I have 1 hrs to do 5 question
Final exam equation sheet ?!"#$ = −?? ?? ?? , ?!"#% = ℎ?(?& − ?'), ?()$ = ???(?*+ − ?&,(+ ) ?(?) = 0?*,. − ?*,/1 ? ? + ?*,/, 0 <>< (?)="�̇�?." 2?="" 81="" −="" .="" .="" :="" +="" *,.="" −="" *,/="" 2="" +="" *,/="" +="" *,.="" 2="" :="" −=""><>< plane="" wall="" +="" generation="" (?)="�̇�?." 2?="" 81="" −="" .="" .="" :="" +="" *:="" 0=""><>< plane="" wall="" +="" generation="" +="" one="" side="" insulated="" (?)="?*,/" −="" *,.="" ln="" (?//?.)="" ln="" @="" .="" a="" +="" *,.,=""><>< .="" radial="" hollow="" cylinder="" 0="2???0?*,/" −="" *,.1="" ln="" (?./?/)="" ,=""><>< .="" radial="" hollow="" cylinder="" (?)="�̇�?1." 4?="" 81="" −="" .="" 1.="" :="" +="" *,="" 0=""><>< 1="" radial="" solid="" cylinder="" +="" generation="" 2="?" −="" '="" 2="" −="" '="exp@−" 3="" a="exp(−??" ⋅="" )="" ,="" loss="" by="" convection="" 3="@" ∀?="" ℎ?*="" a="" ,="" =="" ℎ?4="" ,="" =="" 4.="" −="" '="" −="" (?/?)="" 2="" −="" '="" −="" (?/?)="exp" (−??),="" =="" 1/?3="" ,="" =="" 567="" ∀?="" loss="" by="" convection="" +="" gain="" by="" generation="" (?,="" )="" −="" *="" 2="" −="" *="erf" @="" √4??="" a="" ,="" *88(?)="?(?*" −="" 2)="" √???="" ,="" 0=""><>< ∞="" erf="" @="" √4??="" a="1" √???="" exp8−="" .="" 4??="" :="" criterion="" for="" semi="" −="" infinite="" domain:="" (?,="" )="" −="" *="" 2="" −="" *="" ≥="" 0.9="" ∗="?" 2="?" −="" '="" 2="" −="" '="d" ="" '="" 7:/="" 7="" exp(−?7.??)="" cos="" i?7="" j="" ,="" 0=""><>< ,="" 88(0,="" )="0" material="" constants="" will="" be="" provided="" if="" needed.="" charts="" and="" graphs="" will="" be="" provided="" if="" needed.="" heat="" transfer="" –="" mass="" transfer="" analogy="" !="" ""="−?" ∂?="" ∂?="" (="" #$%="ℎ(?!" −="" &)="" '!="" ""="−?'(" ∂?'="" ∂?="" (="" #$%="ℎ)/?!,'" −="" &,+0="" !,="" evap="" ""="?+,!""" ℎ,-="" ./0="?ℎ" ./0="" =="" ℎ?="" ℎ="ℎ)?" +1="" =="" =="" +1="" ℎ="" ℎ)="?" +1="" :="" +1="" ;="" ./0="" flat="" plate="" boundary="" layer="" laminar="" 2="">< 5="" ×="" 103="" 2="0.332??2" ./4??./0="" (eq.="" 7.23)="" jjjj2="0.664??2" ./4??./0="" (eq.="" 7.30)="" turbulent="" 2="0.0296??2" 5/3??./0="" (eq.="" 7.36)="" jjjj6="N0.037??6" 5/3="" −="" p??./0="" (eq.="" 7.38)="" =="" 0.037??2,7="" 5/3="" −="" 0.664??2,7="" ./4="" (eq.="" 7.39)="" pipe="" flow="" !""="ℎ(?!" −="" ))="" (eq.="" 8.27)="" =="" �̇�?8/?),9="" −="" ),:0="" (eq.="" 8.34)="" )="?),:" +="" !""?="" �̇�?8="" ,="" !""="constant" (eq.="" 8.40)="" !="" −="" )(?)="" !="" −="" ),:="expZ−" �̇�?8="" ℎ‾\="" ,="" !="constant" (eq.="" 8.42)="" δ?9="" δ?:="?&" −="" ),9="" &="" −="" ),:="exp" z−="" _?!="" �̇�?8="" \="" (eq.="" 8.45?)="" cylinder="" in="" cross="" flow="" (eq.