Add a new example of a cascade refrigeration cycle that uses EE units

Also fix a typo in the Rankine cycle example

docs/cascade-refrigeration-cycle-example.ipynb

@@ -0,0 +1,327 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Cascade Refrigeration Cycle Example\n",
+    "\n",
+    "## Imports"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from thermostate import State, Q_, units, EnglishEngineering as EE"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "---\n",
+    "\n",
+    "## Definitions"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "sub_hiT = 'R22'\n",
+    "sub_loT = 'R134A'\n",
+    "\n",
+    "T_1 = Q_(-30.0, 'degF')\n",
+    "x_1 = Q_(1.0, 'dimensionless')\n",
+    "\n",
+    "p_2 = Q_(50.0, 'psi')\n",
+    "\n",
+    "p_3 = p_2\n",
+    "x_3 = Q_(0.0, 'dimensionless')\n",
+    "\n",
+    "delta_T_5 = Q_(-5.0, 'delta_degF')\n",
+    "x_5 = Q_(1.0, 'dimensionless')\n",
+    "\n",
+    "p_6 = Q_(250.0, 'psi')\n",
+    "\n",
+    "p_7 = p_6\n",
+    "x_7 = Q_(0.0, 'dimensionless')\n",
+    "\n",
+    "Qdot_in = Q_(20.0, 'refrigeration_tons')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "---\n",
+    "\n",
+    "## Problem Statement"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "A two-stage cascade vapor-compression refrigeration system operates with R22 as the working fluid in the high-temperature cycle and R134A in the low-temperature cycle. For the R134A cycle, the working fluid enters the compressor as a saturated vapor at -30.0 fahrenheit and is compressed isentropically to 50 lbf/in.<sup>2</sup> Saturated liquid leaves the intermediate heat exchanger at 50 lbf/in.<sup>2</sup> and enters the expansion valve. For the R22 cycle, the working fluid enters the compressor as saturated vapor at a temperature 5 fahrenheit below that of the condensing temperature of the R134A in the intermediate heat exchanger. The R22 is compressed isentropically to 250 lbf/in.<sup>2</sup> Saturated liquid then enters the expansion valve at 250 lbf/in.<sup>2</sup>. The refrigerating capacity of the cascade system is 20 ton_of_refrigeration. Determine\n",
+    "\n",
+    "1. the power input to each compressor, in BTU/min\n",
+    "2. the overall coefficient of performance of the cascade cycle"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "---\n",
+    "\n",
+    "## Solution"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 1. the power input to each compressor"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The power input to each compressor can be found by\n",
+    "\n",
+    "$$\\dot{W}_{cv} = \\dot{m}(h_i - h_e)$$\n",
+    "\n",
+    "To find the enthalpies, we need to fix all of the states."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "st_1 = State(sub_loT, x=x_1, T=T_1)\n",
+    "h_1 = st_1.h.to(EE.h)\n",
+    "s_1 = st_1.s.to(EE.s)\n",
+    "p_1 = st_1.p.to(EE.p)\n",
+    "\n",
+    "s_2 = s_1\n",
+    "st_2 = State(sub_loT, p=p_2, s=s_1)\n",
+    "h_2 = st_2.h.to(EE.h)\n",
+    "T_2 = st_2.T.to(EE.T)\n",
+    "\n",
+    "st_3 = State(sub_loT, p=p_3, x=x_3)\n",
+    "T_3 = st_3.T.to(EE.T)\n",
+    "h_3 = st_3.h.to(EE.h)\n",
+    "s_3 = st_3.s.to(EE.s)\n",
+    "\n",
+    "h_4 = h_3\n",
+    "p_4 = p_1\n",
+    "st_4 = State(sub_loT, p=p_4, h=h_4)\n",
+    "T_4 = st_4.T.to(EE.T)\n",
+    "s_4 = st_4.s.to(EE.s)\n",
+    "x_4 = st_4.x\n",
+    "\n",
+    "T_5 = T_3 + delta_T_5\n",
+    "st_5 = State(sub_hiT, T=T_5, x=x_5)\n",
+    "h_5 = st_5.h.to(EE.h)\n",
+    "p_5 = st_5.p.to(EE.p)\n",
+    "s_5 = st_5.s.to(EE.s)\n",
+    "\n",
+    "s_6 = s_5\n",
+    "st_6 = State(sub_hiT, s=s_6, p=p_6)\n",
+    "h_6 = st_6.h.to(EE.h)\n",
+    "T_6 = st_6.T.to(EE.T)\n",
+    "\n",
+    "st_7 = State(sub_hiT, p=p_7, x=x_7)\n",
