Connecting Transient and Steady-State Analyses Using Heat Transfer and Fluids Examples ABSTRACT. Nomenclature

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1 Connecing ransien and Seady-Sae Analyses Using Hea ransfer and Fluids Examples Washingon Braga Mechanical Engineering Deparmen Ponifical Caholic Universiy of Rio de Janeiro, PUC-Rio Rio de Janeiro, RJ, Brazil ABSRAC. In many undergraduae courses, emphasis is given o he analysis of seady sae siuaions, in spie of he fac ha unseady siuaions are quie common in engineering problems. For insance, while discussing hea ransfer, unseady sae opics are inroduced o sudens wihou connecion o seady sae siuaions, as naure could handle unseadiness separaely. his paper presens a few common siuaions, some of hem ofen reaed only a heir seady counerpar, ha have been used o offer sudens an ineresing and pedagogically rich unseady and seady analysis. he mehodology proposed herein, bridging unseady o seady sae siuaions, helps subjec inegraion, presens some crieria for model simplificaion and allows furher discussion on ransien opics. he curren paper is mainly focused on Hea Conducion. However, similar analysis may be made for siuaions involving Radiaion and Convecive Hea ransfer. Nomenclaure A area, m AR aspec raio, dimensionless B parameer, defined in Eq. (5), s - Bi Bio number, dimensionless C parameer, defined in Eq. (5), K/s C Consan, defined in Eq. (9) c specific hea, J/kg.K D parameer, defined in Eq. (5), s - E parameer, defined in Eq. (5), K/s F dimensionless parameer, Eq. (5) Fo Fourier number, dimensionless H heigh, m h convecive hea ransfer coefficien, W/m K k hermal conduciviy, W/m.K lengh, m m fin parameer, m - P rae of energy generaion inside he sphere, W P e fin perimeer, m - Q rae of ransfer of energy, W q" hea flux, W/m h convecion hea ransfer coefficien inside he recipien, W/m.K; heigh of he channel, m V volume, m 3 R radius of a cylindrical rod, m

2 S dimensionless parameer, Eq. (50) U u x emperaure, K or C ime, s wall velociy, m/s velociy componen coordinae, m Greek eers α hermal diffusiviy, m /s η dimensionless lengh θ emperaure difference, K ρ mass densiy, kg/m 3 φ dimensionless emperaure Subscrips b base of an exended surface c characerisic f fluid H relaive o he heigh i iniial condiion relaive o he lengh R radiaion SS seady-sae s surface; sphere sup superficial ransversal or cross secional free sream condiions Superscrip * seady sae condiion. INRODUCION Engineering problems occur boh during unseady and seady sae siuaions. In fac, experience indicaes ha many problems only occur or are more demanding during ransien siuaions. Noneheless, in mechanical engineering courses, quie ofen, emphasis is given o seady sae analyses. Furhermore, mos undergraduae hea ransfer books inroduce seady and unseady problems as wo separae opics, perhaps because hey require differen analyical mehods: a simple one for seady siuaions and a quie demanding one for unseady problems. As i may be seen, mos unseady sae examples discussed leads o he same seady sae soluion: hermal equilibrium wih he ambien, a siuaion no always found. Pedagogically speaking, he classical approach does no allow knowledge consrucion, and does no help sudens o build up connecions among he new maerial and he many ransien siuaions hey me in heir lives. o some sudens, his procedure may indicae separae roues, soluions and physics and ofen, hey end up believing ha here is no unseadiness behind a seady sae. Consequenly, i may be concluded ha a more deailed analysis should be made linking boh siuaions. his paper inends o presen a few classroom-ype siuaions in which a simple and ye ineresing analysis may be made o inroduce appropriaely he pah from unseady o seady siuaions: he lumped formulaion cooling problem, he hea ransfer problem in a - D slab, a fin (exended surface), a couple of -D siuaions and he flow of fluid inside a channel, he Couee problem. In all such problems, he unseady and seady pars are analyzed in order o obain he ime necessary o reach seady sae as funcion of he relevan physical parameers, an imporan piece of informaion no only for many

