Heat transfer and pressure drop correlations for laminar flow in an in-line and staggered array of circular cylinders

Figures & data

Figure 1. Concept of a flat tube heat exchanger with plate-fin wire structure.

Figure 2. Different wire structure heat exchangers with high aspect ratio tested for thermal–hydraulic performance: (a) continuous wire structure; (b) pin fin structure; and (c) woven wire structure (adopted from [18,21,22]).

Figure 3. Cross-section through a wire structure heat exchanger.

Figure 4. Boundary conditions of the in-line (a) and staggered (b) cross-section model [18].

Table 1. Definition of nondimensional input parameters to simulation model with minimal and maximal values in parametric study.

Symbol	Definition	Description	Min.	Max.
$a$	$\frac{l_{\text{lat}}}{d_{\text{wire}}}$	Nondimensional lateral distance of the wires	2	12
$b$	$\frac{l_{\text{long}}}{d_{\text{wire}}}$	Nondimensional longitudinal distance of the wires	1.3	8
${\text{Re}}_{\text{air},\mathrm{s}\mathrm{t}}$	$\frac{v_{\text{air},\mathrm{i}\mathrm{n},\mathrm{s}\mathrm{t}}{\mathrm{ }d}_{\text{wire}}\mathrm{ }{\rho }_{\text{air}}}{{\mu }_{\text{air}}\mathrm{ }}$	Reynolds number	3	60
$n_{\text{wires}}$	$\frac{L_{\text{st}}}{l_{\text{long}}}\mathrm{ }$	Number of wires in flow direction	2	200
${\text{Pr}}_{\text{air}}\mathrm{ }$	$\frac{v_{\text{air},\mathrm{i}\mathrm{n},\mathrm{s}\mathrm{t}}{c}_{p,\text{air }}{\rho }_{\text{air}}}{k_{\text{air}}}$	Prandtl number	0.71	0.71

Figure 5. Meshing of 2D cross-section with different levels of grid refinement.

Table 2. Grid Convergence Index (GCI) based on Richardson method [27] for Nusselt number and friction factor.

Solution	$r_{\text{mesh}}$	$p_{\text{mesh}}$	${\text{GCI}}_{12}$ in %	${\text{GCI}}_{23}$ in %
Nusselt number	4	0.245	0.032	0.045
Friction factor	4	1.013	0.009	0.038

Geometry consists of five wires in row ($a$ = 8, $b$ = 3) at ${\text{Re}}_{\text{st}}$ = 11.25. Index 1 represents finer mesh, index 3 coarser mesh.

Figure 6. Pressure difference, velocity, and temperature of a 2D in-line wire structure simulation with $a$ = 10, $b$ = 3, ${\text{Re}}_{\text{st}}$ = 20, and$\mathrm{ }{n}_{\text{wires}}=20$. Contour lines for pressure difference are equally distributed. Velocity streamlines are colored based on the temperature scale.

Figure 7. Nusselt number and Fanning friction factor of an in-line wire structure for a developed flow as a function of Reynolds number and geometry parameters $a$ and $b$.

Figure 8. Correlated global (solid line) and local (dashed line) Nusselt number, ${\text{Nu}}_{\text{st}}$ and ${\text{Nu}}_{\mathrm{s}\mathrm{t},\text{local}}$, respectively, for an in-line wire arrangement, as a function of the number of wires based on the simulated global data points (squares) for ${\text{Nu}}_{\text{st}}$ and fixed values for$\mathrm{ }a=10,b=3,{\text{Re}}_{\text{st}}=20$. Downstream of the thermal entrance length (dotted line), the flow is declared as thermally developed.

Figure 9. Correlated global (solid line) and local (dashed line) friction factor,$\mathrm{ }{\mathrm{f}}_{\text{st}}$ and ${\mathrm{f}}_{\mathrm{s}\mathrm{t},\text{local}}$, respectively, for an in-line wire arrangement as a function of the number of wires based on the simulated global data points (squares) for ${\mathrm{f}}_{\text{st}}$ and fixed values for$\mathrm{ }a=10,b=3,{\text{Re}}_{\text{st}}=20$. Downstream of hydraulic entrance length (dotted line), the flow is declared as hydraulically developed; in-line arrangement.

