Original Article Heat transfer characteristics of slip flow over solid spheres Morteza Anbarsooz 1 and Hamid Niazmand 2 Abstract In this study, heat transfer characteristics of slip flow over an isolated impermeable solid sphere are investigated numerically. An isothermal solid sphere is considered at intermediate Reynolds numbers (0 4 Re 4 50) for Prandtl numbers in the range of 0.7–7.0. The Navier–Stokes and energy equations are solved by a control volume technique in conjunction with the velocity slip and temperature jump boundary conditions. It was found that the size of the thermal wake region according to the Knudsen number depends on the Prandtl number. At lower Prandtl numbers (0.7 4 Pr 4 2.0), the thermal wake region shrinks as the Knudsen number increases, while at higher Prandtl num- bers, it grows as the Knudsen number increases. The maximum temperature jump occurs at the front stagnation point where the local Nusselt is itself maximum, owing to the maximum temperature gradient at this point. The results show that due to the opposing effects of the velocity slip and temperature jump, the average Nusselt number variation with the Knudsen number depends nonlinearly on both the Prandtl and Reynolds numbers. Furthermore, for the limiting case of Re ! 0, an analytical solution for the problem is presented which has also served as a validation case. Keywords Slip flow, temperature jump, thermal wake, Nusselt number Date received: 30 November 2014; accepted: 22 September 2015 Introduction Fluid flow and especially heat and mass transfer from spheres are essential issues in many engineering and environmental applications such as spray drying, extraction, humidification, aerosol scrubbing and evaporation, and fuel droplets heating and evapor- ation, among others. The classical problem of heat transfer from a sphere has been the subject of several investigations in the past. Heat transfer from spheres is governed by two independent dimensionless parameters, the Reynolds number and the Peclet number, which account for the velocity and temperature fields, respectively. The available solutions for the flow and temperature fields over spheres can be classified based on the Reynolds number. At creeping flow regime (Reynolds numbers less than one), analytical solution of viscous flow over spherical particles exist for both rigid spheres 1,2 and liquid spheres. 3,4 At Reynolds numbers higher than one, however, stream function form of the momentum equations have been solved numerically for flow over solid spheres. 5–13 A solution of the transient heat transfer from a solid sphere at creeping flow regime is pre- sented by Carslaw and Jaeger. 14 Also, an analytical expression for the heat transfer from a small spherical particle can be found in Feng and Michaelides 15 at low Peclet numbers assuming a Stokesian velocity distribution around the sphere. Solutions to the stream function form of the heat or mass transfer equation have also been obtained by many researchers using the finite-difference schemes. 11,16–19 In all of the above mentioned studies, the bound- ary condition applied to the solid sphere surface is the well-known no-slip condition. This boundary con- dition, however, can be violated in many practical applications, which can be classified into two main categories: (a) Slip flow regime, 20 where the sphere diameter is comparable with the mean free path of its 1 Mechanical Engineering Department, Quchan University of Advanced Technology, Quchan, Iran 2 Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran Corresponding author: Morteza Anbarsooz, Mechanical Engineering Department, Quchan University of Advanced Technology, Quchan 94771-67335, Iran. Email: anbarsouz@qiet.ac.ir Proc IMechE Part C: J Mechanical Engineering Science 2016, Vol. 230(19) 3431–3441 ! IMechE 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954406215612829 pic.sagepub.com by guest on November 29, 2016 pic.sagepub.com Downloaded from