CIRCLE YOUR DIVISION: Div. 1 (9:30 am) Div. 2 (11:30 am) Div. 3 (2:30 pm) Prof. Ruan Prof. Naik Mr. Singh

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1 Nae: CIRCLE YOUR DIVISION: Div. 1 (9:30 a) Div. (11:30 a) Div. 3 (:30 p) Prof. Ruan Prof. Nai Mr. Singh School of Mechanical Engineering Purdue Univerity ME315 Heat and Ma Tranfer Exa # edneday, October 0, 010 Intruction: rite your nae on each page Cloed-boo exa a lit of equation i given Pleae write legibly and how all wor for your own benefit. ide of the page only. Keep all page in order rite on one You are aed to write your auption and anwer to ub-proble in deignated area. Only the wor in it deignated area will be graded. Perforance Total 100

2 Proble 1 [35 pt] Nae: A phere of diaeter 18 ha theral conductivity of 10 /-K, denity of 7800 g/ 3, and pecific heat of 400 J/g-K. Initially the phere i at a unifor teperature of 7C. The phere i expoed to air at T = 000C with convective heat tranfer coefficient h conv = 100 / -K. The eiivity of the phere urface (conidered to be gray) i 0.8. The wall of the furnace are at T urr = 000C and can be conidered large relative to the ize of the phere. (a) Neglecting radiation effect, calculate the tie required (econd) for the teperature of the phere to reach 500C. (b) Now conider that radiation fro the furnace wall cannot be neglected. How uch tie (econd) i required to reach 500C at the center of the phere? Coent on the tie to reach 500C in part (a) and (b). In part (b), aue that radiation can be treated linearly and that the radiative heat tranfer coefficient can be calculated baed on initial teperature of the phere, neglecting the variation of radiative heat tranfer coefficient with changing urface teperature. Lit your auption here [3 pt]: One-dienional radial conduction in the phere Contant propertie Unifor radiative heat tranfer coefficient (reain contant) on the urface Start your anwer to part (a) here [1 pt]: hconvl c -K 1000 V D Biot nuber: Bi 0.03; Lc olid 10 A 6 -K luped capacitance approach i valid (Bi < 0.1) T T Teperature ditribution through the phere: expbifo Ti T t exp0.03fo Solving we get: Fo L Theral diffuivity: C p 6 Therefore, the tie required to reach 500C: t 5.7 econd c

3 Proble 1 - cont. Nae: Start your anwer to part (b) here [0 pt]: h T T T T Radiative heat tranfer coefficient: rad urr urr 8 h rad K K K -K Convection and radiative effect are additive at the urface: htotal K Biot nuber: htotall c -K 1000 V D Bi ; Lc olid 10 A 6 -K luped capacitance approach i not valid (Bi > 0.1) T T T T Non-dienional teperature at the center of the phere: * o o C1exp 1 Fo htotalr o -K 1000 Uing Table 5.1: C and for Bi olid 10 -K t 1.85exp Fo Solving we get: Fo ro Therefore, the tie required to reach 500C at the center of the phere: t 6.6 econd Since total rate of heat tranfer to the phere increae when conidering radiation fro the furnace wall in addition to convection fro the abient air, le tie i required to attain the ae teperature of 500C at the center of the phere. i

