soln07 - Homework 7 26 October 2006 Problem 1 There are a...

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± ± Homework 7 26 October 2006 Problem 1 There are a few ways to solve this problem, one could formulate it as a ODE or as a DAE (We are going to formulate the problem as a ODE). The governing equation in the problem are given below. ± ± 2 π π ± dh D 2 v 2 = [ D ( h )] 2 ± ± ± ± (1) 4 p p 4 ± dt ² ³ 2 ´ µ · ² ³ · 1 ρ dh + p atm + ρgh 1 ρv p 2 + p atm + ρg ( L p ) = K L π D p 2 v p 2 + f D L p π D p 2 v p 2 2 dt 2 4 D p 4 (2) where the friction factor f D can be calculated as follows. 24 f D = Re ´ µ Re < 2100 1 = 2log 10 ( e/D p ) + 2 . 51 Re > 4000 (3) f D 3 . 7 f D Re In this formulation we evaluate the value of v p using equation (1) in terms of h and dh . The friction factor can be calculated using v p or in terms of h dt and dh . Ofcourse none of this needs to be done analytically. Everything is dt coded as a matlab function. Using this friction factor and v p , which are both functions of h and dh , we can get one single equation which can be written dt as g ( h, dh ) = 0. This equation can be solved to obtain a value of dh for any dt dt given value of h . The scheme outlined here can be used to solve the problem and calculate the height of the tank at any given point in time. We can stop the ode from integrating on after the height has reached 0, we use events function.To calculate the volume of the liquid in tank we use the equation dV π [ D ( h )] 2 = dh 4 1 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].
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and integrate it. For the implimentation of this method see the function problem1 . A sample run with the program is shown below >> [t,h]=problem1; The tank gets empty at time: 2529.8923 s The height and volume of water in the tank are shown below. The height in the tank 0 500 1000 1500 2000 2500 3000 Time (sec) The volume of water in tank 0 500 1000 1500 2000 2500 3000 Time(sec) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Height (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume (m 3 ) 2 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].
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10.34 – Fall 2006 Homework #7 - Solutions Problem 2 – CSTR Optimization (Beers’ text 5.B.4) In this problem, we wished to find the optimal operating condition for a CSTR in order to produce the maximum amount of [C]. We were given a reaction mechanism and the necessary rate constants, and we asked to vary the initial concentrations of A and B, the reactor temperature between 298 K and 335 K, and the reactor volume from 10 L to 10000 L.
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soln07 - Homework 7 26 October 2006 Problem 1 There are a...

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