Hagen-Poiseuille equation laminar flow

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... 

AP for laminar flow in a eireular tube is given by the 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. 

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

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. 

From the Hagen-Poiseuille equation, the pressure drop in a micro fluidic system with laminar flow can be derived (compare [9]) ... 

The influence of pores and leaks in a package on the total permeation depends primarily on whether or not the package is vacuum packed or at atmospheric pressure (Becker, 1965). For the case of vacuum packaging, the Hagen-Poiseuille equation for laminar gas flow can be used as a first approximation ... 

Hagen-Poiseuille equation — predicts the laminar flow of an incompressible and uniform viscous liquid (Newtonian fluid) through a cylindrical pipe of constant cross-section. The rate of movement of a liquid volume V, during a time t, may be predicted using ... 

Effective viscosity r/e was defined by Skelland (1967) as the viscosity that is obtained assuming that the Hagen-Poiseuille equation for laminar flow of Newtonian fluids is applicable ... 

Dispersion in a Tubular Reactor with Laminar Flow. In a laminar flow reaetor we know that the axial veloeity varies in the radial direetion according to the Hagen-Poiseuille equation ... 

The pressure drop Aji generated by capillaries is low compared to the flow resistance which comes from the column. It can be calculated by the Hagen-Poiseuille equation (for laminar flow) which, if expressed in the usual HPLC units, appears as ... 

For laminar flows, the laminar sublayer fills the entire cross section. Thus, the effect of wall drag is influenced to a greater extent by the shape of the cross section. For many noncircular conduits with laminar flows, either theoretical analyses (similar to the Hagen-Poiseuille equation) or numerical analyses for the pressure-flow relation have been conducted. The results can be expressed in dimensionless form as... 

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... 

For laminar flow and in the absence of ultraAliraiion, the pressure drop (AF) on the feed side of a dialyzer is given by the Hagen-Poiseuille equation for hollow fibers,... 

With the average velocity and channel size now expressed in terms of the measurable parameters Vq, Dp, and e, thp channel model can be used to predict the form of the correlation for pressure drop. For flow at very low Reynolds numbers, the pressure drop should vary with the first power of the velocity and inversely with the square of the channel size, in accordance with the Hagen-Poiseuille equation for laminar flow in straight tubes, Eq. (5.16). The equations for V and ), are used in Eq. (5.16), and a correction factor Aj is added to account for the fact that the channels are actually tortuous and not straight and parallel ... 

The flow inside the fibers is laminar, and the pressure gradient caused by skin friction is given by the derivative form of the Hagen-Poiseuille equation, Eq. (5.16) ... 

Verify the claim that both methods of solution produce the same final answers, and hence the same reactor design strategy, when the two alternatives [i.e., stoichiometric (1 1) feed vs. the 3 1 feed ratio] are considered. A more rigorons addendum to both approaches employs the Hagen-Poiseuille equation for laminar flow or the Ergun equation if the tubular reactor is packed with porous solid catalysts to calculate the pressure drop through the reactor instead of assuming that p = constant from inlet to outlet. 

High-performance liquid chromatography requires that liquid be pumped across a packed bed within a tubular configuration. Snyder and Kirkland have used the Hagen-Poiseuille equation for laminar flow through tubes... 

The flow on the shell side is also laminar. Using the hydraulic-radius concept, an estimate can be made of the pressure drop by replacing D in the above Hagen-Poiseuille equation with 4r . Thus, the pressure drop is... 

The specific power dissipation is proportional to the flow rate and the pressure drop. The pressure drop through open channels with laminar flow is given by the Hagen-Poiseuille equation [20] ... 

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