Mean velocity fluid

The effect of pulsating flow on pitot-tube accuracy is treated by Ower et al., op. cit., pp. 310-312. For sinusoidal velocity fluctuations, the ratio of indicated velocity to actual mean velocity is given by the factor /l + AV2, where X is the velocity excursion as a fraction of the mean velocity. Thus, the indicated velocity would be about 6 percent high for velocity fluctuations of 50 percent, and pulsations greater than 20 percent should be damped to avoid errors greater than 1 percent. Tne error increases as the frequency of flow oscillations approaches the natural frequency of the pitot tube and the density of the measuring fluid approaches the density of the process fluid [see Horlock and Daneshyar, y. Mech. Eng. Sci, 15, 144-152 (1973)]. 

For normal velocity distribution in straight circular pipes at locations preceded by runs of at least 50 diameters without pipe fittings or other obstructions, the graph in Fig. 10-7 shows the ratio of mean velocity V to velocity at the center plotted against the Reynolds number, where D = inside pipe diameter, p = flmd density, and [L = fluid viscosity, all in consistent units. Mean velocity is readily determined from this graph and a pitot reading at the center of the pipe if the quantity Du p/ I is less than 2000 or greater than 5000. The method is unreliable at intermediate values of the Reynolds number. 

Piping systems should be designed for an economic flow velocity. For relatively clean fluids, a recommended velocity range where minimum corrosion can be expected is 2 to 10 fps. If piping bores exist, maximum fluid velocities may have a mean velocity of 3 fps for a 3/8-in. bore to 10 fps for an 8-in.-diameter bore. Higher flow velocities are not uncommon in situations that require uniform, constant oxygen supply to form protective films on active/passive metals. 

Detemiine tlie mean (superficial) fluid velocity, u, as tlie volumetric flowrate divided by die flow channel cross-section. 

Mixing Due to Obstructions The tortuosity of the flow channels in a porous medium means that fluid elements starting a given distance from each oilier and proceeding at the same velocity will not reniain tlie same distance apart. 

Elutriation differs from sedimentation in that fluid moves vertically upwards and thereby carries with it all particles whose settling velocity by gravity is less than the fluid velocity. In practice, complications are introduced by such factors as the non-uniformity of the fluid velocity across a section of an elutriating tube, the influence of the walls of the tube, and the effect of eddies in the flow. In consequence, any assumption that the separated particle size corresponds to the mean velocity of fluid flow is only approximately true it also requires an infinite time to effect complete separation. This method is predicated on the assumption that Stokes law relating the free-falling velocity of a spherical particle to its density and diameter, and to the density and viscosity of the medium is valid... 

For a Newtonian fluid both Ry and X are zero and the mean velocity is ... 

Equation 3.152 provides a method of determining the relationship between pressure gradient and mean velocity of flow in a pipe for fluids whose rheological properties may be expressed in the form of an explicit relation for shear rate as a function of shear stress. 

It is shown in Chapter 3, that the mean velocity of a fluid flowing down a surface inclined at an angle cp to the horizontal is given by ... 

For the flow of a vertical film of fluid, the mean velocity of flow is governed by equation 3.87 in which sin 0 is put equal to unity for a vertical surface ... 

The expressions for the shear stress at the walls, the thickness of the laminar sub-layer, and the velocity at the outer edge of the laminar sub-layer may be applied to the turbulent flow of a fluid in a pipe. It is convenient to express these relations in terms of the mean velocity in the pipe, the pipe diameter, and the Reynolds group with respect to the mean velocity and diameter. 

In turbulent flow there is a complex interconnected series of circulating or eddy currents in the fluid, generally increasing in scale and intensity with increase of distance from any boundary surface. If, for steady-state turbulent flow, the velocity is measured at any fixed point in the fluid, both its magnitude and direction will be found to vary in a random manner with time. This is because a random velocity component, attributable to the circulation of the fluid in the eddies, is superimposed on the steady state mean velocity. No net motion arises from the eddies and therefore their time average in any direction must be zero. The instantaneous magnitude and direction of velocity at any point is therefore the vector sum of the steady and fluctuating components. 

Equation 12.37 can be used in order to calculate the friction factor

turbulent flow of fluid in a pipe, It is first necessary to obtain an expression for the mean velocity u of the fluid from the relation ... 

In fully developed flow, equations 12.102 and 12.117 can be used, but it is preferable to work in terms of the mean velocity of flow and the ordinary pipe Reynolds number Re. Furthermore, the heat transfer coefficient is generally expressed in terms of a driving force equal to the difference between the bulk fluid temperature and the wall temperature. If the fluid is highly turbulent, however, the bulk temperature will be quite close to the temperature 6S at the axis. 

