# 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
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.
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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