Parabolic-shaped velocity gradient laminar flow

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

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... 

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. 

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. 

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