D3-Distillation: McCabe-Thiele

[@qusaiamer6853]

This is way clearer than my university instructor who spent on this topic 4 weeks. I hate how we pay a looooooot of money for schooling but we end up teach ourselves every single thing!

[@PLE_LU]

Sorry to hear about your struggle but happy to hear that this old video I created for my students helped you. It is now a rather long time since I created this video and there is more recent playlist of videos on distillation that I personally think I did a better job with. In case you're interested you can find it at kzbin.info/www/bejne/p3eWYoJ6oM6Dr8U

[@fiddlepants5947]

Such a lovely video. I've got me midterm tomorrow morning and this is a 10/10 relaxing review. Greetings from Drexel University!

[@Miliccie]

Amazing job! You went through details that might not seem important, but they definitely help to understand what is happening!

[@PLE_LU]

Thank you! By the way, there is a more recently created playlist on distillation. The video in that playlist that derives McCabe-Thiele is kzbin.info/www/bejne/b5i9lpmqirOYoM0 but there is plenty more

[@kingshurehmohamed2778]

Thank u so much for u kindness I understood everything I also appreciate how u answered everyone's question in the comment because I benefited as well continue with this great job you are really a super hero.

[@majedaljaber134]

a highly beneficial video, thanks a lot: from Syria

[@mandilakhemfoboza9432]

This is a very informative video. Thanks for clear explanations.From: South Africa

[@PLE_LU]

Nice to hear, thank you!

[@PLE_LU]

By the way, there is a more recently created playlist on distillation. The video in that playlist that derives McCabe-Thiele is kzbin.info/www/bejne/b5i9lpmqirOYoM0 but there is plenty more

[@MyRandomLife247]

I just want to thank you for making this video, finally a well thought out video for using this method. I'm a mechanical engineer but I have been given the task of designing a distillation column. Great video it really did help :).

[@PLE_LU]

+MyRandomLife247 Thank you, glad it helped you

[@oioiyouthere]

Best video I have seen, thank you so much!!

[@PLE_LU]

Thank you, nice to here you liked it. I have published a much more recent playlist with 14 videos on distillation, see kzbin.info/www/bejne/p3eWYoJ6oM6Dr8U In my personal opinion, the newer video e.g. on McCabe-Thiele kzbin.info/www/bejne/b5i9lpmqirOYoM0 is better than the old one you now have watched.

[@sarimiari9450]

Thanks for showing how to calculate Rm, unfortunately my prof did not explain it on a graph but only said that Rm=(L/V)min so I was confused and lost. Turns out it is very simple

[@iftikharkhan6668]

thank you sir,for valuable lecture

[@rezakhanbani]

wow i am just speechless about how many things we should have considered but the professor never told us to

[@PLE_LU]

Hi, maker of the video here: There is a more recent version of this video if you are interested (kzbin.info/www/bejne/b5i9lpmqirOYoM0) and we do have a playlist on distillation Regarding what we teachers don't mention, I often say to my students "Nature is a bitch" as a short for "reality is usually way more complicated than our models of reality". Deciding what to not mention and how drastic simplifications to make when designing a course is a rather tough task. Jack Cohen and Ian Stewart describes this kind of problem in the drastic term "Lie-to-children" (en.wikipedia.org/wiki/Lie-to-children), a term I personally came across when reading the book "The science of Discworld" that they co-authored with late fantasy writer Terry Pratchett. Other famous quotes on this topic include "The best material model of a cat is another, or preferably the same, cat" (Norbert Wiener) and "All models are wrong. Some models are useful" (G E P Box). Let me take two examples: Should we include in our teaching that the Celsius scale was redefined in 1990 (ITS-90) or that the SI-unit system was redesigned in 2019? Both these facts have interesting consequences for how we do science, but with the limited time we have in a specific course, it might not be the most important for the student to understand. And just a final example, since this is a topic that fascinates me: I recently went through all chemistry text books in our local university library. All but one had an incorrect definition of reaction rate. They define it as a change in concentration over time. You find the same definition all over the internet and in all kinds of books. Even IUPAC (goldbook.iupac.org/terms/view/R05156) states that this is the definition. That does not, however, make the definition correct. In fact rather than a definition, that the reaction rate _equals_ the time derivative of concentration is what you get if all of the following criteria are met * You have only one reaction * That reaction is irreversible * You have a closed system * The volume is constant over time If any of the criteria are not valid, the reaction rate does not equal the time derivative of concentration. If you compress air, the concentration of air molecules increases (PV=nRT => C=n/V=P/RT), but there is no reaction. If you add more water to salt solution, the concentration decreases, but there is no reaction. The oxygen level in the blood of your brain needs to stay fairly constant (the time derivative of concentration being essentially zero) for you to survive, but there is a reaction consuming oxygen. If that reaction stops, you die. Nearly all naturally occurring systems are flow systems and the so called definition of reaction rate that is used by almost everyone is obviously invalid in such systems. Yet the faulty definition is still used and taught and will continue to be used and taught because it is easier to explain for beginners. I myself often say "my jacket is warm" although that's physically incorrect. The correct term is that my jacket is a good insulator. Sorry, went of on a tangent here :D

