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hw1_sol - PROBLEM 1.5 KNOW Width height thickness and...

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Unformatted text preview: PROBLEM 1.5 KNOW: Width height, thickness and thermal conductivity of a single pane window and the air space of a double pane window. Representative winter surface temperatures of single pane and. air space- FIND: Heat loss through single and. double pane windows. SCHEMATIC: —>1 |(—L=5mm —:-| |<— L=1flmm T1 = 15": T2=-20°C n=1ne u=45c Glass pans fiu‘r spans kg=1-4WFm-K k,,=o.u24 wrm-K W=1m.H=2m W=1m.H=2m Glass Glass ASSMIONS: {1) I[l'Ine-dimensional conduction through glass or air, {2) Steady—state conditions, (3) Enclosed air of double pane window is stagnant (negligible buoyancyr induced motion]. ANALYSIS: From Fourier’s law, the heat losses are 35”c onosm Etvvi‘fifluvfi: q _k AT1_T2:I4WIm-K(2m2) _ 219,600 W g E L floufli'afluvs: qa=kaAT1—T2=flfl24(2m2) 25 C =120W L 0-010111 COMNTS: Losses associated with a single pane are unacceptable and would remain excessive, even if the thickness of the glass were doubled to match that of the air space. The principal advantage of the double pane construction resides with the low doormal conductivity of air (_ 60 times smaller than that of glass]. For a fixed ambient outside air temperature, use of the double pane construction would also increase the surface temperature of the glass exposed to the room {inside} air. PROBLEM 1.13 ENO‘WN: Hand experiencing convection heat transferwith moving air and water. FIND: Determine which condition feels colder. Contrast these results 1with a heat loss of 30 mel under normal room conditions. scnssnmo: the T -10 we? V= I12 mts n = 900 thlk / —" Hand —* r5: so Dc gig :finm n 40 thi-K ASSUMPTIONS: (1} Temperature is uniform over the hand's surface, (2) Convection coeflicieot is uniform over the hand, and (3} Negligflale radiation exchange between hand and surroundings in the case of air flow. ANALYSIS: The hand will feel colder for the eondition which results in the larger heat: loss. The heat loss can be determined l'romNewton’s law of cooling, Eq. 1.3a, written as q" =h(Ts 4m) Forthe air stream: qgfi =4DW/m2 .K[3n—(—5)]K =1,4oow/m2 a; Ffll'flilfl water stream: {lam = snow/m3 -K(3o—1D)K = laminar/m? -:: CDMLIIENTS: The heat loss for the hand in the water stream is an order of magnitude larger than when in the air stream for the given temperature and convection eoefficient: conditions. In contrast, the heat: loss inanormalroomenvironmentis only 30 Warm: whichis a factorof400 times less thanthe loss in the air stream In the room environment; the hand would feel cranfortable; in the air and water streams, as you probany know from experience, the hand would feel uncomfortath cold since the heat loss is excessiver high. PROBLEM 1.62 KNOWN: Duct wall of prescribed thickness and thermal conductivity experiences prescribed heat flux <13 at outer surface and convection at inner surface with known heat transfer coefficient. FIND: (a) Heat flux at outer surface required to maintain inner surface of duct at Ti = 85°C, (b) Temperature of outer surface, To , (c) Effect of h on T0 and q; _ SCHEMATIC: Tm=so°c —'* .= o n=1oow.'m2-K _}—. fr," T, as c ASSUMPTIONS: (1) Steady—state conditions, (2) One—dimensional conduction in wall, (3) Constant properties, (4) Backside of heater peifectly insulated, (5) Negligible radiation. ANALYSIS: (a) By performing an energy,r balance onthe wall, recognize that q: = qund _ From an energy balance on the top suffice, it follows that qgond = (1201“, = qg _ Hence, using the convection rate equation, q; =quv = h(Ti —T,,., ) =100WIm2 -K(35—3o)° C = ssoowxm? <1 (b) Considering the duct wall and applying Fouriei"s Law, 9 AT T — T' qo = k_ = k C- 1 AX L , 2 L 55(10me X0.0lOm TuzTi+qL285°C+— :(35+2.3)°c287_3°c. < k 20Wfim - K (c) For T1” = 85°C, the desired results may be obtainedby simultaneously solving the energy balance equations T —T- T —T- (13:1: ° 1 and k 0 1=h(T,-—Tm) L L Using the ]HT mp“ A“ Model for a NW“) "1“)“ Elma" n. “M, the following results are obtained. 12DDD 9] 1mm g on '1 EDDD 89 ‘u- :- fiDDD HE E now g a? ZDDD 86 D 85 fl AID 8D 12D 18D EDD 0 JD 3D 12B 150 EDD Convection beam-.1531. hfllll'tlnm Convection ouefiieien‘t. htwmm Since {1301“, increases linearly,r with increasing h, the applied heat flux q; must be balanced by an increase in qgond , which, with fixed k, Ti and L, necessitates an increase in To. COMMENTS: The temperature difference across the wall is small, amounting to a maximum value of (To —Ti) = 55°C for h = 200 me2-K. If the wall were thinner (L < 10 min) or made from a material with higher conductivity (k )~ 20 me-K), this difference would be reduced. ...
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