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In this article we will discuss about the fourier equation and thermal conductivity of materials.

#### Fourier Equation:

Conduction is primarily a molecular phenomenon requiring temperature gradient as the driving force. Experimental evidence does indicate that steady-state one-dimensional flow of heat by conduction through a homogeneous material is given by the Fourier Law-

Q = – k A dt/dx

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q = Q/A = – k dt/dx …. (2.1)

The heat flux q (heat conducted per unit time per unit area) flows along normal to area A in the direction of decreasing temperature.

**The units on each term are: **

Q – rate of heat flow, kJ/hr

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A – area perpendicular to the direction of heat flow, m^{2 }

dx – thickness of material along the path of heat flow, m

dt – temperature difference between the two surfaces across which heat is passing, degree kelvin K or degree centigrade C.

The ratio dt/dx represents the change in temperature per unit thickness, i.e., the temperature gradient. The negative sign indicates that the heat flow is in the direction of negative temperature gradient, and that serves to make the heat flow positive.

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The proportionality factor k is called the heat conductivity or thermal conductivity of the material through which the heat propagates. Thermal conductivity provides an indication of the rate at which heat energy is transferred through a medium by the diffusion (conduction) process. For a prescribed temperature gradient and geometric parameters, the heat flow rate increase with increasing thermal conductivity.

**The Fourier law is essentially based on the following assumptions: **

i. Steady state conduction which implies that the time rate of heat flow between any two selected points is constant with time. This also means that the temperature of the fixed points within a heat conducting body does not change with time: t ≠ *f*(τ).

ii. One-directional heat flow; only one space co-ordinate is required to describe the temperature distribution within the heat conducting body; t = *f *(x). The surfaces in the y-and z-directions are perfectly insulated.

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iii. Bounding surfaces are isothermal in character, i.e., constant and uniform temperatures are maintained at the two faces.

iv. Isotropic and homogeneous material, i.e., thermal conductivity has a constant value in all the directions.

v. Constant temperature gradient and a linear temperature profile.

vi. No internal heat generation.

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**Some essential features of the Fourier relation are enumerated below: **

i. Fourier law predicts how heat is conducted through a medium from a region of high temperature to a region of low temperature.

ii. Fourier law is valid for all matter regardless of its state- solid, liquid or gas.

iii. Fourier law is a vector expression indicating that heat flow rate is normal to an isotherm and is in the direction of decreasing temperature.

iv. Fourier law cannot be derived from first principle; it is a generalization based on experiment evidence.

v. Fourier law helps to define the transport property k, i.e., the thermal conductivity of the heat conducting medium.

Assuming dx = 1m; A = 1m^{2} and dt = 1°, we obtain-

Q = k

Hence thermal conductivity may be defined as the amount of heat conducted per unit time across unit area and through unit thickness, when a temperature difference of unit degree is maintained across the bounding surface. The magnitude of thermal conductivity tells us how well a material transports energy by conduction.

**The units of thermal conductivity are worked out from the Fourier law written in the form: **

k = – Q/A dx/dt

Thus [k] = hJ/hr 1/m_{2} m/deg = kJ/m-hr-deg

The unit kJ/m-hr-deg could also be specified as J/m-s-deg or W/m-deg and this is actually done while quoting the numerical values of thermal conductivity.

**Thermal Resistance of Materials: **

Observations indicate that in systems involving flow of fluid, heat and electricity, the flow quantity is directly proportional to the driving potential and inversely proportional to the flow resistance. In a hydraulic circuit, the pressure along the path is the driving potential and roughness of the pipe is the flow resistance.

The current flow in a conductor is governed by the voltage potential and electrical resistance of the material. Likewise, temperature difference constitutes the driving force for heat conduction through a medium.

From Ohm’s law-

Current (i) = voltage potential (dV)/electrical resistance (Re)

and from Fourier’s law-

Heat flow rate (Q) = temperature potential (dt)/thermal resistance (dx/kA)

Obviously there is a one-one correspondence between the flow of electric current and heat, i.e., –

i. Electric current (amperes) is analogous to thermal heat flow rate (kJ/hr).

ii. Electric voltage (volts) corresponds to thermal temperature difference (degree Kelvin).

iii. Electric resistance (ohms) is analogous to quantity dx/kA. This quantity is called thermal resistance.

Thermal resistance, R_{t} = (dx/kA), is expressed in the units s-deg/J or deg/W. The reciprocal of thermal resistance is called thermal conductance and it represents the amount of heat conducted through a solid wall of area A and thickness dx when a temperature difference of unit degree is maintained across the bounding surfaces.

Sometimes the heat conducting capacity of a given physical system is expressed in terms of unit thermal resistance r_{f} and unit thermal conductance c-

The concept of thermal resistance is advantageously applied while making computations for heat flow.

**Thermal Conductivity of Materials:**

Thermal conductivity is a property of the material and it depends essentially upon the material structure (chemical composition, physical state and texture), moisture content and density of the material, and operating conditions of pressure and temperature. The value of thermal conductivity may range from 0.0083 W/m-deg for gases such as Freon-12 to as great as 410 W/m-deg for metals such as silver. Apparently, the pure silver has conductivity almost 50,000 times as great as that of Freon-12.

**Following remarks apply to the thermal conductivity and its variation for different materials and under different conditions: **

(1) Metals are the best conductors while liquids are generally poor conductors.

(2) Thermal conductivity is always higher in the purest form of a metal. Alloying of metals and presence of other impurities cause an appreciable decrease in thermal conductivity.

(3) Mechanical forming (i.e., forging, drawing and bending) or heat treatment of metal cause considerable variation in thermal conductivity. For instance, thermal conductivity of hardened steel is lower than that of annealed steel.

