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Vertical plate ous en t flu ua ra ding flow properties. 2015 Elsevier Ltd. All rights reserved. ral con gy and bined wide ture m investigated the radiation effects on the boundary-layer flow of a gray medium using the Rosseland approximation. Results suggested that radiation enhances shear stress and surface heat transfer rate. Hossain and Takhar  and Hossain and Alim  examined the radiation effects on natural convection past an isothermal horizontal plate and a thin vertical cylinder, respectively. The results indicated that key flow and heat transfer eraction between died with other osures by rtical chan
Sparrow et al. .
The majority of previous studies have focused on the co ous flow regime. With the increasing interest in micro-sca transfer in advanced industrial applications, simultaneous natural convection and radiation heat transfer in slip regime is worth in-depth understanding. As the mean free path of the flow approaches the characteristic length scale of the problem, flows will demonstrate non-continuum behavior due to fewer molecular collisions within the dimension of interest. The degree of the rarefaction for gaseous flows is usually characterized by Knudsen ⇑ Corresponding author. Tel.: +1 205 348 4997; fax: +1 205 348 6419.
E-mail address: John.Baker@eng.ua.edu (J. Baker).
International Journal of Heat and Mass Transfer 90 (2015) 26–33
Contents lists availab
H .eatively small importance of conduction compared to radiation.
Arpaci  qualitatively analyzed the interaction between radiation and natural convection from a heated vertical plate, obtaining integral forms of radiant flux for both thin and thick gases. Ali et al.  research done on semi-infinite flat plates, the int natural convection and radiation was also stu geometry configurations, such as square encl et al. , and Akiyama and Chong , and vehttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.014 0017-9310/ 2015 Elsevier Ltd. All rights reserved.Yucel nels by ntinule heatas oxide crystal melt flow and glass production, high temperature heat exchangers, and so forth . The interaction between radiation and natural convection has been studied by many researchers in the past with a major focus on the configuration of semi-infinite flat plates. Cess  studied the natural convection of an absorbing-emitting fluid along a vertical flat plate, assuming a relmethod, Martynenko et al.  investigated the radiation–convection interaction in nonparticipating air over a vertical surface with a uniform heat flux. Numerical results were shown to agree well with known experimental data. Webb  conducted experiments to validate the numerical results obtained for buoyance-driven flow coupled by radiation boundary conditions. In addition to the1. Introduction
Thermal radiation effects on natu tant in the context of space technolo high temperatures. Examples of com convection problems are found in a applications, including high temperavection flow are imporapplications involving radiation and natural variety of engineering aterial processing such characteristics are altered within the boundary layer by interacting with radiation. Chamkha et al.  considered the impact of mass transfer studying the radiation–natural convection interaction, and transformed the boundary-layer equations to non-similar forms. Employing the normal-mode-expansion technique, Cheng and Ozisik  showed that the radiation effects enhance heat transfer and boundary layer thickness. Using the perturbationNatural convection
Radiation presented graphically and discussed. In addition, an integral correlation is presented for the average
Nusselt number as a function of the non-continuum conditions, radiation–conduction parameter, andNon-continuum effects on natural conve flow from a heated vertical plate
Kang Cao a, John Baker b,⇑ aMcKinsey & Company, 17/F Platinum Building, 233 Tai Cang Road, Shanghai 200020, C b The University of Alabama, Department of Aerospace Engineering and Mechanics, Tusc a r t i c l e i n f o
Received 25 July 2014
Received in revised form 15 January 2015
Accepted 1 May 2015
Keywords: a b s t r a c t
Heat transfer by simultane heated vertical plate has be conditions. The radiant hea local non-similarity two-eq on the interaction between ation effects. Results inclu
International Journal of journal homepage: wwwon–radiation boundary layer a sa, AL 35487, United States radiation and natural convection through an optically thick fluid over a studied with first-order momentum and thermal non-continuum boundary x was treated using the Rosseland diffusion approximation. By solving the tion model, numerical solutions were obtained to examine the slip effects diation and natural convection for a range of rarefied conditions and radislip velocity, temperature jump, skin friction, and heat transfer rate are le at ScienceDirect eat and Mass Transfer l sevier .com/locate / i jhmt g pseudo-similarity variable h dimensionless temperature f Henumber defined as the ratio of molecular mean free path to the characteristic length of the problem of interest, given by 
Kn ¼ k=L ð1Þ
Based on the Knudsen number value, flows can be classified into three regimes: continuum flow (Kn < 0.01), slip flow (0.01 6 Kn 6 0.1), and transitional flow (0.1 < Kn < 10) . As the
Cf skin friction coefficient cp specific heat at constant pressure f momentum accommodation coefficient
F transformed stream function
G n-derivative of F g gravitational acceleration
Gr Grashof number, Gr = g bDTL3/m2 h heat transfer coefficient k thermal conductivity
Kn Knudsen number, Kn = k/L
L characteristic length
Nu Nusselt number, Nu = hL/k p pressure
Pr Prandtl number, Pr = l cp/k q heat flux
T temperature u streamwise velocity v normal velocity x coordinate along the plate y coordinate normal to the plate
Greek symbols a thermal accommodation coefficient b volumetric thermal expansion coefficient