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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

  • 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.

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

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B1 中級

自由度入門 (Introduction to Degrees of Freedom)

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    羅紹桀 に公開 2021 年 01 月 14 日
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