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Deal with the closed three-tank system of Fig. 5.2.5, which is described by the equations in (24, Mixed brine flows from tank 1 into tank 2, from tank 2 int $operat...

Question

Deal with the closed three-tank system of Fig. 5.2.5, which is described by the equations in (24, Mixed brine flows from tank 1 into tank 2, from tank 2 int $operatorname{tank} 3$, and from tank 3 into tank 1, all at the given flow rate gallons per minute. The initial amounts $x_{1}(0)=x_{0}$ (pounds, $x_{2}(0)=0$, and $x_{3}(0)=0$ of salt in the three tanks are giver as are their volumes $V_{1}, V_{2}$, and $V_{3}($ in gallons $) .$ First solve for the amounts of salt in the three tanks at time

Deal with the closed three-tank system of Fig. 5.2.5, which is described by the equations in (24, Mixed brine flows from tank 1 into tank 2, from tank 2 int $operatorname{tank} 3$, and from tank 3 into tank 1, all at the given flow rate gallons per minute. The initial amounts $x_{1}(0)=x_{0}$ (pounds, $x_{2}(0)=0$, and $x_{3}(0)=0$ of salt in the three tanks are giver as are their volumes $V_{1}, V_{2}$, and $V_{3}($ in gallons $) .$ First solve for the amounts of salt in the three tanks at time $t$, then determine the limiting amount (as $t ightarrow+infty)$ of salt in each tank Finally, construct a figure showing the graphs of $x_{1}(t), x_{2}(t)$ and $x_{3}(t)$. $$ r=10, x_{0}=18, V_{1}=20, V_{2}=50, V_{3}=20 $$



Answers

Water flows steadily from an open tank as in $\textbf{Fig. P12.81.}$ The elevation of point 1 is 10.0 m, and the elevation of points 2 and 3 is 2.00 m. The cross-sectional area at point 2 is 0.0480 m$^2$; at point 3 it is 0.0160 m$^2$. The area of the tank is very large compared with the cross-sectional area of the pipe. Assuming that Bernoulli's equation applies, compute (a) the discharge rate in cubic meters per second and (b) the gauge pressure at point 2.

So here we're going to apply for a new lease equation and we're going to apply the equation of continuity. We can say that the speed of the influx is V, which will equal the speed. This will equal square root of two G h. And we can say that H would be the distance, uh, of the whole we can say below surface of the of the liquid or the fluid. And so, for part A, we can save these of three a sub three. This would essentially be equaling the square root of two g times y sub one minus y sub three multiplied by a sub three and this would be equaling square root of two times 9.80 meters per second squared multiplied by 8.0 meters, extend the square root and then multiplied by point 0160 meters. Quantity squared, and we find that the volumetric flow rate at 1600.3 would be equal in point 200 meters, cubed for a second For Part B. We can say that since piece of threes atmospheric pressure, we can say that the gauge pressure at 0.2 piece of two would be equal in 1/2 times row. The density times these of three squared minus Visa two squared and this would be equal in 1/2 times row. These of three squared, multiplied by one minus base of three. Excuse me, divide by a set to quantity squared, and this is I'm going to be a piece of two again Piece of three is equaling the atmospheric pressure so we can still say that piece of two is a over nine multiplied by the density times G multiplied by Weiss of one minus Y sub three. And so we're going to use the equation rather the expression for Avi's of three. And we can then substitute in. So piece of two is found to be 6.97 times 10 to the fourth pass cows. This would be our final answer for a piece of two. That is the end of the solution. Thank you for watching

