THERMAL FUNDAMENTALS understanding Heat Transfer Coefficient When there is a motion of fluid with respect to a surface or a gas with heat generation, the transport of heat is referred to as convection [1]. There are three modes of convection. If the motion of flow is generated by external forces, such as a pump or fan, it is referred to as forced convection. If it is driven by gravity forces due to temperature gradients, it is called natural (or free) convection. When the external means are not strong and gravitational forces are strong, the resulting convection heat transfer is called mixed convection. When there is a motion of fluid with respect to a surface Figure 1. or a gas with heat generation, the transport of heat is Velocity and thermal boundary layer growth on referred to as convection [1]. There are three modes of a heated flat plate [4]. convection. If the motion of flow is generated by external forces, such as a pump or fan, it is referred to as forced δ convection. If it is driven by gravity forces due to tem- perature gradients, it is called natural (or free) convection. δ V , T t When the external means are not strong and gravitational ∞ ∞ q forces are strong, the resulting convection heat transfer is T =T T >T s called mixed convection. s ∞ s ∞ A heat transfer coefficient h is generally defined as: ξ x q Q= hA() T − T s ∞ To relate these two parameters, an important equation known as the Reynolds Analogy can be derived from Where Q is the total heat transfer, A is the heated conservation laws that relate the heat transfer coefficient surface area, T is the surface temperature and T s ∞ to the friction coefficient, C : is the approach fluid temperature. For the convection f heat transfer to have a physical meaning, there must re be a temperature difference between the heated surface CL = N u and the moving fluid. This phenomenon is referred to as f 2 the thermal boundary layer that causes heat transfer from the surface. In addition to the thermal boundary Where Nu is the Nusselt number, ReL is the Reynolds layer, there is also a velocity boundary layer due to the number based on a length scale of L and Cf is defined as: friction between the surface and the fluid induced as the result of the fluid viscosity. The combination of the thermal τ and viscous boundary layers governs the heat transfer = s cf 2 from the surface. Figure 1 shows velocity boundary layer ρV / 2 growth ( δ ) that starts from the leading edge of the plate. The thermal boundary layer (δ ) starts after a distance t Where τ is the shear stress and V is the reference ( ξ ) from where the temperature of the plate changes s velocity, the shear stress τs can be calculated as: from ambient temperature to a different temperature (Ts) , causing convection heat transfer. COPYRIGHT© ADvaNCED THERMAL SOLUTIONS, INC. | 89-27 ACCESS ROAD NORWOOD, MA 02062 USA | T: 781.769.2800 WWW.QatS.COM PaGE 11 THERMAL FUNDAMENTALS dV higher than in the fully τ= µ s dy y =0 developed region. Figure 2 shows the Nusselt number Nu, as a function of the inverse of the Graetz number. Where µ is the fluid viscosity, and the derivative of the The Graetz number is defined as: velocity (V) is calculated at the wall. The above equation is significant because it allows the engineer to obtain D G z = ( )re Pr information about the heat transfer coefficient by knowing D x D the surface friction coefficient and vice versa. To calculate Where x is the distance from the leading edge, and ReD the friction coefficient, the velocity gradient at the wall is is the Reynolds number based on the duct diameter. This needed. In a simple flat plate, the definition of the heat figures shows the Nu in an entry length region where the transfer coefficient is well defined, if we assume the plate velocity is already fully developed, and the combined entry has a constant temperature and the fluid temperature length which both the velocity and the temperature are outside the boundary layer is fixed. developing. It shows that if the flow and the temperature are developing at the same time, the Nusselt number From the definition above, h purely depends on the fluid would have been higher than the thermal entry length. and surface reference temperatures. In simple geometries, h can be analytically derived based on the conservation equations. For example, for the laminar flow over a flat 100 plate, analytical derivation for the local heat transfer Thermal Entry Length coefficient yields Combined Entry Length Constant Surface (Pr = 0.7) Heat Flux 20 0.332 1/ 2 1/ 3 Entrance Region Fully Developed Region hx = k f rex Pr x NuD 10 for Prandtle numbers above 0.6. The Prandtl number, Pr, 5 4.36 is the ratio of momentum diffusivity (viscosity) to thermal 4 Constant Surface 3.66 Temperature diffusivity. The Pr for liquid metals is much less than 1, for 3 2 gases it is close to 1, and for oils it is much higher than 1. 