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Fixed name of python_residual function and introduced Contributions in notebooks
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‎examples/core_dual_numbers.ipynb‎

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@@ -9,7 +9,7 @@
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"In `FeOs`, we use [generalized dual numbers](https://www.frontiersin.org/articles/10.3389/fceng.2021.758090/full) to compute partial derivatives of the Helmholtz energy.\n",
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"In this notebook, we take a closer look at how that works.\n",
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"\n",
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"We will make use of the `EquationOfState.python()` feature: we use a simple version of the Peng-Robinson equation of state implemented as Python class which is registered in `FeOs` as equation of state. \n",
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"We will make use of the `EquationOfState.python_residual()` feature: we use a simple version of the Peng-Robinson equation of state implemented as Python class which is registered in `FeOs` as equation of state. \n",
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"If you want to learn how to implement an equation of state as Python class in conjunction with `FeOs`, take a look at the example implementation in the examples section.\n",
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"In short - a class that implements a `helmholtz_energy` function that takes a `StateD` (or *any* state object, where the internal data types can be any generalized dual number)\n",
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"and returns the Helmholtz energy (plus some minor functions), can be used with `FeOs`. We use the equation of state implemented in Python simply because we can add `print` statements and inspect the data types at runtime.\n",
@@ -170,7 +170,7 @@
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"molar_weight = SIArray1(44.0962 * GRAM / MOL)\n",
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"\n",
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"# create an instance of our python class and hand it over to rust\n",
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"eos = EquationOfState.python(PyPengRobinson(tc, pc, omega, molar_weight))"
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"eos = EquationOfState.python_residual(PyPengRobinson(tc, pc, omega, molar_weight))"
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]
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},
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{
@@ -237,10 +237,18 @@
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"A/kT : [5.06008546]\n"
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]
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},
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/tmp/ipykernel_3483/3728344996.py:1: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)\n",
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" state.molar_helmholtz_energy(Contributions.Residual) / (RGAS * 300*KELVIN)\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"-6.554978406564657"
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"5.060085461999784"
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]
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},
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"execution_count": 4,
@@ -249,7 +257,7 @@
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}
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],
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"source": [
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"state.helmholtz_energy() / (KB * 300*KELVIN)"
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"state.molar_helmholtz_energy(Contributions.Residual) / (RGAS * 300*KELVIN)"
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]
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},
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{
@@ -279,11 +287,19 @@
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"text": [
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"\n",
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"data type : <class 'builtins.PyDual64'>\n",
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"temperature: 300 + [0]ε\n",
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"volume : 40744 + [1]ε\n",
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"moles : [1 + [0]ε]\n",
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"density : 0.000024543491066169253 + [-0.00000000060238295371513]ε\n",
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"A/kT : 5.060085461999783 + [0.00000040024326944247174]ε\n"
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"temperature: 300 + 0ε\n",
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"volume : 40744 + 1ε\n",
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"moles : [1 + 0ε]\n",
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"density : 0.000024543491066169253 + -0.00000000060238295371513ε\n",
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"A/kT : 5.060085461999783 + 0.00000040024326944247174ε\n"
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]
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},
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/tmp/ipykernel_3483/2329888183.py:65: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)\n",
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" ak = ((1.0 - np.sqrt(tr)) * self.kappa + 1.0)**2 * self.a_r\n"
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]
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},
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{
@@ -292,7 +308,7 @@
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"$100\\,\\mathrm{kPa}$"
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],
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"text/plain": [
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"100.00005278190909 kPa"
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"100.00005278190908 kPa"
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]
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},
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"execution_count": 5,
@@ -323,97 +339,37 @@
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"text": [
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"\n",
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"data type : <class 'builtins.PyDual64'>\n",
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"temperature: 300 + [1]ε\n",
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"volume : 40744 + [0]ε\n",
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"moles : [1 + [0]ε]\n",
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"density : 0.000024543491066169253 + [0]ε\n",
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"A/kT : 5.060085461999783 + [-0.025513116823522065]ε\n"
