Overview of Convection Heat Transfer

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1 Overview of Convection Heat Transfer Lecture Notes for ME 448/548 Gerald Recktenwald March 16, Introduction CFD is a powerful tool for engineering design involving fluid flow and heat transfer. Obtaining meaningful CFD solutions requires proper specification of boundary condition.s This document provides an overview of the basic features of forced convective heat transfer in confined spaces (ducts or enclosures). Boundary conditions applicable to convective heat transfer in a duct are identified. For each boundary condition, an appropriate heat transfer coefficient is defined. 1.1 Governing Equations In general the velocity field will be a vector valued function u = ê x u+ê y v+ê z w. The temperature is a scalar function of space, T = T (x, y, z). The governing equations for incompressible flow in Cartesian coordinates are the continuity equation u x + v y + w z = 0 (1) the momentum equations (in conservative form, no body forces) t (ρu) + x (ρuu) + y (ρvu) +» p 2 (ρwu) = z x + µ u x + 2 u 2 y + 2 u 2 z 2 t (ρv) + x (ρuv) + y (ρvv) +» p 2 (ρwv) = z y + µ v x + 2 v 2 y + 2 v 2 z 2 t (ρw) + x (ρuw) + y (ρvw) +» p 2 (ρww) = z z + µ w x + 2 w 2 y + 2 w 2 z 2 and the thermal energy equation (for incompressible, constant property flow) (ρcpt )+ t x (ρucpt )+ y (ρvcpt )+» 2 z (ρwcpt ) = k T x + 2 T 2 y 2 (2) (3) (4) + 2 T +S z 2 T (5) Boundary conditions for each of these equations must be specified. In this paper the focus is on the choice of boundary condition for the energy equation. Mechanical and Materials Engineering Department Portland State University, Portland, Oregon, gerry@me.pdx.edu 1

2 1 INTRODUCTION 2 U, T U in, T in Figure 1: External flow (top) and internal flow (bottom) 1.2 External and Internal Flows Wall-bounded flows can be classified as either external or internal flows. An external, wall-bounded flow is often treated as a boundary layer. In the boundary layer model, the fluid passing over an object has a free stream condition far from the wall of the object. Typically the fluid temperature does not change appreciably in the free stream. Characterization of the convective heat transfer in a boundary layer involves determining the variation (if any) of wall temperature and wall heat flux. In an internal, wall-bounded flow, there is no free stream condition. Heat addition or removal at the walls changes the energy content of the flowing stream, and the bulk fluid temperature changes along the flow direction. Therefore, in addition to determining the variation of wall temperature and wall heat flux, we must also quantify the variation in bulk fluid temperature in the duct. In many complex internal flows, it may not be possible to identify a meaningful bulk temperature. This is the situation, for example, in a flow with strong recirculation. Any time the flow is not predominantly unidirectional, the definition of bulk temperature is problematic. 1.3 Continuity of Heat Flux at the Wall Figure 2 depicts the heat flux at a solid wall. In the absence of phase change or chemical reaction at the interface, the heat flux must be continuous. Because the fluid obeys the no-slip condition at the wall, the heat flux normal to the wall on the fluid side of the interface is also governed by Fourier s law of heat conduction. In other words, continuity of heat flux requires k s T y = k f y=0 T y (6) y=0

3 2 HEAT TRANSFER FOR FLOW IN DUCTS 3 y fluid solid T(y + ) T y y = 0 T y y = 0 + T(y ) q + = k f q = s T y y = 0 + T y y = 0 Figure 2: Continuity of heat flux at a fluid-solid interface. 2 Heat Transfer for Flow in Ducts 2.1 Definitions In an internal wall-bounded flow, we define the average velocity to be V = 1 u ˆn da (7) A A where A is the area of the duct cross-section (which may vary along the flow direction), ˆn is the normal to the duct cross section, and u is the fluid velocity vector. The definition of average velocity in Equation (7) guarantees that the total mass flow rate for an incompressible flow through area A is ṁ = ρva. In an analogous way we define the bulk or mixing cup temperature as (u ˆn) T da A T b = (8) (u ˆn) da A The bulk temperature is a measure of the total thermal energy transfer through a plane area. The average velocity is defined so that ṁ = ρva gives the correct mass flow rate. The bulk temperature is defined so that the energy flowing through a cross section is ṁc p T b. Figure 3 depicts the bulk properties for convective heat transfer in a duct. The flow is assumed to be steady and incompressible. The total heat transfer across the wall of the duct is Q. By definition of T b, an energy balance on the T in T b,out ṁ Q Figure 3: Energy balance for flow through a duct with heat transfer across the walls of the duct.

