Cross Flow Heat Exchanger

Cross-Flow Heat Exchanger

ParametersAir Temperature 20 C Ta

Heat transfer coefficient 40000 UPipe diameter 0.1 m D

Liquid density 1000 rho

Specific heat capacity 4200 Cp

Liquid flowrate 1 FInlet liquid temperature 100 C T0Time step 0.01 s dtSpace step 0.02 m dx

Distance x0 0.02 0.04 0.06 0.08

Time t 0 100.00 100.00 100.00 100.00 100.000.01 100.00 99.70 99.70 99.70 99.700.02 100.00 99.41 99.39 99.39 99.390.03 100.00 99.15 99.09 99.09 99.090.04 100.00 98.90 98.79 98.79 98.790.05 100.00 98.67 98.50 98.49 98.490.06 100.00 98.45 98.21 98.19 98.190.07 100.00 98.25 97.93 97.89 97.890.08 100.00 98.07 97.65 97.60 97.590.09 100.00 97.89 97.38 97.31 97.30

0.1 100.00 97.73 97.12 97.02 97.010.11 100.00 97.58 96.87 96.73 96.710.12 100.00 97.44 96.62 96.45 96.420.13 100.00 97.30 96.38 96.17 96.130.14 100.00 97.18 96.15 95.89 95.840.15 100.00 97.07 95.92 95.62 95.560.16 100.00 96.96 95.71 95.35 95.270.17 100.00 96.86 95.50 95.08 94.990.18 100.00 96.77 95.30 94.82 94.710.19 100.00 96.68 95.10 94.57 94.43

0.2 100.00 96.60 94.92 94.32 94.160.21 100.00 96.52 94.74 94.07 93.890.22 100.00 96.45 94.57 93.83 93.620.23 100.00 96.39 94.40 93.60 93.350.24 100.00 96.33 94.25 93.37 93.090.25 100.00 96.27 94.10 93.15 92.830.26 100.00 96.22 93.95 92.93 92.570.27 100.00 96.17 93.82 92.72 92.320.28 100.00 96.12 93.68 92.51 92.07

http://excelcalculations.blogspot.com

W m-2 K-1

kg m-3

J kg-1 K-1

kg s-1

http://excelcalculations.blogspot.com/

0.29 100.00 96.08 93.56 92.31 91.820.3 100.00 96.04 93.44 92.11 91.58

0.31 100.00 96.00 93.32 91.92 91.340.32 100.00 95.97 93.22 91.74 91.100.33 100.00 95.93 93.11 91.56 90.870.34 100.00 95.90 93.01 91.38 90.650.35 100.00 95.87 92.92 91.22 90.420.36 100.00 95.85 92.83 91.05 90.210.37 100.00 95.82 92.74 90.90 89.990.38 100.00 95.80 92.66 90.74 89.780.39 100.00 95.78 92.59 90.60 89.58

0.4 100.00 95.76 92.51 90.45 89.380.41 100.00 95.74 92.44 90.32 89.180.42 100.00 95.72 92.38 90.18 88.990.43 100.00 95.71 92.31 90.06 88.800.44 100.00 95.69 92.25 89.93 88.620.45 100.00 95.68 92.20 89.81 88.440.46 100.00 95.66 92.14 89.70 88.270.47 100.00 95.65 92.09 89.59 88.100.48 100.00 95.64 92.05 89.48 87.940.49 100.00 95.63 92.00 89.38 87.78

0.5 100.00 95.62 91.96 89.29 87.620.51 100.00 95.61 91.92 89.19 87.470.52 100.00 95.60 91.88 89.10 87.320.53 100.00 95.59 91.84 89.01 87.180.54 100.00 95.59 91.81 88.93 87.040.55 100.00 95.58 91.77 88.85 86.900.56 100.00 95.57 91.74 88.78 86.770.57 100.00 95.57 91.71 88.70 86.650.58 100.00 95.56 91.69 88.63 86.520.59 100.00 95.56 91.66 88.57 86.40

0.6 100.00 95.55 91.63 88.50 86.290.61 100.00 95.55 91.61 88.44 86.180.62 100.00 95.54 91.59 88.38 86.070.63 100.00 95.54 91.57 88.32 85.960.64 100.00 95.53 91.55 88.27 85.860.65 100.00 95.53 91.53 88.22 85.770.66 100.00 95.53 91.51 88.17 85.670.67 100.00 95.53 91.49 88.12 85.580.68 100.00 95.52 91.48 88.08 85.490.69 100.00 95.52 91.46 88.04 85.41

