Analytical determination of the temperature distribution and Nusselt numbers in rectangular ducts with constant axial heat flux

International Journal of Heat and Mass Transfer 43 (2000) 3259±3261

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Errata

Analytical determination of the temperature distribution and Nusselt numbers in rectangular ducts with constant axial heat ¯ux [International Journal of Heat and Mass Transfer 43 (2000) 741±755] p Gian Luca Morini The publishers regret that Tables 3±6 (pp. 745 and 746) of the above article were incorrectly published. The correct versions of those tables are printed below. Table 3 The coecients tn, m of Eq. (13): versions 1L, 1S, 2L, 2S Version 1L

tn, m mÿ1

ÿ

1 1 X 128b X dn …ÿ1† 2 p4 A…b† kˆ1, odd jˆ1, odd …4n 2 b 2 ‡ m 2 †…b 2 k 2 ‡ j 2 †…k 2 ÿ n 2 †…4j 2 ÿ m 2 †

for n = 0 or even, and m odd 1S

nÿ1

ÿ

1 1 X 128b X dm …ÿ1† 2 4 p A…b† kˆ1, odd jˆ1, odd …n 2 b 2 ‡ 4m 2 †…b 2 k 2 ‡ j 2 †…4k 2 ÿ n 2 †… j 2 ÿ m 2 †

for n odd and m = 0 or even 2L

ÿ

4b p3 A…b†

1 X kˆ1, odd m…n

2b 2

dn ‡ m 2 †…b 2 k 2 ‡ m 2 †…k 2 ÿ n 2 †

for n = 0 or even, and m odd 2S

ÿ

4b p3 A…b†

1 X

dm 2 b 2 ‡ m 2 †…b 2 n 2 ‡ j 2 †… j 2 ÿ m 2 † n…n jˆ1, odd

n odd and m = 0 or even di=1 if i$0, di=1/2 if i = 0

p

PII of original article: S0017-9310(99)00188-X

0017-9310/00/$ - see front matter 7 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 7 - 9 3 1 0 ( 0 0 ) 0 0 0 8 3 - 1

3260

Errata / Int. J. Heat Mass Transfer 43 (2000) 3259±3261

Table 4 The coecients tn, m of Eq. (13): versions 2C, 3L, 3S, 4 2C

nÿ1

ÿ

mÿ1

1 1 X 256b X …ÿ1† 2 …ÿ1† 2 p4 A…b† kˆ1, odd jˆ1, odd …n 2 b 2 ‡ m 2 †…b 2 k 2 ‡ j 2 †…4k 2 ÿ n 2 †…4j 2 ÿ m 2 †

for n and m odd 3L

ÿ

nÿ1

1 X

32b p3 A…b†

…ÿ1† 2 2 b ‡ 4m 2 †…b 2 k 2 ‡ m 2 †…4k 2 ÿ n 2 † m…n kˆ1, odd 2

for n and m odd 3S

mÿ1

ÿ

1 32b X …ÿ1† 2 p3 A…b† jˆ1, odd n…4n 2 b 2 ‡ m 2 †…b 2 n 2 ‡ j 2 †…4j 2 ÿ m 2 †

for n and m odd 4

ÿ

b p 2 A…b†mn…n 2 b 2 ‡ m 2 † 2

for n and m odd

Table 5 The bulk temperature for the H1 problem: versions 1L, 1S, 2L, 2S Version 1L

1S

2L

2S

Tb ÿ

1 1 1 1 1 1 X X X X X 1024b X dn p6 A2 …b† nˆ0, even mˆ1, odd lˆ1, odd pˆ1, odd kˆ1, odd jˆ1, odd …l 2 b2 ‡ p2 †…4b2 n2 ‡ m2 †…l 2 ÿ n2 †…4p2 ÿ m2 †…b2 k2 ‡ j2 †…k2 ÿ n2 †…4j2 ÿ m2 †

ÿ

1 1 1 1 1 1 X X X X X 1024b X dm 6 2 p A …b† nˆ1, odd mˆ0, even lˆ1, odd pˆ1, odd kˆ1, odd jˆ1, odd …l 2 b2 ‡ p2 †…b2 n2 ‡ 4m2 †…4l 2 ÿ n2 †… p2 ÿ m2 †…b2 k2 ‡ j2 †…4k2 ÿ n2 †… j2 ÿ m2 †

ÿ

ÿ

4b p4 A2 …b†

1 X

1 X

1 X

1 X

dn 2 …l 2 b2 ‡ m2 †…b2 n2 ‡ m2 †…l 2 ÿ n2 †…b2 k2 ‡ m2 †…k2 ÿ n2 † m nˆ0, even mˆ1, odd lˆ1, odd kˆ1, odd

1 1 1 1 X X X X 4b dm p4 A2 …b† nˆ1, odd mˆ0, even pˆ1, odd jˆ1, odd n2 …n2 b2 ‡ m2 †…b2 n2 ‡ p2 †… p2 ÿ m2 †…b2 n2 ‡ j2 †… j2 ÿ m2 †

di=1 if i$0, di=1/2 if i = 0

Errata / Int. J. Heat Mass Transfer 43 (2000) 3259±3261

3261

Table 6 The bulk temperature for the H1 problem: versions 2C, 3L, 3S, 4 Version 2C

3L

3S

4

Tb ÿ

ÿ

ÿ

ÿ

1 1 1 1 1 1 X X X X X 4096b X 1 p6 A2 …b† nˆ1, odd mˆ1, odd lˆ1, odd pˆ1, odd kˆ1, odd jˆ1, odd …l 2 b2 ‡ p2 †…b2 n2 ‡ m2 †…4l 2 ÿ n2 †…4p2 ÿ m2 †…b2 k2 ‡ j2 †…4k2 ÿ n2 †…4j2 ÿ m2 †

64b p4 A2 …b†

64b p4 A2 …b†

1 X

1 X

1 X

1 X

1 X

1 X

1 X

1 X

1 2 … j2 b2 ‡ m2 †…b2 n2 ‡ 4m2 †…4j2 ÿ n2 †…b2 k2 ‡ m2 †…4k2 ÿ n2 † m nˆ1, odd mˆ1, odd jˆ1, odd kˆ1, odd

1 2 …4n2 b2 ‡ m2 †…b2 n2 ‡ j2 †…4j2 ÿ m2 †…b2 n2 ‡ l 2 †…4l 2 ÿ m2 † n nˆ1, odd mˆ1, odd jˆ1, odd lˆ1, odd

1 1 X X b 1 4p2 A2 …b† nˆ1, odd mˆ1, odd n2 m2 …b2 n2 ‡ m2 †3