# STEADY AND UNSTEADY STATE HEAT CONDUCTION CLASS NOTES

## on Fourier and Biot numbers.

### Main Cause of Conduction

#### Heat transfer by conduction is slightly due to the vibration of the lattice. Mainly it is due to movement of free electrons. Fourier’s Law is the basic equation for conduction. This equation is an empirical and can be verified  experimentally . It applies to all the three states of matter. But mainly it takes place in solids. It can be 1-2-3 dimensional. Conduction can be steady state or unsteady state.

General 3-D Unsteady State Heat Conduction equation in Cartesian Coordinates

#### Fig. 3-D Elemental Volume in Cartesian Co-ordinates

Size of element is dx, dy and dz

#### ∂2T/∂x2 +∂2T/∂y2 + ∂2T/∂z2 +q./k = (1/α) ∂T/ (∂t)

(b) Without internal heat generation

#### ∂2T/∂x2 +∂2T/∂y2 + ∂2T/∂z2 = (1/α) ∂T/ (∂t)

Where k is thermal conductivity of the solid

α is the thermal diffusivity

### General 3-D unsteady state heat conduction equation in cylindrical co-ordinates Fig. 3-D Elemental Volume in Cylindrical Co-ordinates

Size of the element is dr, rdθ and dz

(a) With Internal heat generation

#### ∂2T/∂r2 +(1/r) ∂T/∂r + (1/r2)∂2T/∂θ2 + ∂2T/∂z2 +q./k = (1/α) ∂T/(∂t)

(b) Without internal heat generation in a Cylindrical Co-ordinates

### General 3-D Unsteady State Heat Conduction Equation in Spherical Co-ordinates Fig. 3-D Elemental Volume in Spherical Co-ordinates

Size of the element is dr, rdθ and rSinθ dΦ

(a) With internal heat generation

#### +q./k = (1/α) ∂T/(∂t)

(b) Without internal heat generation

#### = (1/α) ∂T/(∂t)

Simplified heat conduction equation in Cartesian co-ordinates

(i) Without internal heat generation

#### ∂2T/∂x2 +∂2T/∂y2 + ∂2T/∂z2 = (1/α) ∂T/ (∂t)      Fourier equation

(ii) Steady State Heat Conduction Equation with internal heat generation

#### ∂2T/∂x2 +∂2T/∂y2 + ∂2T/∂z2 +q./k =0        (Poisson’s equation)

(iii) Steady State Heat Conduction Equation without internal heat generation

#### ∂2T/∂x2 +∂2T/∂y2 + ∂2T/∂z2 = 0                   (Laplace equation)

(iv) Unsteady State 1-D without internal heat generation

#### ∂2T/∂x2 + q./k = 0

(vi) Steady State 1-D Conduction equation without internal heat generation

#### ∂2T/∂x2  = 0

(vii) Steady State 2-D Conduction equation Without internal Heat Generation

### 1-D Steady State Heat Conduction Through Plane and Composite Walls

#### Fig. 1-D Steady State Heat Conduction Through a Composite Wall

Integration of equation 2T/∂x2  gives the following relations .

∂T/∂x =C1

T = C1 x +C2

Constants are found by using the boundary conditions.

At x = 0 T =t1,   gives    t1 =C2

At x = L, T=t2      gives t2=C1L +C2

C2 =t1 and C1 =(t2-t1)/L

t=[(t2-t1)/L] x + t1

Therefore, the temperature variation is linear and independent of thermal conductivity.

Using Fourier equation

### Heat Conduction Through Hollow and Composite Cylinders

#### Temperature variation through a Cylinder is logarithmic. Fig. 1-D Steady State Heat Conduction Through a Cylinder

#### ∂2T/∂r2 +(1/r) ∂T/∂r  = 0 gives

(1/r) d (r dt/dr)/dr  =0

1/r ≠0

∴ d (r dt/dr)/dr =0

On integration, we get

r dt/dr =c1

dt/dr =c1/r

Further integration gives

t = c1 ln r +c2

Using the boundary conditions

(i) At r=r1, t=t1

(ii0 At r=r2, t=t2

We get

t=t1 + (t1-t2)/ln(r2/r1)

Therefore, the temperature variation is logarithmic.

