Fuel cells similar to the PEM cell of Example 1.5 oper- ate with a mixture of liquid water and methanol instead of hydrogen; the anode is placed in direct contact with the liquid fuel. Oxygen (species A) is delivered to the exposed cathode by free convection. Hence, no fans or

(a) Estimate the heat loss from the surface of the bath by radiation exchange with the surroundings.

(b) Calculate the Grashof number using Equation 9.65, which can be applied to natural convection flows driven by temperature and concentration gradients. Use a characteristic length L that is appropriate for the exposed surface of the water bath.

(c) Estimate the free convection heat transfer coeffi- cient using the result for GrL obtained in part (b).

(d) Invoke the heat and mass transfer analogy and use an appropriate correlation to estimate the mass transfer coefficient using GrL. Calculate the water evaporation rate on a daily basis and the heat loss by evaporation.

(e) Calculate the total heat loss from the surface, and compare the relative contributions of the sensible, latent, and radiative effects. Review the assumptions made in your analysis, especially those relating to the heat and mass transfer analogy.

Pumps are needed to operate the device. The power output of passive, direct methanol fuel cells (DMFCs) can become mass transfer limited, since the electric current produced by the DMFC is related to the rate at which oxygen is consumed at the cathode by the expres- sion I = 4nAFMA, where F is Faraday's constant, F= 96,489 coulombs/mol. Consider a passive DMFC with a 120 mm X 120 mm membrane. Determine the maxi- mum possible electric current produced by the DMFC when the oxygen mass fraction at the cathode is mA,s = 0.10 for cases where the cathode is facing up or is vertical. As a first approximation and to illustrate the sensitivity of the device to its orientation relative to the vertical direction, assume buoyancy forces are dom- inated by the difference in density associated with the change in the oxygen mass fraction between the cath- ode surface and the quiescent environment, which is atmospheric air at Too = 25°C. Assume the quiescent air is composed of nitrogen and oxygen, with an  oxygen mass fraction mA,oo = 0.233.