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Integral Method - Chemical Reaction Engineering - Lecture Slides, Slides of Engineering Chemistry

Chemical Reaction Engineering covers Heat Effects, Isothermal Design, Stoichiometry, Rate Laws, Mole Balance and many other topics. This lecture includes: Integral Method, Nonlinear Least Regression, Rate Law, Stoichiometry, Mole Balances, Batch Reactor, Integral Method, Differential Method, Non-Linear Least-Square Analysis, Residuals

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2012/2013

Uploaded on 08/20/2013

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Lecture 11

Block 1: Mole Balances

Block 2: Rate Laws

Block 3: Stoichiometry

Block 4: Combine

Determining the Rate Law from Experimental Data

Integral Method

Differential (Graphical) Method

Nonlinear Least Regression

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Lecture 11 ^ Block 1:^ Mole Balances ^ Block 2:^ Rate Laws ^ Block 3:^ Stoichiometry ^ Block 4:^ Combine ^ Determining the^ Rate Law 1

from Experimental Data ^ Integral Method ^ Differential (Graphical) Method ^ Nonlinear Least Regression

Integral Method Consider the following reaction that occurs in a constantvolume^ Batch Reactor 2

: (We will withdraw samples and record the concentration of A as a function of time.)^ A

^ Products^ dN^ A ^ rVA^ dt^

Mole Balances:

  r  kC (^) A A

Rate Laws:

V^ ^ V^0

Stoichiometry:

dC^^  A  ^ kC^ Adt^

Combine:

Integral Method Guess and check for 4

α^ = 0, 1, 2 and check against experimental plot.

ktktCC CAktCrAA CAAA        0 0 0

(^11) (^210)    ln t C^ A ln(C^ /C^ )A0^ A^ t

1/C^ A t docsity.com

Differential Method 5

dC   A^  kC^ A  dt  dC  A Ck lnlnln ^ Adt  Taking the natural log of

 dC^ A^ ^ vs^ C^ ^ ^ A  dt The reaction order can be found from a ln-ln plot of:dC^ A ^ dt C C AP^ A dCA  dtSlope =P

α ln ln

dC  A ^  dt  pk ^  C^ Ap docsity.com

Three ways to determine (-dC

/dt) from concentration-time dataA^ ^ Graphical differentiation ^ Numerical differentiation formulas ^ Differentiation of a polynomial fit to the data 1. Graphical  C^ A^ ^  t 7

 C^ A ^  t dC^  A   dt ^0 dC^  A  dt  t^1 dC^  A  dt  t^20 The method accentuates measurement error!

t^ t 21

t

C^  A Find f(t) of using equal area differentiation^ t  C 1 0.7A

0.5^ 0.

-dC^ /dt^ 0.35^ 0.25A^

0.175^ 0.

Plot (–dC^ /dt) as a function of CA^

A ln Slope =C^ A

α dC^ /dtA^

Example – Finding the Rate Law 10

Example – Finding the Rate LawChoose a point, p, and find the concentration andderivative at that point to determine k. dC  A ^  dt  p k ^  C^ Ap 11

ln C dC^ /dtA^ Slope =^ α p^ lnA dC^  A  dt  CA^ p

Non-Linear Least-Square Analysis^ For concentration-time data, we can combine themole balance equation for 13

to obtain: Rearranging to obtain the calculated concentrationas a function of time, we obtain:

  r  kC (^) A A dC  AkC   Adt (^0) t C C   (^0) A A 1 1  (^)  (1 ) C C kt    (^0) A A 1 1/(1^ ) ^  [ (1 )^ ] C C C kt     (^0) Ac A A

Non-Linear Least-Square Analysis Now we could use Polymath or MATLAB to find the values of 14

α^ and k that would minimize the sum of squares of differences between themeasured (C) and calculated (CAm^

) concentrations.Ac That is, for N data points,Similarly one can calculate the time at a specified concentration, t

c and compare it with the measured time, t

, at that same concentration.m^ That is, we find the values of k and

α^ that minimize:

Non-Linear Least Squares Analysis

N^22 s  C^ ^ C^^ ^  ^ Ami^ Aci i ^1

1 ^ C  C ^1 ^  ktAmi A^^0

1 1   i

^

N  i ^1

We find the values of alpha and k which minimize s