### Heat Conduction Through Hollow and Composite Spheres

Temperature Variation Through a Sphere is Hyperbolic. Fig. 1-D Steady State Heat Conduction Through a Sphere (Temperature variation is Hyperbolic)

#### ∴ ∂(r2∂r/∂r)/∂r =0

Integration gives

#### ∂r/∂r =c1/r2

r= -c1/r +c2

Using the boundary conditions

(i) At r=r1, t=t1

(ii0 At r=r2, t=t2

We get  c1= (t1–t2)r1r2/(r2-r1) and c2 =t1+ (t1–t2)r1r2/[r1(r2-r1)]

t = (t1–t2)r1r2/[r(r2-r1)] +t1+ (t1–t2)r1r2/[r1(r2-r1)]

Simplification gives

(t-t1)/(t2-t1) =(r2/r)[(r-r1)/(r2-r1)] dimensionless form

Temperature variation is hyperbolic.

### Applications of Steady State Heat Conduction

#### negative shows that the internal energy of the solid is decreasing on cooling

-ρ V Cp dt/dτ =hA (t–t)

Rearranging

dt/(t-t)  = -(h A/ρ V Cp) dτ

On integration, we get

ln (t-t)  = -(h A/ρ V Cp) τ +C1

### At τ=0  t=ti (Initial surface temperature)

∴ C1 = ln (ti-t)

Substituting the value of C1

### ln (t-t∞)  = -(h A/ρ V Cp) τ + ln (ti-t∞)

Taking antilog, we get

### (t-t∞)/(ti-t∞) =θ/θi = exp [(-(h A/ρ V Cp) τ]

This represents exponential variation of temperature with time. It is named as Newtonian heating or Newtonian cooling of solids. Fig. Temperature variation in unsteady state heat conduction Fig. Transient Temperature Response (Response slows down with increase of  Rth . Cth)

The quantity (ρ V Cp/h A) is named as THERMAL TIME CONSTANT. It is represented by τth. It represents the rate of response of a system to a sudden change in its environmental temperature.

∴  τth  =(1/h A) (ρ V Cp) =Rth Cth

Rth =(1/h A) =  resistance to convective heat transfer

Cth = ρ V Cp =Lumped thermal capacitance of the solid

CHARACTERISTIC LENGTH, Lc

### The ratio of volume of solid to surface area of solis is named as characteristic length.

Characteristic length for simple geometric shapes

Flat plate =Lc =LBH/2BH=L/2                 L thickness of flat plate

Cylinder   = Lc = πR2L/2π RL =R/2             R radius of cylinder

Sphere       =Lc = (4/3) πR3/4 πR2 = R/3     R radius of sphere

Cube          = Lc = a3/6a2 = a/6     a is the side of the cube

### Derivation for Temperature Distribution

#### θ/θi = e(– h As / ρ c V)τ

The following points are worth noting:

(i) Equation for θ/θi gives the temperature distribution in a solid body for Newtonian heating or cooling. It also indicates that the temperature variation is exponential with time.

(ii) The quantity ρ c V/hAs has the units of time and is named as THERMAL TIME CONSTANT. It is denoted by zth.

Its value indicates the rate of response of a system to a sudden change in its environmental temperature.

#### zth  = (1/hAs)(ρ c V) = Rth Cth

Rth = (1/h As) = Resistance to convection heat transfer

Cth = (=ρ c V) = Lumped Thermal Capacitance of a solid

Fig below shows that  any increase in Rth or Cth will cause a solid to respond more slowly to changes in its thermal environmental. It increases the time to attain thermal equilibrium. Fig. Newtonian Heating or Cooling of Solids

Fig. shows the lumped heat capacity systemin which Cth =ρcV represents the thermal capacity of the system. It can be obtained from the following equation

Q = (ρcV) t = Cth t Fig. Transient Temperature Response

### PRACTICAL APPLICATIONS OF 1-D UNSTEADY STATE CONDUCTIVE HEAT TRANSFER

8. #### Quenching and other heat treatment processes

3-D Unsteady State Heat Conduction

#### ∂2T/∂x2 +∂2T/∂y2 + ∂2T/∂z2 +q./k = (1/α) ∂T/(∂t)

(b) Cylindrical Co-ordinates

#### ∂2T/∂r2 +(1/r) ∂T/∂r + (1/r2)∂2T/∂θ2 + ∂2T/∂z2 +q./k = (1/α) ∂T/(∂t)

(c) Spherical Co-ordinates