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Question 33 & dz= 2Lefl end (1-0) of a copper rod of length ]OOmm is kept constant lemperalure 0f Temp = 10+d2 degrees and the right end and sides are insulated...

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Question 33 & dz= 2Lefl end (1-0) of a copper rod of length ]OOmm is kept constant lemperalure 0f Temp = 10+d2 degrees and the right end and sides are insulated, so that the temperature in the Ou 3 u rod, u(x,t) obeys - the heat partial DE; where D = H mm-/s for copper: 8x'Write the boundary conditions for U(x,t) of the problem above: Note that for the left end the condition is not zero, which presents problem for the separation of variable method. Use u(x,t) = Const + ((x,t) and select

Question 3 3 & dz= 2 Lefl end (1-0) of a copper rod of length ]OOmm is kept constant lemperalure 0f Temp = 10+d2 degrees and the right end and sides are insulated, so that the temperature in the Ou 3 u rod, u(x,t) obeys - the heat partial DE; where D = H mm-/s for copper: 8x' Write the boundary conditions for U(x,t) of the problem above: Note that for the left end the condition is not zero, which presents problem for the separation of variable method. Use u(x,t) = Const + ((x,t) and select Const appropriately to have the zero boundary condition for ((x,t) atx 0. Check that if o(x,t) is solution to the heat equation; u(x,t) = Const + ((x,t) is also solution. Find the boundary condition for ((x,t) al the right end of the rod_ (b) Find the general solution to heat equation for ((x, satislying the boundary conditions found in (a) and then delermine the temperature in the rod, u(x,t) Using the solution in (b) find u(x,t) that satisfies the initial conditions: 3T 5t u(x,0) =10+dz+sin d, sin +dz sin 200 200 200 (d) Using the solution in (c) determine the temperature in the middle of the rod at t60, record your answer t0 significant figures (this is not three decimal places)



Answers

Consider a uniform rod of material whose temperature varies only along its length, in the $x$ direction. By considering the heat flowing from both directions into a small segment of length $\Delta x,$ derive the heat equation, $$ \frac{\partial T}{\partial t}=K \frac{\partial^{2} T}{\partial x^{2}}
$$ where $K=k_{t} / c \rho, c$ is the specific heat of the material, and $\rho$ is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that $K$ is independent of temperature, show that a solution of the heat equation is $$ T(x, t)=T_{0}+\frac{A}{\sqrt{t}} e^{-x^{2} / 4 K t} $$ where $T_{0}$ is a constant background temperature and $A$ is any constant. Sketch (or use a computer to plot) this solution as a function of $x,$ for several values of $t$ Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.

Hi, friends. This is the problem based on transfer our feet by conduction. Here it is, given the area perception of the rock. All right, Toby. Full into 10 to the par minus four. Military square. Cold temperature. Toby 65 Researches and heart temperature. To me. 100 degree services. Cooper is knowledge made of copper, often thermal conductivity. 3 85 4 permit er per Kermit. And that office state is 15.2. War permitted per capita. In the first part, we have to fight. The rate off flew off it. And second part, we have to find the state for right off the office to be saved. Rate of flow off it through the copper rod and parliament off couple Substituting the metal. Yeah. So it is to be 5.39 words in the second part for the same floor. Freed. Still are floor length. Yeah. Care for students. 50.2 area For him to come to the department store 65 minus zero. You could get 0.242 m. That's all. Thanks for watching it

