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[r_,s_,t_]:=θ(T0-Tm)+Tm(*transformingback*)
(*above,wehaveobtainedafunctionoftheformTl=f(r,t,s(t))*)
(*tandsaresetasfollows*)
T[r,s,t]
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

T3.5,
2.25
100
,15*60*60
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

T[a,b,c]
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
