Incompressible fluid, flow measurement

This chapter has the following structure in Sect. 3.2 the common characteristics of experiments are discussed. Conditions that are needed for proper comparison of experimental and theoretical results are formulated in Sect. 3.3. In Sect. 3.4 the data of flow of incompressible fluids in smooth and rough micro-channels are discussed. Section 3.5 deals with gas flows. The data on transition from laminar to turbulent flow are presented in Sect. 3.6. Effect of measurement accuracy is estimated in Sect. 3.7. A discussion on the flow in capillary tubes is given in Sect. 3.8. 

PoiseuilWs Law Poiseuille flow is the steady flow of incompressible fluid parallel to the axis of a circular pipe or capillary. Poiseuille s law is an expression for the flow rate of a liquid in such tubes. It forms the basis for the measurement of viscosities by capillary viscometry. 

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. 

The mass flow rate under adiabatic conditions is always somewhat greater than that under isothermal conditions, but the difference is normally <20%. In fact, for long piping systems (L/D > 1000), the difference is usually less than 5% (see, e.g., Holland, 1973). The flow of compressible (as well as incompressible) fluids through nozzles and orifices will be considered in the following chapter on flow-measuring devices. 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

An important application of Bernoulli s equation is in flow measurement, discussed in Chapter 8. When an incompressible fluid flows through a constriction such as the throat of the Venturi meter shown in Figure 8.5, by continuity the fluid velocity must increase and by Bernoulli s equation the pressure must fall. By measuring this change in pressure, the change in velocity can be determined and the volumetric flow rate calculated. 

The dynamics of the incompressible fluid flow depend on small changes in the pressure through the flowfield. These changes are negligible compared to the absolute value of the thermodynamic pressure. The reference value can then be taken as some pressure at a fixed point and time in the flow. Changes in pressure result from fluid dynamic effects and an appropriate pressure scale is where Vmax is a measure of the maximum velocity in... 

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... 

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 Vl is given and the pressures pl and p. are measured. Selecting the inlet and outlet surfaces 1 and 2 as shown, the continuity equation Eq. (6-9) can be used to find the exil velocity Vz = VjA/A. The mass flow rate is obtained by m = p V. A. ... 

Suppose the flow rate of an incompressible fluid is to be measured in a venturi meter in which the cross-sectional area at point 1 is four times that at point 2. 

The Bernoulli theorem can also be applied to the measurement of flow rate. The passage of an incompressible fluid through a constriction results in an increase in velocity from Ui to uj, which is associated with a decrease in pressure from Pi to P2, which can be measured directly. The volumetric flow rate Q = Mifli = M2fl2 by algebraic rearrangement (which is not shown). The final linear velocity M2 can be described by Eq. (5). 

The opportunity to measure the dilute polymer solution viscosity in GPC came with the continuous capillary-type viscometers (single capillary or differential multicapillary detectors) coupled to the traditional chromatographic system before or after a concentration detector in series (see the entry Viscometric Detection in GPC-SEC). Because liquid continuously flows through the capillary tube, the detected pressure drop across the capillary provides the measure for the fluid viscosity according to the Poiseuille s equation for laminar flow of incompressible liquids [1], Most commercial on-line viscometers provide either relative or specific viscosities measured continuously across the entire polymer peak. These measurements produce a viscometry elution profile (chromatogram). Combined with a concentration-detector chromatogram (the concentration versus retention volume elution curve), this profile allows one to calculate the instantaneous intrinsic viscosity [17] of a polymer solution at each data point i (time slice) of a polymer distribution. Thus, if the differential refractometer is used as a concentration detector, then for each sample slice i. 

VELOCITY GRADIENT AND RATE OF SHEAR. Consider the steady one-dimensional laminar flow of an incompressible fluid along a solid plane surface. Figure 3.1a shows the velocity profile for such a stream. The abscissa u is the velocity, and the ordinate y is the distance measured perpendicular from the wall and therefore at right angles to the direction of the velocity. At y = 0, u = 0, and u increases with distance from the wall but at a decreasing rate. Focus attention on the velocities on two nearby planes, plane A and plane B, a distance Ay apart. 

Darcy Equation. The bulk resistance to flow of an incompressible fluid through a solid matrix, as compared to the resistance at and near the surfaces confining this solid matrix, was first measured by Darcy [23]. Since in his experiment the internal surface area (interstitial area) was many orders of magnitude larger than the area of the confining surfaces, the bulk shear stress resistance was dominant. 

In the equation of motion for the endolymph fluid, the force Tf dA acts on the fluid at the fluid-otoconial layer interface (Figure 64.2). This shear stress tf is responsible for driving the fluid flow. The linear Navier-Stokes equation for an incompressible fluid are used to describe this endolymph flow. Expressions for the pressure gradient, the flow velocity of the fluid measured with respect to an inertial reference frame, and the force due to gravity (body force) are substituted into the Navier-Stokes equation for flow in the x-direction yielding... 

Example 6.4. An orifice meter is installed in a 4 in. Sch40 pipeline to measure the mass flow rate of butane up to 10000 kg h . Consider that butane is an incompressible fluid with a density of 600 kg m and a viscosity of 0.230 cP. The orifice diameter is exactly half of the pipe diameter ... 

Pitot tube A device for measuring the speed of a fluid. It consists of two tubes, one with an opening facing the moving fluid and the other with an opening at 90 to the direction of the flow. The two tubes are connected to the opposite sides of a manometer so that the difference between the dynamic pressure in the first tube and the static pressure in the second tube can be measured. The speed V of the flow of an incompressible fluid is then jpven by. = 2(P2 where P2 is the dynamic pressure, P, is the static pressure, and p is the density of the fluid. 

By considering the flux of gas through porous membranes, a the volume V measured at a pressure jp, passing through the membrane in time t, in relation to the equation governing the flow (19,20), one can define permeability constants of various types, and having various dimensions. The equation of flow for an incompressible fluid in a pore system may be written ... 

Any rheometric technique involves the simultaneous assessment of force, and deformation and/or rate as a function of temperature. Through the appropriate rheometrical equations, such basic measurements are converted into quantities of rheological interest, for instance, shear or extensional stress and rate in isothermal condition. The rheometrical equations are established by considering the test geometry and type of flow involved, with respect to several hypotheses dealing with the nature of the fluid and the boundary conditions the fluid is generally assumed to be homogeneous and incompressible, and ideal boundaries are considered, for instance, no wall slip. 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

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