HW2+09F+solution - ME487 HW#2 Solutions 1 Set a control...

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Unformatted text preview: ME487 HW#2 Solutions 1. Set a control volume as shown below on a fixed coordinate system. The rate of change of internal energy for the element ΔT ˙ Qv = ρcAΔx Δt x: (1) Heat flux across the upper face of the element is: dT ˙ Qu = −kA dx That across the lower face is: dT ˙ Ql = −kA dx x x+Δx Thus net heat flux for the element becomes: dT ˙ ˙ ˙ Q = Qu − Ql = kA dx -1x+Δx − kA dT dx (2) x Internal heat source due to Joule heating is: ρr Δx ˙ Qs = I 2 R = I 2 A (3) From first principle, ˙ ˙ ˙ Qv = Q + Qs (4) Thus plug Eqs. (1), (2) and (3) into (4), ρcAΔx dT ΔT = kA Δt dx x+Δx − dT dx + I2 x ρr Δx A (5) Divide Eq. (5) by A x ΔT =k ρc Δt dT dx x+Δx − Δx dT dx x + I2 ρr A2 Taking the limit as x 0 and t 0 becomes: ρc ∂ 2T ρr dT = k 2 + I2 2 dt ∂x A (6) But ∂T ∂T dx dT = + dt ∂t ∂x dt (7) and dx = Vw dt (8) Plug in Eqs. (7) & (8) into (6) ρc ∂T ∂T ∂2T I 2 ρr = k 2 − ρcVw +2 ∂t ∂x ∂x A -2- 2. Approximated value: qa = (θm + 273)2 3 = 2.9 J/mm 300000 Theoretic value: 3 qt = ρ[c(θm − θ0 ) + L] = 2700 · [1.028 · 630 + 398] = 2.823 J/mm Thus, the error can be obtained as: %error = 2.9 − 2.823 · 100 = 2.79 % 2.823 3. (a) r= κ= ξ2 + y2 = (−8)2 + 52 = 9.43 mm k 0.05 = = 7.35 mm2 /s ρc 8 × 10−6 · 850 Qi = EI = 30 · 200 = 6000 W for 100% efficiency 10 · 9.43 ux r = = 6.41 2κ 2 · 7.35 Therefore, K0 ux r 2κ = 0.81 × 10−3 10 · (−8) ux ξ = = −5.44 2κ 2 · 7.35 Now, T = = = T0 + ux r ux ξ q · exp − · K0 2πkh 2κ 2κ 6000 · exp(−(−5.44)) · 0.81 × 10−3 25 + 2π · 0.05 · 8 470 ◦ C -3- (b) 1 4.13 × ρcux h × Y 1 = + Tp − T0 Qi Tm − T0 w 2 1 ux w + 5 4κ 1 10 · w + 5 4 · 7.35 where, Y =y− For 2D (thin) plate, Qi = 8k (Tm − T0 ) · h Thus, 6000 = 8 · 0.05 · (1500 − 25) · 8 w = 3.15 mm and 3.15 = 3.43 mm Y =5− 2 Therefore, Thus, 4.13 · 8 × 10−6 · 850 · 10 · 8 · 3.43 1 1 = + Tp − 25 6000 1500 − 25 ∴ Tp = 535 ◦ C -4- ...
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This note was uploaded on 10/20/2009 for the course MECHENG mecheng495 taught by Professor Sastry during the Spring '09 term at University of Michigan.

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HW2+09F+solution - ME487 HW#2 Solutions 1 Set a control...

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