{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "27313f1e",
   "metadata": {},
   "source": [
    "# 2D heat transfer: hot water dipped into liquid nitrogen\n",
    "\n",
    "This notebook simulates heat transfer in a two-dimensional rectangular container.\n",
    "\n",
    "The initial conditions are:\n",
    "\n",
    "- water inside the container: **100 °C**\n",
    "- left, right, and bottom boundaries: **−196 °C** (liquid nitrogen)\n",
    "- top boundary: **22 °C** (room temperature)\n",
    "\n",
    "The temperature field is discretized on a rectangular grid. At each iteration, the\n",
    "temperature of every interior point is replaced by the average of itself and its four\n",
    "nearest neighbours.\n",
    "\n",
    "This is a simple finite-difference relaxation model. It is useful for illustrating\n",
    "diffusion-like heat flow, although it is not yet a quantitatively calibrated solution\n",
    "of the physical heat equation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "480dea4a",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.animation import FuncAnimation\n",
    "from IPython.display import HTML"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35ea1ad4",
   "metadata": {},
   "source": [
    "## The temperature grid\n",
    "\n",
    "The array contains an extra one-cell boundary around the simulated water:\n",
    "\n",
    "```text\n",
    "          22 °C\n",
    "     ┌─────────────┐\n",
    "-196 │             │ -196\n",
    " °C  │   water     │  °C\n",
    "     │             │\n",
    "     └─────────────┘\n",
    "         -196 °C\n",
    "```\n",
    "\n",
    "The array index order is `[y, x]`, which is also the natural order for displaying\n",
    "the field as an image."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "89e328ba",
   "metadata": {},
   "outputs": [],
   "source": [
    "class Heat2D:\n",
    "    def __init__(self, height, width):\n",
    "        # Index order is [y, x].\n",
    "        # The extra two rows/columns contain the fixed boundary temperatures.\n",
    "        self.heat_map = np.full((height + 2, width + 2), 100.0)\n",
    "\n",
    "        # Liquid-nitrogen boundaries\n",
    "        self.heat_map[:, 0]  = -196.0\n",
    "        self.heat_map[:, -1] = -196.0\n",
    "        self.heat_map[-1, :] = -196.0\n",
    "\n",
    "        # Top of the container remains at room temperature\n",
    "        self.heat_map[0, :] = 22.0\n",
    "\n",
    "    def step(self):\n",
    "        # Perform one temperature-relaxation step.\n",
    "        mid   = self.heat_map[1:-1, 1:-1]\n",
    "        above = self.heat_map[:-2, 1:-1]\n",
    "        below = self.heat_map[2:, 1:-1]\n",
    "        left  = self.heat_map[1:-1, :-2]\n",
    "        right = self.heat_map[1:-1, 2:]\n",
    "\n",
    "        mid[:] = (mid + above + below + left + right) / 5.0\n",
    "\n",
    "        return self.heat_map"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5795102a",
   "metadata": {},
   "source": [
    "## Update rule\n",
    "\n",
    "For an interior grid point $(i,j)$, one iteration uses\n",
    "\n",
    "\\[\n",
    "T_{i,j}^{\\,\\mathrm{new}}\n",
    "=\n",
    "\\frac{\n",
    "T_{i,j}\n",
    "+T_{i-1,j}\n",
    "+T_{i+1,j}\n",
    "+T_{i,j-1}\n",
    "+T_{i,j+1}\n",
    "}{5}.\n",
    "\\]\n",
    "\n",
    "So each point moves toward the average temperature of its local neighbourhood.\n",
    "\n",
    "The NumPy implementation updates the whole interior grid without explicit Python\n",
    "loops."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7b0ebdf6",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initialize a 100 x 100 element water container\n",
    "heat = Heat2D(100, 100)\n",
    "\n",
    "print(\"Array shape:\", heat.heat_map.shape)\n",
