Annaratone D. Handbook for Heat Exchangers and Tube Banks design
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162 Appendix B Factor ψ or Corrective Factors for Verification Computation
Table B.24 – (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
0.1 0.325 0.274 0.233 0.199 0.171 0.139 0.115 0.097 0.084 0.071 0.063
0.2 0.348 0.301 0.262 0.231 0.206 0.177 0.155 0.139 0.127 0.115 0.107
0.3 0.370 0.326 0.291 0.263 0.240 0.214 0.194 0.180 0.169 0.159 0.152
0.4 0.391 0.351 0.319 0.294 0.273 0.249 0.232 0.220 0.211 0.202 0.196
0.5 0.412 0.375 0.346 0.323 0.304 0.284 0.269 0.258 0.251 0.243 0.239
0.6 0.432 0.398 0.372 0.351 0.335 0.317 0.304 0.295 0.288 0.283 0.279
0.7 0.451 0.420 0.396 0.378 0.364 0.348 0.337 0.330 0.324 0.320 0.317
0.8 0.470 0.442 0.420 0.404 0.391 0.377 0.368 0.362 0.358 0.355 0.352
0.9 0.488 0.462 0.443 0.428 0.417 0.405 0.398 0.393 0.390 0.387 0.385
1.0 0.505 0.481 0.464 0.451 0.442 0.432 0.426 0.422 0.419 0.417 0.416
1.1 0.521 0.500 0.484 0.473 0.465 0.457 0.452 0.448 0.446 0.445 0.444
1.2 0.537 0.518 0.504 0.494 0.487 0.480 0.476 0.473 0.472 0.471 0.470
1.3 0.552 0.534 0.522 0.514 0.508 0.502 0.498 0.497 0.495 0.494 0.494
1.4 0.566 0.550 0.540 0.532 0.527 0.522 0.520 0.518 0.517 0.516 0.516
1.5 0.580 0.566 0.556 0.550 0.545 0.541 0.539 0.538 0.537 0.537 0.536
1.6 0.593 0.580 0.572 0.566 0.563 0.559 0.558 0.557 0.556 0.555 0.555
1.7 0.605 0.594 0.587 0.582 0.579 0.576 0.575 0.574 0.573 0.573 0.572
1.8 0.617 0.607 0.601 0.597 0.594 0.592 0.591 0.590 0.590 0.589 0.588
1.9 0.629 0.620 0.614 0.611 0.608 0.607 0.606 0.605 0.605 0.604 0.603
2.0 0.640 0.632 0.627 0.624 0.622 0.620 0.620 0.619 0.619 0.618 0.617
2.1 0.650 0.643 0.639 0.636 0.635 0.633 0.633 0.632 0.632 0.631 0.630
2.2 0.660 0.654 0.650 0.648 0.646 0.645 0.645 0.644 0.644 0.643 0.642
2.3 0.670 0.664 0.661 0.659 0.658 0.657 0.656 0.656 0.655 0.654 0.653
2.4 0.679 0.674 0.671 0.669 0.668 0.668 0.667 0.667 0.666 0.665 0.664
2.5 0.688 0.683 0.681 0.679 0.678 0.678 0.677 0.677 0.676 0.675 0.674
2.6 0.696 0.692 0.690 0.689 0.688 0.687 0.687 0.686 0.686 0.685 0.684
2.7 0.704 0.700 0.699 0.697 0.697 0.696 0.696 0.695 0.695 0.694 0.692
2.8 0.712 0.708 0.707 0.706 0.705 0.705 0.704 0.704 0.703 0.702 0.701
2.9 0.719 0.716 0.715 0.714 0.713 0.713 0.713 0.712 0.711 0.710 0.709
3.0 0.726 0.723 0.722 0.721 0.721 0.721 0.720 0.720 0.719 0.718 0.717
B.4 Coils 163
B.4 Coils
B.4.1 Coils with Fluids in Parallel Flow (Fig. 2.8)
Table B.25 – Coils – parallel flow – 2 sections – Fig. A.23 – Corrective factor ϕ
p
β
γ
0.70 0.80 0.90 1.00
0.7 ……. ……. ……. 0.990
0.8 ……. ……. 0.991 0.990
0.9 ……. ……. 0.992 0.990
1.0 ……. 0.993 0.992 0.991
1.1 ……. 0.993 0.992 0.991
1.2 ……. 0.993 0.992 0.992
1.3 0.995 0.994 0.993 0.992
1.4 0.995 0.994 0.993 0.993
1.5 0.995 0.994 0.994 0.994
1.6 0.995 0.994 0.994 …….
