Wolfram Cloud Document

ClearAll["Global`*"];(*Thisfunctionclearsanypastdefinitionsofthevariables*)(*THERMOPHYSICALPROPERTIESOFTHEPARAFFIN*)Tm=303;(*meltingtempreratureoftheparaffin*)ρs=880;ρl=760;ρ=ρl;(*densityatsolidandliquidstates*)ks=0.24;kl=0.15;(*thermalconductivityatsolidandliquidstates*)cps=2.4*

3

10

;cpl=1.8*

3

10

;(*specificheatatsolidandliquidstates*)q=179*

3

10

;(*latentheatoffusion*)vd=3.42;vk=

vd

ρ

;(*dynamicviscosity(vd)andkinematicviscosity(vk)ofparaffin*)(*FORMULASANDTRANSFORMATIONS*)αs=

ks

ρscps

;αl=

kl

ρlcpl

;(*thermaldiffusivityatsolidandliquidstates*)pr=

vk

αl

;(*Prandtlnumber*)gdot=

Gkl(T0-Tm)

2

R

;(*heatsinkparameter*)ξ=

gdot

ρq

2

R

τ

αl

;(*massproportionofliquidinthemixture*)T0=Tm+

qste

cpl

;(*temperatureoutsidethesphere*)γ=

ks

kl

;Γ=

αs

αl

;θi=

Ti-Tm

T0-Tm

;(*dimensionlessinitialtemperature*)λn=

nπ

splus

;keq=1;(*equivalentthermalconductivity*)(*PARAMETERVALUES*)R=0.05;(*radiusofthesphere*)Ti=295;(*initialtemperatureoftheparaffin*)ste=0.05;(*Stefannumber*)ra=0;(*Rayleighnumber*)bi=10;(*Biotnumber*)G=0;(*dimensionlessheatsinkparameter*)

θ=-bi

keqτ

Exp

-

2

η

4keqτ



η

-

Exp

-

2

splus

4keqτ



splus

+

π

2

Erfc

splus

2

keqτ

-Erfc

η

2

keqτ

keq

keqτ

Exp

-1

4keqτ

-bi

keqτ

Exp

-1

4keqτ

-

1

splus

Exp

-

2

splus

4keqτ

+

π

2

Erfc

splus

2

keqτ

-Erfc

1

2

keqτ

-

G

6keq

2

splus

-

2

η

+

splus

η

-1

bi

2

splus

-1-2keq

bi+splus(keq-bi)

-

10-

-

2

splus

4τ



splus

+

-

2

η

4τ



η

τ

+

1

2

π

Erfc

splus

2

τ

-Erfc

η

2

τ



-

1

4

τ



τ

-10

-

1

4

τ



-

2

splus

4τ



splus

τ

+

1

2

π

-Erfc

1

2

τ

+Erfc

splus

2

τ



τ=

αlt

2

R

;splus=s/R;η=r/R;T=θ(T0-Tm)+Tm(*transformingback*)

303-

49.72220.00662266

0.05

-

2.28×

6

10

2

r

t



r

-

0.05

-

2.28×

6

10

2

s

t



s

t

+

1

2

π

-Erfc

1509.97r

t

+Erfc

1509.97s

t



0.00662266

-5700./t



t

-100.00662266

-5700./t



-

0.05

-

2.28×

6

10

2

s

t



s

t

+

1

2

π

-Erfc

75.4983

t

+Erfc

1509.97s

t



(*above,wehaveobtainedafunctionoftheformTl=f(r,t,s(t))*)

(*tandsaresetasfollows*)

t=5(*h*)*60*60;s(*i.e.s(t)*)=3.8(*cm*)/100;(*thisisthescorrespondingtothetasseeninthebelowgraph*)

Plot[T,{r,0,R}]

ManipulatePlot[T,{r,0,R}],{{t,15(*h*)*60*60},0,24(*h*)*60*60},s,

2.25

100

,0,

5

100



t

s

Plot[T,{r,0,R}]

(*theslidersdonotseemtobeworkinghere.WhatamImissinghere?*)

(*relationshipbetweensandt*)