1. Heat Transfer Su Yongkang School of Mechanical Engineering # 1 HEAT TRANSFER CHAPTER 8 Internal flow

2. Heat Transfer Su Yongkang School of Mechanical Engineering # 2 Internal Flow Heat Transfer Where we’ve been …… Introduction to internal flow, boundary layer growth, entry effects Where we’re going: Developing heat transfer coefficient relationships and correlations for internal flow roro

3. Heat Transfer Su Yongkang School of Mechanical Engineering # 3 Internal Flow Heat Transfer KEY POINTS THIS LECTURE Energy balance for internal flow in a tube Temperature and heat transfer relations for two cases: –Constant surface heat flux –Constant surface temperature

4. Heat Transfer Su Yongkang School of Mechanical Engineering # 4 Basic concepts—thermal considerations 1. The mean temperature For the internal energy So For incompressible flow in a circular tube, roro and

5. Heat Transfer Su Yongkang School of Mechanical Engineering # 5 Basic concepts—thermal considerations 2. Newton’s law of cooling Here, the mean T plays the same role as the free stream for external flows. 3. Fully developed conditions because of heat transfer, T(r) is continuously changing with x. can fully developed conditions be reached?? For thermally fully developed Although the temperature profile T(r) changes with x, the relative shape of the profile no longer changes. Not constant!

6. Heat Transfer Su Yongkang School of Mechanical Engineering # 6 Basic concepts—thermal considerations In the thermally fully developed flow of a fluid with constant properties, the local convection coefficient is a constant, independent of x. For the special case of uniform surface heat flux For the case of constant surface temperature Axial T gradient Independent of radial location Depends on the radial coordinate

7. Heat Transfer Su Yongkang School of Mechanical Engineering # 7 Example: Velocity and temperature profiles for laminar flow in a tube of radius have the form with units of m/s and K, respectively. Determine the corresponding value of the mean (or bulk) temperature,, at this axial position.

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9. Heat Transfer Su Yongkang School of Mechanical Engineering # 9 Review: Energy Balance Analysis x x + dx Energy Balance Change in energy in the control volume = energy input – work out + energy in by advection – energy carried out by advection (flow) or Two Special Cases: 1.Constant surface heat flux 2.Constant surface temperature

10. Heat Transfer Su Yongkang School of Mechanical Engineering # 10 Case 1: Constant Surface Heat Flux Example application –Electrical heater element around a pipe Recognize that: From the energy balance equation if q” is constant If q” is a known function of x instead, must integrate the above to obtain T m,x

11. Heat Transfer Su Yongkang School of Mechanical Engineering # 11 Example: Consider flow in a circular tube. Within the test section length (between 1 and 2) a constant heat flux is maintained. For the following two cases, sketch the surface temperature and the fluid mean temperature as a function of distance along the test section x. In case A flow is hydrodynamically and thermally fully developed. In case B flow is not developed. Solution:

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13. Heat Transfer Su Yongkang School of Mechanical Engineering # 13 Example: Pre-heating Water Given: An industrial process requires 0.2 kg/s of pre- heated water. The average tap water temperature during winter is known to be 10º C. You are considering using insulated electrical heater elements wrapped around the water pipe to heat the water. –Electrical heaters provide 5 kW / m² to the pipe –Existing pipe is 0.05m I.D., 0.055m O.D. –Internal h = 500 W / m²K –Available pipe length = 10 m 1.How hot can the outlet water temperature get in these conditions? 2.What is wall temperature at exit?

14. Heat Transfer Su Yongkang School of Mechanical Engineering # 14 Example (Cont’d) Solution: At the inlet condition: Perimeter of pipe: Calculations: If the internal convection coefficient = what is the inner wall T at the exit?

15. Heat Transfer Su Yongkang School of Mechanical Engineering # 15 Case 2: Constant Surface Temperature Example application –Steam condensation on the outer surface of the tube Start with the same general energy balance equation Separate variables and integrate from inlet to outlet 

16. Heat Transfer Su Yongkang School of Mechanical Engineering # 16 Constant Surface Temperature Temperature asymptotically approaches surface temperature Ts as you go downstream At any location ‘x’: T x T x

17. Heat Transfer Su Yongkang School of Mechanical Engineering # 17 Log Mean Temperature Difference Define: Log mean temperature difference (LMTD) as: Convective heat transfer is defined by: and then

18. Heat Transfer Su Yongkang School of Mechanical Engineering # 18 Overall Heat Transfer Coefficient, Recall from the discussion in conduction heat transfer, in general: Put energy balance equation in terms of ambient temperature, T  Define

19. Heat Transfer Su Yongkang School of Mechanical Engineering # 19 Back to Pre-heating Water Example Revised Problem Statement: You decide to also investigate using available steam to heat the water. –Assume pipe wall temperature at 100 º C (condensing steam T) Find: How long would the pre-heating pipe length have to be to give an outlet temperature = 50º C? Solution method??

20. Heat Transfer Su Yongkang School of Mechanical Engineering # 20 Pre-heating Water Example – Part 2 Solution: Fluid properties at a mean T = 30ºC Reynolds number: For turbulent flow, heating mode Log mean temperature difference

21. Heat Transfer Su Yongkang School of Mechanical Engineering # 21 Pre-heating Water Example – Part 2 Solution (Cont’d): Compute the pipe wall surface area required Required length to get this area is: What would be the pro and cons for choosing electrical heater versus heating with steam in this case?

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