[@Khaielaash94]

This was a very informative video. Helped so much! Thank you! :D

[@PLE_LU]

+Khaielaash Manokaran You're wecome, glad it helped

[@Sashra]

So I followed your explanations (which are really good) and did the exercise before you showed the solution being happy about the solution you followed up with but there is one thing different in my solution: The angle you measured for the q-line cannot be -0.333°, but from what i measure is more like -0.19°. Now, I'd understand if you had done that on purpose, for making the diagram look more like anything I would encounter in an exam, but i'd still like a note just to eplain possible confusion. Alternatively i probably have to get a new protractor^^

[@PLE_LU]

It is the _slope_ that should be -0.333, not the angle. Try to think of the q-line as a simple straight line y=kx+m where k is the slope, the slope being k=∆y/∆x. Compare this with the equation for the q-line: y=q/(q-1)x - zF/(q-1) Thus the slope is q/(q-1) and the q-line cuts the y-axis in zF/(q-1) (which, however, might lie far outside your graph).

[@yogiyogi6768]

Thank you very very much sir such a perfect explanation

[@sulatsampah]

hi I am a chem engineering grad and going for an interview soon as production engineer. although I had completed my study, I am still in doubt on how actually distillation works. hope you can explain in theory and not calculation form. thank you

[@PLE_LU]

You can always check my newest playlist on distillation kzbin.info/aero/PLvpgTFzUKO49J3fwJF9y61q17FSNDdoGN

[@ayaalhusseini4995]

How did it go

[@hunadishabz1238]

thank you so much i know understand better ..my lecture confuses me alot

[@PLE_LU]

Glad to help. There are newer videos on distillation on our channel. We have kept this old one since this is the video that get found by people while the newer videos for some reason are not as visible on KZbin. You find the newer playlist on distillation through kzbin.info/www/bejne/p3eWYoJ6oM6Dr8U and the new video on deriving McCabe-Thiele on kzbin.info/www/bejne/b5i9lpmqirOYoM0

[@leventali5491]

Thank you! It is a very useful video.

[@PLE_LU]

Your welcome. Please note that there is a more recently playlist on distillation. The video in that playlist that derives McCabe-Thiele is kzbin.info/www/bejne/b5i9lpmqirOYoM0 but there is plenty more

[@soulaimanebalhan9730]

thank You a lot it was a very interesting explanation ... from Morocco .. even though we study with french system

[@PLE_LU]

Now I get curious. What is the French system in this case? Do you mean Ponchon-Savarit Method or something else? PS We have given up on teaching Ponchon-Savarit Method, as we don't see the point from a learning perspective nowadays when the realistic stuff is done using computers anyway. After the students learn the basics of how distillation works by using McCabe-Thieles method in the basic course we go straight to computer simulations with Aspen (or similar) for multicomponent systems (in advanced courses). DS

[@soulaimanebalhan9730]

Department of Chemical Engineering/Lund University nope ... I mean just the language of education ... we also use the same method ...thanks alot of guys

[@PLE_LU]

Aha, ok, thanks for your answer! Yeah, I teach distillation in English in one course and in Swedish in another but the students in both have to live with the English versions of distillation screencasts (there is a new version by the way). French I can't speak a single word unfortunately :-)

[@soulaimanebalhan9730]

Department of Chemical Engineering/Lund University hahahaha yeah same to me i cannot understand a single word of sweedish language .... we're study using the French language cuz we are colonized by French ... I wish i could study with the English language but I can't unfortunately. .. btw I study bachelor of food engineering and protection of the environment ... and i didn't understood this course at the class .. but thanks to you I passed the exam

[@soulaimanebalhan9730]

Department of Chemical Engineering/Lund University there are also countries in north Africa study in French language like : Algeria, Tunisia

[@chickletsushi2761]

Greetings. Thank you very much. I was wondering if you had a video that demonstrates the plotting of the middle operating line in case of two feeds.

