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 nonmetals.
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 123 dimensional. Conduction can be steady state or unsteady state.
General 3D Unsteady State Heat Conduction equation in Cartesian Coordinates
Fig. 3D Elemental Volume in Cartesian Coordinates
Size of element is dx, dy and dz
(a) With internal heat generation
∂^{2}T/∂x^{2} +∂^{2}T/∂y^{2} + ∂^{2}T/∂z^{2} +q^{.}/k = (1/α) ∂T/ (∂t)
(b) Without internal heat generation
∂^{2}T/∂x^{2} +∂^{2}T/∂y^{2} + ∂^{2}T/∂z^{2} = (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 3D unsteady state heat conduction equation in cylindrical coordinates
Fig. 3D Elemental Volume in Cylindrical Coordinates
Size of the element is dr, rdθ and dz
(a) With Internal heat generation
∂^{2}T/∂r^{2} +(1/r) ∂T/∂r + (1/r^{2})∂^{2}T/∂θ^{2} + ∂^{2}T/∂z^{2} +q^{.}/k = (1/α) ∂T/(∂t)
(b) Without internal heat generation in a Cylindrical Coordinates
∂^{2}T/∂r^{2} +(1/r) ∂T/∂r + (1/r^{2})∂^{2}T/∂θ^{2}+ ∂^{2}T/∂z^{2} =(1/α) ∂T/(∂t)
General 3D Unsteady State Heat Conduction Equation in Spherical Coordinates
Fig. 3D Elemental Volume in Spherical Coordinates
Size of the element is dr, rdθ and rSinθ dΦ
(a) With internal heat generation
(1/r^{2 }sin^{2}θ)∂^{2}T/∂φ^{2} +(1/r^{2} sin θ) ∂(sinθ ∂T/∂θ)/∂θ + (1/r^{2})∂(r^{2}∂T/∂r)/∂r
+q^{.}/k = (1/α) ∂T/(∂t)
(b) Without internal heat generation
(1/r^{2 }sin^{2}θ)∂^{2}T/∂φ^{2} +(1/r^{2} sin θ) ∂(sinθ ∂T/∂θ)/∂θ + (1/r^{2})∂(r^{2}∂T/∂r)/∂r
= (1/α) ∂T/(∂t)
Simplified heat conduction equation in Cartesian coordinates
(i) Without internal heat generation
∂^{2}T/∂x^{2} +∂^{2}T/∂y^{2} + ∂^{2}T/∂z^{2} = (1/α) ∂T/ (∂t) Fourier equation
(ii) Steady State Heat Conduction Equation with internal heat generation
∂^{2}T/∂x^{2} +∂^{2}T/∂y^{2} + ∂^{2}T/∂z^{2} +q^{.}/k =0 (Poisson’s equation)
(iii) Steady State Heat Conduction Equation without internal heat generation
∂^{2}T/∂x^{2} +∂^{2}T/∂y^{2} + ∂^{2}T/∂z^{2} = 0 (Laplace equation)
(iv) Unsteady State 1D without internal heat generation
∂^{2}T/∂x^{2} = (1/α) ∂T/ (∂t)
(v) Steady state 1D with heat generation
∂^{2}T/∂x^{2} + q^{.}/k = 0
(vi) Steady State 1D Conduction equation without internal heat generation
∂^{2}T/∂x^{2} = 0
(vii) Steady State 2D Conduction equation Without internal Heat Generation
∂^{2}T/∂x^{2} + ∂^{2}T/∂y^{2 }= 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 3D unsteady state conduction. Analysis under 23D steady as well as unsteady state is tedious and cumbersome. In theory, it is 1dimensional steady state heat conduction. Fourier Law gives the rate of conduction heat transfer in one dimension as below:
1D Steady State Heat Conduction Through Plane and Composite Walls
Fig. 1D Steady State Heat Conduction Through a Single Wall ( Linear Temperature Variation)
Fig. 1D Steady State Heat Conduction Through a Composite Wall
Integration of equation ∂^{2}T/∂x^{2} gives the following relations .
∂T/∂x =C_{1}
T = C_{1} x +C_{2}
Constants are found by using the boundary conditions.
At x = 0 T =t_{1, }gives t_{1} =C_{2}
At x = L, T=t_{2 }gives t_{2}=C_{1}L +C_{2}
C_{2} =t_{1} and C1 =(t_{2}t_{1})/L
t=[(t_{2}t_{1})/L] x + t_{1}
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= 1m^{2} and ∂T/∂x =1 m
Thus thermal conductivity is rate of heat transfer through 1m^{2} area with a unit temperature gradient
Heat Conduction Through Hollow and Composite Cylinders
Temperature variation through a Cylinder is logarithmic.
Fig. 1D Steady State Heat Conduction Through a Cylinder
∂^{2}T/∂r^{2} +(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 =c_{1}
dt/dr =c_{1}/r
Further integration gives
t = c_{1} ln r +c_{2}
Using the boundary conditions
(i) At r=r_{1}, t=t_{1}
(ii0 At r=r_{2}, t=t_{2}
We get
t=t_{1} + (t_{1}t_{2})/ln(r_{2}/r_{1})
