Hydraulics incompressible fluids

We begin the comparison of experimental data with predictions of the conventional theory for results related to flow of incompressible fluids in smooth micro-channels. For liquid flow in the channels with the hydraulic diameter ranging from 10 m to 10 m the Knudsen number is much smaller than unity. Under these conditions, one might expect a fairly good agreement between the theoretical and experimental results. On the other hand, the existence of discrepancy between those results can be treated as a display of specific features of flow, which were not accounted for by the conventional theory. Bearing in mind these circumstances, we consider such experiments, which were performed under conditions close to those used for the theoretical description of flows in circular, rectangular, and trapezoidal micro-channels. 

Consider the fully developed steady flow of an incompressible fluid through an annular channel, which has an inner radius of r, and an outer radius of r0 (Fig. 4.27). The objective is to derive a general relationship for the friction factor as a function of flow parameters (i.e., Reynolds number) and channel geometry (i.e., hydraulic diameter Dh and the ratio f A friction factor /, which is a nondimensional measure of the wall... 

Let us write the model of nonstationary flow distribution as applied to the problem of search for the maximum pressure rise at a given node of the hydraulic circuit at a fast cut off of the flow in one of its branches (or the largest drop at pipe break) provided that there is an isothermal motion of viscous incompressible fluid subjected to the action of the pressure, friction, and inertia forces (Gorban et al., 2006). find... 

The analysis of stationary and nonstationary flow distributions in multiloop hydraulic systems with lumped, regulated, and distributed parameters and in heterogeneous systems was given in (Gorban et al., 2001, 2006 Kaganovich et al., 1997). In the concluding section of Section 5 the abundant capabilities of the flow MEIS are illustrated by the simplest example of stationary isothermal flow distribution of incompressible fluid in the three-loop circuit. It is shown how the degrees of order (laminar or turbulent modes) on the branches of this circuit are determined from calculation of the final equilibrium. 

It has been suggested that the way to solve the Los Angeles Basin smog problem is to construct fans to pump the air out of the basin and discharge it into the Mohave Desert. Let us assume that we wish to remove a layer 2000 ft thick every day from an area 70 mi by 60 mi. We also make a first guess that the pressure drop necessary from the flow is 1 psi. For this small a pressure drop, air may be considered an incompressible fluid. What hydraulic horsepower is required ... 

For hydraulic subsystems, it is common to choose the atmospheric pressure as reference. After elimination of its associated 0-zero junction along with all incident bonds, 0-junctions represent gage pressures. This results in a simplification of the construction of bond graphs for hydraulic systems. Gage pressures are represented by 0-junction, C elements are attached directly to a proper 0-junction. TF elements in bond graphs of hydraulic systems relate a pressure, p, to its associated mechanical force, F, and a volume flow rate, V, of incompressible fluid flow to its associated translational velocity v. 

The problems of micro-hydrodynamics were considered in different contexts (1) drag in micro-channels with a hydraulic diameter from 10 m to 10 m at laminar, transient and turbulent single-phase flows, (2) heat transfer in liquid and gas flows in small channels, and (3) two-phase flow in adiabatic and heated microchannels. The smdies performed in these directions encompass a vast class of problems related to flow of incompressible and compressible fluids in regular and irregular micro-channels under adiabatic conditions, heat transfer, as well as phase change. 

Obot NT (1988) Determination of incompressible flow friction in smooth circular and nondrcular passages. A generaUzed approach including validation of the century old hydraulic diameter concept. Trans ASME J Fluid Eng 110 431-440... 

As will be outlined below, the computation of compressible flow is significantly more challenging than the corresponding problem for incompressible flow. In order to reduce the computational effort, within a CED model a fluid medium should be treated as incompressible whenever possible. A rule of thumb often found in the literature and used as a criterion for the incompressibility assumption to be valid is based on the Mach number of the flow. The Mach number is defined as the ratio of the local flow velocity and the speed of sound. The rule states that if the Mach number is below 0.3 in the whole flow domain, the flow may be treated as incompressible [84], In practice, this rule has to be supplemented by a few additional criteria [3], Especially for micro flows it is important to consider also the total pressure drop as a criterion for incompressibility. In a long micro channel the Mach number may be well below 0.3, but owing to the small hydraulic diameter of the channel a large pressure drop may be obtained. A pressure drop of a few atmospheres for a gas flow clearly indicates that compressibility effects should be taken into account. 

