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HEAT TRANSFER

CHAPTER 7 External flow. HEAT TRANSFER. External Flow: Flat Plate. Topic of the Day. External Flow: Flat Plate. Where we’ve been …… General overview of the convection transfer equations.

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HEAT TRANSFER

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  1. CHAPTER 7 External flow HEAT TRANSFER # 1

  2. External Flow: Flat Plate Topic of the Day # 2

  3. External Flow: Flat Plate Where we’ve been …… • General overview of the convection transfer equations. • Developed the key non-dimensional parameters used to characterize the boundary layer flow and convective heat and mass transfer. Where we’re going: • Applications to external flow • Flat plate  Today • Other shapes  Next time Then onto internal flow …… # 3

  4. Differences between external and internal flow • External flow:Boundary layer develops freely, without constraints • Internal flow:Boundary layer is constrained and eventually merges # 4

  5. How this impacts convective heat transfer • Recall the boundary layer convection equations: • As you go further from the leading edge, the boundary layer continues to grow. Assuming the surface and freestream T do not change:with increasing distance ‘x’: • Boundary layer thickness, ,  • so • and wall temperature gradient fluid thermal conductivity # 5

  6. Methods to evaluate convection heat transfer • Empirical (experimental) analysis • Use experimental measurements in a controlled lab setting to correlate heat and/or mass transfer in terms of the appropriate non-dimensional parameters • Theoretical or Analytical approach • Solving of the boundary layer equations for a particular geometry. • Example: • Solve for T* • Use evaluate the local Nusselt number, Nux • Compute local convection coefficient, hx • Use these (integrate) to determine the average convection coefficient over the entire surface • Exact solutions possible for simple cases. • Approximate solutions also possible using an integral method # 6

  7. Empirical method to obtain heat transfer coefficient • How to set up an experimental test? • Let’s say you want to know the heat transfer rate of an airplane wing (with fuel inside) flying at steady conditions…………. • What are the parameters involved? • Velocity, –wing length, • Prandtl number, –viscosity, • Nusselt number, • Which of these can we control easily? • Looking for the relation:Experience has shown the following relation works well: # 7

  8. Empirical method to obtain heat transfer coefficient • Experimental test setup • Measure current (hence heat transfer) with various fluids and test conditions for • Fluid properties are typically evaluated at the mean film temperature L insulation # 8

  9. Analytical Solution – Laminar Flow • Assume: • Steady, incompressible, laminar flow • Constant fluid properties • For flat plate, • Boundary layer equations • Blasius developed a similarity solution to the hydrodynamic equations in 1908 based on the stream function, (x,y) y Continuity Momentum Energy # 9

  10. Analytical Solution – Laminar Flow (Cont’d) • Define new dependent and independent variables, • The momentum equation can be rewritten as • And the boundary conditions are and and # 10

  11. Analytical Solution – Laminar Flow (Cont’d) • Blasius solution summary: • Conclusions from the Blasius solution: • Solution for the thermal boundary layer: • For Pr  0.6 • Expressing the local convection coefficient as: • Then the Local Nusselt number is: Eq. 7.21 For 0.6  Pr  50 # 11

  12. y x Analytical Solution – Laminar Flow (Cont’d) • The Average Nusselt number over the whole plate found by integrating: • Ratio of velocity to thermal boundary layer thickness: Eq. 7.25 For large Pr (oils): For small Pr (liquid metals): y x Pr > 1000 Pr < 0.1 Fluid viscosity greater than thermal diffusivity Fluid viscosity less than thermal diffusivity # 12

  13. Analytical Solution – Laminar Flow (Cont’d) • Solution for friction factor • Textbook contains Nusselt number correlations for low Pr (liquid metals) and large Pr (oils) # 13

  14. Analytical Solution – Turbulent Flow • For flat plate in turbulent flow (more common) Important point: • Typically a turbulent boundary layer is preceded by a laminar boundary layer first upstream •  need to consider case with mixed boundary layer conditions! # 14

  15. Analytical Solution – Mixed Boundary Layer • Integrating Equations 7.33 and 7.34 # 15

  16. Analytical Solution – Special Cases • The existence of unheated starting length. • When the boundary condition is a uniform surface heat flux. For laminar flow, For turbulent flow, # 16

  17. Methodology for a Convection Calculation • Become immediately cognizant of the flow geometry. • Specify the appropriate reference temperature and evaluate the fluid properties. • Calculate the Reynolds number • Decide whether a local or surface average coefficient is required. • Select the appropriate correlation. # 17

  18. Example – Cooling of automobile crankcase • Given: • Automobile crankcase with approximate dimensions of 0.6 m long, 0.2 m wide and 0.1 m deep. • Surface temperature of 350 K • Ambient temperature of 300 K • Vehicle velocity of 30 m/s • Find: • Heat loss from bottom surface exposed to air stream • What other information or assumptions needed? # 18

  19. Example – Cooling of automobile crankcase (Cont’d) • Determine air properties at an average film temperature • Calculate Reynolds # • Calculate average Nusselt number (mixed b.l.) • Average convection coefficient is • BOTTOM SURFACE HEAT LOSS: # 19

  20. Example – Cooling of automobile crankcase (Cont’d) • How to determine the heat loss from the other surfaces? • Assumptions ………….. • Analysis procedure ……… # 20

  21. C L Example: Cooling air over electronic chips • Given:Cooling air drawn over electronic devices mounted on board. • Devices are 4 x 4 mm in size, spacing = 0.25 mm • Find the surface temperature of the fourth device, assumed uniform surface T. • Assumptions? • Solution Method? T = 27 º C V = 10 m/s Q = 40 mW each device “turbulator” 15 mm # 21

  22. Example: Consider atmospheric air at 25℃ and a velocity of 25 m/s flowing over both surfaces of a 1-m long flat plate that is maintained at 125 ℃. Determine the rate of heat transfer per unit width from the plate for values of the critical Reynolds number corresponding to , , and . # 22

  23. # 23

  24. External Flow: Flat Plate KEY POINTS THIS SECTION • What key characteristic of external flow compared to internal flow? • Heat transfer rate generally decreases with increasing distance from leading edge. • Turbulent convective heat transfer generally higher than laminar due to mixing effect within boundary layer. • Experimental tests indicate that heat transfer coefficient will generally vary like: • Concept of transition Re number. • Difference in boundary layer growth for high and low Pr number fluids. • General correlation for Nusselt number for flow over flat plate in laminar, turbulent and mixed flows. # 24

  25. Have a good time! Go back and review lecture notes! # 25

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