Okay, So what we have here is a mixing problem in all being a town with an initial volume of my pounded liters and the brine solution coming into the dunk with a concentration, appoint two kilogram per liter and read five liters per minute. So the town is uniformly sphere, and the solution exits the tower at five litres per minute. A swell so the initial amount of salt inside the town, he said. Five kilograms. So we are required to ask to get the value off the concentration of the mixture. After then we needs and let there be the concentration of the mixture. After 20 minutes when the league start that he so for our solution, we know that looks be a over. The P is equal to bean, minus out so we can make use of the sport. Miller, which is the A over a BP, is acquitted off. Be thanks he in kilogram for immediate my nuts. So this is the in a over be not plus e minus if be times in kilogram for a minute. So we know that the rail of which no solution the mixture enters, the donkey said. Five liters per minute and it also exits the town about five liters per minute. So this will be equal to zero. So, substituting the values that we have from our given, we now have ve over d p is equal. Do going to times five minus. Okay. Over 500 names fight often computing we now have the A over DB is equal to one minus a over 100 which is also equals toe 100 minus a over 100. So this is a separate herbal differential equation so we can express this equation as the A over 100 minus a is equal toe baby over 100. So taking the integral of both sides, three have negative end off 100 minus e. Is it Well, though, be over 100 plus C one So multiplying both sides by negative one we have and off 100 minus a is equal to negative be over 100 minus C one. So the further simply by our equation, we want to race it do. Okay, So each raised to L and off 100 minus a is equal to erase the negative p over 100 minus. He want. So this will just be equal 100 minus a and knowing that C is a concept that you can take anybody. We can just express it us. See, erase the negative. Be over 100. So further simplifying our agree Sean, we now have a is equal. Go 100 mine us See the race to the negative Be over 100. So this is now our general solution for the first problem. So we want to find the value of the Constand and we can do this by making use of the initial ah initial conditions that were given in the problem. So at the is equal to zero, we have a is equal to fight to substituting this Our problem we have five is equal toe 100 minus See He raised the negative zero over 100. So this will just be equal The one and our value of C is 95 to substituting this over equation We now have a is equal to 100 minus 95 The race to the negative be over 100. So this is now our particularly single shot. So now we're asked to find for the concentration of the solution. After then we need so we can make use of this formula. So it be looks at the is equal to 10. We have a is equal 100 minus 95 he raised to the negative than over 100 this is approximately equal. This is approximately equal to 14.4 kilogram. So this is just the amount of salt in the tank after 10 minutes, but were asked to get for the wanton three shows. Therefore, seeing I'll spend is a little 14.4 kilogram over 500 leaders is equal to zero point 0 to 81 He looked Graham. Really? The CBC's our final answer. Okay, so for the second question there has to get for the consideration off the top off the mixture in the town after 20 minutes. So it was mentioned that after that Mini, it's the leak upward, and we are asked to get for the concentration after the pending after 20 minutes when the leak first part that so the Dari a wish the Bryant enters the tongue is still the same. So we have, uh, our entry is still equals the one, Our being is equal 2.2. You know the number and either. And now the initial value that we're going to use is a off. Then we just won't be computed a while ago, which is 14.4 so far. The ball you. So after the league develops, the system satisfies a new differential, a quick shot so well that it off the input remains the same, which is one kilogram per minute. The rape of exit It's no different and so, as mentioned, of the problem every minute five leaders of the solution is coming in. So that would be like Plus, one is a cult of six liters which is going up so the value of the solution in the town decreases by one litre per minute. Therefore, we can express our volume as 500 minus one things being minus. Then where in this shows that after then May needs in which the leg started, that how close this one leader every minute past the first that you need So substituting these to our equation, which is input minus output, we now have be a over B B is equal to one minus a over 500 then minus p times the six leave their sperm in you. We cheese The value of our m our new It so began Just right here. F a sequel. Now the six lead 33 minutes. So pump you thing we now have being over the B is equal to Guan minus six a over 500 then minus speak. So this is now a lean your differential equation. So did I think the problem. We have be over the MP plus six a over 500 then minus B is equal. What? So our field be therefore sequel to six over 500 then minus T and computing for our integrating larvae have I s she called the mu off B is equal to e raised the integral of six over 510 minus steal the b So computing for that our word integrating doctor is equal to e today's to the negative six Ellen off 500 then minus t, it's just just he will do 500 then minus t raising the negative six cbc's all we're integrating someone replying the equation to both sides by the integrating popular we now have 500 then minus T place of the negative six D A over a DP plus six a over 500 then minus T times 500 then minus theories of the negative, six is equal to 500 then minus t raised to the negative six. Okay, so we know that we can experience the left side of the equation toe just the over the B off 500 then minus t raised the negative six. A itwas do 500 then minus be raised to the negative six. So taking the integral of both sides, we have a times 500 then minus be raised to the negative. Six is equal to one since times 500 then dynasty, placing the negative five plus c so multiplying both sides by 500 then minus 30 days to six, 500 then minus D books. I'm just going to these can fit on the screen anymore. We just want to show that we multiply this by 500 then minus theories six k So these will cancel out and we now have a is equal to 1/5 one seed times 500 then minus T plus seeing times 500 then minus be face to six. So we're almost close. Our vital answer, I hope yourself following. So this is now our general. So you, Sean, who could, before the value of see, we make use of all word initial value, which is at B is equal. But then we have a is equal to 14.4 and substituting this our equation, the important 0.4 is equal to 1/5 times 500 then minus B plus C times 500 then minus 30 days to six. Oops. Sorry. We should be then that thief, since we our time is at that minute. This is 500 then my nose, my understand. And it should be also then. Okay, so apologies for that built in. Okay, So, solving for the value of C Florida solving on your old we have see basic well too negative. 85.96 over 500. Raise the six. Now. Our particular solution is okay. Off the music with one Felber. Five times. 500 then minus thing minus 85 point 96. Names 500 then minus t over 500. Raise six. So we want to get the concentration on. So we're going toe, uh, divide. Will they play both sides by 500 then one over 500 then minus steep. Therefore, our concentration is now at the off being over 500 then minus T is equal to 1/5 minus 85.96 over 500 then minus thing times 500 then minus T over 500 raised the six. So at being is equal to 30 minutes, which is the time when the when that that was initially filled by the brine solution over concentration then is equal to zero point 0598 kilogram for the This is our find that answer.

