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Rewrite example notebooks to use Simulation and Connection
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‎docs/source/examples/equation_of_state.ipynb‎

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@@ -6,16 +6,16 @@
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"source": [
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"# Cubic Equations of State\n",
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"\n",
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"Computing compressibility factors and molar volumes using the Peng-Robinson and Soave-Redlich-Kwong cubic equations of state for pure methane and a methane-ethane mixture."
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"Simulating the compressibility factor of methane across a pressure sweep using the Peng-Robinson and Soave-Redlich-Kwong equations of state wired into a PathSim simulation."
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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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"## Setup\n",
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"Cubic equations of state compute the compressibility factor $Z = Pv/(RT)$ by solving a cubic polynomial at each $(T, P)$ condition. The EoS blocks take two inputs (temperature and pressure) and produce two outputs (molar volume and compressibility factor).\n",
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"\n",
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"Import the equation of state blocks."
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"At low pressure $Z \\to 1$ (ideal gas). At moderate pressure attractive forces cause $Z < 1$, and at high pressure excluded-volume effects push $Z > 1$."
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]
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},
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{
@@ -24,23 +24,21 @@
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"\n",
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"from pathsim import Simulation, Connection\n",
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"from pathsim.blocks import Source, Constant, Scope\n",
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"\n",
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"from pathsim_chem.thermodynamics import PengRobinson, RedlichKwongSoave"
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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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"## Pure Methane\n",
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"\n",
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"The Peng-Robinson EoS solves a cubic equation in the compressibility factor $Z = Pv/(RT)$:\n",
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"## System Definition\n",
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"\n",
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"$$Z^3 - (1-B)Z^2 + (A - 3B^2 - 2B)Z - (AB - B^2 - B^3) = 0$$\n",
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"\n",
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"where $A$ and $B$ are dimensionless parameters computed from the critical properties. Each block takes $(T, P)$ as inputs and returns $(v, Z)$."
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"Create Peng-Robinson and SRK blocks for pure methane. A `Constant` block supplies a fixed temperature while a `Source` sweeps pressure from 0.1 MPa to 30 MPa."
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]
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},
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{
@@ -50,38 +48,29 @@
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"outputs": [],
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"source": [
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"# Critical properties of methane\n",
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"Tc_CH4 = 190.6 # K\n",
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"Pc_CH4 = 4.6e6 # Pa\n",
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"omega_CH4 = 0.011 # acentric factor\n",
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"\n",
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"# Create EoS blocks\n",
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"pr = PengRobinson(Tc=Tc_CH4, Pc=Pc_CH4, omega=omega_CH4)\n",
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"rks = RedlichKwongSoave(Tc=Tc_CH4, Pc=Pc_CH4, omega=omega_CH4)\n",
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"\n",
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"def eval_eos(block, T, P):\n",
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" \"\"\"Evaluate an EoS block and return (v, Z).\"\"\"\n",
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" block.inputs[0] = T\n",
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" block.inputs[1] = P\n",
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" block.update(None)\n",
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" return block.outputs[0], block.outputs[1]\n",
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"\n",
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"# Evaluate at 300 K, 1 atm\n",
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"v_pr, Z_pr = eval_eos(pr, 300, 101325)\n",
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"v_rks, Z_rks = eval_eos(rks, 300, 101325)\n",
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"\n",
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"print(f\"Methane at 300 K, 1 atm:\")\n",
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"print(f\" PR: Z = {Z_pr:.6f}, v = {v_pr:.6e} m³/mol\")\n",
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"print(f\" RKS: Z = {Z_rks:.6f}, v = {v_rks:.6e} m³/mol\")\n",
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"print(f\" Ideal gas: Z = 1.0, v = {8.314462 * 300 / 101325:.6e} m³/mol\")"
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"Tc, Pc, omega = 190.6, 4.6e6, 0.011\n",
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"\n",
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"# EoS blocks\n",
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"pr = PengRobinson(Tc=Tc, Pc=Pc, omega=omega)\n",
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"rks = RedlichKwongSoave(Tc=Tc, Pc=Pc, omega=omega)\n",
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"\n",
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"# Fixed temperature, logarithmic pressure sweep\n",
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"import numpy as np\n",
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"T_const = Constant(250) # 250 K (above Tc, supercritical)\n",
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"P_src = Source(func=lambda t: 10**(4 + t * 0.035)) # 10 kPa to ~30 MPa over 100s\n",
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"\n",
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"# Scopes: record Z from both EoS (output port 1)\n",
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"scp_pr = Scope(labels=[\"Z_PR\"])\n",
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"scp_rks = Scope(labels=[\"Z_RKS\"])"
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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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"## Compressibility Factor vs Pressure\n",
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"## Wiring\n",
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"\n",
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"At low pressure, $Z \\to 1$ (ideal gas). As pressure increases, intermolecular forces cause $Z$ to deviate. At very high pressures, repulsive forces dominate and $Z > 1$."