+    "T_7 = st_7.T.to(EE.T)\n",
+    "h_7 = st_7.h.to(EE.h)\n",
+    "s_7 = st_7.s.to(EE.s)\n",
+    "\n",
+    "h_8 = h_7\n",
+    "p_8 = p_5\n",
+    "st_8 = State(sub_hiT, p=p_8, h=h_8)\n",
+    "T_8 = st_8.T.to(EE.T)\n",
+    "s_8 = st_8.s.to(EE.s)\n",
+    "x_8 = st_8.x"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "variables": {
+     "format(T_1, '.2F')": "-30.00 degF",
+     "format(T_2, '.2F')": "58.23 degF",
+     "format(T_3, '.2F')": "40.27 degF",
+     "format(T_4, '.2F')": "-30.00 degF",
+     "format(T_5, '.2F')": "35.27 degF",
+     "format(T_6, '.2F')": "146.75 degF",
+     "format(T_7, '.2F')": "112.76 degF",
+     "format(T_8, '.2F')": "35.27 degF",
+     "format(h_1, '.2F')": "162.30 btu / pound",
+     "format(h_2, '.2F')": "176.42 btu / pound",
+     "format(h_3, '.2F')": "88.65 btu / pound",
+     "format(h_4, '.2F')": "88.65 btu / pound",
+     "format(h_5, '.2F')": "174.42 btu / pound",
+     "format(h_6, '.2F')": "187.12 btu / pound",
+     "format(h_7, '.2F')": "110.14 btu / pound",
+     "format(h_8, '.2F')": "110.14 btu / pound",
+     "format(p_1, '.2F')": "9.86 pound_force_per_square_inch",
+     "format(p_2, '.2F')": "50.00 pound_force_per_square_inch",
+     "format(p_3, '.2F')": "50.00 pound_force_per_square_inch",
+     "format(p_4, '.2F')": "9.86 pound_force_per_square_inch",
+     "format(p_5, '.2F')": "76.59 pound_force_per_square_inch",
+     "format(p_6, '.2F')": "250.00 pound_force_per_square_inch",
+     "format(p_7, '.2F')": "250.00 pound_force_per_square_inch",
+     "format(p_8, '.2F')": "76.59 pound_force_per_square_inch",
+     "format(s_1, '.4F')": "0.4196 btu / degR / pound",
+     "format(s_2, '.4F')": "0.4196 btu / degR / pound",
+     "format(s_3, '.4F')": "0.2442 btu / degR / pound",
+     "format(s_4, '.4F')": "0.2482 btu / degR / pound",
+     "format(s_5, '.4F')": "0.4175 btu / degR / pound",
+     "format(s_6, '.4F')": "0.4175 btu / degR / pound",
+     "format(s_7, '.4F')": "0.2834 btu / degR / pound",
+     "format(s_8, '.4F')": "0.2876 btu / degR / pound",
+     "format(x_1.magnitude, '.2%')": "100.00%",
+     "format(x_3.magnitude, '.2%')": "0.00%",
+     "format(x_4.magnitude, '.2%')": "22.96%",
+     "format(x_5.magnitude, '.2%')": "100.00%",
+     "format(x_7.magnitude, '.2%')": "0.00%",
+     "format(x_8.magnitude, '.2%')": "26.55%",
+     "st_1.phase": "twophase",
+     "st_2.phase": "gas",
+     "st_3.phase": "twophase",
+     "st_4.phase": "twophase",
+     "st_5.phase": "twophase",
+     "st_6.phase": "gas",
+     "st_7.phase": "twophase",
+     "st_8.phase": "twophase"
+    }
+   },
+   "source": [
+    "Summarizing the states:\n",
+    "\n",
+    "\n",
+    "| State |           T            |           p            |           h            |           s            |           x            |     phase      |\n",
+    "|-------|------------------------|------------------------|------------------------|------------------------|------------------------|----------------|\n",
+    "|     1 | -30.00 degF | 9.86 pound_force_per_square_inch | 162.30 btu / pound | 0.4196 btu / degR / pound | 100.00% | twophase |\n",
+    "|     2 | 58.23 degF | 50.00 pound_force_per_square_inch | 176.42 btu / pound | 0.4196 btu / degR / pound |          ---           | gas |\n",
+    "|     3 | 40.27 degF | 50.00 pound_force_per_square_inch | 88.65 btu / pound | 0.2442 btu / degR / pound | 0.00% | twophase |\n",
+    "|     4 | -30.00 degF | 9.86 pound_force_per_square_inch | 88.65 btu / pound | 0.2482 btu / degR / pound | 22.96% | twophase |\n",
+    "|     5 | 35.27 degF | 76.59 pound_force_per_square_inch | 174.42 btu / pound | 0.4175 btu / degR / pound | 100.00% | twophase |\n",
