Poiseuille’s equation

For example, consider an open tubular column with the dimensions previously defined, operated at constant mass flow rate of helium (which is normal for temperature programming purposes), then, from Poiseuille s equation, after time (t),... 

The radius of the column cannot be reduced indefinitely due to the pressure difference that will increase as (R) is reduced. From Poiseuille s equation the pressure drop across the tube is given by... 

Thus he modified Poiseuille s equation (Eq. 23) to describe the penetration of such a liquid into a closed pore of length L ... 

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. 

Cf., Poiseuille s equation, according to which the fluid velocity through a capillary is linearly dependent on the pressure gradient along it. 

From Section 7.5.4, Volume 3, for a manometer in which fractional damping is four times that predicted by Poiseuille s equation ... 

For a capillary of circular cross-section, fRe has a value of 16, and dhyd is equal to the capillary diameter. Equation (5.9) then reduces to Poiseuille s equation. 

For streamline fluid flow through a circular pipe where Poiseuille s equation applies (given in Volume 1, Chapter 3), K0 is equal to 2.0, and for streamline flow through a rectangle where the ratio of the lengths of the sides is 10 1, Ko = 2.65. Carman(14) has listed values of K0 for other cross-sections. From equation 4.22 it can be seen that if, say,... 

Several modifications of Poiseuille s equation have been attempted by various authors to describe permeability in the transitional region between viscous and diffusional flow. The assumptions underlying these modifications are often questionable and the results obtained offer little or no theoretical or experimental advantage over the BET theory for surface area measurements. Allen" " discusses these modifications as well as diffusional flow at low pressures. 

This is the two-dimensional equivalent of Poiseuille s equation. All of the other quantities besides 175 in Equation (28) are measurable, so 17s can be evaluated by measuring the rate at which the monolayer flows through the channel. In practice, a second barrier is moved along in front of the advancing interface to maintain a constant film pressure for an insoluble monolayer. 

The analysis of this effect in a closed cylindrical cell is obtained by subtracting from the electroosmotic velocity vEO the velocity of flow vP through a capillary given by Poiseuille s equation (Equation (4.18)) ... 

This technique is by far the easiest for the characterization of polymers in solution. This can be seen from the simplicity of the typical (glass) viscometer shown in Figure 2.7. It is used to obtain the viscosity of a liquid by the use of Poiseuille s equation, which is... 

Let s be the potential difference developed between the ends of a capillary tube of radius a and length / for an applied pressure difference p. Assuming laminar flow, the liquid velocity vx at a distance x measured from the surface of shear and along a radius of the capillary is given by Poiseuille s equation, which can be written in the form... 

Poiseuille s equation applies to laminar flow only. Let s now try some examples. 

This is a fairly straight forward plug-and-chug Poiseuille s equation problem. Using SI values only, we obtain ... 

This is another application of Poiseuille s equation. Notice in this case Pt-P2 = 1000 Pa. 

The conversion of the throughput qpV into volume flow yields the values listed in Table 1. Since in the capillaries laminar flow prevails (Reynolds number RE 2,300), the pressure drop along the capillaries follows Hagen-Poiseuille s equation ... 

Poiseuille s equation is in some respects closely analogous to Darcy s. 

The relative viscosity is easily obtained by rearranging Poiseuille s equation to express the viscosity in terms of the rate of flow, realizing that the pressure, P, maintaining the flow is simply proportional to the density of the solutions or pure solvent, while the rate of flow is simply proportional to the time taken to go between the two marks, as shown in Equation 12-41 ... 

From Poiseuille s Equation, describing the flow of liquid through an open tube,... 

It is assumed in the deduction of Poiseuille s equation that no slip occurs on the walls of the tube ( 12.VII F). In the case of a liquid such as mercury which does not wet glass, this is not obviously valid. Tammann and Hinniiberii found for mercury in glass tubes 77=1 609 cp. and in amalgamated copper tubes 4 931 cp., and they supposed that there was slip on the glass. Bingham... 

The opportunity to measure the dilute polymer solution viscosity in GPC came with the continuous capillary-type viscometers (single capillary or differential multicapillary detectors) coupled to the traditional chromatographic system before or after a concentration detector in series (see the entry Viscometric Detection in GPC-SEC). Because liquid continuously flows through the capillary tube, the detected pressure drop across the capillary provides the measure for the fluid viscosity according to the Poiseuille s equation for laminar flow of incompressible liquids [1], Most commercial on-line viscometers provide either relative or specific viscosities measured continuously across the entire polymer peak. These measurements produce a viscometry elution profile (chromatogram). Combined with a concentration-detector chromatogram (the concentration versus retention volume elution curve), this profile allows one to calculate the instantaneous intrinsic viscosity [17] of a polymer solution at each data point i (time slice) of a polymer distribution. Thus, if the differential refractometer is used as a concentration detector, then for each sample slice i. 

Now, as the flow is increased, the inlet pressure will also increase and, thus, the inlet/outlet pressure ratio (y) wUl change progressively during the program the mean flow rate will be reduced according to the pressure correction function and the decrease in elution rate will not be that which would be expected. Consider an open-tubular column from Poiseuille s equation. 

---