Parabolic-shaped velocity gradient

Figure 48.5(a) Parabolic-shaped velocity gradient laminar flow fluid flows fastest at the center of the pipe. 

In the FFF a field or gradient is applied in a direction, perpendicular to the axis of a narrow flow channel. At the same time a solvent is forced steadily through the channel forming a cross-sectional flow profile of parabolic shape. When a polymer sample is injected into the channel, a steady state is soon reached in which the field induced motion and the opposed diffusion are exactly balanced. The continuous size-distribution of the polymer will migrate with a continuous spectrum of velocities and will emerge at the end of the flow channel with a continuous time distribution. When processed through a detector and its associated electronics, the time distribution becomes an elution (retention) spectrum. 

General model. This is the best alternative as It describes in the most accurate way possible the behaviour of a solute injected into an FIA system. It is based on the general expression describing convective-diffusional transport, which takes account of both axial and radial concentration gradients, the parabolic shape of the velocity profile corresponding to a laminar flow regime and the contribution of convective transport... 

TEMPERATURE AND VELOCITY PROFILES. The radial temperature profile for an exothermic reaction in a packed tube has the shape shown in Fig. 15.18c. There is a steep gradient near the inside wail and a nearly parabolic temperature profile over the rest of the catalyst bed. The velocity profile (Fig. 15,18b) has a peak near the wall, since the particles are packed more loosely in this region than in the rest of the tube. The temperature and velocity profiles for an empty tube with turbulent flow and a homogeneous reaction would have almost all of the gradient near the wall. 

Viscous flow This mode of transport is due to a total pressure gradient of a continuum fluid mixture (Figure 7.3-2). Hence, there is no separation of species due to the viscous flow. The driving force is the total pressure gradient and the parameter characterizing the transport is the mixture viscosity, p, and the viscous flow parameter, B, which is a function of solid properties only. The flow inside the pore is assumed laminar, hence the velocity profile is parabolic in shape. 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

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