Heat Transfer Concept Page - 6

Definition

Spectral absorptive power

Spectral absorptive power al$a_l$: The absorptive power refers to radiations of all wavelengths (or the total energy) while the spectral absorptive power is the ratio of radiant energy absorbed by a surface to the radiant energy incident on it for a particular wavelength . It may have different values for different wavelengths for a given surface.

Definition

Perfectly black body

A body which absorbs all the radiant energy incident upon it and reflects or transmits none is called a perfectly black body.

• It absorbs all incident energy. ∴a=1$\therefore a=1$
• It does not reflect or transmit heat. ∴t=0,r=0$\therefore t=0,r=0$
• At a given temperature, emissive power (Eb$E_b$) of a black body is greater than the emissive power (E) of any other body. (Eb$E_b$ > E)
• No perfectly black body exists.
• Practical black body - Lamp black or platinum black- (absorbs 95% - 98% of the incident radiant energy).

Definition

Ferry's black body

A cavity with walls made of any material, with a small opening, is an excellent black body. This is Ferry's black body. It also absorbs all electromagnetic radiation incident on it irrespective of wavelength. A hollow enclosure maintained at a uniform temperature, with a small opening compared to its size and a conical projection, is an excellent approximation to a black body. The opening acts as a perfect absorber. Radiation entering the hole suffers innumerable absorptions and diffuse reflections at the interior walls and has negligible chance of coming out. It is eventually absorbed fully, whatever be the material of the walls of the enclosure.

Formula

Energy emitted per unit time by a body

u=eσAT4$u=e\sigma A{T}^4$
where u is the energy emitted by a body per unit time,
e is the emissivity
σ$\sigma$ is the Stefan-boltzmann constant
A is the surface area and
T is the temperature

Definition

thermal energy lost per unit time due to radiation

Suppose a body of surface area A at an absolute temperature T$T$ is kept in a surrounding having a lower temperature  T0$T_0$. The net rate of loss of thermal energy from the body due to radiation is 
Δu1=eσA(T4−T40)$\mathrm{\Delta }{u}_1=e\sigma A{(T}^4-T_0^4)$
=4eσAT30(T−T0)$=4e\sigma A{T}_0^3(T-T_0)$     if the temperature difference is small

Definition

Intensity at a finite distance from the source

Intensity of thermal radiation decays as:
I=Io/r2$I=I_o/r^2$
where
I:$I:$ Intensity at a distance r$r$ from the reference point
Io:$I_o:$ Intensity at the reference point

Example:
Find the ratio of amplitudes of radiation emitted by a cylindrical source at distances 2r$2r$ and 18r$18r$ from its axis.Solution:
Let the intensity of the cylindrical source be Io$I_o$
The intensity at a distance 2r$2r$ is given by: I2r=Io4r2$I_{2r}={\displaystyle \frac{I_o}{4r^2}}$
The intensity at a distance 18r$18r$ is given by: I18r=Io182r2$I_{18r}={\displaystyle \frac{I_o}{{18}^2{r}^2}}$
I2rI18r=18222=81${\displaystyle \frac{I_{2r}}{I_{18r}}}={\displaystyle \frac{{18}^2}{2^2}}=81$
I∝A2$I\propto A^2$ where A:$A:$ Amplitude
∴A2rA18r=9$\therefore {\displaystyle \frac{A_{2r}}{A_{18r}}}=9$

Definition

The construction and working of Angstrom pyrheliometer

A pyrheliometer is used to measure direct solar radiation from the sun and its marginal periphery. Tomeasure direct solar radiation correctly, its receiving surface must be arranged to be normal to the solardirection. 
The structure of an Angstrom electrical compensation pyrheliometer is shown in figure.This is a reliable instrument used to observe direct solar radiation, and has long been accepted as a working standard. However, its manual operation requires experience.
A copper-constantan thermocouple is attached to the rear of each sensor strip, and the thermocouple is connected to a galvanometer. The sensor strips also work as electric resistors and generate heat when a current flows across them.(see the figure). 
When solar irradiance is measured with this type of pyrheliometer, the small shutter on the front face of the cylinder shields one sensor strip from sunlight, allowing it to reach only the other sensor. A temperature difference is therefore produced between the two sensor strips because one absorbs solar radiation and the other does not, and a thermo electromotive force proportional to this difference induces current flow through the galvanometer.
Then, a current is supplied to the cooler sensor strip (the one shaded from solar radiation) until the pointer in the galvanometer indicates zero, at which point the temperature raised by solar radiation is compensated by Joule heat. A value for direct solar irradiance is obtained by converting the compensated current at this time. If S$S$ is the intensity of direct solar irradiance and i$i$ is the current, then S=ki2$S=k{i}^2$.

Definition

Coefficient of absorption

Coefficient of Absorption: The ratio of the quantity of radiant energy absorbed by a body per unit area per unit time to the total quantity of radiant energy incident upon it per unit area per unit time is called the coefficient of absorption.

Q$Q$ = quantity of incident radiant energy per unit area per unit time.Qa$Q_a$ = quantity of energy absorbed by body per unit area per unit time.

Coefficient of absorption a=QaQ$a={\displaystyle \frac{Q_a}{Q}}$

or absorptive power or absorptivity

Example

Modes of heat transfer

The three modes of heat transfer are conduction, convection and radiation. 

Law

Kirchoff's Law of Radiation

At a given temperature, the ratio of the emissive power of a body to its absorptive power is constant and is equal to the emissive power of a black body at the same temperature.
Ea=Eb${\displaystyle \frac{E}{a}}=E_b$
But EEb=e${\displaystyle \frac{E}{E_b}}=e$
∴a=e$\therefore a=e$
Alternative statement of Kirchhoff's law: At any given temperature, the emissivity of a body is equal to its coefficient of absorption.

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