0.7 100.00 95.52 91.45 87.99 85.330.71 100.00 95.51 91.44 87.96 85.250.72 100.00 95.51 91.42 87.92 85.170.73 100.00 95.51 91.41 87.88 85.100.74 100.00 95.51 91.40 87.85 85.030.75 100.00 95.51 91.39 87.82 84.960.76 100.00 95.51 91.38 87.79 84.89

0.77 100.00 95.50 91.37 87.76 84.830.78 100.00 95.50 91.36 87.73 84.770.79 100.00 95.50 91.35 87.70 84.71

0.8 100.00 95.50 91.35 87.68 84.650.81 100.00 95.50 91.34 87.65 84.600.82 100.00 95.50 91.33 87.63 84.550.83 100.00 95.50 91.33 87.61 84.500.84 100.00 95.50 91.32 87.59 84.450.85 100.00 95.50 91.31 87.57 84.410.86 100.00 95.49 91.31 87.55 84.360.87 100.00 95.49 91.30 87.53 84.320.88 100.00 95.49 91.30 87.51 84.280.89 100.00 95.49 91.29 87.50 84.24

0.9 100.00 95.49 91.29 87.48 84.200.91 100.00 95.49 91.29 87.47 84.170.92 100.00 95.49 91.28 87.45 84.130.93 100.00 95.49 91.28 87.44 84.100.94 100.00 95.49 91.27 87.43 84.070.95 100.00 95.49 91.27 87.41 84.040.96 100.00 95.49 91.27 87.40 84.010.97 100.00 95.49 91.27 87.39 83.980.98 100.00 95.49 91.26 87.38 83.950.99 100.00 95.49 91.26 87.37 83.93

1 100.00 95.49 91.26 87.36 83.90

0.1 0.12 0.14 0.16 0.18 0.2 0.22

100.00 100.00 100.00 100.00 100.00 100.00 100.0099.70 99.70 99.70 99.70 99.70 99.70 99.7099.39 99.39 99.39 99.39 99.39 99.39 99.3999.09 99.09 99.09 99.09 99.09 99.09 99.0998.79 98.79 98.79 98.79 98.79 98.79 98.7998.49 98.49 98.49 99.92 99.92 99.92 99.9298.19 98.19 98.19 99.53 99.62 99.62 99.6297.89 97.89 97.89 99.14 99.31 99.32 99.3297.59 97.59 97.59 98.76 99.00 99.01 99.0197.30 97.30 97.30 98.38 98.68 98.71 98.7197.00 97.00 97.00 98.02 98.36 98.41 98.4196.71 96.71 96.71 97.66 98.04 98.11 98.1196.42 96.42 96.42 97.30 97.72 97.81 97.8296.13 96.13 96.13 96.95 97.40 97.50 97.5295.84 95.84 95.84 96.60 97.07 97.20 97.2295.55 95.55 95.55 96.26 96.75 96.90 96.9395.26 95.26 95.26 95.93 96.43 96.60 96.6394.98 94.97 94.97 95.60 96.10 96.29 96.3494.69 94.69 94.69 95.27 95.78 95.99 96.0494.41 94.40 94.40 94.94 95.46 95.69 95.7594.13 94.12 94.12 94.62 95.14 95.39 95.4693.85 93.84 93.84 94.31 94.82 95.08 95.1793.57 93.56 93.56 93.99 94.50 94.78 94.8893.29 93.28 93.28 93.68 94.19 94.48 94.5893.02 93.00 93.00 93.38 93.87 94.18 94.2992.74 92.72 92.72 93.07 93.56 93.87 94.0092.47 92.45 92.44 92.77 93.25 93.57 93.7192.20 92.17 92.17 92.48 92.94 93.27 93.4291.93 91.90 91.89 92.18 92.63 92.97 93.13

