This screencast is going to provide an

overview of degree of freedom analysis. 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.

+LearnChemE very good explanation. I am a fan of yours now 😉

Thank you so much!

very good explanation (:

hi, i'm a bit confused why you haven't taken 1-z as one of the equations

Owk, understand now. Thanks

For example2 ,there's no m1 on the chart and I wonder if this makes the example unsolvable ?

sorry I am confused. why didn't u say : y = 1 – x – 0.2. like you did with z, to become z-1 for B????

can 100= m2 + m3 be considered an equation?

very well explained

superb.Well explained

Why not 1 = -0.2-x-y? On the second example?

if we use y in terms of x for the second equation, wouldnt that make one less unknownvariable?

You guys are saving a life.Thank you so much.

Awesome!

How is m3 an unknown here? it is given that it's 0.1m1 and m1 is known so how is m3 an unknown?

can we get a word problem solved, cause the part on stream 3 were (1-x) is the mass of c/kg confuses me.