Annaratone D. Handbook for Heat Exchangers and Tube Banks design
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172 Appendix B Factor ψ or Corrective Factors for Verification Computation
Table B.32 – Tube banks – parallel flow – 3 passages of external fluid – Fig. A.33 – Corrective
factor ϕ
p
β
γ
1.4 1.6 1.8 2.0 2.3 2.6 2.9 3.2 3.6 4.0
1.0 ……. ……. ……. ……. ……. ……. …… . ……. ……. 0.990
1.1 ……. ……. ……. ……. ……. ……. …… . ……. 0.991 0.988
1.2 ……. ……. ……. ……. ……. ……. …… . ……. 0.989 0.986
1.3 ……. ……. ……. ……. ……. ……. …… . 0.991 0.988 0.983
1.4 ……. ……. ……. ……. ……. ……. 0. 992 0.990 0.986 0.981
1.5 ……. ……. ……. ……. ……. 0.993 0.991 0.989 0.984 0.979
1.6 ……. ……. ……. ……. ……. 0.993 0.990 0.987 0.982 0.977
1.7 ……. ……. ……. ……. 0.994 0.992 0.989 0.986 0.981 0.975
1.8 ……. ……. ……. ……. 0.993 0.991 0.988 0.985 0.979 0.973
1.9 ……. ……. ……. 0.995 0.993 0.991 0.987 0.984 0.978 0.971
2.0 ……. ……. ……. 0.994 0.993 0.990 0.986 0.982 0.976 0.970
2.1 ……. ……. 0.995 0.994 0.992 0.9 89 0.985 0.981 0.975 0.968
2.2 ……. ……. 0.995 0.994 0.992 0.9 88 0.984 0.980 0.974 0.967
2.3 ……. ……. 0.995 0.994 0.991 0.9 88 0.984 0.979 0.973 0.966
2.4 ……. 0.996 0.995 0.993 0.990 0 .987 0.983 0.978 0.972 0.965
2.6 ……. 0.996 0.994 0.993 0.989 0 .986 0.981 0.977 0.970 0.964
2.8 0.996 0.995 0.994 0.992 0.988 0.984 0.980 0.975 0.969 0.963
3.0 0.996 0.995 0.993 0.991 0.988 0.983 0.979 0.974 0.968 0.963
If β<1.0orγ < 1.4 use factor. ψ
p
without the corrective factor
B.5 Tube Banks with Several Passages of External Fluid 173
B.5.2 Tube Banks with Fluids in Counter Flow (Fig. 2.11)
Table B.33 – Tube banks – counter flow – 2 passages of external fluid Fig. A.34 – Corrective
factor ϕ
c
β
γ
0.5 0.6 0.7 0.8 0.9 1.0
0.5 ……. ……. ……. ……. ……. 1.013
0.6 ……. ……. ……. ……. 1.011 1.014
0.7 ……. ……. ……. ……. 1.012 1.016
0.8 ……. ……. ……. 1.009 1.013 1.016
0.9 ……. ……. ……. 1.010 1.013 1.017
1.0 ……. ……. 1.007 1.010 1.014 1.0 17
1.1 ……. ……. 1.008 1.011 1.014 1.0 18
1.2 ……. ……. 1.008 1.011 1.014 1.0 18
1.3 ……. 1.006 1.008 1.011 1.014 1 .018
1.4 ……. 1.006 1.008 1.011 1.014 1 .017
1.5 ……. 1.006 1.008 1.011 1.014 1 .017
1.6 ……. 1.006 1.008 1.011 1.014 1 .016
1.7 ……. 1.006 1.008 1.011 1.013 1 .016
1.8 ……. 1.006 1.008 1.010 1.013 1 .015
1.9 1.004 1.006 1.008 1.010 1.013 1.015
2.0 1.004 1.006 1.008 1.010 1.012 1.014
2.2 1.004 1.006 1.008 1.010 1.011 1.013
2.4 1.004 1.006 1.007 1.009 1.010 1.012
2.6 1.004 1.005 1.007 1.008 1.010 1.011
2.8 1.004 1.005 1.007 1.008 1.009 1.009
3.0 1.004 1.005 1.006 1.007 1.008 1.008
If β <0.5orγ < 0.5 use factor ψ
c
without corrective factor
174 Appendix B Factor ψ or Corrective Factors for Verification Computation
Table B.33 – (continued)
β
γ
1.2 1.4 1.6 1.8 2.0 2.3 2.6 2.9 3.2 3.6 4.0
0.2 ……. ……. ……. ……. ……. ……. 1. 088 1.118 1.154 1.212 …….
