mb4 mass balance in transient state
TRANSCRIPT
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Mass Balance: Transient State
MB4
Chemical Engineering
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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)
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Section 1
• Theory
– Transient State Review
– Accumulation Concept
– General Methodology for Transient State
– Differential and Integral MB
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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
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Review: Transient State
• 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
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Review: Transient State
• Typical processes for Transient State:– Batch
– Semi-batch
• But NOT ALL Transient State are batch/semi
• Even Continuous Process start as Transient State
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Transient State: Continuous Processes
• Continuous processes are always transient when:
– Started-up
– Modified in operating conditions (increase/decrease)
– Shut-down
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Transient State: Continuous Processes
Viscosity of a Reactor…
More Heat Increase in Temperature Increase in Reaction Products are more viscous
Concentration of X
Time
Time secondsVs. Time days
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Transient State: Continuous Processes
Viscosity of a Reactor…
More Heat Increase in Temperature Increase in Reaction Products are more viscous
Time
Time secondsVs. Time days
Concentration of X
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The Concept of Accumulation
• The Accumulation is not “0”
• Since it’s a change in time, it generally turns to be a derivative!
Inlet – Outlet + Production – Consumption = Accumulation
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The Concept of Accumulation
• 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
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The Concept of Accumulation
• 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)
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The Concept of Accumulation
• 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
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The Concept of Accumulation
• Water flows in; flows out• 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
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The Concept of Accumulation
• Water flows in; flows out• 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
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The Concept of Accumulation
• Analyze:
– Global System
– Tank #1
– Tank #2
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The Concept of Accumulation
• Analyze:
– Global System
– Tank #1
– Tank #2
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The Concept of Accumulation
• Global System
– No change in mass content
– No change in volume
– Ignore level content (not relevant)
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The Concept of Accumulation
• Tank #1
– Transient State
– Outlet > Inlet
– Will get empty in time
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The Concept of Accumulation
• Tank #2
– Transient State
– Outlet < Inlet
– Will get full in time
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The Concept of Accumulation
• We got various systems
– Some may be modeled as transient state
– Others may be modeled as steady state
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The Concept of Accumulation
• Suppose we fill/empty a tank
• Now we add “salt”
• Initial Salt content• Initial water content (level)
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The Concept of Accumulation
• 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
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The Concept of Accumulation
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 sameDilutes
Level stays the sameConc. = initial
Level stays the sameConcentrates
Water+Salt < Solution Empties, dilutes Empties, conc. = initial Empties, concentrates
Spot the Continuous Process!
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The Concept of Accumulation
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 sameDilutes
Level stays the sameConc. = initial
Level stays the sameConcentrates
Water+Salt < Solution Empties, dilutes Empties, conc. = initial Empties, concentrates
Spot the Continuous Process!
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The Concept of Accumulation
• 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)
Inlet – Outlet + Production – Consumption = Accumulation
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Differential Mass Balance
• Differential rates
• We will need time intervals!
– Boundary Conditions
– t0 to tf
• Mass/time x time Mass
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Differential Mass Balance
• 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]
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Differential Mass Balance
• From this equation:
• We divide by Δt
• We let Δt 0 (derivative definition)
• We need a “Boundary Condition” to solve the system
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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
Inlet – Outlet + Production – Consumption = Accumulation
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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
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Differential MBExercise
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Differential MBExercise
1) Do an Overall MB
• Include Inlet, outlet and accum
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Differential MBExercise
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 MBExercise
5) Sketch a plot (Not included)
6) It can be solved with analytical approaches solve
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Differential MBExercise
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
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Integral Mass Balance
• From:
• We get by multiplying dt
• Then Integrate!
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Integral Mass BalanceExercise
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Integral Mass BalanceExercise
1) Get MB equation
2) Set the accumulation term in derivative form
3) Substitute data and equations: inlet, outlet, accum
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Integral Mass BalanceExercise
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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Integral Mass BalanceExercise
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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
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Section 2
• Exercises
– Mass Balances in Transient State
– Computer solved Mass Balances in Transient State
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Mass Balances in Transient StateExercise 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?
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Mass Balances in Transient StateExercise 1
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Mass Balances in Transient StateExercise 1
Do NOT use time here… it must be done in the Overall Mass Balance!
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Mass Balances in Transient StateExercise 1
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Mass Balances in Transient StateExercise 1
Negative time makes no sense for the problem
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Mass Balances in Transient StateExercise 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)
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Mass Balances in Transient StateExercise 2
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Mass Balances in Transient StateExercise 2
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Computer solved Mass Balances
• Simultaneous Transient State Balances
– Simultaneous Differential Equations!
• Use Software!
– Excel
– Mathematica
– Matlab/Scilab
– Maple
– Polymath
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Computer solved Mass Balances
• Typical set of Differential Equations
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Computer solved Mass Balances
• Typical set of Differential Equations
N equationsM Variables
M = N it can be solved
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Computer solved Mass Balances 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..
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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)
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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
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Elementary Principles in Chemical Processes
What topics did we covered from the book?
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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
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Bibliography
• Elementary Principles in Chemical Processes. Felder, R; Rousseau, R. 3rd edition.
• Basic Principles and Calculation in Chemical Engineering. Himmelblau, D. 7th edition.