SUPPLEMENT LECTURE ON NATURAL CONVECTION OF HEAT TRANSFER

1. SUPPLEMENT LECTURE ON NATURAL CONVECTION OF HEAT TRANSFER ON HEAT TRANSFER COEFFICIENT (h) OF FLUIDS IN CONTACT WITH SOLIDS OF DIFFERENT TEMPERATURE IN NATURAL CONVECTIVE HEAT TRANSFER (for Bonus Quiz No. 6 of ME 130 Course on Applied Engineering Analysis) Complied by: Tai-Ran Hsu Professor of Mechanical Engineering San Jose State University San Jose, California, USA October 1, 2017 Reference books: [1] Frank Kreith and Mark S. Bohn “Principles of Heat Transfer,” Chapters 4 & 5 Natural Convection, 5th Edition, PWS Publishing Company, Boston, ISBN 0-534-95420-0, 1997. [2] Tai-Ran Hsu “Applied Engineering Analysis” Fall 2017, Chapter 3, Printed lecture notes for ME 130 class at Department of Mechanical Engineering, San Jose State University, 2017. File name: On natural convection for Q6/ME 130/Bonus quizzes

2. What is Natural Heat Convection? • It relates to heat transfer (transmission) in fluids in MOTION. The fluid motion occurs by natural means such as buoyancy. • The MOTION of fluid can be created by a fan or pump with noticeable velocity (called Forced convection), or by difference of densities created by temperature differences in fluid in its natural state without external force applied to the fluid to cause the motion of the fluid. • The latter case of heat flow in the fluid is called “NATURAL CONVECTION.” Since the fluid velocity associated with natural convection is relatively low, the heat transfer coefficient encountered in natural convection is also low Convective heat transfer in the present analysis is related to the Boundary Layer that exists between the submerged solids and the surrounding fluids: • Boundary layer exists between a solid and its submerged fluid in motion. • Examples of boundary layers exist between the wings of airfoil cross sections of aircrafts, and also at the interfaces of hot or cold solid surface submerged in surrounding fluids at different temperatures. • Boundary layers act as “barriers” in air flow in the case of aircraft wings, and as a thermal barrier in the case of solids submerged in fluids • In the case in which a solid is submerged in a “virtual standstill” fluid but is subject to difference in temperatures such as illustrated in the figure to your right. What is Illustrated in the figure are the solid arrows indicate a hot solid submerged in a colder fluid at temperature Tf. The arrows in dotted lines shows the case with reversed temperature variations. • Because of the boundary layer at the Interface of the solid and fluid acts like a thermal barrier, the solid surface temperature T(rs, t) ≠ Tf– the bulk fluid temperature!!

3. Heat Transmission in Fluids by Convection – mathematically modelled by the Newton’s Cooling Law [2]: Heat flux between Point A and B = q Ta and Tb = temperatures at 2 locations (A&B) in the fluid with Ta > Tb where h = heat transfer coefficient The above mathematical equation of Newton’s cooling law, though is useful, but is not practical in engineering analysis, in which engineers desire to predict how this law can be used in the design analysis of heat transfer equipment such as heat exchangers and also in thermal management of heat generating equipment such as electronic systems. Different forms of math formulations with the presence of both solids and fluids are needed. Heat transfer coefficient (h) in the fluid with solids submerged in fluids: Physical meaning of h is that 1/h represents the resistance to heat transmission between the solid and the surrounding fluid, as result of the thermal barrier by the boundary layer developed at the interface of the solid and fluid as illustrated. Determination of the h-value is thus an important step in assessing the effectiveness of heat transfer from solid equipmentsuch as heat exchanger with solids submerged in surrounding (cooling or heating) fluids.

4. Determination of Heat Transfer Coefficient (h) for Heat Transmission in Natural Convection One may well appreciate the extreme complexity involved in evaluation of the heat transfer coefficient (h) in convective heat transfer in view of the presence of boundary layer that adds as extra thermal barrier for heat transmission in the fluids. The thickness of the boundary layer theory is a stand-alone complex theory in thermofluid science. Five (5) general methods are available to determine the h-value in convective heat transfer as mentioned in Reference [1] (Section 4.6, P.254). Among these five general methods, the one that is commonly used by engineers is the “Dimensional analysis combined with experiments.” We will highlight this quasi-empirical method in this class, but with our focus on natural convection only. Dimensional analysis is a convenient way to determine the values of critically important parameters involved in engineering problems such as the heat transfer coefficient (h) in convective heat transfer. This technique contributes little to our understanding of the physical nature of the problem, but it facilitates the interpretation and extends the range of experimental data by correlating them in terms of the involved dimensionless groups. (Section 4.6 of [1]). Following are the dimensionless groups used in determining the heat transfer coefficient(h) in convective heat transfer analysis: Nusseltt Number (Nu) = hL/k where h=heat transfer coefficient, L=characteristic length, and k=thermal conductivity of the fluid. PrantlNumbe(Pr) = Cpμ/k where Cp = specific heats of fluid, μ=dynamic viscosity of fluid, k=thermal conductivity of fluid. Grashof Number (Gr) = [gβ(Ts-Tf)L3]/(μ/ρ)2 where β=coefficient of thermal expansion of the fluid, ρ=mass density of the fluid. Reynolds Number (Re)=(ρU∞L)/μ where U∞=velocity of bulk fluid (not used in natural convection) Reyleigh Number (Ra) = (Gr)(Pr) We recognize that the heat transfer coefficient h is embedded in the Nusseltt Number. Readers are encouraged to also study Section 4.7 of Reference [1] for the principle of “Dimensional analysis,” which is a useful method for complex engineering analyses in other applications.