0.3 ……. ……. 1.032 1.044 1.058 1.0 84 1.116 1.155 1.200 1.273 1.361
0.4 1.018 1.027 1.039 1.053 1.069 1.098 1.134 1.177 1.227 1.306 1.400
0.5 1.021 1.031 1.043 1.058 1.076 1.107 1.144 1.188 1.238 1.316 1.407
0.6 1.023 1.034 1.047 1.062 1.080 1.111 1.147 1.189 1.236 1.308 1.390
0.7 1.024 1.035 1.048 1.064 1.081 1.110 1.144 1.182 1.225 1.288 1.358
0.8 1.025 1.036 1.049 1.064 1.080 1.107 1.138 1.172 1.208 1.261 1.318
0.9 1.026 1.037 1.049 1.063 1.078 1.102 1.129 1.158 1.188 1.230 1.274
1.0 1.026 1.036 1.048 1.060 1.074 1.096 1.118 1.142 1.166 1.199 1.232
1.1 1.026 1.036 1.046 1.058 1.070 1.088 1.107 1.126 1.144 1.169 1.192
1.2 1.026 1.035 1.044 1.054 1.065 1.080 1.095 1.110 1.124 1.141 1.157
1.3 1.025 1.033 1.042 1.051 1.060 1.072 1.084 1.095 1.105 1.117 1.126
1.4 1.024 1.032 1.040 1.047 1.054 1.065 1.074 1.082 1.088 1.096 1.101
1.5 1.023 1.030 1.037 1.043 1.049 1.057 1.064 1.069 1.074 1.077 1.080
1.6 1.023 1.029 1.034 1.040 1.044 1.051 1.055 1.059 1.061 1.062 1.062
1.7 1.022 1.027 1.032 1.036 1.040 1.044 1.047 1.049 1.050 1.050 1.049
1.8 1.020 1.025 1.029 1.033 1.036 1.039 1.040 1.041 1.041 1.040 1.038
1.9 1.019 1.023 1.027 1.029 1.032 1.033 1.034 1.034 1.033 1.031 1.029
2.0 1.018 1.022 1.024 1.026 1.028 1.029 1.029 1.028 1.027 1.025 1.022
2.1 1.017 1.020 1.022 1.024 1.025 1.025 1.024 1.023 1.022 1.019 1.017
2.2 1.016 1.019 1.020 1.021 1.022 1.021 1.020 1.019 1.017 1.015 1.013
2.3 1.015 1.017 1.018 1.019 1.019 1.018 1.017 1.016 1.014 1.012 1.010
2.4 1.014 1.016 1.016 1.017 1.016 1.016 1.014 1.013 1,011 1.009 1.008
2.5 1.013 1.014 1.015 1.015 1.014 1.013 1.012 1.010 1.009 1.007 …….
2.6 1.012 1.013 1.013 1.013 1.012 1.011 1.010 1.008 1.007 ……. …….
2.8 1.010 1.011 1.011 1.010 1.009 1.008 1.007 1.006 ……. ……. …….
3.0 1.009 1.009 1.008 1.008 1.007 1.006 ……. ……. ……. ……. …….
B.5 Tube Banks with Several Passages of External Fluid 175
Table B.34 – Tube banks – counter flow – 3 passages of external fluid – Fig. A.35 – Corrective
factor ϕ
c
β
γ
0.9 1.1 1.4 1.7 2.0 2.3 2.6 2.9 3.2 3.6 4.0
0.3 ……. ……. ……. ……. ……. ……. …… . ……. 1.123 1.174 1.238
0.4 ……. ……. ……. ……. ……. 1.056 1.079 1.107 1.141 1.198 1.269
0.5 ……. ……. ……. 1.027 1.042 1.061 1.085 1.115 1.151 1.208 1.279
0.6 ……. ……. ……. 1.029 1.044 1.064 1.088 1.118 1.152 1.207 1.273
0.7 ……. ……. 1.018 1.030 1.046 1.0 65 1.088 1.116 1.148 1.198 1.256
0.8 ……. ……. 1.019 1.031 1.046 1.0 64 1.085 1.110 1.139 1.182 1.232
0.9 ……. ……. 1.019 1.031 1.045 1.0 62 1.081 1.103 1.127 1.163 1.203
1.0 ……. ……. 1.019 1.030 1.043 1.0 58 1.075 1.094 1.114 1.143 1.174
1.1 ……. 1.011 1.019 1.029 1.041 1 .054 1.069 1.084 1.101 1.123 1.146