Figure 10. Predicted (correlated) values versus simulated values for (a) the Nusselt number ${\text{Nu}}_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}$ and (b) the Fanning friction factor $f_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}$. Data are based on Eqs. (13)(13) $${\text{Nu}}_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}=\mathrm{N}{\mathrm{u}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}+\frac{C_{1,\mathrm{N}\mathrm{u}}}{C_{2,\mathrm{N}\mathrm{u}}{(y}^{\mathrm{*}}-1)}\left(1-{y^{\mathrm{*}}}^{{-C}_{2,\mathrm{N}\mathrm{u}}}\right)\mathrm{ }$$(13) and (19)(19) $$f_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}=f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}+\frac{C_{1,f}}{C_{2,f}{(y}^{\mathrm{*}}-1)}\left(1-{y^{\mathrm{*}}}^{{-C}_{2,f}}\right)\mathrm{ }$$(19) . The predicted values are correlated via the number of wires $n_{\text{wires}}$ (see Table 1) for specific Reynolds numbers ${\text{Re}}_{\text{st}}$ and geometry parameters $a$ and $b$ for an in-line arrangement.

Table 3. Derived correlations for $\mathrm{N}{\mathrm{u}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$ and ${\mathrm{f}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$ for an in-line wire structure.

Coefficients	Auxiliary coefficients
$\mathrm{N}{\mathrm{u}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}={\tilde{A}}_{\text{Nu}}+0.021\mathrm{ }{\text{Re}}_{\text{st}}^{{\tilde{B}}_{\text{Nu}}}$	${\tilde{A}}_{\text{Nu}}=\mathrm{ }\frac{2.16\mathrm{ }b}{a\mathrm{ }+\mathrm{ }b\mathrm{ }-\mathrm{ }{\left(\frac{7.68\mathrm{ }b}{{\left(3.56\mathrm{ }+\mathrm{ }b\right)}^{0.5}}\right)}^{0.5}}$ ${\tilde{B}}_{\text{Nu}}=\frac{b^{0.345}}{\text{exp}\left(0.25\mathrm{ }a\right)\mathrm{ }}$
$f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}=\text{exp}\left({\tilde{A}}_f+{\tilde{B}}_f\log \left(\mathrm{R}{\mathrm{e}}_{\text{st}}\right)\right)$	${\tilde{A}}_f=0.005531\mathrm{ }a\mathrm{ }b\mathrm{ }+\mathrm{ }0.005751\mathrm{ }{a}^2\mathrm{ }+\frac{8.6054\mathrm{ }-\mathrm{ }0.2222\mathrm{ }b}{a\mathrm{ }+\mathrm{ }0.1311\mathrm{ }a\mathrm{ }b}\mathrm{ }-\mathrm{ }0.9024\mathrm{ }-\mathrm{ }0.2985\mathrm{ }a\mathrm{ }-0.008528\mathrm{ }{b}^2$ ${\tilde{B}}_f=0.07776\mathrm{ }b\mathrm{ }+\mathrm{ }0.01624\mathrm{ }a\mathrm{ }+\mathrm{ }0.0001427\mathrm{ }a\mathrm{ }{b}^2\mathrm{ }-0.06345\mathrm{ }\log \left(a\right)-0.03442b\log \left(a\right)-1.02134$

Table 4. Derived correlations for coefficients of ${\text{Nu}}_{\text{st}}$ and ${\mathrm{f}}_{\text{st}}$ for in-line wire structure based on the Eqs. (13)(13) $${\text{Nu}}_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}=\mathrm{N}{\mathrm{u}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}+\frac{C_{1,\mathrm{N}\mathrm{u}}}{C_{2,\mathrm{N}\mathrm{u}}{(y}^{\mathrm{*}}-1)}\left(1-{y^{\mathrm{*}}}^{{-C}_{2,\mathrm{N}\mathrm{u}}}\right)\mathrm{ }$$(13) and (19)(19) $$f_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}=f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}+\frac{C_{1,f}}{C_{2,f}{(y}^{\mathrm{*}}-1)}\left(1-{y^{\mathrm{*}}}^{{-C}_{2,f}}\right)\mathrm{ }$$(19) .