1.7 0.995 0.995 ……. …….
1.8 0.995 0.995 ……. …….
If γ < 0.7 use factor ψ
p
without corrective factor
164 Appendix B Factor ψ or Corrective Factors for Verification Computation
Table B.25 – (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
0.3 ……. ……. ……. 0.977 0.974 0.972 ……. ……. ……. ……. …….
0.4 ……. 0.983 0.980 0.978 0.976 0 .977 ……. ……. ……. ……. …….
0.5 0.987 0.983 0.981 0.980 0.980 ……. ……. ……. ……. ……. 1.024
0.6 0.987 0.984 0.983 0.983 0.984 ……. ……. ……. ……. 1.023 1.035
0.7 0.987 0.985 0.985 0.986 ……. … …. ……. ……. 1.019 1.032 1.044
0.8 0.988 0.987 0.987 ……. ……. ……. ……. 1.017 1.027 1.040 1.053
0.9 0.989 0.988 0.990 ……. ……. ……. 1.014 1.024 1.034 1.047 1.060
1.0 0.990 0.990 ……. ……. ……. ……. 1.019 1.029 1.040 1.053 1.066
1.1 0.991 ……. ……. ……. ……. 1.013 1.024 1.035 1.045 1.059 1.071
1.2 0.992 ……. ……. ……. ……. 1.017 1.028 1.039 1.050 1.063 1.075
1.3 0.993 ……. ……. ……. 1.011 1.021 1.032 1.043 1.054 1.067 1.078
1.4 ……. ……. ……. 1.007 1.014 1.024 1.035 1.046 1.057 1.070 1.081
1.5 ……. ……. ……. 1.010 1.016 1.027 1.038 1.049 1.060 1.072 1.083
1.6 ……. ……. ……. 1.012 1.019 1.030 1.041 1.052 1.062 1.074 1.085
1.8 ……. ……. 1.009 1.016 1.023 1.0 34 1.045 1.056 1.066 1.077 1.087
2.0 ……. 1.006 1.012 1.019 1.027 1 .038 1.049 1.059 1.068 1.079 1.088
2.2 ……. 1.009 1.015 1.022 1.030 1 .041 1.051 1.061 1.069 1.079 1.088
2.6 1.007 1.013 1.019 1.027 1.034 1.044 1.054 1.062 1.070 1.079 1.086
3.0 1.009 1.016 1.023 1.029 1.036 1.046 1.054 1.062 1.069 1.076 1.083
B.4 Coils 165
Table B.26 – Coils – parallel flow – 3 sections – Fig. A.24 – 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
1.2 ……. ……. ……. ……. ……. ……. …… . ……. 0.990 0.987
1.3 ……. ……. ……. ……. ……. ……. …… . 0.992 0.989 0.986
1.4 ……. ……. ……. ……. ……. ……. 0. 992 0.991 0.987 0.984
1.5 ……. ……. ……. ……. ……. ……. 0. 992 0.989 0.986 0.982
1.6 ……. ……. ……. ……. ……. 0.993 0.991 0.988 0.985 0.980
1.8 ……. ……. ……. ……. 0.994 0.992 0.989 0.986 0.982 0.977
2.0 ……. ……. ……. 0.995 0.993 0.990 0.987 0.984 0.979 0.975
2.2 ……. ……. 0.995 0.994 0.992 0.9 89 0.986 0.982 0.977 0.973
2.6 ……. 0.996 0.994 0.993 0.990 0 .987 0.983 0.979 0.974 0.970
3.0 0.996 0.995 0.993 0.992 0.988 0.985 0.981 0.977 0.973 0.969
If β <1.2orγ < 1.4 use factor ψ
p
without corrective factor
B.4.2 Coils with Fluids in Counter Flow (Fig. 2.9)
Table B.27 – Coils – counter flow – 2 sections – Fig. A.26 – Corrective factor ϕ
c
β
γ
0.5 0.6 0.7 0.8 0.9 1.0
0.6 ……. ……. ……. ……. 1.012 1.016
0.8 ……. ……. ……. 1.010 1.014 1.019
1.0 ……. ……. 1.008 1.011 1.015 1.020
1.2 ……. 1.006 1.009 1.012 1.016 1.020
1.4 ……. 1.006 1.009 1.012 1.016 1.020
1.6 ……. 1.006 1.009 1.012 1.015 1.019
1.8 1.004 1.006 1.009 1.012 1.015 1.018
2.0 1.004 1.006 1.009 1.011 1.014 1.016
2.2 1.004 1.006 1.008 1.011 1.013 1.015
2.4 1.004 1.006 1.008 1.010 1.012 1.014
2.6 1.004 1.006 1.008 1.009 1.011 1.012
2.8 1.004 1.006 1.007 1.009 1.010 1.011
