Fourier law of heat conduction is essentially valid for heat flow under uni-directional and steady state conditions. However in many practical cases the temperature may be a function of space co-ordinate as well as time.
1. Heat Conduction Equation in Cartesian Co-Ordinates:
Consider the flow of heat through an infinitesimal volume element oriented in a three- dimensional co-ordinate system. The sides dx, dy and dz have been taken parallel to the x, y and z axis respectively.
Let t represent the temperature at the left face of the differential control volume. Since area of this face can be made arbitrarily small, the temperature t may be assumed uniform over the entire surface. The temperature changes along the x-direction and the rate of change is given by ∂t/∂x. Then change of temperature through distance dx will be (∂t/∂x) dx.
This temperature change has been graphically illustrated in Fig. 2.3. Therefore the temperature on the right face, which lies at a distance dx from the left face will be [f + (∂t/∂x) dx]. For non-isotropic materials there will also be a change in thermal conductivity as heat flows through the control volume.
The general conduction equation can be set up by applying Fourier equation in each Cartesian direction, and then applying the energy conservation requirement. If kx represents the thermal conductivity at the left face, then quantity of heat flowing into the control volume through this face during time interval dx is given by-
Heat influx Qx = – kx (dy dz) ∂t/∂x dτ … (2.2)
During the same time interval the heat flow out of the right face of the control volume will be,
Heat efflux Qx + dx = Qx + ∂/∂x (Qx) dx ….. (2.3)
Equation 2.7 simply states that the x-component of heat transfer rate at (x + dx) is equal to value of this component at x plus the amount by which it changes with respect to x times dx.
Accumulation of heat in the elemental volume due to heat flow in the x-direction is given by the difference between heat influx and heat efflux. Thus the heat accumulation due to heat flow in x-direction is-
Likewise the heat accumulation in the control volume due to heat flow along the y-and z-directions will be-
Sum of heat accumulations as prescribed by equations 2.4, 2.5 and 2.6 gives the total heat stored in the elemental volume due to heat flow along all the co-ordinate axes.
Total or net accumulation of heat is equal to-
There may be heat sources inside the control volume due to nuclear fission, flow of electric current in the coils of electric motors and generators, and ohmic heating of the material. If qg is the heat generated per unit volume and per unit time, then the total heat generated in the control volume equals to-
qg dx dy dz dτ …(2.8)
The total heat accumulated in the lattice due to heat flow along all the co-ordinate axes (Eq. 2.7) and the heat generated within the lattice (Eq. 2.8) together serve to increase the thermal energy of the lattice. This increase in thermal energy is reflected by the time rate of change in the heat capacity of the control volume and is given by-
ρ (dx dy dz) c — ∂t/∂τ dτ … (2.9)
where, ρ is the density and c is the specific heat of the material.
Thus from energy balance considerations:
Equation 2.10 represents a volumetric heat balance which must be satisfied at each point for self-generating, unsteady state three-dimensional heat flow through a non-isotropic material. This expression, known as the general heat conduction equation, establishes in differential form the relationship between the time and space variation of temperature at any point of the solid through which conduction takes place. It should be noted that the heat generation term qg may be a function of position or time, or both.
Homogeneous and Isotropic Material:
A homogeneous material implies that the properties, i.e., density, specific heat and thermal conductivity of the material are same everywhere in the material system. Isotropic means that these properties are not directional characteristics of the material, i.e., they are independent of the orientation of the surface. Therefore for an isotropic and homogeneous material, thermal conductivity is same at every point and in all directions.
In that case kx = ky = kz = k and the differential equation 2.10 takes the form:
The quantity a = fc/pc is called the thermal diffusivity, and it represents a physical property of the material of which the solid element is composed. Thermal diffusivity is an important characteristic quantity for unsteady conduction situations.
By using the Laplacian operator ▽2, the equation 2.11 may be written as:
Equation 2.11 governs the temperature distribution under unsteady heat flow through a homogeneous and isotropic material.
Different Cases of Particular Interest are:
(i) In many situations there is no dependence of temperature on time.
Conduction then occurs in the steady state, and the heat flow equation reduces to:
In the absence of internal heat generation or release of energy within the body, equation 2.16 further reduces to:
(ii) Unsteady state heat flow with no internal heat generation gives-
(iii) For one-dimensional and steady state heat flow with no internal heat generation, the general conduction equation takes the form-
Solution of these equations for any specific boundary conditions will yield the temperature distribution in the conducting material.
