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

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

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

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

without corrective factor

174 Appendix B Factor ψ or Corrective Factors for Veriﬁcation Computation

Table B.33 – (continued)

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

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

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

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

without corrective factor