Homework 06

From CBE179 Wiki
Jump to: navigation, search

Homework 06 - Chemical Vapor Deposition

Due Wednesday, 2014.10.22.

File:CBE179-Fall2014-Hwk06.doc

Thiele Modulus and Effectiveness Factors

Like most kinds of film growth (e.g. the Deal-Groves model of silicon oxidation), chemical vapor deposition is always a coupled reaction-diffusion problem. Make sure you understand the dimensionless groups that describe the relative rates in these kinds of processes:

  1. http://en.wikipedia.org/wiki/Thiele_modulus
  2. http://en.wikipedia.org/wiki/Damkohler_numbers

When calculating the Thiele modulus, watch your units!

# Φ² = (k/D)(R²/l)              # unicode 03A6
R = 0.15                        # [=] m
l = 0.0019                      # [=] m
k = 0.001                       # [=] m²/s
D(T) = 0.000062*(T/273.15)^1.5  # [=] m²/s

psi(T) = sqrt((k/D)*((R^2)/l))
show('the Thiele modulus is: ' + str(psi(896)))

	the Thiele modulus is: 5.67007117112554

Need to evaluate a Bessel function? Sage Math is an all-purpose open source computer algebra system for doing every kind of math. It's easy to manipulate Bessel functions with the Bessel module:

  1. http://en.wikipedia.org/wiki/Bessel_function
I = Bessel(1,typ='I')
eta = (1/3)*(1+2*I(0.5*psi(896))/I(psi(896)))
show('the effectiveness factor is: ' + str(eta))

	the effectiveness factor is: 0.383623051697767

Regardless of the value you calculated for the Thiele modulus, the effectiveness factor must logically be somewhere between 0 and 1. Using a computer algebra system, it's trivial to check that the range and functional dependence of your functions make sense:

eta(T) = (1/3)*(1+2*I(0.5*psi(T))/I(psi(T)))
plot(eta,T,0,10000)

Effectiveness.png