{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 4.6 – Steam Tables\n",
"\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4.6.0 – Learning Objectives\n",
"\n",
"By the end of this section you should be able to:\n",
"\n",
"1. Understand what steam tables tell us.\n",
"2. Use steam tables to solve thermodynamic problems.\n",
"3. Interpolate between data points.\n",
"\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4.6.1 – Introduction\n",
"\n",
"Water, and in specific, steam is used in many processes. Most commonly, it is used in the transfer of energy. Because of this, scientist and engineers have created extensive data tables on a variety of conditions.\n",
"\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4.6.2 – Table Description\n",
"\n",
"Shown below is a saturated steam table. A **saturated steam table** will include data of both the liquid and vapour phases of water at a given temperature and pressure."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"text/html": [
"
\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" Temperature (C) | \n",
" Pressure (MPa) | \n",
" Volume (l, m3/kg) | \n",
" Volume (v, m3/kg) | \n",
" Internal Energy (l, kJ/kg) | \n",
" Δ Internal Energy of Vapourization (kJ/kg) | \n",
" Internal Energy (v, kJ/kg) | \n",
" Enthalpy (l, kJ/kg) | \n",
" Δ Enthalpy of Vapourization (kJ/kg) | \n",
" Enthalpy (v, kJ/kg) | \n",
" Entropy (l, J/g*K) | \n",
" Δ Entropy of Vapourization (kJ/kg) | \n",
" Entropy (v, J/g*K) | \n",
"
\n",
" \n",
" \n",
" \n",
" 0 | \n",
" 0.01 | \n",
" 0.000612 | \n",
" 0.001000 | \n",
" 205.9900 | \n",
" 0.000 | \n",
" 2374.900 | \n",
" 2374.9 | \n",
" 0.000 | \n",
" 2500.900 | \n",
" 2500.9 | \n",
" 0.000000 | \n",
" 9.155500 | \n",
" 9.1555 | \n",
"
\n",
" \n",
" 1 | \n",
" 5.00 | \n",
" 0.000873 | \n",
" 0.001000 | \n",
" 147.0100 | \n",
" 21.019 | \n",
" 2360.781 | \n",
" 2381.8 | \n",
" 21.020 | \n",
" 2489.080 | \n",
" 2510.1 | \n",
" 0.076254 | \n",
" 8.948546 | \n",
" 9.0248 | \n",
"
\n",
" \n",
" 2 | \n",
" 10.00 | \n",
" 0.001228 | \n",
" 0.001000 | \n",
" 106.3000 | \n",
" 42.020 | \n",
" 2346.580 | \n",
" 2388.6 | \n",
" 42.021 | \n",
" 2477.179 | \n",
" 2519.2 | \n",
" 0.151090 | \n",
" 8.748710 | \n",
" 8.8998 | \n",
"
\n",
" \n",
" 3 | \n",
" 15.00 | \n",
" 0.001706 | \n",
" 0.001001 | \n",
" 77.8750 | \n",
" 62.980 | \n",
" 2332.520 | \n",
" 2395.5 | \n",
" 62.981 | \n",
" 2465.319 | \n",
" 2528.3 | \n",
" 0.224460 | \n",
" 8.555840 | \n",
" 8.7803 | \n",
"
\n",
" \n",
" 4 | \n",
" 20.00 | \n",
" 0.002339 | \n",
" 0.001002 | \n",
" 57.7570 | \n",
" 83.912 | \n",
" 2318.388 | \n",
" 2402.3 | \n",
" 83.914 | \n",
" 2453.486 | \n",
" 2537.4 | \n",
" 0.296480 | \n",
" 8.369520 | \n",
" 8.6660 | \n",
"
\n",
" \n",
" 5 | \n",
" 25.00 | \n",
" 0.003170 | \n",
" 0.001003 | \n",
" 43.3370 | \n",
" 104.830 | \n",
" 2304.270 | \n",
" 2409.1 | \n",
" 104.830 | \n",
" 2441.670 | \n",
" 2546.5 | \n",
" 0.367220 | \n",
" 8.189380 | \n",
" 8.5566 | \n",
"
\n",
" \n",
" 6 | \n",
" 30.00 | \n",
" 0.004247 | \n",
" 0.001004 | \n",
" 32.8780 | \n",
" 125.730 | \n",
" 2290.170 | \n",
" 2415.9 | \n",
" 125.730 | \n",
" 2429.770 | \n",
" 2555.5 | \n",
" 0.436750 | \n",
" 8.015250 | \n",
" 8.4520 | \n",
"
\n",
" \n",
" 7 | \n",