Air at 20 oC

Liquid at 100 oC

91.67 91.63 91.62 91.89 92.33 92.67 92.8491.40 91.36 91.35 91.60 92.02 92.37 92.5691.14 91.09 91.08 91.31 91.72 92.07 92.2790.88 90.82 90.81 91.02 91.42 91.78 91.9890.63 90.56 90.54 90.74 91.12 91.48 91.6990.37 90.29 90.27 90.45 90.83 91.19 91.4190.12 90.03 90.00 90.17 90.53 90.89 91.1289.88 89.77 89.74 89.90 90.24 90.60 90.8489.63 89.51 89.47 89.62 89.95 90.31 90.5589.39 89.25 89.21 89.35 89.67 90.02 90.2789.15 89.00 88.95 89.07 89.38 89.73 89.9888.91 88.74 88.69 88.80 89.10 89.44 89.7088.68 88.49 88.43 88.53 88.81 89.15 89.4288.45 88.24 88.18 88.27 88.53 88.87 89.1488.22 88.00 87.92 88.00 88.26 88.59 88.8688.00 87.75 87.67 87.74 87.98 88.30 88.5887.78 87.51 87.41 87.47 87.70 88.02 88.3087.57 87.27 87.16 87.21 87.43 87.74 88.0287.35 87.03 86.91 86.95 87.16 87.47 87.7487.14 86.80 86.67 86.70 86.89 87.19 87.4786.94 86.57 86.42 86.44 86.62 86.91 87.1986.74 86.34 86.18 86.19 86.36 86.64 86.9286.54 86.11 85.94 85.93 86.10 86.37 86.6586.35 85.88 85.70 85.68 85.83 86.10 86.3786.15 85.66 85.46 85.43 85.57 85.83 86.1085.97 85.44 85.22 85.18 85.31 85.56 85.8485.78 85.23 84.99 84.94 85.06 85.30 85.5785.61 85.01 84.75 84.69 84.80 85.03 85.3085.43 84.80 84.52 84.45 84.55 84.77 85.0385.26 84.60 84.30 84.21 84.30 84.51 84.7785.09 84.39 84.07 83.97 84.05 84.25 84.5184.93 84.19 83.85 83.73 83.80 83.99 84.2484.77 83.99 83.63 83.50 83.55 83.74 83.9884.61 83.80 83.41 83.26 83.30 83.48 83.7284.46 83.61 83.19 83.03 83.06 83.23 83.4784.31 83.42 82.98 82.80 82.82 82.98 83.2184.16 83.23 82.76 82.57 82.58 82.73 82.9584.02 83.05 82.56 82.35 82.34 82.48 82.7083.88 82.87 82.35 82.12 82.10 82.23 82.4583.74 82.70 82.14 81.90 81.87 81.99 82.1983.61 82.53 81.94 81.68 81.63 81.74 81.9483.48 82.36 81.74 81.46 81.40 81.50 81.7083.36 82.19 81.55 81.25 81.17 81.26 81.4583.24 82.03 81.35 81.03 80.94 81.02 81.2083.12 81.87 81.16 80.82 80.72 80.78 80.9683.01 81.71 80.98 80.61 80.49 80.55 80.7182.89 81.56 80.79 80.40 80.27 80.31 80.4782.79 81.41 80.61 80.20 80.05 80.08 80.23

82.68 81.26 80.43 79.99 79.83 79.85 79.9982.58 81.12 80.25 79.79 79.61 79.62 79.7682.48 80.98 80.08 79.59 79.40 79.39 79.5282.38 80.84 79.91 79.40 79.18 79.17 79.2882.29 80.71 79.74 79.20 78.97 78.94 79.0582.20 80.58 79.57 79.01 78.76 78.72 78.8282.11 80.45 79.41 78.82 78.55 78.50 78.5982.03 80.33 79.25 78.64 78.35 78.28 78.3681.95 80.21 79.09 78.45 78.14 78.06 78.1381.87 80.09 78.94 78.27 77.94 77.85 77.9181.79 79.97 78.79 78.09 77.74 77.63 77.6881.72 79.86 78.64 77.91 77.54 77.42 77.4681.64 79.75 78.49 77.74 77.35 77.21 77.2481.57 79.64 78.35 77.57 77.16 77.00 77.0281.51 79.54 78.21 77.40 76.96 76.79 76.8081.44 79.44 78.07 77.23 76.77 76.59 76.5881.38 79.34 77.94 77.07 76.59 76.38 76.3781.32 79.24 77.81 76.90 76.40 76.18 76.1581.26 79.15 77.68 76.75 76.22 75.98 75.9481.20 79.06 77.55 76.59 76.04 75.78 75.7381.15 78.97 77.43 76.43 75.86 75.59 75.5281.10 78.88 77.31 76.28 75.68 75.39 75.3181.05 78.80 77.19 76.13 75.51 75.20 75.1181.00 78.72 77.07 75.99 75.34 75.01 74.90

