Pressure-velocity relationship

Different test results are available to the designer wanting friction and wear data as well as the usual mechanical short and long term data, corrosion resistance, readings, and so on. The data presented include the load and velocity capabilities of a bearing material as expressed by the product of the unit load P based on the projected bearing area and linear shaft velocity V. The symbol PV denotes the important property of the pressure-velocity relationship. 

Shock Relationships and Formulas, which include Changes During Steady Reversible Compressible Flow (61-4) Pressure-Velocity Relationship (65-6) Irreversibility and Degradation (66-8) Derivation of Formulas (68-70) Pressure Efficiency Factor and Recovery Factor (70-2) and Oblique Shocks in Air (72). Shock Wave Interaction, which includes Strong Shock Waves (81) Superposition of Plane Shock Waves (81-2) ... 

Flg.1 Pressure-velocity relationships for a paste passing through an extruder. 

Typical velocities in plate heat exchangers for waterlike fluids in turbulent flow are 0.3-0.9 meters per second (m/s) but true velocities in certain regions will be higher by a factor of up to 4 due to the effect of the corrugations. All heat transfer and pressure drop relationships are, however, based on either a velocity calculated from the average plate gap or on the flow rate per passage. 

Sudden eniargement/contraction, 70, 80 Total line, 64 Two-phase flow, 124-127 Vacuum lines, 128-134 Velocities, 83, 89, 90 Velocities, chart, 91 Velocity head, 71 Water flow calculations, 96 Water flow, table, 93, 97, 98 Pressure level relationships,... 

Recently Bauer (Ref 107) has proposed a method of generating F—u (shock pressure-particle velocity) relationships (non-reactive... 

Velocity-pressure gradient relationships for fluids of specified rheology... 

One major difference between pneumatic transport and hydraulic transport is that the gas-solid interaction for pneumatic transport is generally much smaller than the particle-particle and particle-wall interaction. There are two primary modes of pneumatic transport dense phase and dilute phase. In the former, the transport occurs below the saltation velocity (which is roughly equivalent to the minimum deposit velocity) in plug flow, dune flow, or sliding bed flow. Dilute phase transport occurs above the saltation velocity in suspended flow. The saltation velocity is not the same as the entrainment or pickup velocity, however, which is approximately 50% greater than the saltation velocity. The pressure gradient-velocity relationship is similar to the one for hydraulic transport, as shown in... 

The Richardson-Zaki equation has been found to agree with experimental data over a wide range of condifions. Equally, if is possible fo use a pressure drop-velocity relationship such as Ergun to determine minimum fluidization velocity, just as for gas-solid fluidizafion. An alternative expression, which has the merit of simplicify, is fhaf of Riba ef al. (1978)... 

Valve trays. Figure 6.216 illustrates the dry pressure drop of a typical valve tray as a function of vapor velocity. At low vapor velocities, all valves are closed (i.e., seated on the tray deck). Vapor rises through the crevices between the valves and the tray deck, and friction losses through these crevices constitute the dry pressure drop. Once the closed balance point (CBP) is reached, there is sufficient force in the rising vapor to open some valves. A further increase in vapor velocity opens more valves. Since vapor flow area increases as valves open, pressure drop remains constant until all valves open. This occurs at the open balance point (OBP). Further increases of vapor velocity cause the dry pressure drop to escalate in a similar manner to a sieve tray. When two weights of valves are used in alternate rows on the tray, a similar behavior applies to each valve type. The result is the pressure drop-vapor velocity relationship in Fig. 619e. 

Ratio of wake volume to bubble volume Constant in pressure drop-velocity relationship, Eq. (101) Constant in pressure drop-velocity relationship, Eq. (102) Constant in pressure drop-velocity relationship, Eq. (101) Constant in pressure drop-velocity relationship, Eq. (102) Proportionality constant in Eq. (31)... 

Whether a pressurized structure will leak or burst when suddenly cracked depends on the material s resistance to rapid crack propagation and its ability to arrest a crack. Thus, the governing material property, namely the stress intensity-crack velocity relationship, is necessary input for designing pressurized pipes and vessels and establishing safe operating conditions. Unfortunately, this material property is not easily measured, and reliable data do not generally exist. 

Figure 3.36. Pressure drop-gas velocity relationship and characteristic of fluidised-bed reactors [63]...

Case II refers to situations where the particle-wall interactions are purely repulsive. The particles are separated from the wall by a thin layer of solvent, even in the absence of any motion. Slip is thus possible for very slow flows, indicating that the sticking yield stress is vanishingly small. The residual film thickness for weak flows corresponds to a balance between the osmotic forces and the short-range repulsive forces, independently of any elastohydrodynamic contribution. This is clearly reflected in Fig. 16c, d, where we observe that the particle facet is nearly flat and symmetric. Since tire pressure in the leading and rear regions of the facet are equal and opposite, the lift force is very small. The film thickness, which is set by the balance of the short-range forces, is constant so that the stress/velocity relationship is linear. 

The velocity corrections in (12.247) were then substituted by the pressure corrections employing appropriately defined pressure-velocity correction relationships. The pressure-velocity correction relationships were constructed by subtracting the discrete momentum equations with the pressure at the old time level n and the preliminary estimates for velocity fields (12.242) from the semi-implicit discretization of the momentum equation with the corrected pressure and the estimates for the velocity fields that satisfy the continuity equations. The desired relationships are the given on the form ... 

A Poisson equation (12.244) was then obtained by inserting the pressure-velocity correction relationships into (12.247). 

FIGURE 6.20 Enthalpy and velocity relationship within similar boundary layers on a body with pressure gradient P(, defined by Eq. 6.103, / = 0,1, = 1 [42]. 

FIGURE 56.2 Input impedance derived from the pressure and velocity data in the ascending aorta of Figure 56.3. The top panei contains the modulus and the bottom panel the phase, both plotted as functions of frequency. The peripheral resistance (DC impedance) for this plot was 16,470 dyne sec/cm. (From Mills C.J., Gabe I.T., Gault J.N. et al. 1970. Pressure-flow relationships and vascular impedance in man. Cardiovasc. Res. 4 405. With permission.)... 

The process of gas entrainment and circulation is complicated and not easily quantified. Problems arise from an abstract relationship between the liquid and gas phases. On the one hand, the gas flow rate affects the liquid flow rate through the gas holdup and hydraulic pressure differential relationship. As the gas flow rate increases, larger bubbles rise faster and increase the circulation velocity. A higher circulation velocity, in turn, would decrease the slip velocity and make entrainment easier. On the other hand, if the liquid velocity is higher than the bubble rise velocity, bubbles would experience a drag (lift) force, which would aid entrainment. 

For fine particles the pressure drop-velocity relationship will be given by the Carman-Kozeny equation, which will take the following form for incipient fluidization ... 

ANS Yes, the time varying behavior of elastance will mathematically result in an inverse force-velocity relationship of muscle. However, as I have just shown there is an additional dependence of pressure on flow that is independent of volume and it is this additional pressure loss that must be accounted for by a resistance term. Furthermore, Dr. Suga recently published the results of a study which indicated a correction term had to be added to his time varying elastance model in order for the isovolumetric and ejecting pressure-volume relationships to coincide. This correction term was of the same magnitude as our resistance term. So you cannot just use a time-varying elastance to describe the dynamics of the left ventricle. 

Note that for n = 1 and k = /i, Equations (4.7) and (4.8) reduce to the familiar Hagen-Poiseuille equation which describes the pressure drop-velocity relationship for the laminar flow of a Newtonian fluid. 

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