" 35.00 | \n",
" 0.005629 | \n",
" 0.001006 | \n",
" 25.2050 | \n",
" 146.630 | \n",
" 2276.070 | \n",
" 2422.7 | \n",
" 146.630 | \n",
" 2417.870 | \n",
" 2564.5 | \n",
" 0.505130 | \n",
" 7.846570 | \n",
" 8.3517 | \n",
"
\n",
" \n",
" 8 | \n",
" 40.00 | \n",
" 0.007385 | \n",
" 0.001008 | \n",
" 19.5150 | \n",
" 167.530 | \n",
" 2261.870 | \n",
" 2429.4 | \n",
" 167.530 | \n",
" 2405.970 | \n",
" 2573.5 | \n",
" 0.572400 | \n",
" 7.683100 | \n",
" 8.2555 | \n",
"
\n",
" \n",
" 9 | \n",
" 45.00 | \n",
" 0.009595 | \n",
" 0.001010 | \n",
" 15.2520 | \n",
" 188.430 | \n",
" 2247.670 | \n",
" 2436.1 | \n",
" 188.430 | \n",
" 2393.970 | \n",
" 2582.4 | \n",
" 0.638610 | \n",
" 7.524690 | \n",
" 8.1633 | \n",
"
\n",
" \n",
" 10 | \n",
" 50.00 | \n",
" 0.012352 | \n",
" 0.001012 | \n",
" 12.0270 | \n",
" 209.330 | \n",
" 2233.370 | \n",
" 2442.7 | \n",
" 209.340 | \n",
" 2381.960 | \n",
" 2591.3 | \n",
" 0.703810 | \n",
" 7.370990 | \n",
" 8.0748 | \n",
"
\n",
" \n",
" 11 | \n",
" 55.00 | \n",
" 0.015762 | \n",
" 0.001015 | \n",
" 9.5643 | \n",
" 230.240 | \n",
" 2219.060 | \n",
" 2449.3 | \n",
" 230.260 | \n",
" 2369.840 | \n",
" 2600.1 | \n",
" 0.768020 | \n",
" 7.221780 | \n",
" 7.9898 | \n",
"
\n",
" \n",
" 12 | \n",
" 60.00 | \n",
" 0.019946 | \n",
" 0.001017 | \n",
" 7.6672 | \n",
" 251.160 | \n",
" 2204.740 | \n",
" 2455.9 | \n",
" 251.180 | \n",
" 2357.620 | \n",
" 2608.8 | \n",
" 0.831290 | \n",
" 7.076810 | \n",
" 7.9081 | \n",
"
\n",
" \n",
" 13 | \n",
" 65.00 | \n",
" 0.025042 | \n",
" 0.001020 | \n",
" 6.1935 | \n",
" 272.090 | \n",
" 2190.310 | \n",
" 2462.4 | \n",
" 272.120 | \n",
" 2345.380 | \n",
" 2617.5 | \n",
" 0.893650 | \n",
" 6.935950 | \n",
" 7.8296 | \n",
"
\n",
" \n",
" 14 | \n",
" 70.00 | \n",
" 0.031201 | \n",
" 0.001023 | \n",
" 5.0395 | \n",
" 293.030 | \n",
" 2175.870 | \n",
" 2468.9 | \n",
" 293.070 | \n",
" 2333.030 | \n",
" 2626.1 | \n",
" 0.955130 | \n",
" 6.798870 | \n",
" 7.7540 | \n",
"
\n",
" \n",
" 15 | \n",
" 75.00 | \n",
" 0.038595 | \n",
" 0.001026 | \n",
" 4.1289 | \n",
" 313.990 | \n",
" 2161.210 | \n",
" 2475.2 | \n",
" 314.030 | \n",
" 2320.570 | \n",
" 2634.6 | \n",
" 1.015800 | \n",
" 6.665400 | \n",
" 7.6812 | \n",
"
\n",
" \n",
" 16 | \n",
" 80.00 | \n",
" 0.047414 | \n",
" 0.001029 | \n",
" 3.4052 | \n",
" 334.960 | \n",
" 2146.640 | \n",
" 2481.6 | \n",
" 335.010 | \n",
" 2307.990 | \n",
" 2643.0 | \n",
" 1.075600 | \n",
" 6.535500 | \n",
" 7.6111 | \n",
"
\n",
" \n",
" 17 | \n",
" 85.00 | \n",
" 0.057867 | \n",
" 0.001032 | \n",
" 2.8258 | \n",
" 355.950 | \n",
" 2131.850 | \n",
" 2487.8 | \n",
" 356.010 | \n",
" 2295.290 | \n",
" 2651.3 | \n",
" 1.134600 | \n",
" 6.408800 | \n",
" 7.5434 | \n",
"
\n",
" \n",
" 18 | \n",
" 90.00 | \n",
" 0.070182 | \n",
" 0.001036 | \n",
" 2.3591 | \n",
" 376.970 | \n",
" 2117.030 | \n",
" 2494.0 | \n",
" 377.040 | \n",
" 2282.460 | \n",
" 2659.5 | \n",
" 1.192900 | \n",
" 6.285200 | \n",
" 7.4781 | \n",
"
\n",
" \n",
" 19 | \n",
" 95.00 | \n",
" 0.084608 | \n",
" 0.001040 | \n",
" 1.9806 | \n",
" 398.000 | \n",
" 2102.000 | \n",
" 2500.0 | \n",
" 398.090 | \n",
" 2269.510 | \n",
" 2667.6 | \n",
" 1.250400 | \n",
" 6.164700 | \n",
" 7.4151 | \n",
"
\n",
" \n",
"
\n",
"
"
],
"text/plain": [
