STEADY AND UNSTEADY STATE HEAT CONDUCTION CLASS NOTES

STEADY & UNSTEADY STATE

HEAT CONDUCTION CLASS NOTES 

Conduction is one of the three modes of

heat transfer. Other modes of heat transfer

are convection and radiation. Rate of heat

transfer is directly proportional to the

temperature difference. Conduction takes

place in solids by physical contact. It is

more in metals and least in non-metals.

The main property linked with it is

thermal conductivity of the solids. The

thermal conductivity of liquids and

gases is very small as compared to

that of solids. There are two types of

conduction heat transfer. These are

steady and unsteady state heat

conduction. Unsteady state heat

conduction is also called a transient

heat conduction. Temperature is a

function of time and space. It depends

on Fourier and Biot numbers.

CONDUCTION

Conduction is one of the three basic modes of heat transfer. The two other modes are convection and radiation. Thermal conduction takes place in solids and stationary fluids adhered to the solid surface. But the heat transfer by conduction in fluids is small because of smaller value of conductivity. Temperature difference is the driving force for conduction. Thus it is a sensible heat transfer.

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

(a) With internal heat generation

∂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

q^. is heat generated per unit volume per unit time

∂T/ (∂t) is rate of change of temperature

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

∂²T/∂r² +(1/r) ∂T/∂r + (1/r²)∂²T/∂θ²+ ∂²T/∂z²  =(1/α) ∂T/(∂t)

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

(1/r² sin²θ)∂²T/∂φ² +(1/r² sin θ) ∂(sinθ ∂T/∂θ)/∂θ + (1/r²)∂(r²∂T/∂r)/∂r

+q^./k = (1/α) ∂T/(∂t)

(b) Without internal heat generation

(1/r² sin²θ)∂²T/∂φ² +(1/r² sin θ) ∂(sinθ ∂T/∂θ)/∂θ + (1/r²)∂(r²∂T/∂r)/∂r

 = (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   = (1/α) ∂T/ (∂t)

(v) Steady state 1-D with 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

∂²T/∂x² + ∂²T/∂y² = 0  

Steady state heat Conduction

Steady state means бT/бz = 0, Change of temperature with respect to time is zero. In nature, conduction is only 3-D unsteady state conduction. Analysis under 2-3-D steady as well as unsteady state is tedious and cumbersome. In theory, it is 1-dimensional steady state heat conduction. Fourier Law gives the rate of conduction heat transfer in one dimension as below:

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

 

Fig. 1-D Steady State Heat Conduction Through a Single Wall ( Linear Temperature Variation)

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 =t_1,   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

    Q_x^. = −κA∂T/∂x           Watts

— Negative sign indicates heat transfer is opposite to the temperature gradient.

∂T/∂x is the temperature gradient

To find the rate of heat transfer  Q_x^., Q_y^. and Q_z^. .

Fourier law gives the definition of the thermal conductivity, k.

k = Q^./ A∂T/∂x    or k = Q^. = rate of heat transfer   for A= 1m² and ∂T/∂x =1 m

Thus thermal conductivity is rate of heat transfer through 1m² area with a unit temperature gradient

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)

(1/r2)∂(r²∂r/∂r)/∂r =0

∴ ∂(r²∂r/∂r)/∂r =0

Integration gives

r²∂r/∂r =c1

∂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

(i)             Fins

(ii)            Hot plate

(iii)          Walls, floors and roofs

(iv)         Cylindrical surfaces like pipes and tubes in

condensers, evaporators and radiators

(v)          Spherical surfaces like spherical tanks 

 

CONDUCTION-UNSTEADY STATE (TRANSIENT)

This is of two types.

(a) Periodic type

In a car engine, processes are repeated under unsteady state conduction

(b) Non periodic type

(i) A hot ball is placed in a cooled fluid

(ii) A quenching process

(iii) A heat treatment process

During unsteady state heat conduction

(i)  the temperature varies with time

(ii0 Temperature varies with x.

(iii)Therefore temperature varies with x and time, T=f(x, τ).

