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• 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

• 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.

This screencast is going to provide an overview of degree of freedom analysis.

# 自由度介紹 (Introduction to Degrees of Freedom)

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羅紹桀 發佈於 2021 年 01 月 14 日