" Temperature (C) Pressure (MPa) Volume (l, m3/kg) Volume (v, m3/kg) \\\n",
"0 0.01 0.000612 0.001000 205.9900 \n",
"1 5.00 0.000873 0.001000 147.0100 \n",
"2 10.00 0.001228 0.001000 106.3000 \n",
"3 15.00 0.001706 0.001001 77.8750 \n",
"4 20.00 0.002339 0.001002 57.7570 \n",
"5 25.00 0.003170 0.001003 43.3370 \n",
"6 30.00 0.004247 0.001004 32.8780 \n",
"7 35.00 0.005629 0.001006 25.2050 \n",
"8 40.00 0.007385 0.001008 19.5150 \n",
"9 45.00 0.009595 0.001010 15.2520 \n",
"10 50.00 0.012352 0.001012 12.0270 \n",
"11 55.00 0.015762 0.001015 9.5643 \n",
"12 60.00 0.019946 0.001017 7.6672 \n",
"13 65.00 0.025042 0.001020 6.1935 \n",
"14 70.00 0.031201 0.001023 5.0395 \n",
"15 75.00 0.038595 0.001026 4.1289 \n",
"16 80.00 0.047414 0.001029 3.4052 \n",
"17 85.00 0.057867 0.001032 2.8258 \n",
"18 90.00 0.070182 0.001036 2.3591 \n",
"19 95.00 0.084608 0.001040 1.9806 \n",
"\n",
" Internal Energy (l, kJ/kg) Δ Internal Energy of Vapourization (kJ/kg) \\\n",
"0 0.000 2374.900 \n",
"1 21.019 2360.781 \n",
"2 42.020 2346.580 \n",
"3 62.980 2332.520 \n",
"4 83.912 2318.388 \n",
"5 104.830 2304.270 \n",
"6 125.730 2290.170 \n",
"7 146.630 2276.070 \n",
"8 167.530 2261.870 \n",
"9 188.430 2247.670 \n",
"10 209.330 2233.370 \n",
"11 230.240 2219.060 \n",
"12 251.160 2204.740 \n",
"13 272.090 2190.310 \n",
"14 293.030 2175.870 \n",
"15 313.990 2161.210 \n",
"16 334.960 2146.640 \n",
"17 355.950 2131.850 \n",
"18 376.970 2117.030 \n",
"19 398.000 2102.000 \n",
"\n",
" Internal Energy (v, kJ/kg) Enthalpy (l, kJ/kg) \\\n",
"0 2374.9 0.000 \n",
"1 2381.8 21.020 \n",
"2 2388.6 42.021 \n",
"3 2395.5 62.981 \n",
"4 2402.3 83.914 \n",
"5 2409.1 104.830 \n",
"6 2415.9 125.730 \n",
"7 2422.7 146.630 \n",
"8 2429.4 167.530 \n",
"9 2436.1 188.430 \n",
"10 2442.7 209.340 \n",
"11 2449.3 230.260 \n",
"12 2455.9 251.180 \n",
"13 2462.4 272.120 \n",
"14 2468.9 293.070 \n",
"15 2475.2 314.030 \n",
"16 2481.6 335.010 \n",
"17 2487.8 356.010 \n",
"18 2494.0 377.040 \n",
"19 2500.0 398.090 \n",
"\n",
" Δ Enthalpy of Vapourization (kJ/kg) Enthalpy (v, kJ/kg) \\\n",
"0 2500.900 2500.9 \n",
"1 2489.080 2510.1 \n",
"2 2477.179 2519.2 \n",
"3 2465.319 2528.3 \n",
"4 2453.486 2537.4 \n",
"5 2441.670 2546.5 \n",
"6 2429.770 2555.5 \n",
"7 2417.870 2564.5 \n",
"8 2405.970 2573.5 \n",
"9 2393.970 2582.4 \n",
"10 2381.960 2591.3 \n",
"11 2369.840 2600.1 \n",
"12 2357.620 2608.8 \n",
"13 2345.380 2617.5 \n",
"14 2333.030 2626.1 \n",
"15 2320.570 2634.6 \n",
"16 2307.990 2643.0 \n",
"17 2295.290 2651.3 \n",
"18 2282.460 2659.5 \n",
"19 2269.510 2667.6 \n",
"\n",
" Entropy (l, J/g*K) Δ Entropy of Vapourization (kJ/kg) Entropy (v, J/g*K) \n",
"0 0.000000 9.155500 9.1555 \n",
"1 0.076254 8.948546 9.0248 \n",
"2 0.151090 8.748710 8.8998 \n",
"3 0.224460 8.555840 8.7803 \n",
"4 0.296480 8.369520 8.6660 \n",
"5 0.367220 8.189380 8.5566 \n",
"6 0.436750 8.015250 8.4520 \n",
"7 0.505130 7.846570 8.3517 \n",
"8 0.572400 7.683100 8.2555 \n",
"9 0.638610 7.524690 8.1633 \n",
"10 0.703810 7.370990 8.0748 \n",
"11 0.768020 7.221780 7.9898 \n",
"12 0.831290 7.076810 7.9081 \n",
"13 0.893650 6.935950 7.8296 \n",
"14 0.955130 6.798870 7.7540 \n",
"15 1.015800 6.665400 7.6812 \n",
"16 1.075600 6.535500 7.6111 \n",
"17 1.134600 6.408800 7.5434 \n",
"18 1.192900 6.285200 7.4781 \n",
