Power laminar

Vilotijevic, M., Markovic, E, Zee, S., Marinkovic, S., and Jokanovic, V. (2011) Hydroxyapatite coatings prepared by a high power laminar jet. J. Mater. Proc. Technol., 211, 996-1004. 

Friction Coefficient. In the design of a heat exchanger, the pumping requirement is an important consideration. For a fully developed laminar flow, the pressure drop inside a tube is inversely proportional to the fourth power of the inside tube diameter. For a turbulent flow, the pressure drop is inversely proportional to D where n Hes between 4.8 and 5. In general, the internal tube diameter, plays the most important role in the deterrnination of the pumping requirement. It can be calculated using the Darcy friction coefficient,, defined as... 

The power number depends on impeller type and mixing Reynolds number. Figure 5 shows this relationship for six commonly used impellers. Similar plots for other impellers can be found in the Hterature. The functionality between and Re can be described as cc Re in laminar regime and depends on p. N in turbulent regime is constant and independent of ]1. 

The criterion of maintaining equal power per unit volume has been commonly used for dupHcating dispersion qualities on the two scales of mixing. However, this criterion would be conservative if only dispersion homogeneity is desired. The scale-up criterion based on laminar shear mechanism (9) consists of constant > typical for suspension polymerization. The turbulence model gives constant tip speed %ND for scale-up. 

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter AIChE J., 7, 445 48 [1961]), Wheeler and Wissler (AJChE J., 11, 207-212 [1965]), Bird, Armstrong, and Hassager Dynamics of Polymeric Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977), and Skelland Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). 

Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant. 

In the region of laminar flow (Vr < 10), the same power is consumed by the impeller whether baffles are present or not, and they are seldom required The flow pattern may be affected by the baffles, but not always advantageously. When they are needed, the baffles are usually placed one or two widths radially off the tank wall, to allow fluid to circulate behind them and at the same time produce some axial deflection of flow. 

In laminar flow < 10), 1/A Re nd P c< [LN D. Since shear stress is proportional to rotational speed, shear stress can be increased at the same power consumption by increasing N proportionally to as impeller diameter is decreased. 

Fluid circulation probably can be increased at the same power consumption and viscosity in laminar flow by increasing impeller diameter and decreasing rotational speed, but the relationship between Q, N, and for laminar flow from turbines has not been determined. 

Laminar or power law velocity distribution in which the linear velocity varies with radial position in a cylindrical vessel. Plug flow exists along any streamline and the mean concentration is found by integration over the cross section. 

Laminar Flow A mathematically simple deviation from uniform flow across a cross section is that of power law fluids whose linear velocity in a tube depends on the radial position = r/R, according to the equation... 

For laminar flow, the velocity at the centerline is Uq = 2u. For a power law rate equation = kO, the differential material balance on a streamline is... 

Experimental evidence regarding the power law is somewhat contradictory. A constant value of 3 = 0..5 is considered to give a good fit to experimental data by many authors. According to Awbi, p depends on the flow regime and has a value of 0..5 for fully turbulent flow and 1,0 for laminar flow.- fn practice the value of P tends to be between 0.6 and 0.7. 

At high the power number, P , stays reasonably constant, thus, viscosity has little effect on the power requirements. Wdien moving to lower through the laminar region into the viscous region, the viscosity effect increases. In the laminar range [29]... 

Figure 5-18. Laminar flow mixing. For known impeller type, diameter, speed, and viscosity, this nomograph will give power consumption. Connect RPM and diameter, also viscosity and impeller scale. The intersection of these two separate lines with alpha and beta respectively is then connected to give horsepower on the HP scale. By permission, Quillen, C. S., Chem. Engr., June 1954, p. 177 [15].

The flow changes from laminar to turbulent in the range of Reynolds numbers from 2,100 to 4,000 [60]. In laminar flow, the friction pressure losses are proportional to the average flow velocity. In turbulent flow, the losses are proportional to the velocity to a power ranging from 1.7 to 2.0. 

