Analysis of pin fins with radiation

The design of fins for heat transfer enhancement remains a topic of great interest in a number of engineering areas and applications, despite a broad and deep prior literature on the subject. Rapid prediction of the effects of convection, conduction and radiation is still an area of concern. For hot-flow conditions, the fin is normally mounted in a cooled surface, leading to substantial axial conduction. Also, radiation plays a very important role in hot flow conditions. One can apply detailed computational methods for simultaneous convection, conduction and radiation heat transfer, but such approaches are not suitable for rapid, routine design studies. So, there is still a place for approximate analytic methods, and that is the subject of this paper. We have extended the traditional pin fin analysis to include a more realistic radiation treatment and also considered variable thermal conductivity, variable heat transfer coefficients over the tip and sides of the fin with variable area distribution, variable internal heat generation and then produced a MATLAB solution procedure for routine use by designers and analysts. INTRODUCTION The use of fins to enhance heat transfer is ubiquitous in industrial and consumer applications and even in nature,[1], [2]. So, the design and analysis of fins for heat transfer enhancement remains a topic of great interest, despite a broad and deep prior literature on the subject (e.g. [3], [4]). Prediction of the effects of convection, conduction and radiation remains an area of concern. For hot-flow conditions, the fin is normally mounted in a cooled surface, leading to substantial axial conduction. Also, radiation plays a very important role in hot conditions. One can apply detailed computational methods for simultaneous convection, conduction and radiation heat transfer. We have used ANSYS Fluent [5] for related studies, but such approaches are not suitable for rapid, routine design studies. So, there is still a place for approximate analytic methods, and that is the topic of this paper. Some useful, early analytical methods treated a socalled “pin fin” (a straight or tapered rod projecting from a wall) with combined convection and conduction. Reference [3] provides a very thorough review of the literature. While useful, these treatments were quite restricted in a number of ways. Some, but not all, of the restrictions are covered in the so-called: Murray-Gardner-Kern Assumptions [3]: (1) The heat flow and temperature distribution are steady in time, (2) The fin material is homogeneous and isotropic, (3) There are no heat sources in the fin, (4) The heat flow to or from the fin surface is proportional to the temperature difference between the surface and the surrounding fluid, (5) The thermal conductivity of the fin is constant, (6) The heat transfer coefficient is the same over the fin surface, (7) The temperature of the surrounding fluid is uniform, (8) The temperature of the base of the fin is uniform, (9) Temperature gradients normal to the surface may be neglected (1D assumption), (10) The heat transferred through the tip of the fin is negligible compared to that passing through the sides, (11) The joint between the fin and the prime surface offers no bond or contact resistance. One can add an additional common assumption: (12) Radiation is neglected, or radiation is treated without convection. NOMENCLATURE A(x) [m2] area of fin d(x) [m] diameter of fin G [1/m2K4] =2/5(σεP/kA) h(x) [W/m2K] convection heat transfer coefficient on fin side htip [W/m2K] convection heat transfer coefficient on fin tip i [-] node index k(T) [W/mK] thermal conductivity k0 [W/mK] reference thermal conductivity k1 [W/mK2] slope of linear thermal conductivity L [m] length of the fin m [1/m] =(hP/kA)1/2 n [-] number of nodes P(x) [m] perimeter of fin ?̇?q(x) [W/m3] internal heat generation Qb [W/m2] base heat transfer T(x) [K] temperature T0 [K] reference temperature Tf [K] fluid temperature Tb [K] base temperature Tsurr [K] surroundings temperature x [m] base to tip coordinate ΔΔΔΔ [m] discretized element length α [-] absorptivity σ [WK4/m2] Stefan-Boltzmann constant ε [-] emissivity 13th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics