Thermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report

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1 Therml iffusivity Pul Hughes eprtment of Physics nd Astronomy The University of nchester nchester 3 9PL Second Yer Lbortory Report Nov 4 Abstrct We investigted the therml diffusivity of cylindricl block of resin by observing the temperture s function of time t it s centre with respect to n outside chnge of temperture. This ws done in two different wys; step method (involving single chnge from ice wter to boiling wter) nd periodic method (involving multiple chnges from ice wter to boiling wter). The result ws ( 9.45 ±.) m s.

2 . Introduction In n electricl conductor, the resistnce of the conductor, mongst other things, depends upon the shpe of the conductor nd the resistivity of the mteril from which it is mde from. This is kin to the properties displyed by therml conductor. However the conductivity of the therml conductor is nlogous to the resistivity of the electricl conductor. But neither of these situtions depend on time. To include time dependence into the sitution we must define new quntity; diffusivity. The study of therml diffusivity is useful in wide rnge of fields including building science, mteril science, meteorology, the design of het flu sensors nd hs even been linked to the effect dentures hve on some people s sense of tste.. Theory..Step ethod The conduction Eqution is ; t θ θ Where θ is the temperture s function of position nd time. By epnding this in cylindricl polr coordintes nd neglecting φ nd z θ θ θ θ ) dependence; + t r r r (i.e. θ ( r,t) The solution to this long the is of the cylinder is series of eponentil terms: θ n (, t) θ + ( θ + θ ) e n n λ t Where λ n re the positive roots of the Bessel function, i.e. λ. 45, λ 5. 5, λ etc. By substituting these vlues it is observed tht close pproimtion is to neglect the n > terms: θ θ + ( θ + θ ) e t e 3.47t e t... This pproimtion gives; θ θ e t

3 θ θ i.e θ λ t e [Eqution ] Where : the therml diffusivity of the block t : time/ s : rdius of the cylindricl block/ m λ.45 Eqution cn be rerrnged to; θ θ θe Hence using nturl logrithms : ln λ t ( θ θ) ln θ + ln e ln θ λ t λ t [Eqution ] Hence plotting the logrithm of the difference of the current temperture with the eternl temperture ginst time will yield liner plot with grdient m; λ m [Eqution 3] This cn be rerrnged to find the diffusivity of the block; m [Eqution 4] λ However, using vernier cllipers gives mesurement of the dimeter not the rdius, hence substituting for the dimeter, d, gives: md [Eqution 5] 4λ

4 .. Periodic ethod Using similr set up to the Step ethod the resin is cooled to be ºC nd then heted to ºC. However this time the resin is spends certin mount of time in the ice nd certin mount of time in the boiling wter. This cycle of ice nd boiling wter is repeted few times to give periodic chnge in the temperture of the resin. Using this method the diffusivity of the resin cn be found. 4 As the temperture is function of rdius nd the periodic chnge of the eternl iωt temperture; θ ( r, t) f ( r) e Substituting this into the conduction eqution; Chnging the vrible to f iω f r iω f f z r gives; + + f z z z f + r r The solution to this eqution is the Kelvin function, ( ). As our periodic chnge of eternl temperture is ctully squre wve (i.e. it is not sinusoidl chnge in θ r,t cn be epnded s Fourier series to give; temperture), ( ) (, t) (, t) ( ) ω θ [Eqution 6] θ This cn be rerrnged to give two different equtions, one concerning the mplitudes of the il temperture nd the eternl temperture: ( ) 4 θ θ [Eqution 7] π B And the other concerning the phse difference between the il temperture nd the eternl temperture; t φ π rg[ ( ) ] T [Eqution 7b] Where; π T [Eqution 8] Notice tht is unitless quntity. Where B : The difference between the mim temperture nd the minim temperture : The Kelvin function θ : The temperture of the boiling wter θ : The temperture of the ice wter φ : The phse difference between the il nd eternl tempertures t : The time difference between pek of the il nd eternl tempertures Thus by plotting the Kelvin function, the vlue of the diffusivity cn be found.

5 3. ethod 3.3. Step ethod To find the diffusivity of the smple of resin, it ws cooled using ice to the point where it s temperture ws minimum. Inside the resin is temperture probe which gives plot of the il temperture (the temperture t the centre of the resin) with time. At this point the il temperture ws ssumed to be ºC. The resin ws then plced in boiling wter nd the il temperture incresed until it reched mim, this temperture ws ssumed to be ºC Periodic ethod Using the sme setup s for the Step ethod, the resin ws cooled to ºC nd ws then plced in boiling wter. However for the periodic method, the resin ws not llowed to rech mimum il temperture, but insted the resin ws left in the boiling wter for set mount of time. At the end of this time, the resin ws once gin plced in the ice wter for the sme mount of time s before. This process ws repeted until few periods of the il temperture hd been completed.

