164
Analytical Heat Transfer
determine the “x” variations of the heat transfer coefficient and
wall temperature for the case of laminar flow over a flat plate with
a uniform wall heat flux (q/A) w . Compare the result of the heat
transfer coefficient to the solution obtained in the textbook for the
case of uniform wall temperature.
7.16. Laminar air flow at 1 atm pressure over a flat plate at a uniform
T w . Assume that U = 3 m/s, T = 20 ◦ C, and T w = 100 ◦
∞
∞
C, and
determine local u and T at x = 3 cm and y = 0.2 cm by using the
similarity solution. Also determine the heat transfer coefficient at
the same x = 3 cm location. Will the heat transfer coefficient be
increased or decreased with increasing x for the same U ∞ ? Why?
Will the Nusselt number be increased or decreased with increasing
x for the same U ∞ ? Why?
7.17. Use similarity solutions: air at 300 K and 1 atm flows along a flat
plate at 5 m/s. At a location 0.2 m from the leading edge, plot the
u and v velocity profiles using the exact solution to the Blasius
equation Also, determine the boundary-layer thickness, if it is
defined as the location where u = 0.99u ∞ . Plot the temperature
profile and determine the thermal boundary thickness if the plate
temperature is 500 K.
7.18. Using integral method solutions: air at 300 K and 1 atm pressure flows along a flat plate at 5 m/s. For x < 10 cm, T s = 300 K,
whereas for 10 cm < x < 20 cm, T s = 500 K.
Calculate the heat loss from the plate and compare the result
with the heat loss if the plate were isothermal at 500 K. Assume
that there exists a laminar boundary layer. Compare and discuss
the two cases.
7.19. Consider a boundary-layer flow over a flat plate.
a. Using the integral method derive the continuity equation in
the boundary layer for flow over a flat plate.
b. Using the integral method derive the energy conservation
equation in the boundary layer for flow over a flat plate.
c. Using appropriate BCs and boundary-layer theory, show that
for an inviscid fluid,
Nu = 0.564 Pe 1/2
where Nu is the Nusselt number and Pe is the Peclet number.
(Hint: Use a plug flow model for velocity.)
7.20. Consider a fluid approaching the leading edge of a flat plate
with uniform velocity and temperature profiles U ∞ and T ∞ . The
flat plate is frictionless and is held at a constant heat flux of q ""
s .
The temperature profile at a distance x from the leading edge is
given by
T = a + a y + a y 2 + a y 3
0
1
2
3
a. Using appropriate BCs evaluate a 0 , a 1 , a 2 , and a 3 .
b. Qualitatively sketch velocity and temperature profiles at a
distance x 1 and x 2 , respectively, from the leading edge.
Analytical Heat Transfer
determine the “x” variations of the heat transfer coefficient and
wall temperature for the case of laminar flow over a flat plate with
a uniform wall heat flux (q/A) w . Compare the result of the heat
transfer coefficient to the solution obtained in the textbook for the
case of uniform wall temperature.
7.16. Laminar air flow at 1 atm pressure over a flat plate at a uniform
T w . Assume that U = 3 m/s, T = 20 ◦ C, and T w = 100 ◦
∞
∞
C, and
determine local u and T at x = 3 cm and y = 0.2 cm by using the
similarity solution. Also determine the heat transfer coefficient at
the same x = 3 cm location. Will the heat transfer coefficient be
increased or decreased with increasing x for the same U ∞ ? Why?
Will the Nusselt number be increased or decreased with increasing
x for the same U ∞ ? Why?
7.17. Use similarity solutions: air at 300 K and 1 atm flows along a flat
plate at 5 m/s. At a location 0.2 m from the leading edge, plot the
u and v velocity profiles using the exact solution to the Blasius
equation Also, determine the boundary-layer thickness, if it is
defined as the location where u = 0.99u ∞ . Plot the temperature
profile and determine the thermal boundary thickness if the plate
temperature is 500 K.
7.18. Using integral method solutions: air at 300 K and 1 atm pressure flows along a flat plate at 5 m/s. For x < 10 cm, T s = 300 K,
whereas for 10 cm < x < 20 cm, T s = 500 K.
Calculate the heat loss from the plate and compare the result
with the heat loss if the plate were isothermal at 500 K. Assume
that there exists a laminar boundary layer. Compare and discuss
the two cases.
7.19. Consider a boundary-layer flow over a flat plate.
a. Using the integral method derive the continuity equation in
the boundary layer for flow over a flat plate.
b. Using the integral method derive the energy conservation
equation in the boundary layer for flow over a flat plate.
c. Using appropriate BCs and boundary-layer theory, show that
for an inviscid fluid,
Nu = 0.564 Pe 1/2
where Nu is the Nusselt number and Pe is the Peclet number.
(Hint: Use a plug flow model for velocity.)
7.20. Consider a fluid approaching the leading edge of a flat plate
with uniform velocity and temperature profiles U ∞ and T ∞ . The
flat plate is frictionless and is held at a constant heat flux of q ""
s .
The temperature profile at a distance x from the leading edge is
given by
T = a + a y + a y 2 + a y 3
0
1
2
3
a. Using appropriate BCs evaluate a 0 , a 1 , a 2 , and a 3 .
b. Qualitatively sketch velocity and temperature profiles at a
distance x 1 and x 2 , respectively, from the leading edge.