Idealized incompressible fluid

The partial differential equations system for steady flows of Maxwell type (i.e., with = 0) is of composite type, neither elliptic, nor hyperbolic. This is not surprising, the same being true for instance for the stationary system of ideal incompressible fluids. The new feature, discovered in [8], is that some change of type may occur. In fact an easy but tedious calculation shows that three types of characteristics axe present ... 

Let us consider a plane problem on heat exchange between a cylinder with arbitrary boundary T of the cross-section in a plane-parallel flow of an ideal incompressible fluid with velocity U normal to the cylinder axis. The temperature of the cylinder is supposed to be constant and equal to Ts, and the temperature of the fluid at infinity is equal to T. We shall use rectangular coordinates X, Y with the X-axis directed along the flow (Figure 4.3). 

The effect of pressure on the properties of an incompressible fluid, an ideal gas. and a non-ideal gas is now considered. 

Methods have been given for the calculation of the pressure drop for the flow of an incompressible fluid and for a compressible fluid which behaves as an ideal gas. If the fluid is compressible and deviations from the ideal gas law are appreciable, one of the approximate equations of state, such as van der Waals equation, may be used in place of the law PV = nRT to give the relation between temperature, pressure, and volume. Alternatively, if the enthalpy of the gas is known over a range of temperature and pressure, the energy balance, equation 2.56, which involves a term representing the change in the enthalpy, may be employed ... 

The general approach for kinetic optiaization of open i tubular columns has been to adopt the familiar Golay equation T (equation 1.34) and to assuae that the aobile phase can be approximated by an incompressible fluid with ideal gas properties, (44-50). Circumstances that are approximate at best but serve adequately to demonstrate some of the fundamental characteristics of open tubular columns operated at low fluid densities. The column plate height equation can be written in the form given in M equation (6.1)... 

Equation (10-14) applies to incompressible fluids, such as liquids. For an ideal gas under adiabatic conditions, Eq. (10-12) gives... 

The theorem is easily generalized also to the case when the fluid density is p = p(z) or p = p(r). Indeed, in this case, the continuity equation implies that for vz = 0 we have div v = 0, i.e., the problem reduces to that already considered. We note that in an ideally conducting medium, for arbitrary density, the quantity Hz/p is conserved (instead of Hz in the case of an incompressible fluid), since instead of (4) we have in this case the equation... 

Equations (8-111) to (8-115) are restricted to incompressible fluids. For gases and vapors, the fluid density is dependent on pressure. For convenience, compressible fluids are often assumed to follow the ideal gas law model. Deviations from ideal behavior are corrected for, to first order, with nonunity values of compressibility factor Z (see Sec. 2, Physical and Chemical Data, for definitions and data for common fluids). For compressible fluids... 

Strictly speaking, most of the equations that are presented in the preceding part of this chapter apply only to incompressible fluids but practically, they may be used for all liquids and even for gases and vapors where the pressure differential is small relative to the total pressure. As in the case of incompressible fluids, equations may be derived for ideal frictionless flow and then a coefficient introduced to obtain a correct result. The ideal conditions that will be imposed for a compressible fluid are that it is frictionless and that there is to be no transfer of heat that is, the flow is adiabatic. This last is practically true for metering devices, as the time for the fluid to pass through is so short that very little heat transfer can take place. Because of the variation in density with both pressure and temperature, it is necessary to express rate of discharge in terms of weight rather than volume. Also, the continuity equation must now be... 

Example 13.1 Flow in an Idealized Runner System We consider a straight tubular runner of length L. A melt following the Power Law model is injected at constant pressure into the runner. The melt front progresses along the runner until it reaches the gate located at its end. We wish to calculate the melt front position and the instantaneous flow rate as a function of time. We assume an incompressible fluid in isothermal and fully developed flow, and make use of the pseudo-steady state approximation. 

