Free convection in a slot, a fluid of constant viscosity, with density given by Eq. 1 1.3-1, is confined in a rectangular slot. The slot has vertical walls at x = + B, y = + W, and a top and bottom at z = + H, with H > > W > > B. The walls are non isothermal, with temperature distribution Tw = T + Ay, so that the fluid circulates by free convection. The velocity profiles are to be predicted, for steady laminar flow conditions and small deviations from the mean density p.
(a) Simplify the equations of continuity, motion, and energy according to the postulates: v = δzvz(x, y), ∂2vz/∂y2 << ∂2vz/∂x2 and T = T(y). These postulates are reasonable for slow flow, except near the edges y = + W and z = + H.
(b) List the boundary conditions to be used with the problem as simplified in (a).
(c) Solve for the temperature, pressure, and velocity profiles.
(d) When making diffusion measurements in closed chambers, free convection can be a serious source of error, and temperature gradients must be avoided. By way of illustration, compute the maximum tolerable temperature gradient, A, for an experiment with water at 20°C in a chamber with B = 0.1 mm, W = 2.0 mm, and H = 2 cm, if the maximum permissible convective movement is 0.1% of H in a one-hour experiment.