Poiseuille constant, flow-through

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 ... 

Two common types of one-dimensional flow regimes examined in interfacial studies Poiseuille and Couette flow [37]. Poiseuille flow is a pressure-driven process commonly used to model flow through pipes. It involves the flow of an incompressible fluid between two infinite stationary plates, where the pressure gradient, Sp/Sx, is constant. At steady state, ignoring gravitational effects, we have... 

Next we consider a fluid flowing through a circular tube with material at the wall diffusing into the moving fluid. This situation is met with in the analysis of the mass transfer to the upward-moving gas stream in wetted-wall-tower experiments. Just as in the discussion of absorption in falling films, we consider mass transfer to a fluid moving with a constant velocity profile and also flow with a parabolic (Poiseuille) profile (see Fig. 5). 

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. 

If a liquid with constant viscosity r flows through a tube, the P(Q) function is described by the well known Poiseuille equation, which can be written as... 

In this section we consider a second problem involving heat transfer in Poiseuille flow through a circular tube. In this case, we assume that the fluid in the region — 8 < z < 8 is initially heated to a temperature 0, while the temperature elsewhere, i.e., z > 8, is held at a constant temperature 60. The wall of the tube is insulated for all z so that there is no heat loss or gain to the surroundings. Precisely the same problem could be formulated as a mass transfer problem for the redistribution of a solute in a solvent with an initial solute concentration C for —5 < z < 5 and concentration Co for z > 8, with tube walls that are impermeable to the solute. The only difference is that the thermal diffusivity k is replaced with the species diffusion coefficient/). However, to make the discussion as straightforward as possible, the analysis in this section is presented as a heat transfer problem. 

Accordingly, for rods, the maximum velocity of rotation occurs at t/t = 0, 1/2, 3/4, 5/4,... For spheres with p = 1, ( )j = 2jtt/tp, i.e., a constant rotational velocity. In non-uniform shear fields, such as that observed during flow through a capillary (Poiseuille flow), the particles rotate with velocity predicted by Eqs 7.31 and 7.32, according to the value of the shear rate existing at the radial location of the sphere in the capillary. Near the wall, for finite diameter spheres, the immobile layer of the suspending medium causes a reduction of rotational and translational velocity. The effect scales with the square of the sphere diameter. 

As mentioned above, the first experimental work in which water through glass microtubes with an inner diameter down to 15 pm was tested was due to Poiseuille in 1840 [2]. The experimental data obtained by Poiseuille were used to state the famous Poiseuille law for laminar flows through circular tubes, according to which for Stokes flows the product of the frictirMi factor times the Reynolds number is a constant (equal to 16 for circular tubes) that depends on the cross-sectional geometry only. 

The single-capillary viscometer (SCV) is represented in Fig. la. Its design is a direct extrapolation of classical viscometry measurement. It is composed of a small capillary, through which the solvent flows at a constant flow rate, and a differential pressure transducer (DPT), which measures the pressure drop across the capillary. SCV obeys Poiseuille s law and the pressure drop AP across the capillary depends on the geometry of the capillary, on flow rate Q, and on viscosity of the fluid rj according to... 

The most basic state of motion for fluid in a pipe is one in which the motion occurs at a constant rate, independent of time. The pressure flow relation for laminar, steady flow in round tubes is called Poiseuille s Law, after J.L.M. Poiseuille, the French physiologist who first derived the relation in 1840 [12]. Accordingly, steady flow through a pipe or channel that is driven by a pressure difference between the pipe ends of just sufficient magnitude to overcome the tendency of the fluid to dissipate energy through the action of viscosity is called Poiseuille flow. 

Most rheological measurements measure quantities associated with simple shear shear viscosity, primary and secondary normal stress differences. There are several test geometries and deformation modes, e.g. parallel-plate simple shear, torsion between parallel plates, torsion between a cone and a plate, rotation between two coaxial cylinders (Couette flow), and axial flow through a capillary (Poiseuille flow). The viscosity can be obtained by simultaneous measurement of the angular velocity of the plate (cylinder, cone) and the torque. The measurements can be carried out at different shear rates under steady-state conditions. A transient experiment is another option from which both y q and ]° can be obtained from creep data (constant stress) or stress relaxation experiment which is often measured after cessation of the steady-state flow (Fig. 6.10). 

The pressure-driven, steady-state flow of an incompressible fluid through a straight channel known as Poiseuille flow provides the basic fundamentals about the flow through channels relevant to liquid handling in a lab-on-a-chip (LOC) system. A constant pressure drop AP results in a constant flow rate Q. This result is summarized as the Hagen-Poiseuille law ... 

When a liquid is made to flow through a capillary tube by a pressure gradient, and the time for a given volume to flow along the tube is measured, then at a constant temperature, the coefficient of viscosity, r, is given by Poiseuille s equation... 

In Chap. 9 we shall examine the flow of a solution through a capillary tube. The rate of volume delivery in that case is given by Poiseuille s law [Eq. (9.29)], which states that the time required for a constant volume of liquid to drain out of the capillary is proportional to r jp. Accordingly, the viscosity is proportional to the product pt, and when the delivery times for two liquids are compared in the same capillary. 

Perhaps the most simple flow problem is that of laminar flow along z through a cylindrical pipe of radius r0. For this so-called Poiseuille flow, the axial velocity vz depends on the radial coordinate r as vz (r) — Vmax [l (ro) ] which is a parabolic distribution with the maximum flow velocity in the center of the pipe and zero velocities at the wall. The distribution function of velocities is obtained from equating f P(r)dr = f P(vz)dvz and the result is that P(vz) is a constant between... 

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 ... 

The steady gas flow in a long macroscopic channel with impermeable walls is basically a Poiseuille flow with the constant velocity determined by the pressure gradient. However, the velocity of the flow in the fuel cell channel varies since there is mass and momentum transfer through the channel/backing layer interface. 

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. 

The various constants defined above have, it will be seen, different dimensions. It is important therefore to specify accurately what constants are being employed. When, as is usually done, the permeability constant is formally defined as the number of unit volumes passing in unit time through unit cube having unit pressure difference between its faces, its dimensions are cm. sec. g.-, which are those of the permeability coefficient of Table 1, for Poiseuille flow. The dimensions of the diffusion constant D defined by Pick s law... 

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