Therefore, the temperature variation is logarithmic.
Heat Conduction Through Hollow and Composite Spheres
Temperature Variation Through a Sphere is Hyperbolic.
Fig. 1D Steady State Heat Conduction Through a Sphere (Temperature variation is Hyperbolic)
(1/r2)∂(r^{2}∂r/∂r)/∂r =0
∴ ∂(r^{2}∂r/∂r)/∂r =0
Integration gives
r^{2}∂r/∂r =c1
∂r/∂r =c_{1}/r^{2}
r= c_{1}/r +c_{2}
Using the boundary conditions
(i) At r=r_{1}, t=t_{1}
(ii0 At r=r_{2}, t=t_{2}
We get c_{1}= (t_{1}–t_{2})r_{1}r_{2}/(r_{2}r_{1}) and c_{2} =t_{1}+ (t_{1}–t_{2})r_{1}r_{2}/[r_{1}(r_{2}r_{1})]
t = (t_{1}–t_{2})r_{1}r_{2}/[r(r_{2}r_{1})] +t_{1}+ (t_{1}–t_{2})r_{1}r_{2}/[r_{1}(r_{2}r_{1})]
Simplification gives
(tt_{1})/(t_{2}t_{1}) =(r_{2}/r)[(rr_{1})/(r_{2}r_{1})] 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
CONDUCTIONUNSTEADY 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/(tt_{∞}) = (h A/ρ V C_{p}) dτ
On integration, we get
ln (tt_{∞}) = (h A/ρ V C_{p}) τ +C_{1}
Boundary conditions for finding the value of constant C_{1}
At τ=0 t=t_{i} (Initial surface temperature)
∴ C_{1} = ln (t_{i}t_{∞})
Substituting the value of C_{1}
ln (tt_{∞}) = (h A/ρ V C_{p}) τ + ln (t_{i}t_{∞})
Taking antilog, we get
(tt_{∞})/(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} = πR^{2}L/2π RL =R/2 R radius of cylinder
Sphere =L_{c} = (4/3) πR^{3}/4 πR^{2} = R/3 R radius of sphere
Cube = L_{c} = a^{3}/6a^{2} = 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
∂^{2}T/∂x^{2} = (1/α) ∂T/ (∂t)
(ii) CYLINDRICAL COORDINATES
∂^{2}T/∂r^{2} + ∂T/∂r = (1/α) ∂T/ (∂t)
(iii) SPHERICAL COORDINATES
∂^{2}T/∂r^{2} = (1/α) ∂T/ (∂t)
Three Cases Of Unsteady State One Dimensional Conductive Heat Transfer
Three cases of are commonly studied for 1D 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
(∂^{2}T/∂x^{2} =(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 thermocouples
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} (TT_{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 + c_{1}
Using the boundary condition at τ = 0, T=T_{i} initial surface temperature
We get c_{1}= ln (T_{i}—Ta)
Therefore solution becomes ln (T–T_{i}) = (–h A_{s} / ρ c V) τ + ln (T_{i}—T_{a})
Taking antilog, we get (T–T_{i}) / (T_{i}—T_{a}) = e^{(– h As / ρ c V)τ}
θ/θ_{i} = e^{(– h As / 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/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}
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 As / 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) xcoordinate as Fourier number
(ii) various curves of 1/Bi
(iii) Determines temperature from the ycoordinate. It is the temperature at the center of the solid at any time ‘t’.
Second Heisler chart has
(i) xcoordinate as 1/Bi
(ii)various curves on this chart for x/L_{c}
(iii) ycoordinate 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) xcoordinate Bi^{2}Fo
(ii) various curves of Bi
(iii) ycoordinate gives Q_{a}/Q_{0.}
Q_{a}/Q_{0} = 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 1D UNSTEADY STATE CONDUCTIVE HEAT TRANSFER

Cooling of I.C. Engines

Annealing

Cooling and freezing

Brick burning

Vulcanization of rubber

Heating of an ingot in a furnace

Explosion

Quenching and other heat treatment processes
3D 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 3dimensional unsteady heat conduction equation with internal heat generation in
(a) Cartesian coordinate
∂^{2}T/∂x^{2} +∂^{2}T/∂y^{2} + ∂^{2}T/∂z^{2} +q^{.}/k = (1/α) ∂T/(∂t)
(b) Cylindrical Coordinates
∂^{2}T/∂r^{2} +(1/r) ∂T/∂r + (1/r^{2})∂^{2}T/∂θ^{2} + ∂^{2}T/∂z^{2} +q^{.}/k = (1/α) ∂T/(∂t)
(c) Spherical Coordinates
(1/r^{2 }sin^{2}θ)∂^{2}T/∂φ^{2} +(1/r^{2} sin θ) ∂(sinθ ∂T/∂θ)/∂θ + (1/r^{2})∂(r^{2}∂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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