ENGR. ANDREW C.
MERAFUENTES
andrew.merafuentes@vsu.edu.ph
T
HEA 09391347145
RANSFER CONDUCTION
MEng 131n
Module 2
CONDUCTION RATE EQUATION
Lesson 2.1
Lesson 2.1
Steady and Transient Heat Conduction
Transient Steady
Transient or Unsteady
Transient implies variation with time.
Lesson 2.1
Steady and Transient Heat Conduction
Transient Steady
Steady
Implies no change with time at any point within
the medium.
Lesson 2.1
Steady and Transient Heat Conduction
Temperature
Time
Cooling of an apple inside a fridge
Most heat transfer problems encountered are transient in nature, but they are usually
analyzed under presumed steady conditions to make it easier.
Lesson 2.1
Multidimensional Conduction
One-dimensional Two-dimensional Three-dimensional
Heat transfer is said to be one
dimensional if the temperature
in the medium varies in one
direction only.
Two-dimensional heat transfer in a
Heat transfer through a glass
long rectangular bar.
window can be considered to be
one-dimensional since heat will flow
predominantly in one direction
(perpendicular to the glass)
Lesson 2.1
Heat Generation
Internal heat generation involves the conversion of
electrical, nuclear, or chemical energy into heat. In the special case of uniform heat generation,
as in the electric resistance heating throughout
the homogenous material, the total amount of
heat generation is given as
Heat generation is a volumetric phenomena, that is, it
occurs throughout the body of a medium.
Lesson 2.1
General Heat Conduction Equation
The general heat conduction equation in Cartesian
coordinates is given by:
heat transfer in the x-direction
heat transfer in the y-direction
heat transfer in the z-direction
rate of internal heat generation
thermal diffusivity
variation with time
Lesson 2.1
General Heat Conduction Equation
In the case when no internal heat generations is present,
and the equation reduces to
Further, when temperature does not depend with time, the
conduction takes place in steady state;
and the equation reduces to
STEADY STATE CONDUCTION OF
PLANE AND COMPOSITE WALLS
Lesson 2.2
Lesson 2.2
Conduction of a Plane Wall
Consider a plane wall of homogenous material through
which heat is flowing in the x direction:
The conduction equation will be:
Integrating the above differential twice:
Lesson 2.2
Conduction of a Plane Wall
The values of these constants may be calculated from
the known boundary conditions as follows;
Substituting these values;
This indicates that the temperature distribution across a
wall is independent of thermal conductivity.
Lesson 2.2
Conduction of a Plane Wall
From Fourier’s equation:
where = temperature gradient.
From Eq. 1:
Back to Fourier’s equation:
Or it could be written as:
Lesson 2.2
Conduction Through a Composite Wall
Since the quantity of heat transmitted per unit time
through each slab/layer is the same, we have
(Assuming there is perfect contact between layers and no temperature
drop occurs across the interface between the materials)
Rearranging the above expressions, we get
Lesson 2.2
Conduction Through a Composite Wall
or
Equivalent thermal resistance circuit.
Lesson 2.2
Thermal Contact Resistance
In real systems, due to surface roughness and void spaces (usually filled with air) the contact
surfaces touch only at discrete locations. This means that the area available for the flow of heat
at the interface will be small compared to the geometric face area.
Lesson 2.2
Thermal Contact Resistance
Due to this reduced are and presence of air voids, a large
resistance to heat flow at the interface occurs.
Lesson 2.2
Overall Heat Transfer Coefficient
The equations of heat flow through the fluid and the
metal surface are;
Lesson 2.2
Overall Heat Transfer Coefficient
Lesson 2.2
EXAMPLE 1
A reactor’s wall, 320 mm thick, is made up of an inner layer of fire brick (k = 0.84 W/m-°C) covered with a layer of
insulation (k = 0.16 W/m-°C). The reactor operates at a temperature of 1325°C and the ambient temperature is 25°C.
Calculate the thickness of fire brick and insulation and the heat loss if the insulating material has a maximum
temperature of 1200°C. Also state if the addition of another layer of insulation is feasible.
Solution
The heat flux is constant throughout the wall and is the same for each layer. For
the unit area of the wall,
Considering the first two quantities, we have
Lesson 2.2
EXAMPLE 1
Lesson 2.2
EXAMPLE 1
Lesson 2.2
EXAMPLE 2
Find the heat flow rate through the composite wall as shown. Assume one dimensional flow.
The thermal circuit for heat flow in the given composite system is
shown below.
Lesson 2.2
EXAMPLE 2
Thermal resistances are;
Lesson 2.2
EXAMPLE 2
The total thermal resistance is given by:
For the heat flow rate
Lesson 2.2
EXAMPLE 3
Two walls of cold storage plant are composed of an insulating material (k = 0.25 kJ/hr-m-°C), 100 mm thick
at the outer layer and material (k = 3.5 kJ/hr-m-°C), 15 mm thick at the inner layer. If the surface
temperature at the cold side is 30°C and hot side is 250°C, find the heat transmitted per square meter.
Solution
Convert k to W/m-°C:
For the heat transmitted per square meter:
Lesson 2.2
EXAMPLE 4
Solution
Lesson 2.2
EXAMPLE 4