0.24 0.26 0.28 0.3 0.32 0.34 0.36

100.00 100.00 100.00 100.00 100.00 100.00 100.0099.70 99.70 99.70 99.70 99.70 99.70 99.7099.39 99.39 99.39 99.39 99.39 99.39 99.3999.09 99.09 99.09 99.09 99.09 99.09 99.0998.79 98.79 98.79 98.79 98.79 98.79 98.7999.92 99.92 99.92 99.92 99.92 99.92 99.9299.62 99.62 99.62 99.62 99.62 99.62 99.6299.32 99.32 99.32 99.32 99.32 99.32 99.3299.01 99.01 99.01 99.01 99.01 99.01 99.0198.71 98.71 98.71 98.71 98.71 98.71 98.7198.41 98.41 98.41 98.41 98.41 98.41 98.4198.11 98.11 98.11 98.11 98.11 98.11 98.1197.82 97.82 97.82 97.82 97.82 97.82 97.8297.52 97.52 97.52 97.52 97.52 97.52 97.5297.22 97.22 97.22 97.22 97.22 97.22 97.2296.93 96.93 96.93 96.93 96.93 96.93 96.9396.64 96.64 96.64 96.64 96.64 96.64 96.6496.34 96.35 96.35 96.35 96.35 96.35 96.3596.05 96.05 96.05 96.05 96.05 96.05 96.0595.76 95.76 95.77 95.77 95.77 95.77 95.7795.47 95.48 95.48 95.48 95.48 95.48 95.4895.19 95.19 95.19 95.19 95.19 95.19 95.1994.90 94.90 94.90 94.90 94.90 94.90 94.9094.61 94.62 94.62 94.62 94.62 94.62 94.6294.33 94.33 94.33 94.33 94.33 94.33 94.3394.04 94.05 94.05 94.05 94.05 94.05 94.0593.76 93.77 93.77 93.77 93.77 93.77 93.7793.47 93.48 93.49 93.49 93.49 93.49 93.4993.19 93.20 93.21 93.21 93.21 93.21 93.21

92.91 92.92 92.93 92.93 92.93 92.93 92.9392.62 92.64 92.65 92.65 92.65 92.65 92.6592.34 92.37 92.37 92.37 92.37 92.37 92.3792.06 92.09 92.10 92.10 92.10 92.10 92.1091.78 91.81 91.82 91.82 91.82 91.82 91.8291.50 91.54 91.55 91.55 91.55 91.55 91.5591.23 91.26 91.27 91.28 91.28 91.28 91.2890.95 90.99 91.00 91.00 91.01 91.01 91.0190.67 90.72 90.73 90.73 90.73 90.73 90.7390.39 90.44 90.46 90.46 90.47 90.47 90.4790.12 90.17 90.19 90.20 90.20 90.20 90.2089.84 89.90 89.92 89.93 89.93 89.93 89.9389.57 89.63 89.65 89.66 89.66 89.66 89.6689.29 89.36 89.39 89.40 89.40 89.40 89.4089.02 89.09 89.12 89.13 89.13 89.13 89.1388.74 88.83 88.86 88.87 88.87 88.87 88.8788.47 88.56 88.59 88.60 88.61 88.61 88.6188.20 88.29 88.33 88.34 88.34 88.35 88.3587.93 88.03 88.07 88.08 88.08 88.09 88.0987.66 87.76 87.80 87.82 87.82 87.83 87.8387.39 87.50 87.54 87.56 87.57 87.57 87.5787.12 87.23 87.28 87.30 87.31 87.31 87.3186.85 86.97 87.02 87.04 87.05 87.05 87.0586.58 86.71 86.76 86.79 86.80 86.80 86.8086.32 86.44 86.51 86.53 86.54 86.54 86.5486.05 86.18 86.25 86.28 86.29 86.29 86.2985.78 85.92 85.99 86.02 86.03 86.04 86.0485.52 85.66 85.74 85.77 85.78 85.79 85.7985.26 85.40 85.48 85.52 85.53 85.53 85.5484.99 85.14 85.23 85.26 85.28 85.28 85.2984.73 84.89 84.97 85.01 85.03 85.04 85.0484.47 84.63 84.72 84.76 84.78 84.79 84.7984.21 84.37 84.47 84.51 84.53 84.54 84.5483.95 84.12 84.22 84.27 84.29 84.29 84.3083.69 83.86 83.97 84.02 84.04 84.05 84.0583.44 83.61 83.72 83.77 83.79 83.80 83.8183.18 83.36 83.47 83.52 83.55 83.56 83.5682.93 83.10 83.22 83.28 83.31 83.32 83.3282.67 82.85 82.97 83.03 83.06 83.08 83.0882.42 82.60 82.72 82.79 82.82 82.83 82.8482.17 82.35 82.47 82.55 82.58 82.59 82.6081.92 82.10 82.23 82.30 82.34 82.35 82.3681.67 81.85 81.98 82.06 82.10 82.12 82.1281.42 81.61 81.74 81.82 81.86 81.88 81.8981.17 81.36 81.50 81.58 81.62 81.64 81.6580.92 81.11 81.25 81.34 81.38 81.41 81.4180.68 80.87 81.01 81.10 81.15 81.17 81.1880.43 80.62 80.77 80.86 80.91 80.94 80.95