3 indusrial problems bu also o an adequae undersanding of he physical siuaion. I is expeced ha following a similar analysis, sudens may sar o visualize correcly ha seady sae may occur in some siuaions, no always, bu only a he end of an unseady siuaion. Sure, he ransien phase may or may no be fas, depending on fluid hermal properies and physical geomery, bu now sudens may undersand why.. SIMPE COOING PROBEM Usually discussed in an inroducory hea ransfer course, his problem describes he cooling (or heaing) of a small diameer, high hermal conduciviy sphere (or cylinder), iniially a emperaure i, ha is dropped inside a pool, conaining some non-idenified fluid, ha far from he ho sphere is mainained a some uniform emperaure. he hea ransfer coefficien, h, assumed o be uniform, akes care of convecion and/or radiaion beween he sphere and he inner walls of he recipien conaining boh he fluid and he sphere. A more ineresing siuaion occurs when i is considered ha he emperaure of he fluid may also change according o hea ransfer no only o he sphere bu also o an exernal environmen []. Considering he presence of some inernal source of hermal energy inside he sphere, Joule heaing, for insance, say P, and aking ino accoun some hermal energy los o ha exernal environmen, Q R, he lumped formulaion balance of energy for boh fluid and sphere, neglecing he paricipaion of he recipien walls, may be wrien as: d ρ cv = ha f sup s f QR () d f fluid: ( ) [ ] d ρ = s sup s f () d s sphere: ( ρ cv) = P ha [ ] his sysem of equaions mus be solved considering as iniial condiions: ( = ) = 0 s ( = ) = 0 f s,i f,i o avoid solving he above sysem, one usually neglecs he energy equaion for he fluid and considers only he hermal profile for he sphere. Naurally, such approximaion simplifies he problem bu, on my accoun, i also reduces he chances of a beer undersanding of is physics. herefore, solving he sysem is recommended. Is soluion [] is: (3) (4) Acually, his erm depends on he emperaure difference beween he recipien and he environmen. I s aken herein as a fixed eniy for simpliciy. 3

4 B(s,i f,i ) (B D) CD BE () s = e s,i B+ D B+ D D (B D) CD BE () = ( ) e B+ D B+ D f s,i f,i f,i where he following definiions apply: B D ha = ρcv sup ( ) ha = ρcv s sup ( ) f C E P = ρcv ( ) s QR = ρcv ( ) f (5) Several ineresing resuls may be found: () (). Afer some ime, s f, ha is, whenever he non-linear drops o zero, he emperaure difference beween sphere and fluid drops o zero, as i may be readily verified.. he muliplier consan CD BE, ha appears on he linear erm on he righ hand side of boh equaions, is in fac a relaion beween he inernal source of hermal energy inside he sphere, P, and he hea los o he exernal environmen, Q R. Whenever boh erms are equal, we may have a seady sae. 3. Whenever P= QR, he final, i.e. he seady sae emperaure may be obained from simple hermodynamics argumens, and i is: or = ( ρ ) s,i + ( ρ ) ( ρ cv) + ( ρ cv) cv cv * s f * Bf,i + Ds,i = B+ D s f f,i 4. We ofen assume ha he fluid has a much larger hermal ineria (or capaciance), ha is, a larger ( ρ cv) han he corresponding value for he solid sphere. In his case, any emperaure variaion for he fluid may be negleced o obain: D= 0, herefore, ( ) * = f,i = f () = e + B s f s,i s,i = = (8) (6) (7) (9) 4

