# Page 7 in the transient case the fourth order runge

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In the transient case, the fourth order runge-kutta method was used to ensure error minimization. None of the equations contain a time variable, so the runge-kutta method uses the function vector from the steady-state solution to output the transient concentration profile. Results and Discussion Solutions to the Steady-State Case For the steady-state problems, 40 nodes were used to determine the solutions with dx = 1/(40-1) = 1/39 because this step size produces and error of = 6 x 10 4 which is less than 0.5% of the average value of y A and y B for each of the parts. A larger number of nodes will take more time for computation, and a smaller number of nodes will produce less accurate graphs. The tolerance for convergence of Newton’s method was set to 10 -5 because MATLAB displays values to 10 -4 , so that no additional error of notable size will incur from not allowing Newton’s method to converse fully to a solution. Page 8
Case A: Graph 1 below displays the solution for the steady-state case A obtained via Newtonian method iteration, as plot of concentration of A vs position. Graph 1: Concentrations at Steady-State (Case A)
Case B: Graph 2 below displays the solution for steady-state case B obtained via Newtonian method iteration, as a plot of concentration of B vs position. Page 9
Graph 2: Concentrations at Steady-State (Case B) Now, y A doesn’t decline as quickly as the previous case since gamma is smaller in this case, but y B declines quicker than in the previous case because it is being turned into species U, but there is no conversion of U to B since zeta is equal to zero. Case C: Graph 3 below displays the solution for steady-state case C obtained via Newtonian method iteration, as a plot of concentration of U vs position. Page 10
Graph 3: Concentrations at Steady-State (Case C) The difference between the solution in Graph 3 and that in Graph 2 is that species U can indeed react to form species B in Case C. This can be seen in the curve for y U that has a final vale at position=1, which is slightly smaller that its final value in case B, and y B ’s final value, which is slightly higher than in the graph of case C than it is in case B. Solutions to the Transient Case The runge-kutta method was used to integrate the system of ordinary differential equations in time, and the number of nodes used here was also 40. The smallest time step that is possible here is determined as: with and set to 2. So, the step size is 6.57 x 10 -4 and = 1.3 x 10 -3 ; a step size smaller than this would increase computational time without a substantial increase in accuracy. To show when the transient solution reached steady-state, one has to track when the transient solution was within 99% of the known steady-state solution. A while loop was used to Page 11