Coefficients	Auxiliary coefficients
$C_{1,\mathrm{N}\mathrm{u}}=\mathrm{ }\begin{array}{c}{\tilde{A}}_{1,\mathrm{N}\mathrm{u}}\text{ if }{\text{Re}}_{\text{st}}^{0.8}{a}^{2.0}{b}^{-1.1}>24\\ 0\text{ if }{\text{Re}}_{\text{st}}^{0.8}{a}^{2.0}{b}^{-1.1}\le 24\end{array}$	${\tilde{A}}_{1,\mathrm{N}\mathrm{u}}=1.6896\mathrm{ }+\mathrm{ }0.03636\mathrm{ }b\mathrm{ }+\mathrm{ }0.02745\mathrm{ }\mathrm{R}{\mathrm{e}}_{\text{st}}+\mathrm{ }\frac{0.5171\mathrm{ }a\mathrm{ }\mathrm{N}{\mathrm{u}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}}{b}\mathrm{ }+\mathrm{ }\frac{-4.4822}{2.0963\mathrm{ }+\mathrm{ }0.1266\mathrm{ }\mathrm{R}{\mathrm{e}}_{\text{st}}}\mathrm{ }-\mathrm{ }0.6369\mathrm{ }\mathrm{N}{\mathrm{u}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$
$C_{1,f}=\mathrm{ }\begin{array}{c}{\tilde{A}}_{1,f}\text{ if }R{e}_{st}^{0.8}{a}^{1.8}{b}^{-1.0}>12\\ 0\text{ if }R{e}_{st}^{0.8}{a}^{1.8}{b}^{-1.0}\le 12\end{array}$	${\tilde{A}}_{1,f}=2.9\cdot {10}^{-5}\mathrm{ }-\mathrm{ }0.428\mathrm{ }{f}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}\mathrm{ }+\frac{f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}^{0.5}}{{\tilde{B}}_{1,f}}$ ${\tilde{B}}_{1,f}=1.631\mathrm{ }b\mathrm{ }+\mathrm{ }0.064\mathrm{ }a\text{ log}\left(\mathrm{R}{\mathrm{e}}_{\text{st}}\right)\mathrm{ }-\mathrm{ }0.62\mathrm{ }-\mathrm{ }0.0562\mathrm{ }a\mathrm{ }{b}^{0.5}-\mathrm{ }0.155\mathrm{ }b\text{ log}\left(\mathrm{R}{\mathrm{e}}_{\text{st}}\right)$
$C_2={\mathrm{C}}_{2,f}=C_{2,\mathrm{N}\mathrm{u}}=\frac{1}{1+{\tilde{A}}_2}$	${\tilde{A}}_2=3.77\cdot {10}^{-6}\mathrm{ }{\text{Re}}_{\text{st}}^{1.95}\mathrm{ }{a}^{3.81}\mathrm{ }{b}^{-0.68}$

Figure 11. Auxiliary coefficients$\mathrm{ }{\tilde{A}}_{\text{Nu}}\mathrm{ }$(a), ${\tilde{B}}_{\text{Nu}}$ (b), ${\tilde{A}}_f$ (c), and ${\tilde{B}}_f$ (d) needed for calculation of correlated Nusselt number and friction factor based on Table 3 for in-line arrangement. Geometry parameter $a$ is shown on the contour lines.

Figure 12. Nondimensional entrance lengths $L_{\text{th}}^{\mathrm{*}}$ and $L_{\text{hy}}^{\mathrm{*}}\mathrm{ }$for an in-line arrangement based on the Reynolds number ${\text{Re}}_{\text{st}}$ and geometry parameters $a$ and $b$. Entrances lengths below 0.1 are not shown.

Figure 13. Predicted (correlated) values versus simulated values for (a) the Nusselt number ${\text{Nu}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$ and (b) the Fanning friction factor ${\mathrm{f}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$ for a developed flow. Data are based on Table 3. The predicted values are correlated via the Reynolds number ${\text{Re}}_{\text{st}}$ and geometry parameters $a$ and $b$ for an in-line wire arrangement.

Figure 14. Predicted (correlated) values versus simulated values for (a) the Nusselt number ${\text{Nu}}_{\mathrm{s}\mathrm{t},{\mathrm{y}}^{\mathrm{*}}}$ and (b) the Fanning friction factor ${\mathrm{f}}_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}$. Data are based on Eqs. (13)(13) $${\text{Nu}}_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}=\mathrm{N}{\mathrm{u}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}+\frac{C_{1,\mathrm{N}\mathrm{u}}}{C_{2,\mathrm{N}\mathrm{u}}{(y}^{\mathrm{*}}-1)}\left(1-{y^{\mathrm{*}}}^{{-C}_{2,\mathrm{N}\mathrm{u}}}\right)\mathrm{ }$$(13) and (19)(19) $$f_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}=f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}+\frac{C_{1,f}}{C_{2,f}{(y}^{\mathrm{*}}-1)}\left(1-{y^{\mathrm{*}}}^{{-C}_{2,f}}\right)\mathrm{ }$$(19) (Tables 3 and 4). The predicted values are correlated via the Reynolds number ${\text{Re}}_{\text{st}}$, geometry parameters $a$ and $b$, and the number of wires for an in-line wire arrangement.