We have a difference off 100 degrees passing to three different materials. And we want to know where is the temperature at its junction. So between the breasts and copper and the copper and aluminium and in the end, the hit current and only in the aluminium You know the temperature at the junction. We need to know how much heat is bath in true. So they hate off each dress for we can do this because we have the information for each material. Ah, to calculate the Timorese activity on DDE the heat passing to be just delta t divided by the sum off all thermal relativity. Let's go play this some off the temporal activity off the three materials. This is our off the breasts plus are of the copper. Both are off the aluminium and by definition, disease. That's right here just to remember our is you go toe. I'll divided by cuppa, eh? So l is the thickness off the material that is here to have seven years, 18 centimeters, 24 centimeters. Copper depends on the material and our is the cross sectional area that they give you. And the cross sectional air is the same thing. All of them. So let's write this as one over. Okay? Times I'll be divided by a cup of tea, both o que ever Catholic You blows l a divided by couple. Okay. Oh, now we can just black values the memories of 60 off the three materials you can look in all I know and the book is given and the cross sectional area It's zero point. There are 00 25 meters l b is tough. So So I'm stand to the managed to because everything's in meters in here a 375 for the breasts. Here is 18 time stands we managed to and for the copper is 290 the temple resistive ity And for the aluminum you is 24 time stand to the managed to and the temperature acidity is the biggest. Them always 1980. This gives us total turmeric festivity off 4.54 me wrists were divided by Mira Scare Kelvin divided by what's so With this information, we can find the total rate flow off this piece. Let's do this The totally hate flow. It's just Delta t divided by the total are there's this 100. If I did by 4.54 there's this 22 went there. Six. What, then I think about the hit was that is the same in all material. So from the breast section, the procession on the album in a session will be the same value. So because of that, you can find the temperature using this fillets. Find Francis the 1st 1 Let's find the breasts called projection and the breasts cop Rejection we have that's right in the breath. Proper jumps and show the heat Moving in the breasts is the heart. That is 100 ways. My honesty off the junction divided by are only for the breasts because it's only this region here, and this gives us 100. That's right here we can first, let's reorganize here so T is equal T hot, minus the heat's bursting through the size off the breasts world. The demo resistive off the brass and the area and these guys are, as we said as I said, is the same in our material. So garbage is 2200 or six watts, so we can just find temperature is equal to 100 minus. So I need to 0.6 times this size of the breast. Good 12 by the by 3 75 times the area there's 0.0 23 and this gives us a 69 0.3 degrees cells. So this is the temperature in this first junction here. So this guy here, 69 point to you agree, cells. So let's find off. This guy here now is the same thing. But now they hated with flowing through the aluminium and the difference from this Do you bid from zero toe some value that we don't know. So let's do this. Hey, Passing through the aluminium is equal to the temperature of a T junction minus the temperature could divided by the restive ity off the palomino. These guys D'oh! So we can write directly that the temperature at the junction is the hit passing to the Dominion. Hurtful Look, Palomino, The left off the aluminium divided by there is a city of the amino in the area against a positive values. So 22.6 time 0.24 divided by my hand and 10 that This is the timorous activity off the aluminium that the cross sectional area 0 23 and this gives us 25.62 years. What we have here 25.6 the recesses. The last part. Ask you what is the hit current in the aluminium? We already calculated this because from this value here, the heat flow off the L Amina will be the equal to hit flow off the copper equal to hit flow off the breast is equal to the total Hateful. There is trying to 0.6 What?