1.2 ……. 1.011 1.019 1.028 1.039 1 .050 1.062 1.075 1.087 1.104 1.121
1.3 ……. 1.011 1.018 1.027 1.036 1 .046 1.055 1.065 1.075 1.087 1.098
1.4 ……. 1.011 1.017 1.025 1.033 1 .041 1.049 1.057 1.064 1.072 1.079
1.5 ……. 1.011 1.017 1.023 1.030 1 .037 1.043 1.048 1.053 1.059 1.063
1.6 ……. 1.010 1.016 1.022 1.027 1 .033 1.037 1.041 1.044 1.048 1.050
1.7 1.007 1.010 1.015 1.020 1.025 1.029 1.032 1.035 1.037 1.038 1.039
1.8 1.007 1.010 1.014 1.018 1.022 1.025 1.027 1.029 1.030 1.031 1.030
1.9 1.006 1.009 1.013 1.017 1.020 1.022 1.023 1.024 1.025 1.024 1.024
2.0 1.006 1.009 1.012 1.015 1.018 1.019 1.020 1.020 1.020 1.019 1.018
2.1 1.006 1.008 1.011 1.014 1.016 1.016 1.017 1.017 1.016 1.015 1.014
2.2 1.006 1.008 1.011 1.013 1.014 1.014 1.014 1.014 1.013 1.012 1.011
2.3 1.006 1.008 1.010 1.011 1.012 1.012 1.012 1.011 1.011 1.009 1.008
2.4 1.005 1.007 1.009 1.010 1.011 1.010 1.010 1.009 1.008 1.007 …….
2.6 1.005 1.006 1.008 1.008 1.008 1.008 1.007 ……. ……. ……. …….
2.8 1.005 1.006 1.006 1.006 1.006 1.005 ……. ……. ……. ……. …….
3.0 1.004 1.005 1.005 1.005 ……. … …. ……. ……. ……. ……. …….
If β <0.3orγ < 0.9 use factor ψ
c
without corrective factor
176 Appendix B Factor ψ or Corrective Factors for Verification Computation
Table B.35 – Tube banks – counter flow – 4 passages of external fluid – Fig. A.36 – Corrective
factor ϕ
c
β
γ
1.4 1.6 1.8 2.0 2.3 2.6 2.9 3.2 3.6 4.0
0.4 ……. ……. ……. ……. ……. ……. …… . ……. ……. 1.170
0.5 ……. ……. ……. ……. ……. ……. 1. 072 1.094 1.132 1.179
0.6 ……. ……. ……. ……. 1.039 1.055 1.074 1.096 1.133 1.178
0.7 ……. ……. ……. 1.028 1.040 1.055 1.073 1.094 1.128 1.169
0.8 ……. ……. ……. 1.028 1.040 1.054 1.070 1.090 1.120 1.155
0.9 ……. ……. 1.021 1.028 1.038 1.0 51 1.066 1.083 1.108 1.138
1.0 ……. ……. 1.021 1.027 1.037 1.0 48 1.061 1.075 1.096 1.119
1.1 ……. 1.016 1.020 1.025 1.034 1 .044 1.055 1.066 1.083 1.101
1.2 ……. 1.015 1.019 1.024 1.032 1 .040 1.049 1.058 1.071 1.084
1.3 ……. 1.014 1.018 1.022 1.029 1 .036 1.043 1.050 1.059 1.068
1.4 ……. 1.014 1.017 1.021 1.026 1 .032 1.037 1.042 1.049 1.055
1.5 1.010 1.013 1.016 1.019 1.023 1.028 1.032 1.036 1.040 1.044
1.6 1.010 1.012 1.015 1.017 1.021 1.024 1.027 1.030 1.032 1.034
1.7 1.009 1.011 1.014 1.016 1.018 1.021 1.023 1.024 1.026 1.027
1.8 1.009 1.011 1.012 1.014 1.016 1.018 1.019 1.020 1.021 1.021
1.9 1.008 1.010 1.011 1.013 1.014 1.015 1.016 1.016 1.016 1.016
2.0 1.008 1.009 1.010 1.011 1.012 1.013 1.013 1.013 1.013 1.012
2.1 ……. 1.008 1.009 1.010 1.011 1 .011 1.011 1.011 1.010 1.009
2.2 ……. 1.007 1.008 1.009 1.009 1 .009 1.009 1.009 ……. …….
2.3 ……. 1.007 1.007 1.008 1.008 1 .008 ……. ……. ……. …….
If β <0.4orβ >2.3orγ < 1.4 use factor ψ
c
without corrective factor