5. Determination of Heat Transfer Coefficient (h) for Heat Transmission in Natural Convection – con’t General expression for determining h-values: For forced convection: For natural convection: where constants a, b and c are usually determined by experiments. These constants (a,b,c) are dependent to the fluid velocities in either laminar or turbulent state, the geometry of the solids, the direction fluid’s motions. So, the procedure of determining these constants are a complicated process. There are a number of ways the engineers can find the “ball park” values of h through Internet search, such as the following chart posted in the website of the Engineering Toolbox: We will present a few cases with special geometry of the solids submerged in fluids in natural convection in the following slides cited from Reference [1].

6. Determination of Heat Transfer Coefficient (h) for Heat Transmission in Natural Convection – con’t h-values for vertical plates and cylinders [1] section 5.3.1: Vertical cylinder: Vertical Plate: Bulk Fluid Bulk Fluid L Ts Ts Ts Ts Ts L Bulk Fluid x Leading edge Local and average values of h for laminar natural convection from an ISOTHERMAL vertical plate or cylinder: h-values at local x: Average value over the length L: In dimensionless form the average Nusselt number is:

7. Determination of Heat Transfer Coefficient (h) for Heat Transmission in Natural Convection – con’t B. Local and average values of h for laminar natural convection from an ISOTHERMAL horizontal plates: B.1 Upper surface hot or Lower surface cool: (Face-down case: Ts < Tf) Surface temperature Ts Surface Area A = LxW Subscript L = Surface Area/Perimeter Insulation L L (Face-up case: Ts > Tf) Surface temperature Ts for 105<RaL<107 for 107<RaL<1010 B.2 Lower surface heated or upper surface cooled: Surface temperature Ts Surface Area A = LxW (Face-down case: Ts > Tf) Insulation L L (Face-up case: Ts < Tf) Surface temperature Ts for 105<RaL<1010

8. Heating or Cooling of Small Solids by Surrounding Fluids in Enclosures The physical situation we have is illustrated in the figure below, in which a small solid at a uniform temperature T(t) is submerged in a fluid with a bulk temperature at Tf at time t=0+. Heat will be transmitted between the solid and the surrounding fluid due to the difference of temperatures of the solid temperature T(t) and the surrounding bulk fluid temperature Tf. The heat transmission between the solid and the fluid can be expressed by the expression: Heat conduction along the normal line to the surface Solid temp. at interface Bulk fluid temperature This equation shows that heat flux from the solid equals to the heat available for transmission in the surrounding fluid. However, it does not provide solution to a crucial question on “How long would it take to have the solid initially at temperature Toto be cooled down (or heated up) to a desired temperature T, with the surrounding bulk fluid maintained at a temperature Tf”.

9. Computation of continuous Temperature Variation in a Small Solid Submerged in an Enclosure Filled with Cool Air Maintained at a Lower Temperature Tf Physical situation: A small solid plate (or other geometry) with initial Uniform temperature T0 is submerged into cold dry air in a refrigerator (or a cool box) maintained at a constant temperature Tf at time t = 0. The submerge of the warm (or hot) solid in the cool (or cold) air will result in dropping of solid temperature with time t. Solution sought: Compute the continuous temperature variation T(t) in the solid, and compare the computed temperature variation with the measure values. T(t), Initial temp=T0 Area A Ts Bulkair temp., Tf Heat transfer Coefficient h Differential equation for the solution of T(t) as derived in ME 130 class (with the postulation that T(t) = Ts(t) for small solids): with ρs, cs and vs = the respective mass density, specific heats and volume of the solid and h is the heat transfer coefficient. where the constant α can be found to be: Given conditions in this analysis are the initial temperature of the solid T0, i.e., T(0) = T0 and the bulk air temperature Tf, as well as the contact surface area A of the solid with the air. Thermophysical properties of dry air at different temperatures are available in the next slide.