c- ( ( a) 0 o) / 6, t ) U I-i Problem 1. (Heat Conduction in a Rod, Ends at Different Temperatures)

Size: px

Start display at page:

Download "c- ( ( a) 0 o) / 6, t ) U I-i Problem 1. (Heat Conduction in a Rod, Ends at Different Temperatures)"

Constance Little
9 years ago
Views:

1 Throughout, k is the mean thermal diffusivity, usually written as Fourier s constant K0 divided by specihe heat c and Problem 1. (Heat Conduction in a Rod, Ends at Different Temperatures) 19 Feb, S2014 Math 3150 Midterm 1 _ I C c ( ( a) 0 o ) U <U () (3 o Ii C) o) 6, t Evaluate all Fourier coefficients. Then display the final answer u(x, t). Solve the rod problem for u(x. t). It is necessary to derive the product solutions. Provide Fourier coefficient formulas. u(x,0) 1!x, 0 <z <1. = kn. 0 <x < 1, t > 0, u(0,t) = 0, t >0, u(1,t) = 1, t >0, (b) [60%] Consider tile heat conduction problem in a laterally insulated rod of length 1 with one end at zero Celsius The answer u(x, t) to this problem is exactly the steady state tenipero.ture. Find tile answer u(x, t) and display a complete answer check. =.r, 0<x<1. u(1,t) = 1, t >0, u(0.t) = 0, t>0, Ut ku,. 011(1 the other end at one Celsius. The initial temperature along the rod is given by function [(c) = 1 + z. a 2., 0 <a; < 1, t> 0, the other end at 1 Celsius. The initial temperature along the rod is given by function [(3;) a;. j (a) [40%] Consider the heat coadiictioii problem in a laterally iiisulated rod of ieugth 1 with (1 at zero Celsius and moss density per unit voliune p. Name KE

Then display the final answer u(x, t). Solve the rod problem for u(x. t). It is necessary to derive the product solutions. Provide Fourier coefficient formulas. u(x,0) 1!x, 0 <z <1. = kn.

2 t F 3 3 ii _ ( 0 1,, x r 1 r E r I, C, fr 0

3 At 4 IlA 4> ], f k [ A (, b) 4 1)d) _;fa ar(, t) ur(df) to obtain v, t) dx t ioiiah itito A is ioblem 2. (Total Thermal Energy in a Rod) (lorgy br all I. c Fc 17 ]1 Thç ) (±\, ç i)imaf r?7y K o i;i1 e IdfrAl 1i&ra1 CfM(X 1 v() (c) [20%] Explain the meanilig of the equation in (b) in terms of heat flux and Fourierh ow. = r(,t) Ur(0,t).,. (ii) (i0%1 Differentiate on t across the equation of (a). Simplify the resulting ecluation usilig U) k7trr and v1 = kv1.. A (a) j30%] Explain why J u,(x. t) dx f :::w Au expussom ior the tiuit d peiidtiit total tilernial energy contained in a rod i: = 0 to x =, with mutorni cross T1 i becriuice i tj OYV1e) v& an A (:( drf 11(111 iquitl 1011 l( kit,... Assmniii ( 011(1 f) are coiistaiits. J cpu(xj)adx. 2,.i e bev c Xy iihol ( is flit specific heat, p is the moss density per unit volume and u(x, t) is the rod temperature, satisf ing the I (_,r,.,l.,)...c. I. _c Ø i oi: i (,) (ô, jo Jo v(i I V() vw),) Suppose o(,r 1) and i (x t) are two temperature distributions for the sonic rod which supply the saute total thermal

Simplify the resulting ecluation usilig U) k7trr and v1 = kv1.. A (a) j30%] Explain why J u,(x.

4 . Problem 2. (Total rrhei1nal Energy in a Rod) An uxplessioli for the ti111c dpe11deht total tlwriiial energy coiit,aiued in a rod c = 0 to.: =, with uniform cross s 1 n i titi 01(0 A is J I lou ( is f l1( 5l)(Clh( 1iett [) IS tlw (lensity per unit volitnie and u(x, t) is the rod teniperature, satisfying the hieiit ( (flliltiofl I( h0,pr. Asslilne ( 011(1 [1 1U0 coiistaiits. Snppose o(r, t) 011(1 cimergy for all 1. e(x. t) are two temperature distributions for the same rod winch supply the same total tliernial J (a) [30%] Explain wily u,(:v. t) dx = v(x, t) (c. (h) [60%] Differentiate on across the equation of (a). Simplify the resulting equatioli using u, to obt,aii r(.t) u = v(,t) lr(0,t). j 1(0t) (c) [20%] Explain the mueanhllg of the equation in (b) in terms of heat flux and Foi.irier s ow. ku audi m = kerr Jc1 ( c.i% t i sj) 11 A i J) A ( J I j J jf.j (i I;, b \I 4,. Jo. i I lj ii! 4 i 104+,, J \ (I i Tkc*cr,H V Ot) 9

I( h0,pr. Asslilne ( 011(1 [1 1U0 coiistaiits. Snppose o(r, t) 011(1 cimergy for all 1. e(x.

5 1 IC t ) Y. ;:; 7.. ) I I \ Ji 4..c....