"19 1.250400 6.164700 7.4151 "
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import pandas as pd\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"\n",
"df = pd.read_excel('../figures/Module-4/SatTandPofSteam.xlsx', sheet_name='Sheet2', index_col=None, na_values=['NA'])\n",
"df.head(20)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Another type of table is a super heated steam table at a given pressure. Here is an example of this type table."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
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" \n",
" \n",
" | \n",
" Temperature (C) | \n",
" Pressure (MPa) | \n",
" Volume (m3/kg) | \n",
" Internal Energy (kJ/kg) | \n",
" Enthalpy (kJ/kg) | \n",
" Entropy (J/g*K) | \n",
"
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" \n",
" \n",
" \n",
" 0 | \n",
" 233.85 | \n",
" 3 | \n",
" 0.066664 | \n",
" 2603.2 | \n",
" 2803.2 | \n",
" 6.1856 | \n",
"
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" 1 | \n",
" 250.00 | \n",
" 3 | \n",
" 0.070627 | \n",
" 2644.7 | \n",
" 2856.5 | \n",
" 6.2893 | \n",
"
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" 2 | \n",
" 300.00 | \n",
" 3 | \n",
" 0.081179 | \n",
" 2750.8 | \n",
" 2994.3 | \n",
" 6.5412 | \n",
"
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" 3 | \n",
" 350.00 | \n",
" 3 | \n",
" 0.090556 | \n",
" 2844.4 | \n",
" 3116.1 | \n",
" 6.7449 | \n",
"
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" 400.00 | \n",
" 3 | \n",
" 0.099379 | \n",
" 2933.5 | \n",
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" 6.9234 | \n",
"
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" 5 | \n",
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" 3 | \n",
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" 7.0856 | \n",
"
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" 6 | \n",
" 500.00 | \n",
" 3 | \n",
" 0.116200 | \n",
" 3108.6 | \n",
" 3457.2 | \n",
" 7.2359 | \n",
"
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" 7 | \n",
" 550.00 | \n",
" 3 | \n",
" 0.124370 | \n",
" 3196.6 | \n",
" 3569.7 | \n",
" 7.3768 | \n",
"
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" 8 | \n",
" 600.00 | \n",
" 3 | \n",
" 0.132450 | \n",
" 3285.5 | \n",
" 3682.8 | \n",
" 7.5103 | \n",
"
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" 9 | \n",
" 650.00 | \n",
" 3 | \n",
" 0.140450 | \n",
" 3375.6 | \n",
" 3796.9 | \n",
" 7.6373 | \n",
"
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" \n",
" 10 | \n",
" 700.00 | \n",
" 3 | \n",
" 0.148410 | \n",
" 3467.0 | \n",
" 3912.2 | \n",
" 7.7590 | \n",
"
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" 11 | \n",
" 750.00 | \n",
" 3 | \n",
" 0.156320 | \n",
" 3559.9 | \n",
" 4028.9 | \n",
" 7.8758 | \n",
"
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" 12 | \n",
" 800.00 | \n",
" 3 | \n",
" 0.164200 | \n",
" 3654.3 | \n",
" 4146.9 | \n",
" 7.9885 | \n",
"
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" 13 | \n",
" 850.00 | \n",
" 3 | \n",
" 0.172050 | \n",
" 3750.3 | \n",
" 4266.5 | \n",
" 8.0973 | \n",
"
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" 14 | \n",
" 900.00 | \n",
" 3 | \n",
" 0.179880 | \n",
" 3847.9 | \n",
" 4387.5 | \n",
" 8.2028 | \n",
"
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" \n",