5 Criical o he presen analysis is he evaluaion of he ime needed o allow body and fluid o reach a common emperaure ha may or may no be, he final seady sae emperaure (ha is, provided P= QR). hrough observaion of Equaion (5), we may conclude ha such siuaion happens whenever he exponenial erm drops o zero. Noicing he characerisic of he exponenial funcion, a sufficien accurae esimae of he ime necessary o aain he desired siuaion is obained assuming a large number for he exponen. I usually consider 8, bu any oher suiable number may be obained (see a more horough discussion on his below). herefore, i may be concluded ha seady sae is obained whenever: * (B+ D) = 8 (0) or ha may be wrien as: Fo * 8 = Bi + ( ρcv) ( ρcv) s f ( cv) 8 ρ D = ha + B * s In he above equaion, he Fourier Number, defined as defined as Bi h sup Fo α s c () =, and he Bio number, c =, are inroduced in erms of a characerisic lengh, c k s, defined as he raio of he sphere s volume o surface area. Considering he sandard siuaion in which he fluid is handled as having an infinie hea capaciy, we obain a simpler resul: ( ρcv) 8k 8 = 8 Fo* = = () ha h(v/a ) Bi * s s sup sup As expeced, he larger he hermal ineria (or he convecive hermal resisance), he longer i will ake o reach seady sae. In any even, wih such expression, sudens may observe clearly he influence of a large hea ransfer coefficien or a small mass on he aainabiliy of a seady sae siuaion, eiher for he cooling of hermal sysems bu also for effecive emperaure measuremen using a ermocouple. Figures, and 3 illusrae some of he analysis ha may be done. Figure describes he emperaure profiles for he fluid and he sphere for he case in which he hermal capaciances of boh bodies are of he same order of magniude (considered as equal, for simpliciy). Figure is obained considering a 5

6 much larger hermal capaciance (aken as infiniy) for he fluid han for he sphere. In such siuaion, no significan emperaure drop is observed for he fluid. Finally, Fig. 3 indicaes he siuaion in which he emperaure difference beween fluid and sphere drops o zero afer a while bu no seady sae is achievable. In such case, he hea generaed inside he sphere is larger han he hea los o he exernal environmen. Mahemaically, his indicaes ha he exponenial erm in Equaion (5) drops o zero bu no he linear one. Fig.. emperaure Profiles for same order hermal capaciance bodies. Seady sae is aainable, Fig. emperaure Profiles for Sandard Cooling Problem. Fluid has infinie hermal capaciy. Seady sae is aainable. 6

7 Fig. 3. emperaure Profiles for a siuaion in which no seady sae is aainable emperaure difference beween bodies drops o zero. Anoher way o pick a reasonable number involves he aainable precision on emperaure measuremens. ha is, assuming ha i is reasonable o measure unseady emperaure differences in he order of some small number, say 0.5% of he oal variaion, bu nohing less, i is sufficien o consider seady sae whenever: B e = 0005, e As i is seen, here is no significan difference beween such approaches FA PAE, UMPED FORMUAION Consider a fla plae, having a (relaively) small hickness, subjeced o a radian hea flux on he lef surface, q " R, and a convecive hea ransfer o an ambien fluid a emperaure and having a convecive hea ransfer coefficien,h, on he oher surface. A ime = 0, he iniial emperaure is i, considered consan. Assuming consan hermodynamical properies, he Firs aw of hermodynamics on such siuaion, may be wrien as: " ρ cv = qr h( ) A sup = i a = 0 x (4) ha has an exac soluion, he following emperaure profile: (3) 7