A power law fluid is flowing under laminar conditions through a pipe of circular cross-section. At what radial position is the fluid velocity equal to the mean velocity in the pipe Where does this occur for a fluid with an n-value of 0.2 ... 

Water, of viscosity 1 mN s/m2 flowing through the pipe at the same mean velocity gives rise to a pressure drop of I04 N/m2 compared with 105 N/m2 for the non-Newtonian fluid. What is the consistency ("k vaiuei of the non-Newtonian fluid ... 

Derive a relationship between the pressure difference recorded between the two orifices of a pitot tube and the velocity of flow of an incompressible fluid. A pitot tube is to be situated in a large circular duct in which fluid is in turbulent flow so that it gives a direct reading of the mean velocity in the duct. At what radius in the duct should it be located, if the radius of the duct is r l... 

The ratio of Ihe mixing length to the distance from the pipe wall has a constant value of 0.4 for the turbulent flow of a fluid in a pipe. What is the value of the pipe friction factor if the ratio of the mean velocity to the... 

Thus, the measurements of integral flow characteristics, as well as mean velocity and rms of velocity fluctuations testify to the fact that the critical Reynolds number is the same as Rccr in the macroscopic Poiseuille flow. Some decrease in the critical Reynolds number down to Re 1,500— 1,700, reported by the second group above, may be due to energy dissipation. The energy dissipation leads to an increase in fluid temperature. As a result, the viscosity would increase in gas and decrease in liquid. Accordingly, in both cases the Reynolds number based on the inlet flow viscosity differs from that based on local viscosity at a given point in the micro-channel. 

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine... 

The stress acting on particles is due to a relative velocity between the particles and the fluid. If their mean velocities also differ, contact between the particles or between a particle and the tank wall or the impeller elements leads to impact stress. However, this impact stress is negligible if the density differences and the particle concentrations are low. 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

Figure 4.3. Velocity profiles of a power law fluid flowing between parallel plates, u is the mean velocity.

It follows that if an element of fluid moves in they-direction in a region where the mean velocity gradient dvjdy is zero, a fluctuation v y gives rise, on average, to a zero fluctuation v x. The time-average product of the fluctuations (the Reynolds stress) is zero and the fluctuations are said to be uncorrelated. 

In equation 1.3, p is the density, p. the dynamic viscosity, and the mean velocity of the fluid d, is the inside diameter of the tube. Any consistent system of units can be used in this equation. The Reynolds number is also frequently written in the form... 

For the steady turbulent flow of a Newtonian fluid at high values of Re in a pipe of circular cross section, the mean velocity u is related to the maximum velocity vmix by the equation... 

Equation 2.73 is another way of writing equation 2.13 where, in this case, the pressure drop is expressed in height of fluid instead of in force per unit area. In equation 2.73, de is the equivalent diameter defined as four times the cross-sectional flow area divided by the appropriate flow perimeter,/is the Fanning friction factor for flow in an open channel and u is the mean velocity. Combining equations 2.72 and 2.73, and solving for u gives... 

Consider points 1 and 2 in Figure 8.2. At point 1 in the pipe, the fluid flow is undisturbed by the orifice plate. The fluid at this point has a mean velocity ut and a cross-sectional flow area Si. At point 2 in the pipe the fluid attains its maximum mean velocity w2 and its smallest cross-sectional flow area S2. This point is known as the vena contracta. It occurs at about one half to two pipe diameters downstream from the orifice plate. The location is a function of the flow rate and the size of the orifice relative to the size of the pipe. Let the mean velocity in the orifice be u0 and let the diameter and cross-sectional flow area of the orifice be da and S0 respectively. 

Orifice meters, Venturi meters and flow nozzles measure volumetric flow rate Q or mean velocity u. In contrast the Pitot tube shown in a horizontal pipe in Figure 8.7 measures a point velocity v. Thus Pitot tubes can be used to obtain velocity profiles in either open or closed conduits. At point 2 in Figure 8.7 a small amount of fluid is brought to a standstill. Thus the combined head at point 2 is the pressure head P/( pg) plus the velocity head v2/(2g) if the potential head z at the centre of the horizontal pipe is arbitrarily taken to be zero. Since at point 3 fluid is not brought to a standstill, the head at point 3 is the pressure head only if points 2 and 3 are sufficiently close for them to be considered to have the same potential head... 

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