" 15 | \n",
" 950.00 | \n",
" 3 | \n",
" 0.187690 | \n",
" 3947.0 | \n",
" 4510.1 | \n",
" 8.3051 | \n",
"
\n",
" \n",
" 16 | \n",
" 1000.00 | \n",
" 3 | \n",
" 0.195490 | \n",
" 4047.7 | \n",
" 4634.1 | \n",
" 8.4045 | \n",
"
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" \n",
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"
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],
"text/plain": [
" Temperature (C) Pressure (MPa) Volume (m3/kg) Internal Energy (kJ/kg) \\\n",
"0 233.85 3 0.066664 2603.2 \n",
"1 250.00 3 0.070627 2644.7 \n",
"2 300.00 3 0.081179 2750.8 \n",
"3 350.00 3 0.090556 2844.4 \n",
"4 400.00 3 0.099379 2933.5 \n",
"5 450.00 3 0.107890 3021.2 \n",
"6 500.00 3 0.116200 3108.6 \n",
"7 550.00 3 0.124370 3196.6 \n",
"8 600.00 3 0.132450 3285.5 \n",
"9 650.00 3 0.140450 3375.6 \n",
"10 700.00 3 0.148410 3467.0 \n",
"11 750.00 3 0.156320 3559.9 \n",
"12 800.00 3 0.164200 3654.3 \n",
"13 850.00 3 0.172050 3750.3 \n",
"14 900.00 3 0.179880 3847.9 \n",
"15 950.00 3 0.187690 3947.0 \n",
"16 1000.00 3 0.195490 4047.7 \n",
"\n",
" Enthalpy (kJ/kg) Entropy (J/g*K) \n",
"0 2803.2 6.1856 \n",
"1 2856.5 6.2893 \n",
"2 2994.3 6.5412 \n",
"3 3116.1 6.7449 \n",
"4 3231.7 6.9234 \n",
"5 3344.8 7.0856 \n",
"6 3457.2 7.2359 \n",
"7 3569.7 7.3768 \n",
"8 3682.8 7.5103 \n",
"9 3796.9 7.6373 \n",
"10 3912.2 7.7590 \n",
"11 4028.9 7.8758 \n",
"12 4146.9 7.9885 \n",
"13 4266.5 8.0973 \n",
"14 4387.5 8.2028 \n",
"15 4510.1 8.3051 \n",
"16 4634.1 8.4045 "
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import pandas as pd\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"\n",
"df = pd.read_excel('../figures/Module-4/Steam_3MPa.xlsx', sheet_name='Sheet2', index_col=None, na_values=['NA'])\n",
"df"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"---\n",
"## 4.6.3 – How to Use Steam Tables\n",
"\n",
"Steam tables are typically used to calculate the change in energy in a specific process, whether it be energy generation or cooling. By measuring the changes in pressure or temperature of steam, you can calculate changes in enthalpy or internal energy. For example, if you have 1 kg of steam at $800.00 ^{\\circ} C$ and 3.0000 MPa and you cool it to $600.00 ^{\\circ} C$ at a constant pressure you change in enthalpy $\\Delta H = H_2 - H_1 =(3682.8 - 4146.9) \\space kJ/kg = -464.1 \\space kJ/kg$. \n",
"\n",
"Be aware of keywords such as superheated steam, superheated liquid, and saturated steam.\n",
"__Saturated steam__ is the least amount of heat required to bring water to steam at a certain pressure. Concurrently, there will be a set temperature associated with it as well. __Superheated steam__ is saturated steam that has much more heat added, raising the temperature and enthalpy considerably. __Superheated liquid__ is water under high pressure being heated. \n",
"\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4.6.4 – Interpolation\n",
"\n",
"Let's suppose you wanted to find out what the internal energy of steam is at $725.00 ^{\\circ} C$ and $3.0000 \\space \\text{MPa}$. Since the value is not given in the table, you must interpolate. The formula for interpolation is: \n",
"\n",
"$$ y = \\frac{y_2 - y_1}{x_2 - x_1} (x - x_1) + y_1 $$\n",
"\n",