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"temperature: 300 + 1ε\n",
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"volume : 40744 + 0ε\n",
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"moles : [1 + 0ε]\n",
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"density : 0.000024543491066169253 + 0ε\n",
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"A/kT : 5.060085461999783 + -0.025513116823522065ε\n"
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]
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},
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{
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"data": {
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"text/latex": [
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"$118.14\\,\\mathrm{\\frac{ J}{molK}}$"
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],
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"text/plain": [
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"118.13947975470876 J/mol/K"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"state.molar_entropy()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let's take a look at a more involved property.\n",
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"For the Joule-Thomson coefficient, we need multiple second partial derivatives:\n",
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"\n",
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"$\\mu_{JT}=\\left(\\frac{\\partial T}{\\partial p}\\right)_{H,N_i} = -\\frac{1}{C_p} \\left(V + T \\left(\\frac{\\partial p}{\\partial T}\\right)_{\\mathbf{n}, V} \\left(\\frac{\\partial p}{\\partial V}\\right)_{\\mathbf{n}, T}^{-1} \\right)$\n",
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"\n",
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"We need three evaluations of the Helmholtz energy (that are actually occuring in the computation of $C_p$):\n",
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"\n",
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"1. $\\left(\\frac{\\partial p}{\\partial T}\\right)_{\\mathbf{n}, V} \\rightarrow -\\left(\\frac{\\partial A}{\\partial T \\partial V}\\right)_\\mathbf{n}$: second partial derivative w.r.t volume and temperature,\n",
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"2. $\\left(\\frac{\\partial p}{\\partial V}\\right)_{\\mathbf{n}, T} \\rightarrow -\\left(\\frac{\\partial^2 A}{\\partial V^2}\\right)_{\\mathbf{n}, T}$: second derivative w.r.t volume,\n",
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"3. $\\left(\\frac{\\partial^2 A}{\\partial T^2}\\right)_{\\mathbf{n}, V}$: 2nd partial derivative w.r.t temperature.\n",
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"\n",
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"Since we compute second partial derivatives, the data type is now `HyperDual64` (one real, 3 non-real values)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"\n",
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"data type : <class 'builtins.PyHyperDual64'>\n",
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"temperature: 300 + [0]ε1 + [1]ε2 + [0]ε1ε2\n",
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"volume : 40744 + [1]ε1 + [0]ε2 + [0]ε1ε2\n",
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"moles : [1 + [0]ε1 + [0]ε2 + [0]ε1ε2]\n",
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"density : 0.000024543491066169253 + [-0.00000000060238295371513]ε1 + [0]ε2 + [0]ε1ε2\n",
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"A/kT : 5.060085461999783 + [0.00000040024326944247174]ε1 + [-0.025513116823522065]ε2 + [-0.0000000023037315630585307]ε1ε2\n",
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"\n",
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"data type : <class 'builtins.PyDual2_64'>\n",
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"temperature: 300 + [0]ε1 + [0]ε1²\n",
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"volume : 40744 + [1]ε1 + [0]ε1²\n",
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"moles : [1 + [0]ε1 + [0]ε1²]\n",
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"density : 0.000024543491066169253 + [-0.00000000060238295371513]ε1 + [0.00000000000002956916128583988]ε1²\n",
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"A/kT : 5.060085461999783 + [0.00000040024326944247174]ε1 + [-0.000000000019592452372041545]ε1²\n",
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"\n",
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"data type : <class 'builtins.PyDual2_64'>\n",
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"temperature: 300 + [1]ε1 + [0]ε1²\n",
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"volume : 40744 + [0]ε1 + [0]ε1²\n",
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"moles : [1 + [0]ε1 + [0]ε1²]\n",
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"density : 0.000024543491066169253 + [0]ε1 + [0]ε1²\n",
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"A/kT : 5.060085461999783 + [-0.025513116823522065]ε1 + [0.0001919137873859282]ε1²\n"
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"/tmp/ipykernel_3483/2329888183.py:65: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)\n",
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" ak = ((1.0 - np.sqrt(tr)) * self.kappa + 1.0)**2 * self.a_r\n"
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]
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},
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{
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"data": {
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"text/latex": [
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"$-1.4939\\times10^{-4}\\,\\mathrm{\\frac{ms^{2}K}{kg}}$"
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"$21.566\\,\\mathrm{\\frac{ J}{molK}}$"
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],
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"text/plain": [
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"-1.4938695195180555e-4 m kg^-1 s^2 K"
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"21.566465412067362 J/mol/K"
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]
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},
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"execution_count": 7,
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"state.joule_thomson()"
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"state.molar_entropy(Contributions.Residual)"
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]
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},
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{
@@ -430,7 +386,7 @@
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
@@ -444,7 +400,7 @@
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.6"
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"version": "3.9.12"
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}
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},
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"nbformat": 4,

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