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    "Jochen Kieninger<br>\n",
    "**Electrochemical Methods for the Micro- and Nanoscale**<br>\n",
    "www.electrochemical-methods.org\n",
    "\n",
    "## Task 6.1 (Rotating ring disk electrode - collection efficiency)\n",
    "\n",
    "The collection efficiency of an RRDE depends on geometrical parameters only, assuming stable reaction products. The radii are $r_1$ for the disk, $r_2$ the inner and $r_3$ the outer of the ring."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "moved-library",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "characteristic-registration",
   "metadata": {},
   "outputs": [],
   "source": [
    "r1 = 2.5  # mm (disk radius)\n",
    "r2 = 3  # mm (inner ring radius)\n",
    "r3 = 4  # mm (outer ring radius)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "hybrid-privilege",
   "metadata": {},
   "source": [
    "- For the collection efficiency $N$, Albery and Bruckenstein (https://dx.doi.org/10.1039/tf9666201920) proposed a generally accepted solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "surgical-little",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = (r2/r1)**3 -1\n",
    "b = (r3/r1)**3 - (r2/r1)**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "portuguese-sixth",
   "metadata": {},
   "outputs": [],
   "source": [
    "def FF(x):\n",
    "    result = np.sqrt(3)/4/np.pi\n",
    "    result *= np.log((1+x**(1/3))**3/(1+x))\n",
    "    result += 3/2/np.pi*np.arctan((2*x**(1/3)-1)/np.sqrt(3))\n",
    "    result += 0.25\n",
    "    return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "endangered-silence",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 1-FF(a/b) + b**(2/3)*(1-FF(a))\n",
    "N = N - (1+a+b)**(2/3)*(1-FF(a/b*(1+a+b)))\n",
    "round(N, 2)"
   ]
  }
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