Heating of a solid cylinder immersed in an insulated bath. Thermal diffusivity and heat capacity experimental evaluation.

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Adrian Underwood
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1 Hatng of a sold cylndr mmrsd n an nsulatd bath. Thrmal dffusvty and hat capacty xprmntal valuaton. Žtný R., CTU FE Dpartmnt of Procss Engnrng, arch. Introducton Th problm as ntatd by th follong E-mal from Cartr Tchnologs Co.: Dar Srs, W ar skng a mthod for masurmnt of th thrmal conductvty of a sand and cmnt grout usd to grout hat transfr pps nto th arth. Th applcaton s for ground loop hat pumps. Prsntly fnd that dffrnt labs gv dffrnt radngs. On uss a /8 nch damtr hatd thrmo coupl prob nsrtd nto a 4 nch by 8 nch cylndr of th grout. Anothr uss som knd of rctangular sampl and a rfrnc matral th a Shothrm QT-D nstrumnt. Our applcaton s ovr a tmpratur rang of 4 dgrs F to dgrs F. Could a accurat tst b dvsd basd on coolng a 3 gram. spcfc dnsty cylndr sampl to 4 dgrs F and thn mmrsng t n an nsulatd atr bath havng cc of atr at dgr F. and masurng th tmpratur at mnut ntrvals for mnuts? Would ths ork and do you hav any nstrumnts to do ths. Ho ould you calculat t? Rsults of suggstd xprmnts (tm courss of atr tmpratur) can b usd for valuaton of hat capacty and thrmal dffusvty of matral f th follong smplfd dscrpton of xprmnt s accptd: Cylndr (radus R, hght H, thrmal dffusvty a) of a unform tmpratur T at ntal tm t= s submrgd nto lqud (mass, spcfc hat capacty c ) havng dffrnt ntal tmpratur T. It s supposd that th facs of cylndr ar nsulatd and th hat s xchangd only through th cylndrcal surfac S=πRH. It s also assumd that th lqud s so ntnsvly mxd, that t has a unform tmpratur T, th sam as th surfac of cylndr.. Soluton basd upon lnar tmpratur profl thn a thn pntraton dpth Lt us assum that th tmpratur s changng lnarly from T to T thn a thn layr of thcknss δ(t) at th all of cylndr (or any othr form of body, havng surfac S). Takng th nthalpy of sold at ntal tmpratur T as a rfrnc (zro), th total nthalpy of body at a tm t s H S t c T t T () = δ() ρ p () Ths nthalpy s changng accordng to th hat balanc dh S ρ c p = ( dδ( T T ) + δdt). () Th nthalpy ncras of body must b th sam as th nthalpy dcras of atr bath dh =cdt (3) At th sam tm th nthalpy chang s gvn by Fourr la as dh S = λ dt. (4) δ Thr unknons, pntraton dpth δ(t), nthalpy H(t) and T (t) ar dtrmnd unquly by thr quatons (,3,4). Enthalpy H can b lmnatd by quatng (-4) and (3-4) carpapr.doc