Table 5. Percentage of correlated data which satisfy a relative residual error below 5% and 10% for ${\text{Nu}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$, ${\mathrm{f}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$,$\mathrm{ }{\text{Nu}}_{\mathrm{s}\mathrm{t},{\mathrm{y}}^{\mathrm{*}}}$, and ${\mathrm{f}}_{\mathrm{s}\mathrm{t},{\mathrm{y}}^{\mathrm{*}}}$.

		${\text{Nu}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$	$f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$	${\text{Nu}}_{\mathrm{s}\mathrm{t},{\mathrm{y}}^{\mathrm{*}}}$		$f_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}$
		$n_{\text{wires}}$
		$>L_{\text{th}}^{\mathrm{*}}$	$>L_{\text{hy}}^{\mathrm{*}}$	$>2$	> 5	> 2	> 5
Relative residual error	<5%	81 %	94 %	76 %	79 %	89 %	92 %
	<10%	94 %	99 %	93 %	95 %	97 %	99 %

The percentage is specified for different number of wires in an in-line wire arrangement.

Table B1. Predicted correlation for coefficients of ${\text{Nu}}_{\text{st}}$ and ${\mathrm{f}}_{\text{st}}$ for staggered wire structure based on the Eqs. (13)(13) $${\text{Nu}}_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}=\mathrm{N}{\mathrm{u}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}+\frac{C_{1,\mathrm{N}\mathrm{u}}}{C_{2,\mathrm{N}\mathrm{u}}{(y}^{\mathrm{*}}-1)}\left(1-{y^{\mathrm{*}}}^{{-C}_{2,\mathrm{N}\mathrm{u}}}\right)\mathrm{ }$$(13) and (19)(19) $$f_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}=f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}+\frac{C_{1,f}}{C_{2,f}{(y}^{\mathrm{*}}-1)}\left(1-{y^{\mathrm{*}}}^{{-C}_{2,f}}\right)\mathrm{ }$$(19) .