1 o The Pr for air at atmospheric pressure and 27 C is 0.707. 0.001 0.005 0.01 0.05 0.1 0.5 1 x / D -1 Re Pr = Gz For a fully developed laminar flow in a circular tube, the D analytical form for the Nusselt number is: hd Figure 2. nu ≡ = 4. 36 the nusselt number as a function of the inverse of the graetz k number in a duct [1]. for a constant temperature surface, and nu = 3.66 The heat transfer coefficient also depends on the flow regime. Figure 3 shows the flow over a flat surface. The for a constant heat flux. laminar boundary layer starts the transition at a Reynolds number around 5 x 105 with a sudden jump in the heat The above values for a circular duct are not valid in the transfer coefficient, and then gradually coming down in the entry region of the tube, where the velocity or thermal turbulent region, but still above the laminar regime. For a boundary layer is still developing (ξ length in figure 1). smooth circular tube, the transition from laminar to turbu- In the entry region, the heat transfer coefficient is much lent starts at a Reynolds number around 2300. COPYRIGHT© ADvaNCED THERMAL SOLUTIONS, INC. | 89-27 ACCESS ROAD NORWOOD, MA 02062 USA | T: 781.769.2800 WWW.QatS.COM PaGE 12 THERMAL FUNDAMENTALS 3- Tad (adiabatic temperature) as the gas temperature measured by a device with no power. In their experiment, they used an array of 3 x 3 components and measured the heat transfer coefficients for different powering schemes. The results revealed that the hin and hm based on Tin and Tm resulted in 25% variation between the mean value, but the variation of had based on Tad V∞, T∞ was about 5% around the mean. The authors concluded that hin and hm depend on the other component powering schemes and the temperature distribution of the system. But, had depends on the aerodynamics of the flow near the component and is independent of the powering scheme. This same argument applies to a heat sink. One has to be careful how the heat transfer coefficient is defined. It can be defined in reference to inlet fluid temperature, Figure 3. or as some mean value between the inlet and outlet. If Heat transfer coefficient on a flat plate for different flow the heat transfer coefficient is defined based on the inlet regimes [4]. temperature, the thermal resistance of the heat sink is calculated as: An important issue is the definition of the ambient and 1 fluid temperatures. The user must be careful when using r = heat transfer coefficient correlations by knowing where ηhA the reference ambient temperature is defined. For exam- Where A is the total surface area, and η is the fin ple, in [2] Azar and Moffat (figure 4.) considered the flow efficiency defined as: between circuit boards in an electronic enclosure as the reference ambient fluid temperature. In this case, there tanh()m× H and h× P η = m = are many choices for defining the heat transfer coeffi- m× H KA× cient, depending on how the reference temperatures are defined. The surface temperature is more fully defined, Where H is fin height, h is the heat transfer coefficient, P as it can be assumed to be either the hottest point on is the fin perimeter, A is the fin cross sectional area and K the chip or some area average of the surface tempera- is the fin thermal conductivity. ture. The fluid temperature, however, is harder to define because there are many choices. The authors assumed But, if the heat transfer coefficient is defined based on the three choices for the fluid temperature: temperature in between the fins, the thermal resistance calculation also involves the addition of the heat 1- T as the inlet fluid temperature in capacitance of the flow as: 2- Tm based on the inlet temperature and upstream heat 1 1 r = + . dissipation .. ηhA 2m c T =T + q / m c p m in ∑ p Where C is the fluid heat capacitance. Where m is the mass flow rate and q is the upstream p heat dissipation. COPYRIGHT© ADvaNCED THERMAL SOLUTIONS, INC. | 89-27 ACCESS ROAD NORWOOD, MA 02062 USA | T: 781.769.2800 WWW.QatS.COM PaGE 13 THERMAL FUNDAMENTALS In summary, before using any heat transfer correlation Convection Radiation from the literature, the engineer must determine the flow conditions and use the appropriate equations.