1. Fourier’s Law and the Heat Equation
Chapter Two
Lecture 3

2. Fourier’s Law
Fourier’s Law
A rate equation that allows determination of the conduction heat flux
from knowledge of the temperature distribution in a medium
Its most general (vector) form for multidimensional conduction is:
Implications:
Heat transfer is in the direction of decreasing temperature
(basis for minus sign).
Fourier’s Law serves to define the thermal conductivity of the
medium
Direction of heat transfer is perpendicular to lines of constant
temperature (isotherms).
Heat flux vector may be resolved into orthogonal components.

3. Cartesian Coordinates:
Heat Flux Components
Cartesian Coordinates:
(2.3)
Cylindrical Coordinates:
(2.24)
Spherical Coordinates:
(2.27)

4. Heat Flux Components (cont.)
In angular coordinates , the temperature gradient is still
based on temperature change over a length scale and hence has
units of C/m and not C/deg.
Heat rate for one-dimensional, radial conduction in a cylinder or sphere:
Cylinder
or,
Sphere

5. Heat Equation
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:
(2.19)
Change in thermal
energy storage
Thermal energy
generation
Net transfer of thermal energy into the
control volume (inflow-outflow)

6. Heat Equation (Radial Systems)
Cylindrical Coordinates:
(2.26)
Spherical Coordinates:
(2.29)

7. Heat Equation (Special Case)
One-Dimensional Conduction in a Planar Medium with Constant Properties
and No Generation
becomes

8. Boundary and Initial Conditions
Boundary Conditions
Boundary and Initial Conditions
For transient conduction, heat equation is first order in time, requiring
specification of an initial temperature distribution:
Since heat equation is second order in space, two boundary conditions
must be specified. Some common cases:
Constant Surface Temperature:
T(0, t) = Ts
Constant Heat Flux:
Applied Flux
Insulated Surface
Convection:

9. 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:
Solids: Tables A.1 – A.3
Gases: Table A.4
Liquids: Tables A.5 – A.7

10. Properties (Micro- and Nanoscale Effects)
Conduction may be viewed as a consequence of energy carrier (electron or
phonon) motion.
For the solid state:
average energy carrier velocity,
energy carrier
specific heat per
unit volume.
mean free path → average distance
traveled by an energy carrier before
a collision.
(2.7)
Energy carriers also collide with
physical boundaries, affecting
their propagation.
External boundaries of a film of material.
thick film (left) and thin film (right).

11. Properties (Micro- and Nanoscale Effects)
(2.9b)
Grain boundaries within a solid
Measured thermal conductivity of a ceramic material vs. grain size, L.
Fourier’s law does not accurately describe the finite energy carrier propagation
velocity. This limitation is not important except in problems involving extremely
small time scales.

12. Typical Methodology of a Conduction Analysis
Consider possible microscale or nanoscale effects in problems involving very
small physical dimensions or very 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

13. Problem: Thermal Response of Plane Wall
Problem Thermal response of a plane wall to convection heat transfer.

14. Problem: Thermal Response (cont).
<
<

15. Problem: Thermal Response (Cont).
<
d) The total energy transferred to the wall may be expressed as
Dividing both sides by AsL, the energy transferred per unit volume is
<

16. Problem: Non-uniform Generation due to Radiation Absorption
Problem Surface heat fluxes, heat generation and total rate of radiation
absorption in an irradiated semi-transparent material with a
prescribed temperature distribution.

17. Problem : Non-uniform Generation (cont.)
<

18. Problem : Non-uniform Generation (cont.)

19. Example 2.3 (pages 75-76)
The temperature distribution across a wall 1m thick at a certain instant of time is given as:
where T is in degree Celsius and x is in meters, while a=900C, b=-300C/m, c=-50C/m2. A uniform heat generation, =1000 W/m3, is present in the wall of area 10 m2 having the property of =1000 kg/m3, k=40W/mK, and cP=4 kJ/kgK.
Determine the rate of heat transfer entering the wall (x=0) and leaving the wall (x=1m).
Determine the rate of change of energy storage in the wall.
Determine the time of temperature change at x =0, 0.25 and 0.5m.

20. Example 2.2 (pages 75-76) Solution
Known: Temperature distribution T(x) at an instant of time t in a 1-D wall with uniform generation
Find: 1. Heat rates entering and leaving;
2. Rate of change of energy storage in the wall,
3. Time rate of temperature change at several locations
Schematic:

21. Example 2.2 (pages 75-76)
Schematic:

22. Example 2.2 (pages 75-76)
Assumptions: 1. 1-D conduction in x-direction
2. Isotropic medium with constant properties
3. Uniform internal heat generation, (W/m3)
Analysis:
1. For steady state 1-D conduction, Fourier’s can be applied to calculate qin and qout. (qin= 120 kW ; qout = 160 kW)

23. Example 2.2 (pages 75-76) Analysis:
2. The rate of change of energy storage in the wall ( ) can be calculated by applying an overall energy balance to the wall. Using Equation 1.1 for control volume about the wall,

24. Example 2.2 (pages 75-76) Analysis:
3. The time rate of change of the temperature at any point in the medium may be determined from the heat Equation 2.19, rewritten as:
From the prescribed temperature distribution, it follows:

25. Example 2.2 (pages 75-76)
Analysis:
Lecture 3