[@PLE_LU]

Our playlist on distillation, kzbin.info/www/bejne/p3eWYoJ6oM6Dr8U, is what we will cover on graphical solution using McCabe-Thiele. In our courses this is as far as we take graphical solutions before we go into advanced computer simulation. Our pedagogic idea is that the fundamental understanding that learning McCabe-Thieles method can give you is what you need to be able to grasp and use advanced models but that it is the advanced computer models that you will need to use in any professional endeavor. BTW, the video you commented on is old. You find a newer version of this video in the playlist on distillation

[@tobiehl7130]

Seems to me like there is a mistake. At 9:07 we get the equation xD/(R+1) = 0.28 and at 15:31 wie get for Rmin "xD/(R+1) = 0.32" - means two time the same equation but two different outcomes? So is this wrong or am I wrong?

[@PLE_LU]

Not an error. If you have different values of R, you get different values for xD/(R+1), right? At 9:07 we solve the problem of a realizable distillation column, with R=1.5. At 15:31 we are looking for the limit, how small R can be for it still to be possible to build a distillation column that carries out the specified separation. We get the result that the limit is R=1.19. The R value that gives you xD/(R+1) is the reflux ratio that would require an infinite number of trays (i.e. an infinitely high distillation column) to carry out he specified separaion PS Please note that there is a more recent video on the same topic. You can find it in this playlist: kzbin.info/aero/PLvpgTFzUKO49J3fwJF9y61q17FSNDdoGN DS

[@rajeshr8755]

Please clarify the q line value is 0.33 Then feed line towards down ward , some what confused

[@PLE_LU]

It is not 0.33. It is -0.333, see text 9:59 into video

[@ChiNguyen-np5wv]

@@PLE_LU if you got -0.333 why did you draw a line at the 0.56 point on the y axis?

[@PLE_LU]

@@ChiNguyen-np5wv a straight line has the mathematical form y=kx+m. In this case, if you do the calculation you end up with m=0.56 thus y=0.56 when x=0. I don't do that calculation here though, I just take the fact that the q-line goes through the point (x=zF, y=zF) and draw the slope -1/3 and end up at 0.56

[@JadAzido]

How do you sketch the curve above the diagonal? My assumption is it is a function y(x) that is either give or based on the volatility of the substance.

[@PLE_LU]

For a beginners intro, see 4.48 into the video kzbin.info/www/bejne/oquQg4qpgr6rapo For more detailed info, search the web for acitivity coefficients such as van Laar, Margule, NRTL, UNIQUAC, …

[@hamadalmehrizi6078]

Excellent video. I didn’t get why you added a/A, shouldn’t we add a alone?

[@PLE_LU]

? I'm not sure what you mean by that. We're trying to count number of equilibrium stages, but a is in the unit mm so how could we add them together? We would get things like 4 eq. stages + 2 mm = ?? if we draw on a small paper and 4 eq. stages + 8 mm = ?? if we used a paper that was 4 times larger. Adding things that have different units never makes sense. a/A, on the other hand, is a fraction, an estimate of how large fraction of an equilibrium (b/B is also such an estimate so you could take either, or, as some text books do, take an average of the two)

[@minhduong3881]

May I ask how can u do it without an equilibrium line ? or how to graph an equilibrium graph ?

[@PLE_LU]

You need the equilibrium line. You can either interpolate from literature data, or use one of many "curve fits" (in which case you need literature data for the parameters). What curve fit to use depend on the system, van Laar is one of those "curve fits". Compare the video studio.kzbin.infolz-RWsMYx48/edit which explains how you can take the first steps if the system is ideal (which is usually never true). From there you have to find ways to calculate activity coefficients (using e.g. van Laar).

[@minhduong3881]

@@PLE_LU Thank you for the advice .

[@najwaeliva8296]

well explained! thank u so much this helps me ALOT

[@vivaldikonzsmith7291]

Would you please explain the -0.33 where does it come from? I dont get at all.

