Modelling and scientific computing

Process modelling slides

<pdfreflow> class_date = 13 September 2010 (slides 1 to 8)
15 September 2010 (slides 9 to 15)
16 September 2010 (slides 16 to 19)
20 September 2010 (slides 20 to the end) button_label = Create my course notes! show_page_layout = 1 show_frame_option = 1 pdf_file = A-Modelling-20-Sept-2010.pdf </pdfreflow>

Approximation and computer representation

<pdfreflow> class_date = 22 and 23 September button_label = Create my course notes! show_page_layout = 1 show_frame_option = 1 pdf_file = A-Approximation-and-roundoff-22-Sept-2010.pdf </pdfreflow>

• More about the Ariane 5 rocket 16-bit overflow
• Python code used in class for ...

calculating relative error	working with integers
import numpy as np y = 13.0 n = 3 # number of significant figures rel_error = 0.5 * 10 ** (2-n) # relative error calculation x = y / 2.0 x_prev = 0.0 iter = 0 while abs(x - x_prev)/x > rel_error: x_prev = x x = (x + y/x) / 2.0 print(abs(x - x_prev)/x) iter += 1 print('Used %d iterations to calculate sqrt(%f) = %.20f; ' 'true value = %.20f\n ' % (iter, y, x, np.sqrt(y)))	import numpy as np print(np.int16(32767)) print(np.int16(32767+1)) print(np.int16(32767+2)) # Smallest and largest 16-bit integer print(np.iinfo(np.int16).min, np.iinfo(np.int16).max) # Smallest and largest 32-bit integer print(np.iinfo(np.int32).min, np.iinfo(np.int32).max)

Practice questions

1. From the Hangos and Cameron reference, (available here] - accessible from McMaster computers only)
   • Work through example 2.4.1 on page 33
   • Exercise A 2.1 and A 2.2 on page 37
   • Exercise A 2.4: which controlling mechanisms would you consider?
2. Homework problem, similar to the case presented on slide 18, except
   • Use two inlet streams \(\sf F_1\) and \(\sf F_2\), and assume they are volumetric flow rates
   • An irreversible reaction occurs, \(\sf A + 3B \stackrel{r}{\rightarrow} 2C\)
   • The reaction rate for A = \(\sf -r_A = -kC_\text{A} C_\text{B}^3\)
   1. Derive the time-varying component mass balance for species B.
      • \( V\frac{dC_B}{dt} = F^{\rm in}_1 C^{\rm in}_{\sf B,1} + F^{\rm in}_2 C^{\rm in}_{\sf B,2} - F^{\rm out} C_{\sf B} + 0 - 3 kC_{\sf A} C_{\sf B}^3 V \)
   2. What is the steady state value of \(\sf C_B\)? Can it be calculated without knowing the steady state value of \(\sf C_A\)?
      • \( F^{\rm in}_1 C^{\rm in}_{\sf B,1} + F^{\rm in}_2 C^{\rm in}_{\sf B,2} - F^{\rm out} \overline{C}_{\sf B} - 3 k \overline{C}_{\sf A} \overline{C}^3_{\sf B} V \) - we require the steady state value of \(C_{\sf A}\), denoted as \(\overline{C}_{\sf A}\), to calculate \(\overline{C}_{\sf B}\).

More exercises to come