103
Transient Heat Conduction
generated uniformly in the wall to prevent heat transfer from
the fluid at T ∞1 to the wall. Sketch the instantaneous temperature distributions in the fluids at several different times near
the wall and in the wall, before and after heat is generated uniformly in the wall to prevent heat transfer from the hot fluid
to the wall, until a steady state is reached.
c. Consider a large wall with a thickness of 2L its surfaces maintained at T 1 and T 2 (T 2 = T 1 ). Heat is generated uniformly in
the wall. Beginning with the steady-state, 1-D, heat conduction
equation and appropriate BCs (T = T 1 at x = −L and T = T 2
at x = +L), derive an expression for the steady-state 1-D temperature distribution in the wall, T(x). Using the expression,
show that the rate of heat generation is equal to the sum of the
rates of heat transfer from the two surfaces.
4.15. The plane wall has constant properties and no internal generation
and is initially at a uniform temperature T i . Suddenly, the surface
x = L is exposed to a heating process with a fluid at T ∞ having a
convection coefficient h. At the same instant, the electrical heater
""
is energized providing a constant heat flux q 0 at x = 0.
a. On T − x coordinates, sketch the temperature distributions for
the following conditions: initial condition (t < 0), steady-state
condition (t → ∞), and for two intermediate times.
""
b. On q − x coordinates, sketch the heat flux corresponding to
x
the four temperature distributions of (a).
""
c. On q − t coordinates, sketch the heat flux at the locations x = 0
x
""
""
and x = L. That is, show qualitatively how q (0,t) and q (L, t)
x
x
vary with time.
d. Derive an expression for the steady-state temperature at the
""
heater surface, T(0, ∞), in terms of q 0 , T ∞ , k, h, and L.
4.16. The plane wall has constant properties and a uniform internal
generation of q ˙(W/m 3 ) that activates only when the electric heater
is energized. The wall is initially at a uniform temperature T i .
Suddenly, the surface x = L is exposed to a cooling process with a
fluid at T ∞ having a convection coefficient h. At the same instant,
""
the electrical heater is energized providing a constant heat flux q 0
at x = 0.
a. On T − x coordinates, sketch the temperature distributions for
the following conditions: initial condition (t ≤ 0), steady-state
condition (t → ∞), and for two intermediate times.
""
b. On q − x coordinate, sketch the heat flux corresponding to
x
the four temperature distributions of (a).
""
c. On q − t coordinates, sketch the heat flux at the locations x = 0
x
""
""
and x = L. That is, show qualitatively how q (0,t) and q (L,t)
x
x
vary with time.
d. Derive an expression for the steady-state temperature
""
at the heater surface, T(0,∞), in terms of q 0 , q ˙, T ∞ , k, h,
and L.
4.17. A 10 m-long 2 cm-diameter copper rod is immersed in a heating bath at a uniform temperature of 100 ◦ C. This rod is suddenly
Transient Heat Conduction
generated uniformly in the wall to prevent heat transfer from
the fluid at T ∞1 to the wall. Sketch the instantaneous temperature distributions in the fluids at several different times near
the wall and in the wall, before and after heat is generated uniformly in the wall to prevent heat transfer from the hot fluid
to the wall, until a steady state is reached.
c. Consider a large wall with a thickness of 2L its surfaces maintained at T 1 and T 2 (T 2 = T 1 ). Heat is generated uniformly in
the wall. Beginning with the steady-state, 1-D, heat conduction
equation and appropriate BCs (T = T 1 at x = −L and T = T 2
at x = +L), derive an expression for the steady-state 1-D temperature distribution in the wall, T(x). Using the expression,
show that the rate of heat generation is equal to the sum of the
rates of heat transfer from the two surfaces.
4.15. The plane wall has constant properties and no internal generation
and is initially at a uniform temperature T i . Suddenly, the surface
x = L is exposed to a heating process with a fluid at T ∞ having a
convection coefficient h. At the same instant, the electrical heater
""
is energized providing a constant heat flux q 0 at x = 0.
a. On T − x coordinates, sketch the temperature distributions for
the following conditions: initial condition (t < 0), steady-state
condition (t → ∞), and for two intermediate times.
""
b. On q − x coordinates, sketch the heat flux corresponding to
x
the four temperature distributions of (a).
""
c. On q − t coordinates, sketch the heat flux at the locations x = 0
x
""
""
and x = L. That is, show qualitatively how q (0,t) and q (L, t)
x
x
vary with time.
d. Derive an expression for the steady-state temperature at the
""
heater surface, T(0, ∞), in terms of q 0 , T ∞ , k, h, and L.
4.16. The plane wall has constant properties and a uniform internal
generation of q ˙(W/m 3 ) that activates only when the electric heater
is energized. The wall is initially at a uniform temperature T i .
Suddenly, the surface x = L is exposed to a cooling process with a
fluid at T ∞ having a convection coefficient h. At the same instant,
""
the electrical heater is energized providing a constant heat flux q 0
at x = 0.
a. On T − x coordinates, sketch the temperature distributions for
the following conditions: initial condition (t ≤ 0), steady-state
condition (t → ∞), and for two intermediate times.
""
b. On q − x coordinate, sketch the heat flux corresponding to
x
the four temperature distributions of (a).
""
c. On q − t coordinates, sketch the heat flux at the locations x = 0
x
""
""
and x = L. That is, show qualitatively how q (0,t) and q (L,t)
x
x
vary with time.
d. Derive an expression for the steady-state temperature
""
at the heater surface, T(0,∞), in terms of q 0 , q ˙, T ∞ , k, h,
and L.
4.17. A 10 m-long 2 cm-diameter copper rod is immersed in a heating bath at a uniform temperature of 100 ◦ C. This rod is suddenly