163
External Forced Convection
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) = (1 cm, 1/2δ) and (x, y) = (1 cm, 1/4δ)
= (3 cm, 1/2δ)
= (3 cm, 1/4δ)
= (9 cm, 1/2δ)
= (9 cm, 1/4δ)
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?
7.13. Consider the development of velocity and thermal boundary layers on a porous flat plate where air passes into the flat plate at a
velocity V o .
a. Assume that no pressure gradient exists in either the x- or the
y-direction and that all fluid properties are constant. Derive
the differential equation that relates boundary-layer thickness
δ to distance x. A linear profile may be assumed.
b. The exact solution of the boundary-layer equations with the
BCs of part (a) shows that δ approaches a constant value for
large x, and that for large x, V x and V y are given by
�
V o y
V x = V ∞ 1 − exp
�
v
��
V y = −V o
Suppose now that at some large x = � (i.e., where δ has
become constant and where V x and V y are given above), a
step change in wall temperature occurs. Using the integral
technique, derive a differential equation relating the thermal
boundary-layer thickness δ T to x (x > �). Again assume that
fluid properties are constant. A linear temperature profile may
be assumed. Integrals and derivatives need not be evaluated.
7.14. Air at 1 atm and at a temperature of 30 ◦ C flows over a 0.3-mlong flat plate at 100 ◦ C with a free-stream velocity of 3 m/s. At
the position x = 0.05 m, determine the values of the boundarylayer thickness, displacement thickness, moments thickness, wall
stress, and heat transfer coefficient. Determine the values of the
velocity parallel and normal to the plate surface, the values of
the shear stress, and the values of temperature in the fluid at the
positions (x = 0.05 m, y = 0.002 m); (x = 0.05 m, y = 0.004 m).
7.15. Using the integral method and assuming that velocity and temperature vary as
u = a + by + cy 2 + dy 3
T = a + by + cy 2 + dy 3
External Forced Convection
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) = (1 cm, 1/2δ) and (x, y) = (1 cm, 1/4δ)
= (3 cm, 1/2δ)
= (3 cm, 1/4δ)
= (9 cm, 1/2δ)
= (9 cm, 1/4δ)
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?
7.13. Consider the development of velocity and thermal boundary layers on a porous flat plate where air passes into the flat plate at a
velocity V o .
a. Assume that no pressure gradient exists in either the x- or the
y-direction and that all fluid properties are constant. Derive
the differential equation that relates boundary-layer thickness
δ to distance x. A linear profile may be assumed.
b. The exact solution of the boundary-layer equations with the
BCs of part (a) shows that δ approaches a constant value for
large x, and that for large x, V x and V y are given by
�
V o y
V x = V ∞ 1 − exp
�
v
��
V y = −V o
Suppose now that at some large x = � (i.e., where δ has
become constant and where V x and V y are given above), a
step change in wall temperature occurs. Using the integral
technique, derive a differential equation relating the thermal
boundary-layer thickness δ T to x (x > �). Again assume that
fluid properties are constant. A linear temperature profile may
be assumed. Integrals and derivatives need not be evaluated.
7.14. Air at 1 atm and at a temperature of 30 ◦ C flows over a 0.3-mlong flat plate at 100 ◦ C with a free-stream velocity of 3 m/s. At
the position x = 0.05 m, determine the values of the boundarylayer thickness, displacement thickness, moments thickness, wall
stress, and heat transfer coefficient. Determine the values of the
velocity parallel and normal to the plate surface, the values of
the shear stress, and the values of temperature in the fluid at the
positions (x = 0.05 m, y = 0.002 m); (x = 0.05 m, y = 0.004 m).
7.15. Using the integral method and assuming that velocity and temperature vary as
u = a + by + cy 2 + dy 3
T = a + by + cy 2 + dy 3