Consideration is given to the effects of viscous dissipation on the developing heat transfer between a fully developed laminar non-Newtonian fluid flow and a concentric annular geometry with a moving heated core. In this report, the results with the third and fourth kinds of thermal boundary condition are presented. Applying the shear stress described by the modified power-law model, the energy equation including the viscous dissipation term is solved numerically. The effects of radius ratio, flow index, relative core velocity, dimensionless shear rate parameter and Brinkman number on temperature distribution and Nusselt numbers are discussed

Davaa Ganbat

Momoki Satoru

Shigechi Toru

Davaa, Ganbat

Shigechi, Toru

Momoki, Satoru

Nagasaki university's Academic Output SITE

Laminar heat transfer in the thermal entrance region of concentric annuli with moving heated cores : Part II : The cases with the third and fourth kinds of thermal boundary condition

Nagasaki University's Academic Output SITE: NAOSITE

Reports of the Faculty of Engineering, Nagasaki University, Vo1.32, No.59Laminar heat transfer in the thermal entrance regionof concentric annuli with moving heated cores(Part II: The cases with the third and fourth kindsof thermal boundary condition)byGanbat DAVAA*, Toru SHIGECHI** and Satoru MOMOKI**51Consideration is given to the effects of viscous dissipation on the developing heat transfer betweena fully developed laminar non-Newtonian Huid How and a concentric annular geometry with a movingheated core. In this report, the results with the third and fourth kinds of thermal boundary conditionare presented. Applying the shear stress described by the modified power-law model, the energyequation including the viscous dissipation term is solved numerically. The effects of radius ratio, Howindex, relative core velocity, dimensionless shear rate parameter and Brinkman number on temperaturedistribution and Nusselt numbers are discussed.1. IntroductionIn the previous report(l), numerical solutionsthe heat transfer of a non-Newtonian Huid in aconcentric annulus with an axially moving corewere obtained for thermal entrance region withfirst and second boundary conditions.In this paper, the entrance-region heat transferbetween a fully developed laminar fluid flow anda concentric annular geometry with a movingheated core is studied numerically. Applyingthe fully developed velocity profile reported forthe modified power-law model in the previousreport(2) , the energy equation including theviscous dissipation term is solved numericallyusing the finite difference method for the thermalboundary conditions of third kind and fourthkind. The effects of the radius ratio, relativevelocity of the core, the flow index and dimen-sionless shear rate parameter and Brinkmannumber on the temperature distribution andNusselt numbers are discussed.NomenclatureBr Brinkman numberCp specific heat at constant pressureDb hydraulic diameter == 2(Ro - Ri)k thermal conductivitym consistency indexn How indexPe Peclet numberr radial coordinateReceived on April 19, 2002• Graduate