Heat Conduction Differential Equation of Heat Conduction Temperature is not varied with one direction Temperature is varying with time Arbitrarily solid Need differential equation
Heat Conduction
Differential Equation of Heat Conduction Temperature is not varied with one direction
Temperature is varying with time
Arbitrarily solid
Need differential equation
Differential Equation of Heat Conduction
Element inside the solid
1) Thermal conductivity is not directly dependent at any point (Isotropic material).
2) Assume that heat generated = (W/m3)
Fourier’s law of heat conduction - For first one dqx
dz.dy.x
Tkdq x
dz.dydx.x
Tk
xx
Tkdq dxx
q
Differential Equation of Heat Conduction
- For first one dqy
....dqy
....dq dyy
- For first one dqz
....dqz
....dq dzz
Similarly
Now, what is the net rate at which heat is being conducted into the element?
Differential Equation of Heat Conduction
1) Net amount of heated conducted into dxdydz per unit time
)dqdqdq()dqdqdq( dzzdyydxxzyx
dz.dy.dxz
Tk
zy
Tk
yx
Tk
x
2) Net amount of heated conducted into dxdydz per unit time
)dz.dy.dx(q
3) Rate of change of energy of the element
t
TC)dz.dy.dx.( p
Differential Equation of Heat Conduction
1 + 2 = 3
t
TCq
z
Tk
zy
Tk
yx
Tk
xp
Therefore, the required differential equation is
Note: dx.dy.dz is cancelled
Apply the 1st law of Thermodynics of closed system
This is the most general form of differential equation for Cartesian coordinate system
Differential Equation of Heat Conduction
t
TCq
z
T
y
T
x
Tk p2
2
2
2
2
2
If k is constant
In the absence of heat generation
t
T1
k
qT2
is called the thermal diffusivity (unit: m2/s)
t
T1T2
If there is a steady state
0T2 Depending of the nature of the problem
(Laplace’s equation)
Differential Equation of Heat Conduction
Cylindrical Spherical
cosrx
sinry
zz
cossinrx
sinsinry
cosrz
Zenith angleAzimuth angle
Differential Equation of Heat ConductionDerivation of different equation of heat conduction in a 2D polar coordinate system ),r(
r
dr drrdq
d
rdq
dq
ddq
1.d.r.r
Tkdq r
ddr.
r
T.r.k
rr
T.r.kdq drr
1.dr.T
r
kdq
drd.r.T
kr
T
r
kdq d
Cylindrical
A
Adistance
Differential Equation of Heat Conduction
1) Rate at which heat is conducted into the element
d.dr
Tk
r
1
r
Tr.k
r
2) Heat generated per unit time
1drdrq
3) Rate of change of energy of the element
t
TC)1.dr.d.r.( p
volume
Differential Equation of Heat Conduction
1 + 2 = 3
t
TCq
Tk
r
1
r
Tkr
rr
1p2
Therefore, the required differential equation is
Differential Equation of Heat Conduction
t
TCq
T
r
1
r
Tr
rr
1k p2
2
2
If k is constant
In the absence of heat generation
If there is a steady state
t
TC
T
r
1
r
Tr
rr
1k p2
2
2
0T
r
1
r
Tr
rr
1k
2
2
2
Differential Equation of Heat Conduction)z,,r( Derive the differential equation for the 3-D cylindrical system
For constant k
2
2
z
Tk
Additional term on left-hand side
t
TCq
z
TT
r
1
r
Tr
rr
1k p2
2
2
2
2
Differential Equation of Heat Conduction
Spherical
t
TCq
T
sinr
1Tsin
sinr
1
r
)rT(
r
1k p2
2
2222
2
Now, what do we know?
Coordinate: Cartesian, cylindrical, spherical coordinatesState: steady state, unsteady state Direction: one, two, three directionsMaterial: isotropic material constant k, not constant k
Differential Equation of Heat Conduction
Solving a problem needs initial and boundary conditions
1. Initial condition
At t = 0, temperature distribution in body is specified
2. Boundary condition1) Prescribed surface temperature2) Prescribed heat flux incident on the surface3) Prescribed heat transfer coefficient at the surface
Differential Equation of Heat Conduction
1. Prescribed surface temperatureFor example, let the surface be a plane face (at x = L)
T0At x = L T = T0
2. Prescribed heat flux incident on the surface
0Lx A
q
x
Tk
3. Prescribed heat transfer coefficient at the surface
)TT(hx
Tk fLx
Lx
0A
q
hTf
x
x directionOpposite x direction
x = L
Differential Equation of Heat Conduction
Heat generation
1. Electrical conductor2. Nuclear fuel element3. Setting of concrete4. Agricultural product
Differential Equation of Heat Conduction
Heat generation (Infinite slab)
Heat generated at uniform rate
x2b
q
We would like to know the temperature distribution at steady state, or equation describing this temperature distribution.
Differential Equation of Heat ConductionLet put down differential equation (k is constant)
t
TCq
z
T
y
T
x
Tk p2
2
2
2
2
2
• One direction, infinite slab • Steady state
k
q
dx
Td2
2
Differential Equation of Heat ConductionBoundary conditions
x2b
q
Tf h
hTf
)TT(hdx
dTk fbx
bx
)TT(hdx
dTk bxf
bx
0dx
dT
0x
Differential Equation of Heat ConductionSolving
1Cxk
q
dx
dT
Integrating it twice
21
2
CxC2
x
k
qT
k
q
dx
Td2
2
Differential Equation of Heat ConductionTemperature distribution
h
bq)xb(
k2
qTT 22f
h
1
k2
bbqTT fmax
Maximum Temperature (x = 0)
Differential Equation of Heat Conduction
Heat generation (Infinite solid cylinder)
t
TCq
Tk
r
1
r
Tkr
rr
1p2
rk
q
dr
dTr
dr
d
Reducing the form to
0dr
dT
0r
fRrRr
TThdr
dTk
Boundary condition
Differential Equation of Heat Conduction
Integrating differential equation and using boundary conditions, we get
h2
Rq)rR(
k4
qTT 22f
h
1
k2
R
2
RqTT fmax
Differential Equation of Heat ConductionProblemA nuclear fuel element is in the form of a long solid rod (k = 0.85 W/mK) of diameter 14 mm. It generates heat at the uniform rate of 0.45 x 108 W/m3 because of nuclear fusion. The heat is transferred to pressurized cooling water at 300 ºC and the surface heat transfer coefficient is 4500 W/m2K. Calculate the maximum temperature in fuel rod in the steady state.
Differential Equation of Heat ConductionProblemHeat generated in a slab of 120 mm thickness with a conductivity of 200 W/mK at a rate of 106 W/m3. Determine the temperature at the mid and quarter planes if the surface of the solid on both sides are exposed to convection at 30ºC with a convective coefficient of 500W/m2K. Also find heat flow rate at these planes and the temperature gradients at these planes
Differential Equation of Heat Conduction
Differential Equation of Heat Conduction
Differential Equation of Heat Conduction
Differential Equation of Heat Conduction
Differential Equation of Heat Conduction
Differential Equation of Heat Conduction