MB4 Mass Balance in Transient State

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Mass Balance: Transient State

MB4

Chemical Engineering


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 secondsVs. Time days



Viscosity of a Reactor…

More Heat Increase in Temperature Increase in Reaction Products are more viscous

Time

Time secondsVs. Time days

Concentration of X


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



• 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



• 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



• 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



• Analyze:

– Global System

– Tank #1

– Tank #2



• 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 sameDilutes

Level stays the sameConc. = initial

Level stays the sameConcentrates

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 sameDilutes

Level stays the sameConc. = initial

Level stays the sameConcentrates

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 MBExercise



1) Do an Overall MB

• Include Inlet, outlet and accum

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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










5) Sketch a plot (Not included)

6) It can be solved with analytical approaches solve










Set of equations for solution









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


Integral Mass Balance

• From:

• We get by multiplying dt

• Then Integrate!


Integral Mass BalanceExercise



1) Get MB equation

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

5) Sketch a plot (Not included)

6) It can be solved with analytical approaches solve










6) Continued…Restricted Content for Slide Share Users!








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 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?












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



















Negative time makes no sense for the problem










• 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)




















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 equationsM Variables

M = N it can be solved


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..

http://www.ChemicalEngineeringGuy.com/courses


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


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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.