[@PLE_LU]

Hi. 1. Please note that you're watching an older version of the video Newer video on the derivation of McCabe-Thiele you can find here: kzbin.info/www/bejne/b5i9lpmqirOYoM0 ditto solving an exam task here kzbin.info/www/bejne/bZecfXyhbtanapo and the entire newer playlist on distillation kzbin.info/www/bejne/p3eWYoJ6oM6Dr8U 2. As I replied on a earlier very similar question: From the task description and the definition of q, 25% is saturated liquid and 75% saturated vapour, thus q=0.25 The slope is q/(q-1) since the equation is y=q/(q-1)x - z_F/(q-1). When you enter q=0.25 you get 0.25/(0.25-1)=0.25/(-0.75)=-0.33333

[@kpoff5128]

Great video, what program are you using to draw the McCabe Thiele diagram in?

[@PLE_LU]

Depending on the purpose I have been shifting between programs. * Matlab when I need to do further calculations (e.g. to draw McCabe-Thiele solutions automatically) * Kaleidagraph from Synergy Software I like since it makes making nice looking graphs so easy (but I actually do the calculation in matlab so that I can easily shift between different types of curve fits for activity coefficients like van Laar, Margule, etc.) In this video (and in the newer version kzbin.info/www/bejne/b5i9lpmqirOYoM0) I've taken a plot from Kaleidagraph and added a "solution" as an animation in PowerPoint.

[@shnoalimhamadkaram1594]

Hi sir ,the explanatian is very amazing ,but generaly l do not know a thing ,when you wanna draw the trigonal between operating and equilibrium cerve if you draw large trigonal you will get a few amount of number of the tray but if you draw the trigonal by the small way you will get a more number of the trays , please explain for me how can l draw the trigonal😪

[@PLE_LU]

I don't think I understand your question. Drawing these lines between the operating lines and the equilibrium curve is analogous to alternating between solving a mass balance and solving the equation for the equilibrium between the phases. Please note that there is a newer playlist on distillation: kzbin.info/aero/PLvpgTFzUKO49J3fwJF9y61q17FSNDdoGN

[@ungkuamirularifungkua.5292]

Hi. I don't get the part that you calculate the slope for q-line. So slope=0.333, and then you drew a line. I don't get where did the line come from.

[@PLE_LU]

Remember: 1. q-line goes through (z_F, z_F) 2. upper op. line goes through (x_D, x_D) 3. lower operating line goes throug (x_W, x_W) To see that for yourself, choose x=z_F, x=x_D, x=x_W in the respective equation

[@ungkuamirularifungkua.5292]

Institutionen för kemiteknik/LTH Thank you so much! :D

[@TheWatchmanontheWall783]

Still don't get it

[@wayne31011]

I'm not sure about it, but isn't it just y=(q/q-1) * x - xF/(q-1), so as you have x=0 it is just xF/(q-1) = -0,56 (which seems to be your point on y) I dont really get it how you get from the -0.333 to the point on the y axis, would be nice if you coult explain Thanks!

[@lorenepecci1367]

@@wayne31011 yes. it's wrong!

[@Ulpons]

Hello, I have to do a research job at the university on a distiller and I need to explain the liquid-vapor equilibrium curve, what would be the correct way to interpret this curve?

[@PLE_LU]

? It is the curve describing the liquid-vapor equilibrium, that is the interpretation. We cover the basics in other videos on this channel and all text books on distillation and phase equilibria explain this curve

[@ajitsinghrathore2838]

why do we start making steps from top and not from bottom? What difference will it make if we start making steps from bottom to up?

[@PLE_LU]

A very good question indeed! Short answer: What it all boils (pun intended) down to is tradition. The most common tradition is to start from above Long answer: There are at least three different ways to start making the steps: 1. From above (the step at the bottom as well as optimum feed location might be fractions of an ideal stage) 2. From below (the step at the top as well as optimum feed location might be fractions of an ideal stage) 3. From the optimum feed location (the step at the bottom as well as the step at the top might be fractions of an ideal stage) The three different methods might give slightly different results, but they are all equally correct. The fact that the three different methods might give slightly different results is a reminder that what we are doing here is an _estimation_ of the number of ideal stages (and number of physical trays) required. Most textbooks I have seen start numbering trays from the top and also start making the steps at the top. I have, however, seen a few that start numbering the trays from below and start making the steps at the bottom.