6 4. Results 4.. Results from the Step ethod 4... Tril For this tril the resin ws cooled to the minimum il temperture nd ws then left in the boiling wter for 9 minutes to give the following plot; 6 Step ethod - Tril 4 y , m dev:.395, r E-4, b.5 ln( θ) Time/ s Figure Results of Tril of the Step ethod Using Eqution 5;.6 m s Using the stndrd methods 3 the vlue of Chi Squred is found to be; χ 88.. This is high vlue but cn esily be rectified by removing the first point. This point hs tht smllest error nd so will hve the gretest effect on reducing the vlue of Chi Squred. 6 Step ethod - Tril (with first point removed 4 y , m dev:.8, r.99.9e-4, b.4 ln( θ) Time/ s Figure Results of Tril of the Step ethod (with first point removed) Removing this point, reduces the vlue of Chi Squred to. 537, which is fr more cceptble. This gives; m.87s 3.7 m s. (By Eqution 5)

7 4... Tril Unlike the first tril, the resin ws heted to the mimum il temperture of ºC in the boiling wter nd then plced in the ice wter to observe how the il temperture vries s the resin is cooled for minutes to give the following plot; 5 Step ethod - Tril y , m dev:.56, r.979.5e-4, b ln( θ) From Eqution 5 we find tht: Time/ s Figure 3 Results of Tril of the Step ethod 5.6 m s Using the stndrd methods 3 the vlue of Chi Squred is found to be; χ 387. However this vlue of Chi squred is too lrge, it cn be decresed to more cceptble vlue by discounting the first five nd lst si points from the eperiment. 4 Step ethod - Tril (with first five nd lst si results omitted) y , m dev:.76, r.988.e-4, b ln( θ) Time/ s Figure 4 Results of Tril of the Step ethod (with first five nd lst si points removed) Removing these points gives more cceptble vlue of for Chi Squred. This gives: m.35s 5.5 m s (By Eqution 5)

8 4.. Results from the Periodic ethod Amplitude 4... etermining the il temperture For the periodic method the plot oscillted between the mimum nd minimum il tempertures. At the end of the eperiment (fter few periods hd been completed), the il temperture ws llowed to rech ºC nd then ºC. This ws done so tht the mimum nd minimum tempertures could be determined. The equivlent temperture for ech of the smllest divisions is mrked on ech plot Tril For the first tril time period of 36s (i.e. leving the resin in the ice wter for 3 minutes then in the boiling wter for 3 minutes, then repeting) The following ws recorded: θ C θ C B C T 36s ( ). 94 (By Eqution 7) To determine the vlue of the diffusivity, plot of ginst the Kelvin function ws used; 3.5 Step ethod - Kelvin Plot for Tril 3. y , m dev:.767, r.99.33, b.468 () Figure 5 Kelvin plot for tril of the Periodic ethod - Amplitude From this plot we find tht π 3.69 (By Eqution 8) T.6 m s

9 4..3. Tril For this second tril time period of s (i.e. leving the resin in the ice wter for minute then in the boiling wter for minute, then repeting) The following ws recorded: θ C θ B 6.8 C T s C ( ) (By Eqution 7) As before plot of the Kelvin function is required: 7 Step ethod - Kelvin plot for Tril 6 y , m dev:.43, r.99.48, b. () Figure 6 Kelvin plot for tril of the Periodic ethod - Amplitude From this we find tht π 4.49 (By Eqution 8) T 3.6 m s

10 4.3. Results from the Periodic ethod Phse ifference Tril As for the first tril of the Amplitude method, time period of 36s ws used first; The following ws recorded: T 36s t 6s φ π.47 rg[ ( ) ] (By Eqution 7b) 3 To determine the vlue of the diffusivity, plot of ginst the rgument of the Kelvin function ws used;.4 Argument of Kelvin Function plot. y , m dev:.3, r , b.4 rg [ ()] Figure 7 Argument of Kelvin plot for tril of the Periodic ethod - Phse From this plot we find tht π.7 (By Eqution 8) T 33.8 m s

11 4.3.. Tril Agin for this second tril time period of s ws used The following ws recorded: The following ws recorded: T s t 6s [ ( ) ] φ π 3.4 rg (By Eqution 7b) As before plot of the Kelvin function is used: 3.5 Argument of Kelvin Function plot 3.5 y , m dev:.3, r , b.985 rg [ ()] Figure 8 Argument of Kelvin plot for tril of the Periodic ethod - Phse From this we find tht π 5. (By Eqution 8) T 8.95 m s

