160
Analytical Heat Transfer
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 that 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 air. (i.e., sketch f "
versus η for both water and air on the same plot; and θ versus η
for both water and air on the same plot). Explain why they
have differences, if any. You do not have 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
air, from the sketches in (c)?
e. Explain briefly how to determine the local heat transfer coefficient from the sketches in (c)? Answer whether water or air
will provide a higher convective heat transfer coefficient from
the surface, if both are at the same free-stream velocity (U ∞ )?
Why?
7.6. Consider a steady, incompressible, low-speed 2-D laminar
boundary-layer flow (at U , T ) over a flat plate at a uniform wall
heat flux q ""
∞ ∞
w
. Assume that there exist no body force and constant
thermal and fluid properties.
a. Based on the integral method, sketch and write down the
momentum balance as well as the energy balance across the
boundary layers? (If you cannot remember, derive it.)
b. 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).
c. Based on (b), determine the local Nusselt number distribution
along the flat plat (i.e., Nu versus x).
d. Consider a parabolic velocity and temperature profile inside
the thermal boundary layer as u = a + b × y + c × y 2 , T =
a + b × y + c× y 2 , and comment on whether the local Nusselt number distribution along the flat plate will be higher, the
same, or lower than those of a uniform velocity profile and a
linear temperature profile as indicated in (b)? Explain why.
e. 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
that without boundary-layer suction? Explain why.
7.7. 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
Analytical Heat Transfer
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 that 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 air. (i.e., sketch f "
versus η for both water and air on the same plot; and θ versus η
for both water and air on the same plot). Explain why they
have differences, if any. You do not have 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
air, from the sketches in (c)?
e. Explain briefly how to determine the local heat transfer coefficient from the sketches in (c)? Answer whether water or air
will provide a higher convective heat transfer coefficient from
the surface, if both are at the same free-stream velocity (U ∞ )?
Why?
7.6. Consider a steady, incompressible, low-speed 2-D laminar
boundary-layer flow (at U , T ) over a flat plate at a uniform wall
heat flux q ""
∞ ∞
w
. Assume that there exist no body force and constant
thermal and fluid properties.
a. Based on the integral method, sketch and write down the
momentum balance as well as the energy balance across the
boundary layers? (If you cannot remember, derive it.)
b. 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).
c. Based on (b), determine the local Nusselt number distribution
along the flat plat (i.e., Nu versus x).
d. Consider a parabolic velocity and temperature profile inside
the thermal boundary layer as u = a + b × y + c × y 2 , T =
a + b × y + c× y 2 , and comment on whether the local Nusselt number distribution along the flat plate will be higher, the
same, or lower than those of a uniform velocity profile and a
linear temperature profile as indicated in (b)? Explain why.
e. 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
that without boundary-layer suction? Explain why.
7.7. 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