="" 7.54)="" jjjj;="0.3" +="" 0.62??;="" ./4="" ./0="" [1="" +="" (0.4/??)4/0]./5="" d1="" +="" :="" ;="" 282,000;="">< e="" 5/3="" flow="" through="" bank="" of="" tubes="" jjjj;="?.??;,=">?) ??%.0A : ?? ??B ; ./5 (Eq. 7.58) ??JJJJ;|(D"E4%) = ?4??JJJJ;|(D"G4%)(Eq. 7.59) ?1 and ? from Table 7.5, ?2 from Table 7.6 Table 7.6 will be provided when required ?=>? = ?H ?H − ? ?(Eq. 7.60) ?! − ?9 ?! − ?: = exp Z− ???ℎ‾ ???H?H?8 \ (Eq. 7.63) Δ?I= = (?! − ?:) − (?! − ?9) ln N?! − ?:?! − ?9 P (Eq. 7.62) ?" = ?/ℎ‾??Δ?I=0(Eq. 7.64) Pipe Flow Laminar Flow (??; < 3000)="" –="" thermal="" entry="" jjjj;="3.66" +="" 0.0668??;="" 1="" +="" 0.04??;="" 4/0="" (eq.="" 8.57)="" 1/??;="?/(???;??)">< 0.05="" (???;)=""> 0.05 Laminar Flow – Combined Entry ??JJJJ; = A + 0.0499??;tanh(??;J.) tanh N2.432??./A??; J./AP (Eq. 8.58) ? = 3.66 tanh s2.264??; J./0 + 1.7??; J4/0t ?/(???;??) < 0.05="" (???;)="">< 0.05="" turbulent="" flow="" ;="0.023??;" 5/3??%.5="" (eq.="" 8.60)="" (heating)="" ;="0.023??;" 5/3??%.0="" (eq.="" 8.60)="" (cooling)=""> 10 ??; = 4�̇� ??? ??2 = ??&? ? ??2 = ?&? ? ??;,=>? = ?=>?? ? ℎ = − ???/??|#$% ?! − ?& ?? = ??∗ ??∗ ( #∗$% ? ∗(?∗) = ? − ?! ?& − ?! ? = 5 x?&/?? = 5? x??2 ?L ≈ 5? ??2 ./4??⬚ ./0 ? ?L = ??./0 ? = z +! ?""??! ℎJ = ? ?!(?! − ?&) ℎ‾ = 1 ? z 6 % ℎ?? = 1 ? dz 2$ % ℎNO)?? + z 6 2$ ℎturb ??e ℎ) = −?+1? ?+ ??|#$%⁄ ?+,! − ?+,& ?ℎ = ??+∗ ??∗ ( #∗$% ?/?7 = ??./0 ?L ?7 = ??./0 Convection ?? = ?? ?? ? = ℎ?!??N) ? = ?_?!Δ?I= ? = ??N) ?L9L Δ?I= = Δ?9 − Δ?: ln(Δ?9/Δ?:) ? ?? d ?!(?) − ?(?, ?) ?!(?) − ?)(?) e PQ,L = 0 Fully developed thermal (fd,t) ?? ?? ( PQ,L = = ??! ?? ( PQ,L Constant Surface Heat Flux (fd,t) ??) ?? ( PQ,L Constant Surface Temperature (fd,t) ?? ?? ( ,R,L = (?! − ?) (?! − ?)) ??) ?? ,R,L ??; = 4.36 ??; = 3.66 ? = ??Δ?!" Δ?!" = Δ?# − Δ?$ ln Δ ⁄?# Δ?$ Δ?$ = ?%,' − ?(,' Δ?# = ?%,) − ?(,) Δ?$ = ?%,' − ?(,) Δ?# = ?%,) − ?(,' ? = �̇�%?*,% ?%,' − ?%,) = ?%Δ?% ? = �̇�(?*,( ?(,) − ?(,' = ?(??( Heat Exchangers ? = ? ?"+, ?"+, = ?"-.Δ?"+, Parallel flow Counter flow NTU = ?? ?"-. ? = ? ?? ?"-. , ?"-. ?"+, ? = 5.67×10/0W/m# ⋅ K1 Radiation ?'2 = ?' − ?2 ?'?'2 /$ Reciprocity relation ?'?'2 = ?2?2' Summation rule for enclosures Composite surfaces E 23$ 4 ?'2 = 1 ?(2)' = 1 ∑73$8 ?7 E 73$ 8 ?7?7' ?' = ?9' − ?' ⁄1 − ?' ?' ?' =E 23$ 4 ?' − ?2 ?'?'2 /$ ?' = ?9' − ?' ⁄1 − ?' ?' ?' =E 23$ 4 ?'2 ?'(2) = E 73$ 8 ?'7 ?"-. = min(?(, ?%) ?":; = max(?(, ?%) ? = ??9 = ???1 ?9 = ??1 Radiation exchange Blackbody ? = ?9 = ??1 ? = 1 ?$# = ? ?$1 − ?#1 1 − ?$ ?$?$ + 1?$?$# + 1 − ?#?#?# Two surface enclosure Charts and graphs will be provided if needed. All material constants will be provided. Free Convection ??; = ?? ?< −="" ==""> ?? Vertical Plate ??? = 0.825 + 0.387??? $/A 1 + (0.492/??)B/$A 0/#C # Hot upper or cold lower horizontal surface ??? = 0.54??? $/1 , 101 ≤ ?? ≤ 10C, ?? ≥ 0.7 ??? = 0.15??? $/> , 10C ≤ ?? ≤ 10$$, all ?? Hot lower or cold upper horizontal surface ??? = 0.52??? $/D, 101 ≤ ?? ≤ 10B, ?? ≥ 0.7 Horizontal Cavity Critical Rayleigh number, ??