1. Chapter 2: Steady-State One-Dimensional Heat Conduction
2.1 …………. Conduction through a plane wall & composite wall
2.2 …………. Conduction from fluid to fluid through a composite wall
2.3 …………. Overall Heat Transfer Coefficient
2.4 …………. Conduction through a hollow cylinder & composite cylinder
2.5 …………. Conduction through a sphere
2.6 …………. Boundary and Initial Condition
2.7 …………. Critical thickness of Insulation

2. 2.1 … Steady State Conduction through a plane wall
T2 T1 x Q . k L
Thermal resistance (in k/W)
(which opposing heat flow)
1/22/05
ME 259

3. Thermal Analogy to Ohm’s Law
Electrical Analogy
Thermal Analogy to Ohm’s Law
OHM’s LAW :Flow of Electricity
V=IR elect
Temp Drop=Heat Flow × Resistance
Voltage Drop = Current flow ×Resistance

4. Steady State Conduction through a composite wallQ .
ΔxA A B C
ΔxB
ΔxC T1 Q . T2 T3 kA kB T4 kA kB kC kC x

5. Steady State Conduction through a composite wall

6. 1 D Heat flow from fluid to fluid through plain wall
Ts,1
Ts,2
T∞,2 … … k
x=0
x=L
Hot fluid
Cold fluid qx A h 2 1 A k L A h 1
(Thermal Resistance )

7. 2.2…1-D Heat flow from fluid to fluid through composite wallB C
T∞,1
T∞,2 h1 h2
K A
K B
K C
L A
L B
L C
q x
T∞,1
T∞,2 A h 1 2

8. Parallel composite The heat transfer rate in the network is parallel
Alternatively, the heat transfer rate can be calculated as the sum of heat transfer rates in the individual materials, i.e.

9. Series composite Contact Resistance
In composite systems, the interface between two layers is usually not perfect. This is due to surface roughness effect.
Contact spots between the two layers are interspersed with gaps that are, in most instances, air filled.
The additional resistance between the two layers, called thermal contact resistance Rt, results in temperature drop across the interface.
Thermal contact resistance is dependent upon the solid materials, surface roughness, contact pressure, temperature, and interfacial fluid.

10. 2.3 … Overall Heat Transfer Coefficient
A modified form of Newton’s Law of Cooling to encompass multiple resistances
to heat transfer.
In above equation,
Overall heat transfer coefficient

11. 2.4 … Conduction through hollow cylinder
Consider a hollow cylinder of length L (shown below), whose inner and outer surfaces are exposed to fluids at different temperatures. The system is analyzed by the standard method as follows:

12. Conduction through hollow cylinder (continue….)
For steady-state conditions with no heat generation, the heat equation for the system is
The boundary conditions are
Integrating the heat equation (assuming constant k) and using the boundary conditions yield
Therefore, the temperature distribution associated with radial conduction through a cylindrical wall is logarithmic.
The heat transfer rate is obtained by using the temperature distribution with Fourier’s law:
where,

13. Conduction through hollow cylinder (continue….)
For pure conduction, we considered resistances in series in which our primary
interest was in the temperatures at the inner and outer walls (not the interior walls).
For convection, a similar principle arises.
Consider a pipe filled with hot fluid at temperature T1. Define intermediate
temperatures as follows: T1 T2
Insulation
Air T5
Hot liquid T3
Pipe T4

14. Conduction through Composite Cylinder
Consider a composite cylindrical wall of length L shown below.

15. Conduction through Composite Cylinder (continue)
Assumption: Neglecting interfacial contact resistances
The heat transfer rate may be expressed as
Above equation may be expressed in terms of an overall heat transfer coefficient as
If U is defined in terms of the inside area A1, above equations may be equated to yield
Note: Similar equations could be written for U2, U3, etc.

16. Conduction through Composite Cylinder (continue)
For pure conduction, we considered resistances in series in which our primary
interest was in the temperatures at the inner and outer walls (not the interior walls).
For convection, a similar principle arises.
Consider a pipe filled with hot fluid at temperature T1. Define intermediate
temperatures as follows: T1 T2
Insulation
Air T5
Hot liquid T3
Pipe T4

17. The heat transfer rate for each "step":
q1->2 = conduction or convection?
q2->3 = conduction or convection?
q3->4 = conduction or convection?
q4->5 = conduction or convection?
Convection
Conduction
Conduction
Convection