8 φ (Fo) = + F exp Bi Fo Bi Bi In such equaion, Bi h =, k { } Fo α =, φ= q /k φ= and " R F q /k i " R (5) =. As i may be seen, he assympoic characerisic exponenial funcion precludes a sharp definiion for he final condiion, ha is he seady sae. o overcome such difficuly, one may use he Riz Inegral Mehod [3], in which a rial profile such as: (6) φ (Fo) =φ SS f(fo) is used, where φ is an approximae profile, φ SS is he seady sae soluion and f(fo) is some arbirary (usually polinomial) funcion such as a quadraic profile, f(fo) a afo afo = + + 3, or a cubic one, 3 = he consans, a,a,a 3 and a 4 are f(fo) a afo afo afo deermined according o he condiions: f(fo= 0) = 0 f(fo= Fo*) = f'(fo= Fo*) = 0 f''(fo= Fo*) = 0 In such equaions, Fo*(or *) indicaes he necessary ime o reach seady sae, he only unknown remaining in Equaion (6) subjeced o Equaion (7). According o he inegral mehod, he rial profile φ, wrien in erms of he unknown Fo*, mus be inroduced in he dimensionless form of he inegral equaion obained from he energy equaion: Fo* φ Fo Fo* d( Fo) = [ Biφ ] d(fo) (8) 0 0 Depending on he order of he approximaion, differen resuls may be obained, such as: quadraic profile: cubic profile: 3 3 Fo* = * = Bi Bi α 4 4 Fo* = * = Bi Bi α Previous experience wih inegral mehods does no allow us o conclude which approximaion gives beer resuls. Oher opions may be chosen: for insance, considering (7) 8

9 ha he hickness of he boundary layer over a fla plae is arbirarily defined as he heigh in which he velociy reaches 99% of he exernal uniform velociy, we may use a similar descripion o obain a value such as: 5 5 = = Bi Bi α Fo* * As a maer of fac, he imporan fac is ha all previous opions indicae ha he correc answer may be generically represened as: C C = = Bi Bi α Fo* * in which he consan, C, is o be deermined following any suiable crieria. In shor, higher Fo* indicaes longer ime o reach seady sae. According o he previous analysis given herein, I ve been using a number such as 8. (9) 4. FA PAE, DISRIBUED FORMUAION In he lieraure, he sudy of unseady sae in siuaions in which he inernal (conducive) resisance is no negligible compared o he exernal (convecive) resisance is usually inroduced using a -D fla plae as model. he physical siuaion is such ha he iniial condiion is such ha he plae has an uniform emperaure, say, and suddenly i is dropped inside a non specified medium having uniform emperaure. he convecive hea ransfer coefficien, h, is considered consan. In such siuaion, i is expeced ha afer some ime, he wall reaches hermal equilibrium wih he medium, having he same final emperaure. he (classical) soluion is obained using he mehod of separaion of variables, a lenghy procedure ha involves eigenvalues and eigenfuncions (sine or a cosine series expansion - Fourier s series - for a Caresian slab). I have been using anoher problem, wih a beer resul, perhaps because of he sudens greaer familiariy. Recognizing ha on he iniial sages of a junior level hea ransfer course, a seady sae -D fla plae is horoughly sudied, I have been discussing he unseady sae associaed o i, rying o infer how long i akes o develop. According o a balance of energy and Fourier s law of hea conducion, sudens learn fundamenals abou a seady profile such as: (x) = cx+ c in which consans c and c will be deermined following some boundary condiions. he simples case involves specified surface emperaures, a x = 0, = and x =,=. o analyze he unseadiness of such problem, we may consider he i Mos ineresing cases involving convecive or radiaive hea flux boundary condiions are easily handled. 9

10 following model: α subjec o: = x = 0,(x, 0) = i (a) (0) (x = 0,) = (b) (x =,) = (c) Applying he superposiion mehod, we may wrie ha a enaive soluion may be found according o he emperaure profile given by: (x,) = (x) + (x,) () SS For he case under analysis, he firs erm is simply he seady profile: = + (3) SS(x) = x+ and he erm(x,) indicaes he unseady par of our problem, ha is expeced o drop o zero evenually. Doing so, our problem is reduced o: = α x and he boundary condiions are: a x = 0: (x = 0,) = (x = 0) + (x = 0) = However, by definiion: SS SS (x = 0) =, so we may say ha (x = 0) = 0 (5a) and a x = : Similarly, SS (x=,) = (x = ) + (x = ) = SS (x= ) =, herefore, we have ha: (x= ) = 0 (5b) he iniial condiion deserves a similar analysis. A = 0,(x, 0) = i. herefore, (x, = 0) = (x) + (x, = 0) = (5c) Consequenly, o SS i (4) 0