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"Both EoS blocks receive the same $(T, P)$ inputs. We record the compressibility factor (output port 1) from each."
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]
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},
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{
@@ -90,25 +79,40 @@
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"metadata": {},
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"outputs": [],
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"source": [
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"T_fixed = 250 # K (above Tc for methane, supercritical)\n",
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"P_range = np.logspace(4, 7.5, 100) # 10 kPa to ~30 MPa\n",
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"\n",
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"Z_pr_arr = []\n",
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"Z_rks_arr = []\n",
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"\n",
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"for P in P_range:\n",
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" _, Z = eval_eos(pr, T_fixed, P)\n",
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" Z_pr_arr.append(Z)\n",
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" _, Z = eval_eos(rks, T_fixed, P)\n",
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" Z_rks_arr.append(Z)\n",
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"sim = Simulation(\n",
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" blocks=[T_const, P_src, pr, rks, scp_pr, scp_rks],\n",
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" connections=[\n",
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" # Temperature -> both EoS (input port 0)\n",
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" Connection(T_const, pr, rks),\n",
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" # Pressure -> both EoS (input port 1)\n",
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" Connection(P_src, pr[1], rks[1]),\n",
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" # Z output (port 1) -> scopes\n",
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" Connection(pr[1], scp_pr),\n",
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" Connection(rks[1], scp_rks),\n",
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" ],\n",
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" dt=1.0,\n",
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")\n",
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"\n",
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"sim.run(100)"
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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": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"time, Z_pr = scp_pr.read()\n",
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"_, Z_rks = scp_rks.read()\n",
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"P_vals = 10**(4 + time * 0.035) / 1e6 # MPa\n",
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"\n",
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"fig, ax = plt.subplots(figsize=(7, 5))\n",
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"ax.semilogx(P_range / 1e6, Z_pr_arr, label=\"Peng-Robinson\")\n",
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"ax.semilogx(P_range / 1e6, Z_rks_arr, \"--\", label=\"Soave-Redlich-Kwong\")\n",
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"ax.semilogx(P_vals, Z_pr[0], label=\"Peng-Robinson\")\n",
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"ax.semilogx(P_vals, Z_rks[0], \"--\", label=\"Soave-Redlich-Kwong\")\n",
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"ax.axhline(1.0, color=\"gray\", linestyle=\"-.\", alpha=0.5, label=\"Ideal gas\")\n",
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"ax.set_xlabel(\"Pressure [MPa]\")\n",
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"ax.set_ylabel(\"Compressibility Factor Z\")\n",
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"ax.set_title(f\"Methane at T = {T_fixed} K\")\n",
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"ax.set_title(\"Methane at T = 250 K\")\n",
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"ax.legend()\n",
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"ax.grid(True, alpha=0.3)\n",
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"plt.tight_layout()\n",
@@ -126,9 +130,9 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Z vs Reduced Temperature\n",
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"## Mixture\n",
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"\n",
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"Vary the temperature at a fixed pressure to see how the compressibility factor changes with reduced temperature $T_r = T / T_c$."
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"The EoS blocks also support mixtures through van der Waals one-fluid mixing rules. Here we set up a methane-ethane mixture and sweep pressure."
