In this study, a closed-form analytical solution is presented for fully developed forced convection in a cylindrical tube partially filled with open-cell metal foam. The flow field is modeled using the Brinkman–Darcy approach, while heat transfer between the solid and fluid phases is analyzed based on the local thermal non-equilibrium (LTNE) model. Compared with the local thermal equilibrium (LTE) assumption, LTNE provides more accurate predictions, particularly when the solid-to-fluid thermal conductivity ratio is high. The validity of the model is confirmed through comparison with two limiting cases: a hollow tube and a fully filled tube. Owing to the closed-form nature of the solution, system performance can be evaluated continuously over the entire porosity range of (0,1).
The results show that, as the dimensionless interfacial radius increases, the friction factor decreases and varies inversely with the Reynolds number. The Nusselt number exhibits a non-uniform dependence on porosity, and for each pore density there exists an optimal porosity that maximizes heat transfer; for ω=30 PPI, this optimum is obtained at ε=0.87. In addition, the dependence of Nu on the interfacial radius is nonlinear, and in some ranges the thermal performance becomes even weaker than that of the hollow tube. Overall, the findings indicate that a partially filled configuration, with proper selection of geometric and morphological parameters, can achieve an effective balance between heat transfer enhancement and pressure-drop control.
1]V.V. Calmidi, R.L. Mahajan, "Forced convection in high porosity metal foams", ASME J. Heat Transfer, vol. 44 pp 827–836,2011doi:10.1115/1.1287793
[2]C.Y. Zhao, T. Kim, T.J. Lu, H.P. Hodson,"Thermal transport phenomena in porvair metal foams and sintered beds", Technical report, University of Cambridge, 2001.
[3]W. Lu, C.Y. Zhao, S.A. Tassou, " Thermal analysis on metal-foam filled heat exchangers. Part I: Metal-foam filled pipes", Int. J. Heat Mass Transfer, vol. 49 pp.2751–2761, 2006.Doi:10.1016/j.ijheatmasstransfer.2005.12.012
[4]C.Y. Zhao, W. Lu, S.A. Tassou, " Thermal analysis on metal-foam filled heat exchangers. Part II: Tube heat exchangers, " Int. J. Heat Mass Transfer, vol. 49 pp 2762–2770,2006.Doi:10.1016/j.ijheatmasstransfer.2006.01.03
[5]N. Dukhan, K.-C. Chen, " Heat transfer measurements in metal foam subjected to constant heat flux, " Exp. Thermal Fluid Sci, vol.32 pp624–631,2007.
[7]Y.P. Du, Z.G. Qu, C.Y. Zhao, W.Q. Tao, " Numerical study of conjugated heat transfer in metal foam filled double-pipe," Int. J. Heat Mass Transfer, vol. 53 pp 4899–4907,2010.
[8]S. Mahjoob, K. Vafai, " A synthesis of fluid and thermal transport models for metal foam heat exchangers, " Int. J. Heat Mass Transfer, vol. 51 pp3701–3711,2008.
[9]S. Mahjoob, K. Vafai, " Analysis of bioheat transport through a dual layer biological media, " ASME J. Heat Transfer , vol.132 pp 031101.1–031101.14,2010.
[10]K. Yang, K. Vafai, " Analysis of temperature gradient bifurcation in porous media – An exact solution, " Int. J. Heat Mass Transfer, vol. 53, pp4316–4325,2010.
[11]K. Vafai, R. Thiyagaraja, " Analysis of flow and heat transfer at the interface region of a porous medium, " Int. J. Heat Mass Transfer, vol. 30 pp 1391–1405,1987.
[12]D. Poulikakos, M. Kazmierczak, " Forced convection in duct partially filled with a porous material, " ASME J. Heat Transfer, vol. 109 pp653–662,1987.
Doi:10.1115/1.3248147
[13]S. Chikh, A. Boumedien, K. Bouhadef, G. Lauriat, "Analytical solution of nonDarcian forced convection in an annular duct partially filled with a porous medium, " Int. J. Heat Mass Transfer, vol.38 pp 1543–1551,1997.
[14]D.Y. Lee, K. Vafai, " Analytical characterization and conceptual assessment of solid and fluid temperature differentials in porous media, " Int. J. Heat Mass Transfer, vol.42 pp 423–435,1999.
[15] P. Gangapatnam, R. Kurian, and S. P. Venkateshan, “Numerical simulation of heat transfer in metal foams,” Heat and Mass Transfer, vol. 54, pp. 553–562, 2018.
[16] Hajiqoli, M., et al. "On the effects of porosity and structural properties on the thermal-hydraulic performance of open-cell metal foams: A CFD study." Applied Thermal Engineering, vol. 181, pp. 115999,2020.
[17] Mancin, S., et al. "Experimental analysis of partial filling in a channel with metal foam for electronics cooling." Experimental Thermal and Fluid Science, vol. 118, pp 110167,2020.
[18]Y. Wang, "Comprehensive analysis of forced convection heat transfer characteristics of gradient pore density metal foam," International Communications in Heat and Mass Transfer, vol. 148, Art. no. 107150, Jun. 2025,
[19]J. Torres-Madroñero et al., "Numerical analysis of non-regular and regular metal foam configurations on thermo-hydraulic performance," Applied Thermal Engineering, vol. 252, Art. no. 124154, Aug. 2025,
[20]V.V. Calmidi, " Transport phenomena in high porosity fibrous metal foams", Ph.D. Thesis, University of Colorado, 1998.
[21]K. Boomsma, D. Poulikakos, " On the effective thermal conductivity of a threedimensionally structured fluid-saturated metal foam", Int. J. Heat Mass Transfer, vol. 44 pp.827–836,2001.
doi:10.1016/S0017-9310(00)00123-X[22]J.H. Lienhard IV, J.H. V Lienhard, " A Heat Transfer Textbook, third ed., Phlogiston Press, Cambridge, 2006.
mahmoudi,M . (2026). Investigation of friction coefficient and convective heat transfer in a metal foam-filled porous tube based on a two-equation model. Modares Mechanical Engineering, 26(10), 831-840. doi: 10.48311/mme.2026.118915.82961
MLA
mahmoudi,M . "Investigation of friction coefficient and convective heat transfer in a metal foam-filled porous tube based on a two-equation model", Modares Mechanical Engineering, 26, 10, 2026, 831-840. doi: 10.48311/mme.2026.118915.82961
HARVARD
mahmoudi M. (2026). 'Investigation of friction coefficient and convective heat transfer in a metal foam-filled porous tube based on a two-equation model', Modares Mechanical Engineering, 26(10), pp. 831-840. doi: 10.48311/mme.2026.118915.82961
CHICAGO
M mahmoudi, "Investigation of friction coefficient and convective heat transfer in a metal foam-filled porous tube based on a two-equation model," Modares Mechanical Engineering, 26 10 (2026): 831-840, doi: 10.48311/mme.2026.118915.82961
VANCOUVER
mahmoudi M. Investigation of friction coefficient and convective heat transfer in a metal foam-filled porous tube based on a two-equation model. Modares Mechanical Engineering. 2026;26(10):831-840 (In Persian). doi: 10.48311/mme.2026.118915.82961