"In this case the formula for internal energy at $725.00 ^{\\circ} C$ is:\n",
"\n",
"$$ U_{3 MPa, 725 C} = \\frac{U_{3 MPa, 750 C} - U_{3 MPa, 700 C}}{T_2 - T_1} (T - T_1) + U_{3 MPa, 700 C} $$\n",
"\n",
"and the answer is:\n",
"\n",
"$$ U_{3 MPa, 725 C} = \\frac{(3559.9 - 3467.0 ) \\space kJ/kg}{(750 - 700) \\space C} (725 - 700) \\space C + 3467.0 \\space kJ/kg = 3513.45 \\space kJ/kg $$\n",
"\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4.6.5 – Problem Statement\n",
"\n",
"### Question\n",
"Using the steam tables below, find the change in internal energy when steam is first cooled isobarically at $750.00 ^{\\circ} C$ and $5.0000 \\space \\text{MPa}$ to $725.00 ^{\\circ} C \\space$ and $5.0000 \\space \\text{MPa}$ and then expanded isothermally at $725.00 ^{\\circ} C$ from $5.0000 \\space \\text{MPa}$ to $1.0000 \\space \\text{MPa}$."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
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"\n",
"\n",
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\n",
" \n",
" \n",
" | \n",
" Temperature (C) | \n",
" Pressure (MPa) | \n",
" Volume (m3/kg) | \n",
" Internal Energy (kJ/kg) | \n",
" Enthalpy (kJ/kg) | \n",
" Entropy (J/g*K) | \n",
"
\n",
" \n",
" \n",
" \n",
" 0 | \n",
" 263.94 | \n",
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" 6.4516 | \n",
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\n",
" \n",
" 3 | \n",
" 400.00 | \n",
" 5 | \n",
" 0.057837 | \n",
" 2907.5 | \n",
" 3196.7 | \n",
" 6.6483 | \n",
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\n",
" \n",
" 4 | \n",
" 450.00 | \n",
" 5 | \n",
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" 3000.6 | \n",
" 3317.2 | \n",
" 6.8210 | \n",
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\n",
" \n",
" 5 | \n",
" 500.00 | \n",
" 5 | \n",
" 0.068583 | \n",
" 3091.7 | \n",
" 3434.7 | \n",
" 6.9781 | \n",
"
\n",
" \n",
" 6 | \n",
" 550.00 | \n",
" 5 | \n",
" 0.073694 | \n",
" 3182.4 | \n",
" 3550.9 | \n",
" 7.1237 | \n",
"
\n",
" \n",
" 7 | \n",
" 600.00 | \n",
" 5 | \n",
" 0.078704 | \n",
" 3273.3 | \n",
" 3666.8 | \n",
" 7.2605 | \n",
"
\n",
" \n",
" 8 | \n",
" 650.00 | \n",
" 5 | \n",
" 0.083639 | \n",
" 3365.0 | \n",
" 3783.2 | \n",
" 7.3901 | \n",
"
\n",
" \n",
" 9 | \n",
" 700.00 | \n",
" 5 | \n",
" 0.088518 | \n",
" 3457.7 | \n",
" 3900.3 | \n",
" 7.5136 | \n",
"
\n",
" \n",
" 10 | \n",
" 750.00 | \n",
" 5 | \n",
" 0.093355 | \n",
" 3551.6 | \n",
" 4018.4 | \n",
" 7.6320 | \n",
"
\n",
" \n",
" 11 | \n",
" 800.00 | \n",
" 5 | \n",
" 0.098158 | \n",
" 3646.9 | \n",
" 4137.7 | \n",
" 7.7458 | \n",
"
\n",
" \n",
" 12 | \n",
" 850.00 | \n",
" 5 | \n",
" 0.102930 | \n",
" 3743.6 | \n",
" 4258.3 | \n",
" 7.8556 | \n",
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" \n",
" 13 | \n",
" 900.00 | \n",
" 5 | \n",
" 0.107690 | \n",
" 3841.8 | \n",
" 4380.2 | \n",
" 7.9618 | \n",
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" 14 | \n",
" 950.00 | \n",
" 5 | \n",
" 0.112420 | \n",
" 3941.5 | \n",
" 4503.6 | \n",
" 8.0648 | \n",
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" \n",
" 15 | \n",
" 1000.00 | \n",
" 5 | \n",
" 0.117150 | \n",
" 4042.6 | \n",
" 4628.3 | \n",
" 8.1648 | \n",
"
\n",
" \n",
