Laminar Versus Turbulent Flow

As mentioned above, two distinct patterns of fluid flow can be identified, namely laminar flow and turbulent flow. Whether a fluid flow becomes laminar or turbulent depends on the value of a dimensionless number called the Reynolds number, (Re). For a flow through a conduit with a circular cross section (i.e., a round tube), (Re) is defined as  

Eigure 2.4a shows the velocity distribution in a steady isothermal laminar flow of an incompressible Newtonian fluid through a straight, round tube. The velocity distribution in laminar flow is parabolic and can be represented by 

From the above relationships it can also be shown that the pressure drop AP (Pa) in the laminar flow of a Newtonian fluid ofviscosity // (Pa s) through a straight round tube of diameter d (m) and length L (m) at an average velocity of v (ms ) is given by Equation 2.9, which expresses the Hagen-Poiseuille law  

the pressure drop AP for laminar flow through a tube varies in proportion to the viscosity n, the average flow velocity v, and the tube length L, and in inverse proportion to the square of the tube diameter d. Since v is proportional to the total flow rate Q (m s ) and to d, C P should vary in proportion to ft, Q, L, and The principle of the capillary tube viscometer is based on this relationship. 

Derive an equation for the shear rate at the tube surface for laminar flow of Newtonian fluids through a tube of radius fj. 

Velocity distributions in turbulent flow through a straight, round tube vary with the flow rate or the Reynolds number. With increasing flow rates, the velocity distribution becomes flatter and the laminar sublayer thinner. Dimensionless empirical equations involving viscosity and density are available which correlate the local fluid velocities in the turbulent core, buffer layer, and the laminar sublayer as functions of the distance from the tube axis. The ratio of the average velocity over the entire tube cross-section to the maximum local velocity at the tube axis is approximately 0.7-0.85, and increases with the Reynolds number. 

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The smallest size pipeline loop usually considered for measurements intended for industrial scale-up is lin. (2.54 cm) inside diameter [178]. The results are used to determine laminar versus turbulent flow regimes and as input in flow models [178]. Nasr-El-Din [182,183] reviews the methods used to predict pressure drops across emulsions flowing in pipelines, as well as those used to sample and measure oil and solid concentrations in pipelines. An example of an equation for the prediction of water-in-crude oil (North Sea crude oil) emulsion viscosity is given in Equation (6.48). 

As the absence of micro-organisms is expected, the question of laminar versus turbulent flow and the strict application of aseptic procedures during operations might be irrelevant. 

From the above the maximum stable drop size can be estimated. There will be smaller drops, but in theory no drops larger than this. No data on distribution as yet exist for laminar breakup. Figure 7-28 compares drop size by laminar mechanisms with those calculated for turbulent flow. Smaller droplets are expected for laminar versus turbulent flow at the same energy dissipation rate. 

Laminar Versus Turbulent Flames. Premixed and diffusion flames can be either laminar or turbulent gaseous flames. Laminar flames are those in which the gas flow is well behaved in the sense that the flow is unchanging in time at a given point (steady) and smooth without sudden disturbances. Laminar flow is often associated with slow flow from small diameter tubular burners. Turbulent flames are associated with highly time dependent flow patterns, often random, and are often associated with high velocity flows from large diameter tubular burners. Either type of flow—laminar or turbulent—can occur with both premixed and diffusion flames. 

Several correlating equations for the friction factor have been proposed for both the laminar and turbulent flow regimes, and plots of fM (or functions thereof) versus Reynolds number are frequently presented in standard fluid flow or chemical engineering handbooks (e.g., 96, 97). Perhaps the most useful of the correlations is that represented by the Ergun equation (98)... 

Sinclair and Jackson (1989) used the kinetic theory of granular flows to simulate gas-solid flows in risers. Their model was found to exhibit extreme sensitivity with respect to the value of restitution coefficient, e, . Nieuwland et al. (1996) also observed such an extreme sensitivity. Bolio et al. (1995) reported that such extreme sensitivity could be overcome by including a gas phase turbulence model. Despite these studies, there are no systematic guidelines available to make appropriate selection of models and model parameters (such as laminar versus turbulent, values of... 

Type of flow regime (laminar versus turbulent, turbulent being better). 

When fluid is flowing in a circular pipe and the velocities are measured at different distances from the pipe wall to the center of the pipe, it has been shown that in both laminar and turbulent flow, the fluid in the center of the pipe is moving faster than the fluid near the walls. These measurements are made at a reasonable distance from the. entrance to the pipe. Figure 2.10-1 is a plot of the relative distance from the center of the pipe versus the fraction of maximum velocity where v is local velocity at the... 

Figure 7-28 Maximum drop size versus energy dissipation for laminar and turbulent flow. (From Streiff et al., 1999.)...

Betsed on laboratory test loop measurements the kaolin and fluidic ash slurries can be determined as time independent non-Newtonian yield pseudo-plastic suspensions. On the contrary a considerable effect of shearing during initial period of flow in pipe was found out for fly ash-gypsum mixture, see Fig. 2. However, after a relative short time about one hour, the decrease of pressure losses finished and slurry became stabilised. This behaviour is probably due to physico-chemical activity of ash and gypsum after mixing with water. The Fig. 3 shows plot of wall shear stress t, versus pseudo-shear rate (8 V/D) for the kaolin slurry flow in three different pipes and for representative ash-water mixtures in pipe of diameter D = 17.5 mm. The transition between laminar and turbulent flow is shown by the steep change in slope of the flow curves. 

