Aerospace Engineering - Heat Transfer and Thermal Analysis

Full exam

Heat Transfer and Thermal Analysis - A.A. 2023/24 - Date 11/06/2024 VERY IMPORTANT: the answers must be written on the \Results" paper sheet, not on this text. Only the \Results" paper sheet must be uploaded and will be evaluated. NOTES: ˆStudents of the Nuclear Engineering programme (attending the 052603 - 5 CFU course) may have some dierent questions in part A. ˆFor the exercise queries, in case none of the multiple-choice answers the exercise matches the result you get, write your own result instead of marking an answer (so that in case we did a mistake and your answer is correct you can get the related points). ˆStudents having right to a test reduction should answer only to: all part A questions; part B questions 1 and 2; queries 1, 2 of the rst part B exercise; queries 1, 2, 3 of the second part B exercise.Part A 1) Which of the following correlations can be for forced convection around a sphere?AN u L= C 1ReC 2 DBiC 3 DB*N u D= C 1ReC 2 DP rC 3CN u D= C 1GrC 2 DP rC 3DN u L= C 1ReC 2 LP eC 32) In which of the following answers is the Wien law correctly written:A*  max emissionT =constB  max emissionT2 =constC  max emissionT3 =constD  max emissionT4 =const3) In steady-state conditions, the general heat ux pro le_ Q00 (r)in a homogeneous, hollow sphere in which a heat source_ U000 is present reads as (C 1> 0):A _ Q00 (r) =_ U000 r=(3) +C 1=r2B* none of the other answers is correctC _ Q00 (r) =_ U000 =3rC 1r2 =D _ Q00 (r) =_ U000 =3r4) The most general form of the conduction equation for homogeneous, isotropic media with conductivity varying with temperature is:A r2 T+_ U000 =c @ T =@ B r2 T+_ Q00 =(c) = @ T =@ C* r(rT) +_ U000 =c @ T =@ D r2 T+_ Q00 =c @ T =@ 5) Which of the following is true for two surfaces that exchange heat by radiation?A F 1!2A 2= F 2!1A 1B F 1!2" 1= F 2!1" 2C* F 1!2A 1= F 2!1A 2D F 1!2" 2= F 2!1" 16) During pool boiling, a typical sequence of conditions is:A* free convection in liquid - nucleate boiling - jets and columns - lm boilingB forced convection in liquid - bubbly ow - plug/slug ow - annular owC free convection in liquid - forced convection in liquid - nucleate boiling - radiation boilingD free convection in liquid - lm boiling - metastable boiling - burnout boiling 7-6CFU) Only for students attending the 6 CFU courseDuring stable operation, in a capillary heat pipe:A the liquid ows in the central core and the vapour in the annular porous region (wick)B the vapour and the liquid share both the central core and the annular porous region (wick), thus minimizing the pressure drop and the entrainmentC the vapour and the liquid share both the central core and the annular porous region (wick), thus maximizing mixing and heat transferD* the vapour ows in the central core and the liquid in the annular porous region (wick)7-5CFU) Only for students attending the 5 CFU courseWhich of the following is true for a very long uniform hollow cylinder, in steady-state conditions, with no heat source and having xed temperatures on the two external surfaces?A the rst integration constant (C 1) is always zeroB the heat transfer rate_ Q(r)is proportional to1=rC* the heat ux_ Q00 (r)is proportional to1=rD the second integration constant (C 2) is always zeroPart B - Questions Question 1 In a CFD simulation, when the mesh is progressively re ned: Aan oscillating behaviour of the solution should be expected, depending on the total number of cells B*the solution should become more and more accurate and tend to vary less and less with further re nements Cthe solution should become initially more accurate then tend to worsen due to numerical noise Dan unstable behaviour is normally obtained if the mesh resolution is too high Question 2 One end of a stainless steel (AISI 316:=14:8 W=m K,=3:6310 6 m2 =s) rod of diameterD=10 mm and lengthL=16 cm is inserted into a xture maintained atT 0= 200°C. The rod, covered with an insulating sleeve, reaches a uniform temperature throughout its length. When the sleeve is removed, the rod is subjected to ambient air at temperatureT 1= 25°C with a convection heat transfer coecienth 1= 30 W=m2 K. Using the FTCS scheme and a space increment
x=1:6 cm, in which of the following the discretized equation for a generic internal node (consideringBi=h 1(
x)2( D=4)) and the maximum allowable time step are correct? ATk +1 i= Tk i(1 + 2
