Fluids incompressible

Geometrically, Liouville s theorem means that if one follows the motion of a small phase volume in Y space, it may change its shape but its volume is invariant. In other words the motion of this volume in T space is like that of an incompressible fluid. Liouville s theorem, being a restatement of mechanics, is an important ingredient in the fomuilation of the theory of statistical ensembles, which is considered next. 

In two classic papers [18, 46], Calm and Flilliard developed a field theoretic extension of early theories of micleation by considering a spatially inliomogeneous system. Their free energy fiinctional, equations (A3.3.52). has already been discussed at length in section A3.3.3. They considered a two-component incompressible fluid. The square gradient approximation implied a slow variation of the concentration on the... 

Kaye, A., 1962. Non-Newtonian Flow in Incompressible Fluids, CoA Note No, 134, College of Aeronautics, Cranfleld. 

The right-hand side of Equation (3,87) is set to zero considering that DA//Dt, DFIDt and the divergence of the velocity field in incompressible fluids are all equal to zero. Therefore, after integration Equation (3.87) yields... 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Wc now obtain the integral of the continuity equation for incompressible fluids with respect to the local gap height hr this flow domain... 

Compressible Vlow. The flow of easily compressible fluids, ie, gases, exhibits features not evident in the flow of substantially incompressible fluid, ie, Hquids. These differences arise because of the ease with which gas velocities can be brought to or beyond the speed of sound and the substantial reversible exchange possible between kinetic energy and internal energy. The Mach number, the ratio of the gas velocity to the local speed of sound, plays a central role in describing such flows. 

Simplified forms of Eq. (6-8) apply to special cases frequently found in prac tice. For a control volume fixed in space with one inlet of area Ai through which an incompressible fluid enters the control volume at an average velocity Vi, and one outlet of area Ao through which fluid leaves at an average velocity V9, as shown in Fig. 6-4, the continuity equation becomes... 

Example 1 Force Exerted on a Reducing Bend An incompressible fluid flows through a reducing elbow (Fig. 6-5) situated in a horizontal plane. The inlet velocity Vj is given and the pressures pi and measured. 

Example 3 Venturi Flowmeter An incompressible fluid flows through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can he solved using the mechanical energy balance. In a well-made venturi, viscous losses are neghgihle, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. 

In the velocity head method, the losses are reported as a number of velocity heads K. Then, the engineering Bernoulh equation for an incompressible fluid can be written... 

Water Hammer When hquid flowing in a pipe is suddenly decelerated to zero velocity by a fast-closing valve, a pressure wave propagates upstream to the pipe inlet, where it is reflected a pounding of the hne commonly known as water hammer is often produced. For an instantaneous flow stoppage of a truly incompressible fluid in an inelastic pipe, the pressure rise would be infinite. Finite compressibility of the flmd and elasticity of the pipe limit the pressure rise to a finite value. The Joukowstd formula gives the maximum pressure... 

Pressure Relief Valve (PRV) A relief valve is a spring-loaded valve actuated by static pressure upstream of the valve. The valve opens normally in proportion to the pressure increase over opening pressure. A relief valve is normally used with incompressible fluids. 

Two more consequences must be recognized about the very low pressure increases of 1 to 200 mm of water column made by the blowers. One is that those piressure surges, common in turbocompressors generating large pressure increases, are absent in these fans therefore, these can be operated at low flows. The second is that, in calculation of energy consumption and dissipation, the expression for incompressible fluids can be used even for gases. 

In incompressible fluids, such as water, the vertical structure of temperature very simply reveals the stability of the fluid. When the lower layer is warmer and thus less dense than the upper layer, the fluid is unstable and convective currents will cause it to overturn. When the lower layer is cooler than the upper layer, the fluid is stable and vertical exchange is minimal. However, because air is compressible, the determination of stability is somewhat more complicated. The temperature and density of the atmosphere normally decrease with elevation density is also affected by moisture in the air. 

The column length, as well as providing the required efficiency, is also defined by the D Arcy equation. The D Arcy equation describes the flow of a liquid through a packed bed in terms of the particle diameter, the pressure applied across the bed, the viscosity of the fluid and the linear velocity of the fluid. The D Arcy equation for an incompressible fluid is given as follows. 

Because there are no moving parts in the Kenics mixer, only the processed materials are in motion. Therefore, the only energy required for the mixer is the energy required to overcome the pressure drop (AP). The general equation for calculating the pressure drop in an empty pipe for isothermal incompressible fluids is given by... 

Safety relief valve A pressure relief valve characterized by rapid opening pop action or by opening generally proportional to tlie increase in pressure over the opening pressure. It may be used for either compressible or incompressible fluids, depending on design, adjustment, or application. 

An incompressible fluid is a fluid whose density remains constant during flow. Liquids are normally treated as being incompressible, as a gas can be when only slight pressure variation occurs. 

This example demonstrates the dimensioning of a duct with a frictional incompressible fluid flow. Now the Bernoulli equation can be written as... 

The steady, laminar, incompressible fluid flow in cyclone collectors is governed by the Navier-Stokes equations ... 

For steady, incompressible fluid flow in a cyclone separator, the governing Navier-Stokes equations of motion are given, in a Cartesian coordinate system, by ... 

Pressure Drop in Straight Pipe Incompressible Fluid... 

For discharging incompressible fluids to atmosphere, take C values from Figures 2-17 or 2-18 if h or AP is taken as upstream head or gauge pressure. 

Scope, 52 Basis, 52 Compressible Flow Vapors and Gases, 54 Factors of Safety for Design Basis, 56 Pipe, Fittings, and Valves, 56 Pipe, 56 Usual Industry Pipe Sizes and Classes Practice, 59 Total Line Pressure Drop, 64 Background Information, 64 Reynolds Number, R,. (Sometimes used Nr ), 67 Friction Factor, f, 68 Pipe—Relative Roughness, 68 Pressure Drop in Fittings, Valves, Connections Incompressible Fluid, 71 Common Denominator for Use of K Factors in a System of Varying Sizes of Internal Dimensions, 72 Validity of K Values,... 

Flow through chokes and nozzles is a special case of fluid dynamics. For incompressible fluids the problem can be handled by mass conservation and Bernoulli s equation. Bernoulli s equation is solved for the pressure drop across the choke, assuming that the velocity of approach and the vertical displacement are negligible. The velocity term is replaced by the volumetric flow rate times the area at the choke throat to yield... 

Pumps are a mechanical device that forces a fluid to move from one position to another. Usually a pump refers to the mechanical means to move incompressible (or nearly incompressible) fluid or liquid. Pumps are our earliest machine and are to this day one of our most numerous mechanical devices. 

Air or gas compressors are very similar in design and operation to liquid pumps discussed earlier. The air and gas compressor is a mover of compressed fluids the pumps are movers of basically incompressible fluids (i.e., liquids). 

The Navier-Stokes equations are the fundatnenta equations describing incompressible, fluid flow see Chapter 9. 

Iadda89] Ladd, A.J.C. and D.Frenkel, Dynamics of colloidal dispersions via lattice-gas models of an incompressible fluid, pages 242-245 in [mann89]. 

Flashing liquids, 134-146 Flow coefficients, Gv, for valves, 81 Friction loss, 68 Incompressible fluid, 71 Laminar flow, 77, 78, 86 Liquid lines, chart, 92 Long natural gas pipe lines, 120 Non-water liquids, 99 Pipe, 71... 

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