1
8. FORCED CONVECTION
1. Correlation Development
- Common Approach
u,
T
Q=IE=hLAT
Ts
Ts
I
› Measure u, T, Ts, I, and E
› Determine hL based on the energy balance (usually account
for the small, but significant heat losses to the “insulation”)
› Calculate the Nu , Re, Pr at the film temperature Tf:
T  T
Tf  s
2
› Determine the “best fit” parameters C, m, n for the
correlation:
Nu  C Re m Pr n
NOTE:
EACH CORRELATION IS ONLY VALID WITHIN A
CERTAIN RANGE OF CONDITIONS!!!
2
2. Parallel Flow on a Flat Plate
- Laminar Flow
› Pr  0.6
  5.0 x ;
C f ,L
Re x
 1.328
Re L
  Pr1 / 3
t
  Sc1 / 3
m
(7.19; 24; 28)
(7.30)
Nu L 
hx L
 0.664(Re L )1 / 2 (Pr)1 / 3
k
(7.31)
ShL 
hm L
 0.664(Re L )1 / 2 ( Sc)1 / 3
D AB
(7.32)
› Pr  0.05 and Pex > 100
Nu x  0.565 Pe1x / 2
(7.33)
where: Pex = Peclet Number (= RexPr)
› Pex > 100 (Churchill and Ozoe Formula)
Nu x 
0.3387 Re x1 / 2 Pr1 / 3
[1  (0.0468 / Pr)
(7.34)
2 / 3 1/ 4
]
- Turbulent Flow
  0.37 x Re x 1 / 5 ;
C f , x  0.0592 Re x 1 / 5
  t   m
Rex  107
(7.36)
(7.35)
Nu x  0.0296 Re x 4 / 5 Pr1 / 3
0.6<Pr <60
(7.37)
Shx  0.0296 Re x 4 / 5 Sc1 / 3
0.6<Pr <300
(7.38)
- Mixed BL Conditions
If transition occurs in the middle of the plate, both laminar and
turbulent flow must be considered!

hL  1 0xc hlam dx  xL 0 htur dx
c
L
See Eqs. 7.39 to 43

3
- Uniform Surface Heat Flux
› Laminar flow
Pr  0.6
Nu x  0.453 Re1x/ 2 Pr1 / 3
› Turbulent flow
0.6  Pr  60
Nu x  0.0308 Re 4x / 5 Pr1 / 3
- Unheated Starting Length
Laminar flow
Nu x 
Nu x
0
3 / 4 1/ 3
[1  ( / x)
]
Turbulent flow
Nu x 
Nu x
[1  ( / x)
0
9 / 10 1 / 9
]
3. Flow around Cylinders
- Drag Coefficient
Eq. 7.54 + Fig. 7.8
- Heat coefficient, hL
› Hilpert Formula (Eq. 7.55b + Tables 7.2 and 7.3)
Note:
Evaluate the fluid properties at Tf
› Zhukauskas Formula (Eq. 7.56 + Table 7.4)
Note:
Evaluate the fluid properties at T
› Churchill and Bernstein Formula (Eq. 7.57)
Note:
Evaluate the fluid properties at Tf
4. Flow around Spheres
Similar idea for cylinders
See Eq. 7.59
Evaluate the fluid properties at T
Summary of hL correlations for external flows: Table 7.9
4
5. Flow across Tube Banks
See Eqs. 7.61 to 7.68
6. Pipe and Tube Flow
- Basic Fluid Dynamics
ReD,c ~ 2,000 to 4,000
› Entry Length
Laminar flow:
Turbulent flow
› Velocity Profile
Laminar flow:
Turbulent flow
› Friction Coefficient
Laminar flow:
Turbulent flow
x fd , h / D lam  0.05 Re D
10  x fd , h / D 
 60
turb
u  1  (r / r ) 2
o
u0
u  1  (r / r ) 0.143
o
u0
f  64 ;
Re D
Cf 
Moody diagram
f
4
5
- Heat Transfer in Pipe Flow
› Newton’s Law of Cooling
q "s  h(Ts  Tm )
where:
Note:
Tm = mean temperature in the pipe
1) Tm is defined in terms of the energy transported
across the pipe section as:
Tm 
 Ac ucvTdAc
 2 2 0ro uTrdr
m cv
u m ro
2) Tm varies with the flow distance x
3) In practice, Tm  (Tm, i  Tm, o ) / 2
› Energy Balance Equation
For incompressible liquids and ideal gases within fully
developed region:
m c p dTm  q "s ( Pdx)  h(Ts  Tm )( Pdx)
or
dTm q "s P h(Ts  Tm ) P


dx
m c p
m c p
where:
P = perimeter of the cross-section.
› Constant Surface Heat Flux, q "s
dTm q "s P

 const
dx
m c p
"
 Tm ( x)  Tm, i  q s P x
m c p
› Constant Surface Temperature, Ts
dTm
d (Ts  Tm ) h(Ts  Tm ) P


dx
dx
m c p
Integrating it from the inlet to outlet,
 TTo
i
d (Ts  Tm )
 P 0L hdx
(Ts  Tm ) m c p
6

(Ts  Tm, o ) To



 exp  PL hL 
(Ts  Tm, i ) Ti
 m c p 
Total heat transfer is:
 c p (Tm, o  Tm, i )  m
 c p (Ti  To )  hL ( PL)Tlm
qconv  m
where: Tlm  To  Ti
ln( To / T )
- Convection Correlations
› Fully Developed, Laminar Flow
q"s  const
Nu D  hD  4.36
Nu D
k
 hD  3.66
k
Ts  const
› Fully Developed, Turbulent Flow
See Eqs. 8.60 to 8.63
Note the fluid properties should be evaluated at Tm
› Entry Length, Laminar Flow with
See Eqs. 8.56 to 8.57
› Noncircular Tubes
Replace D with hydraulic diameter Dh
Dh 
where:
4 Ac
P
Ac = flow cross-section
P = wetted perimeter
Summary of hL correlations for internal flows: Table 8.4