From Eq. (34) a plot of ln[(l - / q)// q] versus (using a as a fitting parameter) would give a straight-line plot with slope of 1/<1> and an ordinate intercept from which / max be determined. Since d) can be seen to be constant by Eq. (33) for any solute—membrane combination of constant cross-sectional area, plots of Eq. (34) should provide better linearity than plots of Eq. (31). Once d) is determined, values ofR can be calculated. A plot for lipase cross-flow filtered at a constant applied pressure through an A002 membrane is shown in Fig. 18. Note that the linearity is almost perfect until the laminar-to-turbulent flow transition occurs in the film. 

Lockhart and Martinelli divided gas-liquid flows into four cases (1) laminar gas-laminar liquid (2) turbulent gas-laminar liquid (3) laminar gas-turbulent liquid and (4) turbulent gas-turbulent liquid. They measured two-phase pressure drops and correlated the value of 0g with parameter % for each case. The authors presented a plot of acceleration effects, incompressible flow (3) no interaction at the interface and (4) the pressure drop in the gas phase equals the pressure drop in the liquid phase. 

Like the von Karman equation, this equation is implicit in/. Equation (6-46) can be applied to any non-Newtonian fluid if the parameter n is interpreted to be the point slope of the shear stress versus shear rate plot from (laminar) viscosity measurements, at the wall shear stress (or shear rate) corresponding to the conditions of interest in turbulent flow. However, it is not a simple matter to acquire the needed data over the appropriate range or to solve the equation for / for a given flow rate and pipe diameter, in turbulent flow. 

The intense heat dissipated by viscous flow near the walls of a tubular reactor leads to an increase in local temperature and acceleration of the chemical reaction, which also promotes an increase in temperature the local situation then propagates to the axis of the tubular reactor. This effect, which was discovered theoretically, may occur in practice in the flow of a highly viscous liquid with relatively weak dependence of viscosity on degree of conversion. However, it is questionable whether this approach could be applied to the flow of ethylene in a tubular reactor as was proposed in the original publication.199 In turbulent flow of a monomer, the near-wall zone is not physically distinct in a hydrodynamic sense, while for a laminar flow the growth of viscosity leads to a directly opposite tendency - a slowing-down of the flow near the walls. In addition, the nature of the viscosity-versus-conversion dependence rj(P) also influences the results of theoretical calculations. For example, although this factor was included in the calculations in Ref.,200 it did not affect the flow patterns because of the rather weak q(P) dependence for the system that was analyzed. 

The efficiency predicted by Eq. 8.15 is only a rough estimate the equation estimates a shape in the efficiency-versus-particle-size curve that is different from what is actually observed. There are a number of factors not considered in this elementary derivation. First, laminar flow is assumed, but turbulent flow is often observed in practice. The effect of turbulence will be to move particles away from the cyclone walls or resuspend deposited ones. Hence, turbulence will decrease cyclone efficiency. Second, the width of the cyclone inlet is not as important a parameter as overall cyclone diameter, since it is the width of an element of gas within the cyclone that determines particle deposi-... 

It can be argued that any turbulent flow correlation should not be applied for Re <10,000. However, in current thin-channel ultrafiltration devices, the entrance geometry is such that fully developed turbulent flow occurs at much lower Reynold s numbers. Measurements of fluid velocity versus pressure drop show a definite transition from laminar to fully developed turbulent flow at Re = 2000. 

Correlations, dimensionless numbers, and regime maps have been developed for many of these applications. The Reynolds number was introduced in Chapter 5 to differentiate flow regimes in pipes—laminar versus transition versus turbulent. In Chapter 6, the notion of particle Reynolds number was mentioned with respect to drag force and rotameters. 

First, if we ignore the filter medium and consider only the cake itself, the pressure drop versus liquid flow relationship is described by the Ergun equation [Equation (6.15)]. The particle size and range of liquid flow and properties commonly used in industry give rise to laminar flow and so the second (turbulent) term vanishes. For a given slurry (particle properties fixed) the resulting cake resistance is defined as ... 

If we compare equation (6-17) and (6-21), we see that the differences occur in the multiplier term (1.62 versus 0.023), Reynolds number power ( versus 0.8), and D/L power ( versus 0). The Prandtl number power (5) is the same for both. A phenomenological approach to these effects indicate that the change in flow from laminar to turbulent is reflected both in the larger power on the Reynolds number and the elimination of the D/L effect (transformation from layers to vortices and eddies). Also, the change in flow does not alter the Prandtl number power. 

The two key properties in single-phase flow are the fluid density and the viscosity. The density is quite straightforward it is the mass per unit volume. In turbulent flow, pressure drop is directly proportional to density, so that the accuracy of the density is the accuracy of the pressure drop prediction. It is easy to get better than 1% accuracy on such values. Viscosity, on the other hand, is a more complex measurement. Low viscosity systems usually run in turbulent flow, where the viscosity has little or no effect on mixing or pressure drop. For low viscosity material the prime use of the viscosity is in calculating a Reynolds number to determine if the flow is laminar or turbulent. If turbulent, little accuracy is needed. An error in viscosity of a factor of 2 will have negligible effect. In laminar flow, however, the viscosity becomes all important and pressure drop is directly proportional to it, so that an accuracy of 10% or less is often required. For laminar processing a complete relation of stress versus strain or shear rate versus shear stress is required. See Chapter 4 for the means and type of data required. 

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