162
Analytical Heat Transfer
inside the thermal boundary layer as T = a + by, determine
the local Nusselt number distribution along the flat plate (i.e.,
Nu versus X).
d. Consider a uniform suction through the wall (i.e., v = −v 0 ),
and comment on whether the local Nusselt number distribution along the flat plate will be higher, the same, or lower than
those without boundary-layer suction? Explain why.
7.10. Consider a laminar air flow (at U ∞ , T ∞ ) over a flat plate at a
uniform wall heat flux q ""
w .
a. Based on the integral method, can you remember to write
down the momentum and energy integral equations? (if you
cannot remember, please derive them).
b. Assuming a linear velocity profile inside the boundary layer
as u = a + b y, derive the velocity boundary-layer thickness
distribution along the flat plate (i.e., δ versus X).
c. Assuming a linear temperature profile inside the thermal
boundary layer as T = c + d y, derive the thermal boundarylayer thickness distribution along the flat plate (i.e., δ T versus
X).
d. Based on (b) and (c), determine the local Nusselt number
distribution along the flat plate (i.e., Nu versus X).
e. Consider a uniform wall temperature (T w ) as a thermal BC at
the wall, and comment on whether the local Nusselt number
distribution along the flat plate will be higher, the same, or
lower than those of uniform wall heat flux as a thermal BC?
Explain why.
f. Consider a uniform suction through the wall (i.e., v = −v o ),
and comment on whether the local Nusselt number distribution along the flat plate will be higher, the same, or lower than
those without boundary-layer suction? Explain why.
7.11. The similarity method for laminar flow over a flat plate:
U ∞ is the free-stream velocity, T ∞ is the free-stream temperature, and T W is the flat plate wall temperature.
a. Write down the similarity variable, differential equations,
and BCs for velocity and temperature, respectively. Then,
determine velocity (u) and temperature (if P r = 1) at
(x, y) = (2 cm, 1/3δ),
= (4 cm, 1/3δ),
= (6 cm, 1/3δ).
b. At any given x, if U ∞ increases, the friction factor will be
increased or decreased. Why? How about shear stress? At any
given U ∞ , if x increases, the heat transfer coefficient will be
increased or decreased. Why? How about heat transfer rate?
7.12. The similarity method for laminar flow over a flat plate:
U ∞ – free-stream velocity, T ∞ – free-stream temperature, and
T W – flat plate wall temperature.
Analytical Heat Transfer
inside the thermal boundary layer as T = a + by, determine
the local Nusselt number distribution along the flat plate (i.e.,
Nu versus X).
d. Consider a uniform suction through the wall (i.e., v = −v 0 ),
and comment on whether the local Nusselt number distribution along the flat plate will be higher, the same, or lower than
those without boundary-layer suction? Explain why.
7.10. Consider a laminar air flow (at U ∞ , T ∞ ) over a flat plate at a
uniform wall heat flux q ""
w .
a. Based on the integral method, can you remember to write
down the momentum and energy integral equations? (if you
cannot remember, please derive them).
b. Assuming a linear velocity profile inside the boundary layer
as u = a + b y, derive the velocity boundary-layer thickness
distribution along the flat plate (i.e., δ versus X).
c. Assuming a linear temperature profile inside the thermal
boundary layer as T = c + d y, derive the thermal boundarylayer thickness distribution along the flat plate (i.e., δ T versus
X).
d. Based on (b) and (c), determine the local Nusselt number
distribution along the flat plate (i.e., Nu versus X).
e. Consider a uniform wall temperature (T w ) as a thermal BC at
the wall, and comment on whether the local Nusselt number
distribution along the flat plate will be higher, the same, or
lower than those of uniform wall heat flux as a thermal BC?
Explain why.
f. Consider a uniform suction through the wall (i.e., v = −v o ),
and comment on whether the local Nusselt number distribution along the flat plate will be higher, the same, or lower than
those without boundary-layer suction? Explain why.
7.11. The similarity method for laminar flow over a flat plate:
U ∞ is the free-stream velocity, T ∞ is the free-stream temperature, and T W is the flat plate wall temperature.
a. Write down the similarity variable, differential equations,
and BCs for velocity and temperature, respectively. Then,
determine velocity (u) and temperature (if P r = 1) at
(x, y) = (2 cm, 1/3δ),
= (4 cm, 1/3δ),
= (6 cm, 1/3δ).
b. At any given x, if U ∞ increases, the friction factor will be
increased or decreased. Why? How about shear stress? At any
given U ∞ , if x increases, the heat transfer coefficient will be
increased or decreased. Why? How about heat transfer rate?
7.12. The similarity method for laminar flow over a flat plate:
U ∞ – free-stream velocity, T ∞ – free-stream temperature, and
T W – flat plate wall temperature.