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" Temperature (C) Pressure (MPa) Volume (m3/kg) Internal Energy (kJ/kg) \\\n",
"0 263.94 5 0.039446 2597.0 \n",
"1 300.00 5 0.045346 2699.0 \n",
"2 350.00 5 0.051969 2809.5 \n",
"3 400.00 5 0.057837 2907.5 \n",
"4 450.00 5 0.063323 3000.6 \n",
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"6 550.00 5 0.073694 3182.4 \n",
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"8 650.00 5 0.083639 3365.0 \n",
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"12 850.00 5 0.102930 3743.6 \n",
"13 900.00 5 0.107690 3841.8 \n",
"14 950.00 5 0.112420 3941.5 \n",
"15 1000.00 5 0.117150 4042.6 \n",
"\n",
" Enthalpy (kJ/kg) Entropy (J/g*K) \n",
"0 2794.2 5.9737 \n",
"1 2925.7 6.2110 \n",
"2 3069.3 6.4516 \n",
"3 3196.7 6.6483 \n",
"4 3317.2 6.8210 \n",
"5 3434.7 6.9781 \n",
"6 3550.9 7.1237 \n",
"7 3666.8 7.2605 \n",
"8 3783.2 7.3901 \n",
"9 3900.3 7.5136 \n",
"10 4018.4 7.6320 \n",
"11 4137.7 7.7458 \n",
"12 4258.3 7.8556 \n",
"13 4380.2 7.9618 \n",
"14 4503.6 8.0648 \n",
"15 4628.3 8.1648 "
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},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import pandas as pd\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"\n",
"df = pd.read_excel('../figures/Module-4/Steam_5MPa.xlsx', sheet_name='Sheet2', index_col=None, na_values=['NA'])\n",
"df"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
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"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" Temperature (C) | \n",
" Pressure (MPa) | \n",
" Volume (m3/kg) | \n",
" Internal Energy (kJ/kg) | \n",
" Enthalpy (kJ/kg) | \n",
" Entropy (J/g*K) | \n",
"
\n",
" \n",
" \n",
" \n",
" 0 | \n",
" 179.88 | \n",
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" 1 | \n",
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" 7.1246 | \n",
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" 4 | \n",
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" 6 | \n",
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" 1 | \n",
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" \n",
" 7 | \n",
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" 1 | \n",
" 0.35411 | \n",
" 3125.0 | \n",
" 3479.1 | \n",
" 7.7641 | \n",
"
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" \n",
" 8 | \n",
" 550.00 | \n",
" 1 | \n",
" 0.37766 | \n",
" 3210.5 | \n",
" 3588.1 | \n",
" 7.9008 | \n",
"
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" 9 | \n",
" 600.00 | \n",
" 1 | \n",
" 0.40111 | \n",
" 3297.5 | \n",
" 3698.6 | \n",
" 8.0310 | \n",
"
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" 10 | \n",
" 650.00 | \n",
" 1 | \n",
" 0.42449 | \n",
" 3386.0 | \n",
" 3810.5 | \n",
" 8.1557 | \n",
"
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" \n",
" 11 | \n",
" 700.00 | \n",
" 1 | \n",
" 0.44783 | \n",
" 3476.2 | \n",
" 3924.1 | \n",
" 8.2755 | \n",
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\n",
" \n",
" 12 | \n",
" 750.00 | \n",
" 1 | \n",
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" 3568.1 | \n",
" 4039.3 | \n",
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" \n",
" 13 | \n",
" 800.00 | \n",
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" 8.5024 | \n",
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" 14 | \n",
