Mechanical Engineering Dept. CEME NUST 1

Ch-3: Steady State Heat Conduction

Mechanical Engineering Dept. CEME NUST 2

Steady State Heat Conduction

Considerable Temperature Difference between the inner and the outer surfaces

of the wall (significant temperature gradient in the x-direction)

Steady Heat Conduction in Plane Walls

The wall surface is nearly isothermal

Steady 1-dimensional modeling approach is justified

Assuming heat transfer is the only energy

interaction and there is no heat generation, the

energy balance is:

Mechanical Engineering Dept. CEME NUST 3

Steady State Heat Conduction

Steady Heat Conduction in Plane Walls

rate of heat transfer through the wall must be

constant, Qcond,wall = constant

⇒ ⇒

Temperature Distribution

general heat conduction equation in Cartesian

coordinates

⇒

⇒

no change in temperature of wall with time at any point

Mechanical Engineering Dept. CEME NUST 4

Steady State Heat Conduction

Steady Heat Conduction in Plane Walls

Temperature Distribution —contd--

⇒

⇒

→

temperature distribution across a wall is

linear and is independent of thermal

conductivity ⇒

heat through the plane wall using Fourier’s Eq.

Mechanical Engineering Dept. CEME NUST 5

Steady State Heat Conduction

Steady Heat Conduction in Plane Walls

Temperature Distribution —contd--

⇒

once the rate of heat conduction is available,

temperature T(x) at any location x can be

determined by replacing T2 in above Eq. by T,

and L by x

Mechanical Engineering Dept. CEME NUST 6

Steady State Heat Conduction

Steady Heat Conduction in Plane Walls

Thermal Resistance Concept

Heat Conduction through a plane wall can be rearranged as:

Where Rwall is the conduction resistance

⇒ Thermal Resistance of a medium depends on the geometry and the

thermal properties of the medium

Mechanical Engineering Dept. CEME NUST 7

Steady State Heat Conduction

Steady Heat Conduction in Plane Walls

Analogy to Electrical Current Flow

Above Eq. is analogous to the relation for electric current flow I, expressed as:

Heat Transfer

Rate of heat transfer

Thermal resistance

Temperature difference

Electrical current flow

←→ Electrical resistance

Electric current

Voltage difference

←→

←→

Thermal Resistance Concept – contd--

Mechanical Engineering Dept. CEME NUST 8

Steady State Heat Conduction

Steady Heat Conduction in Plane Walls

Thermal Resistance Concept – contd--

Convection Resistance

Thermal resistance can also be applied to convection processes

Newton’s Law of Cooling for convection heat transfer rate (Qconv = hAs (Ts-T ))

can be rearranged as:

Rconv is the convection resistance

Mechanical Engineering Dept. CEME NUST 9

Steady State Heat Conduction

Steady Heat Conduction in Plane Walls

Thermal Resistance Concept – contd--

Radiation Resistance

Rate of Radiation Heat Transfer between a surface and the surrounding

Mechanical Engineering Dept. CEME NUST 10

Steady State Heat Conduction

Steady Heat Conduction in Plane Walls

Thermal Resistance Concept – contd--

Radiation and Convection Resistance

A surface exposed to the surrounding might involves convection and radiation

simultaneously

Convection and Radiation Resistances

are parallel to each other

When Tsurr ≈ T∞, radiation effect can

properly be accounted for by replacing h in

the convection resistance relation by

Mechanical Engineering Dept. CEME NUST 11

Steady State Heat Conduction

Heat Conduction Through a Composite Wall

Quantity of heat transmitted per unit

time through each slab/layer is same

Assuming a perfect contact between the

layers and no temperature drop occurs

across the interface between the materials

Mechanical Engineering Dept. CEME NUST 12

Steady State Heat Conduction

Heat Conduction Through a Composite Wall

For composite wall

consists of n layers

Mechanical Engineering Dept. CEME NUST 13

Steady State Heat Conduction

Heat Conduction Through a Composite Wall

Parallel Arrangement

total heat transfer is the sum of the heat

transfers through each layer

⇒

Mechanical Engineering Dept. CEME NUST 14

Steady State Heat Conduction

Heat Conduction Through a Composite Wall

Combined Parallel-Series Arrangement

Total rate of heat transfer through the

composite system

Mechanical Engineering Dept. CEME NUST 15

Steady State Heat Conduction

The Overall Heat-transfer Coefficient

While dealing with the problems of fluid to fluid heat transfer across a metal

boundary, it is usual to adopt an overall heat transfer coefficient U

⇒

Mechanical Engineering Dept. CEME NUST 16

Steady State Heat Conduction

The Overall Heat-transfer Coefficient

⇒ ⇒

⇒

⇒

If U is Overall Coefficient of Heat Transfer

⇒