Laminar-flow

The strength of a flow is generally expressed by the stress it exerts on any plane in the direction of flow. It is simply given by rjG, but it should be realized that the viscosity depends on flow type. Generally, tj is taken to be the shear viscosity, which is the one determined in nearly all viscometers. For elongational flow we then have 

The deformation primarily depends on the ratio of the external stress over the Laplace pressure, expressed in a dimensionless Weber number, in the present case given by 

For instance, for a = 0.2, break-up does not occur at A 15, and the drop attains a fairly small deformation of about 0.1, almost independent of the value of We. Introducing a simple shear component in elongational flow (say, 0.5 a 1), has fairly little effect on the effectiveness of droplet break-up. 

Not all workers in the field have obtained precisely the same results. This is because it is difficult to obtain a true steady state just before break-up. In practice, We depends on dC7/d/ and on irregularities in the flow. Moreover, even trace amounts of surfactants can have a distinct effect this is discussed in Section 2.4. It is generally seen that break-up of drops into two smaller ones also leaves some, generally three, very small satellite drops. In a detailed study of this phenomenon, up to 19 satellite drops have even been observed. However, these drops are of the order of 10 pm in diameter. Consequently, it is very unlikely that satellite drops are formed during practical emulsification, where droplets of about 1 pm diameter are generally desired. Any satellite droplets would then be far smaller than 0.1 m and have an excessive Laplace pressure. 

Another phenomenon often observed is called tip streaming, studied in detail by de Bruijn. The deformed drop attains pointed ends, and small droplets then are shed from these tips, which may go on for a long time. Again, the droplets shed are of some 10 xm in size, and they will not be formed from small drops. Tip streaming is caused by the presence of surfactants, even if present at very low concentration.  

In fully developed laminar flow (Re 4000) in a circular horizontal pipe, the pressure loss and the head loss are given by 

Under laminar flow conditions, the friction factor,/ is directly proporhonal to viscosity and inversely proportional to the velocity, pipe diameter, and fluid density. The friction factor is independent of pipe roughness in laminar flow because the disturbances caused by surface roughness are quickly damped by viscosity [4]. The pressure drop in laminar flow for a circular horizontal pipe is 

When the flow rate and the average velocity are held constant, the head loss becomes proportional to viscosity. The head loss, /zl, is related to the pressure loss by 

In the nineteenth century Darcy (1856) observed that the flow of water through a packed bed of sand was governed by the relationship  

The flow of a fluid through a packed bed of solid particles may be analysed in terms of the fluid flow through tubes. The starting point is the Hagen-Poiseuille equation for laminar flow through a tube  

Consider the packed bed to be equivalent to many tubes of equivalent diameter De following tortuous paths of equivalent length He and carrying fluid with a velocity Hi. Then, from Equation (6.2), 

Introduction to Particle Technology, 2nd Edition Martin Rhodes 2008 John Wiley Sons Ltd. ISBN 978-0-470 -01427-1 

Ui is the actual velocity of fluid through the interstices of the packed bed and is related to superficial fluid velocity by  

Due to the high viscosity, the flow of polymer fluids conventionally appears as laminar flow with a small flow velocity. There are two basic approaches to realize the laminar flow. 

In the shear flow, the shear stress cr(=/M) works on the shear plane along xz directions, as illustrated in Fig. 7.2. When the shear stress j is proportional to the velocity gradient dv/dy, the fluid can be called a. Newtonian fluid, and the proportion factor is defined as shear viscosity rj. The unit of viscosity is thus Pa s or N s/m. In the centimeter-gram-second (CGS) unit system, the unit of viscosity is Poise (P) with 1 Pa s = 10 P. The velocity gradient 

This equation is similar to the second Newton s law (total forces proportional to the acceleration rate), because the shear rate reflects the acceleration rate generated by the shear stress. 

Here dv/dr is the extensional strain rate, and 77 is the extensional viscosity. The correlation between the ideal extensional viscosity and the shear viscosity for a Newtonian fluid can be described by the Trouton s ratio (Trouton 1906), as given by 

Kusakabe, Y. Yamasaki, S. Morooka, 3-D simulation and visualization of laminar flow in a microchannel with hair-pin curves. AIChE J., 2004, 50 (7), 1530-1535. 

Three-dimensional hydrodynamic focusing in two-layer polydimethylsiloxane (PDMS) microchaimels. J. Micromedi. Microeng., 2007, 37(8), 1479-1486. 

Irisawa, M. Ishizuka, K. Hishida, M. Maeda, Visualization of convective mixing in microchannel by fluorescence imaging. Meas. Sci. Technol., 

Hohreiter, S. T. Wereley, M. G. Olsen, J. N. Chimg, Cross-correlation analysis for 

Stemich, H.-J. Wamecke, Fluid mixing in a T-shaped micro-mixer. Chem. Eng. Sci., 2006, 63 (9), 2950-2958. 

