# Coefficient of Thermal Conductivity

Conduction can be defined as the transfer of electrons from one place to another under a potential difference or heat gradient. It is one of three modes of heat transfer between two or more bodies at different temperatures.

The other two modes of heat transfer are convection and radiation. Generally, in solids conduction is due to the vibrations of molecules and motion of free electrons. Metals have high number of free electrons which explains that why they are good conductors.

Thermal conductivity is the extent by which a substance can conduct heat or electricity. Solid, specifically speaking metals, have high thermal conductivity whereas gases have low value of thermal conductivity. The formula for Fourier Law of heat conduction can be given as

Q=kA dT/dx

Where,

Q = Heat flow rate, [Watt]

k = Thermal conductivity of the material, watt/km

A = Cross-sectional area of the conduction, [m2]

dT = Temperature Difference, [K]

dx = Thickness or length of the material specimen, [m]

## Factors Effecting Thermal Conductivity

Thermal conductivity depends upon several parameters.

Material: It highly depends upon the material. As discussed above metals have high thermal conductivity compared to gases. Some non-metal materials can also have higher thermal conductivity compared to metals.

Length: The length of the specimen or material through which heat will flow also affects the thermal conductivity. For a material or specimen whose length is short, the heat will flow easily and faster. But in some cases, thermal conductivity may increase with the increase in length.

Temperature Difference: Thermal conductivity of a material also varies with change in temperature. In some of the cases thermal conductivity increases with increase in temperature while in some cases it decreases with increase in temperature.

Cross-Section Types: Thermal conductivity is also dependent on the shape of material. Materials. The cross-section type like hollow-shaped or C-shaped can affect the thermal conductivity as well.

## Measurement of Thermal Conductivity

There are two types of techniques which are used to measure the thermal conductivity of a material. The first one is steady-state technique and the second one is non-steady or transient technique.

In the steady state technique for measuring thermal conductivity, measurement is taken when the material whose thermal conductivity is measured is in the thermal equilibrium state.

The problem with this method is that it takes a lot of time to achieve the equilibrium. Moreover, this method also includes expensive equipment to take the readings correctly.

In the transient or non-steady method, readings are taken during the heating process. It records the readings of thermal conductivity using transient sensors.

With this method, thermal conductivity can be found relatively quickly.

## Increase and Decrease in Thermal Conductivity

For metals thermal conductivity is mainly the function of free moving electrons. With the increase in temperature, the vibrations of electron increases which decreases the passage for moving electrons.

Hence, it decreases thermal conductivity. For the liquids, thermal conductivity also decreases with increase in temperature.

In the case of gases, increase in temperature will increase the amount of collisions amongst molecules. Thus it increases thermal conductivity.

## Material with Highest Thermal Conductivity

Diamond is considered to have highest thermal conductivity. Solids that have crystalline structures have high thermal conductivity as compared to the solids that have amorphous structures.

Due to the same reason, thermal conductivity of gases is lower because atoms are arranged in any shape. The irregularity in solids can also cause thermal conductivity to decrease.

As diamond is a highly crystalline material so this characteristic of diamond makes it possible to conduct heat more easily than metals.

## Experimental Procedure

Make it sure that the main switch is initially off. Then insert the specimen whose thermal conductivity is to be measured. Then insert the intermediate section into the linear module and clamp it together.

Install the temperature sensors T1 until T4 to the test module and connect the sensor leads to the panel.

Connect the heater supply lead for the linear conduction module to the power supply socket on the control panel.

Turn on the water supply and ensure that water is flowing from the free end of the water pipe to drain. This should be checked at intervals.

Turn the heater power through control knob on control panel to the fully anticlockwise position.

Turn on the power supply and the main switch; the digital readings will be displayed.

Switch on the heater and turn the heater power control and allow sufficient time to achieve steady state condition before recording the temperature at all temperature points as well as the input power reading on the wattmeter (Q).

This procedure can be repeated for other power inputs. After each change, sufficient time must be allowed to achieve steady state conditions again.

Use the formula given above to calculate the value of thermal conductivity for all sets of recorded data.

# Experimental Data

## Calculations of Thermal Conductivity

First of all, we will convert the temperature values given in degree Celsius to Kelvin.

To find the mass flow rate of water, first we will calculate the volume flow rate;
Volume flow rate of water = (0.74 liters )/(398 seconds) = 0.001859 liters/sec
Volume flow rate of water = (0.74 liters )/(398 seconds) = 0.001859 * 10-3 m3/sec

Multiplying with density to find the mass flow rate;

Mass Flow rate of water = Density * Volume Flow Rate
= 1000 * 0.001859 * 10-3 as 1m3=1000Liters
= 0.001859 kg/sec

The equation for thermal conductivity given in the manual is as;
(k A (∆T))/L=(m ) ̇C_p (T_out-T_in)
Now to find the thermal conductivity; rearranging the above equations as;
k=m ̇*c_p*(T_out-T_in )*L/A*1/∆T
Using the value of Cp given, c_p=(4189 J)/(kg.k)
Area=0.00126 m2
Length = 0.065 m
Putting all the values in the formula given above;
k=0.001859*4189*(292-287)*0.065/0.00126*1/(416.4-302.7)