11 = = = (6) (x, = 0) = i SS(x) = (i ) x ha may be wrien as: = 0, (x, 0) = a+ bx Using he sandard procedure for solving unseady D problems [4,5], we obain: ( ) ( ) (x,) = exp( αλ n) dcos λ nx + esin λ nx (7) in which he eigenvalues are given by λ = nπ (8) n Applying he non-homogeneous iniial condiion yields d = 0 and: ( ) (x, 0) = a+ bx= esin λ x (9) n n Using he sandard procedure [3,4,5], we obain ha: e = n n a ( ) ( a b) nπ + + Consequenly, he complee emperaure profile, including boh he unseady and seady soluions, is wrien as (30) n ( ) αλ n n (3) (x,) = a+ bx+ e sin λ x e αλ n= his resul may be presened graphically, as indicaed in Fig. 4. he resuls were obained for he following se of condiions: i = 5C; = 00C; = 30C and are displayed as funcion of he Fourier number, previously defined.

12 Fig. 4. Unseadiness of -D emperaure profile he quesion remaining o be answered is again he ime span necessary for he seady sae. Noicing he ransien emperaure profile, we may conclude ha seady sae is reached whenever he exponenial erm drops significanly o zero. As we are aware, he mos criical eigenvalue is he firs one [4,5]. Consequenly: exp( αλ ) = exp[ Fo ( λ ) ] exp[ ] (3) 8 λ =π In he presen siuaion, =π, see Equaion (8), and herefore, we may conclude ha seady sae happens whenever: Fo = = 08, ( λ ) π 0 in which i was considered ha ð 0. I may be noiced in Fig. 4, ha for Fo 05,, he seady sae (linear) profile is visually obained, indicaing ha his simple analysis is convenien. However, for smaller Fourier numbers, he ransien emperaure profiles are far from he seady sae profile, clearly indicaing he reason why he Fourier s law of hea conducion may no be aken as: q = ka = ka x before he seady sae is achieved, a siuaion usually no clearly undersood. Naurally, he acual profiles depend on he daa used as boundary and iniial condiions and oher siuaions may be easily sudied. For sudens, i may also be ineresing o compare ransien imes for differen maerial, o indicae he influence of he hermal diffusiviy (or he lengh) as shown in (33) (34)

13 able. able : ime for Seady Sae, considering a D fla plae wih hickness = 0, m 5. UNSEADY PROFIE IN EXENDED SURFACES Exended surfaces is one of hose opics ha display an ineresing unseady profile, ofen no discussed among sudens. An energy balance for a consan ransversal area fin, wih consan hermal properies and convecive hea ransfer coefficien is given by: θ m θ= x θ α in which θ (x) = (x), is he fin excess emperaure, m hp e ka (35) = is he fin parameer and α is he hermal diffusiviy of he maerial used on he fin. he iniial and he boundary condiions, chosen for he sake of simpliciy, are expressed by: = 0, θ (x) = 0 (36a) x = 0, θ (x = 0) =θ b, he emperaure a he roo of fin (36b) x =, θ (x = ) = 0 = θ = = (36c) A sraigh forward analysis similar o he one made for he previously discussed example, uses: θ (x,) =θ (x) +θ (x,) (37) where SS SS θ (x) SS is he seady sae emperaure profile for his ype of fin (very long fin) 3 : θ (x) =θ (38) mx be 3 Oher siuaions may be handled similarly. 3