Consider a model of two parallel sheets with one sliding over the other. Common sense teUs us that there is some sort of friction opposing the sliding motion. Viscosity is a drag, literally  

We can develop the idea of a laminar coefficient of viscosity from common experience. First, the force required to move one sheet over the other is proportional to the area A of contact between the sheets. Second, more force will be required to move the upper sheet faster. Third, the actual contact between the sheets depends inversely on the contact distance between the sheets since all 

This leads to the phenomenological units of the coefficient of viscosity in the cgs system as 

While this unit is easy to derive using reasoning from everyday experience, the poise (pwaz) is an ancient unit and viscosity is now measured in (pascal seconds), so that 1 poise = 0.1 Pa s in SI units  

More properly called the Hagen-Poiseuille law, it was developed independently by Gotthilf Heinrich Ludwig Hagen (1797-1884) and Jean Louis Marie Poiseuille. Poiseuille s law was experimentally derived in 1838 and formulated and published in 1840 and 1846 by Jean Louis Marie Poiseuille (1797-1869). Hagen also carried out experiments in 1839. While there are a number of derivations, we follow a simple one here from Physical Chemistry by Castellan [5]. 

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The first term (AQ) is the pressure drop due to laminar flow, and the FQ term is the pressure drop due to turbulent flow. The A and F factors can be determined by well testing, or from the fluid and reservoir properties, if known. 

In general, extreme care has to be taken when LB films are prepared, since tire quality of the resulting films depends cmcially on tire preparation conditions. The best place for an LB trough is a laboratory where tire surroundings, i.e. temperature, humidity and atmosphere, are completely controlled. Often it is placed in a laminar flow box. Also, tire trough should be installed in a shock-free environment. 

Hannart, B. and Hoplinger, E.J., 1998. Laminar flow in a rectangular diffuser near Hele-Sliaw conditions - a two dinien.sioiial numerical simulation. In Bush, A. W., Lewis, B. A. and Warren, M.D. (eds), Flow Modelling in Industrial Processes, cli. 9, Ellis Horwood, Chichester, pp. 110-118. 

It is also possible to simulate nonequilibrium systems. For example, a bulk liquid can be simulated with periodic boundary conditions that have shifting boundaries. This results in simulating a flowing liquid with laminar flow. This makes it possible to compute properties not measurable in a static fluid, such as the viscosity. Nonequilibrium simulations give rise to additional technical difficulties. Readers of this book are advised to leave nonequilibrium simulations to researchers specializing in this type of work. 

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

Lambert s cosine law Lambic beer Lamb modes Lambs Lamb wave Lamellae Lamepon Laminar flow Laminated fabrics Laminated glass... 

Several other factors must also be considered with respect to heating conditions. At the front end of a vehicle, ie, at the nosetip, the heating rate is most severe, generally decreasing toward the aft end of the vehicle in instances of laminar flow. Because of this variation in heating conditions, the nosetip... 

Table 1. Heat of Ablation and Relative Thermal Conductivity for Reentry Vehicle Materials Assuming Laminar Flow...

The concept of the specific resistance used in equation 4 is based on the assumptions that flow is one-dimensional, growth of cake is unrestricted, only soHd and Hquid phases are present, the feed is sufficiently dilute such that the soHds are freely suspended, the filtrate is free of soHds, pressure losses in feed and filtrate piping are negligible, and flow is laminar. Laminar flow is a vaHd assumption in most cake formation operations of practical interest. 

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

Most flow meters are designed and caHbrated for use on turbulent flow, by far the more common fluid condition. Measurements of laminar flow rates may be seriously in error unless the meter selected is insensitive to velocity profile or is specifically caHbrated for the condition of use. 

Enough space must be available to properly service the flow meter and to install any straight lengths of upstream and downstream pipe recommended by the manufacturer for use with the meter. Close-coupled fittings such as elbows or reducers tend to distort the velocity profile and can cause errors in a manner similar to those introduced by laminar flow. The amount of straight pipe required depends on the flow meter type. For the typical case of an orifice plate, piping requirements are normally Hsted in terms of the P or orifice/pipe bore ratio as shown in Table 1 (1) (see Piping systems). 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

Fig. 5. Entrance flows in a tube or duct (a) separation at sharp edge (b) growth of a boundary layer (illustrated for laminar flow).

In configurations more complex than pipes, eg, flow around bodies or through nozzles, additional shearing stresses and velocity gradients must be accounted for. More general equations for some simple fluids in laminar flow are described in Reference 1. 

The phenomena are quite complex even for pipe flow. Efforts to predict the onset of instabiHty have been made using linear stabiHty theory. The analysis predicts that laminar flow in pipes is stable at all values of the Reynolds number. In practice, the laminar—turbulent transition is found to occur at a Reynolds number of about 2000, although by careful design of the pipe inlet it can be postponed to as high as 40,000. It appears that linear stabiHty analysis is not appHcable in this situation. 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

For laminar flow a = 0.5. For fully developed turbulent flowO.88 < a < 0.98, but can be taken to be unity with Httle error. 

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

In the forced convection heat transfer, the heat-transfer coefficient, mainly depends on the fluid velocity because the contribution from natural convection is negligibly small. The dependence of the heat-transfer coefficient, on fluid velocity, which has been observed empirically (1—3), for laminar flow inside tubes, is h for turbulent flow inside tubes, h and for flow outside tubes, h. Flow may be classified as laminar or... 

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 convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

Fig. 5. Moody diagram for Darcy friction factor (13) (-----), smooth flow (----), whoUy turbulent flow ( ), laminar flow.

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