CHAPTER 3_Part 1

One-Dimensional, Steady-State
Conduction without
Thermal Energy Generation
Instructor
Dr. Salman Abbasi
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.
Plane Wall
The Plane Wall
• Consider a plane wall between two fluids of different temperature:

• Heat Equation:
d  dT 
k =0 (3.1)
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
Plane Wall (cont.)

• Heat Flux and Heat Rate:

= (Ts ,1 − Ts ,2 )
dT k
qx = − k (3.5)
dx L
= (Ts ,1 − Ts ,2 )
dT kA
q x = − kA (3.4)
dx L
 T 
• Thermal Resistances  Rt =  and Thermal Circuits:
 q 
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 = + + (3.12)
h1 A kA h 2 A
T,1 − T,2
qx = (3.11)
Rtot
Plane Wall (cont.)

• Thermal Resistance for Unit Surface Area:

L 1
Rt,cond = Rt,conv =
k h
Units: Rt  K/W Rt  m2  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
, =
Rtc 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)
Plane Wall (cont.) • Composite Wall with Negligible
Contact Resistance:

T,1 − T,4
qx = (3.14)
 Rt

For the temperature distribution

shown, kA > kB < kC.

1  1 LA LB LC 1  Rtot
 Rt = Rtot =  + + + + =
A  h1 k A k B kC h4  A
• Overall Heat Transfer Coefficient (U) :
A modified form of Newton’s law of cooling to encompass multiple resistances
to heat transfer.
qx = UAToverall (3.17)

1
Rtot = (3.19)
UA
Plane Wall (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 qx .
Tube Wall (cont.)

• Heat Flux and Heat Rate:

qr = − k
dT
=
k
dr 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] (3.32)

• Conduction Resistance:
ln ( r2 / r1 )
Rt ,cond = [K/W] (3.33)
2 Lk
ln ( r2 / r1 )
Rt,cond = [m  K/W]
2 k
Tube Wall (cont.)

• Composite Wall with

Negligible Contact
Resistance

T,1 − T,4
qr =
Rtot
(
= UA T,1 − T,4 ) (3.35)

Note that
For the temperature distribution
−1
UA = Rtot 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)
Spherical Shell
Spherical Shell

• Heat Equation for Constant k:

1 d  2 dT 
2 dr 
r =0
r  dr 

What does the form of the heat equation tell us about the variation of
qr with r ? Is this result consistent with conservation of energy?

• Temperature Distribution:

T ( r ) = Ts ,1 − (Ts ,1 − Ts ,2 )
( )
1 − r1/ r
(
1 − r1 / r 2 )
Spherical Shell (cont.)

• Heat Flux, Heat Rate and Thermal Resistance:

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

4 k
qr = 4 r 2qr =
(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