2 Sρcp dδ dt S ( ( ) + δ ) = λ dt dt δ (5) dt δ = Sλ. dt c (6) Equatons (5) and (6) can b ntgratd, gvng rlatonshp btn pntraton dpth and tm, unfortunatly n th nvrs form t=f(δ) a Q δ ln( + δ ) = Qt. Q a (7) Only for a vry short tm th pntraton dpth can b xprssd xplctly as a Q Q a Q Q δ ( δ ( δ ) +...) = ( δ ) +... = δ = Qt, gvng δ = at. Q a a Q a 4a (8) Substtutng pntraton dpth Eq. (8) nto Eq.(6), a short tm approxmaton of th lqud tmpratur at tm t can b drvd n th form πr Hρcp T T c = =, T T (9) hr c = πrhρc, = at, R S = π RH (,,) p ar rlatv hat capacty of lqud, dmnsonlss tm and surfac of cylndr, rspctvly. Rsults obtand usng ths "short tm approxmaton" ll b compard th xact soluton n th nxt paragraph. 3. Soluton usng nfnt srs Aftr an nfntly long tm th both tmpraturs of lqud and cylndr achv qulbrum tmpratur ct + πrhct ρ p T + T T = = (3) c + πrhc ρ p + and ths qulbrum tmpratur ll b usd n th dfnton of dmnsonlss tmpratur * T T T =. (4) T T Th tmpratur T * of cylndr s ntally on n th hol cylndr xcpt at ts surfac r= (hr T * =-/()), and tnds to zro at qulbrum. Tmpratur profl n a cylndr s dscrbd by dmnsonlss form of Fourr quaton * T r r r T * = ( ), (5) r hr dmnsonlss radus r s rlatd to th radus of cylndr, and dmnsonlss tm s gvn by Eq.(). Tmpratur of lqud s govrnd by an ordnary dffrntal quaton, xprssng nthalpy balanc of lqud * * ( dt T + ) r= =. (6) d r Dmnsonlss coffcnt s th rato of hat capacts of lqud and cylndr, s Eq.(). Th soluton T * can b xprssd as an nfnt srs carpapr.doc

3 = T * (, r) = AJ ( χr) χ hr th boundary condton at axs (symmtry) s automatcally satsfd for any χ (bcaus J '()=). Boundary condton at surfac of cylndr (r=), Eq.(6), s fulflld only for gnvalus χ hch ar roots of quaton (7) J ( χ ) + χ J ( χ ) = (8) Exampls of rsults, gnvalus χ calculatd numrcally for =.5,, and ar prsntd n Tab.. Tab. Roots of Eq.(3) for =.5,, and χ =.5 = = Th gnvalus nd not b solvd numrcally, bcaus a smpl approxmaton can b drvd from asymptotc proprts of Bssl functons for larg argumnts and usng a mld mprcal corrcton for lo valus of ndx, π χ = arctg[ ] + π, =,,... (9) + π π tg ( ) 4 Comparson of xact valus and approxmaton (9) s shon n Fg. Rlatv rror of (9) thn th rang of from. to. % Fg. Egnvalus χ and χ as a functon of, accordng to xact soluton of Eq.(8) and approxmaton Eq.(9), hch prdcts slghtly hghr valus thn th rang (. ). carpapr.doc 3

4 Th coffcnts A of srs (7) should satsfy ntal condton = AJ = ( χ r), () for r< and T * (, ) = = = AJ ( χ ) () T T = for r= (at surfac). Problm s n th fact, that th systm of functons () s not orthogonal. Hovr, can procd n a standard ay,.. multplyng th srs () by rj and ntgratng rj( χr) dr = A rj( χr) J( χr) dr = For th ntgrals of product of Bssl functons ar, s Jnson (973) χj( χ) J( χ) χj( χ) J( χ) rj( χr) J( χr) dr = + = { us J( χ) = χj( χ)} = χ χ () =J ( χ ) J ( χ ) For = t s ncssary to us a dffrnt formula (b aar of th fact, that th xprsson () s of th form / and for xampl th l'hoptal rul must b usd) rj ( χr ) dr = J J J ( ( χ ) + ( χ )) = ( χ )( + χ ). (3) Intgral on th lft sd of Eq.() follos mmdatly from Eq.() for χ =, χj'( χ) J( χ ) rj( χ r) dr = = = J( χ ). (4) χ χ Substtutng Eqs.(-4) n Eq.() and rarrangng trms obtan A J AJ J ( χ )[ ( χ ) ] = ( χ )( + χ + ) (5) = akng us of ntal condton for tmpratur at surfac, Eq.(), Eq.(5) can b smplfd A J J ( χ )[ ] = ( χ )( + χ + ) (6) and thus th coffcnt A can b valuatd thout ncssty to solv a systm of quatons, gvng th fnal soluton for th dmnsonlss tmpratur fld n a cylndr χ J r T * ( χ ) (, r) = ( + ). (7) J ( χ ) ( + + χ ) = Ths xact soluton can b compard th th short tm approxmaton (9), rarrangd to th form T T T * + ( ) = = = (8) Exampls of rsults ar prsntd n Fgs. and 3 () carpapr.doc 4

