Conduction Formulas

1. Conduction

• In conduction. Molecules do not leave their position.
• The molecules transfer energy to the neighboring molecules due to mutual contact between them.
• The heat conduction i.e. transmission of heat in this mode occurs mainly in solids and mercury.

2. Law of heat transfer through conduction

rate of flow of heat \(\frac{\mathrm{dQ}}{\mathrm{dt}}=-\mathrm{KA} \frac{\mathrm{d} \theta}{\mathrm{dx}}\)
\(\frac{d \theta}{d x}\) → Temp, gradient
K → Coefficient of thermal conductivity conductivity
A → cross sectional Area

3. Comparison of heat conduction with electrical conduction

Electrical Conduction	Thermal Conduction
1. Electric charge flows from higher potential to lower potential	Heat flows from higher temperature to lower temperature
2. The rate flow of charge is called electric current I = \( \frac{d q}{d t}\)	The rate flow of heat is called electric current I = \( \frac{d Q}{d t}\)
3. By Ohm’s law, I = \( \frac{V_{1}-V_{2}}{R}\)	The thermal current is I = \( \frac{\mathrm{T}_{1}-\mathrm{T}_{2}}{\mathrm{R}_{\mathrm{Th}}}\)
4. The electrical resistance is given by R = \( \rho \frac{\ell}{\mathrm{A}}=\frac{\ell}{\sigma \mathrm{A}}\) Where ρ is resistivity and a is conducitividy	The thermal resistance is given by \( \mathrm{R}_{\mathrm{Th}}=\frac{\ell}{\mathrm{K} \mathrm{A}}\) where K is thermal conductivity of the conductor

4. Combination of straight rods (Let θ₁ > θ₂)

Heat current in I^st rod = \(\frac{\mathrm{K}_{1} \mathrm{A}\left(\theta_{1}-\theta\right)}{\ell_{1}}\) … (1)
Heat current in II^st rod = \(\frac{\mathrm{K}_{2} \mathrm{A}\left(\theta-\theta_{2}\right)}{\ell_{2}}\)
In steady state,
\(\frac{\mathrm{K}_{1} \mathrm{A}\left(\theta_{1}-\theta\right)}{\ell_{1}}=\frac{\mathrm{K}_{2} \mathrm{A}\left(\theta-\theta_{2}\right)}{\ell_{2}}\)
θ = \(\frac{\mathrm{K}_{1} \ell_{2} \theta_{1}+\mathrm{K}_{2} \ell_{1} \theta_{2}}{\mathrm{K}_{1} \ell_{2}+\mathrm{K}_{2} \ell_{1}}\)
K_eq = \(\frac{\ell_{1}+\ell_{2}}{\frac{\ell_{1}}{K_{1}}+\frac{\ell_{2}}{K_{2}}}\)

5. Parallel combination

Let \(\frac{\mathrm{d} \mathrm{H}_{1}}{\mathrm{dt}}\) and \(\frac{\mathrm{d} \mathrm{H}_{2}}{\mathrm{dt}}\) be the heat, conducted in first and second rod. The resultant of these is assumed to be \(\frac{\mathrm{d} \mathrm{H}}{\mathrm{dt}}\).
\(\frac{\mathrm{d} \mathrm{H}}{\mathrm{dt}}=\frac{\mathrm{d} \mathrm{H}_{1}}{\mathrm{dt}}+\frac{\mathrm{d} \mathrm{H}_{2}}{\mathrm{dt}}\)
= \(\frac{\mathrm{K}_{1} \mathrm{A}_{1}\left(\theta_{1}-\theta_{2}\right)}{\ell}+\frac{\mathrm{K}_{2} \mathrm{A}_{2}\left(\theta_{1}-\theta_{2}\right)}{\ell}\)
= \(\left(\frac{\mathrm{K}_{1} \mathrm{A}_{1}}{\ell}+\frac{\mathrm{K}_{2} \mathrm{A}_{2}}{\ell}\right)\)(θ₁ – θ₂) …….(1)
K_eq = \(\frac{\mathrm{K}_{1} \mathrm{A}_{1}+\mathrm{K}_{2} \mathrm{A}_{2}}{\mathrm{A}_{1}+\mathrm{A}_{2}}\)

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