In this screencast we will look at approaching a single reaction material balance that involves a recycle and purge stream. We will use the extent of reaction method to help us solve for the unknown flow rate and composition. With that let me introduce the problem. So here we are making methanol by reacting carbon monoxide and hydrogen catalytically in a reactor. We know our fresh feed composition is 30% carbon monoxide. There is a stoichiometric proportion of hydrogen that is added and we have an inert of nitrogen that takes up 10 mole percent. Now the reactor not efficient so we separate the methanol from the unreacted gases using a a condenser. We recycle the gasses back into the main feed line into the reactor.
The recycle ratio something we can control is initially set to 3 moles per hour of recycle stream to every 1 mole per hour that is in the fresh feed. So this recycle stream contains 25% nitrogen gas. So the liquid product from the condenser is essentially 100% methanol. Based on this information, we want to determine the rate of methanol production in moles per hour for every 100 moles per hour of the fresh feed. We want the rate and composition of the purge gas and the overall and single pass conversions for this process. I suggest pausing this and try working it out by yourself before checking the work. We should start by drawing a picture then labeling all streams with known information and unknown variables.
We are told that we have a fresh feed that is coming in to our reactor. Out of the reactor is entering a condenser. I will just put a big C. Out of the condenser we have a gas stream that is split. Part of the gas stream is recycled back to the fresh feed, and the other part is purged out of the process. We also have a liquid product stream that comes out of the condenser. So this is just a really rough schematic with what the process looks like, but now we can start filling in some of the information, and I will draw this a little nicer. So here is my process. I have written the reaction in the reactor.
Now we were not told specifically what the reaction is, but we are told that carbon monoxide and hydrogen form methanol. So using our stoichiometry we can quickly figure out that there should be 2 hydrogen per every mole of carbon monoxide coming in. Now we are told that 100 moles per hour of fresh feed comes into the process. We know it's 30 mole % carbon monoxide, 60 mole % hydrogen, and 10 mole % nitrogen.
We are also told that 3 moles in the recycle stream per one mole in the fresh feed. We are also told that the composition of the recycle stream is 25 mole % of nitrogen gas. So if it is 25 mole % here in the recycle stream then we know nothing changes at the mixing point. So the composition should also be 25 mole % into the purge as well as what leaves the condenser.
So what other information do we have? We know that the composition of the liquid stream is effectively 100 mole percent methanol. So that is pretty much all we know in terms of the streams. So since we don't have flow rates for 1 2, 3, 4, or 5, I am going to put variables associated with those flow rates. We also don't know the composition entering or leaving the reactor. So I will write those out. So now we have the diagram for the process, all streams labelled with our known values and any unknown variables. So looks like there are a lot of unknown variables.
Kind of wonder if we can solve this. So to do a quick analysis to determine if we can solve for this. We will start by doing a degree of freedom analysis. Now I always like starting with an overall balance.
So we will draw our box on the overall process. We only care about the stream entering and the streams leaving. So to do a degree of freedom analysis we determine the number of unknown variables. So we note based on just the overall balance, we know that n5 is unknown. That y5c is unknown. That y5h is an unknown, and n3.
So we have 4 unknowns variables. We then add the number of reactions that are involved in this process, and we have 1 reaction. So we have a total value of 5.
We subtract the number of species balances we can form. So we can do a balance on carbon monoxide, hydrogen, nitrogen, and methanol. We also subtract out any relationships we know.
We know that in our purge stream, the mole compositions must add up to 1. So we have 1 relationship we can use. So that gives us a value of 5. So if we take 5 and subtract what we know. we basically have 5 unknowns and 5 relationships or 5 equations. So we get 0 degrees of freedom and we can solve this problem. So where do we start. Now usually an inert can help us out some. So we have an inert nitrogen that enters in one location and exits in one location.
So using an overall balance on nitrogen. We should be able to calculate the flow rate of the purge stream. So based on whats coming in for nitrogen. We have 10 mole percent times 100 moles per hour of fresh feed. This must be equal to what is leaving the process. We are assuming steady state. No accumulation within the process itself, and nitrogen did not take place in the reaction.
So this must be equal to 0.25, 25 mole percent times n5 the flow rate. So we can solve for n5, and I get 40 moles per hour. So I will plug that information into my diagram. Now we can do the same thing for carbon monoxide, and hydrogen, but we have to use our extent of reaction to help us out.
Since they are involved in the reactor. So for carbon monoxide; one reaction consumes 1 mole of carbon monoxide. So we have our extent of reaction for the number of reactions that take place is equaling the number of moles that come into the process. So we have 0.30 times 100 moles per hour minus what leaves the process. So this is y5c times 40 moles per hour.
