|
1 | 1 | { |
2 | 2 | "cells": [ |
| 3 | + { |
| 4 | + "cell_type": "markdown", |
| 5 | + "metadata": {}, |
| 6 | + "source": [ |
| 7 | + "###### Content under Creative Commons Attribution license CC-BY 4.0, code under BSD 3-Clause License © 2019 Adam Wickenheiser, Lorena Barba" |
| 8 | + ] |
| 9 | + }, |
3 | 10 | { |
4 | 11 | "cell_type": "markdown", |
5 | 12 | "metadata": {}, |
|
359 | 366 | "metadata": {}, |
360 | 367 | "source": [ |
361 | 368 | "**Example:** We would like to design an Otto cycle that produces $900 J$ of energy per cycle. We are constrained by the following design requirements: \n", |
362 | | - "1. $\\dfrac{v_1}{v_2} \\leq 12$. If this **compression ratio** is any larger, the fuel will autoignite, causing undue vibration in the engine (\"knocking\") leading potentially to damage\n", |
| 369 | + "1. $r \\equiv \\dfrac{v_1}{v_2} \\leq 12$. If this **compression ratio** is any larger, the fuel will autoignite, causing undue vibration in the engine (\"knocking\") leading potentially to damage\n", |
363 | 370 | "1. $P_1 = 1 atm$ and $T_1 = 18 ^\\circ C$. The intake and exhaust occur at (roughly) atmospheric pressure and temperature. We will consider ways to circumvent this requirement in a later lesson.\n", |
364 | 371 | "1. $T_3 = 1300 ^\\circ C$. This is the maximum temperature the engine can operate at without suffering fatigue damage.\n", |
365 | | - "1. $m = 0.01 kg$ This limits the volume and hence the physical size of the engine.\n", |
366 | 372 | "1. We will assume the gas in the engine is air. This is a common assumption called the **air standard model**.\n", |
367 | 373 | "1. The two polytropic processes are governed by $Pv^{1.5} = const.$\n", |
368 | 374 | "\n", |
369 | | - "What diameter cylinder is necessary to produce the desired work?\n", |
| 375 | + "What displacement $V_1-V_2$ is necessary to produce the desired work?\n", |
370 | 376 | "\n", |
371 | 377 | "We'd like to make the engine as small as possible (smallest cylinder diameter) and yet still achieve the desired work output. As we previously mentioned, the net work output of the cycle is the area inside the curve on the $P-v$ diagram, so we want to make this area as large as possible. We do this by making $\\dfrac{v_1}{v_2}$ as large as possible, which increases the width of the area, and by making $\\dfrac{T_3}{T_1}$ as large as possible, which increases the height of the area.\n", |
372 | 378 | "\n", |
|
432 | 438 | " plt.ylabel('Pressure [Pa]')\n", |
433 | 439 | " plt.xlabel('Specific volume [m^3/kg]')\n", |
434 | 440 | " \n", |
435 | | - " # compute work using polytropic process integral formulas\n", |
| 441 | + " # compute work/mass using polytropic process integral formulas\n", |
436 | 442 | " if n == 1:\n", |
437 | | - " W = P3*v3*np.log(v4/v3) + P1*v1*np.log(v2/v1)\n", |
| 443 | + " w = P3*v3*np.log(v4/v3) + P1*v1*np.log(v2/v1)\n", |
438 | 444 | " else:\n", |
439 | | - " W = (P2*v2-P1*v1)/(1-n) + (P4*v4-P3*v3)/(1-n)\n", |
440 | | - " return W" |
| 445 | + " w = (P2*v2-P1*v1)/(1-n) + (P4*v4-P3*v3)/(1-n)\n", |
| 446 | + " return w" |
441 | 447 | ] |
442 | 448 | }, |
443 | 449 | { |
|
449 | 455 | "T1 = 18.0 + 273.15 # temperature [K]\n", |
450 | 456 | "P1 = 1.0*101325.0 # pressure [Pa]\n", |
451 | 457 | "T3 = 1300.0 + 273.15 # temperature [K]\n", |
452 | | - "r = 12 # compression ratio\n", |
| 458 | + "r = 12.0 # compression ratio\n", |
453 | 459 | "n = 1.5 # polytropic process exponent\n", |
454 | 460 | "\n", |
455 | 461 | "# plot Otto cycle and compute work/mass\n", |
|
0 commit comments