So we'll provide an explanation of what this means and how it can be used

to help us analyze engineering problems.

So when we're doing a degree of freedom analysis we're basically trying to find out

whether we have enough or too much information to solve a particular problem.

So before we look at an engineering example of doing a degree of freedom analysis

we can kind of get an intuitive sense of what "degree of freedom analysis" means

by looking at a couple systems of algebraic expressions

so we'll look at three scenarios, in the first we have one equation

we have 2X plus Y is equal to 7 and if we were asked to solve this we intuitively know

that we can't because we only have one equation

with two unknowns. So we could rearrange this however we want to solve for X or Y

but we'll never be able to solve for both of these variables.

So in this case when we have the number of unknowns is greater than the number of equations

we have an underspecified system, we can't solve

for the two variables in this equation without more information

in other words, another equation.

That's not the case in our second scenario where we have two unknowns X and Y again

and two equations, in this case the number of unknowns

is equal to the number of equations and this is generally the scenario that

we're looking to find when we're analyzing engineering problems

in this case when the unknowns are equal to the number equations we have

zero degrees of freedom

and we can solve the system of equations this case if we solve these two

equations simultaneously we would find that X is equal to 2

and Y is equal to 3 and in our third scenario

we still have our two unknowns of X and Y but here we have three

equations. So we have a different scenario where the number of unknowns is

actually less than the number of equations

and in this case we are overspecified. If we have an overspecified system we can get

results or answers for our unknowns that are inconsistent. So for example if we use

the first two equations to solve for X and Y as we previously did

we show that X is equal to 2 and Y is equal to 3

if we use the second two equations we would get a different answer for X and Y

so here X is equal to 0.5 and Y is equal to

3.5 so we have a system of equations that aren't consistent with each other and aren't

generating a single unique solution. So when we're analyzing engineering problems we're trying to find

a system that has zero degrees of freedom and provides enough information so that we can solve for

the unknowns for a particular problem. If we're trying to solve an engineering problem

and we're interested in analyzing or modeling

chemical process, doing a degree of freedom analysis is going to be one the first things we want to do

and we can calculate the degrees of freedom by calculating the number of unknowns

for a particular system, and subtracting the number of independent balances that we can write,

whether its mass or energy and then also subtracting the

other equations that we can write that relate those equations.

I'm going to focus on the material balance side of this

and if we want to look at the number of independent balances we can write, it's always going to be

equal to the number species that are present in that particular system

we can always write an independent balance for each species that's present.

We can also write a total balance but the total balance is not independent

from the species balance,

in other words if we sum up all the species balances that are present

that will generate the overall balance, so it's dependent on all the species balances

the other equations can come from a variety of different places, so for example we might have

process specifications, so we might know

the relationship or the ratio between different flow rates in a particular part of the problem

we also could have physical property data

so for example we might know the density or specific gravity

of a liquid stream or we might know the pressure and temperature

of a gas stream which would allow us to use the ideal gas law to figure out a flow rate.

Could also use equilibrium equations, so there's a lot of different equations

that are different than mass balances that will allow us to relate unknowns

and we can account for those as well. When we calculate the degrees of freedom

for a particular system there's three potential outcomes, if we have a

situation where the degrees of freedom

are equal to zero, then we can solve the problem, we have the

necessary equations to relate the unknowns that we have

on the other hand if we have degrees of freedom that's greater than zero

we have more unknowns than we have equations, and we have an

underspecified system, so without more information we can't solve for all the unknowns

and if we have degrees of freedom that's less than zero we're

overspecified, in other words we have more equations than we do

unknowns, similar to what was shown earlier. So let's apply this procedure to

two different examples

of material balances on single units.

In the first example we have a single unit process with two inputs and two outputs

if we want to calculate the degrees of freedom we need to know the number of unknowns,

as well as a number balances that we can write. If we take a look at our flow

chart here we can see that we have one unknown flow rate here on the input side

we also have another unknown composition variable on the output side

and one more unknown flow rate. So with M1,

X, and M2, we have three unknowns. If we want to solve for these three unknowns we can

write out a system of mass balances

and the number of independent balances is always equal to the number of species.

So we have species A, we have species B

and we have species C. So in this case there are three independent

material balances that we can write, and again keep in mind that we can also write the total

but the total will always be equal to the sum of the three species balances

so therefore it is not independent from the other three. We don't have any other information

that's been given to us that can relate the variables in this particular

example so with three unknowns and three balances that we can write, we have zero

degrees of freedom and we can solve for the three unknowns in this case.

Alright, let's look at one more example of a single unit with one input and two outputs

again if we want to do the degrees of freedom analysis we need to know the

number of unknowns

so in this case we have two unknown composition variables on the input side

we have an unknown flow rate on the output side as well as another unknown

composition variable and one more unknown flow rate.

So it looks like we have a total of five unknowns in this particular case

the number of independent material balances is always limited to the number of species that we have

and like the first example we have three, we have A, we have B and we have C.

So with only three balances that we can write and five unknowns

we would have two degrees of freedom and we would have an underspecified system

however we have some more information that we can use. On the right here

we see that we have an equation that relates two of the flow rates

here we have that M3 is equal to 0.1 times M1

so it's not a material balance per se, it is an equation

that relates two variables that are independent from all the balances

that we would write. So we have minus one other equation

for this particular ratio, that leaves us with one degree of freedom left,

if we look at the input side we can show one more relationship that relates these variables

so we know that the sum of all the mass fractions

has to equal one, so we have one more variable on the flow chart than we really need,

we could equivalently write Y is equal to one minus 0.2, which is the mole fraction of A,

minus X which is the mole fraction of B.

So often it's advantageous to write the composition variables

on the flow chart with as few variables as possible, that's actually what was done here

in the second flow rate. So if we keep this constraint in mind, that all the mole fractions sum up to 1

then we have another equation that we can write as we just did,

and that leaves us with zero degrees of freedom in this case, as well

so we could solve for all five variables in this example as well.

So hopefully this shows that through these two examples, the degree of freedom

analysis is a really powerful tool to quickly help us determine

whether we have enough information to solve a problem

it's straight forward on a simple unit, it becomes even more important as we look at more complex processes

with multiple units. So it's often a good place to start with degrees of freedom

analysis for the different systems in a particular problem.