(4) Thermal conductivity of most metals decreases with temperature growth; aluminium and uranium are the exceptions.

Thermal conductivity of gases increases with rise in temperature. This may be attributed to an increase in mean travel velocity and specific heat at elevated temperatures. The increased agitations of gaseous molecules at elevated temperatures also results in greater frequency contact and an attendant increase in molecular exchange rates.

Thermal conductivity for liquids diminishes with rising temperature due to decrease in density with temperature increase.

(5) Thermal conductivity is only very weakly dependent on pressure for solids and for liquids as well and essentially independent of pressure for gases at pressure near standard atmospheric.

(6) For most materials, the dependence of thermal conductivity on temperature is almost linear-

k = k_{0} (1 + βt)

where, k_{0} is the thermal conductivity at 0°C temperature, and β is a constant whose value depends upon the material. This constant may be positive or negative depending on whether thermal conductivity increases or decreases with temperature. The co-efficient β is usually positive for non-metals and insulation materials (exception magnesite bricks) and negative for metallic conductors (exceptions are aluminium and certain non-ferrous alloys).

The value of thermal conductivity increases with temperature for gases while it tends to decrease with temperature for most of liquids; water being notable exception.

(7) Non-metallic solids do not conduct heat as efficiently as metals. For many of the building and insulating materials (concrete, stone, brick, glass wool, cork etc.) the thermal conductivity may vary from sample to sample due to variations in structure, composition, density and porosity.

Thermal conductivity of porous materials depend upon the type of gas or liquid existing in the voids. Presence of air filled pores and cavities reduce thermal conductivity because then the heat has to be transferred across many air spaces and air is known to be a poor heat conductor.

Thermal conductivity of a damp material is considerably higher than the thermal conductivity of the dry material and water taken individually.

Density is another parameter that affects the thermal conductivity of material; thermal conductivity increases with density growth. At densities 400 and 8000 kg/ m^{3}, thermal conductivity values of asbestos are 0.105 and 0.248 W/ m-deg respectively. Thermal conductivity of snow is also proportional to its density.

With a density of 99.8 kg/ m^{3} thermal conductivity is 0.081 W/ m-deg, and with density 598 kg/m^{3} thermal conductivity increases to 0.627 W/m-deg. Since density of ice increases with decrease in temperature the variation of thermal conductivity with temperature would also follow the same pattern. This information helps to estimate the rate of ice formation on a lake or elsewhere.

Materials having a crystalline structure have a high value of thermal conductivity than the substances in amorphous form.

Irregular arrangement of the atoms in case of amorphous solids inhibits the effectiveness of heat transfer by molecular impact.

(8) Majority of engineering materials are isotropic, that is, their properties and constitution in the neighbourhood of any point are invariant with direction from that point. However, some materials exhibit some non-isotropic conductivities due to a directional preferences caused by a fibrous structure (as in the case of wood, asbestos etc.).

Thermal conductivity of most types of wood is large in the direction parallel to the grain compared to that in a direction across the grain. Other materials with such characteristics include crystalline substances, laminated plastics and laminated metals. Conducting materials having this property are called anisotropic materials.

(9) Based on experimental results, Wiedemann and Franz made the following observations concerning thermal and electrical conductivities of a material.

**“The ratio of the thermal and electrical conductivities is same for all metals at the same temperature; and that the ratio is directly proportional to the absolute temperature of the metal.” **

Let k and o be the thermal and electrical conductivities of metal at a temperature T degrees absolute, then-

k/σ ∝ T

or k/σT = constant for all metals

The constant is referred to as Lorenz number with the value L_{0} = 2.45 × 10^{-8} W ohms/K^{2}.

The Wiedemann and Franz law does suggest that materials that are good electrical conductors (pure metals viz. copper and silver) are good conductors of heat too.

(10) Materials with large thermal conductivity are called thermal conductors, and those with small thermal conductivity are called thermal insulators. Insulating materials are used for obstructing the flow of heat between an enclosure and its surroundings.

(a) Low temperature insulation (cork, rock wool, glass wool, cattle hair, slag wool and thermocol etc.) are used when the enclosure is at a temperature lower than the ambient temperature and it is desired to prevent the enclosure from gaining heat.

(b) High temperature insulations (asbestos, diatomaceous earth, magnesia etc.) are used when it is desired to prevent an enclosure at a temperature higher than the ambient from losing heat to its surroundings.

(c) Super insulators include powders, fibres or multi-layer materials that have been evacuated of all air.

The low conductivity of insulating materials is due primarily to air (a poorly conducing gas) that is contained in the pores rather than the low conductivity of the solid substance.

Substances under low temperature conditions that have exceedingly high thermal conductivity are known as super conductors. For example thermal conductivity of aluminium reaches a value of 20000 W/m-deg at 10°K and this is over 100 times as large as the value that occurs at room temperature.

(11) Pure metals possess the highest thermal conductivity, (k = 10 to 400 W/m-deg). Heat insulating and building materials have a comparatively low thermal conductivity, (k = 0.023 to 2.9 W/m-deg) and it ranges from 0.2 to 0.5 W/m-deg for liquids. Still gases and vapours possess the lowest thermal conductivity within the range 0.006 to 0.05 W/m-deg.

Evidently the ratio of conductivity values between metals and good insulators is about 10^{4} and it causes heat loss to be an important factor in thermal systems. In contrast, the ratio of electrical conductivity between good and poor conductors is about 10^{24} and therefore the current loss through insulation in electrical net-works is negligible.

(12) **Thermal conductivity of different materials decreases in the following order:**

(i) Pure metals

(ii) Alloys

(iii) Non-metallic crystalline and amorphous substances

(iv) Liquids and

(v) Gases