School of Science and Technology•• Department of Mechanical Systems Engineeringr* dimensionless radial coordinate == r / DhR radiusq wall heat fluxT temperatureUrn average velocity of the fluidu* dimensionless velocity == u/urnU* dimensionless relative velocity of themoving core == U/ UrnZ axial coordinatez* dimensionless axial coordinate= Z/(PeDh)Greek Symbolsa radius ratio == RdRo/3 dimensionless shear rate parameter""a apparent viscosity.,.,: dimensionless apparent viscosity == ""a/""*.,.,* reference viscosityp density~ transformed dimensionless radialcoordinate == [2(1 - a)r* - 0]/(1 - a)Subscriptsb bulke inletinner tubeii at the inner wall with the inner heatedo outer tubeoi at the outer wall with the inner heated2. AnalysisThe physical model for the analysis is shown inFig.l. The inner core tube is moves axially at aconstant velocity, U. The assumptions used in theanalysis are:52 Ganbat DAVAA, Torn SHIGECHI and Satorn MOMOKI1. The flow is incompressible, steady-laminar,and fully developed hydrodynamically.2. The fluid is non-Newtonian and the shearstress may be described by the modifiedpower-law model(3), and the physical prop-erties are constant except viscosity.3. The body forces and axial heat conductionare neglected.Heat transferThe energy equation together with the assump-tions above is written asFixed tubeo.... a::Moving CoreFig.l Schematic of a concentric annulus with anaxially moving core(10)(11)(12)(13)(14)[- ar(3) I ]8r R;[~](3) = T(3) - T e() - ---;"::(3""")--Ii -Te(3) _ D hNUii - T(3) T(3)j - b(4) - Dh [- 8T8r(4) IR..,]Nu jo - 7:(4) T.(4)o - bThus, the various Nusselt numbers are calcu-lated as:1 8 (*8()) B * (du*)2 _ *8() (15)-- r - + r· '" - - U -r* 8r* 8r* a dr* 8z*k [T(4) - Te]()(4) == ----= _=_qjDhthe energy equation and the boundary conditionsmay be expressed in the dimensionless forms asIntroducing a dimensionless temperature, (), de-fined as(1) The boundary condition of the third kind:{()(3) = 1 at r* = 2(lC:a )88(S) * 1 (16)a;:;r = 0 at r = 2(1-a)(2) The boundary condition of the fourth kind:{d:C:) = -1 at r* ar = 2(1-a) (17)()(4) = 0 at r* = 12(1-a)(1)(7)(6)(k = 3 or 4) (4)for k = 3 or 4 (5)k!~ (r8T) + T (dU) = ()Cpu 8Tb.r 8r 8r dr 8zz = 0: T(k) = T efor Rj:5 r :5 RoThe velocity, u, and its gradient, t, the shearstress, T, have been evaluated and reported in theprevious report (2) •The thermal boundary conditions:(1) The third kind (constant wall temperature atthe moving core with the outer tube insulated):{T(3) = Ii(3) at r = Rj<lfT'(S) (2)9r,:- = 0 at r = Ro(2) The fourth kind (constant heat flux at themoving core and the temperature of the outer tubeis kept equal to the uniform entering fluid temper-ature):{_k8'Jj~4) = qj at r = Rj (3)T(4) = T e at r = RoThe initial condition is:where the heat transfer coefficients are defined as:k 8T(S) I(3) _ - ar R;hii = T(3) 1'J"l(3)j - ~bThe Nusselt numbers are defined as:(k) h~.k) . DhN U.. = -=:..11-:--_11 k(4) h(~) • DhNu. = ~Ol_---:;:;,.01 k(8)(9)The initial condition is:z* = 0: ()(k) = 0a 1for 2(1 _ a) ~ r* ~ 2(1 _ a) (k = 3 or 4)(18)Laminar heat transfer in