Hydrostatic pressure drop

FIGURE 3.10 Hydrostatic Pressure Drop in a 1-g Inverted Vertical Outflow Configuration. 

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The interfacial area in the contactor, which is directly related to the solids hold-up, strongly influences the mass transfer rate. To maximise the overall mass-transfer rate per unit volume of equipment, a high solids hold-up is necessary. On the other hand, the solids hold-up also influences the pressure drop over the contactor. The pressure drop has a hydrostatic and a dynamic component, both of which rise with increased solids hold-up. Since the adsorbent consists of extremely small particles, fluid friction between liquid and solids may lead to a relatively high dynamic pressure drop. The hydrostatic pressure drop is attributable to the density difference between the suspension in the contact zone and in the liquid. 

The osmotic pressure drop on the filter also is negligible, so that at zero current the liquid level in both vessels is equal (h = 0). The experiment consists of passing an increasing sequence of DC currents with a small increment between the subsequent current values, while observing the evolution in time of the voltage E(t) and of the hydrostatic pressure drop P(t) related to the elevation h(t) as... 

In the current case, slightly more than half of the hydrostatic pressure drop along the phloem is necessary to overcome the resistance of the sieve plate pores. When the end walls are steeply inclined to the axis of the sieve element, the pores of the sieve plate can occupy an area that is greater than the cross section of the sieve tube. This causes Jv in the pores to be less than in the lumen and tends to reduce the resistance to flow in the phloem. 

Bengal), Barnes et al. (2006) found no seasonality. However, N2O emissions were negatively correlated with tidal height indicating that N2O from sediment pore-water was released during hydrostatic pressure drop toward low water. A reahstic estimate of the N2O emissions from mangroves is not possible at the moment because of the very small number of available measurements. 

With respect to pressure drop, the situation is quite different for the MR. As the liquid phase enters the top of the monolith it accelerates/decelerates until it reaches the linear velocity at which the hydrostatic pressure drop balances the frictional force. This velocity is approximately... 

For both cases, idealized situations are sketched in fig. 1.4. In the left case it is assumed that the meniscus is hemispherical. The contact angle of the liquid with respect to the inner side of the capillary is zero on the left hand side in the right hcind case a = 180°. At equilibrium Ap is equal to the hydrostatic pressure drop of the column over the flat reference surface, Apgh (or p gh = pgh if the density of the vapour phase is neglected). The precise meaning of h has to be considered later because the meniscus is not flat. For the spherical meniscus Rj = R2 = a, if a is the inner radius of the capillary, hence... 

In Figure 4.7, the pressure at the liquid level and at point a just above the meniscus is atmospheric. The height of the column of liquid, h, will be such that the pressure at point c in the column is also atmospheric, due to the balanced effects of hydrostatic and capillary pressures in the liquid column. The pressure at point b, just below the meniscus, will be less than atmospheric by an amount (2ylr).kt equilibrium, AP is also equal to the hydrostatic pressure drop in the liquid column in the capillary. Thus,... 

Normal Hydrostatic Pressure Drop. There are two hydrostatic pressure drop terms of interest in heat pipes a normal hydrostatic pressure drop AP+, which occurs only in heat pipes... 

Axial Hydrostatic Pressure Drop. The second hydrostatic pressure drop term is the axial hydrostatic pressure drop, Af n, which results from the component of the body force acting along the longitudinal axis. This term can be expressed as... 

In a gravitational environment, the normal and axial hydrostatic pressure terms may either assist or hinder the capillary pumping process depending on whether the tilt of the heat pipe promotes or hinders the flow of liquid back to the evaporator (i.e., the evaporator lies either below or above the condenser). In a zero-g environment, both this term and the normal hydrostatic pressure drop term can be neglected because of the absence of body forces. 

Using a simple scaling analysis, involving (1) viscous pressure drop, (2) hydrostatic pressure drop, (3) interfacial pressure drop and (4) penetration theory for mass transfer, it has been demonstrated that two-phase laminar bubble-train flow in small channels can exhibit better mass transfer for a given power input than turbulent contactors. 

