Heat Transfer

ME F220
Prof. Satish K Dubey
BITS Pilani
Hyderabad Campus

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 BITS Pilani
Hyderabad Campus

Conduction heat transfer

Heat Equation
HeatEquation

The Heat Equation

• A differential equation whose solution provides the temperature distribution in a stationary medium.

• Based on applying conservation of energy to a differential control volume through which energy transfer is
exclusively by conduction.
• Cartesian Coordinates:

  T    T    T  T
+ +  + q = cp
(2.19)
k  k  k
x  x  y  y  z  z  t

Net transfer of thermal energy into the Change in thermal

Thermal energy
control volume (inflow-outflow) energy storage
generation
HeatEquation(RadialSystems)

• Cylindrical Coordinates:

1   T  1   T    T  T
 kr  +  k  +  k  + q =  c (2.26)
r  r 2     z  z 
p
r r  t

• Spherical Coordinates:

1   2 T  1   T  1   T  T
  + +    + = 
r  r 2 sin 2    
kr k k sin q c (2.29)
   r sin     
p
r 2 r  2 t
Properties

Thermophysical Properties
Thermal Conductivity: A measure of a material’s ability to transfer thermal energy by conduction.

Thermal Diffusivity: A measure of a material’s ability to respond to changes in its thermal environment

Property Tables:
k
 → thermal diffusivity of the medium  m 2 /s  Solids: Tables A.1 – A.3
cp  
Gases: Table A.4
Liquids: Tables A.5 – A.7
HeatEquation(SpecialCase)
  T    T    T  T
  + +   + =  (2.19)
x  x  y  y  z  z 
k k k q c p
t

Constant Properties  2T  2T  2T q  c p T 1 T
+ + + = =
x 2 y 2 z 2 k k t  t

• One-Dimensional Conduction in a Planar Medium with Constant Properties and No Generation

  T  T
 k  =  c p
becomes x  x  t

 2T 1 T
=
x 2  t

k
 → thermal diffusivity of the medium  m 2 /s 
cp  

Thermal Diffusivity: A measure of a material’s ability to respond to changes in its thermal environment.
Boundary Conditions

Typically three types of boundary conditions in heat conduction problem:

1. Given boundary temperature: Dirichlet BC

2. Given boundary heat flux: Neumann BC

3. Boundary heat flux specified by via heat transfer coefficient and temperature
of surrounding fluid: Mixed BC

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BoundaryConditions

Boundary and Initial Conditions

• For transient conduction, heat equation is first order in time, requiring
specification of an initial temperature distribution: T ( x,t )t=0 = T ( x,0 )
• Since heat equation is second order in space, two boundary conditions
must be specified for each coordinate direction. Some common cases:
Constant Surface Temperature:

T ( 0,t ) = Ts

Constant Heat Flux:

Applied Flux Insulated Surface

T T
-k |x=0= qs |x=0= 0
x x

Convection:

T
-k
x T - T ( 0,t ) 
|x=0= h  
ConductionAnalysis

Typical Methodology of a Conduction Analysis

• Consider possible microscale or nanoscale effects in problems involving small physical dimensions or rapid
changes in heat or cooling rates.

• Solve appropriate form of heat equation to obtain the temperature distribution.

• Knowing the temperature distribution, apply Fourier’s law to obtain the heat flux at any time, location and direction of
interest.

• Applications:

Chapter 3: One-Dimensional, Steady-State Conduction

Chapter 4: Two-Dimensional, Steady-State Conduction
Chapter 5: Transient Conduction
Problem-1
Solution-1

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Solution-1
One-Dimensional, Steady-State
Conduction without Thermal Energy Generation

Chapter Three
Methodology

Methodology of a Conduction Analysis

• Specify appropriate form of the heat equation.
• Solve for the temperature distribution.
• Apply Fourier’s law to determine the heat flux.

Simplest Case: One-Dimensional, Steady-State Conduction with No Thermal Energy Generation.

• Common Geometries:
– The Plane Wall: Described in rectangular (x) coordinate. Area
perpendicular to direction of heat transfer is constant (independent of x).
– The Tube Wall: Radial conduction through tube wall.
– The Spherical Shell: Radial conduction through shell wall.
The Plane Wall
3D General Heat Conduction Equation

  T    T    T  T
k +  k  + k  + q = cp
x  x  y  y  z  z  t
1D Steady State Conduction
GDE:

Boundary Conditions: T ( 0) = Ts,1, T ( L ) = Ts,2

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PlaneWall
The Plane Wall
• Consider a plane wall between two fluids of different temperature:

• Heat Equation:
d  dT 
k =0
dx  dx 

• Implications:
Heat flux ( qx ) is independent of x.
Heat rate ( qx ) is independent of x.
• Boundary Conditions: T ( 0 ) = Ts,1, T ( L ) = Ts,2

• Temperature Distribution for Constant : k

T ( x ) = Ts ,1 + (Ts ,2 − Ts ,1 )
x
(3.3)
L
PlaneWall(cont.)