This case for this problem, you need to realize some communities in the beginning. So x we use X to denote the man Maso sought in 10-K And then we use why? To denote the mess off salt. You know, tank me since both tanks has half born in hungry. Laters. So the concentrate in tank A we will be x over 100 and the concentrating. Thank be be a wild 100. So once you realized this quantity is we can direct. You write out the season of all of these. So the wreck of change of eggs, Because the 1.2 plus why over 100 times one mine was X over 100 times seven. And if you wire with the key, you closed Teoh X over 100 times, three minus. Why? Over 100 times? Three. Okay, so we used an operator the requested your batikhi to rewrite video the assistant in the following way. And that way we'll do the elimination for this off the east by, um, acting operator. The process is 0.7 to the 2nd 1 And there for the first time, I just multiply Rating by minus 0.3 after doing this way can eliminates the function X. Okay, so now we are ready to eat. Eliminates the first part. And that gives us he square plastic going one b last year 0.18 you know, on while equals 20 Ponzio 36 which is a no homogeneous o e. So the homogeneous part has a characteristic quick Arctic you question and we can solve this by using a quadratic formula so are running close to minus five minus foot off seven divided by I have great r two equals two a minus five passport off seven divided by 100. So we decided to use this our heart we instead off this particular numbers here for our convenience and the further so for the homogeneous heart. Why h equals two c 13 r one p plus c two into the opportunity In the Furneaux homogeneous parts, the right hand side is a constant. So we guess that particular solution is a constant and we're just plugging back. That gives us you coast to 20. So in the end, the solution for why? Because to see one you three are one p process. See to you tree our duty prosperity. Once we have use for white, we go back to the young You know of the X He closed toe 100 my best three times. Do you y plus y reaches 100 you about three hours. One pass one times See one into the are won t plus the 100 year. I got three our tool plus one See to it really hard to key past 20. So this is not enough because we have some initial conditions. So x zero equals zero that gives us, um this equation and the Y zero because 20 years us c one paralysis e two equals zero. So if they saw me as a system of equations that we have see one you close to 30 over wrote off seven c chewy coastal minus 30 overrode off seven. So this complete our solution. So we have toe constant C one and C two and then we have two accounts in the hour in our to And, uh, this is our final solution for this system of these and the represent Head to the moment t cause dio any constant me Positive, constant The mess off salt in tank I 10 a.m. Thank me