3.0 1.004 1.005 1.007 1.008 1.009 1.010
If γ < 0.5 use factor ψ
c
without corrective factor
166 Appendix B Factor ψ or Corrective Factors for Verification Computation
Table B.27 – (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 ……. ……. ……. 1.042 1.057 1.087 1.127 1.179 1.246 1.363 1.520
0.3 ……. 1.027 1.040 1.056 1.076 1.115 1.167 1.233 1.316 1.460 1.651
0.4 1.021 1.033 1.048 1.067 1.090 1.134 1.191 1.264 1.354 1.507 1.705
0.5 1.024 1.037 1.054 1.074 1.099 1.145 1.204 1.277 1.366 1.514 1.701
0.6 1.027 1.040 1.057 1.078 1.104 1.150 1.207 1.276 1.360 1.494 1.659
0.7 1.028 1.042 1.060 1.080 1.105 1.149 1.202 1.266 1.340 1.456 1.595
0.8 1.030 1.043 1.060 1.080 1.104 1.145 1.193 1.249 1.312 1.410 1.522
0.9 1.030 1.044 1.060 1.079 1.100 1.137 1.180 1.228 1.281 1.360 1.447
1.0 1.030 1.044 1.059 1.076 1.096 1.129 1.165 1.205 1.248 1.310 1.376
1.1 1.030 1.043 1.057 1.073 1.090 1.119 1.149 1.182 1.216 1.264 1.312
1.2 1.030 1.042 1.055 1.069 1.084 1.108 1.134 1.160 1.186 1.222 1.257
1.3 1.029 1.040 1.052 1.064 1.078 1.098 1.118 1.139 1.159 1.185 1.209
1.4 1.029 1.038 1.049 1.060 1.071 1.088 1.104 1.120 1.134 1.153 1.169
1.5 1.028 1.036 1.046 1.055 1.065 1.078 1.091 1.102 1.113 1.126 1.136
1.6 1.026 1.034 1.043 1.051 1.058 1.069 1.079 1.087 1.095 1.103 1.109
1.7 1.025 1.032 1.039 1.046 1.052 1.061 1.068 1.074 1.079 1.084 1.087
1.8 1.024 1.030 1.036 1.042 1.047 1.053 1.058 1.062 1.065 1.068 1.070
1.9 1.023 1.028 1.033 1.038 1.042 1.047 1.050 1.053 1.054 1.055 1.056
2.0 1.022 1.026 1.031 1.034 1.037 1.041 1.043 1.044 1.045 1.045 1.044
2.2 1.019 1.023 1.025 1.028 1.029 1.030 1.031 1.031 1.030 1.029 1.028
2.4 1.017 1.019 1.021 1.022 1.023 1.023 1.022 1.021 1.021 1.019 1.018
2.6 1.015 1.016 1.017 1.017 1.017 1.017 1.016 1.015 1.014 1.013 1.011
2.8 1.013 1.013 1.014 1.014 1.013 1,012 1,011 1.010 1.009 1.008 1.007
3.0 1.011 1.011 1.011 1.011 1.010 1.009 1.008 1.007 1.006 1.005 -----
B.4 Coils 167
Table B.28 – Coils – counter flow – 3 sections – Fig. A.27 – Corrective factor ϕ
c
β
γ
1.2 1.4 1.6 1.8 2.0 2.3 2.6 2.9 3.2 3.6 4.0
0.3 ……. ……. ……. ……. ……. ……. …… . 1.100 1.135 1.194 1.270
0.4 ……. ……. ……. ……. 1.040 1.059 1.085 1.116 1.155 1.221 1.306
0.5 ……. ……. ……. 1.033 1.044 1.065 1.092 1.125 1.166 1.233 1.317
0.6 ……. ……. 1.026 1.035 1.047 1.0 68 1.095 1.128 1.167 1.231 1.311
0.7 ……. 1.019 1.027 1.037 1.048 1 .069 1.095 1.126 1.162 1.221 1.291
0.8 ……. 1.020 1.028 1.037 1.048 1 .068 1.092 1.120 1.153 1.203 1.263
0.9 1.014 1.020 1.028 1.037 1.047 1.066 1.087 1.112 1.140 1.182 1.230
1.0 1.014 1.020 1.027 1.036 1.046 1.062 1.081 1.102 1.126 1.160 1.197
1.1 1.014 1.020 1.027 1.035 1.043 1.058 1.074 1.092 1.111 1.138 1.166
1.2 1.014 1.019 1.026 1.033 1.041 1.054 1.067 1.081 1.096 1.117 1.137
1.3 1.014 1.019 1.025 1.031 1.038 1.049 1.060 1.071 1.083 1.097 1.112
1.4 1.013 1.018 1.024 1.029 1.035 1.044 1.053 1.062 1.070 1.080 1.090
