A MATHEMATICAL MODEL FOR NATURAL COOLING OF A CUP OF TEA

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1 MTHEMTICL MODEL FOR NTURL COOLING OF CUP OF TE 1 Mrs.D.Kalpana, 2 Mr.S.Dhvarajan 1 Snior Lcurr, Dparmn of Chmisry, PSB Polychnic Collg, Chnnai, India. 2 ssisan Profssor, Dparmn of Mahmaics, Dr.M.G.R Educaional and Rsarch Insiu, Chnnai, India. bsrac Hr i is proposd h rsuls of h simpl xprimn o masur h mpraur a any im in a cup of ho a. This aciviy offrs svral opporuniis o xplor mahmaics. In his invsigaion, h following mahmaical aciviis and concps ar applid: (i) horical principls modld by mahmaical xprssions (ii) xprssing mpirical rsuls in graphical form (iii) fiing horical prdicions o masurd daa (iv) comparing h qualiy of h horical prdicion wih masur daa. Hr w s our H o, h null hypohsis as; hr is no diffrnc bwn h rcordd valus and horically calculad valus. Kywords: Thrmodynamics, a, mpraur, ambin mpraur. Inroducion: n xprssion for h cooling of a cup of a is dvlopd from basic hrmodynamic principls. This hory is sd agains a simpl sysm, a cooling cup of a. In his xploraion an Excl spradsh was usd o graph h mpirical rsuls and h horical prdicion. Th acual daa and h mahmaical rprsnaion of h mpraur of a cooling of a 250ml cup of ho a is graphd vrsus im. = 0, h ha was rmovd from h a and h mpraur of h a was masurd a 1 minu inrvals. Room ambin mpraur was moniord a ½ minu inrvals o valida h assumpion ha hr was no significan im dpndnc in h room ambin mpraur. Th mach bwn h prdicd and acually valus suggss hr is somhing missing in h modl. Hnc a conjcur was mad as o h caus of his dispariy. I is proposd o us chi squar s o s h diffrnc bwn h obsrvd and horical prdicions. Thorical Modling: Th firs law can b xprssd as an quaion known as ra quaion. de W Qs I dscribs ha h ra of chang of nrgy in a sysm de is qual o h work, W, don on h sysm or powr nring h sysm minus ha ransfr ra Qs ou of h sysm. In our xprimn, 1

2 a =0, w assum ha all of h nrgy has bn impard o h sysm. Thrfor w s W=0 and his quaion bcoms: de Q s This quaion sas ha h ra of chang of nrgy in h sysm is qual o h ha ransfr ra. So a his poin, i is ncssary o raliz ha h nrgy of h sysm is dircly rlad o mpraur lvaion or chang and ha h ha ransfr ra can also b masur in rms of a mpraur diffrnc ovr im. Thrfor, his quaion can b wrin in rms of a mpraur diffrnc, T, vrsus im: dt () kt (), whr K is a consan Finding soluion: By solving h abov quaion, w g, Ingraing on boh sids, w g, dt () T () dt () T () k k log T( ) k C T () k C k T () T ( ) k, whr his im, i is convnin o impar physical maning o h consans usd in his soluion. lso no ha T() was dfind as a mpraur diffrnc, so ha h acual a mpraur is his mpraur diffrnc abov an ssnially im indpndn ambin mpraur: Thn h a mpraur T w can b xprssd as: C k T () T W Thr ar wo characrisics of h sysm and K. Th consan is rlad o h iniial mpraur ris ovr h ambin mpraur. Thrfor, T0 T, T0 is h iniial mpraur and T is h ambin mpraur (i..) room mpraur. Th consan K is rlad o h ha capaciy of h sysm and o h insulaing characrisics. So call h insulaing characrisics a hrmal rsisanc, q, C 2

3 and h ha capaciy of h a C. Sinc h grar h insulaion, or hrmal rsisanc, and h grar h ha capaciy, h slowr w would xpc h mpraur o chang, hs consans ar invrsly proporional o K, hus: c T () T T T W 0 Now ha a gnral xprssion for h a mpraur has bn dvlopd, do a quick saniy chck on how his quaion bhavs. = 0, h xponnial is uniy, and: T (0) T T T T W 0 0 This maks sns, sinc T O b h iniial mpraur of h ho a. s approachs zro and T ( ) w T, h room ambin mpraur as w would xpc. h xponnial Fiing Thorical Prdicions o Masurd Daa his poin w hav nough informaion from h mpirical daa o fill in h consans for our quaion for T W (). =T O -T = 69 C C = 38 C; his is h diffrnc bwn h saring mpraur and h final mpraur, which is h avrag valu a ambin h a mpraur is: w( ) 38 T o C =. So our quaion o prdic Th abl of daa of obsrvd and prdicions ar shown blow: SL NO Tim Room Tmpraur Obsrvd (O) Thorical (E) w( ) 38 T

4 Toal vrags Th graph bwn im and obsrvd mpraur, im and xpcd mpraur and im and room mpraur is shown blow. T m p r a u r Tim Vs Tmpraurs Tim Room Tmpraur Obsrvd Tmprur Expcd Tmprur Comparing h Qualiy of h Thorical Prdicion wih Masurd: Hr, in ordr o compar h qualiy of h prdicd and obsrvd, i is usd paird s. calculaing valu for h abov daa, on can g calc Dgrs of frdom = (n-1) = (13-1) =12. By abl valu for 12 dgrs of frdom is calc abl Hnc our null hypohsis is accpd. Thr is no diffrnc bwn h avrags of wo ss of valus. 4

5 Conclusion: In his aricl, w prdicd ha, hr is no significan diffrnc bwn h hory and obsrvd valus of a cooling sysm; w ook ha as Ho a for a cas sudy. Th mos obvious of his is h procss of vaporaion. This xprimn has bn conducd in Chnnai, India. Th acual rsuls may vary wih diffrn gographical rgions. Hr, i is akn undr crain room condiion mpraurs, in which hr was no allowanc of cooling sysms in h room. If so, h rsuls may vary. par from hs poins hr is apparnly an xcssivly rapid cooling, compard o h prdicion, followd by a slowing in h cooling. Hr w assum ha hr is no such rapidnss. Hr w omid fw variabls, lik h conainr. W ook h conainr as insulaor bu in som ohr siuaions, i may no hold. Bcaus of his, his rial is dubbd an opn sysm. Th sam concp can b xprimnd wih diffrn liquids o s h hory. Rfrncs: 1. Van Wyln, G. J. & Sonnag, R. E. (1973). Fundamnals of Classical Thrmodynamics (pp. 94). Nw York: John Wily and Sons, Inc. 2. Exrnal link: Chmical Thrmodynamics - Univrsiy of Norh Carolina 3. Hrbr B. Calln (1960). Thrmodynamics. Wily & Sons. Th clars accoun of h logical foundaions of h subjc. ISBN Library of Congrss Caalog No Wb basd rfrncs: 4. hp://n.wikipdia.org/wiki/thrmodynamics 5

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