In-Class Activity #6_HT-Spring 2023
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IN-CLASS ACTIVITY #6
(Heat Transfer, MECE:315:001, Spring 2023)
Name: ____________
(March 8, Wednesday)
1. Introduction to Convective Heat Transfer and
convective heat transfer coefficient, h:
Convection ≡ Heat transfer between a surface and a fluid
moving over the surface
Newton’s Law of Cooling:
q′′ = h (Ts − T∞ )
h – convection heat transfer coefficient, (W/m2-K)
1.1 A Velocity Boundary Layer:
Thickness of boundary layer: δ(x) at (u/u∞=0.99)
Surface shear stress: τ = µ ∂u
s
∂y
y =0
µ
Local friction coefficient, Cf:
Cf =
∂u
∂y
y =0
ρ u∞2 / 2
1.2 A Thermal Boundary Layer:
Thickness of thermal boundary layer, δt(x):
Ts − T (δ t )
= 0.99
Ts − T∞
Energy balance over the solid/fluid interface:
∂T
−k f
q′′( x) =
hx (Ts − T∞ ) =
∂y y =0
Therefore one has the local heat transfer coefficient, hx,
hx =
− k f (∂T / ∂y ) y = 0
Ts − T∞
One can see, h x = f ( x, u , T , u ∞ , T∞ , Ts , fluids, geometry )
1.3 Local or Average Convection Heat Transfer Coefficient:
Local heat flux and local heat transfer coefficient, hx:
=
q′′( x) hx [Ts ( x) − T∞ ]
Total rate of heat transfer and an average heat transfer coefficient, h :
q = ∫ q′′dAs = ∫ hx (Ts − T∞ )dAs = hAs (Ts − T∞ )
As
with,
h=
As
∫A hx (Ts − T∞ )dAs 1
q
= s
=
∫ hx dAs
As (Ts − T∞ )
As (Ts − T∞ )
As A
s
For a flat plate with a uniform width of W, ( As = W L ),
=
q
∫
As
=
q′′dAs
∫
x
0
h(Ts − T∞ )Wdx
→
h=
1 L
∫ hx ( x)dx
L 0
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2. Navier-Stokes Equations and Energy Equation for 2-D Convective Heat Transfer:
The governing equations, from the conservations laws of mass, momentum, and energy provides the
following partial differential equations for a two-dimensional (2-D) flow and convective heat transfer of
incompressible fluids in a Cartesian coordinate system:
∂u ∂v
Mass:
+
=0
∂x ∂y
Momentum:
u
1 ∂p µ  ∂ 2u ∂ 2u 
∂u
∂u
+
+
=−
+v
ρ ∂x ρ  ∂x 2 ∂y 2 
∂y
∂x
∂v
1 ∂p µ  ∂ 2v ∂ 2 v 
∂v
+
+
=−
+v
∂y
ρ ∂y ρ  ∂x 2 ∂y 2 
∂x
k  ∂ 2T ∂ 2T 
q
∂T
∂T
+
u
+v
=
+
2
2


∂x
∂y ρc p  ∂x
∂y  ρc p
u
Energy:
x
Note: 4 equations for 4 unknowns (u, v, p, T)
3. Laminar Boundary Layer Flow over a Surface:
For a 2-D flow over a flat place, one has,
δ , δ T  L, v  u ,
Then the governing equation becomes,
∂ 2u ∂ 2u ∂ 2T
∂ 2T


,
,
∂x 2
∂y 2
∂x 2
∂y 2
p = patm
∂u ∂v
0
 ∂x + ∂y =

 ∂u
∂u
∂ 2u
+v
=
ν 2
u
∂y
∂y
 ∂x
 ∂T
∂T
∂ 2T
+v
=
α 2
u
∂y
∂y
 ∂x
where the kinematic viscosity, v = µ / ρ and thermal diffusivity α = k / ( ρ c p ) are used. It is noticed that
both ν and α have a unit of m2/s.
4. Non-dimensionalization of the laminar boundary layer equation over a flat plate:
Introduce the following dimensionless variables,
x*
=
T − Ts
x
y
u
v
y*
, u*
, v*
, T*
=
=
=
=
L
L
u∞
u∞
T∞ − Ts
Then the above boundary-layer equations can be normalized,
∂u * ∂u *
+
=
0
∂x * ∂y *
u*
∂u *
∂u *
1 ∂ 2u *
+ v*
= 2
∂x *
∂y * Re L ∂y *
u*
∂T *
∂T *
1 ∂ 2T *
+ v*
=
∂x *
∂y * Re L Pr ∂y *2
Dimensionless (Group)
numbers
Reynolds
u L
Re L = ∞
number
ν
Prandtl
number
Pr=
ν µcp
=
k
α
One finds two dimensionless groups (numbers) after normalization, Reynolds number and a new Prandtl
numbers. Therefore, the functional forms of dimensionless velocity and temperature should be:
u* = f ( x*, y*, Re L )
T * = f ( x*, y*, Re L , Pr)
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5. Surface (local) friction coefficient Cf

