159
External Forced Convection
e. Explain briefly how to determine the local heat transfer coefficient from the sketches in (c)? Sketch the local heat transfer
coefficient over the flat plate for both water and liquid metal
(i.e., sketch h versus x for both water and liquid metal on the
same plot), if both are at the same free-stream velocity (U ∞ )?
Explain why they have differences, if any.
7.4. 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 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) at a uniform wall temperature T w condition. (Note: a and b are unknown constants that need to be
determined.)
c. Based on (b), determine the local Nusselt number distribution along the flat plate (i.e., Nu versus x) at a uniform wall
temperature.
d. 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 = c + d × y, determine the local thermal boundary-layer growth along the flat
plate (i.e., δ T versus x) at a uniform wall heat flux q ""
w condition. (Note: c and d are unknown constants that need to be
determined.)
e. Based on (d), determine the local Nusselt number distribution
along the flat plate (i.e., Nu versus x) at a uniform wall heat flux.
Explain and comment on whether the local Nusselt number
distribution along the flat plate will be higher, the same, or
lower than that in (c)?
7.5. 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 distributions, respectively for both water and air flowing
over the flat plate? (i.e., sketch δ versus x and δ T versus x on
the same plot for water and for air, respectively). Explain why
they have differences, if any.
External Forced Convection
e. Explain briefly how to determine the local heat transfer coefficient from the sketches in (c)? Sketch the local heat transfer
coefficient over the flat plate for both water and liquid metal
(i.e., sketch h versus x for both water and liquid metal on the
same plot), if both are at the same free-stream velocity (U ∞ )?
Explain why they have differences, if any.
7.4. 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 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) at a uniform wall temperature T w condition. (Note: a and b are unknown constants that need to be
determined.)
c. Based on (b), determine the local Nusselt number distribution along the flat plate (i.e., Nu versus x) at a uniform wall
temperature.
d. 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 = c + d × y, determine the local thermal boundary-layer growth along the flat
plate (i.e., δ T versus x) at a uniform wall heat flux q ""
w condition. (Note: c and d are unknown constants that need to be
determined.)
e. Based on (d), determine the local Nusselt number distribution
along the flat plate (i.e., Nu versus x) at a uniform wall heat flux.
Explain and comment on whether the local Nusselt number
distribution along the flat plate will be higher, the same, or
lower than that in (c)?
7.5. 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 distributions, respectively for both water and air flowing
over the flat plate? (i.e., sketch δ versus x and δ T versus x on
the same plot for water and for air, respectively). Explain why
they have differences, if any.