Correction to: thermal conductivity of porous sintered metal powder and the Langmuir shape factor

CORRECTION

Correction to: thermal conductivity of porous sintered metal powder and the Langmuir shape factor

Osama M. Ibrahim¹ &Ahmed H. Al-Saiafi²&Sorour Alotaibi¹

#Springer-Verlag GmbH Germany, part of Springer Nature 2021

Correction to: Heat and Mass Transfer

https://doi.org/10.1007/s00231-021-03032-x 1)On the first page, in the Nomenclatures section, the overbar symbol is missing. Add ¯.

Overbar

¯ normalized

2)On Page 3, at the beginning of the second column, correct q:;q:p

s;and q:p

f in the text and Eq.1; the dots, subscripts, and superscripts are misaligned.

At steady state, the total heat flow (q:) in a parallel configura- tion is the sum of the heat flow in the solid-phase (q:p

s) plus the heat flow in the fluid-phase (q:p

fÞ, q:¼q:p

sþq:p

f ð1Þ

3)On Page 4, at the beginning of the second column, correct q:;q:s

s;and q:s

f in the text and Eq.8; the dots, subscripts, and superscripts are misaligned.

At steady state, the total heat flow (q:) in a series configuration is equal to the heat flow in the solid-phase (q:s

s) and the heat flow in the fluid-phase (q:s

fÞ:

q:¼q:s s¼q:s

f ð8Þ

4)On Page 5, the second column starting with the sentence before Eq.19a, change Eq. 20 to Eq. 17. Correct, Eqs.19aand 19bby adding overbars, onS^p_s;A^p_s;S^p_f;and A^p_f.

The normalized Langmuir shape factors in Eq. 17 are related to the normalized cross-sectional areas and poros- ity as follows:

S^p_s ¼S^p_s

S ¼ A^p_s=L^p_s A=L

ð Þ ¼ A^p_s²=A²

L^p_sA^p_s=LA¼ A^p_s 2

1−P

ð Þ ð19aÞ

S^p_f ¼S^p_f

S ¼A^p_f=L^p_f A=L

ð Þ ¼ A^p_f²=A² L^p_fA^p_f=LA

¼ A^p_f²

P ð19bÞ

5)On Page 6, Correct Eqs.29aand29bby adding overbars on S^s_s;A^s;S^s_f;and A^s.

S^s_s¼S^s_s

S ¼ðA^s=AÞ

L^s_s=L¼ A^s2=A² L^s_sA^s=LA¼ A^s

2

1−P

ð Þ ð29aÞ

S^s_f ¼S^s_f

S ¼ ðA^s=AÞ L^s_f=L

¼ A^s2=A² L^s_fA^s=LA ¼A^s

2

P ð29bÞ

6)On Page 7, the second paragraph on the second column, correct q:;q:p

s;q:p

f; and q:s

sf in the text and Eq.35; the dots, subscripts, and superscripts are misaligned.

The online version of the original article can be found at https://doi.org/

10.1007/s00231-021-03032-x

* Osama M. Ibrahim osama.ibrahim@ku.edu.kw Ahmed H. Al-Saiafi Asaiafi@kockw.com Sorour Alotaibi sr.alotaibi@ku.edu.kw

1 Department of Mechanical Engineering, College of Engineering and Petroleum, Kuwait University, P. O. Box 5969, 13060 Safat, Kuwait

2 Kuwait Oil Company (K.S.C.), P.O. Box 9758, 61008 Ahmadi, Kuwait

https://doi.org/10.1007/s00231-021-03064-3

Published online: 16 April 2021

Heat and Mass Transfer (2021) 57:1561–1563

Assuming the solid and fluid phases of the porous medium are transferring heat in parallel and series at the same time. Simple superpositioning of the parallel and series heat transfer con- figuration results in an energy balance where the total heat flow (q:), at steady state is the sum of the parallel heat flow in the solid-phase (q:p

s ), the parallel heat flow in the fluid- phase (q:p

fÞ;and the series heat flow through the solid-phase and fluid-phase q:s

sf . q:¼q:p

sþq:p f þq:s

sf ð35Þ

7) On Page 7, Eq. 36, correct ke;kf; S^p_s;S^s_s; and S^s_f by adding the overbars,

ke¼S^p_s þkf S^p_f þ kfS^s_sS^s_f S^s_sþkf S^s_f

ð36Þ

8)On Page 8, Eq. 41, correct, ke and kf; by adding the overbars,

ke¼ð1−PÞ²ⁿ⁻¹þkfð1−ð1−PÞⁿÞ²

P þkfðaP^mð1−PÞÞ² Pþkfð1−PÞ ð41Þ 9)On Page 8, the beginning of the second column, the last sentence before Section 4, change“Equations 40 to 43”to

“Equations 38 to 41”,

Equations. 38 to 41 represent the re-examined parallel-series model.

10)On Page 12, Fig. 13 caption, correct (q:p s=q:Þ; q:p

f=q:

, and (q:p

sf=q:Þ; the dots, subscripts, and superscripts are misaligned.