5 Fg. Dmnsonlss tmpratur of all as a functon of at/r for =. Curvs rprsnt th short tm soluton Eq.(8), and th srs (7) for 3 trms or ust trm. Fg.3 Tm courss of T * for =.5,.5,.5,, and 4. Curvs rprsnt th short tm soluton Eq.(8), and srs (7) r calculatd th trms usng approxmaton Eq.(9). Comparson of mthods rsults n th concluson that th "short tm approxmaton" s sutabl for dmnsonlss tm <.3 at =.5. Ths valu s not so lo, bcaus.g. for a= -7 m /s and for radus.5 m th corrspondng tm s 75 s (mor than mnuts). Rducng srs (8) only to th frst trm of xpanson s accptabl for >.. carpapr.doc 5

6 4. Applcaton, xprmntal procdur Lt us assum that th hol tmpratur cours T (t) has bn rcordd, and thrfor th qulbrum tmpratur nd not b calculatd, s Fg.4. T T Fg.4 Exprmntal stup. Knong ntal tmpratur of sold T and ntal tmpratur of atr bath T th dmnsonlss coffcnt, s Eq.(), can b calculatd as T T =. (9) T T For ths th smallst postv gnvalu χ hav to b solvd numrcally from Eq.(8) or xprssd from approxmaton (9). Th tmpratur of atr can b dscrbd by only th frst trm of xpanson (7) at a suffcntly long tm (>.) as T( ) T + χ = (3) T T + + χ Ths rlatonshp nabls to valuat thrmal dffusvty a (.g. from th slop of T(t) n a smlogarthmc plot). Thrmal conductvty s rlatd to th thrmal dffusvty as a = λ (3) ρc p Bcaus th mass and thrmal capacty of atr s knon, th product ρc p can b valuatd from th masurd valu of c ρcp =. (3) πrh Thus th xprmnt rcordng ntal tmpratur of cylndrcal sampl T and th tm cours of lqud tmpratur T (t) ylds th ρc p valu from Eq.(3), and thrmal conductvty from Eq.(3). Rfrncs Jnson,V.G., Jffrys,G.V.: atmatcké mtódy v chmckom nžnrstv Alfa Bratslava. Přklad z orgnálu athmatcal thods n Chmcal Engnrng, Acadmc Prss Inc. Ltd., London, 963. carpapr.doc 6

7 Lst of symbols A coffcnt of xpanson (7) a thrmal dffusvty [m.s - ] c spcfc hat capacty of lqud [J.kg -.K - ] c p spcfc hat capacty of cylndr [J.kg -.K - ] H hght of cylndr [m] J (x) Bssl functon rlatv hat capacty of lqud, Eq.() mass of lqud [kg] R radus of cylndr [m] r dmnsonlss radal coordnat (r/r) S surfac of sampl [m ] T tmpratur of cylndr [K] T tmpratur of lqud [K] T qulbrum tmpratur [K] T * dmnsonlss tmpratur, s Eq.(4) [-] t tm [s] χ gnvalu, s Eq.(3) δ pntraton dpth [m] λ thrmal conductvty of cylndr [W.m -.K - ] ρ dnsty of cylndr [kg.m -3 ] dmnsonlss tm (Fourr numbr) carpapr.doc 7

COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP

ISAHP 00, Bal, Indonsa, August -9, 00 COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP Chkako MIYAKE, Kkch OHSAWA, Masahro KITO, and Masaak SHINOHARA Dpartmnt of Mathmatcal Informaton Engnrng