The following reflections can be made with regard to this physical property of the conducting material:
(i) Thermal diffusivity of a material is the ratio of its thermal conductivity k to the thermal storage capacity pc. The heat storage capacity essentially represents thermal capacitance or thermal inertia of the material, i.e., its sluggishness to conduct heat. A high value of thermal diffusivity could result either from a high value of thermal conductivity or from low value of thermal capacity. Liquids have a low thermal conductivity, high thermal inertia and hence a small thermal diffusivity. Metals possess high thermal conductivity, low thermal inertia and hence a large thermal diffusivity.
(ii) Thermal diffusivity indicates the rate at which heat is distributed in a material, and this rate depends not only on the conductivity but also on the rate at which heat energy can be stored. An insight into equation 2.11 would reveal that larger the thermal diffusivity, higher would be the rate of change of temperature at any point of the material. Equalisation of temperature would then proceed at a higher rate in materials having large thermal diffusivity.
(iii) Temperature distribution in the unsteady state is being governed both by thermal conductivity as well as by thermal storage capacity. In contrast, during steady state heat condition (Eq. 2.12), thermal conductivity is the only property of the medium which influences the temperature distribution.
(iv) The relative magnitude of thermal diffusivity is a measure of the rapidity with which the material responds to sudden temperature changes in the surrounding. Metals and gases have relatively large value of a and their response to temperature changes is quite rapid. The non-metallic solids and liquids respond slowly to temperature changes because of their relatively small value of thermal diffusivity.
2. Heat Conduction Equation in Cylindrical Co-Ordinates:
When heat conduction occurs through systems having cylindrical geometries (e.g., conduction through rods and pipes) it is considered more convenient to work in the cylindrical co-ordinates. The general heat equation can be set up by considering an infinitesimal cylindrical volume element-
dV = (dr rdɸ dz)
and writing energy balance equations in the radial, tangential and axial directions.
(i) Thermal conductivity k, density ρ and specific heat c for the material do not vary with position.
(ii) Uniform heat generation at the rate of per qg unit volume per unit time,
(a) Radial direction (x – ɸ plane)
Heat stored in the element due to flow of heat in the radial direction
(b) Tangential direction (r-z plane)
Heat stored in the element due to flow in the tangential direction,
(c) Axial direction (r – ɸ) plane)-
Heat stored in the element due to heat flow in axial direction,
(d) Heat generated within the control volume
= qg dV dτ … (2.19)
(e) Rate of change of energy within the control volume
= p dV c ∂t/∂τ dτ … (2.20)
From energy balance considerations, the rate of change of energy within the control volume equals the total heat storage plus the heat generated. Therefore,
which is the general heat conduction equation in the cylindrical co-ordinates.
For steady-state uni-direction heat flow in the radial direction, and with no internal heat generation, equation 2.21 reduces to-
3. Heat Conduction Equation in Spherical Co-Ordinates:
The general heat conduction equation in spherical co-ordinates can be set up by considering an infinitesimal spherical volume element-
dV = (dr × rdɸ × r sin ϴ dɸ)
and writing the heat balance equation for the r, ϴ and ɸ directions.
(i) Thermal conductivity k, density p and specific heat c for the material do not vary with position,
(ii) Uniform heat generation at the rate of qg per unit volume per unit time.
(a) Heat flow through r-ϴ plane; ɸ-direction
(b) Heat flow through r-ɸ plane; ϴ direction
(c) Heat flow in the ϴ-ɸ plane; r-direction
(d) Heat generated within the control volume
= qg dV dτ … (2.26)
(e) Rate of change of energy within the control volume
= ρ dV c ∂t/∂x dτ … (2.27)
From energy balance consideration, the rate of change of energy within the control volume equals the total storage plus the internal heat generation. Therefore,
which is the general heat conduction equation in spherical co-ordinates.
For steady-state, uni-direction heat flow in the radial direction for a sphere with no internal heat generation, equation 2.31 can be rewritten as-
The one-dimensional time dependent heat conduction equation can be written more compactly as a simple equation
where n = 0, 1 and 2 for rectangular, cylindrical and spherical co-ordinates respectively. Further, while using rectangular co-ordinates it is customary to replace the r-variable by the x-variable.