|
| 1 | +--- |
| 2 | +category: basic |
| 3 | +intro: | |
| 4 | + # On the evaluation of thermal expansion coefficients |
| 5 | + |
| 6 | + When solving thermal-mechanical problems it is customary to use |
| 7 | + thermal expansion coefficients in order to take into account the mechanical |
| 8 | + strains induced by changes in the material temperature with respect to |
| 9 | + a reference temperature where the deformation is identically zero. |
| 10 | + These coefficients $\alpha$ are defined as the partial derivative |
| 11 | + of the strain $\epsilon$ with respect to temperature $T$ such that |
| 12 | + differential relationships like |
| 13 | + |
| 14 | + $$ |
| 15 | + d\epsilon = \frac{\partial \epsilon}{\partial T} \, dT = \alpha \cdot dT |
| 16 | + $$ |
| 17 | + |
| 18 | + hold. This derivative $\alpha$ is called the _instantaneous_ |
| 19 | + thermal expansion coefficient. |
| 20 | + For finite temperature increments, one would like to be able to write |
| 21 | + |
| 22 | + $$ |
| 23 | + \Delta \epsilon = \alpha \cdot \Delta T |
| 24 | + $$ |
| 25 | + |
| 26 | + But if the strain is not linear with respect to the temperature---which |
| 27 | + is the most common case---then $\alpha$ depends on $T$. |
| 28 | + Therefore, when dealing with finite temperature increments $\Delta T = T-T_0$ |
| 29 | + where the thermal expansion coefficient $\alpha(T)$ depends on the temperature |
| 30 | + itself then _mean_ values for the thermal expansion ought to be used: |
| 31 | + |
| 32 | + $$ |
| 33 | + \Delta \epsilon = \int_{\epsilon_0}^{\epsilon} d\epsilon^{\prime} |
| 34 | + = \int_{T_0}^{T} \frac{\partial \epsilon}{\partial T^\prime} \, dT^\prime |
| 35 | + = \int_{T_0}^{T} \alpha(T^\prime) \, dT^\prime |
| 36 | + $$ |
| 37 | + |
| 38 | + We can multiply and divide by $\Delta T$ to obtain |
| 39 | + |
| 40 | + $$ |
| 41 | + \int_{T_0}^{T} \alpha(T^\prime) \, dT^\prime \cdot \frac{\Delta T}{\Delta T} |
| 42 | + = \bar{\alpha}(T) \cdot \Delta T |
| 43 | + $$ |
| 44 | + |
| 45 | + where the mean expansion coefficient for the temperature range $[T_0,T]$ is |
| 46 | + |
| 47 | + $$ |
| 48 | + \bar{\alpha}(T) = \frac{\displaystyle \int_{T_0}^{T} \alpha(T^\prime) \, dT^\prime}{T-T_0} |
| 49 | + $$ |
| 50 | + |
| 51 | + This is of course the classical calculus result of the mean value of a |
| 52 | + continuous one-dimensional function in a certain range. |
| 53 | + |
| 54 | + Let $\epsilon(T)$ be the linear thermal expansion of a given material in a |
| 55 | + certain direction when heating a piece of such material from an initial |
| 56 | + temperature $T_0$ up to $T$ so that $\epsilon(T_0)=0$. |
| 57 | + |
| 58 | + .](asme-expansion-table.png){#fig:table-asme-expansion} |
| 59 | + |
| 60 | + From our previous analysis, we can see that in @fig:table-asme-expansion: |
| 61 | + |
| 62 | + $$ |
| 63 | + \begin{aligned} |
| 64 | + A(T) &= \alpha(T) = \frac{\partial \epsilon}{\partial T} \\ |
| 65 | + B(T) &= \bar{\alpha}(T) = \frac{\epsilon(T)}{T-T_0} = \frac{\displaystyle \int_{T_0}^{T} \alpha(T^\prime) \, dT^\prime}{T - T_0} \\ |
| 66 | + C(T) &= \epsilon(T) = \int_{T_0}^T \alpha(T^\prime) \, dT^\prime |
| 67 | + \end{aligned} |
| 68 | + $$ |
| 69 | + |
| 70 | + Therefore, |
| 71 | + |
| 72 | + i. $A(T)$ can be computed out of $C(T)$ |
| 73 | + ii. $B(T)$ can be computed either out of $A(T)$ or $C(T)$ |
| 74 | + iii. $C(T)$ can be computed out of $A(T)$ |