( = 1708 ??? = ‾ℎ? ? = 0.069??? $/>??E.EC1 Vertical Cavity ??! = 0.22 Pr 0.2 + Pr??! ".$% ? ? &'/) 2 ≤ ? ? ≤ 10 ??! = 0.18 ?? 0.2 + ????! ".$* 1 ≤ ? ? ≤ 2 ??! = 0.42??! '/)Pr"."'$ ? ? &".+ 10 ≤ ? ? ≤ 40 ?$# = ?$? ?$1 − ?#1 1 ?$ + 1?# + 1 − ?>,$ ?>,$ + 1 − ?>,# ?>,# Two surface enclosure with radiation shield 1/?? = 1/ℎ(?( + ?G + 1/ℎ%?% Eq. 11.1a List of tables and graphs that will be provide in the exam if needed Heat exchangers Figure 11.10 Parallel flow Figure 11.11 Counter flow Figure 11.12 Shell and tube (1 shell) Figure 11.13 Shell and tube (2 shells) Figure 11.14 Cross-flow - unmixed Figure 11.15 Cross-flow – mixed/unmixed Radiation exchange Table 13.1 a) Parallel plates with midlines connected by perpendicular b) Inclined parallel plates of equal width and a common edge c) Perpendicular Plates with a Common Edge Figure 13.4 View factor for aligned parallel rectangles. Figure 13.5 View factor for coaxial parallel disks. Figure 13.6 View factor for perpendicular rectangles with a common edge. Conduction Table 5.1 Coefficients used in the one-term approximation to the series solutions for transient one-dimensional conduction Appendix B.2 Gaussian Error Function Convection Table 7.5 Constants of Equation 7.58 for the tube bank in cross flow Table 7.6 CorrecWon factor ?# of EquaWon 7.59 for ?? < 20="" figure="" 8.10="" (a)="" local="" nusselt="" numbers.="" final="" exam="" review="" transport="" processes="" enme332="" fall="" 2023="" final="" exam="" review="" •="" 120="" mins.="" in="" class="" •="" upload="" to="" canvas="" assignments="" (10="" mins)="" –="" submit="" hardcopy="" •="" chapters="" 5,="" 7,="" 8,="" 9,="" 11,="" 13="" •="" closed="" books,="" closed="" notes="" •="" equation="" sheet="" provided="" material="" reviewed="" in="" this="" review="" today="" is="" not="" be="" exhaustive="" of="" all="" topics="" covered="" on="" the="" final="" exam="" topics="" –="" transient="" conduction="" 5.1="" the="" lumped="" capacitance="" method="" 5.2="" validity="" of="" the="" lumped="" capacitance="" method="" 5.3="" general="" lumped="" capacitance="" analysis="" 5.3.1="" radiation="" only="" 5.3.2="" negligible="" radiation="" (convection="" +="" generation)="" 5.5="" the="" plane="" wall="" with="" convection="" 5.5.1="" exact="" solution="" 5.5.2="" approximate="" solution="" 5.7="" the="" semi-infinite="" solid="" topics="" –="" boundary="" layer="" flow="" 7.1="" the="" empirical="" method="" 7.2.1="" laminar="" flow="" over="" an="" isothermal="" plate:="" a="" similarity="" solution="" 7.2.2="" turbulent="" flow="" over="" an="" isothermal="" plate="" 