Modern Engineering Thermodynamics

128 CHAPTER 4: The First Law of Thermodynamics and Energy Transport Mechanisms
spontaneously moving down a potential gradient (such as from high to a low temperature) and the reverse cannot
spontaneously occur, no heat transfer process can be reversed in any way whatsoever.Therefore,allfinite
heat transfer processes are irreversible.
4.13 HEAT TRANSFER MODES
Heat transfer is such a large and important mechanical engineering topic that most curricula have at least one
required course in it. Heat transfer equations are always cast as heat transfer rate (i.e., _Q) equations. To determinetheamountofheatenergytransport
that occurs as a system undergoes a process from one equilibrium
statetoanotheryoumustintegrate _Q over the time interval of the process, or 1 Q 2 =
Z 2
1
_Q dt. Normally, we
choose processes in which _Q is constant in time so that the integral becomes simply 1 Q 2 = _Q ðt 2 − t 1 Þ = _QðΔtÞ,
where Δt is the time required for the process to occur.
Historically, the field has been divided into three heat transfer modes: conduction, convection, and radiation.
These three modes are briefly described next.
4.13.1 Conduction
The basic equation of conduction heat transfer is Fourier’s law:
_Q cond = −k t A dT
dx
(4.73)
where _Q cond is the conduction heat transfer rate, k t is the thermal conductivity of the material, A is the crosssectional
area normal to the heat transfer direction, and dT/dx is the temperature gradient in the direction of
heat transfer. The algebraic sign of this equation is such that a positive _Q cond always corresponds to heat transfer
in the positive x direction, and a negative _Q cond always corresponds to heat transfer in a negative x direction.
Since this is not the same sign convention adopted earlier in this text, the sign of the values calculated from
Fourier’s law may have to be altered to produce a positive when it enters a system and a negative when it leaves
asystem.
For steady conduction heat transfer through a plane wall (Figure 4.20), Fourier’s law can be integrated to give
ð _Q cond Þ plane = − k t A T
2 − T 1
(4.74)
x 2 − x 1
and for steady conduction heat transfer through a hollow cylinder of length L, Fourier’s law can be integrated to give
T
ð _Q cond Þ cylinder
= −2πLk inside − T outside
t
(4.75)
lnðr inside /r outside
Table 4.7 gives thermal conductivity values for various materials.
Q
T 1
Q
Q
r i
Q
T 2
T i
r o
x 1
x 2
T o
x = 0
(a) Plane wall
(b) Cylindrical and spherical
FIGURE 4.20
Thermal conduction notation in plane, cylindrical, and spherical coordinates.