ME 259
Heat Transfer
Lecture Slides III
Dr. Gregory A. Kallio
Dept. of Mechanical Engineering,
Mechatronic Engineering &
Manufacturing Technology
California State University, Chico
3/7/05
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1
Transient Conduction
Reading: Incropera & DeWitt
Chapter 5
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2
Introduction


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Transient = Unsteady (time-dependent )
Examples
– Very short time scale: hot wire anemometry
(< 1 ms)
– Short time scale: quenching of metallic
parts (seconds)
– Intermediate time scale: baking cookies
(minutes)
– Long time scale – daily heating/cooling of
atmosphere (hours)
– Very long time scale: seasonal
heating/cooling of the earth’s surface
(months)
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3
Governing Heat Conduction
Equation

Assuming k = constant,
q
1 T
 T 
k  t
2

For 1-D conduction (x) and no internal heat
generation,
 2T 1 T

2
x
 t

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Solution, T(x,t) , requires two BCs and an
initial condition
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4
The Biot Number

The Biot number is a dimensionless
parameter that indicates the relative
importance of of conduction and
convection heat transfer processes:
hL
Bi 
k
Rt ,cond
L kA


1 hA
Rt ,conv

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Practical implications of Bi << 1: objects
may heat/cool in an isothermal manner if
– they are small and metallic
– they are cooled/heated by natural
convection in a gas (e.g., air)
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5
The Lumped Capacitance Method
(LCM)



LCM is a viable approach for solving
transient conduction problems when Bi<<1
Treat system as an isothermal,
homogeneous mass (V) with uniform
specific heat (cp)
For suddenly imposed, uniform convection
(h, T) boundary conditions, solution
yields:
T (t )  T
 exp  t  t 
Ti  T
where
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t 
c p V
hAs
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6
General LCM

LCM can be applied to systems with
convection, radiation, and heat flux
boundary conditions and internal heat
generation (see eqn. 5.15 for ODE)
–
–
–
–
Forced convection, h = constant
Natural convection, h = C(T-T)1/4
Radiation (eqn. 5.18)
Convection and radiation (requires
numerical integration)
– Forced convection with constant surface
heat flux or internal heat generation (eqn.
5.25)
(Note that the Biot number must be redefined when
effects other than convection are included)
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Spatial Effects (Bi  0.1)

LCM not valid since temperature gradient
within solid is significant

Need to solve heat conduction equation
with applied boundary conditions

1-D transient conduction “family” of
solutions:
– Uniform, symmetric convection applied
to plane wall, long cylinder, sphere
(sections 5.4-5.6)
– Semi-infinite solid with various BCs
(section 5.7)
– Superposition of 1-D solutions for
multidimensional conduction (section
5.8)
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8
Convection Heat Transfer:
Fundamentals
Reading: Incropera & DeWitt
Chapter 6
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9
The Convection Problem

Newton’s law of cooling (convection)
qs  h(Ts  T )

“h” is often the controlling parameter in heat
transfer problems involving fluids; knowing its
value accurately is important

h can be obtained by
– theoretical derivation (difficult)
– direct measurement (time-consuming)
– empirical correlation (most common)

Theoretical derivation is difficult because
– h is dependent upon many parameters
– it involves solving several PDEs
– it usually involves turbulent fluid flow, for which no
unified modeling approach exists
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10
Average Convection Coefficient

Consider fluid flow over a surface:

Total heat rate:
q   qdAs  (Ts  T )  hdAs
As
As
 1
 (Ts  T ) As 
 As

 h As (Ts  T )

If h = h(x), then
1
h 
L
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
A hdAs 
s


L
0
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h dx
11
The Defining Equation for h

The thermal boundary layer:

From Fourier’s law of conduction:
qs   k solid
  k fluid

y 0
T fluid
y
y 0
Setting qconv = qcond and solving for h:
h
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Tsolid
y
T
k
y
y 0
Ts  T
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Determination of T y y 0