14 and indicaes ha he eigenvalues are given by: nπ λ = + m herefore, he complee emperaure profile is given by: (39) n αλ ( ) (40) θ (x) =θ (x) + ce sin λ x SS n n in which he inegraion consans are given by: c n = φ π SSsin( n xdx ) (4) 0 Repeaing he previous analysis, we obain ha seady sae is reached when ha is: exp( αλ ) = exp[ Fo ( λ ) ] exp[ ] Fo = 0 8 ( λ ) ( π + m ) ( + m ) I is now simpler o undersand how he cooling rae is affeced by a higher hea ransfer coefficien, obained for insance increasing he velociy of he cooling fluid, bu also by he hermal conduciviy, he cross secion and he perimeer of he fin. See Fig. 5 for a graphical display of he ransien behavior. I may be noiced he quick ime evoluion of he dimensionless emperaure profile, comparing for insance he soluions for Fo = 0,03 and Fo = 0,08 and hen he soluions for Fo = 0,08 and Fo* = 0,4, ha corresponds o he seady sae. (4) Fig. 5. Unseady emperaure Profiles in Fins for m = 3,0. 4

15 6. A WO DIMENSIONA FA PAE Consider a plane wall of hickness, iniially a a uniform emperaure i. A ime = 0, he wall is placed in a medium ha is a some emperaure, far from he wall. Hea ransfer occurs by convecion wih a uniform and consan hea ransfer coefficien h. Mahemaically, he problem may be defined according o he following energy balance: + = α x y he sandard procedure o obain an analyical soluion o his problem sars wih he proposiion ha variables can be separaed: (x,y,) (43) = X(x,)Y(y,) (44) Generalizing previous resuls, we may wrie ha he criical Fourier Number is given by: Fo in which α 8 = = ( ) ( H) /AR λ + µ AR H eigenvalues are given by: (45) = is he (geomerical) aspec raio of he D plae and he wo ses of Bi an( λ ) = λ and Bi an( µ H) = H µ H Naurally, i is ineresing o have a general crieria o jusify he usage of a D problem insead of a much simpler D problem, as before. A simple one may be proposed using Equaion (45). For insance, i is obvious ha whenever: ( λ ) >> ( µ H) /AR one may neglec he influence of he hea ransferred along he horizonal surfaces and rea he problem as a sandard D verical fla plae. For he presen purpose, a raio of 0 will be considered reasonable for ha. herefore, whenever: ( ) ( µ H) ( ) λ = 0 AR o make hings easier, we may consider as before ha π 0, resuling in: 5

16 µ H λ =π AR o generalize hose conceps, we may define a physical aspec raio (as i akes ino accoun he influence of he boundary condiions hrough he eigenvalues) as: AR* = λ herefore: Fo ( µ H) ( ) α 8 = = ( ) ( AR* ) /AR λ + Consequenly, whenever AR >π( AR* ) (46) (47) >π, ha is, he geomerical aspec raio is greaer han π imes he physical aspec raio, he D problem may be reduced o a D verical plae problem, possessing a much simpler soluion. Similarly, if AR* >π AR, he D problem may be reaed as a D horizonal plae problem. For oher values, he D model becomes relevan. o illusrae such effec, some resuls are shown in able, in which he nondimensional emperaure differences beween he D and he D approximaions are displayed. he chosen geomerical aspec raio is,0. he ambien emperaure is 40 C and he iniial uniform emperaure is 450 C. he hermophysical properies are he hermal conduciviy (=,3 W/mK) and he hermal diffusiviy (=,E-6 m /s). he convecive hea ransfer coefficien is assumed o be 00 W/m K along he verical surface. Along he horizonal surface, he corresponding coefficien is reaed as parameer and handled as he Bio number defined as BiH = hhh/k. he resuls are displayed as funcion of ime. able. emperaure Differences beween D and D Approximaions for a Fla Plae. Geomerical Aspec Raio (AR) =.0 As i can be seen, he emperaure differences are reduced significanly whenever 6

