Tutorial 2 - 2010

From Process Model Formulation and Solution: 3E4
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Due date(s): 22 September 2010
Nuvola mimetypes pdf.png (PDF) Tutorial questions (updated)
Other instructions Hand-in at class.

Tutorial objectives

  • Some questions to help you feel comfortable deriving model equations for actual chemical engineering systems.
  • Brute force solving of equation systems. The rest of the course will focus on better ways to solve these equations.
  • Interpreting source code written on paper.

Recap of tutorial rules

  • Tutorials can be done in groups of two - please take advantage of this to learn with each other.
  • Tutorials must be handed in at the start of class on Wednesday. No electronic submissions - thanks!

Question 1 [2]

Note

Parts 1 and 2 of this question were from the 2006 final exam (slightly modified). It was worth 10% of the 3 hour exam

Consider a mixing tank with fluid volume \(V\) where fluids A and B with densities \(\rho_A\) and \(\rho_b\), specific heat capacities \(C_{p,A}\), \(C_{p,B}\) and temperatures \(T_A\) and \(T_B\) are the inlet streams at volumetric flow rates \(F_A\) and \(F_B\). The outlet stream is at a flow rate \(F_A+F_B\). The specific heat capacity and density of the outlet streams is given by \(C_p=(C_{p,A}F_A+C_{p,B}F_B)/F\) and \(\rho=(\rho_AF_A+\rho_BF_B)/F\). The fluid also looses heat from the tank at a rate \(q=k_T(T-T_\text{wall})\) where \(T_\text{wall}\) is the constant tank wall temperature, \(k_T\) is a constant and \(T\) denotes the current fluid temperature.

  1. Using 3-step modelling approach shown in class, derive a dynamical balance describing the time-dependent exit stream temperature.

  2. Can the steady state exit stream temperature be higher than both \(T_A\) and \(T_B\)? Explain.

  3. Calculate, by-hand, the steady-state exit temperature, using that

    • \(V\) = 10 m3
    • \(\rho_A\) = 1200 kg/m3 and \(\rho_b\) = 950 kg/m3
    • \(C_{p,A}\) = 2440 J/(kg.K) and \(C_{p,B}\) = 3950 J/(kg.K)
    • \(T_A\) = 320 K and \(T_B\) = 350 K and \(T_\text{wall}\) = 300K
    • \(F_A = F_B\) = 0.05 m3/s
    • \(k_T\) = 200 W/(m2.K) \(\times\) 24 m2 = 4800 W/K

Question 2 [2]

Note

You don't need to write any code for this question.

Consider the reaction

\[{\sf P_2I_4} + n\,{\sf P_4} + p\,{\sf H_2O} \longrightarrow 4\,{\sf PH_4I} + q\,{\sf H_3PO_4}\]

where \(n\), \(p\) and \(q\) denote the stoichiometric coefficients for \(\sf P_4\), \(\sf H_2O\) and \(\sf H_3PO_4\) respectively.

  1. Derive the equations necessary to solve for \(n, p\), and \(q\) by equating atoms of \({\sf P}\), \({\sf H}\), and \({\sf O}\) on the reactant and product sides.

  2. In the next section of the course we will use Guass Elimination to solve these equations. For now though, let's describe a brute force approach. First, complete the two lines of this MATLAB function:

    function total_error = equation_error( n, p, q )
    
    % Given the values of n, p, and q, calculate the error of each balance equation.
    % Returns the sum of squares of the errors.
    
    error_1 = _______________  % from the P-balance
    error_2 = 2*p - 3*q - 16;  % from the H-balance
    error_3 = _______________  % from the O-balance
    
    total_error = (error_1)^2 + (error_2)^2 + (error_3)^2;
    
    end  % end of function
    

    or complete this Python function:

    def equation_error(n, p, q ):
        """
        Given the values of n, p, and q, calculate the error of each of the 3 equations.
        Returns the sum of squares of the errors.
        """
        error_1 = _______________
        error_2 = 2*p - 3*q - 16
        error_3 = _______________
    
        return error_1**2 + error_2**2 + error_3**2
    
  3. Since we known that \(n, p\), and \(q\) must be positive, we can construct a set of 3 nested for-loops, as shown below in MATLAB and Python. Describe in plain English what the code does.

    In MATLAB:

    smallest = 0.0;
    largest = 14.9;
    step_size = 0.1;
    vector = smallest : step_size : largest;
    
    % How many elements in each vector?
    num = length(vector);
    
    errors = zeros(num, num, num);
    
    index_n = 0;
    index_p = 0;
    index_q = 0;
    for n = smallest : step_size : largest
        index_n = index_n + 1;
        for p = smallest : step_size : largest
            index_p = index_p + 1;
            for q = smallest : step_size : largest
                index_q = index_q + 1;
    
                % Calculate the error at this value of n, p and q:
                errors(index_n, index_p, index_q) = equation_error(n, p, q);
    
            end
            index_q = 0;
        end
        index_p = 0;
    end
    [min_error, min_index] = min(errors(:))
    [index_n, index_p, index_q] = ind2sub([num, num, num], min_index);
    disp(['Solution at ', num2str([vector(index_n), vector(index_p), vector(index_q)])])
    

    In Python:

    import numpy as np
    
    smallest = 0.0
    largest = 15.0
    step_size = 0.1
    vector = np.arange(smallest, largest, step_size)
    
    # How many elements in each vector?
    num = len(vector)
    
    errors = np.zeros( (num, num, num) )
    
    for index_n, n in enumerate(vector):
        for index_p, p in enumerate(vector):
            for index_q, q in enumerate(vector):
    
                # Calculate the error at this value of n, p and q:
                errors[index_n, index_p, index_q] = equation_error(n, p, q)
    
    # Which combination had the smallest error?
    min_index = np.argmin(errors)
    index_n, index_p, index_q = np.unravel_index(min_index, (num, num, num))
    n, p, q = vector[index_n], vector[index_p], vector[index_q]
    print(n, p, q)
    
  1. How many times will the function equation_error be called?
  2. What will this function output be if \((n, p, q) = (1.0, 9.2, 2.5)\)?

Bonus question [0.5]

Using the code given in question 2, report what the min_index variable is and what are the values of \(n, p\), and \(q\) which give minimum error to the set of equations. How long did it take to find the solution of this simple linear equation system?