[@ateja8853]

why the feed tray should be drawn horizontal and vertical why not in vertical and horizontal direction

[@PLE_LU]

Some suggest that you should calculate it using the horizontal _and_ the vertical line and take the average of the two. In whatever way you do it, the result you get is an approximation and in my courses I suggest my students to only calculate one value and the only way I teach them is by using the horizontal line. If you look at my matlab distillation demo video (from 5:18) kzbin.info/www/bejne/n6bciYaZaK2VY7M you see there the matlab calculation programme I've made for educational purposes. In that programme the feed location is calculated using both horizontal and vertical lines. In that programme I also implemented three methods of drawing the triangles: From above, from below and from the feed location. All three methods give slightly different results I haven't gotten around to publish my matlab code. I find compiling stand alone matlab things rather tedious, not at all as simple as making stand alone programmes based on LiveCode or standard programming languages.

[@alinedeboleto]

Hi, what books do you recommend on this subject?

[@PLE_LU]

Mattias here: To be honest, I personally feel that most text books in this discipline are not designed to help new students learn but rather designed to help engineers who already know the field to refresh their memories. The most famous books were designed in the 1940-ies and the 1960-ies and once you already know quite a lot they are very, very good sources of facts and explanations, but from a pedagogic point of view I find them rather useless in my courses. We have used some of them in introductory courses at least since the time I was a student back in the 1980:ies, but most students did not read them and learnt better without reading them, so I gave up on them some years back. (The material I'm currently using for mass transfer, heat transfer, distillation, evaporation, filtration/membrane filtration, adsorption/chromatography and drying is currently only available to our students)

[@anthonygrant5396]

i am really not understanding how you drew that q line

[@PLE_LU]

+Anthony Grant See previous answer below. If you don't have my text book, you'll likely need a text book on distillation and Mc Cabe Thieles graphical method

[@vivekkathula92]

isnt the slope of q line, -q/(1-q) which means it will become -0.33, can you please clarify??

[@PLE_LU]

The slope is q/(q-1) since the equation is y=q/(q-1)x - z_F/(q-1). When you enter q=0.25 you get 0.25/(0.25-1)=0.25/(-0.75)=-0.33333 Perhaps you are mislead by the font I used, there _is_ a small minus sign in front of 0.333 (at 10:01) Please also note that we have published a newer series of distillation videos, collected in the playlist: kzbin.info/www/bejne/p3eWYoJ6oM6Dr8U Deriving of McCabe-Thiele is done in kzbin.info/www/bejne/b5i9lpmqirOYoM0 and an exam example (Ethanol-Water) is done in kzbin.info/www/bejne/bZecfXyhbtanapo

[@rajeshr8755]

@@PLE_LU Where are get 0.25 please explain

[@PLE_LU]

@@rajeshr8755 From the task description and the definition of q, 25% is saturated liquid and 75% saturated vapour, thus q=0.25

[@rajeshr8755]

@@PLE_LU thanks sir

[@nurasyiqin7787]

Department of Chemical Engineering/Lund University why did you line the graph towards between 0.5 and 0.6 on y-axis ? How did you know the slope -0.333 is there?

[@ferielbabaabdi4509]

Sir how can one know or is the trays optimal to introduce the load?

[@PLE_LU]

Are you asking how you can know that the feed is placed on the optimal tray? I answer your question in case my students who have an exam in January also are wondering: I need to divide my answer into two sections depending on the context of your question: A) You are looking at a real distillation column in operation B) You are designing a distillation column, e.g. calculating how many trays you need A: The simple answer is: You can't, at least not without taking a lot of measurements and doing a lot of calculations. Or, as an alternative, you can try changing the feed location, keeping all other operating conditions constant and see how xD and xW changes with feed location. The optimal feed location in a given setup for given operating conditions is the location that maximizes (xD-xW) B: At the design state you are free to choose your design, right? So your best choice is to place the feed optimally in your design calculations. You need fewer equilibrium stages (and hence fewer physical trays) if you place the feed at the optimal location. You get the optimal feed location if you switch from drawing your vertical lines (when stepping) to the upper operating line to drawing your vertical lines to the lower operating line immediately after you have passed the point when the two operating lines intersect. You may realize that this is the case by looking at the geometry of the graphical solution. To draw as few triangles as possible you want to draw them between the system curve and the operating line that lies furthest away from the system curve. Clearly, * for x-values larger than the point where the upper and lower operating line intersects, the upper operating line is furthest away from the system curve. * for x-values smaller than the point where the upper and lower operating line intersects, the lower operating line is furthest away from the system curve.