1.5 1.013 1.017 1.022 1.027 1.032 1.039 1.046 1.053 1.059 1.066 1.072
1.6 1.013 1.017 1.021 1.025 1.029 1.035 1.040 1.045 1.049 1.053 1.057
1.7 1.012 1.016 1.019 1.023 1.026 1.031 1.035 1.038 1.040 1.041 1.045
1.8 1.011 1.015 1.018 1.021 1.024 1.027 1.030 1.032 1.033 1.034 1.035
1.9 1.011 1.014 1.016 1.019 1.021 1.024 1.025 1.027 1.027 1.027 1.027
2.0 1.010 1.013 1.015 1.017 1.019 1.020 1.022 1.022 1.022 1.022 1.021
2.2 1.009 1.011 1.013 1.014 1.015 1.015 1.015 1.015 1.015 1.014 1.013
2.6 1.007 1.008 1.008 1.009 1.009 1.008 1.008 1.007 ……. ……. …….
3.0 1.005 1.006 1.005 1.005 1.005 ……. ……. ……. ……. ……. …….
If β <0.3orγ < 1.2 use factor ψ
c
without corrective factor
168 Appendix B Factor ψ or Corrective Factors for Verification Computation
Table B.29 – Coils – counter flow – 4 sections – Fig. A.28 – 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 ……. ……. ……. ……. ……. ……. …… . ……. ……. 1.171
0.5 ……. ……. ……. ……. ……. ……. 1. 071 1.095 1.133 1.180
0.6 ……. ……. ……. ……. 1.039 1.055 1.074 1.096 1.133 1.179
0.7 ……. ……. ……. 1.028 1.040 1.055 1.073 1.095 1.129 1.170
0.8 ……. ……. 1.021 1.028 1.040 1.0 54 1.070 1.090 1.120 1.156
0.9 ……. ……. 1.021 1.028 1.038 1.0 51 1.066 1.083 1.109 1.138
1.0 ……. 1.016 1.021 1.027 1.037 1 .048 1.061 1.075 1.096 1.120
1.1 ……. 1.016 1.020 1.025 1.034 1 .044 1.055 1.067 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 1.011 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
2.2 ……. 1.007 1.008 1.009 1.009 1 .009 1.009 1.008 ……. …….
2.4 ……. ……. 1.007 1.007 ……. ……. ……. ……. ……. …….
If β <0.4orβ >2.4orγ < 1.4 use factor ψ
c
without corrective factor
B.4 Coils 169
Table B.30 – Coils – counter flow – 6 sections – Fig. A.29 – Corrective factor ϕ
c
β
γ
2.9 3.2 3.6 4.0
0.7 ……. ……. ……. 1.077
0.8 ……. ……. 1.055 1.072
0.9 ……. 1.038 1.050 1.065
1.0 ……. 1.035 1.045 1.056
1.1 1.025 1.031 1.039 1.048
1.2 1.023 1.027 1.033 1.040
1.3 1.020 1.023 1.028 1.033
1.4 1.017 1.020 1.023 1.026
1.5 ……. 1.017 1.019 1.021
1.6 ……. ……. 1.015 1.016
If β <0.7orβ >1.6orγ < 2.9 use factor ψ
c
without corrective factor
170 Appendix B Factor ψ or Corrective Factors for Verification Computation
B.5 Tube Banks with Several Passages of External Fluid
B.5.1 Tube Banks with Fluids in Parallel Flow (Fig. 2.10)
Table B.31 – Tube banks – parallel flow – 2 passages of external fluid Fig. A.32 – Corrective
factor ϕ
p
β
γ
0.6 0.7 0.8 0.9 1.0
0.5 ……. ……. ……. ……. 0.989
0.6 ……. ……. ……. 0.991 0.989
0.7 ……. ……. ……. 0.991 0.989
0.8 ……. ……. 0.992 0.990 0.989
0.9 ……. ……. 0.992 0.991 0.989
1.0 ……. 0.994 0.992 0.991 0.989
1.1 ……. 0.994 0.992 0.991 0.990
1.2 ……. 0.994 0.993 0.991 0.991
1.3 ……. 0.994 0.993 0.992 0.991
1.4 ......... 0.994 0.993 0.992 0.992
1.5 0.995 0.994 0.993 0.993 0.993
1.6 0.996 0.995 0.994 0.993 0.994
1.7 0.996 0.995 0.994 0.994 0.994
1.8 0.996 0.995 0.995 0.995 …….
