Frictional and dynamic pressure drop

FIGURE 3.13 Frictional and Dynamic Pressure Losses down Liquid Acquisition Device Channel. 

Stokes hypothesis holds Navier-Stokes equations apply 

Gravity force is accounted for in the hydrostatic term and is therefore ignored 

The first two are the no slip conditions and the second two are symmetry conditions. The method of solution is to break Equation (3.39) into a homogeneous PDE with one non-homogeneous BC and a non-homogeneous ordinary differential equation (ODE) with homogeneous BCs as follows  

The solution of the PDE is straightforward using separation of variables and solving for the eigenvalues. The solution of the ODE is straightforward as well. Details are omitted. The final solution for steady state velocity inside the channel becomes  

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The horizontal LAD testing methodology is depicted in Figure 9.8. To measure the frictional and dynamic pressure drops, the channel is mounted inside a test tank and flow is routed through the screen and out through a port at the end of the channel. Pressure drop as a function of mass flow rate down the LAD is measured at multiple locations to quantify performance. 

A picture of the 325 x 2300 horizontal LAD channel used for testing is shown in Figure 9.9. The same exact LAD channel used from previous 2010 LOX LADs outflow tests was used here. Total channel length was 0.61 m (24 in.), but the available screen area for flow was 0.46 m (18 in.). The width of the screen was 5 cm (2 in.). Four pressure taps were spaced out 15 cm (6 in.) apart to measure pressure inside the channel. To deduce the actual frictional and dynamic pressure drop between any two of the taps, the absolute pressure measurements at successive taps can be corrected for FTS across the screen using CFD simulations from Zhang et al. (2009). The LAD was used to feed the line chill down assembly, so flow was routed out of the channel to the bottom of the tank. A Coreolis FM was used to measure flow rate downstream of the channel outlet just outside the bottom of... 

Substituting the hydrostatic, FTS, and frictional and dynamic models from Chapter 3 into Equation (3.3), the total ID pressure drop for the LAD channel in 1-g steady flow is determined, and the model can be used to predict the breakdown point as a function of the liquid temperature and mass flow rate through the LAD. To demonstrate general model trends and predictions, setting Equation (3.3) equal to the bubble point pressure in Equation (3.16), one can then simulate LAD outflow in an inverted 1 -g configuration in LHa for the LAD channels and test conditions that would be typical of an in-space propellant transfer. For a fixed 325 X 2300 screen mesh and LAD channel geometry, examination of the hydrostatic, FTS, frictional, and dynamic pressure drop equations show that the steady state pressure drop is a function of LAD dimensions, liquid temperature, pressure, and mass flow rate. 

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. 

Meanwhile, the pressure drop for Region 2 is the sum of the hydrostatic, frictional, and dynamic pressure drop terms as defined in the steady state ID pressure drop model. Since there is no injection velocity in this region, there is no FTS pressure loss. 

Examination of Equations (3.57) and (3.58) for the frictional and dynamic pressure losses indicates that both are functions of length and width of the channel, liquid temperature, and demand flow rate. Because both of these contributions are essentially third-order effects in a 1-g environment, the horizontal configuration was chosen to null out hydrostatics to even be able to measure these small pressure drops. At all points within the channel the hydrostatic pressure is constant, and according to the FTS model, pressure drop across the screen would be constant as well. 

Horizontal LAD channel tests show that the frictional and dynamic pressure losses within the channel are quite small. The signals are barely measurable in LH2, and the signals are small for LOX flows in excess of 2.25 kg/s. The ID model correlates with the single set of LOX channel frictional pressure drop data. Disparity is due to higher than expected screen injection velocities at the channel exit and scatter in the data. 

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. 

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 1,120 psi of this pressure drop is a dynamic loss due to the change in velocity, and 505 psi is a frictional loss due to the fitting. 

Kom GA, Korn TM (1968) Mathematical handbook. McGraw-Hill, Boston Landau LD, Lifshitz EM (1959) Fluid mechanics, 2nd edn. Pergamon, London Morijama K, Inoue A (1992) The thermodynamic characteristics of two-phase flow in extremely narrow channels (the frictional pressure drop and heat transfer of boiling two-phase flow, analytical model). Heat Transfer Jpn Res 21 838-856 Ngan CD, Dussan EBV (1982) On the nature of the dynamic contact angle an experimental study. JEluidMech 118 27- 0... 

The reactor is modeled by three partial differential equations component balances on A and B [Eqs. (6.1) and (6.2)] and an energy balance [Eq. (6.3) for an adiabatic reactor or Eq. (6.4) for a cooled reactor]. The overall heat transfer coefficient U in the cooled reactor in Eq. (6.4) is calculated by Eq. (6.5) and is a function of Reynolds number Re, Eq. (6.6). Equation (6.7) is used for pressure drop in the reactor using the friction factor /given in Eq. (6.8). The dynamics of the momentum balance in the reactor are neglected because they are much faster than the composition and temperature dynamics. A constant... 

PRESSURE DROP - Pressure loss in fluid pressure, as from one end of a duct or pipe to the other, due to friction, dynamic losses, and changes in velocity pressure. 

Viscous and inertial frictional pressure losses and dynamic head pressure losses arise due to propellant flowing down the channel to the exit as indicated in the red arrows in Figure 3.13. Consider the vertically oriented LAD channel as shown in Figures 3.1 and 3.13, with the origin attached to the bottom center of the channel. The list of assumptions and corresponding implications used to solve for the viscous pressure drop inside the channel are as follows ... 

Figure 6.9 Pressure required to form and eject an ink drop in inkjet printing. The acoustic pressure generated by an actuator must overcome the viscous pressure drop required to force liquid through the nozzle (frictional energy loss), the surface tension pressure rise to form the drop (surface energy), and the dynamic pressure (kinetic energy) of the liquid.

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