Introduction to Degrees of Freedom


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.

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