Modelling and scientific computing

Process modelling slides

Please download the lecture slides. 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)

Approximation and computer representation

Please download the lecture slides. 22 and 23 September (updated)

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

Calculating relative error

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)))

Working with integers

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)

Working with floats

import numpy as np help(np.finfo) # Read what the np.finfo function does f = np.float32 # single precision, 32-bit float, 4 bytes f = np.float64 # double precision, 64-bit float, 8 bytes print('machine precision = eps = %.10g' % np.finfo(f).eps) print('number of exponent bits = %.10g' % np.finfo(f).iexp) print('number of significand bits = %.10g' % np.finfo(f).nmant) print('smallest floating point value = %.10g' % np.finfo(f).min) print('largest floating point value = %.10g' % np.finfo(f).max) # Approximate number of decimal digits to which this kind # of float is precise. print('decimal precision = %.10g' % np.finfo(f).precision)

Special numbers

import numpy as np # Infinities print(np.inf, -np.inf) # inf, number exceeds maximum value that # is possible with a 64-bit float: overflow print(np.float(1E400)) print(np.inf * -4.0) # -inf print(np.divide(2.4, 0.0)) # inf # NaN's a = np.float(-2.3) print(np.sqrt(a)) # nan print(np.log(a)) # nan # Negative zeros a = np.float(0.0) b = np.float(-4.0) c = a/b print(c) # -0.0 print(c * c) # 0.0 eps = np.finfo(np.float).eps # Create a number smaller than machine precision e = eps/3.0 # Question: why can we create a number smaller than eps? print(e) # Interesting property: non-commutative operations can occur # when working with values smaller than eps. Why? # The print out here should "True", but it prints "False" print((1.0 + (e + e)) == (1.0 + e + e))

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 available here