So basically what comes in minus what comes out, must be what is consumed or the extent of our reaction. We have 2 unknowns, and 1 equation. So we are not going to be able to solve yet, but if we do the same thing for hydrogen, and write out the extent of reaction as equaling what comes in minus what comes out. Now hopefully you have caught that I made a slight misjudgment. For every reaction that occurs we lose 2 moles of hydrogen. So we need to take our extent of reaction and multiply it by 2 to make this accurate. So now we have 2 equations and 3 unknowns.
So we are still missing something. We do know from our relationship that the mole fraction in the purge stream must add up to 1. So know we have 3 equations, 3 unknowns we can solve for the composition of our purge stream and the extent of reaction. So I am going to solve for it and plug those values in now.
So I get the mole fraction of carbon monoxide in our purge is 0.25 and the mole fraction of hydrogen is 0.5. Our extent of reaction is 20 moles per hour. It is important to note that there are units in the extent of reaction.
So we know the composition of the recycle stream. Since it has to be the same as the purge. So I will transfer that those values over to the recycle stream and now we can use this information to do a mole balance on the mixing point to determine the number of moles entering the reactor. Now at this point we have figured out how many moles per hour are leaving in a purge, and we actually know how many moles of methanol are produced. We know from the reactor that 1 mole of methanol is produced per reaction.
And our extent of reaction is 20 moles per hour. So that must mean that what is leaving the condenser is 20 moles per our of methanol. So we have our purge steam our recycle steam and our product stream of methanol, and that almost answers everything we were looking for in the problem, but we want to know what the single pass and the overall conversion was for this process, and the only way to calculate the single pass conversion is to know the amount of moles coming into the reactor and what is leaving the reactor. So as I said we can use the information for the recycle stream to help is do a balance at the mixing point over here to solve for the values entering the reactor. So let's set that up. So with 3 species being involved in this process, we can write 3 material balances.
So I have written an overall balance, which is the easiest one. We know 100 moles of fresh feed plus 300 moles of our recycle must be entering our reactor we did are carbon monoxide balance as I have written here and a hydrogen balance. So we can find the compositions y1c, and y1h. So when I solve these equations I get the following for the composition entering the reactor.
And as I mentioned we need the information leaving the reactor. So we need to do a balance around the condenser. So before we calculate the n2 we need to know n4. So we do know the composition at n4. It must be the same as recycle stream and the purge stream and the flow rate is pretty easy to see. It is just going to be the 300 moles plus the 40 moles. So let me fill in that information in for n4.
So that we can use n4 and our 20 moles of methanol to determine n2 and the composition of that stream. So let me walk through an example balance. Let's do this for carbon monoxide. We know into the condenser we have y2c this is going to be multiplied by n2 and this has to equal what comes out, which is 0.25 times 340 moles per hour.
Now we know we can do an overall balance to figure out n2, n2 is equal to 340 moles per hour plus the 20 moles of our methanol stream. So a quick balance for each component, can give us the composition we are looking for. I will plug those values in. So now we have all the information for our compositions and our molar flow rates. We can calculate the single pass conversion and the overall conversion. We'll do the single pass conversion first. Now the idea of single pass conversion is basically the amount of moles that reacted out of the amount of moles that could have reacted.
So let's look at carbon monoxide. We have 0.26 mole fraction times our 400 entering. We subtract out what is left. So that is 0.24 times 360. So we have a certain amount entering a certain amount leaving. The difference being what is being reacted. That amount of reaction over what could have reacted. So the 0.26 times 400 is going to give us our single pass conversion. So I get 17 percent. 17 percent conversion of carbon monoxide in the reactor for each single pass.
Now to do an overall conversion on this process is the same mentality but we use the values that are entering and leaving the process. So we have 30 moles of carbon monoxide entering, a certain fraction leaving this over what entered. Gives us an overall conversion of 0.667 or roughly 67 percent. So you can see that we get 67 percent of carbon monoxide in our process. Even though or single pass conversion was only 19 percent. Hence the importance of our recycle stream.
It allows us to recycle some of the unused reactants back into the feed of the reactor. So that they do react. Thus increasing our overall conversion of those incoming fresh feed materials.
So to reiterate: Start with the diagram, label your knowns and unknowns, and perform a degree of freedom analysis using one of the 3 methods for reactive balances. Whether it is a molecular species balance, an atomic species balance, or using the extent of reaction as shown here. Then it is just a matter of math. Hope this helped.
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