Conduction through slab

Application of Fourier’s Law of conduction

Assumptions:
→  Steady state Heat Transfer 
→ Temperature, T ≠ f(Time) at a particular point.
→ One dimensional conduction
→Uniform thermal conductivity ‘k’

Case1: Conduction through a slab

T = f(x)
Boundary Conditions
at x = 0, T = T1
at x = b , T = T2
from fourier’s law of conduction 
q = -kA dT/dx
qdx = -kA dT

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To satisfy steady state conditions q ≠ f(x)
qx = q(x+dx)
solving integration
q * b = kA (T1-T2)

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To solve problems on conduction through slab let’s compare results of above-ntegrated equations with electric circuits.

Electrical analogy of Heat Transfer

Electrical
  • i, Ampere →Rate of current 
  • Δv or emf, V → potential gradient
  • Relectric, Ω → electric resistance

Rele. = ΔV/ i  Ω

Thermal
  • q, Watt →Rate of Heat transfer 
  • ΔT , Kelvin → thermal gradient
  • Rthermal,  → thermal resistance

Rthermal = ΔT/ q   K/Watt

For a single slab
more the thickness of the slab and smaller the thermal conductivity of material →  more the thermal resistance offered by slab and lesser will be heat transfer 
i.e. if b↑ and  k↓ →Rth ⇒q↓
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Case2: Conduction through a composite slab

We will use electrical analogy and two adjacent slabs will act like resistance in series

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Rate of Heat Transfer through composite slab
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Heat Transfer per unit area  or Heat Flux = q/A
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by this equation we can find intermediate temperature T2

Till now we have read about conductive thermal resistance. In some cases there is convective thermal resistance.

Convective thermal resistance

Newton’s law of cooling 

qconv. = hA (Tw – T) Watt

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Since, h value is associated with water (liquid) being more the conventional thermal resistance with water shall be lesser

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Case3: Conduction-Convection Heat Transfer through a composite slab

We will use electrical analogy and two adjacent slabs will act like resistance in series

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Rate of Heat Transfer between gases and ambient fluid through composite slab
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Heat Transfer per unit area  or Heat Flux = q/A
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Overall Heat Transfer Coefficient (U)

It is the parameter which takes into account all the modes of heat transfer into a single entity.
q = UAΔT watt

where ΔT Total temp. differnce (TG – T)

q = UA (TG – T)

If the value of U is more, the total thermal resistance in the entire circuit will be lesser and thus, Rate of Heat Transfer will be more.
U and h have the same units (watt/m2k)

 

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