7.2.3="" mixed="" boundary="" layer="" conditions="" 7.4.2="" convection="" heat="" and="" mass="" transfer="" 8.1.1="" flow="" conditions="" 8.1.2="" the="" mean="" velocity="" 8.2="" thermal="" considerations="" 8.3="" the="" energy="" balance="" 8.4="" laminar="" flow="" in="" circular="" tubes:="" thermal="" analysis="" and="" convection="" correlations="" 8.5="" convection="" correlations:="" turbulent="" flow="" in="" circular="" tubes="" topics="" –="" pipe="" flow="" topics="" –="" free="" convection="" 9.1="" physical="" considerations="" 9.2="" the="" governing="" equations="" for="" laminar="" boundary="" layers="" 9.3="" similarity="" considerations="" 9.4="" laminar="" free="" convection="" on="" a="" vertical="" surface="" 9.5="" the="" effects="" of="" turbulence="" 9.6="" empirical="" correlations:="" external="" free="" convection="" flows="" 9.6.1="" the="" vertical="" plate="" 9.6.2="" inclined="" and="" horizontal="" plates="" 9.7="" free="" convection="" within="" parallel="" plate="" channels="" 9.7.1="" vertical="" channels="" 9.8="" empirical="" correlations:="" enclosures="" 9.8.1="" rectangular="" cavities="" topics="" –="" heat="" exchangers="" 11.1="" heat="" exchanger="" types="" 11.2="" the="" overall="" heat="" transfer="" coefficient="" 11.3="" heat="" exchanger="" analysis:="" use="" of="" the="" log="" mean="" temperature="" difference="" 11.4="" heat="" exchanger="" analysis:="" the="" effectiveness–ntu="" method="" 11.5="" heat="" exchanger="" design="" and="" performance="" calculations="" 13.1="" the="" view="" factor="" 13.1.1="" the="" view="" factor="" integral="" 13.1.2="" view="" factor="" relations="" 13.2="" blackbody="" radiation="" exchange="" 13.3="" radiation="" exchange="" between="" opaque,="" diffuse,="" gray="" surfaces="" in="" an="" enclosure="" 13.3.1="" net="" radiation="" exchange="" at="" a="" surface="" 13.3.2="" radiation="" exchange="" between="" surfaces="" 13.3.3="" the="" two-surface="" enclosure="" 13.3.4="" two-surface="" enclosures="" in="" series="" and="" radiation="" shields="" 13.3.5="" the="" reradiating="" surface="" 13.4="" multimode="" heat="" transfer="" topics="" –="" radiation="" free="" convection=""> 0 ?? ?? < 0="">< 0=""> 0 Vertical Surface ??! = ?? ?" − ?# ?$ ?% Grashoff Number Measures the relative strength of buoyancy and viscous forces Rayleigh Number ??! = ??!?? = ?? ?" − ?# ?$ ?? Factors in thermal diffusion in the relative strength of buoyancy and viscous forces ??! = ? ??! , ?? = ?(??!) Nusselt Number ??! = 0.825 + 0.387??! "/$ 1 + (0.492/??)%/"$ &/'( ' (Eq. 9.26) ?& = ?? ?" − ?# ? Horizontal Surface ??! = 0.52??! '/) (Eq. 9.32) ??! = 0.54??! '/* (Eq. 9.30) Upper Surface of Cold Plate Upper Surface of Hot Plate Lower Surface of Cold