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Need to find T(x,y) within fluid boundary layer,
then differentiate and evaluate at y = 0
T(x,y) is obtained by solving the following
conservation equations:
– Mass (continuity) ……. Eqn (6.25)
– Momentum (x,y) ……. Eqn (6.26, 6.27)
– Thermal Energy …….. Eqn (6.28a,b)
Solution of these four coupled PDEs yields the
velocity components (u,v), pressure (p), and
temperature (T)
Exact solution is only possible for laminar flow
in simple geometries
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13
Boundary Layer Approximations

Because the velocity and thermal boundary
layer thicknesses are typically very small, the
following BL approximations apply:
u  v
u
u v v

,
,
y
x y x
T
T

y
x

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The BL approximations with the additional
assumptions of steady-state, incompressible
flow, constant properties, negligible body
forces, and zero energy generation yield a
simpler set of conservation equations given by
eqns. (6.31-6.33).
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Boundary Layer Similarity
Parameters

BL equations are normalized by defining the following
dimensionless parameters:
x   x / L,
y  y / L
u  u / V , v  v / V
T  Ts
p
, p 
T  Ts
V 2
where L is the characteristic length of the surface
and V is the velocity upstream of the surface
T 

The resulting normalized BL equations (Table 6.1)
produce the following unique dimensionless groups:
VL
 Reynoldsnumber (Re)


 Prandtl number (Pr)


k
where  
and  

c p

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Note: all properties are those of the fluid
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15
Boundary Layer Similarity
Parameters, cont.

Recall the defining equation for h:
 k T
h 
Ts  T y

y 0
k T 

L y 
y  0
The normalized temperature gradient at the
surface is defined as the Nusselt number,
which provides a dimensionless measure of
the convection heat transfer:
hL
Nu 
k
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The Nusselt Number

The normalized BL equations indicate that



dp
T   f  x , y  , Re, Pr,  
dx 


Since the Nusselt number is a normalized
surface temperature gradient, its functional
dependence for a prescribed geometry is
Nu  f ( x , Re, Pr)

Recalling that the average convection
coefficient results from an integration over the
entire surface, the x* dependence disappears
and we have:
hL
Nu 
 f (Re, Pr)
k
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17
Physical Significance of the
Reynolds and Prandtl Numbers



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Reynolds number (VL/) represents a ratio of
fluid inertia forces to fluid viscous shear forces
– small Re: low velocity, low density, small
objects, high viscosity
– large Re: high velocity, high density, large
objects, low viscosity
Prandtl number (cp/k) represents the ratio of
momentum diffusion to heat diffusion in the
boundary layer.
– small Pr: relatively large thermal boundary
layer thickness (t > )
– large Pr: relatively small thermal boundary
layer thickness (t < )
Note that Pr is a (composite) fluid property that
is commonly tabulated in property tables
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18
The Reynolds Analogy




The BL equations for momentum and energy
are similar mathematically and indicate
analogous behavior for the transport of
momentum and heat
This analogy allows one to determine thermal
parameters from velocity parameters and viceversa (e.g., h-values can be found from
viscous drag values)
The heat-momentum analogy is applicable in
BLs when dp*/dx*  0 (turbulent flow is less
sensitive to this)
The modified Reynolds analogy states
1
Nu  C f Re Pr1 / 3
2
for 0.6  Pr  60
– where Cf is the skin friction coefficient
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Turbulence





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Turbulence represents an unsteady flow,
characterized by random velocity, pressure,
and temperature fluctuations in the fluid due to
small-scale eddies
Turbulence occurs when the Reynolds number
reaches some critical value, determined by the
particular flow geometry
Turbulence is typically modeled with eddy
diffusivities for mass, momentum, and heat
Turbulent flow increases viscous drag, but may
actually reduce form drag in some instances
Turbulent flow is advantageous in the sense of
providing higher h-values, leading to higher
convection heat transfer rates
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Forced Convection Heat
Transfer – External Flow
Reading: Incropera & DeWitt
Chapter 7
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The Empirical Approach



Design an experiment to measure h
– steady-state heating
– transient cooling, Bi<<1
Perform experiment over a wide range of
test conditions, varying:
– freestream velocity (u )
– type of fluid (,  )
– object size (L)
Reduce data in terms of Reynolds, Nusselt,
and Prandtl
u L
hL

Re L 
, Nu L 
, Pr 

k

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22
The Empirical Approach, cont.