4 Proble [30 pt] Nae: Conider a two-dienional unifor eh with cell width x = y hown below. The olid ha theral conductivity, theral diffuivity, and generate heat at a unifor voluetric rate of q. You need to develop the tranient nodal equation for a corner node (, n) uing energy balance ethod. The top boundary i perfectly inulated, while the inclined boundary i expoed to a fluid at T with a convective heat tranfer coefficient h. (a) Uing the explicit ethod derive an 1 expreion for T p p in ter of T and the teperature of the relevant urrounding node a well a the teperature of abient fluid. Expre your anwer in ter of the finite difference Biot nuber (Bi) the finite difference Fourier nuber (Fo) a well a any other given paraeter. (b) Derive the tability criterion for the nodal equation developed in (a). Start you anwer to part (a) here [0 pt]: Conidering energy balance for the control volue hown, we have: E E E E in out gen t E P P in qe q q where y T1, n T, n P P nw qe qcond1 1 T 1, n T, n x P P T T x q qcond x T 1 Tn, y 1 x y x y E 1 3 gen qv q q x 8 P1 P dt dt 3 Tn, T E t Cp VCp x Cp dt dt 8 t Subtituting in the energy balance, we have: P1 P P P P P hx P P 3 3 Tn, T T1, nt T 1 T T T q x x Cp 8 8 t Dividing throughout by /, we get: 1 P P P P 1 and qnw qconv h 1 T T, n x hx 3 q 3 C T T T T T T x 4 4 t T T hx t where Bi,, and Fo C x P P P P P P p P 1 P 1, n 1 p

5 Proble - cont. Nae: Re-arrangeent of the equation give: q 4 T FoT FoT BiT Fo x 1 BiFo 4Fo T P1 P P P P 1, n 1 Start your anwer to part (b) here [10 pt]: 4 For tability, we ut have: 1 BiFo 4Fo i.e. 1 Bi Fo 3. Thi will give the required liitation on the tie tep to enure 4 tability in explicit ethod.

6 Proble 3 [35 pt] Nae: An open wiing pool of area A = 10 5 i expoed to dry abient air which flow with a velocity u = 10 / at T = 30 o C. The condition at the urface of the pool ay be aued to be aturated. Approxiate thi configuration a flow of air over a flat plate and ue 4/5 1/3 Nu L ReL Pr to calculate the Nuelt nuber. Conider that the water in the pool i to be aintained at a teperature of T = 7 o C and aue that the heat-a tranfer analogy i valid for the given condition. Aue the binary diffuion coefficient of water vapor in air i D AB = /. All other propertie hould be obtained fro the table provided. (a) Calculate the average heat tranfer coefficient (/ -K) and the convective heat tranfer rate (). (b) hat i the rate of evaporation of water (g/) fro the pool? (c) Should the pool be refrigerated or heated in order to aintain it urface teperature at T = 7 o C? Calculate the external power () provided to/withdrawn fro the pool. T =30 o C, φ = 0.0 U =10/ T =7 o C L=10 Lit your auption here [3 pt]: Steady tate Contant propertie Negligible radiation Heat and a tranfer analogy valid Boundary layer auption Lit the propertie ued and clearly ention the teperature for each property [5 pt]: All air propertie at the fil teperature: T K ; 0.06 ; Pr ;.7 10 (Table A-4) -K All water vapor propertie at the urface teperature: T 300 K 1 1 g J ; hfg 438 (Table A-6) g at, 3 vg fil Proble 3 - cont.

7 Nae: Start you anwer to part (a) here [10 pt]: ul 6 Reynold nuber: Re L /5 1/3 hl Average Nuelt nuber: NuL 0.037ReL Pr Average heat tranfer coefficient: h 3.4 -K Convective heat tranfer rate fro the abient air to the pool urface: qconv hat T K qconv 3,510 -K Start you anwer to part (b) here [7 pt]: h Applying heat and a tranfer analogy: h ; Le /3 DABLe DAB Average a tranfer coefficient: h 0.04 Rate of evaporation of water vapor fro the pool urface to the abient air: h A h A g A, A, at, at, 3 g Start your anwer to part (c) here [10 pt]: Heat tranfer aociated with evaporation of water vapor: g J q h q 69, 76.8 evap fg g evap Conidering energy balance at the pool urface, heat leaving the urface (due to evaporation) i higher than heat entering the urface (due to convection) the pool ut be heated in order to aintain it urface at contant teperature of 7C. The external power provided to the pool: Pe qevap qconv P 66,16.8 e