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Unformatted text preview: 10.34 Quiz 1, October 4, 2006 Solution Graded out of a total of 15 points + 1 bonus point 0 2 4 6 8 10 12 14 16 18 20 Frequency Average: 9.25 Std. Dev: 2.54 Histogram < 2 < 4 < 6 < 8 < 10 < 12 < 14 < 16 (a) Write a couple of Matlab functions that together compute the concentrations [P] and [Nutrients] (units: M = moles/liter), as well as the number of cells per liter, in the output stream when the system is operated at steady-state. Give numerical values for all the inputs. Do you think that scaling will be a problem? Explain and give an appropriate scaling factor if necessary. 10 points - 2 pts: general structure of the functions - 0.5 pts: Unit errors/mismatches - 4.5 pts: Each balance equation (1.5 pts each) - 0.5 pts: Writing the fsolve equations correctly as dX/dt = In Out + Gen Consum. This is essential to getting the sign of the Jacobian eigenvalues correct (but not the solution). - 2.5 pts: Scaling assessment (1 pts for scaling could be a problem; 1.5 pts for scaling factor) For this part, you were essentially asked to write a program that can solve the problem, giving all the numerical inputs necessary to run the function. These could be passed to the function as arguments, or included in the function/script body as parameter definitions. Two Matlab functions (one each for scaled and unscaled N cells ) are included at the end as examples of possible solutions (of course, this is not the only way to solve the problem, but it works). Executable .m files are also posted on MIT Se rve r. The basic approach should have involved using fsolve to solve a set of nonlinear equations. Other functions that could have been used were fzero (probably not a good choice), fminsearch , and fmincon . An ODE solver approach is also valid and would generally give you a stable solution (expect with a very unfortunate initial guess that is close to the unstable conditions). The three equations that needed to be solved were the dN cells /dt = 0 , the nutrient balance , and the product balance . These were given to you for the most part in the supplied functions, but you needed to define the nutrient consumption rate and the P production rate in terms of the known system properties; the resulting equations are: (be sure that you convert the flow rate to L/sec ) Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. k N Nutrients dN cells 1 cells [ ] V flow N cells = = dt 1 + c Nutrients 1 + d P V ( 1 [ ])( [ ]) rxtr [ ] 0 V ([ Nutrient ] [ Nutrient k N + c ( d Nutrients = = flow In ]) 2 cells 2 Cell Multiplication ) dt cells d P dt [ ] = = 0 k N ( 3 1 + c [ exp Nutrients ( d P [ ] ]) ) ([ Nutrients ] 0.01 ) 2 V [ ] P flow 1 Scaling could be a problem in this scenario due to the large differences in the inherent system variables, but does not make the system intractable. variables, but does not make the system intractable....
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