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exam1soln

# exam1soln - 1 A water heater shown in the ﬁgure below...

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Unformatted text preview: 1. A water heater shown in the ﬁgure below includes a built-in insulating layer of 5.5 cm thickness on the side, top and bottom. An energy conscious user has added a 5 cm thick insulation blanket to the side. The heat transfer coefﬁcients on the outside surfaces of the side, top, and bottom are respectively 10 W/mzK, 4 W/mzK, and 2 W/mZK. T =21.6C ENKKK‘LXXXWk'xXXXXXXXRCAEﬁLMXX‘KXXXY ., a 55 {A a g; r“ HI 3 a. a 5: '9' ,5 '9' 9" 5-:- .o“ a g d a) (10 pts) Draw a thermal resistance circuit that would enable you to calculate the heat ﬂow from the water to the air in the room. Name each resistance according to its associated mode of heat ﬂow, but do not deﬁne its equation. Also identify the locations of the ﬁve temperatures indicated in the ﬁgure above. You may ignore the heat transfer through the corner rings (square black regions in the diagram). lRtuﬁcqi‘t 1-H)? Raj-Mat w—Aﬂ" W‘W\ T 2. Tags m 12mm“ 1 Raw! 1'1 Rwy} and WH’VWF—ﬁ'" R 126011“; [0 (0(9le IRQSLS'LBTS + 3d5’ Tex/MP S *l '25— b) (5 pts) If the thermal resistance of the blanket is 0.65 K/W, what is its thermal conductivity? 9“" 0V ' ‘“‘*"“’”j> 3% k (W/m-K) 19493; K’ ln( rL/r‘g 21: V2. L f 7, "7’ M0 /w> l t (53/ was) W Z t V\ - - W = - '- M’“ r. . ' . k m— 0 03C) W/m (g Z-rr(1.‘lm>o'(°§l< H c) (5 pts) If the heat ﬂow through the wall is 25 W, determine the temperature Twz at the outer surface of the tank. TWZ (°C)=g/Z+l§__. ‘r’L Twz'Too : 1w (. RCW‘VJ‘” + ZC‘WA’Z') / _ kHz-l4 __ C)” ll; #7, KCcmv)u¢ "' lDW W(Q3W‘qum) " O 56 ‘/W ”ﬂu? ; 20°C, 4 zg w [otogel + 0.559 g/W] : Zo°c + I11; +\ 3 \$7. (S 0C; d) (5 pts) If the temperature at the top outer surface of the tank is 21.6 °C, what is the heat ﬂow out through the top of the tank? +2 | qtop (w) = O. 9577 (in, Z n 00 ; R. = w . <0 Wit “Wit 4W mm: ”80" “/w 3: \ +7, 1. (Zl-éz ’ZO)1¢ w _. ( . 2. A stainless steel spoon (k = 15 W/m—K) sits in a cup of coffee. The cross sectional area of the spoon handle is uniform with width w = 1 cm and thickness t = 2 mm. An 8 cm length of the spoon sticking out from the coffee is exposed to a slight breeze with a heat transfer coefﬁcient of 20 W/mzK. The temperature of the coffee is 80 °C and the room temperature is 20 °C. a) (15 pts) Find the temperature of the spoon at a distance of 5 cm from the top (exposed) end of the spoon. Along the way, identify the following parameters: M(W)=L m(1/m)=i__ T(X)=Mx(1 311.22 are re .9 9/93. : (MK TCT") 5': T43 'i" [poamﬂlixp {‘" “l0 * .03 m\) b) (10 pts) A silver spoon (k = 429 W/m-K) of the same dimensions replaces the stainless steel spoon, and with steady state conditions as deﬁned above, the total heat ﬂow out of the spoon is found to be 2.08 watts. What is the ﬁn efﬁciency of the silver spoon? ﬁfﬂB— L/ ' a: Cl” ”L hApob . - “l; 6’ A5, 3 P L, + W '3 Z :(O.DZ‘+VH /Y.¢_)ym) 4/ leQ’SIMZ/ :_ l.‘i‘f;<(o VW V7 .- 1 _. L or .. istmla (ml—c > “ll VG " m LC 4/2 L( z 08 + _-UO‘Z/Z : 03;, I. l 7.48 1 34‘ T: “id/W'\[( Xe) ) 290833 “V .6209? 