Hagen-Poiseuille flow profile

The more or less laminar flow condition in column chromatography causes a Hagen-Poiseuille flow profile, resulting in peak broadening (see Fig. 35-1 a). The quest is to limit the development of this flow profile. 

You can inject guanidine or urea with your sample. In this case, the high viscosity of the plug injected in front of the sample limits the development of a Hagen-Poiseuille flow profile. The sample will elute in a smaller peak volume, peak concentration increases and the peak height increases too (see Fig. 35-lb). In a reversed-phase system, urea or guanidine elute with the front and do not disturb the separation. 

Fig. 4.10 Instantaneous nondimensional velocity profiles in a circular duct with an oscillating pressure gradient. The mean velocity, averaged over one full period, shows that the parabolic velocity profile or the Hagen-Poiseuille flow. These solutions were computed in a spreadsheet with an explicit finite-volume method using 16 equally spaced radial nodes and 200 time steps per period. The plotted solution is that obtained after 10 periods of oscillation.

Figure 5.16 shows the product Rej/ / as a function of Rev. For Rev less than approximately 2, the wall-injection has very little effect. In this case the wall friction approaches that of the Hagen-Poiseuille flow (i.e., Reyf = 24). For Rev greater than approximately 2, the V velocity serves to skew the axial velocity profile and thus affect the wall stress. For Rev greater than approximately 20, Re / approaches a linear relationship as... 

Show that the limiting case of flow in a planar wedge for a < C 1 (Section 5.2.4) degenerates into the parallel-plate Hagen-Poiseuille flow. Show that the parabolic velocity profile is recovered and that the pressure gradient approaches a constant. 

Velocity Profile and Friction Factor. The velocity profile of fully developed laminar flow of a constant-property fluid in a circular duct with an origin at the duct axis is given by the Hagen-Poiseuille parabolic profile, as follows ... 

Calculate the residence-time distribution (RTD) for a tubular reactor undergoing steady, laminar flow (Hagen-Poiseuille flow). The velocity profile for Hagen-Poiseuille flow is 2, p. 51]... 

We have just discussed several variations of the flow in ducts, assuming that there are no axial variations. In fact there well may be axial variations, especially in the entry regions of a duct. Consider the situation illustrated in Fig. 4.8, where a square velocity profile enters a circular duct. After a certain hydrodynamic entry length, the flow must eventually come to the parabolic velocity profile specified by the Hagen-Poiseuille solution. 

Consider the flow of an incompressible fluid in the entry region of a circular duct. Assuming the inlet velocity profile is flat, determine the length needed to achieve the parabolic Hagen-Poiseuille profile. Recast the momentum equation in nondimensional form, where the Reynolds number is based on channel diameter and inlet velocity emerges as a parameter. Based on solutions at different Reynolds numbers, develop a correlation for the entry length as a function of inlet Reynolds number. 

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-Poiseuille equation, gives the velocity v as a function of radial position r in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity twice the average velocity, is shown in Fig. 6-10. 

In fully developed flow we have wr = 0, so that the second term on the left hand side drops out. The velocity profile wx r) is given by the Hagen-Poiseuille parabola (3.223). 

To assess the physical deviation between the average of products and the product of averages a momentum velocity correction factor can be defined by Cm = vz) / v1)a- By use of the Hagen-Poiseuille law (1.353) and the power law velocity profile (1.354) it follows that at steady state Cm has a value of about 0.95 for turbulent flow and 0.75 for laminar flow [55]. In practice a value of 1 is used in turbulent flow so that v1)a is simply replaced by the averaged bulk velocity vz) - On the other hand, for laminar flows a correction factor is needed. For more precise calculations a simplified (not averaged ) 2D model is often considered for ideal axisymmetric pipe flows [52, 69]. 

Prom the introductory courses in fluid flow one recalls that the simple parabolic profile for laminar flow in a pipe, the Hagen-Poiseuille law, is derived by integration of a sufficiently simplified form of the generalized momentum equation (see e.g., [13], Example 3.6-1) ... 

Similar balance equations with purely laminar diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.353) might suffice. 

Microfluidics is a concept that describes the science and technology of design, fabrication and operation of systems of microchannels that conduct liquids and gases. T q)ically, the channels have widths of tens to hundreds of micrometers and the speed of flow of the fluids is such that the viscous forces dominate over inertial ones. The resulting - linear - equations of flow and its laminar character provide for extensive control the speed of flow obeys the simple Hagen-Poiseuille equation that relates the speed linearly to the pressure drop through the particular channel and to its inverse hydraulic resistance, which in term is a function of the dimensions of the channel and the viscosity of the fluid. This property, when combined with t5q)ically large values of the Peclet number [1] that reflect the fact that diffiisional transport is t5q)ically slow in comparison to the flow, it is possible to control the profiles of concentration [2] of chemicals and... 

Figure 13 shows the axial velocity profiles computed from equation 124. One can observe the variation of the axial velocity with the normalized bed radius, DF1/2/2. When the normalized bed radius is zero, the axial velocity displays a parabolic profile that corresponds to the Hagen-Poiseuille solution. As the normalized bed radius increases, the axial velocity profile flattens. When the normalized bed radius is infinite, the axial velocity corresponds to a unidirectional flow (flat) profile. 

Flow is laminar for Re < 2300, and in the ideal case (no disturbances due to pipe roughness, internals, etc.) can be described by the Hagen-Poiseuille law. In the derivation of this law, equating the surface force with the shear force gives a parabolic velocity profile u(r) (Equation 2.4-9), as shown in Figure 2.5-2 ... 

In a cylindrical tube with diameter D = 2R and length L, the flow due to the pressure difference (Po - T z,) is described by a parabolic profile (see Figure 7.7). The velocity gradient is given by the Hagen-Poiseuille relation ... 

After calibration was measured between the flow velocity and fluorescence intensity, flow velocity profile in a pressure-driven flow was measured. The measured flow velocity profile within the submicrocapiUary and comparison with theoretical prediction based on the Hagen-Poiseuille equation is shown in Fig. 4a, where rectangular dots represent the average velocity of the measurement point. The standard deviatimi error bars are also shown in Fig. 4a. The difference between the experimental data and... 

For turbulent flow in pipes the velocity profile can be calculated from the empirical power law design formula (1.360). Similar balance equations with purely molecular diffusivities can be used for a fully developed laminar flow in tubular reactors. The velocity profile is then parabolic, so the Hagen Poiseuille law (1.359) might suffice. It is important to note that the difference between the cross section averaged ID axial dispersion model equations (discussed in the previous section) and the simplified 2D model equations (presented above) is that the latter is valid locally at each point within the reactor, whereas the averaged one simply gives a cross sectional average description of the axial composition and temperature profiles. 

Initial collection of acetophenone and of various coloured dye tracers indicated that when using a stainless steel cylinder (10 mm x 250 mm, 20 mL volume) as the collection vessel the total volume of the cylinder was not available for the solute. The dye experiments indicated that about half the total volume was available for collection of the solute. The reasons for this are as follows. The Hagen-Poiseuille equation, which describes the velocity profile of a liquid flowing through a tube under laminar flow conditions predicts that the maximum velocity of the head of the parabolic flow profile will be twice the mean velocity of the liquid [23]. Breakthrough of the dye tracer would therefore be expected in about half the volume of the tube. Fluid mechanics predicts that if the flow is turbulent the difference between the mean and maximum flow velocities is much less than in laminar flow [23]. This means that un-... 

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