102
Analytical Heat Transfer
b. Nondimensionalize the governing equations and BCs by
appropriate choice of temperature variable, length scale, and
timescale.
4.11. An infinite body of cold liquid initially at uniform temperature
T s is brought in contact with a heated horizontal wall of infinite
length maintained at a constant temperature (T w ). It is expected
that after infinite time the liquid temperature profile will be linear
within a thermal boundary layer of thickness δ. Neglect gravity or
body forces and liquid convection and assume that heat transfer
in the liquid is by conduction only.
a. Write the governing equations and the BCs.
b. Nondimensionalize the governing equations and BCs by
appropriate choice of temperature variable, length scale, and
timescale.
c. Solve the governing equations to obtain the transient temperature profile. Verify if the solution for the transient
temperature profile satisfies the linear temperature profile when steady-state conditions are reached (at infinite
time).
4.12. A plate is initially at temperature T i when laid on an insulated
surface and cooled by air flow at temperature T ∞ , with the heat
transfer coefficient h. The length of the plate is l, width w, and
thickness b (l » w, l » b, w » b). The thermal diffusivity of the
plate is α and the thermal conductivity of the plate material is k.
The viscosity of air is μ and density is ρ. Estimate the time required
for the bottom surface of the plate to cool to T b , when
a. The plate material is made of copper (the conduction resistance
in the slab is negligible).
b. The plate material is made of plastic (the convection resistance
on the slab is negligible).
4.13. A long metal plane wall with a thickness of 2L is initially at temperature T i and suddenly both sides are heated by a convection
fluid flow at temperature T ∞ . Outline the procedures and solve
the temperature distributions in the plane wall and sketch the
temperature profiles in the plane wall during the heating process
for two different cases.
a. Fluid flow is natural convection air.
b. Fluid flow is forced convection water.
4.14. a. Consider a large wall, separating two fluids at T ∞1 and T ∞2
(T ∞2 < T ∞1 ). To prevent heat transfer from the hot fluid at
T ∞1 to the wall, a thin foil guard heater (of negligible thickness) on the surface of the wall exposed to the hot fluid at
T ∞1 is used to raise the surface temperature to T ∞1 . Sketch
the instantaneous temperature distributions at several different times in the fluids near the wall and in the wall, before and
after the surface temperature is raised with the thin foil guard
heater, until a steady state is reached.
b. Consider a large wall, separating two fluids at T ∞1 and T ∞2
(T ∞2 < T ∞1 ). Instead of the thin foil guard heater in (a), heat is
Analytical Heat Transfer
b. Nondimensionalize the governing equations and BCs by
appropriate choice of temperature variable, length scale, and
timescale.
4.11. An infinite body of cold liquid initially at uniform temperature
T s is brought in contact with a heated horizontal wall of infinite
length maintained at a constant temperature (T w ). It is expected
that after infinite time the liquid temperature profile will be linear
within a thermal boundary layer of thickness δ. Neglect gravity or
body forces and liquid convection and assume that heat transfer
in the liquid is by conduction only.
a. Write the governing equations and the BCs.
b. Nondimensionalize the governing equations and BCs by
appropriate choice of temperature variable, length scale, and
timescale.
c. Solve the governing equations to obtain the transient temperature profile. Verify if the solution for the transient
temperature profile satisfies the linear temperature profile when steady-state conditions are reached (at infinite
time).
4.12. A plate is initially at temperature T i when laid on an insulated
surface and cooled by air flow at temperature T ∞ , with the heat
transfer coefficient h. The length of the plate is l, width w, and
thickness b (l » w, l » b, w » b). The thermal diffusivity of the
plate is α and the thermal conductivity of the plate material is k.
The viscosity of air is μ and density is ρ. Estimate the time required
for the bottom surface of the plate to cool to T b , when
a. The plate material is made of copper (the conduction resistance
in the slab is negligible).
b. The plate material is made of plastic (the convection resistance
on the slab is negligible).
4.13. A long metal plane wall with a thickness of 2L is initially at temperature T i and suddenly both sides are heated by a convection
fluid flow at temperature T ∞ . Outline the procedures and solve
the temperature distributions in the plane wall and sketch the
temperature profiles in the plane wall during the heating process
for two different cases.
a. Fluid flow is natural convection air.
b. Fluid flow is forced convection water.
4.14. a. Consider a large wall, separating two fluids at T ∞1 and T ∞2
(T ∞2 < T ∞1 ). To prevent heat transfer from the hot fluid at
T ∞1 to the wall, a thin foil guard heater (of negligible thickness) on the surface of the wall exposed to the hot fluid at
T ∞1 is used to raise the surface temperature to T ∞1 . Sketch
the instantaneous temperature distributions at several different times in the fluids near the wall and in the wall, before and
after the surface temperature is raised with the thin foil guard
heater, until a steady state is reached.
b. Consider a large wall, separating two fluids at T ∞1 and T ∞2
(T ∞2 < T ∞1 ). Instead of the thin foil guard heater in (a), heat is