[@souravkumar2522]

Sir how to calculate the steam requirement in thermosyphone reboiler ????

[@PLE_LU]

I only deal with simplified models, thus no specific types of reboilers, but for a general estimate to calculate the energy needed in the reboiler, take the vapour flux below the feed and multiply with the evaporation enthalpy Q=V_below*∆Hvap(bottoms) [J/s] (if you want more accuracy, make sure you take the evaporation enthalpy for the composition of the bottoms.) To recalculate that to steam needed, look up the evaporation enthalpy for the steam you use (decreases with the increasing pressure) and calculate the steam needed S as S=Q/∆Hvap(steam) [kg/s] The real steam requirement is always bigger than the value you calculate (since there always are heat losses), but you at least get a rather good estimate.

[@ahiqtidar]

Sir one thing to get stages inside Distillation colounm we will always minus 2 stages of boiler and condensor, Even if boiler or condensor are parital or full we always minus right?

[@PLE_LU]

No, that's actually not correct. A total condensor is _not_ an ideal stage. The composition in the liquid flow out from a total condensor is exactly the same as the composition of the gas flow that went into the total condensor. Compare that with an ideal stage where the two flows leaving the stage (gas up, liquid down) are in chemical equilibrium with each other (hence differnt composition) Furthermore, although this might be omitted in basic courses, there might not be a reboiler. Alternatively, there might be no condensor: If you let the feed be gas that enters at the bottom of the column, no reboiler is needed. If you let the feed be liquid that enters at the top of the column, no condensor is needed

[@ahiqtidar]

+Department of Chemical Engineering/Lund University ok if u are using a Total condensor and boiler and u find 10 theoritical stages . Then how many of them will be inside the colounm?

[@PLE_LU]

10 - 1 = 9 (If you're not taking an advanced course that is. In basic courses it is common to make the assumption that a reboiler is exactly 1.0 ideal stages but if you think carefully about reboiler design it's not entirely certain that the reboiler will be 1.0 ideal stages)

[@ahiqtidar]

+Department of Chemical Engineering/Lund University Sir do have some sort of forum where i can ask frequent Qs?

[@PLE_LU]

Unfortunately we do not have the funds to provide such facilities for other than our own students

[@chinmoydas5464]

5:35

[@infotainment3789]

q = -0.33 tgan at 0.55? how

[@PLE_LU]

See earlier question and answer, I've already answered that question

[@evelynstanley7002]

I still don't get it how you get the q line .... please help ...?

[@PLE_LU]

+Evelyn Stanley I would need a more specific question to be able to help you. What is it more specifically that you don't understand?

[@evelynstanley7002]

+Department of Chemical Engineering/Lund University I mean we got the slope for the Q line but I don't understand how to draw it on the graph ._.

[@PLE_LU]

+Evelyn Stanley Consider the equation for the q-line. It's the equation for a straight line. You can choose any x-value and calculate its corresponding y-value. If you choose x=z_F you get y=z_F which is a point on the diagonal. The two operating lines work the same way. If you set x=x_D in the equation for the upper operating line you end up on the diagonal in (x=x_D; y=x_D) and if you set x=x_W in the lower operating line you end up on the diagonal in (x=x_W; y=x_W)

[@evelynstanley7002]

+Department of Chemical Engineering/Lund University Thank you for the explanation :D

[@lorenepecci1367]

the construction of the q line is wrong guys !

[@PLE_LU]

No, I think you misunderstand something regarding the q-line. With 25% liquid at its boiling point and 75% gas at its condensing point, the energy needed to evaporate the feed is 25% of ∆Hvap which means q=0.25. The equation for the lower operating line is y=q*x/(q-1)-zF/(q-1)=0.25x/(0.25-1)-zF/(0.25-1)=0.25x/(-.75)-zF/(-0.75)=-0.333x+1.333zF Thus the slope is -0.333. The slope can be seen as ∆y/∆x, so with ∆y= one square down you get ∆x=three squares right The q-line always intersect the diagonal in the point (zF, zF) since if you set x=zF you get y=(q*zF-zF)/(q-1)=zF*(q-1)/(q-1)=zF If you have further doubts you can also calculate the intersection with the y-axis. Set x=0 and get and intersection with y-axis is (x=zF=0 =>) y=-0.333*0+1.333*0.42≈0.56 which is the value you see in the diagram PS There is a newer video where a similar task is solved: www.youtube.com/watch?v=7f9LI... DS