80.19 80.38 80.53 80.62 80.68 80.70 80.7179.95 80.14 80.29 80.39 80.44 80.47 80.4879.71 79.90 80.05 80.15 80.21 80.24 80.2579.47 79.66 79.81 79.91 79.97 80.01 80.0279.23 79.42 79.57 79.68 79.74 79.78 79.7978.99 79.18 79.34 79.45 79.51 79.55 79.5678.76 78.94 79.10 79.21 79.28 79.32 79.3378.52 78.71 78.86 78.98 79.05 79.09 79.1178.29 78.47 78.63 78.75 78.82 78.86 78.8878.06 78.24 78.40 78.52 78.59 78.63 78.6577.83 78.00 78.16 78.29 78.36 78.41 78.4377.60 77.77 77.93 78.06 78.14 78.18 78.2177.37 77.54 77.70 77.83 77.91 77.96 77.9877.14 77.31 77.47 77.60 77.68 77.73 77.7676.92 77.08 77.24 77.37 77.46 77.51 77.5476.69 76.85 77.01 77.14 77.23 77.29 77.3276.47 76.63 76.79 76.92 77.01 77.07 77.1076.25 76.40 76.56 76.69 76.79 76.85 76.8876.03 76.18 76.33 76.47 76.57 76.63 76.6675.81 75.95 76.11 76.24 76.34 76.41 76.4475.59 75.73 75.89 76.02 76.12 76.19 76.2275.37 75.51 75.66 75.80 75.90 75.97 76.0175.16 75.29 75.44 75.58 75.68 75.75 75.7974.95 75.07 75.22 75.36 75.46 75.53 75.58

0.38 0.4

100.00 100.0099.70 99.7099.39 99.3999.09 99.0998.79 98.7999.92 98.4999.62 98.2899.32 98.0799.01 97.8598.71 97.6398.41 97.4098.11 97.1797.82 96.9497.52 96.7097.22 96.4696.93 96.2296.64 95.9796.35 95.7296.05 95.4895.77 95.2395.48 94.9795.19 94.7294.90 94.4694.62 94.2194.33 93.9594.05 93.6993.77 93.4493.49 93.1893.21 92.92

92.93 92.6692.65 92.4092.37 92.1492.10 91.8891.82 91.6291.55 91.3691.28 91.1091.01 90.8490.73 90.5890.47 90.3290.20 90.0689.93 89.8089.66 89.5589.40 89.2989.13 89.0388.87 88.7888.61 88.5288.35 88.2688.09 88.0187.83 87.7687.57 87.5087.31 87.2587.05 87.0086.80 86.7486.54 86.4986.29 86.2486.04 85.9985.79 85.7585.54 85.5085.29 85.2585.04 85.0084.79 84.7684.54 84.5184.30 84.2784.05 84.0383.81 83.7883.56 83.5483.32 83.3083.08 83.0682.84 82.8282.60 82.5982.36 82.3582.13 82.1181.89 81.8881.65 81.6481.42 81.4181.18 81.1780.95 80.94