Plot reduced data, NuL vs. ReL, for each
fluid (Pr):

Develop an equation curve-fit to the data; a
common form is
Nu L  C Re mL Pr n
typically:
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<m<1
<n<
0.01 < C < 1
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23
Flat Plate in Parallel, Laminar Flow
(the Blasius Solution)

Assumptions:
– steady, incompressible flow
– constant fluid properties
– negligible viscous dissipation
– zero pressure gradient (dp/dx = 0)

Governing equations
u v

0
x y
(continuit y)
u
u
 2u
u
v
 2
x
y
y
T
T
 2T
u
v
 2
x
y
y
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(momentum)
(energy)
24
Blasius Solution, cont.

Similarity solution, where a similarity
variable () is found which transforms the
PDEs to ODEs:
 y

u
x
Blasius derived:
 
5.0

u /x
5x
Re x
C f , x  0.664 Re x1 / 2
Nu x 
hx x
 0.332 Re1x/ 2 Pr1 / 3
k

 Pr1 / 3
t
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Empirical Correlations

Flat Plate in parallel flow: Section 7.2

Cylinder in crossflow: Section 7.4

Sphere: Section 7.5

Flow over banks of tubes: Section 7.6

Impinging jets: Section 7.7
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Forced Convection Heat
Transfer – Internal Flow
Reading: Incropera & DeWitt
Chapter 8
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27
Review of Internal (Pipe) Flow
Fluid Mechanics








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Flow characteristics
Reynolds number
Laminar vs. turbulent flow
Mean velocity
Hydrodynamic entry region
Fully-developed conditions
Velocity profiles
Friction factor and pressure drop
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Pipe Reynolds Number
Re 


um is mean velocity, given by
m
um 
Ac

um D


4m
for circular pipes 
 
2
 D

Reynolds number for circular pipes:
4m
Re 
D

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Reynolds number determines flow condition:
– ReD < 2300 : laminar
– 2300 < ReD < 10,000 : turbulent transition
– ReD > 10,000 : fully turbulent
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Pipe Friction Factor

Darcy friction factor, f, is a dimensionless
parameter related to the pressure drop:
f 

 (dp / dx ) D
um2 / 2
For laminar flow in smooth pipes,
f  64 Re D

For turbulent flow in smooth pipes,
f  0.790 ln(Re D )  1.64
2
– good for 3000 < ReD < 5x106

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For rough pipes, use Moody diagram (Figure
8.3) or Colebrook formula (given in most fluid
mechanics texts)
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Pressure Drop and Power

For fully-developed flow, dp/dx is constant; so
the pressure drop ( p) in a pipe of length L is
p 

um2 fL
2D
 2 fL
8m
 2
 D5
The pump or fan power (Wp) required to
overcome this pressure drop is
 p
m

Wp 
,
 p
– where p is the pump or fan efficiency
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Thermal Characteristics of Pipe Flow




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Thermal entry region
Thermally fully-developed conditions
Mean temperature
Newton’s law of cooling
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Newton’s Law of Cooling & Mean
Temperature

The absence of a fixed freestream temperature
necessitates the use of a mean temperature,
Tm, in Newton’s law of cooling:
qs  h(Ts  Tm )

Tm is the average fluid temperature at a
particular cross-section based upon the
transport of thermal (internal) energy, Et:
 cvTm 
E t  m

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
Ac
cvuTdAc
Unlike T , Tm is not a constant in the flow
direction; it will increase in a heating situation
and decrease in a cooling situation.
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Fully-Developed Thermal Conditions

While the fluid temperature T(x,r) and mean
temperature Tm(x) never reach constant values
in internal flows, a dimensionless temperature
difference does - and this is used to define the
fully-developed condition:
  Ts  T 

