{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Cascade Refrigeration Cycle Example\n", "\n", "## Imports" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from thermostate import State, Q_, units, EnglishEngineering as EE\n", "from thermostate.plotting import IdealGas, VaporDome" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "---\n", "\n", "## Definitions" ] }, { "cell_type": "code", "execution_count": 2, "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.2 Saturated liquid leaves the intermediate heat exchanger at 50 lbf/in.2 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.2 Saturated liquid then enters the expansion valve at 250 lbf/in.2. 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.\n", "\n", "The cascade refrigeration cycle is consistes of two smaller cycles both with 4 processes:\n", "\n", " R22 cycle:\n", " 1. Isentropic compression\n", " 2. Isobaric heat exchange\n", " 3. Isoenthalpic expansion\n", " 4. Isobaric heat exchange\n", " \n", " R134A cycle:\n", " 1. Isentropic compression\n", " 2. Isobaric heat exchange\n", " 3. Isoenthalpic expansion\n", " 4. Isobaric heat exchange\n", " \n", "The following properties are used to fix the four states:\n", "\n", "State | Property 1 | Property 2 \n", ":-----:|:-----:|:-----:\n", "1|$$x_1 $$|$$T_1 $$\n", "2|$$p_2 $$|$$s_2=s_1 $$\n", "3|$$p_3=p_2 $$|$$x_3 $$\n", "4|$$p_4=p_1 $$|$$h_4=h_3 $$\n", "5|$$T_5=T_3 + \\Delta T_5 $$|$$x_5 $$\n", "6|$$p_6 $$|$$s_6=s_5 $$\n", "7|$$p_7=p_6 $$|$$x_7 $$\n", "8|$$p_8=p_5 $$|$$h_8=h_7 $$" ] }, { "cell_type": "code", "execution_count": 3, "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": {}, "source": [ "Plotting the T-s diagrams of both cycles," ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "R22_vapordome = VaporDome(sub_hiT, (\"s\", \"T\"))\n", "R134A_vapordome = VaporDome(sub_loT, (\"s\", \"T\"))\n", "\n", "R22_vapordome.add_process(st_1, st_2, \"isentropic\")\n", "R22_vapordome.add_process(st_2, st_3, \"isobaric\")\n", "R22_vapordome.add_process(st_3, st_4, \"isoenthalpic\")\n", "R22_vapordome.add_process(st_4, st_1, \"isobaric\")\n", "\n", "R134A_vapordome.add_process(st_5, st_6, \"isentropic\")\n", "R134A_vapordome.add_process(st_6, st_7, \"isobaric\")\n", "R134A_vapordome.add_process(st_7, st_8, \"isoenthalpic\")\n", "R134A_vapordome.add_process(st_8, st_5, \"isobaric\")" ] }, { "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": 5, "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": [ "
\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", "
" ] }, { "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": 6, "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": [ "
\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", "
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