Heat model of an initial liquid freezing
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If i have a region $x>0$ that is initially liquid at constant temp $T_0$ above freezing temp $T_f$ (so $T_0 > T_f$). The surface at $x=0$ stays at $T_1$ below freezing temp (so $T_1<T_f<T_0$)
And $T_s(x,t)$ and $ T_L(x,t)$ satisfy:
$frac{partial {T_s}}{partial{t}} = D_sfrac{partial^2{T_s}}{partial x^2}$ and $frac{partial {T_L}}{partial{t}} = D_Lfrac{partial^2{T_L}}{partial x^2}$
For constants $D_s$ and $D_L$, with $T_s(x,t)=T_1$ and $T_L(x,0)=T_0$.
The boundary between liquid and solid is $x=s(t)$ with $s(0)=0$
So the boundary conditions i found were:
At $x=0$ $T=T_1$ and is maintained at $T_1$ until it starts to change to solid at $x=s(t)$ where $T=T_f$
Using boundary conditions That $T_s$ and $T_L$ must satisfy show the solution can be written as:
$T_s(x,t) = T_1+frac{T_f-T_1}{erf(m)}erf(frac{x}{sqrt{4D_Lt}})$
$T_L(x,t) = T_0-frac{T_0-T_f}{1-erf(msqrt{frac{D_s}{D_L}})}(1-erf(frac{x}{sqrt{4D_Lt}}))$
That took flipping ages to write... anyway i understand how to find the boundary conditions, that's the easy part however i have zero clue how to go about showing these solutions are true, any help would be absolutely amazing! Thanks in advance!
pde mathematical-modeling
asked Jan 14 at 12:22
L GL G
428
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add a comment |
$begingroup$
If i have a region $x>0$ that is initially liquid at constant temp $T_0$ above freezing temp $T_f$ (so $T_0 > T_f$). The surface at $x=0$ stays at $T_1$ below freezing temp (so $T_1<T_f<T_0$)
And $T_s(x,t)$ and $ T_L(x,t)$ satisfy:
$frac{partial {T_s}}{partial{t}} = D_sfrac{partial^2{T_s}}{partial x^2}$ and $frac{partial {T_L}}{partial{t}} = D_Lfrac{partial^2{T_L}}{partial x^2}$
For constants $D_s$ and $D_L$, with $T_s(x,t)=T_1$ and $T_L(x,0)=T_0$.
The boundary between liquid and solid is $x=s(t)$ with $s(0)=0$
So the boundary conditions i found were:
At $x=0$ $T=T_1$ and is maintained at $T_1$ until it starts to change to solid at $x=s(t)$ where $T=T_f$
Using boundary conditions That $T_s$ and $T_L$ must satisfy show the solution can be written as:
$T_s(x,t) = T_1+frac{T_f-T_1}{erf(m)}erf(frac{x}{sqrt{4D_Lt}})$
$T_L(x,t) = T_0-frac{T_0-T_f}{1-erf(msqrt{frac{D_s}{D_L}})}(1-erf(frac{x}{sqrt{4D_Lt}}))$
That took flipping ages to write... anyway i understand how to find the boundary conditions, that's the easy part however i have zero clue how to go about showing these solutions are true, any help would be absolutely amazing! Thanks in advance!
pde mathematical-modeling
asked Jan 14 at 12:22
L GL G
428
$endgroup$
$begingroup$
Maybe my physics is lacking, but I'm very confused by your set-up. Is the temperature distribution $$ T = begin{cases} T_s(x,t), & 0 < x < s \ T_L(x,t), & s < x end{cases} $$ where $s$ changes over time? Is $T(s(t),t)=T_f$? A moving boundary problem is pretty difficult, just so you know. Or are both $T_S$ and $T_L$ defined on $0 < x < infty$? Then what temperature do they represent?
$endgroup$
– Dylan
Jan 14 at 17:26
add a comment |
$begingroup$
If i have a region $x>0$ that is initially liquid at constant temp $T_0$ above freezing temp $T_f$ (so $T_0 > T_f$). The surface at $x=0$ stays at $T_1$ below freezing temp (so $T_1<T_f<T_0$)
And $T_s(x,t)$ and $ T_L(x,t)$ satisfy:
$frac{partial {T_s}}{partial{t}} = D_sfrac{partial^2{T_s}}{partial x^2}$ and $frac{partial {T_L}}{partial{t}} = D_Lfrac{partial^2{T_L}}{partial x^2}$
For constants $D_s$ and $D_L$, with $T_s(x,t)=T_1$ and $T_L(x,0)=T_0$.
