7/24/2019 151Lecture3-24-15

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The Cooper Union for the Advancement of Science and Art

Chemical Engineering Department

ChE 151 Lecture

3/24/15

Prof. Davis

7/24/2019 151Lecture3-24-15

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The Cooper Union for the Advancement of Science and Art

Chemical Engineering Department

Ch. 5.2: Eulers Method

a!lor sa!s "e ca# e$%a#d ! at so&e %oi#t ti:

( ) ( ),dy f t y for a t b with y adt = =

( ) ( ) ( ) ( ) ( ) ( )

2

' ''

2i

i i i i

t ty t y t t t y t y

= + +

'f "e defi#e a ste% si(e h ) ti*1+ tifor a#! i a#d

evaluate at a #ear,! %oi#t t i*1:

( ) ( ) ( )( ) ( )2

''

1 ,

2i i i i i

hy t y t hf t y t y

+ = + +

7/24/2019 151Lecture3-24-15

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The Cooper Union for the Advancement of Science and Art

Chemical Engineering Department

Eulers -Marchi# 0or&ula

'#ori# the error ter& "hich ro"s "ith h2

:( ) ( ) ( )( )1 ,i i i iy t y t hf t y t+ +

i#ce ! is a%%ro$i&ate let the #e" varia,le "i#dicate the a%%ro$i&atio# to !:

( ) ( ) ( )( )1 ,i i i iw y w t w t hf t w t + = +

i&%lif!i# #otatio#:

( ) ( )1 ,i i i i i iw w t w w hf t w+ = +@

7/24/2019 151Lecture3-24-15

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The Cooper Union for the Advancement of Science and Art

Chemical Engineering Department

si# Eulers Method

o i&%le&e#t the &ethod "e "ould defi#e avector of ti&es call it -t a#d the# calculate

a# a%%ro$i&atio# to ! -" ,! -&archi#

for"ards i# -ti&e fro& the -i#itial ti&e t6) a:

( )

( )

( )

( )

( )

00 0

0 0 01 1

2 2 1 1 1

1 1 2 2 2

,

2 , ,

1 ,

,

N N N N N

N

y tt wa

w hf t wt wa h

t t a h w w w hf t w

t wa N h w hf t w

t w

++ = = + = = + + +

MM M M

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The Cooper Union for the Advancement of Science and Art

Chemical Engineering Department

7ctivit!: Euler i# Pseudocode

i&e: 16 &i# to do 16 &i# discuss8rite a# alorith& i# %seudocode "hichi&%le&e#ts Eulers &ethod to solve the '9P

for the 1storder irreversi,le eth!l,e#(e#e

reactor:( ), 0 oEB f EB EB EB

dCk C C C

d

= = =

our out%ut should ,e a vector of uesses forCE;"hich ca# ,e i#ter%olated to et a#

a%%ro$i&ati# fu#ctio#. 7ssu&e !our ste%

si(e is s&all 16

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The Cooper Union for the Advancement of Science and Art

Chemical Engineering Department

7#s"er: Eulers Methodet = ) 166666 > et ? of ti&e %oi#ts = to evaluate

et t6) 6 tfi#al) 16 > et the i#itial/fi#al ti&e %oi#ts

h ) tfi#al+ t6 / = > Calculate the ste% si(e h ) 1e A#o"# i#itial E; co#c. rate co#st.

> Calculate the first a%%ro$i&atio# %oi#ts"1 ) CE;

6 > " vector is a# out%ut/a#s"er

t1 ) t6 > i&e vector is also a# out%ut

0or i ) 1:= + 1 > 8e alread! did the first o#e

"i*1 ) "i * hB@fB"i > 0i#d #e$t uess fro& %rev.

ti*1 ) ti * h > Aee% vector of ti&es

E#d 0or loo%

> 7#s"er is vector of uesses " a#d vector of ti&es t sa&e si(e