Flows are typically considered compressible when the density varies by more than 5 to 10 percent. In practice compressible flows are normally limited to gases, supercritical fluids, and multiphase flows containing gases. Liquid flows are normally considered incompressible, except for certain calculations involved in hydraulic transient analysis (see following) where compressibility effects are important even for nearly incompressible liquids with extremely small density variations. Textbooks on compressible gas flow include Shapiro Dynamics and Thermodynamics of Compressible Fluid Flow, vol. 1 and 11, Ronald Press, New York [1953]) and Zucrow and Hofmann (Gas Dynamics, vol. I and II, Wiley, New York [1976]). 

Power Transmission Hydraulic systems utilize lubricating oils to serve all of the above listed functions plus the transfer of force and motion based on the fluids with general incompressibility. 

A hydraulic lift is shown in Fig. 4.14. The combined mass of the piston, rack, and car is 4000 Ibm. The working fluid is water. There is no heat transfer to or from the water, and the internal energy of the water per unit mass is constant. The water may be considered incompressible. 

Compressible and Incompressible Flow An incompressible flow is one in which the density of the fluid is constant or nearly constant. Liquid flows are normally treated as incompressible, except in the context of hydraulic transients (see followiM). Compressible fluids, such as gases, may undergo incompressible flow if pressure and/or temperature changes are small enough to render density changes insignificant. Frequently, compressible flows are regarded as flows in which the density varies by more than 5 to 10 percent. 

The fluid in these cadence-responsive knee units may be oil (hydraulic) or air (pneumatic). For hydraulic knees, the fluid is incompressible. The resistance to piston motion results from fluid flow through one or more orifices. As such, the resistance is dependent on the fluid viscosity and density, the size and smoothness of the channel, and the speed of movement. In contrast, for pneumatic knees, the fluid is compressible. The resistance is again due to fluid flow through the orifice(s) but is also influenced by fluid compression. Since air is a gas, potential leaks in pneumatic knee units will not result in soiled clothing, unlike what may occur with hydraulic knees. In addition, since air is less dense than oil, pneumatic units tend to be lighter than hydraulic units. However, since air is less dense and less viscous than oil, pneumatic units provide less cadence control than hydraulic units. Note that since viscosity is influenced by temperature, hydraulic (and pneumatic) knee units may perform differently inside and outside in cold weather climates. An example of a hydraulic cadence-responsive knee unit is the Black Max (USMC, Pasadena, Calif.). Additional examples include the Spectrum Ex (pneumatic, Hosmer, Campbell, Calif), Pendulum (pneumatic, Ohio Willow Wood, Mt. Sterling, Ohio), and Total Knee (hydraulic. Model 2000, Century XXII Innovations, Jackson, Mich.), which combine a cadence-responsive resistance swing-phase-control knee with a four-bar polycentric stance control knee. 

The pumps used in handling these high-pressure liquids can suffer considerable damage from cavitation. Incompressible liquids will not compress, nor will they withstand tension thus if the suction inlet to a pump is restricted the fluid will release any contained air to form cavities. This condition seriously affects the performance of the pump, can cause damage to its rotor and generates a great deal of noise. Gas or air entrained in a hydraulic fluid is detrimental to its effectiveness as a power transmission medium. 

Electrical resistance leads to dissipation of electrical energy in the form of Joule heating. Similarly, hydraulic resistance leads to viscous dissipation of mechanical energy into heat by internal friction in the fluid. The role of viscous dissipation can be explained based on the schematic of transient flow behavior shown in Figure 2.9. Let an incompressible Poiseuille fluid flow takes place inside a channel at times t < 0. The constant Poiseuille-type velocity field is maintained by a constant over-pressure AP applied to the left end of the channel. The over-pressure AP is suddenly removed at time, t = 0. However, the fluid flow continues due to the inertia of the fluid. The internal viscous friction of the fluid gradually slows down the motion of the fluid, and eventually in the limit t - c the fluid comes to rest relative to the channel walls. As time passes, the kinetic energy of the fluid at t = 0 is gradually transformed into heat by the viscous friction. 

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