Fig. 13.Fractions of the parallel heat transfer through the solid-phase (q:p

s=q:Þ and the fluid-phase q:p f=q:

; also shown is the fraction of the series heat transfer through the solid and fluid phases (q:p

sf=q:Þ vs. porosity: compar- isons between the re-examined parallel and the re- examined parallel-series models.

11) On Page 13, first column, second paragraph, correct (q:p

s=q:Þ; q:p f=q:

, and (q:p

sf=q:Þ; the dots, subscripts, and super- scripts are misaligned.

Fractions of the parallel heat transfer through the solid-phase (q:p

s=q:Þand through the fluid-phase q:p f=q:

, as predicted by the re-examined parallel and parallel-series model, are shown in Fig. 13; whereq:is the total heat transfer rate. Also shown in Fig. 13 is a fraction of the series heat transfer through the solid and fluid phases (q:p

sf=q:Þ, as predicted by the parallel-series model.

12)On Page 15, Appendix B, Table2; Correct the equations by adding the overbars; they are missing on some variables;

correct the subscripts and superscripts, interference, by adding more space between lines.

Table 2 The classical and re-examined parallel models

Description Classical parallel model Equation Re-examined parallel model Equation

Cross-sectional areas A^p_s¼1−P A^p_f ¼P

(6a)

(6b) A^p_s¼ð1−PÞⁿ A^p_f ¼1−ð1−PÞⁿ

(21b) (21b) Lengths of heat transfer pathway L^p_s¼L^p_f ¼1 ⁽⁴⁾ L^p_s¼ð1−PÞ¹⁻ⁿ

L^p_f ¼P=ð1−ð1−PÞⁿÞ

(22a) (22b) Langmuir shape factors S^p_s¼1−P

S^p_f ¼P

(25a)

(25b) S^p_s¼ð1−PÞ²ⁿ⁻¹ S^p_f ¼ð1−ð1−PÞⁿÞ²=P

(23a) (23b)

Effective thermal conductivity ke¼S^p_sþkfS^p_f ⁽¹⁸⁾ ke¼S^p_sþkfS^p_f ⁽¹⁸⁾

Fitting parameter n=1.567

Coefficient of determination R²=0.93

Root mean square error RMSE=0.045

1562 Heat Mass Transfer (2021) 57:1561–1563

13)On Page 15, Appendix B, Table3; Correct the equations by adding the overbars; they are missing on some variables;

correct the subscripts and superscripts, interference, by adding more space between lines.

14)ON Page 15, Appendix B, Table4; Correct the equations by adding the overbars; they are missing on some variables;

correct the subscripts and superscripts, interference, by adding more space between lines.

The original article has been corrected. ^{Publisher’s note}Springer Nature remains neutral with regard to jurisdic- tional claims in published maps and institutional affiliations.

Table 3 The classical and re-examined series models

Description Classical series model Equation Re-examined series model Equation

Contact area A^s¼1 ⁽¹²⁾ A^s¼1þaP^mð1−PÞ ⁽³²⁾

Lengths of heat transfer

pathway L^s_s¼1−P

L^s_f ¼P

(14a)

(14b) L^s_s¼ð1−PÞ=ð1þaP^mð1−PÞÞ L^s_f ¼P=ð1þaP^mð1−PÞÞ

(33a) (33b) Langmuir shape factors S^s_s¼1=ð1−PÞ

S^s_f ¼1=P

(34a)

(34b) S^s_s¼ð1þaP^mð1−PÞÞ²=ð1−PÞ S^s_f ¼ð1þaP^mð1−PÞÞ²=P

(34a) (34b) Effective thermal conductivity ke¼kf S^s_f S^s_s= S^s_sþkf S^s_f

(28) ke¼kf S^s_f S^s_s= S^s_sþkf S^s_f

(28)

Fitting parameter a=109.1;m=0.513

Coefficient of determination R²=0.89

Root mean square error RMSE=0.054

Table 4 The re-examined parallel-series model

Description Re-examined parallel-series model Equation

Cross-sectional areas

Contact area

A^p_s¼ð1−PÞⁿ A^p_f ¼1−ð1−PÞⁿ A^s¼aP^mð1−PÞ

(38a) (38b) (38c)

Lengths of heat transfer pathway L^p_s¼ð1−PÞ¹⁻ⁿ

L^p_f ¼P=ð1−ð1−PÞⁿÞ L^S_S¼ð1−PÞ=ð1þaP^mð1−PÞÞ L^s_f ¼P=ð1þaP^mð1−PÞÞ

(39a) (39b) (39c) (39d)

Langmuir shape factors

S^p_s¼ð1−PÞ²ⁿ⁻¹ S^p_f ¼ð1−ð1−PÞⁿÞ²=P S^s_s¼ðaP^mð1−PÞÞ²=ð1−PÞ S^s_f ¼ðaP^mð1−PÞÞ²=P

(40a) (40b) (40c) (40d)

Effective thermal conductivity ke¼S^p_sþkf S^p_f þ kfS^s_sS^s_f=S^s_sþkf S^s_f

(36) Fitting parameters

Coefficient of determination Root Mean Square Error

a=179.7;m=2.307;n=1.728 R²=0.95

RMSE=0.038

1563 Heat Mass Transfer (2021) 57:1561–1563