the thermal entrance region of concentric annuli with moving heated cores 53where The effects of the relative velocity, U*, on thedevelopment of temperature profiles are demon-strated in Figs.2a and 2c for the third kind ofboundary condition (k = 3). It is seen that thefluid temperature increase is less rapid for largervalues of the core velocity.The effects of viscous dissipation on the devel-opment of temperature profiles for Q = 0.5, n =0.5, f3 = 105 and U* = 1.0 for the third kind ofboundary condition (k = 3) are shown in Fig.2aand 2d.The effect of the moving core velocity on Nus-selt number at the tube walls are shown in Figs.3ato 3c at given values of Br = 0.0 and a = 0.5 forthree different fluids (n < 1.0 pseudoplastic, n =1.0 Newtonian and n > 1.0 dilatant).The calculation results of the particular case ofNewtonian fluids (n = 1.0) with neglected viscousdissipation (Br = 0.0) are compared with the pre-dictions by Shah and London(4) for the stationarycore (U* = 0) and by Shigechi and Araki(5) forthe moving core (U* = 1.0), respectively. Even atsmall values of z*, it can be seen in Figs.3a, 3band 3c that the agreement is excellent. The effectof the relative velocity of the moving core tubeis always to increase the values of Nusselt num-bers, N 'Uii and to decrease the values of Nusseltnumbers, N Uoi, for the given conditions of a andBr.The viscous dissipation effects on Nusselt num-ber are shown in Figs.4 to 6 for two different flu-ids. With an increase in Brinkman number NUiidecreases for U* = 0.0 and N'Uoi increases for U*= 0.0 and U* = 1.0.For the third kind of boundary condition Breffects strongly on N Uii in both the thermally de-veloping and developed regions when the core isfixed. N Uii decreases with an increase in Br. Forthe moving core in the thermal entrance regionthe effect of the Br on NUii is almost negligiblebut in the fully developed region N Uii increaseswith an increase in Br.For the fourth kind of boundary condition, Brhas a strong effect on N Uii at the thermally fullydeveloped region while the core is fixed. Butfor the case of the moving core the .effect of theBrinkman number is very small at both the ther-mal entrance region and the fully developed re-gion.4. ConclusionsThe heat transfer between a fully developedlaminar fluid flow and concentric annular geom-(19)(20)(21)(22)(23)(24)~z* = 10-9~z* = 10-3NU~.4)1112(1-",)O(k) - 8(1-a) I * o(k) *d *b - U r rl+a2(1~0<)* 2Br(4) == TJ UrnDhqiThe dimensionless apparent viscosity, TJ~, and ref-erence viscosity, TJ*, have been reported in the pre-vious report(2).Nusselt numbers at the tube walls:(3) 1 ao(3) IN uii = - 1 _ 0(3) ar* 0<b 2(1-"')(4) 1 00(4) INUoi = 0(4) _ (J(4) ar* 1o b 2(1-0<)where the dimensionless bulk temperature, ~, isdefined as3. Results and discussionThe calculation has been carried out by usingthe finite difference method. The range of param-eters considered are:The radius ratio: 0.2 ~ a ~ 1.0The relative velocity: 0 ~ U* ~ 1.0The flow index: 0.5 ~ n ~ 1.5Dimensionless shear rate parameter:10-5 ~ f3 ~ 105Brinkman number: 0.0,0.01, 0.05 and 0.1.The mesh