    "print(\"Initial interior temperature:\", heat.heat_map[50, 50], \"°C\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "673c9a2a",
   "metadata": {},
   "source": [
    "## Initial temperature field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "03a4cea7",
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(6, 5))\n",
    "\n",
    "im = ax.imshow(\n",
    "    heat.heat_map,\n",
    "    origin=\"upper\",\n",
    "    cmap=\"cool\",\n",
    "    vmin=-200,\n",
    "    vmax=100,\n",
    ")\n",
    "\n",
    "ax.set_title(\"Initial temperature\")\n",
    "ax.set_xlabel(\"x\")\n",
    "ax.set_ylabel(\"y\")\n",
    "\n",
    "cbar = fig.colorbar(im, ax=ax)\n",
    "cbar.set_label(\"Temperature (°C)\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1a3a764",
   "metadata": {},
   "source": [
    "## Take a few steps manually\n",
    "\n",
    "This is useful for seeing what one call to `step()` does before running an animation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8b69d850",
   "metadata": {},
   "outputs": [],
   "source": [
    "heat_test = Heat2D(100, 100)\n",
    "\n",
    "for _ in range(100):\n",
    "    heat_test.step()\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(6, 5))\n",
    "\n",
    "im = ax.imshow(\n",
    "    heat_test.heat_map,\n",
    "    origin=\"upper\",\n",
    "    cmap=\"cool\",\n",
    "    vmin=-200,\n",
    "    vmax=100,\n",
    ")\n",
    "\n",
    "ax.set_title(\"Temperature after 100 iterations\")\n",
    "ax.set_xlabel(\"x\")\n",
    "ax.set_ylabel(\"y\")\n",
    "\n",
    "cbar = fig.colorbar(im, ax=ax)\n",
    "cbar.set_label(\"Temperature (°C)\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d6d7256",
   "metadata": {},
   "source": [
    "## Animation\n",
    "\n",
    "In a Jupyter notebook it is convenient to display the Matplotlib animation as\n",
    "embedded JavaScript/HTML rather than relying on a separate GUI window.\n",
    "\n",
    "The simulation below starts again from the initial state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "67b3a138",
   "metadata": {},
   "outputs": [],
   "source": [
    "heat = Heat2D(100, 100)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(6, 5))\n",
    "\n",
    "im = ax.imshow(\n",
    "    heat.heat_map,\n",
    "    origin=\"upper\",\n",
    "    cmap=\"cool\",\n",
    "    vmin=-200,\n",
    "    vmax=100,\n",
    ")\n",
    "\n",
    "ax.set_xlabel(\"x\")\n",
    "ax.set_ylabel(\"y\")\n",
    "\n",
    "cbar = fig.colorbar(im, ax=ax)\n",
    "cbar.set_label(\"Temperature (°C)\")\n",
    "\n",
    "\n",
    "def update(frame):\n",
    "    heat.step()\n",
    "    im.set_data(heat.heat_map)\n",
    "    ax.set_title(f\"2D heat-transfer simulation — iteration {frame}\")\n",
    "    return (im,)\n",
    "\n",
    "\n",
    "ani = FuncAnimation(\n",
    "    fig,\n",
    "    update,\n",
    "    frames=500,\n",
    "    interval=20,\n",
    "    blit=False,\n",
    ")\n",
    "\n",
    "plt.close(fig)\n",
    "\n",
    "HTML(ani.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a264380",
   "metadata": {},
   "source": [
    "## A note about the physics\n",
    "\n",
    "This algorithm qualitatively resembles heat diffusion, but the iteration number is\n",
    "not yet a physical time.\n",
    "\n",
    "A standard finite-difference discretization of the heat equation\n",
    "\n",
    "\\[\n",
    "\\frac{\\partial T}{\\partial t}\n",
    "=\n",
    "\\alpha \\nabla^2 T\n",
    "\\]\n",
    "\n",
    "would normally contain the thermal diffusivity $\\alpha$, the spatial grid spacing\n",
    "$\\Delta x$, and the physical time step $\\Delta t$.\n",
    "\n",
    "Those can be added if the goal is to turn this visualization into a physically\n",
    "scaled heat-equation simulation."
   ]
  }
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