• Heat Flux and Heat Rate:

= (Ts ,1 − Ts ,2 )
dT k
qx = −k
dx L
qx = −kA
dT
dx
=
kA
L
( Ts ,1 − Ts ,2 ) (3.4)

 T 
•  t =
q 
Thermal Resistances and Thermal Circuits: R (3.5)

L  
Conduction in a plane wall: Rt ,cond = (3.6)
kA
1
Convection: Rt ,conv = (3.9)
hA
Thermal circuit for plane wall with adjoining fluids:

1 L 1
Rtot = + +
h1 A kA h 2 A
T ,1 − T ,2
qx =
Rtot
PlaneWall(cont.)

• Thermal Resistance for Unit Surface Area:

L 1
Rt,cond = Rt,conv =
k h
Units: Rt  K/W Rt  m 2  K/W
• Radiation Resistance:
1 1
Rt ,rad = Rt,rad =
hr A hr
(
hr =  (Ts + Tsur ) Ts2 + Tsur
2
) (1.9)

• Contact Resistance:

TA − TB Rt,c
Rt,c = Rt ,c =
qx Ac

Values depend on: Materials A and B, surface finishes, interstitial conditions, and contact pressure (Tables 3.1 and 3.2)
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PlaneWall(cont.) • Composite Wall with Negligible
Contact Resistance:

T ,1 − T ,4
qx =
 Rt

For the temperature distribution

shown, kA > kB < kC.

11 LA LB LC 1  Rtot

 Rt = Rtot =  + + + + =
A  h1 kA kB kC h4  A
• Overall Heat Transfer Coefficient (U) :
A modified form of Newton’s law of cooling to encompass multiple resistances
to heat transfer. q = UAT
x overall
1
Rtot =
UA
PlaneWall(cont.)

• Series – Parallel Composite Wall:

Assuming isothermal
surfaces perpendicular
to x-direction.

Assuming adiabatic
surfaces parallel
to x-direction.

• Note departure from one-dimensional conditions for . k F  kG

• Circuits based on assumption of isothermal surfaces normal to x direction or adiabatic surfaces parallel to x
direction provide approximations for q x.
TubeWall
The Tube Wall

• Heat Equation:
1 d  dT 
 kr =0 •Boundary Conditions:
r dr  dr 

• Temperature Distribution for Constant : k

Ts ,1 − Ts ,2  r 
T (r ) = ln   + Ts ,2
ln ( r1 / r2 )   r2


TubeWall(cont.)

• Heat Flux and Heat Rate:

qr = − k
dT
dr
=
k
r ln ( r2 / r1 )
(Ts,1 − Ts,2 ) [W/m2]

2 k
qr = 2 rqr =
ln ( r2 / r1 )
( Ts ,1 − Ts ,2 ) [W/m]

2 Lk
qr = 2 rLqr =
ln ( r2 / r1 )
(Ts ,1 − Ts ,2 ) [W]

• Conduction Resistance:
ln ( r2 / r1 )
Rt ,cond = [K/W]
2 Lk
ln ( r2 / r1 )
Rt,cond = [m  K/W]
2 k
TubeWall(cont.)

• Composite Wall with

Negligible Contact
Resistance

T,1 − T,4
qr =
Rtot
(
= UA T,1 − T,4 )
Note that
UA = Rtot −1
For the temperature distribution
shown, kA > kB > kC.
is a constant independent of radius,
but U itself is tied to specification of an interface.
−1
U i = ( Ai Rtot ) (3.37)
 Tube Wall: Critical Radius To determine whether the foregoing result
maximizes or minimizes the total resistance, the
second derivative must be evaluated. Hence

•Thermal resistance per unit length: •Heat Transfer per unit length:

Since this result is always positive, it

follows that r = k/h is the insulation
radius for which the total resistance is a
minimum, not a maximum

•Critical Radiation of Insulation

Critical Thickness of Insulation

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Critical Thickness of Insulation

BITS Pilani, Hyderabad Campus

Critical Thickness of Insulation

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SphericalShell
Spherical Shell

• Heat Equation for Constant k:

1 d  2 dT 
2 dr 
r =0
r  dr 

• Temperature Distribution:

T ( r ) = Ts ,1 − (Ts ,1 − Ts ,2 )
(
1 − r1/ r )
(
1 − r1 / r 2 )
Critical Thickness of Insulation: Sphere

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SphericalShell(cont.)

• Heat Flux, Heat Rate and Thermal Resistance:

dT
qr = − k = 2
k
( Ts ,1 − Ts ,2 )
dr r (1 / r1 ) − (1 / r2 ) 

4 k
qr = 4 r 2 qr =
(1 / r1 ) − (1 / r2 )
( Ts ,1 − Ts ,2 ) (3.40)

Rt ,cond =
(1 / r1 ) − (1 / r2 ) (3.41)
4 k

• Composite Shell:
Toverall
qr = = UAToverall
Rtot

UA = Rtot −1  Constant

−1
U i = ( Ai Rtot )  Depends on Ai
Summary

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References
• Text Books: T1: Frank P. Incropera, David P. Dewitt, et al.,
Incropera's Principles of Heat and Mass Transfer, Wiley
India Edition, 2018 3.

• Reference Books: R1 : J.P. Holman, Heat Transfer,

McGraw Hill, 2002, 9th Edition.,

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