17 AR* is small compared o AR(he geomerical one). he emperaure differences increase wih ime bu are reduced o zero as he seady sae emperaure is he same in boh cases. 7. Unseady Profile for Shor Cylinders Following a similar analysis, we may conclude ha, for a cylindrical rod of radius R and heigh, seady sae is obained whenever Fo wherein α 8 = = ( ) ( R) /AR λ + µ AR R = and µ is he firs eigenvalue obained as roo of he equaion: µ µ + µ = (49) J( ) Bi J( o ) 0 where Jnindicaes he Bessel funcion of n-h order. As before, a similar crieria may be obained o allow us o neglec he hea ransfer hrough he horizonal surfaces (he infinie cylinder case) or hrough he laeral surface (he infinie fla plae case). (48) 8. ransien Couee Problem A simple soluion o he Navier-Sokes equaion is obained for he flow beween wo parallel fla plaes, one of which is a res, he oher is moving wih consan velociy U. his is he so-called Couee Problem [6]. he seady sae soluion is easily obained as: u =η+ S ( ) U η η (50) in which he following definiions apply: y η=, where h indicaes he disance beween he wo plaes, i.e. he channel S h = h dp µ U dx ypically, for S 0 >, ha is, for a pressure decreasing in he flow direcion, he velociy is posiive over he whole channel. For negaive values of P, however, he velociy may become negaive, indicaing he exisence of a back-flow near he saionary wall. Following a similar procedure close o he ones already shown, we obain he following profile as ransien soluion: 7

18 u(y,) U where: Fo n n ( n ) (5) =η+ S η ( η ) + d exp{ Fo ς }sin ςη n ν = h ς, a dimensionless eigenvalue, is given by λ h=ς= nπ n = + + ς ς ς ς n ( ) S S dn S ( n) 3 n n n For he sake of demonsraion, Fig. 6 indicaes he ransien velociy profile for he special case ha S=. As i may be seen, i is obvious ha he slip condiion is a localized effec while he pressure gradien is a bulk effec. he developing of a boundary layer 4 ype of flow along he moving wall is clearly displayed. I is also shown he final seady sae profile, for comparison purposes. A similar analysis may be done for he Hagen-Poiseuilli flow, as done originally by Szymanski [7]. Fig. 6 ransien Couee Flow, indicaing he developmen of boh a back-flow and a posiive flow regions wih ime. 4 ha is, a region in which he moving wall affecs he flow. 8

19 9. Conclusions his paper presens a few discussions on how o invesigae seady sae siuaions as a final sage for hea ransfer problems, having in mind undergraduae engineering sudens. In all he siuaions reaed here, i is shown how hermal and geomerical parameers affec he ime necessary o reach seady sae, allowing a deeper undersanding of ransien effecs. Considering he sandard approach used in mos exbooks, he procedure discussed here has a leas wo advanages: he sequenial analysis of he problems and he link beween conceps, allowing a clearer view of he evoluion. Alhough no shown here, here are many oher siuaions ha may be handled accordingly, allowing, perhaps, a fuller inegraed engineering course. Acknowledgemens I would like o hank he reviewers for suggesing improvemens and helpful commens o he original manuscrip. 9. References [] PIS, D.R. & SISSOM,.E., Hea ransfer, Schaum Oulines, 977, [] KAPAN, W. Advanced Mahemaics for Engineers, Addison Wesley, 98 [3] ARPACI, V.S., Conducion Hea ransfer, Addison Wesley, 966 [4] BRAGA, W., Hea ransfer, homson earning Pub. Co., 003, in poruguese [5] INCROPERA F.P. & DEWI D.P. Fundamenals of Hea and Mass ransfer, Wiley, N.York, 996 [6] SCHICHING, H., McGraw-Hill Book Company, 6h Ediion, 968. [7] SZYMANSKI, F., Proc. Inern. Congr. Appl. Mech. Sockholm,