| 75 | +terminal: | |
| 76 | + $ cat asme-expansion-table.dat |
| 77 | + # temp A B C |
| 78 | + 20 21.7 21.7 0 |
| 79 | + 50 23.3 22.6 0.7 |
| 80 | + 75 23.9 23.1 1.3 |
| 81 | + 100 24.3 23.4 1.9 |
| 82 | + 125 24.7 23.7 2.5 |
| 83 | + 150 25.2 23.9 3.1 |
| 84 | + 175 25.7 24.2 3.7 |
| 85 | + 200 26.4 24.4 4.4 |
| 86 | + 225 27.0 24.7 5.1 |
| 87 | + 250 27.5 25.0 5.7 |
| 88 | + 275 27.7 25.2 6.4 |
| 89 | + 300 27.6 25.5 7.1 |
| 90 | + 325 27.1 25.6 7.8 |
| 91 | + $ feenox asme-expansion.fee > asme-expansion-interpolation.dat |
| 92 | + $ pyxplot asme-expansion.ppl |
| 93 | + $ |
| 94 | +... |
| 95 | +# just in case we wanted to interpolate with another method (linear, splines, etc.) |
| 96 | +DEFAULT_ARGUMENT_VALUE 1 steffen |
| 97 | + |
| 98 | +# read columns from data file and interpolate |
| 99 | +# A is the instantaenous coefficient of thermal expansion x 10^-6 (mm/mm/ºC) |
| 100 | +FUNCTION A(T) FILE asme-expansion-table.dat COLUMNS 1 2 INTERPOLATION $1 |
| 101 | +# B is the mean coefficient of thermal expansion x 10^-6 (mm/mm/ºC) in going |
| 102 | +# from 20ºC to indicated temperature |
| 103 | +FUNCTION B(T) FILE asme-expansion-table.dat COLUMNS 1 3 INTERPOLATION $1 |
| 104 | +# C is the linear thermal expansion (mm/m) in going from 20ºC |
| 105 | +# to indicated temperature |
| 106 | +FUNCTION C(T) FILE asme-expansion-table.dat COLUMNS 1 4 INTERPOLATION $1 |
| 107 | + |
| 108 | +VAR T' # dummy variable for integration |
| 109 | +T0 = 20 # reference temperature |
| 110 | +T_min = vecmin(vec_A_T) # smallest argument of function A(T) |
| 111 | +T_max = vecmax(vec_A_T) # largest argument of function A(T) |
| 112 | + |
| 113 | +# compute one column from another one |
| 114 | +A_fromC(T) := 1e3*derivative(C(T'), T', T) |
| 115 | + |
| 116 | +B_fromA(T) := integral(A(T'), T', T0, T)/(T-T0) |
| 117 | +B_fromC(T) := 1e3*C(T)/(T-T0) # C is in mm/m, hence the 1e3 |
| 118 | + |
| 119 | +C_fromA(T) := 1e-3*integral(A(T'), T', T0, T) |
| 120 | + |
| 121 | +# write interpolated results |
| 122 | +PRINT_FUNCTION A A_fromC B B_fromA B_fromC C C_fromA MIN T_min+1 MAX T_max-1 STEP 1 |
| 123 | +--- |
| 124 | +figures: | |
| 125 | + {width=100%} |
| 126 | + {width=100%} |
| 127 | + {width=100%} |
| 128 | +
|
| 129 | + > The conclusion (see |
| 130 | + > [this](https://www.linkedin.com/pulse/accuracy-thermal-expansion-properties-asme-bpv-code-angus-ramsay/), |
| 131 | + > [this](https://www.seamplex.com/docs/SP-WA-17-TN-F38B-A.pdf) and |
| 132 | + > [this](https://www.linkedin.com/pulse/ansys-potential-issue-thermal-expansion-calculations-angus-ramsay/) reports) |
| 133 | + > is that values rounded to only one decimal value as presented in the ASME code |
| 134 | + > section II subsection D tables are not enough to satisfy the mathematical relationships |
| 135 | + > between the physical magnitudes related to thermal expansion properties of the materials listed. |
| 136 | + > Therefore, care has to be taken as which of the three columns is chosen when using the data |
| 137 | + > for actual thermo-mechanical numerical computations. |
| 138 | + > As an exercise, the reader is encouraged to try different interpolation algorithms to |
| 139 | + > see how the results change. _Spoiler alert_: they are also highly sensible to the interpolation method used |
| 140 | + > to “fill in” the gaps between the table values. |
| 141 | +... |
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