Convective Heat Transfer

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					                 HEAT    TRANSFER



                Recall: Fluid Mechanics




                           #1
Heat Transfer                   Su Yongkang
                                School of Mechanical Engineering
       What is Convective Heat Transfer?


    ─ You have already experienced it.
    ─ Difficulty lies in generalizing our experience;
      filtering it down to a few laws; learning how to
      apply these laws to systems we engineers design and
      use.
    ─ Here is what I want you to do:
    ─ If a person masters the fundamentals of his subject
      and has learned to think and work independently, he
      will surely find his way and besides will better be able
      to adapt himself to progress and changes than the
      person whose training principally consists in the
      acquiring of detailed knowledge.
    ─                                 – Albert Einstein
    ─ So, please read ahead and come prepared with good
      questions for the class.




                               #2
Heat Transfer                       Su Yongkang
                                    School of Mechanical Engineering
                     1. Introduction to
                 DIMENSIONAL ANALYSIS

        1)      Dimensions
        Each quantitative aspect provides a
        number and a unit.
        For example,
                        V  5m / s
        The three basic dimensions are L, T, and M.
        Alternatively, L, T, and F could be used.

        We can write
                                    1
                         V  LT
                           
                         F  MLT 2
                           
        The notation     
                         
                        is used to indicate
        dimensional equality.


                               #3
Heat Transfer                       Su Yongkang
                                    School of Mechanical Engineering
     2) Dimensional homogeneity

     Fundamental premise:

     All theoretically derived equations are
     dimensionally homogeneous----that is, the
     dimensions of the left side of the equation
     must be the same as those on the right side,
     and all additive separate terms must have the
     same dimensions.
     For example, the velocity equation,

                     V  V0  at
     In terms of dimensions the equation is

                1        1                   1
          LT  LT  LT
             
     ----dimensional homogeneous

                           #4
Heat Transfer                   Su Yongkang
                                School of Mechanical Engineering
    3) Dimensional analysis
    A problem.
    An incompressible, Newtonian fluid, steady
    flow, through a long, smooth-walled, horizontal,
    circular pipe.

                pl  f D,  ,  ,V 
    The nature of function is unknown and
    the experiments are necessary.

    We can recollect these variables into
    dimensionless products,

                  Dpl     VD 
                        
                             
                  V 2
                               
    Variables from 5 to 2.



                             #5
Heat Transfer                     Su Yongkang
                                  School of Mechanical Engineering
    Here,

            Dpl       L( F / L3 )
                 
                                        F 0 L0T 0
                                        
            V 2       4 2
                   ( FL T )(LT )   1 2


            VD ( FL4T 2 )(LT 1 ) L
               
                                      F 0 L0T 0
                                      
                         2
                     ( FL T )
    The results will be independent of the system of
    units.
    This type of analysis is called
    -----dimensional analysis.




                               #6
Heat Transfer                       Su Yongkang
                                    School of Mechanical Engineering
   4) Buckingham Pi theorem

  If an equation involving k variables is
  dimensionally homogeneous, it can be reduced to
  a relationship among k-r independent
  dimensionless products, where r is the minimum
  number of reference dimensions required to
  describe the variables.




                         #7
Heat Transfer                 Su Yongkang
                              School of Mechanical Engineering
      Here, we use the symbol                  to represent a
      dimensionless product.

      For equation,

                u1  f u2 , u3 ,..., uk 
      We can rearrange to,

           1    2 ,  3 ,...,  k r 

  Usually, the reference dimensions required to
  describe the variables will be the basic
  dimensions M, L, and T or F, L, and T. In some
  cases, maybe only two are required, or just one.


         determination of Pi terms???


                             #8
Heat Transfer                     Su Yongkang
                                  School of Mechanical Engineering
         2. The Navier-Stokes Equations

 Combine the differential equations of motion,
 the stress-deformation relationships and the
 continuity equation.


     u u u   u   p            2u  2u  2u 
     u  v  w     g x    2  2  2 
     t
        x y   z 
                      x           x y z 
                                                   
     v v v   v   p            2v  2v  2v 
     u  v  w     g y    2  2  2 
     t
        x y   z 
                      y           x y z 
                                                   
     w w w   w   p           2w 2w 2w 
     u  v  w     g z    2  2  2 
     t
        x  y  z 
                      z           x y z 
                                                




                           #9
Heat Transfer                   Su Yongkang
                                School of Mechanical Engineering
    Here, four unknowns (u, v, w, p.)
    We know the conservation of
    mass equation,

                u v w
                    0
                x y z

    ----------four equations.

    Nonlinear, second order, partial
    differential equations.




                       # 10
Heat Transfer             Su Yongkang
                          School of Mechanical Engineering
      3. general characteristics of pipe flow




                        # 11
Heat Transfer              Su Yongkang
                           School of Mechanical Engineering
 1) Laminar or turbulent flow
 Osborne Reynolds-------flowrates




                        # 12
Heat Transfer              Su Yongkang
                           School of Mechanical Engineering
      And




     The Reynolds number,

                Re  VD / 
     For the flow in a round pipe,

     Laminar flow:         Re<2100
     Transitional flow:    2100<Re<4000
     Turbulent flow:       Re>4000
                           # 13
Heat Transfer                 Su Yongkang
                              School of Mechanical Engineering
      2) Entrance region and fully developed flow




    Typical entrance lengths are given by
                 le
                     0.06 Re for laminar flow
                 D
    and
                le
                    4.4 Re1/ 6 for turbulent flow
                D


                               # 14
Heat Transfer                     Su Yongkang
                                  School of Mechanical Engineering
        3) Fully developed laminar flow




                           # 15
Heat Transfer                 Su Yongkang
                              School of Mechanical Engineering
          4) Fully developed turbulent flow

    Transitional flow:    2100<Re<4000
    Turbulent flow:       Re>4000




                           # 16
Heat Transfer                 Su Yongkang
                              School of Mechanical Engineering
 Turbulent velocity profile
 There is no general accurate expression
 for turbulent velocity profile.




                         # 17
Heat Transfer               Su Yongkang
                            School of Mechanical Engineering
    Nothing Is Impossible To A Willing Heart




                      # 18
Heat Transfer            Su Yongkang
                         School of Mechanical Engineering

				
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