3. The ﬁgure below represents a two-dimensional channel, inﬁnitely long in the direction into the page. The channel is made of a material with thermal conductivity k. A liquid ﬂows along the center of the channel, with temperature mum and heat transfer coefﬁcient huquid. The top of the channel is exposed to air at a temperature Tai, and heat transfer coefﬁcient hair. The bottom of the channel is exposed to a uniform heat transfer rate per unit length q’. For its analysis, a cross section of the channel has been discretized with elements of equal length in the vertical and horizontal directions (Ax = Ay). a. (7 pts) Based on the energy balance equation, derive an expression for the temperature T2 as function of the surrounding temperatures: 57.1 jlmdfm 3,,L—'~E,dl 1T, ‘1‘ 2_ ASC 31? mt m 2” L Alt A Em.- “if k1 lKT+kT +l M L T' g 4' —-’" ' qr \ £4, . _‘_ A71 L‘ L Q l i {2' in 4 "Ea-“ml LT 7; -( “a, 1‘9 [LT *Kﬂ‘li‘ + LVN 511m «- l/uﬂf;%lb%] Z , (K + %l «we,» (Ea/#3“ 5? 1 Ef' 141,01’ 333’“ gm“ :0 b. (6 pts) Derive an expression for the temperature T8 as function of the surrounding temperatures. .4,» 7,5 , 3L __ k A}. (Tls'T?) W__ﬂ%& —”I 328; KE<TL*T33 L '3’?) / AL '313 I 2— A1 T. —-— 113 \$723, ; k AX, (T *T%) gum/5 thl (A? T93 5.2.. 9&3“ K bi (“Fm ’ 2. M. K%( TL”; quwL Tq +Z-E3“4E)+L\L%AL(TA{T"E :0 Ax: c. (6 pts) Derive an expression for temperature T15 as function of the surrounding temperatures. mm Eur/WE??? (ifo‘) M A [—1 a ‘ . d. (6 pts) If the level of the liquid falls below node 4, exposing it to the surrounding aii derive a new expression for the temperature T2. g. ),7, ‘l 44' 7 T 460%“! :5 O 3. The tip of a thermocouple may be approximated as a sphere of constantan (k = 20 W/m-K, p = 8920 kg/ms, cp = 384 J/kg-K, OL = 6.7 x 10'6 mz/s). It is used to measure the temperature of a hot air stream with h = 300 W/mzK. a) (7 pts) When the temperature of the hot air stream changes, it is required that the thermocouple has a 95% reaction time of 10s or less, i.e., it should reach 95% of the change between initial and steady state temperatures within 10s. What is the maximum allowable diameter of the lead? <6;T-T; b) (5 pts) If the thermocouple tip has a diameter of 2mm, prove whether or not it is appropriate to model its transient behavior using the lumped capacitance approach. B; 2L La 1K flT'Y léi: o.oo< 4 m 9m is market)“: “F“ U56 W c) (5 pts) Imagine that the thermocouple is used to measure the temperature of the gases inside a combustion chamber. In this situation, the temperature of the hot gases changes very rapidly (< msec). Using thermocouples has the advantage that they are cheap, but the time response is slow for the application. In this case, the gas temperature may be approximated by the expression 1) UH J\ dt ( where T TC is the temperature Mo‘lf’the thermocouple and ris the time constant of the thermocouple tip If I ~ 1 second, and the temperature of the hot gas follows the simpliﬁed sketch below, where t1 — to = 10 ms, draw the transient temperature response of the thermocouple associated with this event. <9 :54 (53.. awn-male i7 '17 HWN /\~}NJT mmwm TO 31\$ch /5\ d) (8 points) If the time constant is 2s, the initial temperature is 100°C, and the gas temperature jumps suddenly to 1000°C what would be the temperature of the hot gas as calculated with equation (1) immediately after the jump in the gas temperature? 9":- T”Too - .. , A’ Oil“; W; AT hawk Tu * b A J6 56‘ 13 17.. MG? ‘rT THC WK lTyWMW-WJ. r Tips ll 91ng = ‘9 ”fl—J; T ”L“ salmasmol’ T1”); — <65 lam-W4 My W W» l we me my, gm? Km WW “C H4 Um TEEN" It‘:0 all; r- 137; 10 ail ,ﬂ l ...
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