Transport Phenomena · Momentum transport and viscous flow
A spherical crystal of initial radius 6.00 mm and density 1200 kg/m^3 dissolves in a…
Problem
A spherical crystal of initial radius 6.00 mm and density 1200 kg/m^3 dissolves in a well-mixed vessel containing 6.00 × 10^-4 m^3 of liquid. Initially the bulk solute concentration is 0.100 kg/m^3, and the saturation concentration at the crystal surface is 1.80 kg/m^3. The liquid has density 1000 kg/m^3, viscosity 1.00 × 10^-3 Pa·s, and solute diffusivity 1.10 × 10^-9 m^2/s. Agitation produces a constant relative speed of 0.060 m/s between the crystal and liquid. At every radius R, take Sh = k_m(2R)/D = 2 + 0.60 Re^(1/2) Sc^(1/3), with Re = rho(0.060)(2R)/mu and Sc = mu/(rho D). The governing balances are -rho_s 4pi R^2 dR/dt = 4pi R^2 k_m(C_s-C_b) and V dC_b/dt = 4pi R^2 k_m(C_s-C_b). Determine the time for the radius to decrease to 3.00 mm and the nonzero equilibrium radius approached if agitation continues indefinitely. Assume constant properties, a spherical crystal, and no kinetic resistance at the interface.
Hint
Combine the two balances first to express the bulk concentration directly as a function of radius.
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