Flow Around Bodies

Ap is the body s projected area in the direction of flow. For a sphere, Ap = Tidz/4, and for a cylinder Ap = dL (L = length of the cylinder). 

Further discussion on flow around and drag forces on bodies is available in the literature.1,2,4 

2 Flow Through a Packed Bed of Spheres. The Ergun Equation (46), gives the pressure drop through a packed bed of spheres 2 

We see that Apjl, the frictional pressure drop per unit depth of bed, is made up of two components. The first term on the right-hand-side accounts for viscous (laminar) frictional losses, cc pu. and dominates at low Reynolds numbers. The second term on the right-hand-side accounts for the inertial (turbulent) frictional losses, oc pu2, and dominates at high Reynolds numbers. For further information about flow through packed beds, see Chapter 7 An Introduction to Particle Systems . 

The power P to turn an impeller is given by 2nN x M, (angular velocity x torque). This power would be expected to be a function of impeller diameter and speed, and geometry. Gravitational acceleration should also be included in the analysis as there is normally a free liquid surface present and gravity affects its shape and the flow within the vessel. 

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Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

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 recent development of tensorial schemes for characterizing the intrinsic hydrodynamic resistance of particles of arbitrary shape, and the application of singular perturbation techniques to obtain asymptotic solutions of the Navier-Stokes equations at small Reynolds numbers constitute significant contributions to oim understanding of slow viscous flow around bodies. It is with these topics that this review is primarily concerned. In presenting this material we have elected to use Gibbs polyadics in preference to conventional tensor notation. For in our view, the former symbolism— dealing as it does with direction as a primitive concept—is more closely related to the physical world in which we live than is the latter notation. 

The effect of polymeric additives on collective phenomena, associated with the dynamics of bubbles, can be illustrated by the example of hydrodynamic cavitation, caused by abrupt decrease in local pressure (in flows around bodies, after stream contraction, in jets). It has been found that the use of polymers permits to decrease the cavitation noise, lower the cavitation erosion, and delay the cavitation inception. For example, adding a small amount of POE to a water jet issuing from the orifice caused the decrease of the critical cavitation number, K , by 35-40%. In experiments with rotating disk flic value of K, was decreased by 65% with addition of 500 ppm POE. Note, however, that these features are linked not only to the changes in individual bubble dynamics, but to the influence of macromolecules on the total flow regime as well. In particular, phenomena listed above are... 

Relates the nature of the fluid flow in and around bodies. Richardson Number... 

U. Kei.kar. J. V.. Mashelkar, R. a., arid DLBRECHT, J. Trans. Inst. Chem. Eng. 50 (I972i 343. On ihe rotational viscoelastic flows around simple bodies and agitators. 

As the fluid flows over the forward part of the sphere, the velocity increases because the available flow area decreases, and the pressure decreases as a result of the conservation of energy. Conversely, as the fluid flows around the back side of the body, the velocity decreases and the pressure increases. This is not unlike the flow in a diffuser or a converging-diverging duct. The flow behind the sphere into an adverse pressure gradient is inherently unstable, so as the velocity (and lVRe) increase it becomes more difficult for the streamlines to follow the contour of the body, and they eventually break away from the surface. This condition is called separation, although it is the smooth streamline that is separating from the surface, not the fluid itself. When separation occurs eddies or vortices form behind the body as illustrated in Fig. 11-1 and form a wake behind the sphere. 

Large frontal areas create air turbulence and drag. Bodies derived from wind tunnel testing can provide a more smooth air flow around the vehicle. Details such as mirrors, rain gutters, trim, wheel wells and covers can also be more appropriate for air flow. Radial tires can reduce fuel consumption as much as 3%. Puncture-proof tires of plastic could save even more and eliminate the cost and weight of a spare tire and wheel. 

When fluid flows around the outside of an object, an additional loss occurs separately from the frictional energy loss. This loss, called form drag, arises from Bernoulli s effect pressure changes across the finite body and would occur even in the absence of viscosity. In the simple case of very slow or creeping flow around a sphere, it is possible to compute this form drag force theoretically. In all other cases of practical interest, however, this is essentially impossible because of the difficulty of the differential equations involved. 

Modelling Flow Around a Body Immersed in a Fluid... 

Problem. In this example, we consider the flow around a body. Air, at atmospheric pressure, flows at 20 m s 1 across a bank of heat exchanger tubes. A l/10th-scale model is built. At what velocity must air flow over the model bank of tubes to achieve dynamic similarity ... 

For boundary layer flows on bodies of other shape, the transition Reynolds number based on the distance around the surface from the leading edge of the body is usually increased if the pressure is decreasing, i.e., if there is a favorable pressure gradient, and is usually decreased if the pressure is increasing, i.e., if there is an unfavorable pressure gradient. 

Suppose it is desired to visualize the flow around a 1-pm sphere by studying the flow around a 1-cm sphere. One could ask, Under what conditions is it reasonable to assume that a l-pm-diameter sphere moving in a continuous medium will behave in a manner similar to a 1-cm sphere moving in the same medium Or more generally, under what conditions will geometrically similar flow occur around geometrically similar bodies The answer, fundamental to fluid mechanics, is that in similar fields of flow, the forces acting on an element of either body must bear the same ratio to each other at any instant. 

Reynolds number can be applied to either a fluid flowing around a body or a fluid flowing inside a pipe. The transition from laminar to turbulent flow occurs at different Reynolds numbers for these two cases. The Reynolds numbers at which different flow conditions prevail are tabulated in Table 4.2. Since v is the relative velocity between the medium and the body, the Reynolds number is the same whether the body is moving through a stationary fluid or the fluid is flowing around a stationary body. 

Acupuncture An ancient Chinese system of medical treatment that aims to influence energy flow around the body by inserting needles into particular points on the skin,... 

W. Lick, 7. Fluid Mech. 7, 128 (1960) Inviscid Flow Around a Blunt Body of a Reacting Mixture of Gases, Part A, General Analysis Part B, Numerical Solutions, AFOSR Tech. Note No. 58-522 R.P.I. Tech. Rept. No. AE5810) and AFOSR Tech. Note No. 58-1124 R.P.I. Tech. Rept. No. AE 5814), Rensselaer Polytechnic Institute, Troy (1958). 

The temperature of the fluid t F far away from the wall, appears in (1.23), the definition of the local heat transfer coefficient. If a fluid flows around a body, so called external flow, the temperature t F is taken to be that of the fluid so far away from the surface of the body that it is hardly influenced by heat transfer, i) F is called the free flow temperature, and is often written as diDC. However, when a fluid flows in a channel, (internal flow), e.g. in a heated tube, the fluid temperature at each point in a cross-section of the channel will be influenced by the heat transfer from the wall. The temperature profile for this case is shown in Figure 1.8. i) F is defined here as a cross sectional average temperature in such a way that t F is also a characteristic temperature for energy transport in the fluid along the channel axis. This definition of F links the heat flow from the wall characterised by a and the energy transported by the flowing fluid. 

The velocity held is determined by the characteristic length L0, and velocity w0 e.g. the entry velocity in a tube or the undisturbed velocity of a fluid flowing around a body, along with the density g and viscosity rj of the fluid. While density already plays a role in frictionless flow, the viscosity is the fluid property which is characteristic in friction flow and in the development of the boundary layer. The two material properties, thermal conductivity A and specific heat capacity c, of the fluid are important for the determination of the temperature held in conjunction with the characteristic temperature difference Ai 0. The specihc heat capacity links the enthalpy of the fluid to its temperature. 

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