4 2 HEAT TRANSFER FOR FLOW IN DUCTS 4 Temperature profile: T(r)-T in U in T in r x T w > T in T w q w T b (x) Figure 4: Development of the temperature profile for flow through a pipe with uniform wall temperature. The upper plot shows how the temperature profile changes. The lower plot shows how the wall heat flux and bulk temperature vary with position along the pipe. duct shows that the total heat transfer rate is must be equal to Q = ṁc p (T b,out T in ) (9) Equation (9) applies to a duct of any shape, and with any thermal boundary condition. A duct flow (an internal, wall-bounded flow) is hydrodynamically fully developed when the velocity profile no longer changes in the flow direction. Similarly, a duct flow is thermally fully developed when a suitably defined dimensionless temperature no longer changes in the flow direction. The ideas presented in the rest of this paper are not dependent on the flow being either hydrodynamically or thermally fully developed. 2.2 Physical Boundary Conditions We now consider the implications of different thermal boundary conditions on flow in straight duct (i.e. a duct with constant cross section). These boundary conditions represent idealizations of physical situations. Although they are ideal, the different boundary conditions are useful models of practical heat transfer equipment Uniform Wall Temperature Figure 4 depicts the case of heat transfer in a pipe with a uniform wall temperature boundary condition. The wall temperature T w, and the inlet temperature

5 2 HEAT TRANSFER FOR FLOW IN DUCTS 5 T in are specified. T b,out and q w (x w ) are determined by the flow field (Re), fluid properties (Pr), and duct geometry. Without knowing any more details about the flow, we can say that Equation (9) applies. The bulk temperature T b (x) will increase with distance when T w > T in, and it will decrease with distance when T w < T in. If T w > T in, then T b,out < T w, and lim x T b = T w. If T w < T in, then T b,out > T w, and lim x T b = T w. The total heat transfer through the duct wall is Q = q w (x) da (10) A w where the integral is over the surface of the duct walls, not the duct crosssection. Figure 4 shows how the temperature profile changes as the fluid moves through the pipe. Initially the temperature gradient at the wall is very steep. In the ideal (and physically unrealizable case) of a step change in temperature at the entrance x = 0, the heat flux is infinite at x 0. As the fluid moves through the pipe, the bulk temperature rises; the temperature gradient at the wall decreases; and the wall heat flux decreases from its initially infinite value. If the duct were infinitely long, the bulk temperature would eventually equal the wall temperature. Using a CFD model to obtain the velocity and temperature fields allows us to find q w (x w ). The local heat transfer coefficient is computed from the results of the CFD simulation with h(x w ) = q w(x w ) T w T in. (11) Note that the heat transfer coefficient does not come from a correlation. The numerical solution gives the variation of q w (x w ) and the heat transfer coefficient is the ratio of local q w (x w ) to the driving temperature Given h(x w ) the average heat transfer coefficient is h = 1 h(x w ) da. (12) A w A w Correlations for h in heat transfer textbooks are usually obtained from experimental measurements. Such a correlation is merely a summary of the experimental data, not a definition of h. One could also use a CFD program to generate h data. An alternative approach to computing the average heat transfer coefficient uses the overall heat transfer rate. h = Q/A w T w T in (13)

6 2 HEAT TRANSFER FOR FLOW IN DUCTS 6 r x u(r) T(r) q w Figure 5: Flow through a pipe with uniform wall heat flux. Substitution of Equation (10) into Equation (13) shows that Equation (12) and Equation (13) are equivalent. The average or overall Nusselt is Uniform Wall Heat Flux Nu = hl k. (14) Figure 5 depicts the case of constant wall flux boundary condition. The inlet temperature T in is specified. Because q w is known, T b,out can be obtained from a simple energy balance as shown in Equation (15), below. T w (x w ) is determined by the flow field (Re), fluid properties (Pr), and duct geometry. Without knowing any more details about the flow, we can say that Equation (9) applies, but since Q, ṁ, c p, and T in are known, we can compute T b,out = T in + Q (15) ṁc p without knowing the variation of T w. The bulk temperature T b (x) will increase with distance (for q w > 0). The wall temperature T w (x) will increase with distance The total heat transfer through the walls is Q = q w da = q w A w because q w is uniform. A w Using a CFD model to obtain the velocity and temperature fields allows us to find T w (x w ). The local heat transfer coefficient is h(x w ) = q w T w (x w ) T in (16) The average or overall heat transfer coefficient is computed with h = Q/A w T w T in (17)

