1. Heat Transfer In Channels Flow
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2. 6.5 Channels with Uniform Surface Temperature
We wish to determine the following:
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3. 6.5 Channels with Uniform Surface Temperature
Applying conservation of energy to the element dx
Eq. (a) = Eq. (b), we get
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4. (d) 6.5 Channels with Uniform Surface Temperature
From the average heat transfer coefficient over the length x
We get
(d)
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5. 6.5 Channels with Uniform Surface Temperature
Introducing (d) into (6.11) and solving the resulting equation for Tm(x)
Application of conservation of energy between the inlet of the channel and a section x gives
Application of Newton’s law of cooling gives the heat flux q”s(x) at location x gives
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6. 6.5 Channels with Uniform Surface Temperature
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7. 6.5 Channels with Uniform Surface Temperature
Solution
For flow through a tube at uniform surface temperature, applying Eq.(6.13)
At the outlet of the heat section (x=L) and solving for L
Where
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8. 6.5 Channels with Uniform Surface Temperature
The properties of air using at the mean temperature Tm(x)
Check the flow is laminar or turbulent
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9. 6.5 Channels with Uniform Surface Temperature
Since the Reynolds number is smaller than 2300, the flow is laminar. Thus
The mass flow rate.
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10. 6.5 Channels with Uniform Surface Temperature
The perimeter.
Finally, the length of tube
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11. Equating Fourier’s law with Newton’s law
6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
6.6.1 Scale Analysis
Equating Fourier’s law with Newton’s law
A scale for r is
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12. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
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13. From Eq. (6.18) applying thermal thickness of external flow
6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
From Eq. (6.18) applying thermal thickness of external flow
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14. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
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15. (1) Fourier’s law and Newton’s law.
6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
6.6.2 Basic Considerations for the Analytical Determination of Heat Flux, Heat Transfer Coefficient and Nusselt Number
(1) Fourier’s law and Newton’s law.
(6.21)
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16. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
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17. We define h using Newton’s law of cooling
6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
Substituting into (a)
(6.22)
We define h using Newton’s law of cooling
(6.23)
Combining (6.22) and (6.23)
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18. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
Where
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19. The last term in Eq.(6.28) can be neglected for
6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
(2) The Energy Equation
The last term in Eq.(6.28) can be neglected for
where
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20. Thus, under such conditions, Eq.(6.28) becomes
6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
Thus, under such conditions, Eq.(6.28) becomes
3) Mean (Bulk) Temperature, Tm
Where
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21. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD
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22. This section focuses on the fully developed region.
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
This section focuses on the fully developed region.
6.7.1 Definition of Fully Developed Temperature Profile
Far away from the entrance of a channel
We introduce a dimensionless temperature defined as
For fully developed is independent of x. That is
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23. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
Thus.
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24. 6.7.2 Heat Transfer Coefficient and Nusselt Number
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
6.7.2 Heat Transfer Coefficient and Nusselt Number
Equating Fourier’s with Newton’s law
Using Eq.(6.37) in the definition of the Nusselt number, give
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25. For scale analysis of temperature gradient
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
For scale analysis of temperature gradient
Compared Eq.(6.19)
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26. 6.7.3 Fully Developed Region for Tubes at Uniform Surface flux
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
6.7.3 Fully Developed Region for Tubes at Uniform Surface flux
Application of Newton’s law of cooling gives
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27. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
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28. Using energy balance on element dx for detemine eq.(6.41)
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
Using energy balance on element dx for detemine eq.(6.41)
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29. Assume Cp and m constant
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
Assume Cp and m constant
Substituting eq.(6.42) into (6.41)
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30. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
For determine fluid temperature distribution T(r,x) and surface temperature Ts(x), from energy equation
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31. The axial velocity for fully developed flow is
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
The axial velocity for fully developed flow is
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32. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
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33. Substituting eq.(6.46) and (6.49) into (6.32a)
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
Substituting eq.(6.46) and (6.49) into (6.32a)
gives
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34. Substituting T(r,x), Tm(x) and Ts(x) into eq.(6.33) gives
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
Substituting T(r,x), Tm(x) and Ts(x) into eq.(6.33) gives
Differentiating (6.54) and substituting into (6.38) gives the Nusselt number
From scaling analysis
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35. Substituting (6.51) into (6.49)
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
From eq.(6.44) and (6.50), we get
Substituting (6.51) into (6.49)
Surface temperature, by setting r=ro in (6.52)
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36. 6.7.4 Fully Developed Region for Tubes at Uniform Surface Temperature
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
6.7.4 Fully Developed Region for Tubes at Uniform Surface Temperature
By energy equation
- Neglecting axial conduction and dissipation
- vr = 0
Simplifies to
Boundary conditions
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37. Using equation (6.36a) to eliminate
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
Using equation (6.36a) to eliminate
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38. Applied boundary condition to Eq.(6.58)
6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
Applied boundary condition to Eq.(6.58)
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39. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region
6.7.5 Nusselt Number for Laminar Fully Developed Velocity and Temperature in Channels of Various Cross-Sections
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40. Example 6.4: Maximum Surface Temperature in an Air Duct
Solution
Temperature distribution for uniform heat flux, given by eq.(6.10)
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41. Example 6.4: Maximum Surface Temperature in an Air Duct
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42. Example 6.4: Maximum Surface Temperature in an Air Duct
Using Energy conservation to determine L
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43. Example 6.4: Maximum Surface Temperature in an Air Duct
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44. Example 6.4: Maximum Surface Temperature in an Air Duct
Laminar flow
From Table.6.2 for uniform heat flux
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45. Example 6.4: Maximum Surface Temperature in an Air Duct
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46. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
6.8.1 Uniform Surface Temperature: Graetz Solution
Consider laminar flow in Fig Fluid enters a heated or cooled section with a fully developed velocity
We neglect axial conduction (Pe >100)
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47. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
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48. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
Assume product solution as the form
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49. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
Substitution the solution of (b) and (c) into (a)
Where Cn is constant
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50. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
The surface heat flux is given by
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51. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
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52. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
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53. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
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54. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
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55. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
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56. Example 6.5 hot water heater
Solution
For flow through the tube at uniform surface temperature, from Eq.(6.13)
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57. Example 6.5 hot water heater
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58. Example 6.5 hot water heater
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59. Example 6.5 hot water heater
Compute the thermal entrance length, from Eq.(6.6)
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60. Example 6.5 hot water heater
Compute the heat transfer coefficient
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61. Example 6.5 hot water heater
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62. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
6.8.2 Uniform Surface Heat Flux
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63. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
The solution for Nusselt number is
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64. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
The average Nusselt number is given by.
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65. 6.8 Thermal Entrance Region: Laminar Flow through Tubes
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