Flow, cylindrical tube

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

Flow velocity field determined by PIV. Lean limit flames propagating upward in a standard cylindrical tube in methane/air and propane/ air mixtures, (a) Methane/air—laboratory coordinates, (b) propane/air—laboratory coordinates, (c) methane/air—flame coordinates, and (d) propane/air—flame coordinates. 

The small intestine is assumed to be a cylindrical tube with a surface area of 2kRL, where R is the radius and L is the length of the tube (Fig. 4). The rate at which the drug enters the tube is the product of the inlet concentration, C0, and the volumetric flow rate, Q. The rate at which it exits the tube is the product of the outlet concentration, Cout, and the volumetric flow rate, Q. The absorption flux across the small intestinal membrane, , is the product of the effective permeability, Peff, and concentration, C. The total drug loss by absorption from the... 

Figure 4 Schematic of macroscopic mass balance approach. The small intestine is assumed to be a cylindrical tube with a radius of R and a length of L. The inlet and outlet concentrations are C0 and Cout. The inlet or outlet volumetric flow rate is Q. 

The velocity profile during slip flow in a cylindrical tube is shown in Figure 21. As in conventional fluid flow, the flow velocity in the z direction, u(r), is parabolic, but rather than reach zero at the tube wall, slip occurs, and the velocity at the wall is greater than zero. The velocity does not reach zero until distance h from the wall surface. The derivation of the mass flux equation proceeds along the same lines as the derivation of Poiseuille s law in conventional hydrodynamics, but in slip flow, u(r) = 0 at r = a + h instead of reaching zero at r = a. 

Figure 21 Velocity profile during slip flow in a cylindrical tube.

In 1961 Bretherton solved the problem of a long gas bubble, uncontaminated by surface-active impurities, flowing in a cylindrical tube at low capillary numbers, Ca /xU/a (/x is the... 

In this section we illustrate how the proposed theory for single, surfactant-laden bubbles in a cylindrical tube can be extended to predict the hydrodynamic resistance of bubble trains flowing in porous media. Some of the basic ideas are known (7, 23), so the present discussion is brief. 

The effect of a soluble surfactant on the flow of long bubbles in a cylindrical tube has been quantified when the surfactant... 

Fixed Bed Reactors. In its most basic form, a fixed bed reactor consists of a cylindrical tube filled with catalyst pellets. Reactants flow through the catalyst bed and are converted into products. Fixed bed reactors are often referred to as packed bed reactors. They may be regarded as the workhorse of the chemical industry with respect to the number of reactors employed and the economic value of the materials produced. Ammonia synthesis, sulfuric acid production (by oxidation of S02 to S03), and nitric acid production (by ammonia oxidation) are only a few of the extremely high tonnage processes that make extensive use of various forms of packed bed reactors. 

Show how the Hagen-Poiseuille equation for the steady laminar flow of a Newtonian fluid in a uniform cylindrical tube can be derived starting from the general microscopic equations of motion (e.g., the continuity and momentum equations). 

Ingham, D.B., Diffusion of Aerosols from a Stream Flowing Through a Cylindrical Tube, Aerosol Sci. 6 125-132 (1975). 

A catalytic fixed bed reactor is a (usually) cylindrical tube that is randomly filled with porous catalyst particles. These are frequently spheres or cylindrical pellets, but other shapes are also possible. The use of rings or other forms of particles with internal voids or external shaping is on the increase. During single-phase operation, a gas or liquid flows through the tube and over the catalyst particles, and reactions take place on the surfaces, both interior and exterior, of the particles. 

The balance is made with respect to a control volume which may be of finite (V) or of differential (dV) size, as illustrated in Figure 1.3(a) and (b). The control volume is bounded by a control surface. In Figure 1.3, m, F, and q are mass (kg), molar (mol), and volumetric (m3) rates of flow, respectively, across specified parts of the control surface,6 and Q is the rate of heat transfer to or from the control volume. In (a), the control volume could be the contents of a tank, and in (b), it could be a thin slice of a cylindrical tube. 

To determine the pressure gradient along the reactor, the Fanning or Darcy equation for flow in cylindrical tubes may be used (Knudsen and Katz, 1958, p. 80) ... 

In laminar flow through a cylindrical tube of radius R and length L the linear velocity depends on the radial position (3 = r/R according to... 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

As mentioned in Chapter 1, the first published work on fluid flow patterns in pipes and tubes was done by Reynolds in 1883. He observed the flow patterns of fluids in cylindrical tubes by injecting dye into the moving stream. Reynolds correlated his data by using a dimensionless group later known as the Reynolds number Re ... 

Resistance functions have been evaluated in numerical compu-tations15831 for low Reynolds number flows past spherical particles, droplets and bubbles in cylindrical tubes. The undisturbed fluid may be at rest or subject to a pressure-driven flow. A spectral boundary element method was employed to calculate the resistance force for torque-free bodies in three cases (a) rigid solids, (b) fluid droplets with viscosity ratio of unity, and (c) bubbles with viscosity ratio of zero. A lubrication theory was developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions were derived to accurately represent the numerical data over the entire range of particle positions in a tube for all particle diameters ranging from nearly zero up to almost the tube diameter. The resistance functions formulated are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels. 

The modelling of the flow of a non-Newtonian fluid through a packed bed follows a similar, though more complex, procedure to that adopted earlier in this chapter for the flow of a Newtonian fluid. It first involves a consideration of the flow through a cylindrical tube and then adapting this to the flow in the complex geometry existing in a packed bed. The procedure is described in detail elsewhere(24,25). 

For laminar flow of a power-law fluid through a cylindrical tube, the relation between mean velocity u and pressure drop —AP is given by ... 

At steady state, the following mass balance can be applied across a perfused segment of the intestine. The rate of mass entering and exiting the intestinal segment (i.e. cylindrical tube) is the product of the volumetric flow rate (Q) and either inlet concentration (Cin) or outlet concentration (Cout), respectively. Assuming that mass is lost from the tube only by absorption into the blood, the mass absorbed per unit time is the difference between the rates of mass flow in and out of the tube as follows ... 

In this method premixed gases flow up a jacketed cylindrical tube long enough to ensure streamline flow at the mouth. The gas bums at the mouth of the tube, and the shape of the Bunsen cone is recorded and measured by various means and in various ways. When shaped nozzles are used instead of long tubes, the flow is uniform instead of parabolic and the cone has straight edges. Because of the complicated flame surface, the different procedures used for measuring the flame cone have led to different results. 

We first assume that the tube has constant diameter D and also that the density does not vary with position (either liquids or gases with no mole number change, pressure drop, or temperature change with gases). In this case the linear velocity u with which the fluid flows through the tube is equal to the volumetric flow rate t divided by the cross-sectional tube area At (At = 7t j A for a cylindrical tube),... 

Wall Correction Factor K for a Rigid Spherical Particle Moving on the Axis of a Cylindrical Tube in Creeping Flow... 

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