F o+Bi
F o) +F o Tk i1+ Tk i+1
Bi
F o
T 1, max time step 25.1 s B*Tk +1 i= Tk i(1 2
F oBi
F o) +F o Tk i1+ Tk i+1
+Bi
F o
T 1, max time step 31.1 s CTk +1 i= Tk i(1 2
F o+Bi
F o)F o Tk i1 Tk i+1
+Bi
F o
T 1, max time step 18.2 s DTk +1 i= Tk i(1 + 2
F oBi
F o) +F o Tk i1+ Tk i+1
Bi
F o
T 1, max time step 12.7 s Question 3 In which of the following answers is the centered nite dierence approximation of @2 T@ x 2! iwritten correctly? A @2 T@ x 2! i= T i+1+ 2 T i+ T i14
xB* @2 T@ x 2! i= T i+1+ T i1 2T i
x2 C @2 T@ x 2! i= T i+1+ T i1+ 2 T i
x2D @2 T@ x 2! i= T i+1+ 2 T i+ T i12
x Part B - Exercise 1 A system is made by (from left to right): ˆAn active slab A, with thicknesss A= 10 mm, conductivity  A= 0.25 W/mK, uniform heat source_ U000 A= 105 W/m3 . ˆAn MLI with two internal foils, having negligible thickness and emissivity" m= 0.35 for all foils. ˆTwo passive slabs B and C (s B= 20 mm,  B= 1 W/mK, s C= 5 mm,  C= 0.5 W/mK), between which a contact resistance must be considered (h BC= 10 W/m2 K). The left surface of slab A is perfectly adiabatic, while air atT a= 25 °C ows over the right surface of slab C with a convective coecienth A= 12 W/m2 K. Perfect contact can be assumed between the external foils of the MLI and the right surface of slab A / left surface of slab B. Knowing that the system is in steady-state conditions, determine: 1.The temperature of the right surface of slab CA*381:5 KB286:4 KC324:2 KD407:3 K 2.The temperature of the left surface of slab BA537:1 KB418:3 KC279:7 KD*511:5 K 3.The temperature of the right surface of slab AA580:9 KB612:4 KC*750:9 KD811:9 K 4.The maximum temperature reached in the systemA*770:9 KB981:6 KC853:2 KD658:8 K Part B - Exercise 2 Saturated process steam at 1 atm (latent heat of condensation,h lv= 2257 kJ=kg) is condensed in a shell-and- tube heat exchanger (one shell, two tube passes). Cooling water enters the tubes at 15°C with an average velocity u=3:5 m=s. The tubes are thin walled and made of copper with a diameterD=14 mm and lengthL=0:5 m. The convective heat transfer coecient for condensation on the outer surface of the tubes is 21 800 W=m2 K. Find: 1.The overall heat transfer coecientA*7900 W=m2 KB5670 W=m2 KC10 201 W=m2 KD9230 W=m2 K 2.The minimum number of tubes/pass required to condensate_ m=2:3 kg=s of steam; A211B176C*189D102 3.The outlet water temperature;A32:1°CB24:5°CC*27:7°CD22:1°C 4.The maximum possible condensation rate that could be achieved with this heat exchanger using the samewater ow rate and inlet temperature A22 kg=sB*16 kg=sC12 kg=sD10 kg=s Thermophysical properties of water :=0:606 W=m K,=95910 6 Ns=m2 ; P r= 6:62 Nusselt number correlations: ˆN u D= 0 :023Re0 :8 DP r0 :4 ReD> 2300 ˆN u D= 0 :058Re0 :7 DP r0 :3 (=)0 :1 ReD> 2300 ˆN u D= 0 :34Ra0 :6 DP r0 :4 RaD> 108 ˆN u D= 0 :34Ra0 :45 DP r0 :35 (=)0 :1 RaD> 108 Solutions Part B - Q2 1) The discretized equation for a generic interior node is:T k +1 i= Tk i(1 2
F oBi
F o) +F o Tk i1+ Tk i+1 +Bi
F o
T 12) To determine the maximum allowable time step it is necessary to inspect the discretized equations also for the last node (node 10): Tk +1 10= Tk 10(1 2
F o2
N
F oBi
F o) + 2
F o
Tk 9+ T 1(2
N
F o+Bi
F o) whereN=h 1
x=. The stability criterion requires that all the coecients of the unknown terms are positive. Therefore: 12
F o2
N
F oBi
F o0 =)F o0:5(1 +N+Bi=2)
t=F o (
x)2 = 31:1 sPart B - Exercise 1 %INPUT DATA sA=10e-3; lbdA=0.25; Up3A=100000; epsm=0.35; sB=20e-3; lbdB=1; sC=5e-3; lbdC=0.5; hBC=10; Ta=25+273.15; ha=12; sigma=5.67e-8; %SOLUTION Qp2=Up3A*sA; %Q1 TCR=Ta+Qp2/ha; %Q2 TBL=TCR+Qp2*(sB/lbdB+1/hBC+sC/lbdC); %Q3 RtotMLI=6/epsm-3; TAR=(TBL^4+Qp2*RtotMLI/sigma)^(1/4); %Q4 TAL=TAR+Up3A*sA^2/(2*lbdA); Part B - Exercise 2 1) The required heat transfer rate is_ Q= _mh lv= 5:191106 kg=s. The water-side convective heat transfer coecient, using the rst of the provided correlations, is: ReD=uD= = 50993 = 998kg=m3 hi=D N u D= 12 400 W=m2 K The overall HTC is then:U = (1=h i+ 1 =h o) 1 =7900 W=m2 K2) Considering that _ m c= (D2 =4)uN: Cmin= C c= 2248 N _ Q="_ Qmax= "C min( T h;i T c;i) From the provided tables:"= 1exp(N T U) =C r= 0 N T U=U A sC min= 0 :155A s= DLN P;P= 2 Hence"N= 27:17and:N = 27:17="= 1893) The water outlet temperature with C min= 2248 N=424 900 W=m2 K results in:T c;o= T c;i+_ Q=Cmin= 27:7°C4) The maximum condensation rate occurs when _ Q=_ Qmax. Hence:_ m max=_ Qmaxh lv= C min( T h;i T c;i)h lv= 16 kg=s