Okay. So disgusting. It's also Keith. Karen. Okay, so this question requires us to complete how much heat Basic and that fluids from a boiling motel into a nice on what? That mixture to a long road now in the situation this long road it's basically made of made up of two types off meta one is made up off Copa and then the other meet ups too. All right. And his accidents with complete They hit the second that flows from the boiling water into the ice. What are mixture? So basically, what is accidents is to couple in the hit current. If you company the heat current is to populate the hits can't. So let's protect a diagram for question. All right, so this is fine with hot water on. Decide. It's called water. So we have foot. What? I hear way. So OK, so it is a long road. Okay, it's sometimes a meter corpora. So part of it made up off Copa and then the other side were up of still between on the limp. So that's it Helps him let me do that. Suspect is for still all right. And then the the copper end is wonderful. Kalbar the compliant is one talon. So if you bring with an opportunity to sell land called what else? They're devious Celsius. Okay, Now the area of this is giving us four points. Ever say that four planes, They're sensing it A script for their sensing. It's a script. All right, so that's more Want to know how we can solve. So we know he finds time stevia to wreck it. H minus two years. Sorry. There were about a by in the situation. We don't know the length way. We don't know the leg off the steal. However, they see how best we can go above No, What? We need to know you that we can companies, uh, the heat current for Copa and then equated to the hits karent off steal because at the junction where they are connected together, they will have a formal creatively. They will be at the family people so we can see that the heat going from Papa, his parents off steal. All right, So picking the information for come up, we can straighten everyone in conflict with the heat coming off copper Now the family conductivity of copper straight five uh What? Amita Kelly. Okay. And then that off. Still see you watch Meteo. Alright, so that's one. So, considering the hit grants for court, we can see the age or see you immediately. You It's people to the year ago that someone less stuff bring someone connected. If it's fun, it's five times the area we deserve. So so 4 m there. Don't forget that the area is four centimeters committing to meet us. This is what you get to record 100 minus 65 underlying assist. In fact, that is the temperature under between them, it's semi temperature or the city temperature between the joints off the roots. But so all of this divided by the length, which is 1 m. Okay. All right, so at the end of the at the end of the way, we have it. You know what, 539 watts. So that is the heat parent that is associating with rocks. His going associating with both brothers of the junctions are now We should complete the land with the steal. So remember we say the hits guarantor corporate is equal to the heat's guarantor Still. All right, so since we compare the heat going to combatively 5.391 can equate it to the heat gun off. Still, which is 15. The family collectively 50 times the area. Zero point. Those are I'm fearful into bracket the temperature of 65 a zero. Remember, 65 is the steady temperature between the two roles. My zeal in the zero is the other and still which is inside the ice and water mixture. All right, so all of these red about the land to ST A We're committed to this and that they have ltu caught when we compute this to give us 1.305 divided by 5.39 And if she hears about 0.24 m which is also you go to 24 centimeters all right?

Okay, so we know the heat conduction and convenient. Okay, a chance. Dalati over l cases thermal conductive the a C section area and lt's a change in temperature out. It's the origin that in this case we use a thermal conducted off copper, which is 385 watts per meter test album is because you want a in the boiling waters made out with copper, and he's throwing on the boning water park to the eyes. Texture bar. Okay, in the sexual areas, human as a four centimeter square which is four times 10 to apartment, four meters were and changing temperatures is from 65 degrees houses to 100 degrees houses, which is 35 helping OK and the origin and was given a long meter. Therefore can plug in back to the question to determine e t conduction, right? Uh, which is age able to 385. Once premier attends Calvin and in times four times tension partner before Mitre Square and Times 35 happened over 1.0 meter. Okay, and this will give us hope the heat conduction rates. It's about five point 39 watts for the next question. Well, we know the heat conduction commingled. Okay, a $10 T over L Therefore, let's alchemy go to K dollar the older age. So alto chemical okay, to attend a has gelati to over age. In this case, K two is determined canal activity of the steel Since I'll to is is made out with the steal. Okay, in the Democratic people's steer is 50 point to watch. The media has covered and they share the same section area from the last bush which is four times 10 to the power in 84 meters square. And changing temperatures from Jodi Resources to 65 degree shots is because these two apart is made out is in the eye stature. Okay, so in a nice you know, the temptress zero degrees ounces Here we come here to Calvin will have 273 coat and we do some calculation here. We'll find out the changing temperatures 65 Children And from the last question we know the heat conduction raised 5.39 waas because, um, this tool partial is saying he conduction right because they will connect it with each other. OK, so we can just use the sheikh indefinite from the last question. So therefore could determine the AL to which is evil to, um, you see, Okay, to just 15. Going to Watts a meter, has Children in times four times, sends about 94 meters square and has 65 Calvin over 5.39 watts. And this will give us altruism. You go to 0.242 meter and he's an answer.


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