Kun Yang School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, P. R. China; Department of Mechanical Engineering, University of California, Riverside, Riverside, CA 92521-0425 Kambiz Vafai 1 Department of Mechanical Engineering, University of California, Riverside, Riverside, CA 92521-0425 e-mail: vafai@engr.ucr.edu Restrictions on the Validity of the Thermal Conditions at the Porous-Fluid Interface—An Exact Solution Thermal conditions at the porous-fluid interface under local thermal nonequilibrium (LTNE) conditions are analyzed in this work. Exact solutions are derived for both the fluid and solid temperature distributions for five of the most fundamental forms of thermal conditions at the interface between a porous medium and a fluid under LTNE conditions and the relationships between these solutions are discussed. This work concentrates on restrictions, based on the physical attributes of the system, which must be placed for va- lidity of the thermal interface conditions. The analytical results clearly point out the range of validity for each model for the first time in the literature. Furthermore, the range of validity of the local thermal equilibrium (LTE) condition is discussed based on the introduction of a critical parameter. The Nusselt number for the fluid at the wall of a channel that contains the fluid and porous medium is also obtained. The effects of the per- tinent parameters such as Darcy number, Biot number, Bi, Interface Biot number, Bi int , and fluid to solid thermal conductivity ratio are discussed. [DOI: 10.1115/1.4004350] Keywords: thermal condition, porous-fluid interface, local thermal nonequilibrium 1 Introduction Due to its wide range of engineering applications, convective heat transfer in porous media has gained increased interest in recent years. These applications include geothermal engineering, heat pipes, solid matrix heat exchangers, electronics cooling, enhanced oil recovery, thermal insulation, and chemical reactors. Among which, thermal convection in composite systems is an im- portant aspect. This system consists partly of a porous region and partly of an open region. One example is a channel with a partially filled porous medium. Poulikakos and Kazmierczak [1] studied fully developed forced convection in a channel, where the porous matrix was attached at the channel wall but did not extent throughout the channel. The results showed that there was a criti- cal value of porous region thickness at which the Nusselt number reaches a minimum. Chikh et al. [2] investigated forced convec- tion between two concentric cylinders where the inner cylinder is exposed to a constant heat flux, a porous layer is attached to the inner cylinder and the porous material does not extend across the full annulus. It was also found that there exists a critical thickness of the porous layer at which heat transfer is minimum in the case of low thermal conductivity materials; however, this was not observed for the highly conducting materials. Alkam and Al-Nimr [3] presented a method to improve the thermal performance of a conventional concentric tube heat exchanger by inserting high- thermal conductivity porous substrates on both sides of the inner tube wall. Mohamad [4] numerically investigated heat transfer enhancement in a pipe or a channel with the porous medium par- tially filling the core of the conduit. It was found that this method can enhance the rate of heat transfer, while the pressure drop is much less than that for a conduit fully filled with a porous me- dium. Pavel and Mohamad [5] experimentally investigated the problem of air flowing inside a pipe when different porous media are emplaced at the core of the pipe. The results showed that a partial filling has the advantage of a comparable increase in the Nusselt number and a smaller increase in the pressure drop. Kim et al. [6] numerically investigated forced convection in a circular pipe partially filled with a porous medium, which included two types of configurations (in two separate cases). It was found that there exists a critical porous layer thickness where the Nu reaches a minimum in one case and a maximum for another case. Satya- murty and Bhargavi [7] studied forced convection in the thermally developing region of a channel where a partially filled porous me- dium was attached to one wall only. Kuznetsov [8] has obtained some solutions for the velocity and temperature distributions for few composite systems. Different types of interfacial conditions between a porous medium and a fluid layer have been presented in the literature [1–12]. Beavers and Joseph [9] first presented a velocity interfa- cial condition based on a slip velocity proportional to the exterior velocity gradient, which was shown to be in reasonable agreement with experimental results. References [1–8] utilized continuity in both the temperature and heat flux at the interface. Vafai and Thiyagaraja [10] presented a detailed analytical solution for the velocity and temperature distributions, as well as the Nusselt num- ber distribution, for three general and fundamental interfaces, namely, the interface between two different porous media, the interface between a fluid region and a porous medium and the interface between an impermeable medium and a porous medium. Vafai and Kim [11] first derived an exact solution for the fluid mechanics of the interface region between a porous medium and a fluid layer, accounting for both boundary and inertial effects. Alazmi and Vafai [12] comprehensively analyzed five fundamen- tal hydrodynamic interface conditions and four thermal interface conditions. It was shown that the variance within different models have a negligible effect on the results for most practical applications. There are two primary ways for representing heat transfer in a porous medium, LTE model and LTNE model. The LTE model is more convenient to use, and it is utilized by the above-mentioned references [1–8,10,12]. However, the temperature difference 1 Corresponding author. Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 10, 2011; final manuscript received May 22, 2011; published online September 16, 2011. Assoc. Editor: Darrell W. Pepper. Journal of Heat Transfer NOVEMBER 2011, Vol. 133 / 112601-1 Copyright VC 2011 by ASME