τs
1  ∂u
 = 2 ∂u *
µ
Cf =
=
ρ u∞ / 2 ρ u∞ / 2  ∂y y =0  Re L ∂y * y*=0


u* = f ( x*, y*, Re L ) ,
Since
therefore
C f ( x* ) =
Over the entire flat plate, the average friction coefficient will be
L
1
2
Cf =
C f ( x* )dx
f (Re L )
=
∫
Re L
L x =0
6. Heat transfer coefficient and Nusselt number (Nu)
For the local convective heat transfer coefficient
at x,
h( x )
=
One can write it in a dimensionless form as
=
Nu
( x* )
2
f ( x*, Re L )
Re L
−k f (∂T / ∂y ) y =0 k f ∂T *
=
Ts − T∞
L ∂y * y*=0
h( x) L ∂T *
=
= f ( x*, Re L , Pr)
∂y * y *=0
kf
where Nu is non-dimensional heat transfer coefficient and is named Nusselt number.
For the average heat transfer coefficient over the entire plate (boundary layer) of length L,
L
hL 1
Nu=
h( x)dx
=
= f (Re L , Pr)
L
k f k f x∫=0
7. Physical meaning of Prandtl number,
Pr=
Prandtl number is defined as a ratio of the momentum
diffusivity (ν) to the thermal diffusivity (α).
ν µcp
=
α
k
Prandtl number provides a measure of the relative
effectiveness of momentum and energy transport by diffusion
in the velocity and thermal boundary layers.
δ
≈ Pr1 / 3
δt
Air
Water
Mercury(Hg)
k (W/m-K)
0.0263
0.613
8.54
Pr
0.707
5.83
0.0248
8. Momentum and Heat Transfer
(Reynolds) Analogy
When Pr = 1, the momentum and energy
equations are identical,
u*
∂u *
∂u *
1 ∂ 2u *
+ v*
= 2
∂x *
∂y * Re L ∂y *
u*
∂T *
∂T *
1 ∂ 2T *
+ v*
= 2
∂x *
∂y * Re L ∂y *
Indicating a strong analogy between momentum transport and heat transfer. This analogy is represented
by a clear relation between Cf and Nu,
Re L
hL ∂T *
∂u *
Nu
= =
=
=
Cf .
kf =
∂y * y* 0=
∂y * y* 0
2
This relation indicates that the Nueeslt number can be determined from the data of Cf.
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If the effect of Prandtl number is considered, the analogy gives,
Cf
2
= St =
Nu
Re L Pr
A more accurate empirical relation is the Chilton-Colburn analogy relation,
Cf
= St Pr 2 / 3 = jH
0.6 < Pr < 60
2
jH – Colburn j factor.
9. Experimental Method to Determine the Convection Heat Transfer Coefficient:
Most of time, average heat transfer coefficients can be determined
experimentally, e.g., for a flat plate shown here,
q
I ⋅E
,
=
h=
As (Ts − T∞ ) As (Ts − T∞ )
Under various conditions. Then the corresponding Nusselt number
can be calculated and correlated with Reynolds number and
Prandtl number,
Nu =
hL
= f (Re L , Pr) .
kf
Usually, a power law relation is used as
follows,
n
Nu = C Rem
L Pr
where constant C and power components m and
n are obtained by curve-fitting the experimental
data as shown on right.
In the dimensionless numbers, the properties k,
ν (µ, ρ), and Pr of fluids are usually evaluated
at the film temperature:
T f = (Ts + T∞ ) / 2
10. Empirical Correlations for Convective Heat Transfer
over a Flat Plate:
For local heat transfer coefficient,
Laminar flow ( Re
=
u∞ x / ν < 5 × 105 ):
x
hx x
Nu
=
= 0.332 Re1/x 2 Pr1/3
x
kf
Pr ≥ 0.6
Turbulent flow ( =
Re x u∞ x / ν > 5 × 105 ):
Nu x = 0.0296 Re 4x / 5 Pr1 / 3
0.6 < Pr < 60

Re x ≤ 108
For average heat transfer coefficient,
hL =

L
1  xc
hlam dx + ∫ hturb dx 
∫

L 0
xc


0.6 ≤ Pr ≤ 60

5
8
5 ×10 ≤ Re L ≤ 10

5
Re x ,c = 5 ×10
where hL is averaged for a mixed flow (laminar to turbulent) over the isothermal flat plate.
hL
1/3
Nu L =
= (0.037 Re 4/5
L − 871) Pr
kf