The rise or fall of a liquid interface in a capiUaiy can be easily calculated by writing the balance of forces on a cylindrical column of height H (see Fig. 5). The water at the bottom of the tube, leveled with the free surface of the liquid, is at atmospheric pressure. The hydrostatic pressure drop just below the meniscus is balanced by the vertical component of surface tension at the wall. 

Ap, is the hydrostatic pressure drop. This is due to the gas flowing through the two phase regime, or froth section, of height on the tray... 

Van Buel, Van der Wielen, and Luyben have proposed a model to explain the considerable pressure drop arising in the column during CPC separation. The overall pressure drop is the sum of the hydrostatic pressure drop term and the hydrodynamic pressure drop terms over the individual parts of the system. The hydrostatic contribution is caused by the difference in density between the liquids in the ducts and in the channels (APstat = nlApu/R, where n is the number of channels, I the height of stationary phase in the channel, Ap the density difference between the phases. 

For the maximum pressure drop, as specified by the geometry of the ganglion ends, i.e., AP = AP, the hydrostatic pressure drop (as measured normal to the flow direction) is evaluated by ... 

The hydrostatic pressure drop due to the presence of gravity forces is ... 

FIGURE 2.18. The hydrostatic pressure drop between points B and A offsets the Laplace underpressure at point A. 

A/ v = sum of inertial and viscous pressure drops occurring in vapor phase Apg — hydrostatic pressure drop... 

The hydrostatic pressure drop is highlighted in red in Figure 3.10, which for a 1-g inverted vertical outflow configuration can be expressed simply as ... 

Two 91 cm long by 2.54 cm wide by 2.54 cm deep (36.2 in. x 1 in. x 1 in.) LAD channels were custom built, a standard 325 x 2300 channel and a TVS cooled 325 x 2300 channel. The width and depth of both channels were sized based off analysis of the pressure drop model from Chapter 3 to achieve a broad range of breakdown locations and also to maximize the FTS and frictional pressure losses. The 91 cm (36 in.) height was chosen to overcome the relatively large LH2 hydrostatic pressure drop over the range of liquid temperatures tested. 

As shown for the low flow case, hydrostatic pressure drop is the leading order term for the duration of the tank drain, while FTS pressures drop is a second-order effect. Meanwhile at the high flow case, the FTS contribution increases proportionally initially this contribution is higher than the hydrostatic term, but is eventually overcome by hydrostatics. To illustrate further. Table 9.9 shows the percent contribution of hydrostatic, FTS, and the sum of frictional and dynamic pressure drops at LAD breakdown for both of these model runs. 

In addition, the simplified ID steady state pressure drop model for screen channel liquid acquisition devices has been developed and compared to the FTS, horizontal LAD, and full-scale LAD outflow experimental data. Both experimental data and model confirm that, in 1-g outflow from a cryogenic propellant tank, the hydrostatic pressure drop is the leading order term, followed by the FTS pressure drop, and frictional and dynamic losses down the channel. The model qualitatively tracks the LH2 1-g inverted outflow test results the model predicts the breakdown point within 9% for the TVS cooled channel and 18% for the standard channel. Discrepancies between 1-g model and data are primarily attributed to a non-uniform FTS pressure distribution along the channel. Results show that both LAD channels behaved close to anticipated performance and that this simplified ID model can be used to qualitatively track LAD performance in a dynamic outflow environment. 

The most common case is penetration under the action of gravity which affects the shape of the meniscus and the height of rise in the capillary. In a capillary-rise situation, AP is equal to the hydrostatic pressure drop in the liquid column in the capillary, as follows ... 

Consider a separation device, tubular or otherwise, filled with a liquid assume also that the inlet pipe filled with the liquid is connected to a pump (Figure 6.1.1) at the inlet of the separation device, which is assumed to be horizontal. As the pump runs, it introduces mechanical energy into the liquid, which is driven into the device against whatever flow resistance is offered. As the liquid moves through the device, its hydrostatic pressure is reduced due to frictional losses. The relation between the hydrostatic pressure drop AP encountered by the liquid and its volumetric flow rate Q depends on the flow regime, liquid viscosity /i, the flow channel geometry and... 

The normal and axial hydrostatic pressure drops AP+ and AP are the result of the local gravitational force and can be expressed as... 

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