We get;
k=17.66 Watt/m.k

Using the rearranged equation;
k=m ̇*c_p*(T_out-T_in )*L/A*1/∆T
k=0.001859*4189*(292.2-287.5)*0.065/0.00126*1/(416.5-302.6)
k=16.5762 Watt/m.k

Using the rearranged equation;
k=m ̇*c_p*(T_out-T_in )*L/A*1/∆T
k=0.001859*4189*(291.7-287.5)*0.065/0.00126*1/(416.2-302.7)
k=15.4645 Watt/m.k

Using the rearranged equation;
k=m ̇*c_p*(T_out-T_in )*L/A*1/∆T
k=0.001859*4189*(291.7-287.5)*0.065/0.00126*1/(416.2-302.7)
k=15.4645 Watt/m.k

Calculated Results: Results calculated above are given in the table below

In the graph it can be seen that the temperature decreases along the length of the specimen. Moreover, the thermal conductivity is decreasing with the increase in temperature gradient along the surface of the specimen.

The point on the material where thermocouples are inserted or mounted are not known. Therefore the gradient of the temperature is not quite linear. The other reasons of non-linearity will be discussed in detail in the sources of error section.

## Discussion on Coefficient of thermal conductivity

The value of thermal conductivity was highest in the first reading and it decreased in the second reading and became constant after the third reading.

The value of thermal conductivity obtained for stainless steel has a very minute error. It has a maximum error of 8% in the first reading. For the second reading the percentage error is 2% which is minimum. Percentage error for the last two readings is 5%.

### Calculation of Thermal Conductivity of Grease

As the graph of thermal conductivity is a straight-line in the x-direction, this shows the thermal conductivity of grease is very high.
Using the same formula;
k=0.001859*4189*(292-287)*0.065/0.00126*1/(416.4-411.9)
k=446.3422 Watt/m.k

The value of thermal conductivity of a material is the property which can help us in the development of cooling and heating techniques.

Generally a substance with proper arrangement of molecules has higher values of thermal conductivities. A common physical characteristic that effect the thermal conductivity is the porosity of the material.

The thermal conductivity of air is 0.02 watt/m.k which is very low as compared to solids. When air is trapped in the pores of a substance, it acts to decrease the thermal conductivity of a substance.

A material with high porosity have low thermal conductivity. Humidity and direction of flow of heat also effect the value of thermal conductivity.

In some engineering applications, choosing a material with appropriate value of thermal conductivity can increase the efficiency of a product which can save us energy or money.

Phase change material usually called as PCM are the materials which release or absorb splendid amount of latent heat when they change their phase from gas to solid or solid to gas respectively.

Thermal conductivity is the important parameter in the study of phase change materials. The amount of time taken by a PCM material is strictly dependent on its thermal conductivity. Organic PCMs are non-metal mixtures that display very little thermal conductivity values.

In the construction of heat exchangers, the material used to construct the shell and tube has very high thermal conductivity. When the flow inside a heat exchanger is laminar, the heat exchange is relatively lesser.

Therefore, the flow inside a heat exchange is usually kept turbulent so that more amount of heat can be transferred. This happens because the contact between the molecules of hot and cold fluid increases in the case of turbulent flow.

According to the modern researches, Nano materials such as carbon nano-tubes etc. can also be used in the application of heat exchanger to increase the heat transfer. Studies say that nano-fluids have high values of thermal conductivity as compared to simple fluids or liquids.

#### Sources of Error

The method we used in this experiment to measure the thermal conductivity is steady state technique. In this method the achievement of thermal equilibrium is necessary.

As we can see that the error in the first reading for stainless steel is highest. Perhaps, this can be due to the fact that the first value was taken before achieving thermal equilibrium.

This error can be avoided by allowing more time to note make the readings stable.

Conducting material between specimen i.e. grease can also cause a change in the value of thermal conductivity. If the material used between specimens has low value of thermal conductivity, it can cause a huge error in the calculation of thermal conductivity of a material.

The way to avoid this error is that we should use a material of very high conductivity to fill the gap between two specimens.

Heat dissipation due to the convection at the surface of specimen can also cause the error in the value of thermal conductivity.

This error can be avoided if the air around has a low velocity. Convection takes place when a bulk of fluid moves from one location to another due the difference of temperature. If the velocity of air around the apparatus is low, it will slow down the process of convection and hence the error will be low.

## Conclusion on Thermal Conductivity

The thermal conductivity is the ease of material when it comes to conducting heat or electricity. In this experiment we found the value of thermal conductivity of two different materials using steady-state technique.

In the steady-state technique, the material is brought to thermal equilibrium. This process of measuring thermal conductivity is slow. Some materials like metals have high value of thermal conductivity at lower temperatures and it decreases at high temperatures.

Gases have lower thermal conductivity as compared to metals but their thermal conductivity increases with increase in temperature because of increased molecular collisions.