Accounting for Mass

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Accounting for Mass

• Class 21.1

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Accounting for Mass

• Many engineers work with processes or systems
  where materials are mixed, separated or
  distributed, and must be accounted for.
• A fundamental feature of many of these situations
  is conservation of mass.
• The mass of the components remains constant
  during the process.
• Mass can neither be created nor destroyed.

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Accounting for mass

• Note that mass is an extensive quantity
• It can be counted
• It can accumulate or deplete
• The systems youll be dealing with for mass
  accounting are typically open systems, i.e. mass
  is allowed to enter and leave the system

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Accounting for mass Applying the UAE

• Final amount - initial amount input - output
  generation - consumption
• since there is no generation or consumption of
  mass
• final amount - initial amount input - output
• Accumulation (or depletion) is defined as final
  amount - initial amount, thus
• accumulation input - output

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The System

• First, you must define the system.
• Sketch the system
• 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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Accounting Problem 1 (Individual Exercise)

• You are flying a cargo airplane that has 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 found to have a total
  mass of 64,500 lbm before the remainder of the
  freight is unloaded.

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Accounting Problem 1 (contd)

• 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?

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Accounting for Mass Applying the Mass Balance
Approach
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Mass balance procedure

• Describe the system
• Is it a batch or rate-flow process?
• Sketch the system, labeling all inputs and
  outputs
• Identify known quantities and compositions
• Identify and assign a variable to each unknown
  quantity or composition

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Mass balance procedure (continued)

• Write a balance equation for the total mass in
  the system and for each material component
• youll need n independent equations for n
  unknowns
• Solve for the unknown variables
• Check your answer to see if it is reasonable

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Batch example

• 1.0 kg of sugar solution initially contains 2.3
  sugar, and the rest is water. How much dry
  sugar must be added to obtain a solution that is
  18.0 sugar?

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Solution to batch example
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Solution, continued

• 6. Write mass balance equations
• there is no accumulation of mass in mixer, so
  input output
• since we have 2 unknowns, we need 2 equations
• total amount 1.0 kg X kg Y kg
• sugar amount 0.0231.0 1.00X 0.180Y
• Note we could have written the second equation
  for water instead of sugar
• 0.9771.0 0X 0.820Y
• Which makes the solution easier?

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Solution, continued

• 7. Solve for unknown variables
• Y 0.977/0.82 1.191 kg solution, and
• X Y - 1.0 0.191 kg dry sugar
• 8. Check it is not always possible to do an
  exact check on a mass balance problem.
• Here we could substitute values for X and Y in
  the sugar equation and verify the answer.
• The value does seem reasonable?

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Rate-flow processes

• The previous example could be converted to a
  rate-flow example by putting all the amounts on a
  time basis
• 1.0 kg/hr of sugar solution containing 2.3 sugar
  enters a continuous mixer.
• How much dry sugar (kg/hr) must be added to
  obtain a solution that is 18 sugar?

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Rate-flow processes

• The problem solution is identical except that all
  units are kg/hr instead of just kg
• Mathematically, we are taking the derivative with
  respect to time on both sides of the UAE, or
• rate of change of...
• mass in mass out accumulation

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Pairs Exercise (10 minutes)

• This time, make pairs using left and right pairs,
  instead of front and back.
• Lets do problem 18.4 from Foundations, but
  change the numbers
• A grain drier is fed 10,000 lbm/hr of wet corn
  (25 water) which is dried to 14 water by the
  drier.
• How much water (lbm/hr)is removed by the drier,
  and how much (lbm/hr) dried corn (at 14 water)
  exits the drier?

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Hint
QUESTION Why cant we solve this problem by
inspection? We have 25 water in and 14 water
out, so water removed is 11, or 1100 lbm/hr.
No. No. No! Water in is 25 of 10,000, but
water out, in the corn, is 14 of Y, not of
10,000. The correct answer is 12.8 of 10,000, by
using the material balance procedure
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Using Excel to Solve Systems of Equations
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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.

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Ax b

• For A the matrix has 3 rows and 3 columns
• while b has 3 rows and 1 column.
• and x has 3 rows and 1 column.
  

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Axb

• Now consider the operation
• i.e., multiply each row of A by the column x

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Traditional Solution

• The by-hand solution requires that you manipulate
  the rows by multiplying them by a constant or
  subtracting rows etc. to eliminate variables.
• When you get to an equation with one unknown then
  solve and substitute until all 3 unknowns are
  known.

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Team Exercise (5 minutes)

• Solve the system of equations for u, v, and w.
• Use what ever method you prefer.

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Excel Example

• First, enter your matrix into excel...

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Calculating the Inverse in Excel

• Highlight the range where youd 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 CTRLSHIFTENTER

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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
• CTRLSHIFTENTER

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THATS IT!!!
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Exercise for pairs (5 minutes)

• Do the first sugar solution problem setting up
  the solution in matrix form and solving for the
  unknowns using Excel.

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Pairs Exercise Due by the end of class.

• If 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 in
  order to get a final milk which is at 1.6 fat
  and 2.2 protein?
• Use Excel to solve the equations.

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Assignment 9

• INDIVIDUAL ASSIGNMENT
• Due
• FOUNDATIONS 18.2, 18.9, 18.16