7 2 HEAT TRANSFER FOR FLOW IN DUCTS 7 h, T amb U in T in r x T(r) T w (x), q w (x) Figure 6: Heat transfer to a fluid flowing inside a pipe with an external convective boundary conditions. h and T amb are known. T w (x) and q w (x) are unknown. where T w is the average wall temperature T w = 1 T w (x w ) da (18) A w A w The average or overall Nusselt is Convection Boundary Condition Nu = hl k. (19) Figure 6 depicts a fluid moving through the inside of a pipe. The thermal boundary condition at the pipe wall is determined by convection to an external fluid. In the simplest case, we neglect axial heat conduction in the pipe wall. The thermal resistance of the wall to heat flow in the radial direction can be combined with the external heat transfer coefficient to yield an overall external heat transfer coefficient. Both the pipe wall temperature and wall heat flux are unknown until the fluid temperature inside the pipe is determined. Assume T amb > T in. Then the following observations can be made. Equation (9) applies. The bulk temperature T b (x) will increase with x The wall temperature T w (x) will increase with x The wall heat flux q w (x) will decrease with x The total heat transfer through the walls is Q = q w da A w

8 2 HEAT TRANSFER FOR FLOW IN DUCTS 8 Enclosure at T surf ε surf ε w U in T in r x T(r) T w (x), q w (x) Figure 7: Heat transfer to a fluid flowing inside a pipe with an external radiation boundary conditions. T surf and the radiation properties ɛ w and ɛ surf are known. T w (x) and q w (x) are unknown Radiation Boundary Condition Figure 7 depicts a fluid moving through the inside of a pipe. The thermal boundary condition at the pipe wall is determined by radiation to an external surface at temperature T surf. The radiation heat transfer is determined by the variation of the pipe wall temperature, the emissivity of the pipe wall, and the emissivity of the external surface. In the simplest case, we neglect axial heat conduction in the pipe wall. The thermal resistance of the wall can be included as part of an effective external heat transfer coefficient.

9 3 CONJUGATE HEAT TRANSFER 9 U in, T in radiation convection conduction in the board Figure 8: Conjugate heat transfer involving electronic components mounted on a printed circuit board in an enclosure. 3 Conjugate Heat Transfer Conjugate heat transfer occurs when the temperature field in the fluid is coupled to the unknown temperature field in the solid. Figure 8 depicts convective cooling of electronic devices on a printed circuit board (PCB). The electronic devices on the board have a complex internal structure. Heat generated at the transistor junctions is transported by conduction to the surface of the device. Some of the heat is transferred directly to the flowing fluid by convection. Some of the heat is is transferred to the PCB by conduction, and it is then transferred to the air by convection. Some of the heat is transferred to the enclosure by radiation.

10 4 CASE STUDY: ELECTRONICS COOLING 10 What BC should be imposed here? Electronic component dissipating heat External flow due to natural convection T = T amb y x Sealed enclosure Figure 9: An enclosure containing a heat-generating device. 4 Case Study: Electronics Cooling Consider a typical electronic enclosure depicted in Figure 9. If the enclosure does not have a fan or vents, the air inside will circulate due to free convection, i.e., buoyancy effects will cause the air to rise near the warm electronic components. The air loses heat to the walls of the enclosure, which lose heat to the surroundings. In developing a CFD model of the enclosure, the choice of boundary condition on the external wall will have a significant influence on the internal temperature field. For simplicity of exposition only the boundary condition on the far right boundary will be discussed. Extension to other boundaries is straightforward. The choices of boundary condition are 1. Constant temperature on the walls of the enclosure T = T amb 2. Constant heat flux on the walls of the enclosure T x = q w w 3. Convective conditions on the walls of the enclosure T x = h(t T amb ) w Figure?? shows how the choice of boundary condition determines the qualitative variation in wall temperature and heat flux. The convective boundary condition is the least restrictive and provides the most realistic boundary condition. Of course, the quality of the solution depends on the choice of parameters h and T amb.

11 4 CASE STUDY: ELECTRONICS COOLING 11 Constant T (y) Constant q(y) Convective BC y y y T(y) q(y) q(y) q(y) T(y) T(y) Figure 10: Influence of boundary condition model on the temperature of the external wall for the electronics enclosure depicted in Figure 9. Only compare the shapes of the q(y) and T (y). The magnitudes of q(y) and T (y) are controlled by the parameters of each boundary condition specification.