18. (T5–T1) = (T5-T4) + (T4-T3) + (T3-T2) + (T2-T1)
Q = UoAo (T5-T1)
Q = hoAo (T5-T4) Outside film (air)
Q = kinsul Alm (T4-T3) / Drinsul Insulation
Q = kpipe Alm (T3-T2) / Drpipe Pipe
Q = hiAi (T2-T1) Inside film (hot liquid)

19. (T5–T1) = (T5-T4) + (T4-T3) + (T3-T2) + (T2-T1)
(T5-T1) = Q / UoAo Similar substitutions yield:

21. With U known, it is a simple matter to calculate the overall
heat transfer rate given the total temperature difference.
Note that we have used the symbol Uo because the overall heat
transfer coefficient was defined with respect to the outside area
(Ao) of the pipe. This is the most common practice.
We could equally have defined 1/Ui or UiAi. In this case, we
would use Ai as the basis for calculations. In either case, the
values of U would be slightly different, but UA and hence q are
the same.

22. Equation for 1/Ui
Rule of thumb: If the ratio of Do/Di is less than 1.5, then the arithmetic average of
Do and Di is roughly equal to the log mean average (good for ANY log mean average).
Limiting Resistance:
A very useful concept in heat transfer is that of limiting resistance.
What is limiting resistance?
What would the limiting resistance be for a hot liquid flowing inside an uninsulated
pipe?
How would you increase the heat transfer rate?

23. 2.5…Conduction through Sphere
Heat Equation
Temperature Distribution for Constant k:

24. Heat flux, Heat Rate and Thermal Resistance:
Composite Shell:

25. 2.6… Boundary and Initial Conditions
Heat equation is a differential equation:
Second order in spatial coordinates: Need 2 boundary conditions
First order in time: Need 1 initial condition
Boundary Conditions
B.C. of first kind (Dirichlet condition): x
T(x,t) Ts
x=0
Constant Surface Temperature
At x=0, T(x,t)=T(0,t)=Ts

26. Boundary and Initial Conditions (Continue…)
B.C. of second kind (Neumann condition): Constant heat flux at the surface
Finite heat flux x
T(x,t)
qx”
= qs”
Adiabatic surface
qx”=0 x
T(x,t)

27. Boundary and Initial Conditions (continue…)
3) B.C. of third kind: When convective heat transfer occurs at the surface
T(x,t)
T(0,t) x
= q”conduction
q”convection

28. 2.7…Critical Radius of Insulation
We know that by adding more insulation to a wall always decreases heat transfer.
This is expected, since the heat transfer area A is constant, and adding insulation will always increase the thermal resistance of the wall without affecting the convection resistance.
However, adding insulation to a cylindrical piece or a spherical shell, is a different matter.
The additional insulation increases the conduction resistance of the insulation layer but it also decreases the convection resistance of the surface because of the increase in the outer surface area for convection.
Therefore, the heat transfer from the pipe may increase or decrease, depending on which effect dominates.
A critical radius (rcr) exists for radial systems, where:
adding insulation up to this radius will increase heat transfer
adding insulation beyond this radius will decrease heat transfer
For cylindrical systems, rcr = kins/h
For spherical systems, rcr = 2kins/h

29. Critical Radius of Insulation (continue….)
Consider a cylindrical pipe, where, r1 -- outer radius T1 -- constant outer surface temperature k -- thermal conductivity of the insulation r2 -- outer radius - temperature of surrounding medium h - convection heat transfer coefficient Insulated Cylindrical Pipe

30. Critical Radius of Insulation (continue….)
The rate of heat transfer from the insulated pipe to the surrounding air can be expressed as The variation of heat transfer rate with the outer radius of insulation r2 is plotted in Figure The value of r2 at which heat transfer rate reaches maximum is determined from the requirement that (zero slope). Performing the differentiation and solving for r2 gives us the critical radius of insulation for a cylindrical body to be NOTE: The rate of heat transfer from the cylinder increases with the addition of insulation for r2< rcr, reaches a maximum when r2= rcr, and starts to decrease for r2> rcr. Thus, insulating the pipe may actually increase the rate of heat transfer from the pipe instead of decreasing it when r2< rcr .

31. Critical Radius of Insulation (continue….)
good for steam pipes etc.
good for electrical cables
R c r=k/h r0
R t o t
Variation Of Heat Transfer Rate With Radius

32. Temperature Distribution
One-Dimensional Steady State Solutions to the Heat Equation With No Generation
Plane Wall
Cylindrical Wall
Spherical Wall
Heat Equation
Temperature Distribution
Heat Flux (q” )
Heat Rate (q)
Thermal Resistance
(Rt, cond)