Coefficients	Auxiliary coefficients
${\text{Nu}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}={\tilde{A}}_{\text{Nu}}+0.021\mathrm{ }{\text{Re}}_{\text{st}}^{{\tilde{B}}_{\text{Nu}}}$	${\tilde{A}}_{\text{Nu}}=\mathrm{ }0.91\mathrm{ }+\frac{2.57}{b}\mathrm{ }+\mathrm{ }\frac{2.2\mathrm{ }-\mathrm{ }0.78a}{\text{exp}\left(b\right)}$ ${\tilde{B}}_{\text{Nu}}=0.598\mathrm{ }+\mathrm{ }0.065\mathrm{ }\frac{a}{b}\mathrm{ }-\mathrm{ }0.393{\left(\frac{a}{b}\right)}^{0.5}$
$f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}=\text{exp}\left({\tilde{A}}_f+{\tilde{B}}_f\log \left(\mathrm{R}{\mathrm{e}}_{\text{st}}\right)\right)$	${\tilde{A}}_f=1.28\mathrm{ }+\mathrm{ }0.62\mathrm{ }\frac{b}{a}\mathrm{ }+\frac{4.31\mathrm{ }-\mathrm{ }2.05\mathrm{ }{\left(a\right)}^{0.5}}{b}\mathrm{ }-\mathrm{ }0.015\mathrm{ }a\mathrm{ }-\mathrm{ }2.06\text{ log}\left(b\right)$ ${\tilde{B}}_f=-\frac{1.43}{b}\mathrm{ }+\frac{2.65}{a\mathrm{ }b}\mathrm{ }+\mathrm{ }0.078\frac{a}{b^2}\mathrm{ }-\mathrm{ }0.297\mathrm{ }-\mathrm{ }b\mathrm{ }{0.23}^a\mathrm{ }-\mathrm{ }0.025\mathrm{ }a$
$C_{1,\mathrm{N}\mathrm{u}}=\mathrm{ }\begin{array}{ll}{\tilde{A}}_{1,\mathrm{N}\mathrm{u}}& \text{if }{\text{Re}}_{\text{st}}^{0.9}{a}^{2.4}{b}^{-1.2}>240\\ 0& \text{if }{\text{Re}}_{\text{st}}^{0.9}{a}^{2.4}{b}^{-1.2}\le 240\end{array}$	${\tilde{A}}_{1,\mathrm{N}\mathrm{u}}=0.22\mathrm{ }+\mathrm{ }0.16\mathrm{ }\mathrm{R}{\mathrm{e}}_{\text{st}}+\frac{0.89}{b}+\mathrm{ }0.02\mathrm{ }a\mathrm{ }b\mathrm{ }+\mathrm{ }0.000045\mathrm{ }\mathrm{R}{\mathrm{e}}_{\text{st}}^3-\mathrm{ }0.35\mathrm{ }b\mathrm{ }-\mathrm{ }0.0021\mathrm{ }\mathrm{R}{\mathrm{e}}_{\text{st}}^2-\mathrm{ }0.2{\text{ Nu}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}^2$
$C_{1,f}=\mathrm{ }\begin{array}{cc}{\tilde{A}}_{1,\mathrm{f}}& \text{if }\mathrm{R}{\mathrm{e}}_{st}^{0.9}{a}^{2.2}{b}^{-1.1}>255\\ 0& \text{if }\mathrm{R}{\mathrm{e}}_{\mathrm{s}\mathrm{t}}^{0.9}{a}^{2.2}{b}^{-1.1}\le 255\end{array}$	${\tilde{A}}_{1,f}=f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}\mathrm{ }\left[0.15\mathrm{ }\mathrm{R}{\mathrm{e}}_{\text{st}}^{0.5}-\frac{4.97}{a}\mathrm{ }+\mathrm{ }\frac{0.017\mathrm{ }a{\mathrm{ }\left(0.87\mathrm{ }+\mathrm{ }\mathrm{R}{\mathrm{e}}_{\text{st}}\right)}^{0.5}{\left(13.85\text{ log}\left(a\right)\right)}^{0.5}}{b}\mathrm{ }-\mathrm{ }0.46\right]\mathrm{ }$
$C_{2,\mathrm{N}\mathrm{u}}=\mathrm{ }\frac{1.20}{1+{\tilde{A}}_{2,\mathrm{N}\mathrm{u}}}$	${\tilde{A}}_{2,\mathrm{N}\mathrm{u}}=3.070\cdot {10}^{-4}\mathrm{ }{\text{Re}}_{\text{st}}^{0.886}\mathrm{ }{a}^{2.719}\mathrm{ }{b}^{-0.928}$
$C_{2,f}=\mathrm{ }\frac{1.01}{1+{\tilde{A}}_{2,f}}$	${\tilde{A}}_{2,f}=3.286\cdot {10}^{-6}\mathrm{ }{\text{Re}}_{\text{st}}^{1.448}\mathrm{ }{a}^{3.842}\mathrm{ }{b}^{-1.260}$

Correlation is valid for:$\mathrm{ }3\le a\le 12$,$\mathrm{ }1.3\le b\le 8$,$\text{ and }3\le \mathrm{R}{\mathrm{e}}_{\text{st}}\le 60$.

Table B2. Percentage of correlated data which satisfy a relative residual error below 5% and 10% for ${\text{Nu}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$, ${\mathrm{f}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$,$\mathrm{ }{\text{Nu}}_{\mathrm{s}\mathrm{t},{\mathrm{y}}^{\mathrm{*}}}$, and ${\mathrm{f}}_{\mathrm{s}\mathrm{t},{\mathrm{y}}^{\mathrm{*}}}\mathrm{ }$in staggered wire arrangement.

		${\text{Nu}}_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$	$f_{\mathrm{s}\mathrm{t},\mathrm{\infty }}$	${\text{Nu}}_{\mathrm{s}\mathrm{t},{\mathrm{y}}^{\mathrm{*}}}$		$f_{\mathrm{s}\mathrm{t},y^{\mathrm{*}}}$
		$n_{\text{wires}}$
		$>L_{\text{th}}^{\mathrm{*}}$	$>L_{\text{hy}}^{\mathrm{*}}$	$>2$	> 5	> 2	> 5
Relative residual error $3\le a\le 12$	<5%	67	71	68	68	68	69
	<10%	90	89	92	91	87	88
Relative residual error $5\le a\le 12$	<5%	70	80	73	74	78	79
	<10%	88	95	91	90	95	95

The percentage is specified for different number of wires. Correlation is valid for: $3\le a\le 12$, $1.3\le b\le 8$, and $3\le \mathrm{R}{\mathrm{e}}_{\text{st}}\le 60$.

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