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Power,
Here, σ is the Stefan’s constant.
Rate of flow of heat = dQ/dt
Heat current, through a conducting rod, is defined as the amount of heat conducted across any cross-section of the rod in one second.
H= dQ/dt
H depends upon following factors:
(a) Area of cross-section of rod: H ∝ A
(b) Temperature Gradient: H ∝-dθ/dx
(c) Nature of the material
So, H = -KA (dθ/dx)
Here K is called the coefficient of thermal conductivity of the material of rod. It depends upon the nature of material of rod.
(d) The total heat Q crossing from one cross section to the other in time t:-
Q = KA(θ_{1}-θ_{2})t/l
Or K = Ql/A(θ_{1}-θ_{2})t
Coefficient of thermal conductivity of the material of a rod is defined as the heat current (amount of heat flowing per second) flowing per unit area between two cross-section of the rod each of area 1 m^{2} and separated 1 m apart.
Dimension of K:- [K] = [M^{1}L^{1}T^{-3}K^{-1}]
Unit:- C.G.S- cal cm^{-1}s^{-1 }ºC^{-1}
S.I – Wm^{-1}K^{-1}
= KA/l
= H/dθ
Unit- S.I- WK^{-1}
Reciprocal of thermal conductance is known as thermal resistance of the substance.
R_{h}= 1/σ_{H} = l/KA = dθ/H
Units of R_{h}:- S.I – W^{-1}K
H= (θ_{1}- θ_{2})/(l/KA) = (θ_{1}- θ_{2})/R_{h}
K= m(θ_{4}- θ_{3})d/A(θ_{1}- θ)t
K/σ ∝ T or K/ σT = constant
Ingen Hausz Experiment:- K_{1}/K_{2} = l_{1}^{2}/ l_{2}^{2}
Thermal resistance of a conductor of length d:- R_{TH} = d/KA
Flow of a heat through a composite slab:-
(a)Thermal resistance in series:- Thermal resistance of the composite slab is equal to the sum of their individual thermal resistances.
(l_{1} +l_{2})/KA = (l_{1}/K_{1}A) + (l_{2}/K_{2}A)
R_{c}_{omb} = R_{h}+R_{h}^{'}
If l_{1}=l_{2}=l, then, K = 2K_{1}K_{2}/K_{1}+K_{2}
Temperature of the interface:-
θ_{0}= [θ_{1}R_{h}^{'} + θ_{2}R_{h}]/ [R_{h}+ R_{h}^{'}] or θ_{0} = [θ_{1}K_{1}l_{2}+ θ_{2} K_{2}l_{1}]/ [K_{1}l_{2}+ K_{2}l_{1}]
(b) Thermal resistance in parallel:- Reciprocal of the combination thermal resistance is equal to the sum of the reciprocals of individual thermal resistances.
1/R_{comb} = 1/R_{h} + 1/R_{h´}
Convection:- It is the mode of transmission of heat, through fluids, in which the particle of fluids acquire heat from one region and deliver the same to the other regions by leaving their mean positions and moving from one point to another.
Radiation:- Radiation is that process of transmission of heat in which heat travels from one point to another in straight lines, with velocity of light, without heating the intervening medium.
Bolometer:- If R_{t} and R_{0} are the resistances of the conductor at 0ºC and tºC, then, R_{t} = R_{0}(1+αt), Here α is the temperature coefficient of change of resistance with temperature.
Absorptive power (a):- Absorptive power (a) of the substance is defined as the ratio between amounts of heat absorbed by it to the total amount of heat incident upon it.
a = Q_{1}/Q
r= Q_{2}/Q
t = Q_{3}/Q
Unit:- S.I-Jm^{-2}s^{-1}
C.G.S- erg cm^{-2}s^{-1}
It states that at any temperature, the ratio of emissive power e_{λ} of a body to its absorptive power a_{λ}, for a particular wave-length, is always constant and is equal to the emissive power of perfect black body for that wavelength.
e_{λ}/a_{λ} = Constant = E_{λ}
This implies the ratio between e_{λ} and a_{λ} for any body is a constant quantity (=E_{λ}).
Wein’s Displacement Law:-
It states that wavelength of radiation which is emitted with maximum intensity varies inversely as the absolute temperature of the body.
λ_{m}×T = Constant
Radiant emittance or the energy radiated per second per unit area by a perfect black body varies directly as the fourth power of its absolute temperature.
E = σT^{4}
Here σ is the Stefan’s constant and its value is 5.735×10^{-8} Wm^{-2}K^{-4}
e_{λ}= Q/At(dλ)
Emissivity:- ε = e/E, 0 ≤ ε ≤1
Rate of loss of heat:- -dQ/dT = εAσ(θ^{4}- θ_{0}^{4})
For spherical objects:- (dQ/dT)_{1}/(dQ/dT)_{2} = r_{1}^{2}/r_{2}^{2}
e = a_{λ}
(a) Emissivity of body determines the radiant emittance of a body.
(b) Emissivity of a perfect body is always one.
(c) Emissivity of any body other than a perfect black body is less than one.
(d) Emissivity of any body is numerically equal to its absorbing power.
dQ/dt = -K(T-T_{0}) or (T-T_{0}) ∝ e^{-KT}
Wein’s Radiation Law:- E_{λ}dλ = (A/λ^{5}) f(λT) dλ = (A/λ^{5}) e^{-a/λT} dλ
Solar Constant:- S = (R_{S}/R_{ES})^{2} σT^{4}
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Solved Examples on Heat Transfer:- Question 1:-...