$. ) I I \$

6 I 4 1\ u(m y). f\_ (b) (30%] et f(:r) be the smn of the first three eigciifiiiictioos which is the sum of time first three X answers. Find required threc zero boundary conditions. Ala) ir(.:, p) = 0 for :r = (I tuid,r = 1, n(.r, y) 0 for y = 1, u.(,i, j) = [(a) for p = 0. 70%) Find the product solutions a(.r, y) = X (a)y(). Include a cheek that each product solution satisfies Consider apliwes equation + on the rectangle (I <,t < 1, (1 < p < 1 subject to the boundary conditions d,r ()y ( (: (O) 1 C (o (:Q 6 Y 1: )1 w 3 iv7i o CCX ( f( T 0 y:o ch(: O (,c(: 0 i uj 0 0, () r1 Problem 3. (SteadyState Heat Conduction on a Rectangular Plate) 9c Corr! 5( cv Y( ((: ce 7.z (fl: 0 X (o: X (io 1+ :Q X y

7 2 (fl.r)7i 1T1t 1 (1) ) UhI1 q (i:)] 4J d J) T1i 3( 3 J i.10 TTTI I T! I 11T..)TiliJTl) it) )).l1 4H.ITJ )II} 1 1))1Il1))i 1 0.T) )J.II X110tfll)T10.) A.IUplIfl()1 1 P!d [%OI () IIo!4llIo 4311p0.Id i i (1i)(;) x = (Ii ) 411p01c Ii ioj (.i)f = (Ii.r) ii Ii.oj o (Ii.r) 1 = piiu n : oj o = (Ii.r) n _1i()._.r() ))UT(IU J.!J1)pI10) II03U1l1).) HTI0i1)ITO.) n])un()q im14 0J 4)Jql0 I > 1 > > > 0 J14 TIC) (wia J1T Tfl3DOH U OT.)flpt10D WOH spuos) g WOJJ OQr2 o i:ec 9; ),)21i]7 1_i ; 2 josj 11 x (p. O22 K ci o1 (2,3ç)J 4l., }...(1 (i),i 77 P 2 ),) F Q; A c7zj ) ( ) ii ); JI XJ5 (J,,.IJ :()(.2. fr.? )..1 ç J 1,Q :,k (o 9 CQ A)ul, <2 (I 1)ç) C2 0 1 pi),1,nc1 (G() I_JJ <f.

) HTI0i1)ITO.) n])un()q im14 0J 4)Jql0 I > 1 > > > 0 J14 TIC) (wia J1T Tfl3DOH U 110 11OT.)flpt10D WOH spuos) g WOJJ OQr2 o i:ec 9; ),)21i]7 1_i ; 2 josj 11 x2 401 7 (p.

8 NI dc d1j ii..r(,i:, i) 0 = for :i (I and.r 1, = u(, y) (I for p 1, u(.i, = y) = fcc) for p 0. u(.u, ti). (a) [70%] Find the prodmt solutions u(,r, i) (h) [30%] et ICc) be lie suni of tlit first flirci cigeufunctiuns, winch is the suui of the first three X answers. Find required three zero boundary conditions. Prob em 3. (SteadyState Heat Conduction on a Rectangular Plate) Conshkr aplace s equation + on flit rictongle () < i < 1, (1 < p < 1 sub ject to the boundary conditions = X (.c)y (y). Inelwhlk that each product solution satisfies the 11 C (r \\)Hi k I c_) ). 4 3 S I No (IC3 0 N 00 C c q \E l O1flt 1. tt()9 Co 5c co i, ccc,0 A0 0 CotQ. Fo C t, me. r c( ) 2 1.l( o. o NO._ 1O C çr c c D C1 e e y gc 7?

Find required three zero boundary conditions. Prob em 3.

9 c I \ II I 0 4, jc J N t I 4 I

10 4 (5o) f(( (1 (b) [40Yi Calculate u(o, 6) when f(8) = (1 on 0 < 8 <, 1(8) = 50 on 8 < 2. Hint: Tue Poisson integral theorem omit superposition, omit the_series solution and do not develop formulas fur the Fourier coefficients. (a) 1(i0XJ Find the product solutions n = R(r)e(8), then identify tin orthogonal set and the interval, Stop at this stage: mid the Mcmi Value Theorem. I u(2,8) =,f(8), 0<8< 2. 1v(r,8) + v.oo(r,8) = 0, 0 < i <2, 0 <8 <2, 12e M; v f cf!ec. ro, f (p) (: (e.); C) ç ) 7 4 (r,&) fru a), (r, ) Problem 4. (SteadyState Heat Conduction on a Disk) (cushier till stead v state heat conduction problem in polar coordinates 1 9c 7t

(a) 1(i0XJ Find the product solutions n = R(r)e(8), then identify tin orthogonal set and the interval, Stop at this stage: mid the Mcmi Value Theorem.

1.- L a m e j o r o p c ió n e s c l o na r e l d i s co ( s e e x p li c a r á d es p u é s ).

1.- L a m e j o r o p c ió n e s c l o na r e l d i s co ( s e e x p li c a r á d es p u é s ). PROCEDIMIENTO DE RECUPERACION Y COPIAS DE SEGURIDAD DEL CORTAFUEGOS LINUX P ar a p od e r re c u p e ra r nu e s t r o c o rt a f u e go s an t e un d es a s t r e ( r ot u r a d e l di s c o o d e l a