Heat Transfer Concept Page - 7

Example

Problems using Kirchoff's Law

At a particular temperature and wavelength, the spectral emissive power and monochromatic absorptive power of a body are 10 and 8 units respectively. The emissive power of a black body at the same temperature will be:According to Kirchoff's law.
eλaλ=Eλ${\displaystyle \frac{e_{\lambda }}{a_{\lambda }}}=E_{\lambda }$
here, we are given that eλ$e_{\lambda }$ = 10 and aλ$a_{\lambda }$ = 8
So, Eλ$E_{\lambda }$ = 108${\displaystyle \frac{10}{8}}$ = 1.25

Definition

Heat detection devices

Bolometer and Thermopile are two devices to measure radiation.
BolometerIt is based on the working principle of the Wheatstone bridge. A thin foil of platinum is cut into strips in a grid like pattern. Four such identical grids are taken to form a Wheatstone bridge with a battery and a galvanometer connected. Grids 1 and 4 are allowed to fall radiation upon whereas grids 2 and 3 are protected from radiation. At balance condition the resistances of the grids are related as:
R1R2=R3R4${\displaystyle \frac{R_1}{R_2}}={\displaystyle \frac{R_3}{R_4}}$ where R1,R2,R3,R4$R_1,R_2,R_3,R_4$ are the resistances of the grid.
As temperature increases on radiation, R1$R_1$ and R4$R_4$ increases but the product R2.R3$R_2.R_3$ remains constant. This brings about a deflection in the galvanometer which is a measure of radiation.
ThermopileIt works on the principle of Seeback effect. A thermopile s composed of a number of thermocouples made from bismuth and antimony in a series connection to increase the sensitivity of the system. The free ends are connected to a galvanometer. The junctions are arranged in such a way that all the hot junctions lie on a plane face and all the cold junctions lie on the opposite plane face.The face of the hot junctions is blackened and exposed to radiation while the other face is shielded.  A metallic cone generally concentrates the radiation on the hot face. The radiation is measured by the deflection in the galvanometer.

Diagram

Spectrum of blackbody radiation vs wavelength at different temperatures

Intensity increases with temperature and peak wavelength decreases with temperature

Law

Wien's displacement Law

Wien's displacement law states that the black body radiation curve for different temperatures peaks at a wavelength inversely proportional to the temperature.
λmax=bT${\lambda }_{max}={\displaystyle \frac{b}{T}}$
where T is the absolute temperature in kelvin. b is a constant of proportionality called Wien's displacement constant, equal to 2.8977729(17)×10−3mK$2.8977729(17)\times {10}^{-3}mK$

Definition

Stefan Boltzmann law

The Stefan Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the StefanBoltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time, is directly proportional to the fourth power of the black body's thermodynamic temperature T:
j=σT4$j=\sigma T^4$
where σ=5.670373×10−8Wm−2K−4$\sigma =5.670373\times {10}^{-8}{Wm}^{-2}{K}^{-4}$ , Stefan Boltzmann constant

Definition

Solar constant

The amount of energy received by the earth per second per unit surface area placed normally to the sun's ray at the earth's surface is called the solar constant. Its value is about 1340Wm−2$1340W{m}^{-2}$. 
It is useful in determining the temperature of the sun with the help of Stefan-Boltzmann Law.
Temperature of the sun is given by:
T4=Sσ(Rr)2$T^4={\displaystyle \frac{S}{\sigma }}{\left({\displaystyle \frac{R}{r}}\right)}^2$
where
S:$S:$ Solar constant
σ:$\sigma :$ Stefan's constant
R:$R:$ Mean distance of the earth from the sun
r:$r:$ Radius of the sun

Example

Black surfaces are good absorbers of heat

Take two aluminium containers. One painted with black and the other with original colours. Fill them half with water and keep them under the sun. Each one should have a lid and thermometer.Note the temperatures in both the thermometers. Initially both of them will show the same temperature. Later, the thermometer inserted in the blacked can will show higher temperature. The difference between the readings will increase as time advances. This shows that the black surface absorbs more heat and reflects less as compared with the other can.

Example

Applications of black and white surfaces

Following are some applications of the principle that black surfaces are better abosrbers and radiators than polished surfaces.

1. The bottom part of cooking utensils is blackened to absorb the heat from the flame more quickly. The upper part of the utensil and the outer surface are kept polished and shining to reduce heat loss due to radiation.
2. White clothes are preferred for use during summer as they are poor absorbers of heat.
3. A fire fighting suit is bright and shiny, so that it does not take in a lot of heat energy and the fire fighter is kept cool.
4. Coffee and tea stay hot longer in a shiny pot than in a blackened one
5. A light coloured building stays cooler in summer because it reflects most of the heat radiation from the sun. It is also a poor emitter of heat and stays warm in winter.

BookMarks

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