20130211 Problem 4-60

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Problem Statement: Methanol is synthesized from carbon monoxide and hydrogen in a

catalytic reactor. The fresh feed to the process contains 32.0 mol% CO, 64.0 mol% H2, and 4.0

mol% N2. This stream is mixed with a recycle stream in a ratio 5 mol recycle/1 mol fresh feed to

produce the feed to the reactor, which contains 13.0 mol% N2. A low single-pass conversion is

attained in the reactor. The reactor effluent goes to a condenser from which two streams emerge: a liquid product stream containing essentially all the methanol formed in the reactor,

and a gas stream containing all the CO, H2, and N2 leaving the reactor. The gas stream is split

in two fractions: one is removed from the process as a purge stream, and the other is the recycle stream that combines with the fresh feed to the reactor.

Step 1: Write down any reactions taking place.

CO + 2H2 à CH3OH ξ ≡ mol CO/h

Step 2: Draw and label the process flow diagram.

Step 3: Identify any process specifications and write equations in terms of the variables on the

process flow diagram.

5 mol recycle

=

n

5

→ 5n

= n

Reactor n•1= 100 mol / h y_CO,1=0.320 y_H2,1=0.640 y_N2,1=0.040 y_CH3OH,1=0 Fresh Feed (1) Condenser Reactor Feed (2) Reactor Effluent (3) Gas Stream (4) Gas Product (6) Recycle (5) Liquid Product (7) n•2= y_CO,2= y_H2,2= y_N2,2=0.130 y_CH3OH,2=0 n•3= y_CO,3= y_H2,3= y_N2,3= y_CH3OH,3= n•4= y_CO,4= y_H2,4= y_N2,4= y_CH3OH,4=0 n•6= y_CO,6= y_H2,6= y_N2,6= y_CH3OH,6=0 n•CH3OH,7= n•5= y_CO,5= y_H2,5= y_N2,5= y_CH3OH,5=0

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Step 4: Identify the variables to solve for.

Find:

n

_CH3OH,7

n

₆

y

_CO,6

y

_H2,6

y

_N2,6

f

_sp,CO

=

y

CO,2

n

2

− y

CO,3

n

3

y

_CO,2

n

₂

f

_Overall,CO

=

y

CO,1

n

1

− y

CO,6

n

6

y

_CO,1

n

₁

Step 5: Choose to work with atomic balances or extent of reaction balances. Step 6 (atomic): Complete a degree of freedom analysis.

Overall: 5 unknowns (𝑛! 𝑦!",! 𝑦!!,! 𝑦!!,! 𝑛!"!!",!) 3 atom balances (C H N) 1 physical constraint (𝑦!",! + 𝑦!!,! + 𝑦!!,! = 1) 0 process specifications _______________________________________ 1 D of F Mixing: 7 unknowns (𝑛! 𝑦!",! 𝑦!!,! 𝑛! 𝑦!",! 𝑦!!,! 𝑦!!,!)

3 molecular balances (CO H2 N2)

2 physical constraint (𝑦!",! + 𝑦!!,! = 0.87; 𝑦!",! + 𝑦!!,! + 𝑦!!,! = 1) 1 process specifications (5𝑛!= 𝑛!) _______________________________________ 1 D of F Reactor: 8 unknowns (𝑛! 𝑦!",! 𝑦!!,! 𝑛! 𝑦!",! 𝑦!!,! 𝑦!!,! 𝑦!"!!",!) 3 atom balances (C H N) 2 physical constraints (𝑦_!",! + 𝑦_!!,!= 0.87; 𝑦_!",! + 𝑦_!!,! + 𝑦_!!,! = 1) 0 process specifications _______________________________________ 3 D of F Condenser: 10 unknowns (𝑛! 𝑦!",! 𝑦!!,! 𝑦!!,! 𝑦!"!!",! 𝑛! 𝑦!",! 𝑦!!,! 𝑦!!,! 𝑛!"!!",!)

4 molecular balances (CO H2 N2 CH3OH)

2 physical constraint (𝑦_!",!+ 𝑦_!!,!+ 𝑦_!!,! = 1; 𝑦_!",! + 𝑦_!!,! + 𝑦_!!,! = 1) 0 process specifications