80.72 80.7180.49 80.4880.26 80.2580.03 80.0279.80 79.7979.57 79.5679.34 79.3479.11 79.1178.89 78.8978.66 78.6678.44 78.4478.22 78.2277.99 77.9977.77 77.7777.55 77.5577.33 77.3377.11 77.1276.89 76.9076.68 76.6876.46 76.4676.24 76.2576.03 76.0375.81 75.8275.60 75.61

Dynamic Model of a Cross-Flow Heat Exchanger

This article will develop a dynamic model of a cross-flow heat exchanger from first principles, and then discretize the governing partial differential equation with finite difference approximations. It will then demonstrate how this equation can be implemented in Excel (or indeed any other math tool)

First Principles ModelingConsider liquid flowing (at mass flowrate F) through a length Δx of pipe (diameter D), subject to cooling by cross-flow air (at temperature Ta and heat transfer coefficient U)

A heat balance over time Δt gives the following.

Dividing by Δx and Δt and simplifying gives

As Δx and Δt tend to zero, we get the following parabolic partial differential equation

Equation 1

Finite Difference ApproximationA forward difference approximation for the first of temperature with respect to time is

Equation 2

A backward difference approximation for the first of temperature with respect to space is

Introduction

If you just want the Excel implementation, then click here, but I encourage you to read the rest of the article so you understand how the spreadsheet is implemented.

https://sites.google.com/site/simulationsmodelsandworksheets/CrossFlowHeatExchanger.zip?attredirects=0&d=1

Equation 4

Substituting Equation 2 and 3 into Equation 1, and rearranging gives

Equation 4

We only need to know the temperature of the bar at time t (on the RHS of Equation 4) to calculate the temperature at time t + Δt (on the LHS of Equation 4).

Implementating in ExcelThis is how Equation 4 will be implemented in Excel

We will now discuss the individual steps in detail.

Step 1 - Specify your parameters, including your chosen time and space step. I've named the cells in Column C with the names in Column E. I'll use named values when entering Equation 4.

Step 2 - Create a column and row containing your space and time steps

Step 3 - Fill in your initial conditions at time t = 0 (this will be the inlet liquid temperature as specified in the parameters).

Step 4 - Insert your boundary conditions at distance x = 0 (this will be the inlet liquid temperature - the same as the initial condition).

Step 5 - Implement Equation 4 into the first empty cell (at t = Δt and x = Δx)

The techniques I've demonstrated above can be applied to many other challenges in science, engineering and math. If you have any requests, then let me know.

Step 5 - Copy this formula to all other times and positions. For my implementation, I go up to t = 1 and x = 0.4.

Dynamic Model of a Cross-Flow Heat Exchanger


Consider liquid flowing (at mass flowrate F) through a length Δx of pipe (diameter D), subject to cooling by cross-flow air (at temperature Ta and heat transfer coefficient U)

As Δx and Δt tend to zero, we get the following parabolic partial differential equation

A forward difference approximation for the first of temperature with respect to time is

A backward difference approximation for the first of temperature with respect to space is



We only need to know the temperature of the bar at time t (on the RHS of Equation 4) to calculate the temperature at time t + Δt (on the LHS of Equation 4).

- Specify your parameters, including your chosen time and space step. I've named the cells in Column C with the names in Column E. I'll use named values when entering Equation 4.

- Create a column and row containing your space and time steps

- Fill in your initial conditions at time t = 0 (this will be the inlet liquid temperature as specified in the parameters).

- Insert your boundary conditions at distance x = 0 (this will be the inlet liquid temperature - the same as the initial condition).

- Implement Equation 4 into the first empty cell (at t = Δt and x = Δx)

The techniques I've demonstrated above can be applied to many other challenges in science, engineering and math. If you have any requests, then let me know.

- Copy this formula to all other times and positions. For my implementation, I go up to t = 1 and x = 0.4.




- Specify your parameters, including your chosen time and space step. I've named the cells in Column C with the names in Column E. I'll use named values when entering Equation 4.


Documents

Cross Flow Heat Exchanger