Thermodynamics - Heat Transfer Process

Heat Transfer Process

Heat can be transferred from one place to another by three different methods: conduction, convection and radiation. Conduction usually occurs in solids convection in liquids and gases. No medium is required for radiation. 

Thermal conduction if A is area of cross-section of a conductor I its length K thermal conductivity T₁ and T₂temperatures at two ends, ten rate of transfer of heat dQ / dt or thermal current is given by 

dQ / dt = KA(T₂ – T₁) /1 = - KA dT / dx

Temperature gradient dT / dx is negative in the direction of heat flow.

Comparing it with ohm’s law in electricity I = V/R

I thermal = dQ / dt Vthermal = T₁ – T₂ and Rthermal = 1 / KA

Laws of resistance in series and parallel as in electricity are valid in thermal resistance s also, it is believed that current carriers (free electrons) and heat carriers are same because all electrical conductors are also thermal conductors. In general metals are better thermal conductors than liquids and gases as metals have large number of free electrons. 

Thermometric conductivity (D) it is the ratio of thermal conductivity to thermal capacity per unit volume. Thus thermometric conductivity or diffusivity is 

D= K/pC where K -----> thermal conductivity 

P ----> density 

C -----> specific heat Cv

Thermal conductivity K of gases K = 1 / 3 V av λp c_V

= Dp Cv where D = 1 / 3 v av λ is diffusion coefficient and 

λ = 1/2 πd²n is mean free path 

{d---> effective diameter of a molecule 

{n --> number of molecules / volume

Wiedemann-franz law Tiedemann and Franz have shown that at a given temperature T, the ratio of thermal conductivity K) to electrical conductivity (σ) is constant 

That is, 

K/σT = constant 

Ingen’s –housz experiment: Ingen Housz showed that if the number of identical rods of different metals are coated with wax an one of their ends is put in baling water, then in steady state the square of length of the bar over which wax melts is directly proportional to the thermal conductivity of the metal. That is, 

K/L² = constant 

Thermal resistances in series 

R = R₁ + R₂

I₁ + I₂ / K_eff A_eff + I₁ / K₁ A₁ + I₂ / K₂ A₂

If A₁ = A₂ and I₁ = I₁

Then 2/K_eff = 1/K₁ + 1/K₂

Thermal resistances in parallel 

1/R = 1/R₁ + 1/R₂

K eff A_eff/1 = K₁ A₁ / 1 + K₂ A₂ / 1 

Or K_eff A_eff = K₁ A₁ + K₂ A₂ 

And K_eff = K₁ + k₂/2 

If the area of cross-section is equal then 

A_eff = A₁ + A₂ + 2A

Growth of ice in a pond 

dQ₁ = KA 0 – ( - θ) / y dy = dQ₂ = mL = PAdyL

Or dy / dt = Kθ / pL x 1 / y or 1 = pL y² / Kθ²

The ratio of times for thicknesses 0 to y : y to 2y : 2y to 2y : : 1 : 3 : 5 

In a shell of radius r₁ and r₂

dQ / dt = K 4πr₁ r₂ / (r₂ – r₂) (θ₁ – θ₂)

Thickness of the shell (r₂ – r₁) = K4πr²θ / dQ/ dt

Where r₁ = r₂ = r and θ₁ – θ₂ = θ

In a cylinder of length I, radii r₁ and r₂

dQ / dt = 2πK I ( θ₁ –n θ₂)

log e r₂ / r₁ 

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