446_01c_test2_soln

# 446_01c_test2_soln - METR 446 Test 2 Name Solution 1(30 pts...

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METR 446 Test # 2 Name: Solution October 16, 2001 1) (30 pts) If a spherical ice crystal and a spherical liquid drop with the same radius are in a cloud with a supersaturation with respect to water of 0.3% and a temperature of –20 o C, how much faster will the ice crystal grow due to condensation? You can neglect the impact of curvature on equilibrium vapor pressure, and the impact of condensational heating at the crystal and drop surface. The rate of growth due to condensation is directly related to the difference between the surrounding vapor pressure and the equilibrium vapor pressure at the surface of the droplet / crystal. To calculate this difference you have to use the Clausius-Clapyeron Equation for both liquid water and ice. The solution below is lengthy only because I have shown even the relatively obvious intermediate steps, and simplified the final equation as much as possible. Since the ice crystal was assumed to be spherical, its diffusion capacitance is equivalent to its radius (i.e., as in the sample problem). [ ] [ ] [ ] 62 ) 003 . 0 )( 1 . 127 ( 3 . 103 1 . 127 1 1 . 127 253 1 273 1 461 10 5 . 2 exp ) 611 ( 1 1 exp 3 . 103 253 1 273 1 461 10 83 . 2 exp ) 611 ( 1 1 exp 1 1 ) 1 ( ) ( 1 ) 1 ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( 4 ) ( 4 6 6 = + = = × = = = × = = + = + + + + = = = = Pa Pa dt dm dt dm Pa K K Pa T T R L e e Pa K K Pa T T R L e e S e e e S e e e S e e e e e e e e e e e e e e e e e e e r e e C T R e e Dr T R e e DC dt dm dt dm

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