158
Analytical Heat Transfer
have differences, if any. You do not need to solve the above
equations.
b. For a given problem (i.e., U ∞ , T ∞ , ρ, μ, T w , P r are given),
explain briefly how to determine the local velocity u(y) and
temperature T(y) at a specified location x, for both water and
air, from the sketches in (a)?
7.2. Consider a steady, incompressible, low-speed 2-D laminar
boundary-layer flow (at U ∞ , T ∞ ) over a flat plate at a uniform
wall temperature T w or at a uniform wall heat flux q ""
w , respectively. Assume that there exist no body force and constant thermal
and fluid properties.
a. Based on the approximate integral method, assuming a uniform velocity profile inside the boundary layer, that is, u =
U ∞ , and assuming a linear temperature profile inside the
thermal boundary layer as T = a + b × y, determine the local
thermal boundary-layer growth along the flat plate (i.e., δ T
versus x) at a uniform wall temperature T w condition. (Note:
a and b are unknown constants that need to be determined.)
b. Based on (a), determine the local Nusselt number distribution along the flat plate (i.e., Nu versus x) at a uniform wall
temperature.
7.3. Consider a steady, incompressible, low-speed 2-D laminar
boundary-layer flow (at U ∞ , T ∞ ) over a flat plate at a uniform wall temperature T W . Assume that there exist no body
force and constant thermal and fluid properties. The similarity
momentum and energy equations are listed here for reference:
f """ + (1/2)ff "" = 0 and θ "" + (1/2)P r f θ " = 0.
a. Sketch both the velocity and the thermal boundary-layer thickness distribution for both water and liquid metal, respectively,
flowing over the flat plate? (i.e., sketch δ versus x and δ T versus
x on the same plot for water, and δ versus x and δ T versus x
on the same plot for the liquid metal). Explain why they have
differences, if any.
b. Define the similarity variable (η), the similarity function for
temperature (θ), and the derivative of the similarity function
for velocity (f " )? Write the BCs which can be used for this
problem?
c. From the Blasius solution of the above similarity equations,
sketch the relations between the similarity functions (f " , θ) and
the similarity variable (η) for both water and liquid metal (i.e.,
sketch f " versus η for both water and liquid metal on the same
plot; and θ versus η for both water and liquid metal on the
same plot). Explain why they have differences, if any. You do
not need to solve the above equations.
d. For a given problem (i.e., U ∞ , T ∞ , ρ, μ, T w , P r are given),
explain briefly how to determine the local velocity u(y) and
temperature T(y) at a specified location x, for both water and
liquid metal, from the sketches in (c)?
Analytical Heat Transfer
have differences, if any. You do not need to solve the above
equations.
b. For a given problem (i.e., U ∞ , T ∞ , ρ, μ, T w , P r are given),
explain briefly how to determine the local velocity u(y) and
temperature T(y) at a specified location x, for both water and
air, from the sketches in (a)?
7.2. Consider a steady, incompressible, low-speed 2-D laminar
boundary-layer flow (at U ∞ , T ∞ ) over a flat plate at a uniform
wall temperature T w or at a uniform wall heat flux q ""
w , respectively. Assume that there exist no body force and constant thermal
and fluid properties.
a. Based on the approximate integral method, assuming a uniform velocity profile inside the boundary layer, that is, u =
U ∞ , and assuming a linear temperature profile inside the
thermal boundary layer as T = a + b × y, determine the local
thermal boundary-layer growth along the flat plate (i.e., δ T
versus x) at a uniform wall temperature T w condition. (Note:
a and b are unknown constants that need to be determined.)
b. Based on (a), determine the local Nusselt number distribution along the flat plate (i.e., Nu versus x) at a uniform wall
temperature.
7.3. Consider a steady, incompressible, low-speed 2-D laminar
boundary-layer flow (at U ∞ , T ∞ ) over a flat plate at a uniform wall temperature T W . Assume that there exist no body
force and constant thermal and fluid properties. The similarity
momentum and energy equations are listed here for reference:
f """ + (1/2)ff "" = 0 and θ "" + (1/2)P r f θ " = 0.
a. Sketch both the velocity and the thermal boundary-layer thickness distribution for both water and liquid metal, respectively,
flowing over the flat plate? (i.e., sketch δ versus x and δ T versus
x on the same plot for water, and δ versus x and δ T versus x
on the same plot for the liquid metal). Explain why they have
differences, if any.
b. Define the similarity variable (η), the similarity function for
temperature (θ), and the derivative of the similarity function
for velocity (f " )? Write the BCs which can be used for this
problem?
c. From the Blasius solution of the above similarity equations,
sketch the relations between the similarity functions (f " , θ) and
the similarity variable (η) for both water and liquid metal (i.e.,
sketch f " versus η for both water and liquid metal on the same
plot; and θ versus η for both water and liquid metal on the
same plot). Explain why they have differences, if any. You do
not need to solve the above equations.
d. For a given problem (i.e., U ∞ , T ∞ , ρ, μ, T w , P r are given),
explain briefly how to determine the local velocity u(y) and
temperature T(y) at a specified location x, for both water and
liquid metal, from the sketches in (c)?