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Rate of convergence in an algorithm

  1. Oct 13, 2009 #1
    1. The problem statement, all variables and given/known data
    Calculate the operating temperature for a flash unit.

    [tex]osf_2= 0.9[/tex] [tex]P = 750 mmHg[/tex]
    [tex]f_1=30, A_1=15.9, B_1=2788.5, C_1=-52.3[/tex]
    [tex]f_2=50, A_2=16.0, B_2=3096.5, C_2=-53.7[/tex]
    [tex]f_3=40, A_3=16.1, B_3=3096.5, C_3=-59.4[/tex]

    [tex]T_{guess} = 400K[/tex]

    calculate [tex]P_i^0=exp\left(A_i-\frac{B_i}{T_guess+C_i}\right)[/tex] for all components

    calculate [tex]K_i=\frac{P_i^0}{P}[/tex] for all components

    calculate [tex]\alpha_\frac{i}{2}=\frac{K_i}{K_2}[/tex] for all components

    calculate [tex]osf_i=\frac{\alpha_\frac{i}{2}\cdot osf_2}{1+\left(\alpha_\frac{i}{2}-1\right)\cdot osf_2}[/tex]

    calculate
    [tex]v_i=osf_i\cdot f_i[/tex]
    [tex]l_i=(1-osf_i)\cdot f_i[/tex]
    [tex]y_i=\frac{v_i}{V}[/tex]
    [tex]x_i=\frac{l_i}{L}[/tex]
    for all components

    calculate a new [tex]K_2^{new}=\frac{1}{\alpha_\frac{1}{2}\cdot x_1+\alpha_\frac{2}{2}\cdot x_2+\alpha_\frac{3}{2}\cdot x_3}[/tex]

    calculate [tex]K_1^{new}=\alpha_\frac{1}{2}\cdot K_2^{new}[/tex] and [tex]K_3^{new}=\alpha_\frac{3}{2}\cdot K_2^{new}[/tex]

    calculate new [tex]P_i^0=P \cdot K_i^{new}[/tex] for all components

    calculate new [tex]T_i=\frac{B_i}{A_i-\log{P_i^0}}-C_i[/tex]

    When comparing the new temperatures to the guessed temperature, [tex]T_3[/tex] will always be closer. As i understand [tex]T_3[/tex] converges faster.

    Prove this analytical.


    3. The attempt at a solution


    [tex]\frac{K_1^j}{K_1^{j-1}} = \frac{K_2^j}{K_2^{j-1}} = \frac{K_3^j}{K_3^{j-1}}[/tex] where j indicates new value and j-1 from the iteration before

    From the definition of [tex]K_i[/tex], this eq. can be written.

    [tex]\frac{P_1^j}{P_1^{j-1}} = \frac{P_2^j}{P_2^{j-1}} = \frac{P_3^j}{P_3^{j-1}} = Q[/tex]

    I then define this equation, which is the square of the deviation of the new temperature to the previous temperature.


    [tex]y\left( P_i^j, f \right) = \left( T_{guess} - \frac{B_i^j}{A_i^j - \ln(P_i^j)} + C_i^j \right)^2 = \left( T_{guess} - \frac{B_i^j}{A_i^j - \ln(f + P_i^{j-1})} + C_i^j \right)^2[/tex]

    where

    [tex]f = P_i^j - P_i^{j-1} = Q \cdot P_i^{j-1} -P_i^{j-1} = P_i^{j-1} \cdot (Q-1)[/tex]

    My idea was to show that this square of deviation always is smaller for component 3 but I can't get further.
    Im new to comparing rate of convergence so maybe my solution is in the totally wrong direction. I appreciate any help!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
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