One Dimensional Heat Conduction

Steady and Unsteady State Conductions

Steady state conduction is the form of conduction that happens when the temperature difference driving the conduction is constant, so that, the spatial distribution of temperatures in the conducting object does not change any further.

In general, during any period in which temperatures are changing in time at any place within an object, the mode of thermal energy flow is termed transient conduction or non-steady state conduction.

Methods for Determining Heat Transfer Coefficient

The convective heat transfer coefficient in forced flow can be evaluated by:

• Dimensional Analysis combined with experiments;

• Reynolds Analogy – an analogy between heat and momentum transfer;

• Analytical Methods – exact and approximate analyses of boundary layer equations.

1. Only one space coordinate is required to describe the temperature distribution within a heat conducting body;

1. Edge effects are neglected;

1. The flow of heat energy takes place along the coordinate measured normal to the surface.

• Dimensional Analysis combined with experiments;

• Reynolds Analogy – an analogy between heat and momentum transfer;

• Analytical Methods – exact and approximate analyses of boundary layer equations.

1. Only one space coordinate is required to describe the temperature distribution within a heat conducting body;

1. Edge effects are neglected;

1. The flow of heat energy takes place along the coordinate measured normal to the surface.

1. Carterisan Coordinates (side parallel to x, y and z-directions)

• For steady state condition (Poisson’s equation),

• For steady state and absence of internal heat generation (Laplace equation),

• For unsteady heat flow with no internal heat genIntereration,

2. Cylindrical Coordinates

• For homogeneous and isotropic material,

3. Spherical Coordinates

• For homogeneous and isotropic material,

• For steady state unidirection heat flow in radial direction with no internal heat generation,

One-Dimensional Heat Flow

The term ‘one-dimensional’ is applied to heat conduction problem when:

Fourier’s Law of Heat Conduction

The rate of heat flow through homogeneous solid is directly proportional to the area measured normal to the direction of heat flow and the temperature gradient in the direction of heat flow.

Steady and Unsteady State Conductions

Steady state conduction is the form of conduction that happens when the temperature difference driving the conduction is constant, so that, the spatial distribution of temperatures in the conducting object does not change any further.

In general, during any period in which temperatures are changing in time at any place within an object, the mode of thermal energy flow is termed transient conduction or non-steady state conduction.

Methods for Determining Heat Transfer Coefficient

The convective heat transfer coefficient in forced flow can be evaluated by:

One-Dimensional Heat Flow

The term ‘one-dimensional’ is applied to heat conduction problem when:

Fourier’s Law of Heat Conduction

The rate of heat flow through homogeneous solid is directly proportional to the area measured normal to the direction of heat flow and the temperature gradient in the direction of heat flow.

where, Q = Heat transfer in given direction.

A = Cross-sectional area perpendicular to heat flow direction.

dT = Temperature difference between two ends of a block of thickness dx

dx = Thickness of solid body

 = Temperature gradient in direction of heat flow.

General Heat Conduction Equation

Let q_g = Internal heat generation per unit volume per unit time

t = Temperature at left face of differential control volume

k_x, k_y, k_z = Thermal conductivities of the material in x, y and z-directions respectively

c = Specific heat of the material

ρ = Density of the material

α = Thermal diffusivity 

dτ = Instantaneous time.

For homogeneous and isotropic material.

For steady state unidirectional heat flow in radial direction with no internal heat generation,

 Constant

Thermal Resistance of Hollow Cylinders

Let, r₁ = Internal radius of cylinder

r₂ = Outer radius of cylinder

L = Length of cylinder

Thermal resistance of hollow cylinders is given as

Heat transfer