The isotherms for the liquid phase on the left side of Fig. 3.2 are very steep and closely spaced. Thus both (dV/dP)T and dV/dT)P, and hence both /3 and k, are small. This characteristic behavior of liquids (outside the region of the critical point) suggests an idealization, commonly employed in fluid mechanics and known as the incompressible fluid, for which /3 and k are both zero. No real fluid is in fact incompressible, but the idealization is nevertheless useful, because it often provides a sufficiently realistic model of liquid behavior for practical purposes. The incompressible fluid cannot be described by an equation of state relating V to T and P, because V is constant. 

Whenever the internal energy is independent of volume, regardless of the process. This is exactly true for ideal gases and incompressible fluids. 

For liquids not near the critical point, the volume itself is small, as are both ft and k. Thus at most conditions pressure has little effect on the entropy, enthalpy, and internal energy of liquids. For an incompressiblefluid (Sec. 3.1), an idealization useful in fluid mechanics, both ft and k are zero. In this case both (dS/dP)T and (BU/dP)T are zero, and the entropy and internal energy are independent of P. However, the enthalpy of an Incompressible fluid is a function of P, as is evident from Eq. (6.25). 

The expression in the square brackets disappears for ideal gases. For incompressible fluids, g = const, dp/di = 0 and dwi/dxi = 0. The expression is simplified in both cases to... 

Fluid flow may be steady or unsteady, uniform or nonuniform, and it can also be laminar or turbulent, as well as one-, two-, or three-dimensional, and rotational or irrotational. One-dimensional flow of incompressible fluid in food systems occurs when the direction and magnitude of the velocity at all points are identical. In this case, flow analysis is based on the single dimension taken along the central streamline of the flow, and velocities and accelerations normal to the streamline are negligible. In such cases, average values of velocity, pressure, and elevation are considered to represent the flow as a whole. Two-dimensional flow occurs when the fluid particles of food systems move in planes or parallel planes and the streamline patterns are identical in each plane. For an ideal fluid there is no shear stress and no torque additionally, no rotational motion of fluid particles about their own mass centers exists. 

Most chemical engineers relate the term incompressible flow to incompressible fluid systems. For non-reactive ideal liquid mixtures operated at nearly constant temperatures, the incompressible flow limit is obviously a reasonable approximation in practice. 

For inviscid, incompressible fluids (commonly called ideal fluids) (1.236) yields,... 

For a steady flow of an incompressible fluid in a uniform pipe, the only property that varies along the pipe is pressure. However, for a compressible fluid when the pressure varies (i.e., drops), the density also drops, which means that the velocity must increase for a given mass flow. The kinetic energy thus increases, which results in a decrease in the internal energy and the temperature. This process is usually described as adiabatic, or locally isentropic, with the effect of friction loss included separately. A limiting case is the isothermal condition, although special means are usually required to achieve constant temperature. Under isothermal conditions for an ideal gas. 

If w e consider the constant-volume reactor with incompressible fluid (a = 0,Cv = Cp), Equation 6.16 reduces to Equation 6.15 as it should because Equation 6.15 is valid for any reactor operation with an incompressible fluid. We also notice that, in the constant-pressure case, the same energy balance applies for any fluid mixture (ideal gas, incompressible fluid, etc.), and that this balance is the same as the balance for an incompressible fluid in a constant-volume reactor. Although the same final balances are obtained for these two cases, the physical situations they describe are completely different. 

The discussion in the preceding section reveals that n-dimensional ideal fluid flow solutions can be obtained without using the equation of motion. Now the generalized vector force balance is manipulated to calculate dynamic pressure. The starting point is the equation of motion for generalized incompressible fluids, given by equation (8-39) ... 

Obviously, the second-term on the right side of this identity vanishes for ideal fluid flow in which the vorticity vector vanishes. Hence, for incompressible fluids with constant density. 

Potential Flow Solutions for Gas Bubbles Which Rise through Incompressible Fluids That Are Stagnant Far from the Submerged Objects. A nondeformable bubble of radius R rises through an ideal fluid such that... 

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