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]
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},
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{
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"metadata": {},
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"outputs": [],
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"source": [
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"P_fixed = 5e6 # 5 MPa\n",
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"T_range = np.linspace(200, 600, 100)\n",
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"Tr_range = T_range / Tc_CH4\n",
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"\n",
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"Z_vs_T = []\n",
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"for T in T_range:\n",
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" _, Z = eval_eos(pr, T, P_fixed)\n",
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" Z_vs_T.append(Z)\n",
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"\n",
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"fig, ax = plt.subplots(figsize=(7, 5))\n",
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"ax.plot(Tr_range, Z_vs_T)\n",
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"ax.axhline(1.0, color=\"gray\", linestyle=\"-.\", alpha=0.5)\n",
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"ax.axvline(1.0, color=\"red\", linestyle=\":\", alpha=0.5, label=r\"$T_r = 1$ (critical)\")\n",
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"ax.set_xlabel(r\"Reduced Temperature $T_r = T / T_c$\")\n",
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"ax.set_ylabel(\"Compressibility Factor Z\")\n",
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"ax.set_title(f\"Peng-Robinson: Methane at P = {P_fixed/1e6:.0f} MPa\")\n",
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"ax.legend()\n",
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"ax.grid(True, alpha=0.3)\n",
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"plt.tight_layout()\n",
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"plt.show()"
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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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"## Methane-Ethane Mixture\n",
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"\n",
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"The EoS blocks support mixtures through standard van der Waals one-fluid mixing rules. Supply arrays of critical properties and mole fractions."
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"pr_mix = PengRobinson(\n",
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" Tc=[190.6, 305.3],\n",
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" Pc=[4.6e6, 4.872e6],\n",
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" omega=[0.011, 0.099],\n",
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" x=[0.7, 0.3],\n",
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")\n",
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"\n",
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"T_const2 = Constant(300)\n",
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"P_src2 = Source(func=lambda t: 10**(4 + t * 0.035))\n",
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"scp_mix = Scope(labels=[\"Z_mixture\"])\n",
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"\n",
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"sim_mix = Simulation(\n",
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" blocks=[T_const2, P_src2, pr_mix, scp_mix],\n",
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" connections=[\n",
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" Connection(T_const2, pr_mix),\n",
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" Connection(P_src2, pr_mix[1]),\n",
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" Connection(pr_mix[1], scp_mix),\n",
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" ],\n",
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" dt=1.0,\n",
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")\n",
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"\n",
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"sim_mix.run(100)"
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]
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{
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"metadata": {},
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"outputs": [],
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"source": [
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"# Critical properties: methane, ethane\n",
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"Tc = [190.6, 305.3] # K\n",
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"Pc = [4.6e6, 4.872e6] # Pa\n",
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"omega = [0.011, 0.099] # acentric factors\n",
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"\n",
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"# Compare Z at different compositions\n",
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"x1_range = np.linspace(0, 1, 20) # methane mole fraction\n",
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"T_mix, P_mix = 300, 3e6 # 300 K, 3 MPa\n",
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"\n",
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"Z_mix = []\n",
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"for x1 in x1_range:\n",
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" pr_mix = PengRobinson(\n",
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" Tc=Tc, Pc=Pc, omega=omega,\n",
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" x=[x1, 1 - x1],\n",
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" )\n",
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" _, Z = eval_eos(pr_mix, T_mix, P_mix)\n",
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" Z_mix.append(Z)\n",
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"time_m, Z_mix = scp_mix.read()\n",
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"P_mix = 10**(4 + time_m * 0.035) / 1e6\n",
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"\n",
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"fig, ax = plt.subplots(figsize=(7, 5))\n",
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"ax.plot(x1_range, Z_mix, \"o-\")\n",
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"ax.set_xlabel(r\"$x_{\\mathrm{CH_4}}$\")\n",
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"ax.semilogx(P_vals, Z_pr[0], label=\"Pure CH₄ (250 K)\")\n",
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"ax.semilogx(P_mix, Z_mix[0], \"--\", label=\"70/30 CH₄-C₂H₆ (300 K)\")\n",
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"ax.axhline(1.0, color=\"gray\", linestyle=\"-.\", alpha=0.5)\n",
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"ax.set_xlabel(\"Pressure [MPa]\")\n",
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"ax.set_ylabel(\"Compressibility Factor Z\")\n",
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"ax.set_title(f\"PR: Methane-Ethane at T = {T_mix} K, P = {P_mix/1e6:.0f} MPa\")\n",
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"ax.set_title(\"Peng-Robinson: Pure vs Mixture\")\n",
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"ax.legend()\n",
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"ax.grid(True, alpha=0.3)\n",
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"plt.tight_layout()\n",
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"plt.show()"
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The mixture compressibility factor varies smoothly with composition. Pure ethane (right side) has a lower $Z$ at these conditions because it is closer to its critical point ($T_c = 305.3$ K) and therefore deviates more from ideal gas behavior."
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"The mixture shows a deeper dip because ethane ($T_c = 305.3$ K) is near its critical temperature at 300 K, leading to stronger non-ideal behavior."
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]
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}
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