Natural convection, laminar boundary conditions

In electrochemical reactors, the externally imposed velocity is often low. Therefore, natural convection can exert a substantial influence. As an example, let us consider a vertical parallel plate reactor in which the electrodes are separated by a distance d and let us assume that the electrodes are sufficiently distant from the reactor inlet for the forced laminar flow to be fully developed. Since the reaction occurs only at the electrodes, the concentration profile begins to develop at the leading edges of the electrodes. The thickness of the concentration boundary layer along the length of the electrode is assumed to be much smaller than the distance d between the plates, a condition that is usually satisfied in practice. 

Laminar Flow Normally, laminar flow occurs in closed ducts when Npe < 2100 (based on equivalent diameter = 4 x free area h-perimeter). Laminar-flow heat transfer has been subjected to extensive theoretical study. The energy equation has been solved for a variety of boundary conditions and geometrical configurations. However, true laminar-flow heat transfer very rarely occurs. Natural-convection effects are almost always present, so that the assumption that molecular conduction alone occurs is not valid. Therefore, empirically derived equations are most rehable. 

With terms III-V in Eq. 4.7 deleted, Eqs. 4.5-4.7, together with the x and y momentum equations, constitute the simplified equations of motion appropriate to natural convection problems. For constant T . and 7U, the boundary conditions on these equations are 0 = 1 and u = v = w = 0 on the body and 0 = 0 far from the body. Steady-state laminar solutions to these equations are those that are obtained after setting the time partials (i.e., terms containing partial derivatives with respect to t ) in the equations equal to zero. Steady-state turbulent... 

Uniform Heat Flux. For laminar flow in a horizontal tube where uniform heat flux is applied at the outer boundary of the tube, the bulk temperature Tb, increases linearly in the axial direction. To maintain the heat flow to the fluid, the wall temperature must remain higher than the fluid temperature, and under these conditions a fully developed natural convection motion becomes established in which velocity and temperature gradients become independent of the axial location. Because the fully developed Nusselt number for laminar pure forced convection is small (Nuf —> 4.36), the buoyancy-induced mixing motion can greatly enhance the heat transfer. 

As highlighted by Shah and London [2], a natural tendency exists to use in convection problems a large number of different sets of dimensionless groups based on the analyst s particular normalization of the differential equations and boundary conditions. An effort to standardize the definitions of dimensionless groups for laminar flows through channels was made by Shah and London [2] some years ago. In this section, the normalization of the convection problems proposed by Shah and London will be followed. 

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