#### 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 = T_{1}

at x = b , T = T_{2}

from fourier’s law of conduction

**q = -kA dT/dx**

qdx = -kA dT

#### To satisfy steady state conditions q ≠ f(x)

q_{x} = q_{(x+dx)}

solving integration

q * b = kA (T_{1}-T_{2})

##### 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

##### R

_{electric}, Ω → electric resistance

R_{ele.} = ΔV/ i Ω

##### Thermal

##### q, Watt →Rate of Heat transfer

##### ΔT , Kelvin → thermal gradient

##### R

_{thermal}, → thermal resistance

R_{thermal} = 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↓ →R**_{th} ⇒q↓

_{th}⇒q↓

**Case2: Conduction through a composite slab**

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

##### Rate of Heat Transfer through composite slab

##### Heat Transfer per unit area or Heat Flux = q/A

by this equation we can find intermediate temperature T_{2}

##### 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

**q _{conv.} = hA (T_{w} – T_{∞}) Watt**

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

**Case3: Conduction-Convection Heat Transfer through a composite slab**

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

##### Rate of Heat Transfer between gases and ambient fluid through composite slab

##### Heat Transfer per unit area or Heat Flux = q/A

**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 (T_{G} – T_{∞})

q = UA (T_{G} – 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/m**^{2}k)

^{2}k)