(iv) Temperature variation is exponential.

  Now we study the non periodic type unsteady state heat conduction. Energy balance in such cases states that the rate of change of internal energy of the solid =rate of conduction heat transfer from the solid surface to the fluid.

Mathematically,

–m C_p dt/dτ =hA (t–t_∞)

 

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

-ρ V C_p dt/dτ =hA (t–t_∞)

Rearranging

dt/(t-t_∞)  = -(h A/ρ V C_p) dτ

On integration, we get

ln (t-t_∞)  = -(h A/ρ V C_p) τ +C₁

Boundary conditions for finding the value of constant C₁ 

At τ=0  t=t_i (Initial surface temperature)

∴ C₁ = ln (t_i-t_∞)

Substituting the value of C₁

ln (t-t_∞)  = -(h A/ρ V C_p) τ + ln (t_i-t_∞)

Taking antilog, we get

(t-t_∞)/(t_i-t_∞) =θ/θ_i = exp [(-(h A/ρ V C_p) 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  R_th . C_th)

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

 

∴  τ_th  =(1/h A) (ρ V C_p) =R_th C_th   

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

  C_th = ρ V C_p =Lumped thermal capacitance of the solid

CHARACTERISTIC LENGTH, L_c

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

Characteristic length for simple geometric shapes

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

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

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

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

ONE DIMENSIONAL UNSTEADY HEAT CONDUCTION

FOURIER EQUATIONS (With constant thermal conductivity ‘ k ‘and no internal heat generation )

(i)                 CARTESIAN COORDINATES

∂²T/∂x² = (1/α) ∂T/ (∂t)

(ii)               CYLINDRICAL COORDINATES

∂²T/∂r² + ∂T/∂r = (1/α) ∂T/ (∂t)

(iii)             SPHERICAL COORDINATES

∂²T/∂r² =  (1/α) ∂T/ (∂t)

 Three Cases Of Unsteady State One Dimensional Conductive Heat Transfer

Three cases of are commonly studied for 1-D unsteady state conduction heat transfer.

(i) Conduction through solids of infinite conductivity ( when Bi <<0.1 ) (Negligible Internal Resistance)

(ii) Conduction through solids of finite conductivity, Bi > 0.1 to Bi =100

(iii) Conduction through solids of infinite thickness OR when Bi →∞.

For the above three cases, the same differential equation

(∂²T/∂x² =(1/α)  ∂T/ (∂t)) applies but the solution is different because of different boundary conditions.

 VARIOUS METHODS OF ANALYZING 1- D UNSTEADY STATE CONDUCTION

1.Analytical method or Lumped parameter method

2.Graphical method or Heisler charts method

3.Gauss error function method

 Lumped Capacitance method

The outcome of this method is that temperature varies exponentially with time. Lumped capacitance method is a common approximation in transient conduction. Here the heat conduction within is much faster than heat convection across the boundary of the object. It is Lumped capacitance method. Total heat capacity of the system is  one lump. It is a function of temperature only.

It assumes space temperature gradients are zero. Temperature is a function of time only.

I.e. dt/ dx=0, dt/dy=0, dt/dz=0 and t=f (time)

This is applicable in materials with very high thermal conductivity or Biot number << 0.1.

Hence thermal conductive resistance x/k A is very small as compared to convective resistance 1/h A at the surface.

Such a situation is found in the followings;

Heat treatment of metallic parts

Time response of thermo-couples

Further time response of thermometers

 Derivation for Temperature Distribution

Conductive resistance= internal thermal resistance = x / kA

Convective resistance= external resistance=Surface resistance = 1/ h A

Therefore under capacitance method h A_s (T-T_a) = — mc dT/dτ

dT/ (T—Ta) = (–h A_s / ρ c V)  dτ

On integration, we get

ln (T—Ta) = (–h A_s / ρ c V) τ + c₁

Using the boundary condition at τ = 0, T=T_i initial surface temperature

We get    c₁= ln (T_i—Ta)

Therefore solution becomes    ln (T–T_i) = (–h A_s / ρ c V) τ + ln (T_i—T_a)

