Slow frictional flow

In the analysis of flow of granular material, two types of flow can be distinguished. The first is slow frictional flow where the particles remain in continuous contact with each other the internal forces result from Coulomb friction between contacting particles. The second type of flow is much more rapid the particles are not in constant contact with their neighbors. The energy associated with the velocity fluctuations is comparable to that of the mean motion. In this type of flow, the internal forces arise because of the transfer of momentum during collisions between particles. The constitutive relations for this rapid flow are rate-dependent. This type of flow, therefore, is referred to as viscous flow (sometimes just rapid flow). Steady,... 

We also assume isothermal flow. Of course, no viscous flow can be truly isothermal, because the friction between the sliding layers of fluid generates heat, called viscous dissipation. But slow viscous flows in narrow channels can be assumed, at first approximation, to be isothermal. This assumption greatly simplifies the solution and provides simple, useful working equations. 

Other Losses Not only are there friction losses along pipe walls and dynamic losses at the system entry and at each bend or transition, but there are also losses at filters or other air cleaning devices that are part of the system. Filters placed in the air stream will want to slow the flow and create a friction loss. In summary, Bernoulli s equation (Equation 10-8) applies to air flow in ducts. A related form for the equation is the sum of the static pressure and velocity pressure of a point upstream in a ventilation system is equal to the sum of the static pressure, the velocity pressure, friction loss (FL), and dynamic losses (DL) at a point downstream in the system ... 

General laws for the flow of fluids were determined by Reynolds, who recognized two flow patterns, laminar and turbulent. In laminar flow the fluid can be considered as a series of parallel strata, each moving at its own speed, and not mixing. Strata adjacent to walls of the duct will be slowed by friction and will move slowest, while those remote from the walls will move fastest. In turbulent flow there is a general forward movement together with irregular transfer between strata. 

A 4 in. diameter open can has a 1/4 in. diameter hole in the bottom. The can is immersed bottom down in a pool of water, to a point where the bottom is 6 in. below the water surface and is held there while the water flows through the hole into the can. How long will it take for the water in the can to rise to the same level as that outside the can Neglect friction, and assume a pseudo steady state, i.e., time changes are so slow that at any instant the steady state Bernoulli equation applies. 

Currents in rivers and streams are turbulent. Turbulent mixing can be described by the Fickian laws (Eqs.18-6 and 18-14) and by empirical turbulent diffusion coefficients Ea, where a stands for x, y, z (Chapter 22). The main source of turbulence is the friction between the water and the river bed. It can be expected that increasing roughness of the river leads to increasing turbulence, much in the same way as a large roughness causes the mean flow u to become slow (see the effect on Eq. 24-4 if the friction coefficient f increases). In fact, turbulence in rivers can be scaled by the shear velocity, u, defined in Eq. 24-5. 

For non-Newtonian fluids in slow flow, friction loss across a square-woven or full-twill-woven screen can be estimated by considering the screen as a set of parallel tubes, each of diameter equal to... 

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. 

The simplest and often the most cost effective way to combat friction is to reduce flow rate to a minimum. By no coincidence, this often leads to an increase in the efficiency of a separation since in many circumstances for preparative purifications, the less experienced have followed a linear scale-up from analytical column flow rates. In an ideal world each separation should, at some stage, involve a flow rate optimization. The fundamental principles behind this are discussed by JJ van Deemter[52 in what is probably the most cited paper in the history of chromatography. In summary, this suggests doing a graphical plot of separation efficiency versus flow rate and is particularly important for peptide purification where mass transport is comparatively slow. The van Deemter equation in simplified form can be represented as ... 

Phenomenological systems show that in relatively slow processes, the conjugate flow J is largely determined by frictional forces, and is linearly related to the conjugate force X... 

Next, we introduce a dimensionless number that describes flow characteristics, such as whether the flow will be laminar or turbulent. This quantity indicates the ratio of inertial forces (due to momentum, which tends to keep things moving) to viscous forces (due to friction, which tends to slow things down) and is known as the Reynolds number ... 

Equation 9.11 is usually referred to as Poiseuille s law and sometimes as the Hagen-Poiseuille law. It assumes that the fluid in the cylinder moves in layers, or laminae, with each layer gliding over the adjacent one (Fig. 9-14). Such laminar movement occurs only if the flow is slow enough to meet a criterion deduced by Osborne Reynolds in 1883. Specifically, the Reynolds number Re, which equals vd/v (Eq. 7.19), must be less than 2000 (the mean velocity of fluid movement v equals JV, d is the cylinder diameter, and v is the kinematic viscosity). Otherwise, a transition to turbulent flow occurs, and Equation 9.11 is no longer valid. Due to frictional interactions, the fluid in Poiseuille (laminar) flow is stationary at the wall of the cylinder (Fig. 9-14). The speed of solution flow increases in a parabolic fashion to a maximum value in the center of the tube, where it is twice the average speed, Jv. Thus the flows in Equation 9.11 are actually the mean flows averaged over the entire cross section of cylinders of radius r (Fig. 9-14). 

For non-Newtonian fluids in slow flow, friction loss across a square-woven or full-twill-woven screen can be estimated by considering the screen as a set of parallel tubes, each of diameter equal to the average minimal opening between adjacent wires, and length twice the diameter, without entrance effects (Carley and Smith, Polym. Eng. Set, 18, 408-415 [1978]). For screen stacks, the losses of individual screens should be summed. 

Consider a fluid entering a circular pipe at a uniform velocity. Because of ilie no-slip condition, the fluid particles in (he layer in contact with the surface of the pipe come to a complete stop. This layer also causes the fluid particles in l) e adjacent layers to slow down gradually as a result of friction. To make up for this velocity reduction, the velocity of the fluid at the midsection of the pipe has to increase to keep the mass flow rate through the pipe constant. As a result, a velocity gradient develops along the pipe. 

As we have discussed earlier, the buoyancy force is caused by the density difference between the healed (or cooled) fluid adjacent to the surface and tiie fluid surrounding it, and is proportional to this density difference and the volume occupied by the warmer fluid. It is also well knowu ll at whenever Iwc bodies in contact (.solid--solid, solid-fluid, or fluid-fluid) move relative to cacf other, a friction force develops at the contact surface in the direction opposite ic that of the motion. This opposing force slows down the fluid and thus reduce the flow rate of the fluid. Under steady conditions, the airflow rate driven b buoyancy is established at the point where these two effects balance each othet The friction force increases as more and more solid surfaces are introduced, se tiously disrupting the fluid flow and heat transfer. For that reason, heat sink with closely spaced fins are not suitable for natural convection cooling. 

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