 

A. Tubes, laminar, fuUy developed parabolic velocity profile, developing concentration profile, constant wall concentration... 

E. Laminar, fully developed parabolic velocity profile, constant mass flux at wall... 

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

The dispersion that takes place in an open tube, as discussed in chapter 8, results from the parabolic velocity profile that occurs under conditions of Newtonian flow (i.e., when the velocity is significantly below that which produces turbulence). Under condition of Newtonian flow, the distribution of fluid velocity across the tube... 

When the Reynolds number is under 2000, it is shown empirically that the flow in a smooth tube is laminar. This flow has a parabolic velocity profile, as shown in Fig. 4.3. 

Non-slip condition on the wall u = v = 0), and parabolic velocity profile u at the inlet... 

Example 8.1 Find the mixing-cup average outlet concentration for an isothermal, first-order reaction with rate constant k that is occurring in a laminar flow reactor with a parabolic velocity profile as given by Equation (8.1). 

Equation (8.9) can be applied to any reaction, even a complex reaction where ctbatch(t) must be determined by the simultaneous solution of many ODEs. The restrictions on Equation (8.9) are isothermal laminar flow in a circular tube with a parabolic velocity profile and negligible diffusion. 

Example 8.3 The reactor of Example 8.2 is actually in laminar flow with a parabolic velocity profile. Estimate the outlet concentration ignoring molecular diffusion. 

Example 8.6 Generalize Example 8.5 to determine the fraction unreacted for a first-order reaction in a laminar flow reactor as a function of the dimensionless groups and kt. Treat the case of a parabolic velocity profile. 

Figure 8.1 includes a curve for laminar flow with 3>AtlR = 0.1. The performance of a laminar flow reactor with diffusion is intermediate between piston flow and laminar flow without diffusion, aVI = 0. Laminar flow reactors give better conversion than CSTRs, but do not generalize this result too far It is restricted to a parabolic velocity profile. Laminar velocity profiles exist that, in the absence of diffusion, give reactor performance far worse than a CSTR. 

FIGURE 8.3 First-order reaction with fet = 1 in a tubular reactor with a parabolic velocity prohle. 

Solution of Equation (8.63) for the case of constant viscosity gives the parabolic velocity profile. Equation (8.1), and Poiseuille s equation for pressure drop. Equation (3.14). In the more general case of /r = /r(r), the velocity profile and pressure drop are determined numerically. 

Solution The numerical integration techniques require some care. The inlet to the reactor is usually assumed to have a flat viscosity profile and a parabolic velocity distribution. We would like the numerical integration to reproduce the paraboUc distribution exactly when q, is constant. Otherwise, there will be an initial, fictitious change in at the first axial increment. Define... 

Consider an isothermal, laminar flow reactor with a parabolic velocity profile. Suppose an elementary, second-order reaction of the form A -h B P with rate SR- = kab is occurring with kui 1=2. Assume aj = bi . Find Uoutlam for the following cases ... 

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. 

Equation (8.11) gave the differential distribution function that corresponds to a parabolic velocity prohle in a tube. This specihc result is now derived in a more general way. 

Assuming a parabolic velocity profile for laminar conditions, the flow averaged shear stress is calculated as ... 

Fig. 4.2.13 Velocity profile for low density polyethylene melt. The line refers to a fit of the data based on a parabolic velocity profile.

It should be emphasized that these results are applicable only to fully developed flow. However, if the fluid enters a pipe with a uniform ( plug ) velocity distribution, a minimum hydrodynamic entry length (Lc) is required for the parabolic velocity flow profile to develop and the pressure gradient to become uniform. It can be shown that this (dimensionless) hydrodynamic entry length is approximately Le/D = 7VRe/20. 

We consider steady-state, one-dimensional laminar flow (q ) through a cylindrical vessel of constant cross-section, with no axial or radial diffusion, and no entry-length effect, as illustrated in the central portion of Figure 2.5. The length of the vessel is L and its radius is R. The parabolic velocity profile u(r) is given by equation 2.5-1, and the mean velocity u by equation 2.5-2 ... 

Consider an element of fluid ("tracer") entering the vessel at / = 0 with the parabolic velocity profile fully established. The portion at the center travels fastest, and has a residence time t0 = Uu0 = LI2u = 02, since ii = uJ2. That is, no portion of the tracer entering at t = 0 leaves until t = 02. As a result, we conclude that... 

Laminar flow in a cylindrical tube is characterized by a parabolic velocity profile or distribution ... 

With the carrier stream unsegmented by air bubbles, dispersion results from two processes, convective transport and diffusional transport. The former leads to the formation of a parabolic velocity profile in the direction of the flow. In the latter, radial diffusion is most significant which provides for mixing in directions perpendicular to the flow. The extent of dispersion is characterized by the dispersion coefficient/). 

It is assumed that the Reynolds number is sufficiently high for the fluid s momentum to be dominant and consequently the momentum flow rate in the jet will be the same as that in the tube. On emerging from the tube, there is no wall to maintain the liquid s parabolic velocity profile and consequently the jet develops a uniform velocity profile. 

Thus, for a parabolic velocity profile in a pipe, the volumetric average velocity is half the centre-line velocity and the equation for the velocity profile can be written as ... 

---