Hagen—Poiseuille equation

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

Laminar and Turbulent Flow Below a critical Reynolds number of about 2,100, the flow is laminar over the range 2,100 < Re < 5,000 there is a transition to turbulent flow. For laminar flow, the Hagen-Poiseuille equation... 

Simplified theoiy for both batch and continuous filtration is based on the time-honored Hagen-Poiseuille equation ... 

In the case of multiparticle blockage, as the suspension flows through the medium, the capillary walls of the pores are gradually covered by a uniform layer of particles. This particle layer continues to build up due to mechanical impaction, particle interception and physical adsorption of particles. As the process continues, the available flow area of the pores decreases. Denoting as the ratio of accumulated cake on the inside pore walls to the volume of filtrate recovered, and applying the Hagen-Poiseuille equation, the rate of filtration (per unit area of filter medium) at the start of the process is ... 

In the capillary method, the time required for a liquid to flow through a capillary tube is determined. The melt under investigation flows with a constant rate through a tube with a small, definite cross-sectional area, such as a cylindrical capillary. The viscosity can be measured in an absolute way from the pressure drop. This method can yield the most reliable absolute data, the viscosity being given by a modified Hagen-Poiseuille equation ... 

Currently used equations for water flow in unsaturated soil are based on the assumption that soils are similar to a bundle of capillary tubes and that water flow can be approximated by the Hagen-Poiseuille equation.58 While it is obvious that the pore space in soil is not the same as a bundle of capillary tubes, the concept has proven highly useful and is currently used in mathematical descriptions of water flow in soil. 

If the wall stress (rw) in Eq. (6-11) is expressed in terms of the Fanning friction factor (i.e., tw = /pF2/2) and the result solved for/, the dimensionless form of the Hagen-Poiseuille equation results ... 

This is the power law equivalent of the Hagen-Poiseuille equation. It can be written in dimensionless form by expressing the wall stress in terms of the friction factor using Eq. (6-5), solving for /, and equating the result to 16/ N t (i.e., the form of the Newtonian result). The result is an expression that is identical to the dimensionless Hagen-Poiseuille equation ... 

Show how the Hagen-Poiseuille equation for the steady laminar flow of a Newtonian fluid in a uniform cylindrical tube can be derived starting from the general microscopic equations of motion (e.g., the continuity and momentum equations). 

The Hagen-Poiseuille equation [Eq. (6-11)] describes the laminar flow of a Newtonian fluid in a tube. Since a Newtonian fluid is defined by the relation r = fiy, rearrange the Hagen-Poiseuille equation to show that the shear rate at the tube wall for a Newtonian fluid is given by yw = 4Q/nR3 = 8 V/D. 

This is known as Stokes flow, and Eq. (11-3) has been found be accurate for flow over a sphere for NRe < 0.1 and to within about 5% for NRe < 1. Note the similarity between Eq. (11-3) and the dimensionless Hagen-Poiseuille equation for laminar tube flow, i.e.,/ = 16/tVRe. 

Figure 7 illustrates the dynamics of fluid migration through porous carbon electrodes to obey the Hagen-Poiseuille equation that is normally used to describe the transport through membranes having the pores of cylinder-like shape. Therefore, this method can probably be used for express analysis of the electrolyte dynamics in different porous carbon materials. 

Fig. 7. Experimental flow rates plotted vs. theoretical flow rates calculated using the Hagen-Poiseuille equation. The 100 data points include flow rates calculated using five different solvents, four different tube lengths and five different values of overpressure (see Supporting information for data used to generate this plot). The dashed lines indicate the variation expected if the inner diameter was 25% less (1 0.32) or 25% more (1 2.44) than the claimed value.

This last result is known as the Hagen-Poiseuille equation. 

The Hagen-Poiseuille equation for steady laminar flow of Newtonian fluids in pipes and tubes can be written as... 

Pressure (or hydrodynamic) injection involves using pressure to push the sample onto the capillary. The sample loaded onto the column is independent of mobility and is indiscriminant with regard to what is loaded onto the capillary. The quantity injected is given by Hagen-Poiseuille equation ... 

The velocity profile for a Newtonian fluid in a capillary is well-described in most introductory transport texts by the Hagen-Poiseuille equation... 

Flow of a Power Law Fluid in a Straight Circular Tube (Hagen-Poiseuille Equation)... 

Solve for the Hagen-Poiseuille equation in a tube for a Newtonian model. 

For this specific example, the manifold diameter will be kept constant and we will assume a Newtonian isothermal flow, with a constant viscosity fi. The flow of the manifold can be represented using the Hagen-Poiseuille equation as,... 

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