Page 1 PROBLEM 3.1 KNOWN: One-dimensional, plane wall ...

PROBLEM 3.14KNOWN: Composite wall of a house with prescribed convection processes at inner andouter surfaces.FIND: Daily heat loss for prescribed diurnal variation in ambient air temperature.SCHEMATIC:ASSUMPTIONS: (1) One-dimensional, steady-state conduction (negligible change in wallthermal energy storage over 24h period), (2) Negligible contact resistance.PROPERTIES: Table A-3, T ≈ 300 K: Fiberglass blanket (28 kg/m 3 ), k b = 0.038 W/m⋅K;Plywood, k s = 0.12 W/m⋅K; Plasterboard, k p = 0.17 W/m⋅K.ANALYSIS: The heat loss may be approximated as⎡⎤24hT∞,i − T∞,oQ= ∫dt whereR0tot1 1 Lp Lb Ls1Rtot= ⎢ + + + + ⎥A ⎢⎣hi kp kb ks ho⎥⎦1 ⎡ 1 0.01m 0.1m 0.02m 1Rtot =2 ⎢+ + + +2 0.17 W/m K 0.038 W/m K 0.12 W/m K 2200m ⎣30 W/m ⋅K ⋅ ⋅ ⋅ 60 W/m ⋅KRtot= 0.01454 K/W.Hence the heat rate is12h24h1⎧2π2π⎫⎪ ⎡ ⎡ ⎤⎤ ⎡ ⎡ ⎤⎤⎪Q = ⎨ 293 273 5 sin t dt 293 273 11 sin t dtR∫ ⎢ −⎢+ + − +⎬tot24 ⎥⎥ ∫ ⎢ ⎢ 24 ⎥⎥⎪ ⎣ ⎦ ⎣ ⎦⎩ 0⎣ ⎦12⎣ ⎦ ⎪⎭W ⎧⎪⎡ ⎡24⎤ 2πt⎤ 12 ⎡ ⎡24⎤2πt⎤24 ⎫⎪Q = 68.8 ⎨ 20t+5 cos 20t+11 cos K hK⎢ ⎢2π⎥ 24⎥ + ⎬ ⋅0⎢ ⎢2π⎥ 24⎥⎪⎣ ⎣ ⎦ ⎦ ⎣ ⎣ ⎦ ⎦ 12⎩⎪⎭⎧⎡ 60 ⎤ ⎡ 132 ⎤⎫Q = 68.8⎨⎢ 240 + ( −1− 1) + 480 − 240 + ( 1+ 1)⎬W ⋅h⎣ π ⎥⎦⎢⎣ π ⎥⎩⎦⎭{ }Q = 68.8 480-38.2+84.03 W ⋅hQ=36.18 kW ⋅ h=1.302× 108J.