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Math 3150 Midterm 1 19 Feb, S2014

Problem 1.

(Heat Conduction in a Rod, Ends at Different Temperatures)

Throughout, k is the mean thermal diffusivity, usually written as Fourier’s constant K0 divided by specihe heat c and moss density perunit voliune p.

j (a) [40%] Consider the heat coadiictioii problem in a laterallyiiisulated rod ofieugth 1 with 000 011(1 at zero Celsius and /—the other end at 1 Celsius. The initial temperature along the rod is given byfunction [(3;) a;.

Ut — ku,.2., 0 <a; < 1, t> 0,

u(0.t) = 0, t>0,

u(1,t) = 1, t >0,

= .r, 0<x<1.

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.

(b) [60%] Consider tile heat conduction problem in a laterally insulated rod of length 1 with one end at zero Celsius 011(1 the other end at one Celsius. The initial temperature along the rod is given by function [(c) ⁼ 1 +z.

= kn. 0 <x < 1, t >0,

u(0,t) = 0, t >0,

u(1,t) = 1, t >0,

u(x,0) 1-!-x, 0 <z <1.

Solve the rod problem foru(x.t). It is necessary to derive the product solutions. Provide Fourier coefficient formulas.

Evaluate all Fouriercoefficients. Then display the final answer u(x, t).

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2. (Total Thermal Energy in a Rod)

Au expussom ior the tiuit—d’peiidtiit total tilernial energy contained in a rod ^{i: =} 0 to x ⁼ L, with mutorni cross—

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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

11(111 iquitl^{1011 l(} kit,... Assmniii ( 011(1 f) are coiistaiits.

Suppose o(,r 1) and i’(x t) are two temperature distributions for the sonic rod which supply the saute total thermal (lorgy br all I.

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(a) j30%] Explain why

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^{v, t) dx.}

,. (ii) (i0%1 Differentiate on t across the equation of (a). Simplify the resulting ecluation ^{usilig U)} k7trr and v1 ⁼ kv1..

to obtain

‘ar(L, t)^—‘ur(Df) ⁼ Lr(L,t) Ur(0,t).

At”(c) [20%] Explain the meanilig of the equation in (b) in terms of heat flux and Fourierh Low.

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Problem 2. (Total ^rrheI1nal Energy in a Rod)

An uxplessioli for the ti111c—dLpe11deHt total tlwriiial energy coiit,aiued in a rod c = 0 to ^{.: =} L, with uniform cross—

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I lou ( is fl1( 5l)(Clh( 1iett [) IS tlW1111155 (lensity per unit volitnie and u(x,t) is the rod teniperature, satisfying the

hieiit (‘(flliltiofl IL( h0,pr. Asslilne ( 011(1 [1 1U0 coiistaiits.

Snppose o(r,t) 011(1 e(x. t) are two temperature distributions for the same rod winch supply the same total tliernial

cimergy for all 1.

(a) [30%] Explain wily

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^{u,(:v. t)dx=}

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^v(x,t)(Lc.

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(h) [60%] Differentiate on across the equation of (a). Simplify the resulting equatioli usingu, ku audi m’ ⁼ kerr

to obt,aii

r(L.t) —u1(0t) ⁼v(L,t) lr(0,t).

(c) [20%] Explain the mueanhllg of the equation in (b) in terms of heat flux and Foi.irier’s Low.

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Problem

3. (Steady-State Heat Conduction on a Rectangular Plate)

Consider Lapliwes equation _d,r— + _()y— on the rectangle (I < ,t < 1, (1 < p < 1 subject to the boundary conditions

iLr(.:,p) ⁼0 for :r = (I tuid ,r⁼ 1, n(.r, y) 0 for y ⁼ 1,

u.(,i,

^{j) =} [(a) for p ⁼ 0.

Ala) 70%) Find the product solutions a(.r,y) ⁼ X(a)Y(). Include a cheek that each product solution satisfies required threc zero boundary conditions.

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^{(b) (30%] Let}

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^{be the smn of the first three} eigciifiiiictioos which is the sum of time first three X—answers. Find

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Prob’em 3. (Steady-State Heat Conduction on a Rectangular Plate)

Conshkr Laplace’s equation dc— + d1j— on flit rictongle () < i < 1, (1 < p < 1 subject to the boundary conditions

ii..r(,i:,i) ⁼ 0 for :i (I and .r = 1, u(, y) (I for p ⁼ 1, u(.i’,y) ⁼ fCc) for p 0.

(a) [70%] Find the prodmt solutions u(,r,i) ⁼ X (.c)Y (y). Inelwhlk that each product solution satisfies the required three zero boundary conditions.

(h) [30%] Let

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be lie suni of tlit first flirci’ cigeufunctiuns, winch is the suui of the first three X—answers. Find

u(.u,_ti).

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Problem

4. (Steady-State Heat Conduction on a Disk)

(cushier till’ stead v—state heat conduction problem in polar coordinates

1

v(r,8) + v.oo(r,8)

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^{0, 0 < i <2,} 0 <8 <2,

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^u(2,8)

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^{,f(8), 0<8< 2.}

(a) 1(i0XJ Find the product solutions n

=

^{R(r)e(8),then}identify tin orthogonal set and the interval, Stop at this ^stage:

omit superposition, omit the_series solution and do not develop formulas ^{fur the} Fourier coefficients.

(b) [40Yi Calculate ‘u(O, 6) when f(8)

=

^{(1 on 0 < 8 <,} 1(8)

=

⁵⁰ on 8 < 2. Hint: Tue Poisson integral theorem mid the Mcmi Value Theorem.

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