Monte Carlo Integration in Excel

Sean Gant

Jay Lee

A

y

x

B

f(x)

Monte Carlo Integration

Monte Carlo Integration

A

y

x

B

f(x)

Monte Carlo Integration

A

y

x

B

f(x)

B

total

below

N

NABdxxf

0)(

Gas-phase elementary reaction:2A → B

Isothermal reaction in a PFR with no pressure drop

Only A is fed, P0 = 8.2 atm, T0 = 500K, ε = -0.5

CA0 = 0.2 mol/dm3, k = 10 dm3/mol-s, vo = 25 dm3/s

QUESTIONWhat volume is needed to achieve 90% conversion?

Mole Balance:

Rate Law:

Stoichiometry:

Combine:

X

AA r

dXFV

0

0

2AA kCr

)1(

)1(

0

0

Xv

XF

v

FC AAA

dXX

X

kC

FV

X

A

A

0 2

2

20

0

)1(

)5.01(

Setting Up Excel Spreadsheet

• What are your inputs? (Using “Rand()” Function)– X: Your independent variable (range you are

integrating)– Y: Dependent variable ( from 0 to max over X-

range)

• What is your defined area?– L x W of “box” (multiply the two ranges)

• What constitutes a hit?– If random point lies below function then “hit”

Setting up Excel Spreadsheet

Setting up Excel Spreadsheet• Random Number Generator

– Rand()*(b-a)+a • Generate a number between a and b

• The “Status”– If(X<f(X), “below”, “above”)

• If value of f(X) > Y then mark “below”

• The “Count”– If(Status=“below”,1,0)

• If value of status is “below” then count a “hit”

• Making sense of data– Take total number of hits and divide by total number of points to

find fraction under function– Multiply calculated fraction by area of “box” to find the area

under the curve. In this case this value is the volume of your PFR. 44.6 dm3 as compared to 45.3 dm3 found by solving analytically.

Setting up Excel Spreadsheet