The boundary between liquid and solid is $x=s(t)$ with $s(0)=0$
So the boundary conditions i found were:
At $x=0$ $T=T_1$ and is maintained at $T_1$ until it starts to change to solid at $x=s(t)$ where $T=T_f$
Using boundary conditions That $T_s$ and $T_L$ must satisfy show the solution can be written as:
$T_s(x,t) = T_1+frac{T_f-T_1}{erf(m)}erf(frac{x}{sqrt{4D_Lt}})$
$T_L(x,t) = T_0-frac{T_0-T_f}{1-erf(msqrt{frac{D_s}{D_L}})}(1-erf(frac{x}{sqrt{4D_Lt}}))$
That took flipping ages to write... anyway i understand how to find the boundary conditions, that's the easy part however i have zero clue how to go about showing these solutions are true, any help would be absolutely amazing! Thanks in advance!
pde mathematical-modeling
asked Jan 14 at 12:22
L GL G
428
$endgroup$
If i have a region $x>0$ that is initially liquid at constant temp $T_0$ above freezing temp $T_f$ (so $T_0 > T_f$). The surface at $x=0$ stays at $T_1$ below freezing temp (so $T_1<T_f<T_0$)
And $T_s(x,t)$ and $ T_L(x,t)$ satisfy:
$frac{partial {T_s}}{partial{t}} = D_sfrac{partial^2{T_s}}{partial x^2}$ and $frac{partial {T_L}}{partial{t}} = D_Lfrac{partial^2{T_L}}{partial x^2}$
For constants $D_s$ and $D_L$, with $T_s(x,t)=T_1$ and $T_L(x,0)=T_0$.
The boundary between liquid and solid is $x=s(t)$ with $s(0)=0$
So the boundary conditions i found were:
At $x=0$ $T=T_1$ and is maintained at $T_1$ until it starts to change to solid at $x=s(t)$ where $T=T_f$
Using boundary conditions That $T_s$ and $T_L$ must satisfy show the solution can be written as:
$T_s(x,t) = T_1+frac{T_f-T_1}{erf(m)}erf(frac{x}{sqrt{4D_Lt}})$
$T_L(x,t) = T_0-frac{T_0-T_f}{1-erf(msqrt{frac{D_s}{D_L}})}(1-erf(frac{x}{sqrt{4D_Lt}}))$
That took flipping ages to write... anyway i understand how to find the boundary conditions, that's the easy part however i have zero clue how to go about showing these solutions are true, any help would be absolutely amazing! Thanks in advance!
pde mathematical-modeling
pde mathematical-modeling
asked Jan 14 at 12:22
L GL G
428
asked Jan 14 at 12:22
L GL G
428
asked Jan 14 at 12:22
L GL G
428
asked Jan 14 at 12:22
L GL G
428
asked Jan 14 at 12:22
L GL G
428
428
$begingroup$
Maybe my physics is lacking, but I'm very confused by your set-up. Is the temperature distribution $$ T = begin{cases} T_s(x,t), & 0 < x < s \ T_L(x,t), & s < x end{cases} $$ where $s$ changes over time? Is $T(s(t),t)=T_f$? A moving boundary problem is pretty difficult, just so you know. Or are both $T_S$ and $T_L$ defined on $0 < x < infty$? Then what temperature do they represent?
$endgroup$
– Dylan
Jan 14 at 17:26
add a comment |
$begingroup$
Maybe my physics is lacking, but I'm very confused by your set-up. Is the temperature distribution $$ T = begin{cases} T_s(x,t), & 0 < x < s \ T_L(x,t), & s < x end{cases} $$ where $s$ changes over time? Is $T(s(t),t)=T_f$? A moving boundary problem is pretty difficult, just so you know. Or are both $T_S$ and $T_L$ defined on $0 < x < infty$? Then what temperature do they represent?
$endgroup$
– Dylan
Jan 14 at 17:26
$begingroup$
Maybe my physics is lacking, but I'm very confused by your set-up. Is the temperature distribution $$ T = begin{cases} T_s(x,t), & 0 < x < s \ T_L(x,t), & s < x end{cases} $$ where $s$ changes over time? Is $T(s(t),t)=T_f$? A moving boundary problem is pretty difficult, just so you know. Or are both $T_S$ and $T_L$ defined on $0 < x < infty$? Then what temperature do they represent?
$endgroup$
– Dylan
Jan 14 at 17:26
$begingroup$
Maybe my physics is lacking, but I'm very confused by your set-up. Is the temperature distribution $$ T = begin{cases} T_s(x,t), & 0 < x < s \ T_L(x,t), & s < x end{cases} $$ where $s$ changes over time? Is $T(s(t),t)=T_f$? A moving boundary problem is pretty difficult, just so you know. Or are both $T_S$ and $T_L$ defined on $0 < x < infty$? Then what temperature do they represent?
$endgroup$
– Dylan
Jan 14 at 17:26
add a comment |
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$begingroup$
Maybe my physics is lacking, but I'm very confused by your set-up. Is the temperature distribution $$ T = begin{cases} T_s(x,t), & 0 < x < s \ T_L(x,t), & s < x end{cases} $$ where $s$ changes over time? Is $T(s(t),t)=T_f$? A moving boundary problem is pretty difficult, just so you know. Or are both $T_S$ and $T_L$ defined on $0 < x < infty$? Then what temperature do they represent?
$endgroup$
– Dylan
Jan 14 at 17:26