1. Accounting for Mass
Chapter 18

2. Objectives Know that mass is conserved but chemical species are not
Know how to solve mass accounting problems for steady-state and non-steady-state systems.

3. Accounting for Mass
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.
Mass can neither be created nor destroyed.

4. Accounting for Mass 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

5. 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

6. 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?

7. 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.
RAT?

8. Individual Exercise #1 (cont’d)
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?
“You” are the pilot...just a little humor...
final mass (64.5k) – initial mass (74k) = input (0) – output (fuel+3500)
so, output = fuel+3500 = k + 74k = 9500
so, fuel = 6000 lbm

9. Accounting for Mass: Applying the Mass Balance Approach
Input
Output 1
Accumulation
Output 2
System Boundary

10. Mass balance procedure
Describe the system
Is it a batch or flow process?
Sketch the system.
Label all inputs and outputs
Identify known quantities and compositions
Identify and assign a variable to each unknown quantity or composition

11. Mass balance procedure
Batch- no input and output during process
Continuous – input and output during process
Describe the system
Is it a batch or flow process?
Sketch the system.
Label all inputs and outputs
Identify known quantities and compositions
Identify and assign a variable to each unknown quantity or composition

12. Mass balance procedure (continued)
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
Solve for the unknown variables
Check your answer to see if it is reasonable

13. Pairs Exercise #1
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?

14. 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?

15. 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

16. Pairs Exercise #2
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?

17. Using Excel to Solve Systems of Equations

18. 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:

19. Ax = b For A, the matrix has 3 rows and 3 columns
whereas b has 3 rows and 1 column.
and x has 3 rows and 1 column.

20. Ax=b Now consider the operation
i.e., multiply each row of A by the column x

21. Traditional By-Hand Solution
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.

22. Team Exercise (5 minutes)
Solve the system of equations for u, v, and w.
Use what ever method you prefer.

23. Excel Solution Intuitively we can solve for x.
Fortunately Excel has an inverse function:
=MINVERSE(cell range).

24. Excel Example
First, enter your matrix into Excel...

25. Calculating the Matrix Inverse in Excel
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

26. 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

27. THAT’S IT!!!

28. 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.

29. Pairs Exercise #4
We have 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?