In this video to go through the answer to a question about 45 from chapter 9.5. So we're given a model off integrated tanks. Andi asked to find how the quantities of salt in each of the tanks changes three time. Okay, so I need to write this. Formalize this model mathematically. So next one is the value or the massive salts in tango, eh? Okay, So how can we say that? Yeah, what we say about the way that that Kansi changes. Well, there's three liters per minute off pure water community left that's is assaulting that works. That's not gonna change the massive soul, Tuncay. But there's four. Lee is for minute leaving. Holy it involved is of water per minute. Leaving tank, eh? Okay, So is gonna be leaving. Ah, yeah. Whatever. Quantity is in Tank A already, which is X one. We're gonna have four over 50 kilogram this leaving per minute, because that's what's going to be. But then we're also gonna have some amount coming back in from Thank be by the same logic. That's gonna be one over 50. Because technically is also got 50 liters. Andi is coming into tank. A warm liter per minute. That's next to similarly X two s o the quantity that massive so in a V changes by Well, Foley is coming in from Tank A with us 50 leads is from tank, eh? And then we're leaving. We're having one Lito leaving in sank. A of three years of the salt water will be leaving. Ah, on the right. So we're losing four. Lee is out of the 50 that we have for a minute from take two. Okay, so how can we write this matrix? Normal form? Well, if X is the vector X one x two Hey, then we could take a factor off one of 50 other friend and we're gonna have four groups minus for one for minus four. That's my ex. Okay, so let's call this guy without the one over 50. Let's call it a prime. Oh, a star. Uh huh. Then a is one of 50 petty stuff. Okay, so let's first work out. What? The Eigen values off. Ah, of Matrix A star, huh? That's gonna be minus for minus, huh? One minus four, minus four minus R. That's four plus. Ah, squared clothes for It's just kind of a 60. Bring that damn wrong. So this shouldn't be a minus here, which makes that a minus. Robyn an A plus. Okay, that makes more sense. So this is gonna be, ah, squared minus plus eight. Uh, that plus 16 months. Four is plus 12. Fact arise. That's gonna be our poor six are close to then if we said about equal to zero, then we've got Ikon values I was equal to minus T on minus six. Okay, so now that means that the values off, um, off matrix a are gonna be so yeah, let's call these crimes then. Now our stars, then the economics of Matrix A B R, which is minus two over 50. I'm minus six over 50. Okay, so Thea Eigen vectors on your hand off. Uh, a are gonna be the same at the other. Factors off a star, because just multiplying by what must find the Matrix. Try a constant doesn't change the Eiken factors, but it just rings. Aiken values but the Okay, so let's do a star minus, Uh uh, prime. Well, let's use the particular values of our stars. The 1st 1 was minus two plus two losses I times affecting you want it was zero. Was this gonna be did to me to remind us to one for minus two? You want equal zero? So just read off water you want gonna have to be? Well, if the first component it is one than the second component must be too Next. Oh, based, Ah, plus six times I tells you to equals. Ever. So what's this gonna be? That's that's full minus six is I mean, minus four plus six is plus two and one for myself for 62 You too, Because they were again. We enough. Quite simply, you two is equal to with first importance. One certain companies from honesty. It's there are two. I came back to stop a matrix. A star should also directed the matrix A. So that means that the the general solution is gonna bay C one times e to the first wagon back that wagon value, which is to no, uh 50 which is also one but with 25 toasty. That's what the first item vector I just want to for us a second constant C two times e to the minus six over 50 which is three t 25 times by vector one minus two. Okay, but with Okay, so let's let's first figure out. What the Well, you this zero, that Secret service with the initial value, it's gonna be C one and tuck one too close. See? Two times. 1 to 1, minus two. So we want to figure out what, See, Want to see to our little in the question that this is equal to Bye Said I said initial amount of salt initial massive socks in tank A is 2.5 kilograms. Attack B is zero of the grounds. Okay, so this tells is that, uh well, the second bomb components tell us that C two equal see one putting that into the top of the component that houses the two times seem once. It was fine. So C one and C two must be quite stupid. Died? What about city? Okay, therefore, X, it's on t. It's gonna be equal to C 0.5 over too. Times eats the minus t over 25 12 plus 2.5 over two times e to the monastery. Thio. Oh, it's 25 turns one minus two


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