Pressure gradient drag

Sinusoidal oscillations of the continuous phase cause levitation or countergravity motion much more readily for gas bubbles, due to changes in bubble volume which cause a steady component in the pressure gradient drag term (Jl, J2). If the fluid motion is given by Eq. (11-49), the pressure in the vicinity of the bubble also varies sinusoidally. For normal experimental conditions, the resulting volume oscillations are isothermal (P2), and given by (Jl) ... 

In the friction layer where the isobars are curved, the effect of frictional drag is added to the forces discussed under gradient wind. The balance of the pressure gradient force, the coriolis deviating force, the centrifugal force, and the frictional drag in the vicinity of the curved isobars results in wind flow around low pressure and high pressure in the Northern Hemisphere, as shown in Fig. 17-16. 

Simple pressure/drag flow. Here we treat an idealization of the down-channel flow in a melt extruder, in which an incompressible viscous fluid constrained between two boundaries of infinite lateral extent (2). A positive pressure gradient is applied in the X-direction, and the upper boundary surface at y - H is displaced to the right at a velocity of u(H) - U this velocity is that of the barrel relative to the screw. This simple problem was solved by a 10x3 mesh of 4-node quadrilateral elements, as shown in Figure 1. 

Figure 3 illustrates some additional capability of the flow code. Here no pressure gradient is Imposed (this is then drag or "Couette flow only), but we also compute the temperatures resulting from Internal viscous dissipation. The shear rate in this case is just 7 — 3u/3y — U/H. The associated stress is.r — 177 = i/CU/H), and the thermal dissipation is then Q - r7 - i/CU/H). Figure 3 also shows the temperature profile which is obtained if the upper boundary exhibits a convective rather than fixed condition. The convective heat transfer coefficient h was set to unity this corresponds to a "Nusselt Number" Nu - (hH/k) - 1. 

Werner si ey, J. R., Method for the Calculation of Velocity, Rate of Flow and Viscous Drag in Arteries when the Pressure Gradient is Known, . Physiol. 127 553-563 (1955). 

For dense gas-solid two-phase flows, a four-way coupling is required however, the coupling between particles is managed in a natural way in DPMs. The task is, therefore, only to find a two-way coupling between the gas and the solid phases, which satisfies Newton s third law. Basically, the gas phase exerts two forces on particle a a drag force Vda due the fluid-solid friction at the surface of the spheres, and a force Vpa = -Va Vp due to the pressure gradient Vp in the gas phase. We will next describe these forces in more detail, along with the procedure to calculate void fraction, which is an essential quantity in the equations for the gas-solid interaction. 

The terms represent the contributions from the total pressure gradient, the frictional drag of the pipe wall and the hydrostatic head of the two-phase mixture. 

The friction factor, which is plotted against the modified Reynolds number, is Pi/pu, where R is the component of the drag force per unit area of particle surface in the direction of motion. R can be related to the properties of the bed and pressure gradient as follows. Considering the forces acting on the fluid in a bed of unit cross-sectional area and thickness /, the volume of particles in the bed is /(I — e) and therefore the total surface is 5/(1 — e). Thus the resistance force is R SH — e). This force on the fluid must be equal to that produced by a pressure difference of AP across the bed. Then, since the free cross-section of fluid is equal to e ... 

Two driving forces for flow exist in the metering section of the screw. The first flow is due just to the rotation of the screw and is referred to as the rotational flow component. The second component of flow is due to the pressure gradient that exist in the z direction, and it is referred to as pressure flow. The sum of the two flows must be equal to the overall flow rate. The overall flow rate, Q, the rotational flow, 0 and the pressure flow, Qp, for a constant depth metering channel are related as shown in Eq. 1.12. The subscript d is maintained in the nomenclature for historical consistency even though the term is for screw rotational flow rather than the historical drag flow concept. 

The thickness indicated by the red line in Fig. 6.18 is the gap between the solid bed and the screw root. The screw root is moving in the minus z direction while the solid bed is moving in the positive z direction. Melted polymer will thus be dragged into the gap, and there will be a negative pressure gradient dP/dz in the film. This topic will be presented in Section 6.3.1.3. 

The generalized Newtonian model over-predicted the rotational flow rates and pressure gradients for the channel for most conditions. This over-prediction was caused in part by the utilization of drag flow shape factors (FJ that were too large. Then in order for the sum of the rotational and pressure flows to match the actual flow In the channel, the pressure gradient was forced to be higher than actually required by the process. It has been known for a long time [9] that the power law... 

Spalding, M. A. and Campbell, G.A., The Accuracy of Standard Drag Flow and Pressure Gradient Calculations for Singie-Screw Extruders, SPE ANTEC Tech. Papers, 54, 262 (2008)... 

The equation delivers the result that is expected intuitively namely the required force increases with increased rod velocity, increased viscosity, and decreased gap thickness. As seen from Fig. 4.4, the pressure gradient has a large effect on the drag, since it affects the velocity gradient. 

The cell A 17 refers to the pressure-gradient parameter, with the needed to identify the fact that it is always in column A and thus not shifted relatively with dragging commands that will follow. The cells D 14 and D 15 refer to rows containing the values of fj-1/2 and rj+1/2. Here the is needed to fix the row reference in subsequent dragging operations. That is, the values of fj-1/2 and rJ+1/2 are always in rows 14 and 15, but the columns must be allowed to change in a relative dragging operation. Cells C17 and E17 refer to the values of the axial velocity in the adjacent cells (i.e., uj-1 and uj+i). 

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