Taking anti-log, we get            (T–T_i) / (T_i—T_a) = e^{(– h A_s / ρ c V)τ}

                     θ/θ_i = e^{(– h A_s / ρ c V)τ}

The following points are worth noting:

(i) Equation for θ/θ_i 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/hA_s has the units of time and is named as THERMAL TIME CONSTANT. It is denoted by z_th.}

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}

R_th = (1/h A_s) = Resistance to convection heat transfer

C_th = (=_{ρ c V) = Lumped Thermal Capacitance of a solid}

Fig below shows that  any increase in R_th or C_th 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 C_th =ρcV represents the thermal capacity of the system. It can be obtained from the following equation

Q = (ρcV) t = C_th t

Fig. Transient Temperature Response

 Equation θ/θ_i = e^{(– h A_s / ρ c V)τ is} for temperature distribution. It is also the equation for Newtonian heating/cooling of a solid. The TEMPERATURE VARIES EXPONENTIALLY WITH TIME. There are three possibilities.

Fig. Temperature Variation under Lumped Parameter Method

HEISLER CHARTS METHOD

 Heat conduction in solids of finite conductivity, 0.1<Bi<100

In this, method of analysis is graphical. It uses three HEISLER CHARTS for finding temperature distribution. First Heisler chart has

(i) x-coordinate as Fourier number

(ii) various curves of 1/Bi

(iii) Determines temperature from the y-coordinate. It is the temperature at the center of the solid at any time ‘t’.

Second Heisler chart has

(i) x-coordinate as 1/Bi

(ii)various curves on this chart for x/L_c

(iii) y-coordinate gives the temperature at distance ‘x’ from the center at time ‘t’ .

(iv) L_c is the characteristic length equal to volume/surface area.

Firstly L_c = δ/2 for a plane wall

Secondly L_c = R/2 for a cylinder

Thirdly L_c = R/3 for a sphere

Fourthly L_c = L/6 for a cube.

Third Heisler chart OR Grober chart has

(i) x-coordinate  Bi²Fo

(ii) various curves of Bi

(iii) y-coordinate gives Q_a/Q_0.

Q_a/Q₀ = actual rate of heat transfer at time ‘t’/ rate of heat transfer at t=0.

Separate three Heisler charts for each of the following are available

(i)Firstly for a plain wall,

(ii) Secondly for a cylinder

(iii) Thirdly for a sphere

Gaussian Error Function method

When Bi is tending to infinity

Gaussian Error Function is erf =[(x/2 (αt)^0.5].

Values of these functions for different cases.

(i)Firstly for a wall

erf functions from a table.

(ii) Secondly for a cylinder

  erf functions are found from a standard graph for a cylinder

(iii) Thirdly for a sphere

erf functions are found from a standard graph for a sphere

PRACTICAL APPLICATIONS OF 1-D UNSTEADY STATE CONDUCTIVE HEAT TRANSFER

1. Cooling of I.C. Engines

2. Annealing

3. Cooling and freezing

4. Brick burning

5. Vulcanization of rubber

6. Heating of an ingot in a furnace

7. Explosion

8. Quenching and other heat treatment processes

3-D Unsteady State Heat Conduction

Three dimensional unsteady steady state conduction means Temperature T is a function of x, y, z and time i.e. T=f(x,y,z,τ). In nature, conductive heat transfer problems are 3 dimensional. The 3-dimensional unsteady heat conduction equation with internal heat generation in

(a) Cartesian coordinate

∂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

(1/r² sin²θ)∂²T/∂φ² +(1/r² sin θ) ∂(sinθ ∂T/∂θ)/∂θ + (1/r²)∂(r²∂T/∂r)/∂r +q^./k = (1/α) ∂T/(∂t)

In order to solve the equation, it requires a total of six boundary conditions. Two for each direction are required. one initial condition with respect to time at the time of start is required. It accounts for transient behavior. It is highly complex to find solutions for such an equation. Therefore only 1- D unsteady state conduction is being considered. Unsteady state conduction is also said transient heat conduction.

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