101
Transient Heat Conduction
b. Approximately plot the temperature profiles for the following
BCs:
i. Constant q” at the surface.
ii. Constant temperature at the surface, T s (0,t) > T i .
iii. Constant fluid temperature T f and h, T f > T i .
4.6. A semiinfinite carbon steel block is initially at a uniform temperature T i and its surfaces is suddenly exposed to a constant
irradiation flux q "" , simultaneously to an ambient air of h, T ∞ .
a. Determine the temperature history in the solid and sketch the
temperature profiles in the solid assuming T i = T ∞ . Analytical
method.
b. If a semiinfinite pure copper block will be operated at the same
condition, state that the time required for the surface of the
block to reach T s will be longer or shorter than that of the steel
block. Why?
4.7. A liquid confined in a half-space x > 0 is initially at a temperature
T i which is higher than its freezing temperature T m . For times
t > 0, the surfaces at x = 0 are subjected to the following BCs.
Plot the temperature history both in the liquid and the solid.
a. Constant heat flux is q "" ; this q "" is removed away from the
surface.
b. Constant surface temperature T o , T o < T m .
c. If natural convection takes place in the liquid region with
a constant h, plot the temperature profiles and freezing distance δ with time in both case (a) and case (b). Explain the
difference.
4.8. Asolid simulated as a half-space, x > 0, is initially at a temperature
T i which is equal to its melting temperature T m . For time t > 0,
the surface at x = 0 is subjected to the following BCs. Plot the
temperature history both in liquid and solid:
a. Constant heat flux is q "" ; this q "" is applied to the surface.
b. Constant surface temperature T s , T s > T m .
c. Convection to the surface with heating fluid at T ∞ , h.
4.9. Refer to Equation 4.14, and solve the temperature distributions
for the 3-D block with the following BCs:
(1) x = ±a, T = T s
y = ±b, T = T s
z = ±c, T = T s
(2) x = ±a, (−k∂T(±a, t)/∂x) = h[T(±a, t) − T ∞ ]
y = ±b, (−k∂T(±b, t)/∂y) = h[T(±b, t) − T ∞ ]
z = ±c, (−k∂T(±c, t)/∂z) = h[T(±c, t) − T ∞ ]
4.10. A semiinfinite solid is initially at uniform temperature T s . The
surface of the semiinfinite solid is suddenly exposed to a constant
temperature T w .
a. Write the governing equations and the BCs.
Transient Heat Conduction
b. Approximately plot the temperature profiles for the following
BCs:
i. Constant q” at the surface.
ii. Constant temperature at the surface, T s (0,t) > T i .
iii. Constant fluid temperature T f and h, T f > T i .
4.6. A semiinfinite carbon steel block is initially at a uniform temperature T i and its surfaces is suddenly exposed to a constant
irradiation flux q "" , simultaneously to an ambient air of h, T ∞ .
a. Determine the temperature history in the solid and sketch the
temperature profiles in the solid assuming T i = T ∞ . Analytical
method.
b. If a semiinfinite pure copper block will be operated at the same
condition, state that the time required for the surface of the
block to reach T s will be longer or shorter than that of the steel
block. Why?
4.7. A liquid confined in a half-space x > 0 is initially at a temperature
T i which is higher than its freezing temperature T m . For times
t > 0, the surfaces at x = 0 are subjected to the following BCs.
Plot the temperature history both in the liquid and the solid.
a. Constant heat flux is q "" ; this q "" is removed away from the
surface.
b. Constant surface temperature T o , T o < T m .
c. If natural convection takes place in the liquid region with
a constant h, plot the temperature profiles and freezing distance δ with time in both case (a) and case (b). Explain the
difference.
4.8. Asolid simulated as a half-space, x > 0, is initially at a temperature
T i which is equal to its melting temperature T m . For time t > 0,
the surface at x = 0 is subjected to the following BCs. Plot the
temperature history both in liquid and solid:
a. Constant heat flux is q "" ; this q "" is applied to the surface.
b. Constant surface temperature T s , T s > T m .
c. Convection to the surface with heating fluid at T ∞ , h.
4.9. Refer to Equation 4.14, and solve the temperature distributions
for the 3-D block with the following BCs:
(1) x = ±a, T = T s
y = ±b, T = T s
z = ±c, T = T s
(2) x = ±a, (−k∂T(±a, t)/∂x) = h[T(±a, t) − T ∞ ]
y = ±b, (−k∂T(±b, t)/∂y) = h[T(±b, t) − T ∞ ]
z = ±c, (−k∂T(±c, t)/∂z) = h[T(±c, t) − T ∞ ]
4.10. A semiinfinite solid is initially at uniform temperature T s . The
surface of the semiinfinite solid is suddenly exposed to a constant
temperature T w .
a. Write the governing equations and the BCs.