Linear pressure gradient

EXAMPLE 4.3 Plagan Poiseuille flow - laminar, steady incompressible flow in a long pipe with a linear pressure gradient (first-order, nonlinear solution to an ordinary differential equation)... 

An incompressible fluid flows in a laminar steady fashion through a long pipe with a linear pressure gradient. Describe the velocity profile and determine the relationship for the Darcy-Weisbach friction factor. 

The right-hand side of equation (E4.3.1) is a constant because of the linear pressure gradient. The left-hand side can thus be written as an ordinary differential equation ... 

Consider the steady, laminar flow of an incompressible fluid in a long and wide closed conduit channel subject to a linear pressure gradient, (a) Derive the equation for velocity profile, (b) Derive the equation for discharge per unit width and cross-sectional mean velocity, and compare this with the maximum velocity in the channel, (c) Derive the equation for wall shear stress on both walls and compare them. Explain the sign convention for shear stress on each wall. 

In Figure 7.2.18, the contour plot of an extraction of 820 mg of roasted coffee is shown [20]. The contour plot is similar to that given above Figure 7.2.11 where the horizontal axis represents the proton chemical shift and the vertical axis represents the pressure. A linear pressure gradient from 85 to 220 bar over 55 min, and a CO2 flow of 1 ml/min at a temperature of 317 K was applied. One can clearly see the caffeine, which is extracted at a pressure of 105 bar, as... 

Figure 7.2.18 Extraction profile (contour plot) of roasted coffee using a linear pressure gradient from 85 to 220 bar...

HPLC column which is closed with conventional frits on both sides. The extraction cell is kept at a temperature of 315 K, with a CO2 flow rate of l.Oml/min, while for the extraction a linear pressure gradient from 79 to 317 bar over 50 min is employed. The weight reduction after the extraction is 119 mg, corresponding to 18.5%. At the final extraction density of 0.9g/ml (315 bar), the valve at the SFC probe is closed in order to switch to stop-flow conditions. The proton spectrum shown in Figure 7.2.21 is acquired under static conditions. From this, the structure of the extracted phthalate could be assigned as bis(2-ethylhexyl) phthalate. 

The permeability of hardwoods (such as birch treated here) is 10 to 10 times higher in the axial direction than in the radial and tangential directions [14]. Therefore the axial flow was included in the conservation equations (Equation 2, 3 and 4). A discussion on the anisotropy of wood, the inclusion of axial convective terms and the omission of axial diffusive terms are presented in [7], In the conservation of gas species and of liquid phase the axial flow was estimated by Equation 7 and 8 assuming, for simplicity, a linear pressure gradient in the axial direction of the sample. For the energy equation (Equation 4) two cases were studied, q H) and an axial heat loss according to ... 

If there is a steady flow from right to left in a thin, porous stratum (as would exist in a horizontal oil stratum with a linear pressure gradient) and some of the fluid is being withdrawn from a well in a direction perpendicular to the page, then the flow pattern is as shown, and the equipotential lines are isobars. j... 

With regard to pressure gradients, extrusion compounds tested to date all proved to answer one of two type descriptions. Figure 2 depicts the pressure as a function of cylinder length. Type-1 compounds have a linear pressure gradient resulting in constant shear stress. Type-2 compounds fol-... 

Normal pressure regimes follow a hydrostatic fluid gradient from surface, and are approximately linear. Abnormal pressure regimes include overpressured and underpressured fluid pressures, and represent a discontinuity in the normal pressure gradient. Drilling through abnormal pressure regimes requires special care. 

For many practical purposes sufficient accuracy is obtained by neglecting the leakage flow term. In addition the pressure gradient is often considered as linear so... 

It was found that the pressure gradient and flow friction in micro-channels were higher than that predicted by the conventional laminar flow theory. In a low Re range, the measured pressure gradient increased linearly with Re. For Re > 500, the slope of the /(c-Re relationship increases with Re. The ratio C was about 1.3 for micro-channels of hydraulic diameter 51.3-64.9pm and 1.15-1.18 for microchannels of hydraulic diameter 114.5-168.9pm. It was also found that the ratio of C depends on the Reynolds number. 

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

Possible driving forces for solute flux can be enumerated as a linear combination of gradient contributions [Eq. (20)] to solute potential across the membrane barrier (see Part I of this volume). These transbarrier gradients include chemical potential (concentration gradient-driven diffusion), hydrostatic potential (pressure gradient-driven convection), electrical potential (ion gradient-driven cotransport), osmotic potential (osmotic pressure-driven convection), and chemical potential modified by chemical or biochemical reaction. 

Clearly, the pressure itself cannot be periodic even if the geometry of the model is periodic. Instead, the pressure drop is periodic. The local pressure gradient is decomposed into two parts the gradient of a periodic component, superimposed on the gradient of a linearly varying component. The linearly varying component of the pressure corresponds to the familiar packed-tube pressure drop. Its value is not known before the simulation it must be iterated on until the mass flow rate that was specified is achieved in the computational model. 

---