Circular tube laminar flow

Let us briefly consider convective mass transfer accompanied by a surface reaction in a circular tube. Laminar steady-state fluid flow in a circular tube of radius a with Poiseuille velocity profile is outlined in Subsection 1.5-3. For... 

FIGURE 4.18 Laminar flow in circular tubes, volumetric flow rate ... 

Exact Solutions to the Navier-Stokes Equations. As was tme for the inviscid flow equations, exact solutions to the Navier-Stokes equations are limited to fairly simple configurations that aHow for considerable simplification both in the equation and in the boundary conditions. For the important situation of steady, fully developed, laminar, Newtonian flow in a circular tube, for example, the Navier-Stokes equations reduce to... 

In the simple Bunsen flame on a tube of circular cross-section, the stabilization depends on the velocity variation in the flow emerging from the tube. For laminar flow (paraboHc velocity profile) in a tube, the velocity at a radius r is given by equation 20 ... 

For laminar flow in a circular tube, the Leveque relationship is ... 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

Consider isothermal laminar flow of a Newtonian fluid in a circular tube of radius R, length L, and average fluid velocity u. When the viscosity is constant, the axial velocity profile is... 

Equation (8.9) can be applied to any reaction, even a complex reaction where ctbatch(t) must be determined by the simultaneous solution of many ODEs. The restrictions on Equation (8.9) are isothermal laminar flow in a circular tube with a parabolic velocity profile and negligible diffusion. 

The special flow conditions in circular (capillaries, tubes) or rectangular channels cause very different stresses depending on the position of the particles in the flow cross section. With laminar flow, for example the following applies to velocity gradient (see e.g. [37]) ... 

In the steady flow of a Newtonian fluid through a long uniform circular tube, if ARe < 2000 the flow is laminar and the fluid elements move in smooth straight parallel lines. Under these conditions, it is known that the relationship between the flow rate and the pressure drop in the pipe does not depend upon the fluid density or the pipe wall material. 

As will be shown later, the velocity profile for a Newtonian fluid in laminar flow in a circular tube is parabolic. When this is introduced into Eq. (5-38), the result is a = 2. For highly turbulent flow, the profile is much flatter and a 1.06, although for practical applications it is usually assumed that a = 1 for turbulent flow. 

Example 5-2 Kinetic Energy Correction Factor for Laminar Flow of a Newtonian Fluid. We will show later that the velocity profile for the laminar flow of a Newtonian fluid in fully developed flow in a circular tube is parabolic. Because the velocity is zero at the wall of the tube and maximum in the center, the equation for the profile is... 

Laminar flow In circular tubes with parabolic velocity distribution Is known as Poiseuille flow. This special case is found frequently in vacuum technology. Viscous flow will generally be found where the molecules mean free path is considerably shorter than the diameter of the pipe X d. 

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 ... 

When a culture medium flows in a circular tube as a laminar flow, determine the fraction of the medium that passes through the tube with a higher velocity than the averaged linear velocity. 

Circular Cross-section Pipes. In contrast to nozzles and apertures, the calculation of gas flow through tubes (/ d) has to take the effects of gas viscosity into account. For smooth-walled tubes, gas fluxes less than q l3 and laminar flow conditions, the well-known expression for p F-throughput, can be obtained ... 

The Circular Tube Thermal-Entry-Length, with Hydrodynamically Fully Developed Laminar Flow... 

Let us look at an incompressible, constant-property fluid flowing laminarly inside a circular tube in regions away from the inlet where the velocity profile is fully developed. The continuity equation gives... 

In Ulrichson and Schmit s work on laminar flow heat transfer in the entrance region of circular tubes the following results were obtained. 

Consider laminar flow of a high-viscosity oil in a circular tube with a uniform wall temperature. If viscous dissipation effects are significant, determine the temperature profile far from the inlet, i.e., in folly developed flow. Assume constant fluid properties. 

Worsoe-Schmidt, P.M., Heat Transfer and Friction for Laminar Flow of Helium and Carbon Dioxide in a Circular Tube at High Heating Rate , Int. J. Heat-Mass Transfer, Vol. 9, pp. 1291-1295,1966. 

Fig. 5-5 Local and average Nusselt numbers tor circular tube thermal entrance regions in fully developed laminar flow.

The dispersion of a non-reactive solute in a circular tube of constant cross-section in which the flow is laminar is described by the convective-diffusion equation... 

With fully developed laminar flow, the velocity at any point r in a circular tube of radius R can be expressed by the equation... 

Let us imagine a long cylindrical capillary tube, of length L and radius r. Let x denote the distance outward radially from the axis of the tube and z denote the distance along the axis. Under conditions of laminar flow, the laminar surfaces are a family of right circular cylinders coaxial with the tube. 

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