Transcript Accounting for Mass

Accounting for Mass
Chapter 18
Objectives
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Know that mass is conserved but
chemical species are not
Know how to solve mass accounting
problems for steady-state and nonsteady-state systems.
Accounting for Mass
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Many engineers work with processes
where materials are mixed, separated or
distributed, and must be accounted for.
A fundamental feature of these situations
is conservation of mass.
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Mass can neither be created nor destroyed.
Accounting for Mass
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Note that mass is an extensive quantity
It can be counted
It can accumulate or deplete
Many systems are open systems, i.e.,
mass enters or leaves the system
Accounting for Mass:
Applying the UAE
Final - Initial = In - Out + Gen - Cons
Because there is no generation or
consumption of mass:
Final - Initial = In - Out
or
Accumulation = In - Out
The System
Define and sketch the system.
Questions:
What are the boundaries of the system?
 What material components enter and leave
the system?
 Is there an accumulation or depletion of
mass within the system?
 What are the known and unknown material
amounts or composition?
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Individual Exercise #1
You are flying a cargo airplane with a
mass of 60,000 lbm when empty. It is
loaded with 8,000 lbm of fuel and 6,000
lbm of freight in Chicago. It lands in
Detroit and unloads 3500 lbm of freight.
Then the plane flies to Indianapolis
where it is has a total mass of 64,500
lbm before the remainder of the freight is
unloaded.
Individual Exercise #1 (cont’d)
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What is the pilot’s name?
How much fuel was burned between
Chicago and Indianapolis?
Did the amount of airplane fuel in the
universe change?
Did the amount of mass in the universe
change?
Accounting for Mass: Applying
the Mass Balance Approach
Output 1
Input
Accumulation
Output 2
System
Boundary
Mass balance procedure
1. Describe the system
Is it a batch or flow process?
2. Sketch the system.
3. Label all inputs and outputs
4. Identify known quantities and
compositions
5. Identify and assign a variable to each
unknown quantity or composition
Mass balance procedure
Batch- no input and output
during process
1. Describe the system
Continuous – input and
Is it a batch or flow output
process?
during process
2. Sketch the system.
3. Label all inputs and outputs
4. Identify known quantities and
compositions
5. Identify and assign a variable to each
unknown quantity or composition
Mass balance procedure (continued)
6. Write a balance equation for the total
mass in the system and for each
material component

You’ll need n independent equations for
n unknowns
7. Solve for the unknown variables
8. Check your answer to see if it is
reasonable
Pairs Exercise #1
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Into a mixer is placed 1.0 kg of
sugar solution initially containing
2.3% sugar, and the rest water.
How much dry sugar must be
added to withdraw a solution that is
18.0% sugar?
Flow processes
The previous example could be converted
to a flow example by putting all the
amounts on a time basis:
1.0 kg/h of sugar solution containing 2.3%
sugar enters a continuous mixer.
 How much dry sugar (kg/h) must be added
to obtain a solution that is 18% sugar?
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Flow processes
The problem solution is identical except
that all units are kg/h instead of just kg
Mathematically, we take the derivative
with respect to time of both sides of the
UAE, or
rate of change of...
accumulation = mass in – mass out
Pairs Exercise #2
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Do Problem 18.4 from Foundations of
Engineering, but change the numbers:
A grain drier is fed 10,000 lbm/h of wet corn
(25% water) that is dried to 14% water by
the drier.
 How much water (lbm/h)is removed by the
drier, and how much (lbm/h) dried corn (with
14% water) exits the drier?
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Using Excel to Solve Systems of
Equations
Ax = b: System of n Equations
and n Unknowns
A is a square matrix with a row for every
equation and a column for every
variable.
For example consider the system below:
1u  2v  3w  1 4
3u  1v  2 w  1
2u  3v  1w  1
Ax = b
For A, the matrix has 3 rows and 3
columns
1 2
3
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A  3 1  2
2  3 1 
whereas b has 3 rows and 1 column.
and x has 3 rows and 1 column.
14 
b    1
  1
u 
x   v 
 w
Ax=b
Now consider the operation
1
3

2
2
1
3
3   u  1u  2v  3w 14
 2   v   3u  1v  2 w   1
1   w  2u  3v  1w  1
i.e., multiply each row of A by the column
x
1u  2v  3w  14
3u  1v  2 w  1
2u  3v  1w  1
Traditional By-Hand Solution
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Manipulate the rows by multiplying them
by an appropriate constant, subtract
rows to eliminate variables.
When you get to an equation with one
unknown, then solve and substitute until
all 3 unknowns are known.
Team Exercise (5 minutes)
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Solve the system of equations for u, v,
and w.
Use what ever method you prefer.
1u  2v  3w  1 4
3u  1v  2 w  1
2u  3v  1w  1
Excel Solution
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Intuitively we can solve for x.
Ax  b
1
1
x  A b where A
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is theinverseof A.
Fortunately Excel has an inverse function:
=MINVERSE(cell range).
Excel Example
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First, enter your matrix into Excel...
Calculating the Matrix Inverse in
Excel
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Highlight the range
where you’d like the
inverse matrix to go…
Click in the input widow
and type:
=MINVERSE(matrix)
where matrix is the
range of your A matrix
DO NOT HIT ENTER
HIT
CTRL+SHIFT+ENTER
Multiplying Matrices in Excel
Now enter the b matrix
 As in the previous step,
highlight the answer
range
 Use the function
=MMULT(matrix1,matrix2)
where matrix1 and
matrix2 are the ranges
of your inverse and b
matrices, respectively
 CTRL+SHIFT+ENTER
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THAT’S IT!!!
Pairs Exercise #3
Redo Pairs Exercise #1 (the first sugar
problem) setting up the solution in
matrix form and solving for the
unknowns using Excel.
Pairs Exercise #4
We have 100.0 kg of skim milk at 0% fat
and 2.5% protein. How many kg of milk
at 2.0% fat and 2.1% protein, and whole
milk at 3.5% fat and 1.9% protein must
be added to the skim milk to get a final
milk that is 1.6% fat and 2.2% protein?