EXAMPLES of THEORETICAL PROBLEMS in the COURSE MMV031 HEAT TRANSFER, version 2017

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1 EXAMPLES of THEORETICAL PROBLEMS n the COURSE MMV03 HEAT TRANSFER, verson 207 a) What s eant by sotropc ateral? b) What s eant by hoogeneous ateral? 2 Defne the theral dffusvty and gve the unts for the dfferent propertes 3 Derve the general heat conducton equaton n rectangular (x,y,z) coordnates for an sotropc ateral 4 Why s the theral conductvty hgher for lqud etals than for ordnary lquds? 5 Explan what s eant by crtcal thckness of nsulaton 6 Derve the dfferental equaton that governs the teperature dstrbuton n a rectangular fn Also, gve the boundary condtons 7 Derve the dfferental equaton that governs the teperature dstrbuton n a trangular fn Also, gve the boundary condtons 8 Forulate the condton showng when a fnned surface s favorable copared to an unfnned surface (assue rectangular fns and Q known) Defne the propertes and ther lts n ths condton 9 For a rectangular fn the heat transfer rate becoes Q 2 tanh( b Z L) where 2 2 / b Derve the expressons for the fn effcency and the fn effectveness ( and ) 0 For an optal fn the followng relaton apples L c b / 2 2 b What s the optzaton crteron? A plane wall (theral conductvty ) wth the thckness 2b s cooled on both sdes by convecton by a flud wth the teperature t f Heat s generated Q (W/ 3 ) hoogeneously n the wall ateral Derve an expresson for the teperature dstrbuton n the wall when the heat transfer coeffcent s 2 Derve an expresson for the transent teperature dstrbuton for a body wth hgh theral conductvty (unfor teperature wthn the body) The surfaces of the body are cooled by convecton 3 Show that the ter A cv can be wrtten as A cv B Fo 4 Defne the B- and the Fo-nuber and gve the unts for the ncluded quanttes 5 When solvng the equaton of the non-steady and one-densonal heat conducton n a plane plate wth oderate theral conductvty, an nteredary soluton s obtaned as: t t f e 2 ( Acos x Bsn x)

2 2 See what results the boundary condtons at coolng) gves x 0 (syetry) and x L (convectve 6 The teperature dstrbuton for two-densonal transent heat conducton ay under certan condtons be expressed as a product of solutons for two one-densonal probles What are the condtons? 7 Defne the heat transfer coeffcent, and gve the unt 8 What s eant by a Newtonan flud? 9 Defne the Reynolds nuber 20 Derve the teperature (energy) equaton for a flud n oton The flud can be assued to be ncopressble and steady-state condtons apply 2 Defne the Prandtl-nuber and gve the unts for the dfferent quanttes 22 Defne the Nusselt-nuber and gve the unts for the dfferent quanttes 23 Forulate the slarty prncples for forced convectve heat transfer 24 Interpret the results presented n Fg 7-9, and explan what happens when ( v w / ) Re 069 U x u/u (vw/u ) Re x 02 Blow-off at y Re x/2 x 25 What s eant by theral entry length? 26 Defne the concept hydraulc daeter 27 Consder a crcular tube wth the constant wall teperature T w A flud wth a hoogeneous nlet teperature T enters the tube Derve the governng dfferental equaton for the teperature feld for the flowng eda n the tube The velocty feld can be assued to be fully developed as r 2u R u 28 Defne the bulk teperature T B 2

3 3 29 Show that the bulk teperature ncreases lnearly wth the channel length for a constant wall heat flux q w 30 For convectve heat transfer n crcular tubes the followng relatons ay apply NuD = 3656 and NuD = 4364 Gve the condtons for applcablty of these relatons 3 Gve three characterstcs for turbulent flow t u 32 If the heat flux q cp ( / Pr / Prt ) and the stress ( ), derve y y Reynolds analogy n the for St=CF/2 33 Defne the turbulent (eddy) vscosty turbulent Prandtl-nuber 34 Defne the frcton velocty u ( u * ) ( M ), the turbulent dffusvty q ( H ) and the 35 For heat transfer n a turbulent boundary layer a densonless teperature s ntroduced as T ( Tw T) c q w Explan the dfferent quanttes p u * 36 Defne the Stanton-nuber and gve the unts for the dfferent quanttes 37 Descrbe how the heat transfer coeffcent (NuD) vares n the crcuferental drecton of a crcular cylnder at low (< 40) and hgh ReD (>05), respectvely 38 What s eant by n-lne arrangeent and staggered arrangeent regardng flow across tube banks? 39 Defne the velocty uax for flow across tube banks 40 How s the pressure drop deterned for flow across tube banks n an n-lne arrangeent? 4 Defne the Grashof nuber, Gr, for natural convecton at a vertcal wall wth tw = const 42 Show that Gr/Re 2 ay be nterpreted as the rato of gravtatonal to nerta forces 43 Show that the voluetrc theral expanson coeffcent s equal to /T for the deal gases n the case of natural convecton 44 What s the Boussnesqs approxaton? 45 Defne the Raylegh nuber Ra 46 Descrbe how the heat transport between two vertcal plates can be calculated f the edu between these plates has a densty, whch s strongly teperature dependent 47 What s eant by a black body? 48 What s eant by a gray body? 49 a) Defne the radaton shape factor between two surfaces b) Gve the recprocty relaton

4 4 50 Derve the relaton between transssvty, absorptvty and reflectvty 5 Show that, f all the surfaces n a closed roo are dffuse, the net radaton heat transfer for each surface ay be expressed as Q A E Q A F J k k B J J k 52 Derve the Beer's law for gas-radaton, e, I I e a x o 53 Gve the Nusselt's assuptons for fl condensaton 54 Derve the velocty profle n a condensate fl on a vertcal surface 55 Defne the Jacob nuber and gve the physcal sgnfcance of t 56 Menton two ways to facltate droplet condensaton 57 Descrbe how the vapor condensaton can occur nsde a horzontal tube 58 Condensaton occurs anly n two ways Whch? 59 Descrbe Nukyaa's experent 60 Descrbe the bolng curve 6 What does the fl bolng eans? 62 Deterne the radus for a vapor bubble n theral equlbru 63 Deterne the Taylor wavelength T by eans of densonal analyss 64 What s eant by Helholtz nstablty? 65 Defne the Weber nuber 66 Specfy the three types of two-phase flow of gas-lqud xtures, whch can occur n horzontal tubes 67 The sae as 8 but n vertcal tubes 68 Defne, X F, X S, ug, uf, ugs, uf S 69 Descrbe how the pressure drop can be calculated for an sotheral two-phase flow 70 Defne the Martnell paraeter 7 Defne the two-phase frcton ultpler L 2 72 Gve two ways to classfy heat exchangers 73 How s foulng taken nto consderaton n the desgn of heat exchangers? 74 Gve the crteron for a heat exchanger to be classfed as copact (Answer: A/V > / 3 )

5 5 75 Explan the LMTD-ethod for desgn of heat exchangers 76 Descrbe the -NTU-ethod for analyss of heat exchangers 77 Defne, NTU, LMTD 78 Why s t not sutably to choose the correcton factor F < 075? 79 Derve an expresson for the effectveness ( Cn, Cax, NTU), for a counter-flow heat exchanger 80 Explan the flows n the Fgure below 8 What s eant by a regeneratve heat exchanger?