  0
x  Ts  Tm 

the following must also be true:
  Ts  T 
 T r

 
 f ( x)
r  Ts  Tm 
Ts  Tm

therefore, from the defining equation for h:
 k T r r 0
h 
Ts  Tm
 f ( x)
i.e., h = constant in the fully-developed region
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Pipe Surface Conditions

There are two special cases of interest, that
are used to approximate many real situations:
1) Constant surface temperature (Ts = const)
qs  qs(x )
2) Constant surface heat flux (q”s = const)
Ts  Ts ( x ), Tm  Tm ( x )
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Review of Energy Balance Results

For any incompressible fluid or ideal gas
pipe flow,
qconv  m c p (Tmo  Tmi )
1) For constant surface heat flux,
Tmo
qsPL
 Tmi 
 cp
m
2) For constant surface temperature,
 PLh 
Ts  Tmo

 exp  


Ts  Tmi
 m c p 
3) For constant ambient fluid temperature,


T  Tmo
1


 exp 



T  Tmi
m
c
R
p
t
,
tot


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Convection Correlations for
Laminar Flow in Circular Pipes

Fully-developed conditions with Pr > 0.6:
NuD 
hD
 4.36
k
 3.66
( qs  constant)
(Ts  constant)
– Note that these correlations are independent of
Reynolds number !
– All properties evaluated at (Tmi+Tmo)/2

Entry region with Ts = const and thermal
entry length only (i.e., Pr >> 1 or unheated
starting length):
NuD  3.66 
0.0668( D L) Re D Pr
2/3
1  0.04( D L) Re D Pr 
(eqn. 8.56)
– combined entry length correlation given by eqn.
(8.57) in text
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Convection Correlations for
Turbulent Flow in Circular Pipes

Fully-developed conditions, smooth wall,
q”s = const or Ts = const
– Dittus-Boelter equation - fully turbulent flow
only (ReD > 10,000) and 0.7 < Pr < 160
NuD  0.023 Re 4 / 5 Pr n
(eqn. 8.60)
where
n  0.3 for Ts  Tm
(fluid cooling)
n  0.4 for Ts  Tm
(fluid heating)
– NOTE: should not be used for transitional
flow or flows with large property variation
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Convection Correlations for Turbulent
Flow in Circular Pipes, cont.
– Sieder-Tate equation - fully tubulent flow
(ReD > 10,000),very wide range of Pr (0.7 16,700), large property variations
NuD  0.027 Re 4 / 5 Pr1/ 3   s 
0.14
(eqn. 8.61)
– Gnielinski equation - transitional and fullyturbulent flow (3000 < ReD < 5x106), wide
range of Pr (0.5 - 2000)
NuD 
( f 8)(Re D  1000) Pr
1  12.7( f 8)1/ 2 (Pr 2 / 3  1)
(eqn. 8.63)
– f given on a previous slide (eqn. 8.21)
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Convection Correlations for Turbulent
Flow in Circular Pipes, cont.

Entry Region
– Recall that the thermal entry region for
turbulent flow is relatively short, I.e.,only 10
to 60D
– Thus, fully-developed correlations are
generally valid if L/D > 60
– For 20 < L/D < 60, Molki & Sparrow
suggest
Nu
6
D
NuD , fd

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1 
LD
Liquid Metals (Pr << 1)
– Correlations for fully-developed turbulent
pipe flow are given by eqns. (8.65), (8.66)
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Convection Correlations for
Noncircular Pipes

Hydraulic Diameter, Dh
4 Ac
Dh 
P
– where: Ac = flow cross-sectional area
P = “wetted” perimeter

Laminar Flows (ReDh < 2300)
– use special NuDh relations
– rectangular & triangular pipes: Table 8.1
– annuli: Table 8.2, 8.3

Turbulent Flows (ReDh > 2300)
– use regular pipe flow correlations with ReD
and NuD based upon Dh
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Heat Transfer Enhancement

A variety of methods can be used to enhance
the convection heat transfer in internal flows;
this can be achieved by
– increasing h, and/or
– increasing the convection surface area

Methods include
– surface roughening
– coil spring insert
– longitudinal fins
– twisted tape insert
– helical ribs
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