-
1
Thermal and Fluids
in Architectural Engineering
11. Conduction heat transfer
Jun-Seok Park, Dr. Eng., Prof.
Dept. of Architectural Engineering
Hanyang Univ.
-
Where do we learn in this chaper
1. Introduction
2.The first law
3.Thermal resistances
4. Fundamentals of fluid mechanics
5. Thermodynamics
6. Application
7.Second law
8. Refrigeration,
heat pump, and
power cycle
9. Internal flow
10. External flow
11. Conduction
12. Convection
14. Radiation
13. Heat Exchangers15. Ideal Gas Mixtures
and Combustion
-
11.1 Introduction
11.2 Heat Conduction Equation
11.3 Steady one-dimensional Conduction
11.4 Steady multidimensional Conduction
11.6 One-dimensional Transient Conduction
11. Conduction Heat Transfer
-
11.1 Introduction
□ Conduction refers to the transport of energy in solids,
liquids, and gases due to a temperature gradient
- The physical mechanism is atomic or molecular activity
□ Conduction heat transfer is governed by Fourier’s law and that
use of the law to determine
- heat flux depends on temperature varies within the system
□ Fourier’s law is applicable to transient, multidimensional
conduction in complex geometries
W - Q ΔE
-
11.1 Introduction
□ Fourier’s law for heat flux in conduction
W - Q ΔE
dz
dTkq
dy
dTkq
dx
dTkq
kqjqiqq
dx
dTk
A
Qq
dx
dTAkQ
zyx
zyx
"""
""""
"
; ;
vector)isflux (heat dimension multiIn
; Flux Heat
; FlowHeat
-
11.2 Heat Conduction Equation (Heat diffusion)
□ The first law in a system
W - Q ΔE
WQdt
dE
ΔUΔPEΔKEΔE
Q - WΔE
)(
Source: Fundamental of Heat and mass transfer, Wiley, pp61
-
11.2 Heat Conduction Equation (Heat diffusion)
□ The first law in a system
W - Q ΔE
)generation(heat 3)
flux)heat net ( )2
)( )1
W
QQQQ
dt
dTczyx
dt
dTVc
dt
dTmc
dt
dum
dt
dU
dt
dE
WQdt
dE
zyx
ppp
-
11.2 Heat Conduction Equation (Heat diffusion)
□ Heat flux in a system
W - Q ΔE
)()(
)()(
)()(
flux)heat net ( )2
""
""
""
yxqyxq
zxqzxq
zyqzyq
QQQQ
zzz
zz
yyy
yy
xxx
xx
zyx
dx
d z
dy
qx qx+dx
qz
qz+dz
qy
qy+dy
EgEst
Control Volume
-
11.2 Heat Conduction Equation (Heat diffusion)
□ Heat generation in a system
W - Q ΔE
) W/mrate, generationheat :(
)generation(heat )3
3"'
"'
x
x
q
zyxqW
W
dx
d z
dy
qx qx+dx
qz
qz+dz
qy
qy+dy
EgEst
Control Volume
-
11.2 Heat Conduction Equation (Heat diffusion)
□ Conduction Equation (Heat diffusion equation)in a system
W - Q ΔE
"'
""""""
"'""
""""
)()(
)()( )()()(
xzz
zz
zyyx
yy
xxx
xx
p
xzz
zz
z
yyy
yy
xxx
xxp
qz
qq
y
qq
x
qq
dt
dTc
zyxqyxqyxq
zxqzxqzyqzyqdt
dTczyx
WQdt
dE
-
11.2 Heat Conduction Equation (Heat diffusion)
□ Conduction Equation (Heat diffusion equation)in a system
W - Q ΔE
) ; ;("""
"'
"'"""
"'
""""""
dz
dTkq
dy
dTkq
dx
dTkq
qz
Tk
zy
Tk
yx
Tk
xdt
dTc
qz
q
y
q
x
q
dt
dTc
qz
qq
y
qq
x
qq
dt
dTc
zyx
p
zyxp
zzz
zzyy
xy
yxx
xx
x
p
-
11.2 Heat Conduction Equation (Heat diffusion)
□ Boundary and Initial Condition in a system
- To determine the temperature distribution in a system,
it is necessary to solve the appropriate form of the
equation.
- The solution depends on the physical conditions existing
at
the boundaries, if the situation is time dependent,
on conditions at some initial time
W - Q ΔE
"'xp q
z
Tk
zy
Tk
yx
Tk
xdt
dTc
-
11.2 Heat Conduction Equation (Heat diffusion)
□ Boundary and Initial Condition in a system
- Because the equation is second order in the coordinates,
two boundary conditions must be expressed
- Because the equation is first order in time,
the initial condition, must be specified
W - Q ΔE
"'xp q
z
Tk
zy
Tk
yx
Tk
xdt
dTc
-
11.3 Steady One-dimensional Conduction
□Assumption for steady one-dimensional Conduction
- In a one-dimensional system, temperature gradients exist
along only a single coordinate direction
- Heat transfer occurs exclusively in that direction
- Steady-state conditions means that the temperature at each
point is independent of time
W - Q ΔE
-
11.3 Steady One-dimensional Conduction
□ The equation in steady one-dimensional Conduction
- For steady-state conditions with no source or sink
of energy within the system, the appropriate form of
the heat equation is as below
- The equation may be integrated twice to obtain
the general solution.
W - Q ΔE
0 "'
x
Tk
xq
z
Tk
zy
Tk
yx
Tk
xdt
dTc xp
12)( CxCxT
-
11.3 Steady One-dimensional Conduction
□ If there is a heat source (generator) in the system
- The equation may be integrated twice to obtain
the general solution. (if the heat source is constant)
W - Q ΔE
0 "'"'
q
x
Tk
xq
z
Tk
zy
Tk
yx
Tk
xdt
dTc xp
122"')( CxCxqxT x
-
11.4 Steady Multidimensional Conduction
□ The number of boundary and initial conditions areneeded to
solve temperature distribution in a system.
□ For example, two dimensional transient conduction, there are
four boundary conditions and one initial
condition are needed
□Many cases of multidimensional conduction, numerical approaches
is used to solve temperature
distribution
W - Q ΔE
-
11.6 One-dimensional Transient Conduction
W - Q ΔE
□Assumption for steady one-dimensional Conduction
- In a one-dimensional system, temperature gradients exist
along only a single coordinate direction
- Heat transfer occurs exclusively in that direction
- The temperature at each point in the system is changed
dependent of time
-
11.6 One-dimensional Transient Conduction
W - Q ΔE
□ The equation of one-dimensional transient Conduction
- To solve the equation, one initial temperature condition,
and
two boundary condition is needed.
"'xq
x
Tk
xdt
dTcq
z
Tk
zy
Tk
yx
Tk
xdt
dTc pxp
"'
),,,,(),( pfi ckTTftxT
