MB4 Mass Balance in Transient State

Mass Balance: Transient State
MB4
Chemical Engineering
www. Chemical Engineering Guy .com

Content
• Section 1
– Theory
• Transient State Review
• Accumulation Concept
• General Methodology for Transient State
• Differential and Integral MB
• Section 2
– Exercises
• Mass Balances (Analytical, “by hand”)
• Mass Balances (Computer solved)

Section 1
• Theory
– Transient State Review
– Accumulation Concept
– General Methodology for Transient State
– Differential and Integral MB

Review: Transient State
• The value of any system variable changes with
time
– Process Variables such as:
• Pressure, Temperature, Volume
• Concentration, Conversion
• Mass flow, Mole Flow, Volume flow
• Level and many others

• Some examples, Process Variables that change with time:
– Viscosity of mixture in a reactor
– Concentration/Conversion of a product line
– Mass flow of product A
– Total mole flow of a line
– Level of a tank
– Total volume
– Change in pressure of a system
– Temperature increase of a mixture
– Density of feed change
– Type of Reactions (kinetics)
– Velocity in pipes

• Typical processes for Transient State:
– Batch
– Semi-batch
• But NOT ALL Transient State are batch/semi
• Even Continuous Process start as Transient State

Transient State: Continuous Processes
• Continuous processes are always transient
when:
– Started-up
– Modified in operating conditions
(increase/decrease)
– Shut-down

Viscosity of a Reactor…
More Heat  Increase in
Temperature  Increase in
Reaction  Products are more
viscous
Concentration of X
Time
Time  seconds
Vs.
Time  days

Viscosity of a Reactor…
More Heat  Increase in
Temperature  Increase in
Reaction  Products are more
viscous
Time
Time  seconds
Vs.
Time  days
Concentration of X

The Concept of Accumulation
• The Accumulation is not “0”
Inlet – Outlet + Production – Consumption = Accumulation
• Since it’s a change in time, it generally turns to
be a derivative!

• Algebraic Equations  Differential Equations
– First Order Differential equation  “by hand”
– 2nd+ Order Differential Equations  computer
• We will use numerical methods for either
types of differential equations

• Water flows to a Tank
• No water goes out of the
tank
• The next Process Variables
are affected:
– Level (increases)
– Volume (increases)
– Mass content (increases)
– Mole content (increases)

• Water flows in; flows out
• If Qin = Qout
– Steady State (no accumulation)
– No increase in level, mass, volume, etc.
• In time, there is no change in the system

• If Qin < Qout
– Unsteady/Transient State
– We are “empting” the tank
– Level decreases, Volume decreases, Mass decreases
• In time, the tank will get empty
Change of mass vs. time

• If Qin > Qout
– Unsteady/Transient State
– We are “filling” the tank
– Level increases, Volume increases, Mass increases
• In time, the tank will get full and even spill
Change of mass vs. time

• Analyze:
– Global System
– Tank #1
– Tank #2

• Global System
– No change in mass content
– No change in volume
– Ignore level content (not relevant)

• Tank #1
– Transient State
– Outlet > Inlet
– Will get empty in time

• Tank #2
– Transient State
– Outlet < Inlet
– Will get full in time

• We got various systems
– Some may be modeled as transient state
– Others may be modeled as steady state

• Suppose we fill/empty a tank
• Now we add “salt”
• Initial Salt content
• Initial water content (level)

• Water accumulates
• Salt accumulates
Overall Balances
• Water+Salt flows > Solution flow
• Water+Salt flows = Solution flow
• Water+Salt flows < Solution flow
Initial Values
• Salt concentration

Case Initial Conc. > Feed Conc. Initial Conc. = Feed Conc. Initial Conc. < Feed Conc.
Water+Salt > Solution Spills, dilutes Spills, conc. = initial Spills, concentrates
Water+Salt = Solution Level stays the same
Dilutes
Level stays the same
Conc. = initial
Concentrates
Water+Salt < Solution Empties, dilutes Empties, conc. = initial Empties, concentrates
Spot the Continuous Process!

Case Initial Conc. < Feed Conc. Initial Conc. = Feed Conc. Initial Conc. > Feed Conc.
Water+Salt > Solution Spills, dilutes Spills, conc. = initial Spills, concentrates
Water+Salt = Solution Level stays the same
Dilutes
Conc. = initial
Concentrates
Water+Salt < Solution Empties, dilutes Empties, conc. = initial Empties, concentrates
Spot the Continuous Process!

