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150
Analytical Heat Transfer
numbers. Students are expected to calculate heat transfer coefficients from
these relations by giving Reynolds and Prandtl numbers.
For the similarity method, students are expected to know how to derive
the similarity momentum and energy equations with proper velocity and
thermal BCs. Students are also expected to know how to sketch and predict
velocity profiles inside the boundary layer, for a given Reynolds number,
from the velocity similarity solution by using tables or figures; how to sketch
and predict temperature profiles inside the thermal boundary layer, for a
given Reynolds number and Prandtl number, from the temperature similarity
solution by using tables or figures. Here we focus on flow over a flat plate
(zero-pressure gradient flow) with constant surface temperature BC, and do
not include the one at constant surface heat flux BC.
In advanced heat convection, the similarity solution can be extended to
include various constant pressure gradient flows (such as flow acceleration
or deceleration) with variable surface temperature BCs. These can be solved
using the fourth-order Runge–Kutta method in order to obtain the velocity
and temperature profiles across the forced convection boundary-layer flow.
7.2 Laminar Flow and Heat Transfer over a Flat Surface:
Integral Method
The other powerful method to solve boundary-layer flow and the heat transfer
problem is using the integral approximate solution [1–6]. Instead of performing mass, momentum, and energy balance through a differential fluid element
inside the boundary layer as the similarity method, the integral method
performs conservation of mass, momentum, and energy across the boundarylayer thickness at a given differential x-direction. It is noted that, for fluid
with a Prandtl number different from unity, such as gases, water, and oils,
the hydrodynamic boundary-layer thickness is different from the thermal
boundary layer.
7.2.1 Momentum Integral Equation by Von Karman
Employing a control volume that is infinitesimal in the x-direction but finite
in the y-direction across the boundary-layer thickness, as shown in Figure 7.5,
we apply the mass and momentum conservation to the control volume.
From mass conservation,
δ
δ
∂v
∂u
= −
v| 0 −
∂u dy = −
∂u dy
v(δ) =
∂y
∂x
∂x
∂x
0
0
d
ρv dx = dx
(ρu dy) dx
(7.31)
�
�
150
Analytical Heat Transfer
numbers. Students are expected to calculate heat transfer coefficients from
these relations by giving Reynolds and Prandtl numbers.
For the similarity method, students are expected to know how to derive
the similarity momentum and energy equations with proper velocity and
thermal BCs. Students are also expected to know how to sketch and predict
velocity profiles inside the boundary layer, for a given Reynolds number,
from the velocity similarity solution by using tables or figures; how to sketch
and predict temperature profiles inside the thermal boundary layer, for a
given Reynolds number and Prandtl number, from the temperature similarity
solution by using tables or figures. Here we focus on flow over a flat plate
(zero-pressure gradient flow) with constant surface temperature BC, and do
not include the one at constant surface heat flux BC.
In advanced heat convection, the similarity solution can be extended to
include various constant pressure gradient flows (such as flow acceleration
or deceleration) with variable surface temperature BCs. These can be solved
using the fourth-order Runge–Kutta method in order to obtain the velocity
and temperature profiles across the forced convection boundary-layer flow.
7.2 Laminar Flow and Heat Transfer over a Flat Surface:
Integral Method
The other powerful method to solve boundary-layer flow and the heat transfer
problem is using the integral approximate solution [1–6]. Instead of performing mass, momentum, and energy balance through a differential fluid element
inside the boundary layer as the similarity method, the integral method
performs conservation of mass, momentum, and energy across the boundarylayer thickness at a given differential x-direction. It is noted that, for fluid
with a Prandtl number different from unity, such as gases, water, and oils,
the hydrodynamic boundary-layer thickness is different from the thermal
boundary layer.
7.2.1 Momentum Integral Equation by Von Karman
Employing a control volume that is infinitesimal in the x-direction but finite
in the y-direction across the boundary-layer thickness, as shown in Figure 7.5,
we apply the mass and momentum conservation to the control volume.
From mass conservation,
δ
δ
∂v
∂u
= −
v| 0 −
∂u dy = −
∂u dy
v(δ) =
∂y
∂x
∂x
∂x
0
0
d
ρv dx = dx
(ρu dy) dx
(7.31)