homework1solution - MatSc503 Spring 2002 Solutions to...

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MatSc503 – Spring, 2002 Solutions to Homework #1 1) Hydrogen at 1.0 MPa is to be stored at 400 o C in outer space, in a thin wall (0.1 mm) spherical iron tank of 0.1 m radius. Please calculate the rate of pressure drop ( MPa/s ) as a result of diffusion of hydrogen through the wall. Take D = 10 -8 m 2 /s, and assume that the concentration of hydrogen in the iron at either side of the wall is the equilibrium solubility given by the equation c H ( p ) = 10 -5 p 1/2 gm H /gm Fe ( p in MPa ) ( gm – gram ) Solution: This is a quasi-steady state problem, i.e., the local concentration of H in the wall varies with time, but at any given moment of time, the total flux of H is a constant through the wall thickness, ( ) constant 1 4 4 2 = = = r d dc D dr dc D r J H π π Let’s use R 1 and R 2 to denote the inside and outside wall radii, and C 1 and C 2 the inside and outside wall concentrations, respectively, we have 2 1 1 2 2 1 1 2 1 2 4 1 1 4 R R C C DR R R C C D J H = π π From conservation of mass, we have dt dp RT V dt dn R R C C DR J H = = = 2 1 1 2 2 1 4 π where n is the total number of moles of H 2 in the tank, V is the volume of the tank, R is the gas constant, T is temperature, and p is the H 2 gas pressure which decreases slowly as a function of time. In the above equation, the ideal gas law is assumed for H 2 .
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