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" 1 | \n",
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" 3757.0 | \n",
" 4274.6 | \n",
" 8.6103 | \n",
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" 15 | \n",
" 900.00 | \n",
" 1 | \n",
" 0.54083 | \n",
" 3853.9 | \n",
" 4394.8 | \n",
" 8.7150 | \n",
"
\n",
" \n",
" 16 | \n",
" 950.00 | \n",
" 1 | \n",
" 0.56403 | \n",
" 3952.5 | \n",
" 4516.5 | \n",
" 8.8166 | \n",
"
\n",
" \n",
" 17 | \n",
" 1000.00 | \n",
" 1 | \n",
" 0.58721 | \n",
" 4052.7 | \n",
" 4639.9 | \n",
" 8.9155 | \n",
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\n",
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" Temperature (C) Pressure (MPa) Volume (m3/kg) Internal Energy (kJ/kg) \\\n",
"0 179.88 1 0.19436 2582.7 \n",
"1 200.00 1 0.20602 2622.2 \n",
"2 250.00 1 0.23275 2710.4 \n",
"3 300.00 1 0.25799 2793.6 \n",
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"16 950.00 1 0.56403 3952.5 \n",
"17 1000.00 1 0.58721 4052.7 \n",
"\n",
" Enthalpy (kJ/kg) Entropy (J/g*K) \n",
"0 2777.1 6.5850 \n",
"1 2828.3 6.6955 \n",
"2 2943.1 6.9265 \n",
"3 3051.6 7.1246 \n",
"4 3158.2 7.3029 \n",
"5 3264.5 7.4669 \n",
"6 3371.3 7.6200 \n",
"7 3479.1 7.7641 \n",
"8 3588.1 7.9008 \n",
"9 3698.6 8.0310 \n",
"10 3810.5 8.1557 \n",
"11 3924.1 8.2755 \n",
"12 4039.3 8.3909 \n",
"13 4156.1 8.5024 \n",
"14 4274.6 8.6103 \n",
"15 4394.8 8.7150 \n",
"16 4516.5 8.8166 \n",
"17 4639.9 8.9155 "
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import pandas as pd\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"\n",
"df = pd.read_excel('../figures/Module-4/Steam_1MPa.xlsx', sheet_name='Sheet2', index_col=None, na_values=['NA'])\n",
"df"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"### Answer\n",
"\n",
"The internal energy at $750.00 ^{\\circ} C$ and $5.0000 \\space \\text{MPa}$ is\n",
"\n",
"$$ U_{5 MPa, 750 C} = 3511.6 \\space kJ/kg $$\n",
"\n",
"Since $725.00 ^{\\circ} C$ is not present in the first table, we must interpolate\n",
"\n",
"$$ U_{5 MPa, 725 C} = \\frac{U_{5 MPa, 750 C} - U_{5 MPa, 700 C}}{T_2 - T_1} (T - T_1) + U_{5 MPa, 700 C} $$\n",
"\n",
"$$ U_{5 MPa, 725 C} = \\frac{(3511.6 - 3457.7 ) \\space kJ/kg}{(750 - 700) \\space C} (725 - 700) \\space C + 3457.7 \\space kJ/kg $$\n",
"\n",
"$$ U_{5 MPa, 725 C} = 3,484.65 \\space kJ/kg $$\n",
"\n",
"now that we have $U_{5 MPa, 725 C}$, we must find $U_{1 MPa, 725 C}$ using interpolation\n",
"\n",
"$$ U_{1 MPa, 725 C} = \\frac{U_{1 MPa, 750 C} - U_{1 MPa, 700 C}}{T_2 - T_1} (T - T_1) + U_{1 MPa, 700 C} $$\n",
"\n",
"$$ U_{1 MPa, 725 C} = \\frac{(3568.1 - 3476.2 ) \\space kJ/kg}{(750 - 700) \\space C} (725 - 700) \\space C + 3476.2 \\space kJ/kg $$\n",
"\n",
"$$ U_{1 MPa, 725 C} = 3,522.15 \\space kJ/kg $$\n",
"\n",
"Now that we have all the necessary values, we just need to sum the changes in internal energy\n",
"\n",
"$$ \\Delta U_{tot} = \\Delta U_{1} + \\Delta U_{2} = (U_{5 MPa, 725 C} - U_{5 MPa, 750 C}) + (U_{1 MPa, 725 C} - U_{5 MPa, 725 C}) $$\n",
"\n",
"$$ \\Delta U_{tot} = (3,484.65 \\space kJ/kg - \t3551.6 \\space kJ/kg) + (3,522.15 \\space kJ/kg - 3,484.65 \\space kJ/kg) $$\n",
"\n",
"$$ \\Delta U_{tot} = -10.6 \\space kJ/kg $$"
]
},
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