_______________________________________ 4 D of F

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Step 7 (atomic): Write all of the equations for the selected subsystem. Overall C: 𝑦!",!𝑛!− 𝑦!",!𝑛!− 𝑛!"!!",! = 0 H: 2𝑦!!,!𝑛!− 2𝑦!!,!𝑛!− 4𝑛!"!!",! = 0 O: 𝑦_!",!𝑛_!− 𝑦_!",!𝑛_!− 𝑛_!"!!",!= 0 N: 2𝑦!!,!𝑛!− 2𝑦!!,!𝑛!= 0 PC: 𝑦_!",! + 𝑦_!!,! + 𝑦_!!,! = 1 Mixing CO: 𝑦!",!𝑛!+ 𝑦!",!𝑛!− 𝑦!",!𝑛!= 0 H2: 𝑦!!,!𝑛!+ 𝑦!!,!𝑛!− 𝑦!!,!𝑛! = 0 N2: 𝑦!!,!𝑛!+ 𝑦!!,!𝑛!− 𝑦!!,!𝑛! = 0 Total: 𝑛!+ 𝑛!− 𝑛!= 0 PC: 𝑦_!",! + 𝑦_!!,! = 0.87 𝑦!",! + 𝑦!!,! + 𝑦!!,! = 1 PS: 5𝑛_!= 𝑛_!

Step 8 (atomic): Identify a solution strategy, and update the degree of freedom analysis to

account for all variables that can be solved for.

Step 9 (atomic): Select a new subsystem and write equations to solve.

Condenser

CO: 𝑦!",!𝑛!− 𝑦!",!𝑛!= 0

H2: 𝑦!!,!𝑛!− 𝑦!!,!𝑛!= 0

N2: 𝑦!!,!𝑛!− 𝑦!!,!𝑛!= 0

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Step 10 (atomic): Solve 𝑛! = 500 𝑚𝑜𝑙/ℎ 𝑛_! = 600 𝑚𝑜𝑙/ℎ 𝑦!!,!= 0.148 𝑛_! = 27.03 𝑚𝑜𝑙/ℎ 𝑛!"!!",! = 24.33 𝑚𝑜𝑙/ℎ 𝑦_!!,! = 0.568 𝑦!",! = 0.284 𝑦!",!= 0.291 𝑦!",!𝑛!= 149.68 𝑚𝑜𝑙/ℎ 𝑓!",!"= 0.143 𝑓!"#$%&&,!" = 0.760

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Step 6 (extent): Define units for ξ

Step 7 (extent): Complete a degree of freedom analysis.

Overall: 6 unknowns (𝑛! 𝑦!",! 𝑦!!,! 𝑦!!,! 𝑛!"!!",! 𝜉)

1 physical constraint (𝑦!",! + 𝑦!!,! + 𝑦!!,! = 1)

0 process specifications

_______________________________________ 1 D of F

Mixing: 7 unknowns (𝑛_! 𝑦_!",! 𝑦_!!,! 𝑛_! 𝑦_!",! 𝑦_!!,! 𝑦_!!,!)

3 molecular balances (CO H2 N2)

2 physical constraint (𝑦!",! + 𝑦!!,! = 0.87; 𝑦!",! + 𝑦!!,! + 𝑦!!,! = 1)

1 process specifications (5𝑛!= 𝑛!)

_______________________________________ 1 D of F

Reactor: 9 unknowns (𝑛! 𝑦!",! 𝑦!!,! 𝑛! 𝑦!",! 𝑦!!,! 𝑦!!,! 𝑦!"!!",! 𝜉)

2 physical constraints (𝑦_!",! + 𝑦_!!,!= 0.87; 𝑦_!",! + 𝑦_!!,! + 𝑦_!!,! = 1) 0 process specifications

_______________________________________ 3 D of F

Condenser: 10 unknowns (𝑛! 𝑦!",! 𝑦!!,! 𝑦!!,! 𝑦!"!!",! 𝑛! 𝑦!",! 𝑦!!,! 𝑦!!,! 𝑛!"!!",!)

2 physical constraint (𝑦!",!+ 𝑦!!,!+ 𝑦!!,! = 1; 𝑦!",! + 𝑦!!,! + 𝑦!!,! = 1)

0 process specifications

_______________________________________ 4 D of F

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Step 8 (extent): Write all of the equations for the selected subsystem. Overall CO: 𝑦!",!𝑛!− 𝑦!",!𝑛!− 𝜉 = 0 H2: 𝑦!!,!𝑛!− 𝑦!!,!𝑛!− 2𝜉 = 0 N2: 𝑦!!,!𝑛!− 𝑦!!,!𝑛!= 0 CH3OH:−𝑛!"!!",! + 𝜉 = 0 PC: 𝑦_!",! + 𝑦_!!,! + 𝑦_!!,! = 1 Mixing

*Identical to equations on page 3.

Step 9 (extent): Identify a solution strategy, and update the degree of freedom analysis to

account for all variables that can be solved for.

Step 10 (extent): Select a new subsystem and write equations to solve. *Identical to equations on page 3.