12 4.4. Errors There re two types of errors 3 ; Systemtic errors; which re flws in the eperiment. The systemtic errors of this eperiment include:. The ssumption tht the resin would be cooled to ºC or heted to ºC, for this to ctully hppen the temperture of the ice wter would need to be less thn ºC nd the temperture of the boiling wter would need to be greter thn.ºc. The ssumption tht the het density in the ice wter nd the boiling wter were uniform. As they were not, the il temperture would vry long the is of the resin. The wy to eliminte these systemtic errors would be to use more ccurte equipment nd to use different mediums to cool nd to het the resin. For emple lower rnge of tempertures could be chieved if liquid nitrogen were used in plce of the ice wter. Rndom errors; which re imperfect redings due to the error of the equipment we cn use nd the humn error involved. These cn e ccounted for. From our bse mesurements we evluted the errors to be: o Error on dimeter (s given by Vernier Cllipers): ±.5mm o Error on temperture reding ±.5 smll squres this is different for the two trils s the squre equte to differing tempertures Clcultion of errors for the Step ethod Using the stndrd methods 3 the error for the vlue of the diffusivity s given by the Step ethod is given by; md [Eqution 5] 4λ m m + 4 d d + 4 m d [Eqution 9] m d For Tril ; For Tril :.57 m s.9 m s

13 4.4.. Clcultion of errors for the Periodic ethod - Amplitude The error for the vlue of the diffusivity s given by the Periodic ethod is given by 3 ; ( ) 4 θ θ [Eqution 7] π B θ θ B θ θ + 4 B θ θ + 4 θ θ B B [Eqution ] This gives tht. 5 for Tril nd. 3 for Tril. This error in the vlue given by the Kelvin function cn now be clculted using the Kelvin plots (see Figures 5 nd 6) with the outcome being the error in the vlue of For Tril : ( + ) ( ) Hence we cn find men error; For Tril : ( + ) ( ) Agin tking men error; As is unitless quntity, so too is the error on. By substituting nd rerrnging Eqution 8; d π [Eqution ] T

14 It is found tht 3 ; d d This rerrnges to; + d [Eqution ] d For Tril ; For Tril :. m s.5 m s Clcultion of errors for the Periodic ethod Phse ifference The error for the vlue of the diffusivity s given by the Periodic ethod is given by 3 ; t φ π rg[ ( ) ] [Eqution 7b] T rg rg [ ] [ ] t t T + T rg + t T t T [ ] rg[ ] [Eqution 3] Given the following for both trils; ± 3s t ±.5s T This gives tht [ ]. 54 for Tril nd [ ]. 57 for Tril. rg rg This error in the vlue given by the rgument of the Kelvin function cn now be clculted using the rgument Kelvin plots (see Figures 7 nd 8) with the outcome being the error in the vlue of For Tril : ( rg[ ] +.54) ( rg[ ].54) Hence we cn find men error; +. 47

15 For Tril : ( rg[ ] +.57) ( rg[ ].57) Agin tking men error; +. 5 As is unitless quntity, so too is the error on. As for the Amplitude method; + d [Eqution ] d For Tril ; For Tril :.. m s.35 m s 4.5. Finl Answer From the four trils (two in ech method), it ws found tht; Step ethod Tril ( ) 3.7 ±.57 m s Step ethod Tril ( ) 5.5 ±.9 m s Periodic ethod Amplitude Tril ( ).6 ±.3 m s Periodic ethod Amplitude Tril ( ) 3.6 ±.49 m s Periodic ethod Phse Tril ( ) 33.8 ±.. m s Periodic ethod Phse Tril ( ) 8.95 ±.35 m s Using these results weighted men 3 cn be used to rrive t finl nswer for the diffusivity; The weighted men of the diffusivity: 7.86 m s The error ssocited with this weighted men;. m s Hence the finl nswer is ( ) m 9.45 ±. s

16 5. Conclusion From the eperiment it ws found tht the vlue of the diffusivity of our cylindricl piece of resin is ( 9.45 ±.) m s A liner reltionship ws demonstrted, the grph of which hd negtive grdient, this showed greement with Eqution. In the future, the errors could be reduced to show this reltionship more clerly in number of wys. Firstly we could repet the sme method, lthough this would require much time to return ny significnt improvement in the vlue of the diffusivity. The method could be modified such tht the mesurements re more ccurte. For emple greter rnge of tempertures could be chieved by mking use of different cooling methods. Alterntively the diffusivity of the smple could be clculted if the therml conductivity, the density nd the specific het of the resin could be found using the reltionship 3 ; K ρ C 6. References. G. Stephenson, An Introduction to Prtil ifferentil Equtions for Science Students, Longmn, Esse, 988. Young & Freedmn, University Physics, Person Addison Wesley, P John R. Tylor, An Introduction to Error Anlysis: The Study of Uncertinties in Physicl esurements nd edition, University Science Books, Chrles Kittel & Herbert Kroemer, Therml Physics nd Edition, W. H. Freemn, 98

A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW. factor and remainder theorems A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division