10.3: Heat Transfer Simulation
- Page ID
- 7845
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Here we demonstrate how heat transfer through the mould wall determines the temperature change in a casting during solidification. The simulation assumes Newtonian cooling where heat transfer is limited by the interface between the metal and the mould.
The simulation shows the effect of varying parameters such as the interfacial heat transfer coefficient, h, the casting length, L and the amount of superheat (determined by the pouring temperature, Tp). It can be carried out for a wide range of metals to study the effect of properties such melting temperature Tm, latent heat of fusion per unit volume, ΔHf,V, thermal conductivity, K and heat capacity per unit volume, Cp,V. Click here to see how the heat capacity per unit volume is related to specific heats measured relative to other amounts.
The relevant thermal properties of several pure metals are shown below
|
Melting temperature
|
Latent Heat of fusion
|
Volumetric Heat capacity
|
Thermal conductivity
|
|
|---|---|---|---|---|
|
K
|
MJ m-3
|
kJ m-3 K-1
|
W m-1 K-1
|
|
|
Ag
|
1235
|
1100
|
2920
|
422
|
|
Au
|
1337
|
1200
|
2800
|
272
|
|
Al
|
933
|
1070
|
3100
|
240
|
|
Cu
|
1358
|
1842
|
4080
|
395
|
|
Mg
|
922
|
640
|
2240
|
154
|
|
Pb
|
601
|
260
|
1590
|
34
|
|
Sn
|
505
|
440
|
1780
|
63
|
|
Zn
|
693
|
812
|
3070
|
112
|
In the simulation you can select any of these materials and its properties will be displayed. You may then vary parameters such as the casting length, the interfacial heat transfer coefficient between the solid and the mould wall and the pouring temperature in order to see how long it takes for the casting to solidify and cool.
There are a couple of provisos: Firstly because this is a Newtonian cooling simulation you must ensure that the Biot number is low enough for this assumption to be valid (hence for the metals which have poor thermal conductivities, you must keep the casting length relatively low). Secondly, of course, you must ensure that the metal is poured above its melting temperature!
There are a number of assumptions and simplifications we are making in this simulation which may not be the case for a real casting.