sizes used in the numerical calculationare shown below.a. Axial direction (~z*):0< z* ~ 10-3 :10-3 < z* ~ 1 :b. Radial direction (~~)~~ = 1/100.The development of the non-dimensional temper-ature profiles in the thermal entrance region of aconcentric annulus with a heated core for the twokinds of the boundary conditions (k = 3 and 4)are presented in Fig.2a and Fig.2b, respectively,for the same condition ( a = 0.5, n = 0.5, powerlaw fluid (f3 = 105) and U* = 1.0). The figuresillustrate clearly how the temperature profiles de-velop for the two different boundary conditions.54 Ganbat DAVAA, Toru SHIGECHI and Satoru MOMOKIFig.2.a Development of temperature profiles(3rd kind)oooo2'0'""oo2Fig.2.c Relative velocity, U* on the developmentof temperature profiles (3rd kind)o0.. ~0 .•a-O.5.. -0.5U - 1.0II3. M.Capobianchi and T.F.Irvine, "Predictions of pres-sure drop and heat transfer in concentric annularducts with modified power law fluids" Wiinne-undStojfUbertragung, 21, p.209-215, (1992).4. Shah,R.K. and London,A.L., "Laminar flow forcedconvection in ducts" , Advance' in Heat T'ran,fer, Sup-plement 1, Academic Press, (1970).5. K.Araki, "Laminar heat transfer in annuli", Depart-ment of Mechanical Engineering, Nagasaki Univer-sity. Master thesis, (1991) (in Japanese).Reference1. G.Davaa, T.Shigechi and S.Momoki "Laminar heattransfer in the thermal entrance region of concentricannuli with moving heated cores (Part I: the caseswith the first and second kinds of thermal boundaryconditions)" Report, of the Faculty of Engineering,Nagasaki UnitJer"ity, vo1.32, No.59, (2002).2. G.Davaa, T.Shigechi and S.Momoki "Fluid flow formodified power law fluids in concentric annuli withaxially moving cores" Reporta of the Faculty of Engi-neering, Nagasaki Univer"ity, vo1.32, No.58, p.83-90,(2002).etry with a moving heated core of Huid or solidbody is studied with the two different boundaryconditions.For equal conditions, increasing U*, an increasein Nusselt numbers N'Uii as NUoi decreases for allthe cases of 1st, 2nd, 3rd and 4th kind. Br affectsstrongly on Nusselt number at the unheated fixedtube.Fig.2.b Development of temperature profiles(4th kind)Fig.2.d Effect of Br on the developmentof temperature profiles (3rd kind)o0.40 ••a-O.5.. _0.5U- 1.0IIoo0.. ~0.40 ••0.2a-O.5.. -0.5u- 1.0.r-O.05fJ - 10'2oLaminar heat transfer in the thermal entrance region of concentric annuli with moving heated cores 55--_ u=o,o--- -- U = 0,5 -----------«:~<:::~:<:;«,<::'::,:::::::..,a=0.5n =0.5Br=O.Of3 =Iffu=O.O ---___ U = 0.5 --------m<~:::~:::<<:;;:,"<,o""o,<",,.::.':0..:::::.:,.a=0.5n =0.5Br=O.Of3 =lOs100 r--..........,.....,...............,r--.....,......,.....,...............,..--.....,......,.....,................,..--~ ..................~1L...- L...- L....- ........JL....---'- .......J100 r- ~ ,...,..,.r- ...,..... .....,r- ..,....., .....,;--.....,......,....., ~u=O.O--U = 0.5 --------U= 1.0 -.-Shah & London ..Shigechi & Araki ..10--..--<~~>."--<>..--<~~~:::::.,"~""'''''''~ "".,~".",,,,,,.a=0.5n =1.0Br =0.0a=0.5n = 1.0Br=O.O1L....-.....................................JL....