• We analyze now:
– Differential balances  rates/velocities
– Integral Balances  changes in finite
time
• We will need a Methodology for
Transient State (similar to Steady
State)

Differential Mass Balance
• Differential  rates
• We will need time intervals!
– Boundary Conditions
– t0 to tf
• Mass/time  x time  Mass

• Input  mass in / Δtime [kg/s]
• Output  mass out / Δtime [kg/s]
• Production  r prod/ Δtime [kg/s]
• Consumption  r cons / Δtime [kg/s]
• Accumulation  ΔMass/ [kg/s]

• From this equation:
• We divide by Δt
• We let Δt  0 (derivative definition)
• We need a “Boundary Condition” to solve the system

Methodology
1. Write an expression for the amount of the
balanced quantity in the system
1. Set the accumulation term in the balance to
the derivative of mass & time
2. Substitute Inlet, Outlet, Production,
Consumption

Methodology
4. Transform to:
5. Recommended  sketch plot of y vs. t
6. If solvable by analytic approach
– Solve “by hand”
7. If not solvable or numeric method needed
– Solve with Computer Software

Differential MB
Exercise

Differential MB
Exercise
Restricted Content for Slide Share Users!
1) Do an Overall MB
• Include Inlet, outlet and accum
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Differential MB
Exercise
2) Set the accumulation term in
derivative form
3) Substitute data and equations:
inlet, outlet, accum
4) Transform to differential equation;
don’t forget the boundaries
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Differential MB
Exercise
5) Sketch a plot (Not included)
6) It can be solved with
analytical approaches  solve
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Differential MB
Exercise
Set of equations
for solution
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Integral Mass Balance
• Now we proceed to do Integral Mass Balances
• We use finite amount of time
– We get finite amount of mass
• Use/approach of integrals

• From:
• We get by multiplying dt
• Then Integrate!

Exercise

Exercise
1) Get MB equation
2) Set the accumulation
term in derivative form
3) Substitute data and
equations: inlet, outlet,
accum
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Exercise
4) Transform to differential
equation; don’t forget the
boundaries
5) Sketch a plot (Not included)
6) It can be solved with
analytical approaches  solve
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Exercise
Restricted Content for Slide Sha6re) UCsoenrst!inued…
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End of Section 1
• We've seen:
– Review of Transient state
– Unsteady State vs. Steady State
– Accumulation Concept
– Integral vs. Differential MB

Section 2
• Exercises
– Mass Balances in Transient State
– Computer solved Mass Balances in Transient State

Mass Balances in Transient State
Exercise 1
Methanol is being purged. At t0, the tank contains 750
lb of the liquid. The extraction rate is about 50 lb/h.
After 48h the extraction rate is 74 lb/h. Methanol is
added to sum up the lost purge at 90 lb/h.
• a) Math expression for the outlet flow (function of
time)
• b) Calculate the MB for methanol after one week
• c) What is the time needed to empty the tank?

Exercise 1
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Exercise 1
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Do NOT use time here… it must be done in the Overall Mass Balance!

Exercise 1
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Negative time makes no sense for the problem

Exercise 2
• A cubic tank (4 ft each side) and 10 ft height is filled
up. Determine the time required to empty the tank if
there is a 1 square inch orifice.
• You could use Torricelli Law  v = sqrt(2·g·h)

Exercise 2
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Computer solved Mass Balances
• Simultaneous Transient State Balances
– Simultaneous Differential Equations!
• Use Software!
– Excel
– Mathematica
– Matlab/Scilab
– Maple
– Polymath

• Typical set of Differential Equations

• Typical set of Differential Equations
N equations
M Variables
M = N  it can be solved

Exercise
• Go to www.ChemicalEngineeringGuy.com/courses
• The “Computer Solved Transient States” and other
computer related problems in Chemical Engineering
will be displayed in the:
• Chemical Engineering: Computer Aided & Computer Methods
• You will find other programs to work with
– Polymath
– Aspen
– Excel
– Many others..

End of Section 2
• We've seen:
– How to solve MB in Transient State “by hand”
– Software for Numerical methods and solving
differential equations
– How to solve MB in Transient State for complex
problems in computer (Go to Course)

Problems & Exercises
• Pair problems (problems 2-18) of Elementary Principles in
Chemical Processes. Felder, R; Rousseau, R. 3rd edition are
solved in the next videos. (Chapter 11)
• Remember: practice makes the master

Elementary Principles in Chemical
Processes
What topics did we covered from the book?

End of MB4: Transient State
• You should be now able to perform MB of process
• Hopefully:
– You are able to understand Transient State
– You differentiate between Steady vs. Transient State
– You know when and how to apply the Transient State
– You can solve a Transient Problem (simple by hand)
– You know there is software for complex differential
equations systems

MORE INFORMATION
• Get extra information here!
• FB page:
– www.facebook.com/Chemical.Engineering.Guy
• Contact me by e-mail:
– Chemical.Engineering.Guy@gmail.com
• Directly on the WebPage:
– www.ChemicalEngineeringGuy.com/courses

Bibliography
• Elementary Principles in Chemical Processes. Felder, R;
Rousseau, R. 3rd edition.
• Basic Principles and Calculation in Chemical Engineering.
Himmelblau, D. 7th edition.

MB4 Mass Balance in Transient State

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