Accounting for Mass

Chapter 18

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.

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.

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

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 SystemDefine 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?

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)

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

Input Output 1

Output 2

System Boundary

Accumulation

Mass balance procedure1. 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 procedure1. 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

Batch- no input and output during process

Continuous – input and output during process

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

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?

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 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?

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:

1132

1213

14321

wvu

wvu

wvu

132

213

321

A

1

1

14

b

w

v

u

x

Ax = bFor 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.

Ax=b

Now consider the operation

i.e., multiply each row of A by the column x

1

1

14

132

213

321

132

213

321

wvu

wvu

wvu

w

v

u

1132

1213

14321

wvu

wvu

wvu

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.

Team Exercise (5 minutes)

Solve the system of equations for u, v, and w.

Use what ever method you prefer.

1132

1213

14321

wvu

wvu

wvu

Excel Solution

Intuitively we can solve for x.

Fortunately Excel has an inverse function:

=MINVERSE(cell range).

. of inverse theis where 11 AAbAx

bAx

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

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

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

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?