+    "|     6 | 146.75 degF | 250.00 pound_force_per_square_inch | 187.12 btu / pound | 0.4175 btu / degR / pound |          ---           | gas |\n",
+    "|     7 | 112.76 degF | 250.00 pound_force_per_square_inch | 110.14 btu / pound | 0.2834 btu / degR / pound | 0.00% | twophase |\n",
+    "|     8 | 35.27 degF | 76.59 pound_force_per_square_inch | 110.14 btu / pound | 0.2876 btu / degR / pound | 26.55% | twophase |\n",
+    "\n",
+    "The mass flow rate of the R134a is found from the refrigeration capacity\n",
+    "\n",
+    "$$\\dot{m}_{\\text{R134A}} = \\frac{\\dot{Q}_{in}}{h_1 - h_4}$$\n",
+    "\n",
+    "while the mass flow rate of the R22 is found from the intermediate heat exchanger\n",
+    "\n",
+    "$$\\dot{m}_{\\text{R22}} = \\frac{\\dot{m}_{\\text{R134A}}\\left(h_2 - h_3\\right)}{h_5 - h_8}$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "mdot_r134a = (Qdot_in/(h_1 - h_4)).to('lb/min')\n",
+    "Wdot_loT = (mdot_r134a*(h_1 - h_2)).to('BTU/min')\n",
+    "\n",
+    "mdot_r22 = mdot_r134a*(h_2 - h_3)/(h_5 - h_8)\n",
+    "Wdot_hiT = (mdot_r22*(h_5 - h_6)).to('BTU/min')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "variables": {
+     "format(Wdot_hiT, '.2F')": "-941.38 btu / minute",
+     "format(Wdot_loT, '.2F')": "-766.52 btu / minute"
+    }
+   },
+   "source": [
+    "<div class=\"alert alert-success\">\n",
+    "\n",
+    "**Answer:** The compressor work for the low-temperature cycle is $\\dot{W}_{\\text{R134A}} =$ -766.52 btu / minute and the work for the high temperature cycle is $\\dot{W}_{\\text{R22}} =$ -941.38 btu / minute\n",
+    "\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 2. The coefficient of performance"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The coefficient of performance of this refrigeration cycle is defined as\n",
+    "\n",
+    "$$\\beta = \\frac{\\dot{Q}_{in}}{\\lvert\\dot{W}_{\\text{R134A}} + \\dot{W}_{\\text{R22}}\\rvert}$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "beta = (Qdot_in/abs(Wdot_loT + Wdot_hiT)).to('dimensionless')\n",
+    "beta_max = (T_4/(T_7 - T_4)).to('dimensionless')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "variables": {
+     "format(T_4, '.2F')": "-30.00 degF",
+     "format(T_7, '.2F')": "112.76 degF",
+     "format(beta.magnitude, '.2F')": "2.34",
+     "format(beta_max, '.2F')": "3.01 dimensionless"
+    }
+   },
+   "source": [
+    "<div class=\"alert alert-success\">\n",
+    "\n",
+    "**Answer:** The coefficient of performance is $\\beta =$ 2.34, while the maximum coefficient of performance for a refrigeration cycle operating between $T_C =$ -30.00 degF and $T_H =$ 112.76 degF is $\\beta_{\\text{max}} =$ 3.01 dimensionless. The actual cycle has a lower COP, so it is possible.\n",
+    "\n",
+    "</div>"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.0"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

docs/examples.rst

@@ -15,3 +15,4 @@ tables, due to the number of calculations required from the property tables.
    diesel-cycle-example
    rankine-cycle-example
    regen-reheat-rankine-cycle-example
+   cascade-refrigeration-cycle-example

docs/rankine-cycle-example.ipynb

@@ -197,7 +197,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Summarizing the state:\n",
+    "Summarizing the states:\n",
     "\n",
     "| State |               T                |               p                |               h                |               s                |               x                |     phase      |\n",
     "|-------|--------------------------------|--------------------------------|--------------------------------|--------------------------------|--------------------------------|----------------|\n",