While designers of fluid power equipment do what they can to minimize turbulence, it cannot be avoided. For example, in a 4-inch pipe at 68°F, flow becomes turbulent at velocities over approximately 6 inches per second (ips) or about 3 ips in a 6-inch pipe. These velocities are far below those commonly encountered in fluid power systems, where velocities of 5 feet per second (fps) and above are common. In laminar flow, losses due to friction increase directly with velocity. With turbulent flow, these losses increase much more rapidly. 

Using the mathematical technique of dimensionless group analysis, the rate of mass transport (/ m) in terms of moles per unit area per unit time can be shown to be a function of these variables, which when grouped together can be related to the rate by a power term. For many systems under laminar flow conditions it has been shown that the following relationship holds ... 

With respect to general corrosion, once a surface film is formed the rate of corrosion is essentially determined by the ionic concentration gradient across the film. Consequently the corrosion rate tends to be independent of water flow rate across the corroding surface. However, under impingement conditions where the surface film is unable to form or is removed due to the shear stress created by the flow, the corrosion rate is theoretically velocity (10 dependent and is proportional to the power for laminar flow and... 

Tlie power for laminar flow is proportional to agitation rate, N2, and if the flow is turbulent the power is proportional to N3Dt2. Let us assume the mass transfer coefficients remain constant (Kha unchanged) ... 

This relation holds provided that the one-seventh power law may be assumed to apply over the whole of the cross-section of the pipe. This is strictly the case only at high Reynolds numbers when the thickness of the laminar sub-layer is small. By combining equations 3.59 and 3.63, the velocity profile is given by ... 

As in the case of Newtonian fluids, one of the most important practical problems involving non-Newtonian fluids is the calculation of the pressure drop for flow in pipelines. The flow is much more likely to be streamline, or laminar, because non-Newtonian fluids usually have very much higher apparent viscosities than most simple Newtonian fluids. Furthermore, the difference in behaviour is much greater for laminar flow where viscosity plays such an important role than for turbulent flow. Attention will initially be focused on laminar-flow, with particular reference to the flow of power-law and Bingham-plastic fluids. 

Figure 3.39. Fully-developed laminar velocity profiles for power-law fluids in a pipe (from equation 3.134)...

What will be the pressure drop, when the suspension is flowing under laminar conditions in a pipe 200 m long and 40 mm diameter, when the centre line velocity is 1 m/s, according to the power-law model Calculate the centre-line velocity for this pressure drop for the Bingham-plastic model. 

The critical value of the Reynolds number (Remit) for the transition from laminar to turbulent flow may be calculated from the Ryan and Johnson001 stability parameter, defined earlier by equation 3.56. For a power-law fluid, this becomes ... 

Equation 5.2 is found to hold well for non-Newtonian shear-thinning suspensions as well, provided that the liquid flow is turbulent. However, for laminar flow of the liquid, equation 5.2 considerably overpredicts the liquid hold-up e/,. The extent of overprediction increases as the degree of shear-thinning increases and as the liquid Reynolds number becomes progressively less. A modified parameter X has therefore been defined 16 171 for a power-law fluid (Chapter 3) in such a way that it reduces to X both at the superficial velocity uL equal to the transitional velocity (m )f from streamline to turbulent flow and when the liquid exhibits Newtonian properties. The parameter X is defined by the relation... 

Equation 5.2, with the modified parameter X used in place of X, may be used for laminar flow of shear-thinning fluids whose behaviour can be described by the power-taw model. 

For a power-law fluid in laminar flow at a velocity /. in a pipe of length /. the pressure drop -APL will be given by ... 

In a series of experiments on the flow of flocculated kaolin suspensions in laboratory and industrial scale pipelines(26-27-2Sl, measurements of pressure drop were made as a function of flowrate. Results were obtained using a laboratory capillary-tube viscometer, and pipelines of 42 mm and 205 mm diameter arranged in a recirculating loop. The rheology of all of the suspensions was described by the power-law model with a power law index less than unity, that is they were all shear-thinning. The behaviour in the laminar region can be described by the equation ... 

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