---'-............................JL...---'-............................_ ....................................100 r--------,..--------,..--------,-------,---a=0.5n =1.5Br=O.Of3 =10-5u=O.O --U =0.5 --------U= 1.0 .U=O.O --- --" U =0.5 -----------«>._.. U = 1,0 ... --- ............----<:~~:::::~::::<:::::;:""::"o,,...a = 0.5 ----.:..:.::::..-.:.:.:.:.:..-n = 1.5Br = 0.0f3 = 10-51L...- L...- -JL....- -JL...- ---I10-4 10-3 10-2Z-Fig.3.a Nusselt numbers, NUiiat n = 0.5, 1.0 and 1.5 (3rd kind)Fig.3.b Nusselt numbers, NUiiat n = 0.5, 1.0 and 1.5 (4th kind)10a=0.5 a=0.5 a=0.511=0.5 11= 1.0 II =1.5Br=O.O Br=O.O Br=O.O,<-::.:....:-::.::::..:::::.-l/~=1(r p=lO-1 l?1 ,t~ ["• ShIiJ &Lmdon ,~• SHgscN&AtaId lU=O.O- - U=O.O U=O.O-U=O.5 ---. ----. U=O.5 U=O.5 ------ ",~U=1.0 ......... ····u·.· U=1.0 U=1.0 ......... €0.110-1 10-1 10-2I~ 10-1 10-2 10-1 I~ 10-1 10-1 10-4 10-1t t tFig.3.c Nusselt numbers, N Uoi at n = 0.5, 1.0 and 1.5 (4th kind)56 Ganbat DAVAA. Tom SHIGECHI and Salom MOMOKIa=0.5n=0.5If =0.0{3 = IO~10-l 10a=0.2n =0.5If =0.0{3 = IO~Br=O.oo --Br-O.01 --------Br=O.05 .Br=O.10 .a=0.2n =0.5If = 1.0{3 = IO~Br-D.OO --Br-O.01 -------.B(>=0.05 .Br-O.10 .a=0.5n =0.5If = 1.0fJ = 10'Bt' .00--Br-D.01 -------.Bf'2=O.05 .Br-0.10 .B .00--Br>=0.01 --------B(>=0.05 .Bf'2=O. 10 .1'--~-~........-~~~ ......-~~~~""--~~-~.-I100 .,..-------y--------,-----~----_:JBr-O.OO --Br-D.01 --------Br=O.05 .Br-0.10 .-l 10a=O.Bn =0.5If =0.0{3 = IO~Br=.OO --Br-D.01 -------.Br=0.05 -Br=0.10 .Fig.4.a Nusselt numbers, NUii, for n = 0.5, U· = 0.0 and 1.0 (3rd kind)10a=0.2n = 1.0If =0.0B,.,.O.OO--Br=O.01 --------Br-eJ.05 .Br-O.10 ..............•._.,.....~,.,.,.,.,.,.:.:.:.:.:.:.:.:.:.~.: .. :.:.:.::.:.~.::.:.:.:.:.:.:.:.:.:.a=0.2n = 1.0If = 1.0Br-.OO --Br-O.01 -------.Br-O.05 .Br=O.10 - .1L....~~_~.........__~~......_~~_~"'----~_~~....J100 ..-~~~~'"T"""-~~~........,-~~.......~...-~~~~"'"'2Br-O.OO --Br=O.01 --------Br-O.05 .Br=0.10 .~ 10a=0.5n = 1.0If =0.0a-0.5n = 1.0If = 1.0B .00--Br=0.01 --------Br-O.05 .Br=O.10 .1'--~-~........-~~~ ......-~~~~""--~~~~.-I100 .,..-------y--------,-----~----~Br-eJ.OO --Br-O.01 --------Br-D.05 .Br-D.10 .-l 10a=O.Bn = 1.0If =0.0......:.:.~:.:.-_ ... -- .. ------ .----_.a=O.Bn = 1.0If = 1.0B .00--Br>=0.01 -------.B1=0.05 .Br-O.10 .~OL...-:------I....O...-3:------I....O-:-2~----I....LO--:,-------'1 10'" 10-3 10-2 10-1~ ~FigA.b Nusselt numbers, NUii, for n = 1.0, U* = 0.0 and 1.0 (3rd kind)Laminar heat transfer in the thermal entrance region of concentric annuli with moving heated cores 57100 .--~~~~........-~~~~~~~~ ..............-~~~~..,Br-O.OO --Br-D.01 --------Br-D.05 .Br-D.10 .10Bf' .00--Br-D.01 --------Br-o.05 ._ .Br=O.fO .a=0.2n =0.5If =0.0f3 = lOs- . a=0.2n =0.5U· = 1.0f3 = lOsBr-.OO --Br=0.01 --------Br=0.05B1"=0.10a=0.5n =0.5if = 1.0f3 = lOs.. ~.:-:..~....~.. ~-~-...ja=0.5n =0.5If =0.0f3 = lOs1L..-~~~~........---~~...L...-~~-- -~~~~ ...100 .--~~~~~---~~.._~~~ .....,.-~~~~...Br-D.OO ---Br-D.01 --------Br-D.05 .Br-D.10 .-l 10fL-~~~~........---~~...L...-~~-- .........-~~~~ ...100 .---------.-------.-------.