1.9 0.996 0.995 0.995 0.995 …….
2.0 0.996 0.995 0.995 ……. …….
2.1 0.996 0.996 …….. ……. …….
2.2 0.996 0.996 …….. ……. …….
If β <0.5orβ>2.2orγ < 0.5 use factor ψ
p
without corrective factor
B.5 Tube Banks with Several Passages of External Fluid 171
Table B.31 – (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 ……. ……. ……. ……. 0.967 0.960 0.956 0.954 0.955 ……. …….
0.3 ……. 0.981 0.976 0.971 0.967 0 .963 0.962 0.964 ……. ……. …….
0.4 0.985 0.980 0.976 0.972 0.970 0.969 0.971 0.976 ……. ……. …….
0.5 0.984 0.980 0.977 0.974 0.974 0.975 0.980 ……. ……. ……. 1.031
0.6 0.984 0.981 0.979 0.978 0.978 0.982 ……. ……. ……. 1.028 1.047
0.7 0.985 0.982 0.981 0.981 0.983 ……. ……. ……. 1.022 1.041 1.060
0.8 0.986 0.984 0.984 0.985 0.988 ……. ……. 1.018 1.032 1.052 1.072
0.9 0.987 0.986 0.986 0.989 ……. … …. 1.013 1.026 1.041 1.061 1.081
1.0 0.988 0.988 0.989 ……. ……. ……. 1.020 1.034 1.048 1.069 1.089
1.1 0.989 0.990 ……. ……. ……. 1.013 1.026 1.040 1.055 1.075 1.095
1.2 0.990 0.992 ……. ……. ……. 1.018 1.031 1.046 1.061 1.081 1.100
1.3 0.991 ……. ……. ……. 1.010 1.022 1.036 1.050 1.065 1.085 1.104
1.4 0.993 ……. ……. ……. 1.013 1.026 1.040 1.055 1.069 1.089 1.107
1.5 ……. ……. ……. 1.009 1.016 1.029 1.044 1.058 1.073 1.092 1.110
1.6 ……. ……. ……. 1.011 1.019 1.033 1.047 1.061 1.075 1.094 1.111
1.7 ……. ……. 1.007 1.014 1.022 1.0 35 1.049 1.064 1.077 1.095 1.112
1.8 ……. ……. 1.009 1.016 1.024 1.0 38 1.052 1.066 1.079 1.096 1.113
1.9 ……. ……. 1.011 1.018 1.027 1.0 40 1.054 1.067 1.080 1.097 1.113
2.0 ……. 1.006 1.012 1.020 1.029 1 .042 1.055 1.069 1.081 1.098 1.112
2.1 ……. 1.007 1.014 1.022 1.030 1 .043 1.057 1.070 1.082 1.098 1.112
2.2 ……. 1.008 1.015 1.023 1.032 1 .045 1.058 1.070 1.082 1.097 1.111
2.3 ……. 1.010 1.017 1.025 1.033 1 .046 1.059 1.071 1.083 1.097 1.110
2.4 1.004 1.011 1.018 1.026 1.034 1.047 1.059 1.071 1.083 1.096 1.109
2.5 1.005 1.012 1.019 1.027 1.035 1.048 1.060 1.072 1.082 1.096 1.108
2.6 1.006 1.013 1.020 1.028 1.036 1.049 1.060 1.072 1.082 1.095 1.107
2.8 1.008 1.015 1.022 1.030 1.038 1.050 1.061 1.071 1.081 1.093 1.104
3.0 1.009 1.016 1.024 1.031 1.039 1.050 1.061 1.071 1.080 1.091 1.101