HEAT TRANSFER

1. Aim: To determine the thermal conductivity of a metallic rod.

Apparatus Used: Voltmeter, Ammeter, Stop watch, Copper – constantan thermocouple, Power supply, Heating element, Digital temperature indicator and Voltage regulator.

Theory: Thermal conductivity is an important thermo - physical property of conducting materials, by virtue of which the material conducts the heat energy through it. From Fourier’s law of conduction the thermal conductivity is defined as

 K=(Q/A)*(dx/dT)= q(dx/dT)  in W/m.K

Where,

Q = heat transfer rate, watts,

 q = heat flux, W /m2

A = area normal to heat transfer, m2, and

dT / dx = temperature gradient in the direction of heat flow=(T₂-T₁)/(x₂-x₁).

The thermal conductivity for a given material depends on its state and it varies with direction, structure, humidity, pressure and temperature change. Thermal energy can be transported in solids by two means.

1. Lattice vibration,
2. Transport of free electrons.

In good conducting materials, a large number of free electrons  move about their lattice structure of metal. These electrons move from higher temperature region to lower temperature region, thus transport heat  energy.  Further,  the  increased  temperature  increases  vibration  energy  of  atoms  in  the  lattice structure. Thus in hotter portion of the solid, the atoms which have larger vibration energy, transfer a part of its energy to the neighboring low energy molecules and so on throughout the whole length of the body.

The  apparatus  consist  the  heating  elements  fitted  on  a  table  stand.  One  hole  is  made  in  it  to accommodate the metal rod whose thermal conductivity has to be measured. The metal rod of brass is inserted into hole so that when power supplied to heater, heat transfer will take place from the base to another end. The temperature of the metal rod is measured at six positions by using copper constantan thermocouples and digital  temperature indicator. The power supply to heat may be adjusted to desire quantity by means of electronic controlled circuit for that one rotary switch is provided on the control panel. Electric power input can be measured by using digital voltmeter and ammeter multiplier.

Procedure:

1. Switch on the power supply and adjust voltage and current so as to allow some 500 watts input to heater coil.
2. Wait for steady state. Steady state can be observed by the temperature reading at one or all points on the surface of the metal rod. Steady state is reached when these temperatures stop changing with time.
3. Under steady state conditions note the temperature of each point on the surface of the rod as well as temperature of surrounding. Repeat above procedure for several power input.

Observations & Calculations:

Specifications: -

•    Room temperature (T) = --------------------°C

•    Diameter of the metal rod (d) = ------------ m

•    Length of the metal rod (L) = ---------------m

•    Digital voltmeter (0-220) Volt

•    Digital ammeter (0-2)Amp

•    Digital temperature indicator (0-300°C)

•    Thermocouple (copper constantan)

•    Heating filament (500 watts)

•    Variac (voltage regulator) (0-2A, 0-230 V, AC supply )

Observation Table:

The following readings are noted as shown in the table after reaching steady state condition.

Note down x1,x2,x3,x4,x5,x6 i.e the distance between the base plate and corresponding temperature. i.e for x1 is the distance between the base plate and T1  temperature  thermocouple. All temperatures are in °C.

2. Aim: To determine the thermal conductivity of an insulating powder.

Apparatus Used: There are two spherical shells and the insulating material whose conductivity is to be calculated is packed between these two shells. A heating coil is provided in the inner shell. The power is supplied from outside through auto-transformer for heating purpose. A few copper-constantan thermocouples are provided along the radius of the inner and outer spheres, one is fixed on the outer-surface of inner sphere and one is fixed on the outer spheres, one is fixed on the outer-surface of inner sphere and one is fixed on the outer surface of the outer sphere. Sometimes 4 – thermocouples are provided on the inner sphere surface and four are provided on the outer sphere surface for finding the average temperatures on inner and outer sphere surfaces.

Theory: Thermal conductivity is one of the important properties of the materials and its knowledge is required for analyzing heat conduction problems. Physical meaning of thermal conductivity is how quickly heat passes through a given material. Thus the determination of this property is of considerable engineering significance. There are various methods of determination of thermal conductivity suitable for finding out thermal conductivity of materials in the powdered form.

Considering the transfer of heat by heat conduction through the wall of a hollow sphere formed by the insulating powdered layer packed between two thin copper spheres.

For,

ri  = inner radius in meters.

r0  = outer radius in meters.

Ti  = average temperature of the inner surface in °C.

T0  = average temperature of the outer surface in °C.

Where,

Ti  = (T₁+T₂+T₃+T₄)/4

T0  = (T₅+T₆+T₇+T₈+T₉+T₁₀)/6

Note: - That T₁ to T₁₀ denote the temperature of thermocouples (1) to (10). Applying the Fourier law of heat conduction for a thin spherical layer of radius r and thickness dr with temperature difference dT the heat transfer rate.

q = -K(4πr² )(dT/dr)   its units are Kcal / hr ……………..(1)

Where,

K = thermal conductivity.

Separating the variables (q/4π K)(dr/r²)= dT …....(2)

Integrating eq.(2) in the limits of r_i, r₀  and T_i, and T₀

(q/4π K)[(1/ r_i)-(1/ r₀)]=(T_i - T₀)

q=4π K r_i r₀(T_i - T₀)/( r₀ – r_i) …....(3)

K=q( r₀ – r_i)/ 4π r_i r₀(T_i - T₀) …....(4)

 Procedure:

1. Increases slowly the input to heater by the dimmer stat starting from zero volt position.
2. Adjust input equal to 40watt max. by voltmeter and ammeter / Wattage (W= V x I)
3. See that this input remains constant throughout the experiment.
4. Wait till  a  satisfactory  steady  state  condition  is  reached.  This  can  be  checked  by  reading temperatures of thermocouples (1) to (6) and note changes in their readings with time.

3. Aim: To verify the Stefen-Boltzmann constant for thermal radiation

Apparatus Used: Heater, temperature-indicators, box containing metallic hemisphere with provision for water-flow through its annulus, a suitable black body which can be connected at the bottom of this metallic hemisphere.

Fig: Stefen-Boltzmann Experiment

Theory: A black body is an ideal physical body which absorbs all types of electromagnetic- radiation incident on it. Because of its 100% absorptivity, it is also the best emitter of thermal radiation. According to Stefan’s Boltzmann law (formulated by the Austrian physicists, Stefan and Boltzmann), energy radiated by a body per unit area per unit time is given by,

R=єσT⁴

where R =energy radiated per area per time,

Є =emissivity of the material of the body,

σ =Stefan’s constant=5.67x10-8 w/m2/K4,

T is the temperature in Kelvin scale.

For a black body, emissivity Є=1, and hence,

R=σT⁴

In   the given experimental set up, the net heat transferred to the disc per second is,

(∆Q/∆t)=mc_p(dT/dt)= σA(T_h⁴- T_d⁴)

Where,

Where    m = mass of the disc in kg,

C_p=specific heat of the material of the disc,

A=area of the disc,

dT/dt=slope of the temperature –time
T_d= steady state temperature of disc in Kelvin,

T_h=hot temperature in Kelvin.

Form the above expression,

σ= [mc_p(dT/dt)] / [A(T_h⁴- T_d⁴)] 

Procedure:

1. Remove the disc from the bottom of the hemisphere  and , switch on the heater and allow the water to flow through it.

2. Allow the hemisphere to reach the steady state and note down the temperature   T1, T2, T3 .

3. Fit the disc (black body) at the bottom of the hemisphere and note down its rise in temperature, T₄ with respect to time till steady state is reached