Overview of Convection Heat Transfer
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1 Overview of Convection Heat Transfer ME 448/548 Notes Gerald Recktenwald Portland State University Department of Mechanical Engineering February 23, 2012 ME 448/548: Convection Heat Transfer
2 Outline 1. External versus internal flow 2. Definitions: mean velocity and bulk temperature 3. Types of boundary conditions a. Uniform wall temperature b. Uniform wall heat flux c. Convective (external) boundary d. Radiation (external) boundary ME 448/548: Convection Heat Transfer page 1
3 External and internal flow have different modeling concerns External flow: Boundary layers on aerodynamic shapes immersed in a fluid U, T Internal flow: Wall-bounded flows with potentially large pressure drops U in, T in ME 448/548: Convection Heat Transfer page 2
4 Continuity of heat flux at the wall Continuity of heat flux requires T T(y + ) y y = 0 + T k s T y y = k f y=0 y q + T = k f y=0 fluid solid T y y = 0 T(y ) q = s y y = 0 + T y y = 0 ME 448/548: Convection Heat Transfer page 3
5 Definitions (1) Average velocity in the duct: V is the velocity that gives the correct flow rate in ṁ = ρv A. V = 1 u ˆn da (1) A A Similarly, the bulk temperature is defined so that the energy flowing through a cross section is ṁc p T b (u ˆn) T da A T b = (u ˆn) da A (2) ME 448/548: Convection Heat Transfer page 4
6 Overall energy balance for a duct Assume the flow is steady and incompressible T in T b,out ṁ Q The total heat transfer across the wall of the duct is Q. By definition of T b, an energy balance on the duct shows that the total heat transfer rate is must be equal to Q = ṁc p (T b,out T in ) (3) ME 448/548: Convection Heat Transfer page 5
7 Physical Boundary Conditions 1. Uniform wall temperature 2. Uniform wall heat flux 3. Convective (external) boundary 4. Radiation (external) boundary ME 448/548: Convection Heat Transfer page 6
8 Duct with uniform wall temperature (1) Development of the temperature profile for flow through a pipe with uniform wall temperature. U in T in r Temperature profile: T(r)-T in x T w > T in T w q w T b (x) ME 448/548: Convection Heat Transfer page 7
9 Behavior of bulk temperature: Duct with uniform wall temperature (2) T b (x) asymptotically approaches T w Behavior of wall temperature: T w = constant is an imposed constraint Total heat transfer through the duct wall: Q = Aw q w (x) da (4) and, as always Q = ṁc p (T b,out T in ) Remember: this equation defines T b. Also note: T in T b,in. ME 448/548: Convection Heat Transfer page 8
10 Duct with uniform wall temperature (3) Since the wall heat flux varies with position along the duct, the heat transfer coefficient is also varying with position h(x w ) = q w(x w ) T w T in. (5) Note that the heat transfer coefficient does not come from a correlation! Given knowledge of h(x w ), e.g. from computation or experiments, the average heat transfer coefficient is h = 1 h(x w ) da. (6) A w Correlations for h in heat transfer textbooks are usually obtained from experimental measurements. 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. Aw ME 448/548: Convection Heat Transfer page 9
11 Duct with uniform wall temperature (4) An alternative approach to computing the average heat transfer coefficient uses the overall heat transfer rate. h = Q/A w (7) T w T in Substitution of Equation (4) into Equation (7) shows that Equation (6) and Equation (7) are equivalent. The average or overall Nusselt is Nu = hl k. (8) ME 448/548: Convection Heat Transfer page 10
12 Duct with uniform wall heat flux (1) r x u(r) T(r) q w ME 448/548: Convection Heat Transfer page 11
13 Duct with uniform wall heat flux (2) Behavior of bulk temperature: T b,out = T in + Q ṁc p increases with x (assumes q > 0) Behavior of wall temperature: T w increases with x Total heat transfer through the duct wall: Q = q w (x) da = q w A w because q w is uniform. Aw ME 448/548: Convection Heat Transfer page 12
14 Duct with uniform wall heat flux (3) The local heat transfer coefficient is h(x w ) = q w T w (x w ) T in (9) The average or overall heat transfer coefficient is computed with h = Q/A w T w T in (10) where T w is the average wall temperature T w = 1 A w Aw T w (x w ) da (11) The average or overall Nusselt is Nu = hl k. (12) ME 448/548: Convection Heat Transfer page 13
15 Convection boundary condition (1) h, T amb U in T in r x T(r) T w (x), q w (x) Note: In a CFD model, the heat transfer coefficient is applied to determine the thermal resistance from the walls of the domain to the ambient. The heat transfer coefficient is not used internally, i.e., between the walls of the duct and the fluid in the domain. ME 448/548: Convection Heat Transfer page 14
16 Convection boundary condition (2) Assume T amb > T in. Then the following observations can be made. 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 = Aw q w da ME 448/548: Convection Heat Transfer page 15
17 Radiation boundary condition Enclosure at T surf ε surf ε w U in T in r x T(r) T w (x), q w (x) As with the convective boundary condition, the radiation exchange (as a boundary condition) determines the thermal resistance from the walls of the domain to the ambient. Note: It is also possible to include radiation between surfaces inside the domain, but that is another topic. ME 448/548: Convection Heat Transfer page 16
18 Conjugate Heat Transfer U in, T in radiation convection conduction in the board ME 448/548: Convection Heat Transfer page 17
19 Case Study: Electronics Cooling (1) What BC should be imposed here? Electronic component dissipating heat External flow due to natural convection T = T amb y x Sealed enclosure ME 448/548: Convection Heat Transfer page 18
20 Case Study: Electronics Cooling (2) Choices of boundary condition: 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 ME 448/548: Convection Heat Transfer page 19
21 Case Study: Electronics Cooling (3) Constant T (y) Constant q(y) Convective BC y y y T(y) q(y) q(y) q(y) T(y) T(y) ME 448/548: Convection Heat Transfer page 20
Overview of Convection Heat Transfer
Overview of Convection Heat Transfer ME 448/548 Notes Gerald Recktenwald Portland State University Department of Mechanical Engineering gerry@me.pdx.edu February 23, 2012 ME 448/548: Convection Heat Transfer
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