Potential flow fluid pressure

Another pitfall in pressure measurement, partieularly important in flow measurement, is the potential for liquids in gauge lines. All too often gauge lines eoming from overhead pipes have no provision for maintaining a liquid-free status, even though the flowing fluid may be eondensible at gauge-line temperatures. 

Stable Foam. When a well is drilled with stable foam as the drilling fluid, there is a back pressure valve at the blooey line. The back pressure valve allows for a continuous column of foam in the annulus while drilling operations are under way. Thus, while drilling, this foam column can have significant bottom-hole pressure. This bottomhole pressure can be sufficient to counter formation pore pressure and thus control potential production fluid flow into the well annulus. 

Aerated Mud. In aerated mud drilling operations, the drilling mud is injected with compressed air to lighten the mud. Therefore, at the bottom of the well in the annulus, the bottomhole pressure for an aerated mud will be less than that of the mud without aeration. However, an aerated mud drilling operation will have very significant bottomhole pressure capabilities and can easily be used to control potential production fluid flow into the well annulus. 

The flow pattern in this case reflects the distribution of fluid pressure and fluid density it cannot be determined from the gradient of any single potential function. 

For a fluid having a pressure P and flowing at speed v, the quantity ipv2 is known as the dynamic pressure and P+ipv2 is called the total pressure or the stagnation pressure. The pressure P of the flowing fluid is often called the static pressure, a potentially misleading name because it is not the same as the hydrostatic pressure. 

Actual shapes of fluid particles deviate from the idealized shape which leads to Eqs. (8-15) and (8-16). Surface pressure distributions derived from observed shapes (W2) are shown in Fig. 8.3 for spherical-cap bubbles at high Re. It is seen that the pressure variation is well described by Eq. (8-15) for 0 < 0 < while the potential flow pressure distribution, Eq. (1-32), gives good agreement up to about 30° from the nose. 

The use of the compressibility term can be described as follows. The greater the stiffness a system model has, the more quickly the flow reacts to a change in pressure, and vice versa. For instance, if all fluids in the system are incompressible, and quasi-steady assumptions are used, then a step change to a valve should result in an instantaneous equilibrium of flows and pressures throughout the entire system. This makes for a stiff numerical solution, and is thus computationally intense. This pressure-flow solution technique allows for some compressibility to relax the problem. The equilibrium time of a quasi-steady model can be modified by changing this parameter, for instance this term could be set such that equilibrium occurs after 2 to 3 seconds for the entire model. However, quantitative results less than this timescale would then potentially not be captured accurately. As a final note, this technique can also incorporate flow elements that use the momentum equation (non-quasi-steady), but its strength is more suited by quasi-steady flow assumptions. 

Consider a bubble rising in a fluidized bed. It is assumed that the bubble is solids-free, is spherical, and has a constant internal pressure. Moreover, the emulsion phase is assumed to be a pseudocontinuum, incompressible, and inviscid single fluid with an apparent density of pp(l — amf) + pamf. It should be noted that the assumption of incompressibility of the mixture is not strictly valid as voidage in the vicinity of the bubble is higher than that in the emulsion phase [Jackson, 1963 Yates et al., 1994]. With these assumptions, the velocity and pressure distributions of the fluid in a uniform potential flow field around a bubble, as portrayed by Fig. 9.10, can be given as [Davidson and Harrison, 1963]... 

Bernoulli s Equation gives us the total energy in a flowing fluid and sums the energy due to fluid pressure, kinetic energy and potential energy. 

Finally, the last two terms on the r.h.s. are needed If the potential and the pressure gradients In the normal direction are considered, when no fluid flow occurs In that direction. This is often the case because the surface cannot act as a source of liquid (o = 0 at x = 0). Formation of a diffuse layer leads to local excesses of ions exerting a certain osmotic pressure, which Is just equal and opposite to Vp, because otherwise the liquid would start to flow. Equation 14.6.6) then reduces to... 

A.20.2 Diffusion and chemical potential electrical conduction and electrical potential heat flow and temperature fluid flow and pressure. Everyday examples of (1) diffusion car exhaust expanding past a person on the street, ingested medication (such as ibuprofen and aspirin) (2) Electrical conduction electric light bulb, power lines, static electricity (3) Heat flow steaming cup of coffee, space heater, oven (4) Eluid flow tap water, windshield wiper fluid, geyser. 

Although the potential energy provides the flowing fluid with kinetic energy at the pipe entrance, the kinetic energy is later recovered. This indicates that the measured pipe pressure will be lower than the calculated pressure by one velocity head. If the kinetic energy is not recovered at the pipe exit, the exit counts as a loss of one velocity head. Table 3-2 shows how K varies with changes in pipe size. 

The most commonly accepted explanation for separation can be paraphrased in the following way. In a potential flow with no viscosity, there is an exact interchange of kinetic energy and work that is due to the action of the pressure gradient for a fluid element that is near the body surface. On the front portion of the body, the pressure gradient is favorable and... 

Consider a volume element of a stream tube within a larger stream of fluid in steady potential flow, as shown in Fig. 4.4. Assume that the cross section of the tube increases continuously in the direction of flow. Also, assume that the axis of the tube is straight and inclined upward at an angle from the vertical. Denote the cross section, pressure, linear velocity, and elevation at the tube entrance by S, p, u, and Z, respectively, and let the corresponding quantities at the exit be S -t- AS, p -f- Ap, u + Ah, and Z + AZ. The axial length is AL, and the constant fluid density is p. The constant mass flow rate through the tube is m,... 

When the cross section is constant, u does not change with position, the term d(u l 2)ldL is zero, and Eq. (4.23) becomes identical with Eq. (Z3) for a stationary fluid. In unidirectional potential flow at a constant velocity, then, the magnitude of the velocity does not affect the pressure drop in the tube the pressure drop depends only on the rate of change of elevation. In a straight horizontal tube, in consequence, there is no pressure drop in steady constant-velocity potential flow. 

The thin region near the body surface, which is known as the boundary layer, lends itself to relatively simple analysis by the very fact of its thinness relative to the dimensions of the body. A fundamental assumption of the boundary layer approximation is that the fluid immediately adjacent to the body surface is at rest relative to the body, an assumption that appears to be valid except for very low-pressure gases, when the mean free path of the gas molecules is large relative to the body [6]. Thus the hydrodynamic or velocity boundary layer 8 may be defined as the region in which the fluid velocity changes from its free-stream, or potential flow, value to zero at the body surface (Fig. 1.3). In reality there is no precise thickness to a boundary layer defined in this manner, since the velocity asymptotically approaches the free-stream value. In practice we simply imply that the boundary layer thickness is the distance in which most of the velocity change takes place. 

Equation (33.22) is a chemical example of a linear law analogous to those mentioned in Section 30.2. In each of those cases, a flow, such as a heat flow, an electrical current, a fluid flow, or a diffusive flow, was proportional to a driving force such as a temperature gradient, an electrical potential gradient, a pressure gradient or a concentration gradient. In the chemical case, Eq. (33.22), the flow is the rate of the reaction, while the driving force is the affinity of the reaction divided by T. 

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