Question

temperature f(x) °C, where 5. f(x) = sin 0.1 x 6 f(x) = 4 - 08 |x - 5 7. fix) =x(10 - x) 8 Arbitrarytemperatures at ends. If the ends x = 0 and x= Lof the bar in the text are kept at constant 20. CAS PROJECT. Isotherms. Fim solutions (tempe rature s) in the squa with a 2 satisfying the followin tions. Graph isotherms. (a) u80 sin Tx on the upper s urxo) (b) u = 0 on the vertical sides, ass sides are perfectly insulated (c) Boundary conditions of your solution is not identically zero). temperatures U and U2, respective ly, what is the tem- perature u1(x) in the bar after a long time (theoretically, as t0o)? First guess, then calculate. 9. In Prob. 8 find the temperature at any time. 10. Change of end temperatures. Assume that the ends of the bar in Probs. 5-7 have been kept at 100°C for a , )l long time. Then at some instant, call it t 0, the t 0 temperature atx = L is suddenly changed to 0°C and kept at 0°C, whereas the temperature at x = 0 is kept vl0,t)=l at 100°C. Find the temperature in the middle of the bar at t 1, 2, 3, 10, 50 sec. First guess, then calculate. Fig. 297. Square BAR UNDER ADIABATIC CONDITIONS 21, Heat flow in a plate. The faces in Fig. 297 with side a 24 The upper side is kept at 25°C kept at 0°C. Find the steady-st in the plate. "Adiabatic" means no heat exchange with the neigh- borhood, because the bar is completely insulated, also at the ends. Physical Information: The heat flux at the ends is proportional to the value of au/ax there. 11. Show that for the completely insulated bar, ur (0, ) 0, Wr (L, t) = 0, u (x, t) = f(x) and separation of variables gives the following solution, with An given by (2) in Sec. 11.3. 22. Find the steady-state temperatu 21 if the lower side is kept at U°C, and the other sides are into two problems in which th is 0 on three sides for each pr nx -(enm/L)2t L ux, t) = Ao + An cos Mixed boundary value pro state temperature in the plate it and-lower sides perfectly ins at 0°C, and the right side ker 23 n-1 12-15 Find the temperature in Prob. 11 with L = T, c = 1, and 13. f(x) = 1 15, f(x) 1 x/T 12. f(x)= x 24. Radiation. Find steady-sta 14. f(x)= cos 2x rectangle in Fig. 296 with perfectly insulated and the n medium at 0°C according to h>0 constant. (You will ge (16. A bar with heat generation of constant rate H(> 0) is modeled by ut= + H. Solve this problem if L T and the ends of the bar are kept at 0°C. Hint. Set u = v - Hx(x - T/(2c2) condition on the lower side 17./Heat flux. The heat flux of a solution u (x, t) acrossx = 0 25. Find formulas similar to (17 in the rectangle R of the text is kept at temperature f(x) a is defined by (t) = -Kur (0, t). Find (t) for the solution (9). Explain the name. Is it physically under- standable that goes to 0 as t co? at 0°C.

please solve 17 for me thanks~~ :) !

Answer 1

16)

Consider the heat equation u, =c'u + H . The objective of the problem is to solve the given equation if L- and the ends of the bar are kept at 0°C Let -u-Hx(x-x)/2e2 with the initial conditions u(0,1) 0 =u(0,H) and u(r,t)=0= v(,t). Also, v(х,0) - /(*)+v(х,0)- Нк (п-я)/2c? v(x,0) f(x)+Hx(n-r)/2e


Differentiating the equation with respect to t implies u H(2x-/2e дх дх a'u _&v_H]c ax Substituting these values in the heat equation implies av Ov +H ax v Н+Н v ax with the boundary conditions v(0,1)= 0 = v(z,1) ax Thus, the equation reduces to at and the initial condition v(x,0)=f(x)+Hx(n-r)/2c*


with the boundary conditions v(0,1)=0 = v(7,t) i The solution of the equation IS ν(s.) -Σν (h, ) ntx -ΣΒ, sin L where ntx dx L B Hence, the solution of the given equation isBsin an ntx IS -Hx(x-/2c where e L )H()/2 ]sin ntx dx L

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we have nx e -(1) 1 cnt where L and also given condition is )Ku, (0, ) , (0,))


From (1) we get nz nx B. cos L nT Be u, (0, t) -) ) K n B KT ()=- The cooling of the ends led to heat loss andu>0 The temperature at which the ends were kept

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