-----....Br=.OO --Br-O.O 1 -------.Br=O.OS .Br-O.fOa=O.Bn=0.5if = 1.0f3 = lOs.........:.: .a=O.Bn =0.5U· =0.0f3 = lOsBr-D.OO --Br=0.01 --------Br=0.05 .Br=0.10 .-l 101L--------'------...L.-------'----_.....10'" 10-3 10-2 ur' 1 10-4Z"Fig.5.a Nusselt numbers, NUii, for n = 0.5, U· = 0.0 and 1.0 (4th kind)100 .--~~~~~---~~.._~~~~.....,.-~~~~...Br-O.OO --Br=O.Of -------.Br-D.OS .Br=O.10 .10 '""'C:C":>-> :: _.........B1"=.OO --B1"=O.01 --------Br-0.05····_····Br=O.10~:.~.:.:~:-. :.:.~:- --. --_. _. --a=0.2n = 1.0if =0.0a=0.2n = 1.0if = 1.0......~ ~::::::.' ~:;::.:.:.:' ..B1"=.OO --Br-O.Of -------.B1"=0.05 .Br-O.tO.......fL-~~~~ ........ ~~...L..._~~__........._~~~~...100 .-~~~~~---~~.._~~~~.....,.-~~~~..,Br=O.OO --Br-D.Of -------.Br=O.05 .Br-D. fO .-l fOa=0.5n = 1.0if =0.0.............•...••.... a=0.5n = 1.0if = 1.0Br-.OO --Br=O.Of --------B1"=O.05 .. _ .Br-O.tO .1L..--~~~........-~-~~ .........~~-- ........-~~~~ ....100 .------.-------.-------.-----....Br-D.OO --Br=O.01 ------0-Br=O.05 .Br-D.fO .-l fOa=O.Bn = 1.0if =0.0............ a=O.Bn = 1.0if = 1.0Fig.5.b Nusselt numbers, NUjj, for n = 1.0, U· = 0.0 and 1.0 (4th kind)58 Ganbat DAVAA, Toru SHIGECHI and Satoru MOMOKIS,-.o.oo --S,-..0.01 --------Sr=O.06 ---"'---Sr-O.10 .1000 ~~~~~.-rr-~~~~r-~~""""-S~r_~~b"'..-oo~=::===""'"Sr-D.01 --------Sr-O.06 --Sr-0.10 - ..J. 10a=0.2,. = 0.5cr = 0.0J3 = 10'100S .00 ---Sr-O.01 --------Sr-O.06 ------ ...SI'>-O. 10 .......•.........B .00 ---Sr-0.01 --------Sr-O.06 .-------.Sr-0.10 .-----------------------------_.._..._.._~_._--_._...,-,.__._--_..._--__ :::.::~.:.-::-~.:.:":":::". ~ .:::0';;;"';'';-':-';-:-':-':::"'::'a 0.5 ------,. 0.5cr = 1.0J3 = lOSB .00 ---SI'>-O.01 --------S'-'O.06 ----.-.-.Sr=O.10 ............•....Sr-.OO ---Sr-o.01 --------Sr-D.06 -----.-.-Sr-O.10 .................................--------'---------------------------~_ .._.._.__.__..,.._.._--- --~.:.:::-:.:.::::.:.:.:;::-:..:':;.... -:-:..:.~ ..a = 0.8 -----,. == 0.5cr = 1.0J3 == 10'1 10.....Fig.6.a Nusselt numbers, NUoh for n = 0.5, U· = 0.0 and 1.0 (4th kind)SIl .00 ---Sr-0.01 --------Sr-O.06 -----.- ..Sr=O.10 .B .00 ---Sr-O.01 --------Sr-O.06·-·······S-o.10 .a=0.2,. = 1.0c.r = 1.0a=0.8,. = 1.0c.r = 1.0a=0.5,. = 1.0c.r = 1.0........'" ::.::::-.....-:::.::> ........ "0_. -"......-...... ..:: .-------------------------------_.,--,.._.--,..._.•._._..~.._~-~.-.- --- ....:.:..... .: •...;.;.;.;..:..:.;.;.,;.;.;.;.•B .00 ---:~:::::'::::::_:._-.:_:_.:._~"'~.-:~g ~~:~~.~:.:~:~";;":";";-:':0;-;0;'-:::":";":-:":-:-:-:"----------------------------1 a- 0.5,. ... 1.0c.r == 0.01 a=0.2,. = 1.0c.r = 0.01 a=0.8,. -= 1.0c.r =0.0o. ~<r'-:-----1-0L_-:-3----1 ....0...--:-2----1-o_...."..1-----1, 10..... U,3 10-2~ ~Fig.6.b Nusselt numbers, NUoi, for n = 1.0, U· = 0.0 and 1.0 (4th kind)0.1 '--__......_ ~~~ ...L_~~~...........1000 ..--~~~~ --~~~.--~~........~......-~~~.........

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Laminar heat transfer in the thermal entrance region of concentric annuli with moving heated cores : Part II : The cases with the third and fourth kinds of thermal boundary condition

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