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Overview of Forced Convection Heat TransferME 448/548 Notes
Gerald Recktenwald
Portland State University
Department of Mechanical and Materials Engineering
February 24, 2020
ME 448/548: Convection Heat Transfer
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Outline
1. External versus internal flow
2. Definitions: mean velocity and bulk temperature
3. Types of boundary conditions
a. Uniform wall temperature
b. Uniform wall heat flux
c. Convective (external) boundary
d. Radiation (external) boundary
4. Conjugate heat transfer
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External and internal flow have different modeling concerns
External flow: Boundary layers on aerodynamic shapes immersed in a fluid
U , T
Internal flow: Wall-bounded flows with potentially large pressure drops
Uin, Tin
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Continuity of heat flux at the wall
Continuity of heat flux requires
−ks∂T
∂y
∣∣∣∣y=0
= −kf∂T
∂y
∣∣∣∣y=0
fluidsolid
y
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Definitions (1)
Average velocity in the duct: V is the velocity that gives the correct flow rate in
ṁ = ρV A.
V =1
A
∫A
u · n̂ dA (1)
Similarly, the bulk temperature is defined so that the energy flowing through a cross
section is ṁcpTb
Tb =
∫A
(u · n̂)T dA∫A
(u · n̂) dA(2)
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Overall energy balance for a duct
Assume the flow is steady and incompressible
Tin
m.
Tb,out
Q
The total heat transfer across the wall of the duct is Q. By definition of Tb, an energy
balance on the duct shows that the total heat transfer rate is must be equal to
Q = ṁcp(Tb,out − Tin) (3)
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Physical Boundary Conditions
1. Uniform wall temperature
2. Uniform wall heat flux
3. Convective (external) boundary
4. Radiation (external) boundary
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Duct with uniform wall temperature (1)
Development of the
temperature profile for flow
through a pipe with uniform
wall temperature.
x
rUin Tin
Temperature profile: T(r)-Tin
qw
Tw
Tb(x)
Tw > Tin
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Duct with uniform wall temperature (2)
Behavior of bulk temperature:
Tb(x) asymptotically approaches Tw
Behavior of wall temperature:
Tw = constant is an imposed constraint
Total heat transfer through the duct wall:
Q =
∫Aw
qw(x) dA (4)
and, as always
Q = ṁcp(Tb,out − Tin)
Remember: this equation defines Tb. Also note: Tin ≡ Tb,in.
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Duct with uniform wall temperature (3)
Since the wall heat flux varies with position along the duct, the heat transfer coefficient is
also varying with position
h(xw) =qw(xw)
Tw − Tin. (5)
Note that the heat transfer coefficient does not come from a correlation!
Given knowledge of h(xw), e.g. from computation or experiments, the average heat
transfer coefficient is
h̄ =1
Aw
∫Aw
h(xw) dA. (6)
Correlations for h̄ in heat transfer textbooks are usually obtained from experimental
measurements. A correlation is merely a summary of the experimental data, not a
definition of h̄. One could also use a CFD program to generate h̄ data.
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Duct with uniform wall temperature (4)
An alternative approach to computing the average heat transfer coefficient uses the
overall heat transfer rate.
h̄ =Q/Aw
Tw − Tin(7)
Substitution of Equation (4) into Equation (7) shows that Equation (6) and Equation (7)
are equivalent.
The average or overall Nusselt is
Nu =h̄L
k. (8)
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Duct with uniform wall heat flux (1)
u(r)
T(r)
qw
x
r
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Duct with uniform wall heat flux (2)
Behavior of bulk temperature:
Tb,out = Tin +Q
ṁcpincreases with x (assumes q > 0)
Behavior of wall temperature:
Tw increases with x
Total heat transfer through the duct wall:
Q =
∫Aw
qw(x) dA = qwAw
because qw is uniform.
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Duct with uniform wall heat flux (3)
The local heat transfer coefficient is
h(xw) =qw
Tw(xw) − Tin(9)
The average or overall heat transfer coefficient is computed with
h̄ =Q/Aw
T̄w − Tin(10)
where T̄w is the average wall temperature
T̄w =1
Aw
∫Aw
Tw(xw) dA (11)
The average or overall Nusselt is
Nu =h̄L
k. (12)
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Convection boundary condition (1)
x
rUin Tin
Tw(x), qw(x)
T(r)
h, Tamb
Note: In a CFD model, the heat transfer coefficient is applied to determine the thermalresistance from the walls of the domain to the ambient. The heat transfer coefficient is
not used internally, i.e., between the walls of the duct and the fluid in the domain.
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Convection boundary condition (2)
Assume Tamb > Tin. Then the following observations can be made.
• The bulk temperature Tb(x) will increase with x• The wall temperature Tw(x) will increase with x• The wall heat flux qw(x) will decrease with x• The total heat transfer through the walls is
Q =
∫Aw
qw dA
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Radiation boundary condition
x
rUin Tin
Tw(x), qw(x)
T(r)
Enclosure at Tsurf
εw
εsurf
As with the convective boundary condition, the radiation exchange (as a boundary
condition) determines the thermal resistance from the walls of the domain to the ambient.
Note: It is also possible to include radiation between surfaces inside the domain, but thatis another topic.
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Conjugate Heat Transfer
Conjugate heat transfer occurs with multiple modes of heat transfer are happening at the
same time.
A common form of conjugate heat transfer involves conduction in a solid that is immersed
in a flowing fluid.
Example: Heat transfer from electronic devices in an enclosure.
conduction in the board
radiationconvection
Uin, Tin
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Case Study: Electronics Cooling (1)
What BC should be imposed here?
External flow due to natural convection
T = Tamb
y
x
Electronic component dissipating heat
Sealed enclosure
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Case Study: Electronics Cooling (2)
Choices of boundary condition:
1. Constant temperature on the walls of the enclosure
T = Tamb
2. Constant heat flux on the walls of the enclosure
∂T
∂x
∣∣∣∣w
= qw
3. Convective conditions on the walls of the enclosure
∂T
∂x
∣∣∣∣w
= h(T − Tamb)
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Case Study: Electronics Cooling (3)
Physical problem Constant T (y) Constant q(y) Convective BC
What BC should be imposed here?
External flow due to natural convection
T = Tamb
y
x
Electronic component dissipating heat
Sealed enclosureT(y)
q(y)
y
T(y)
q(y)y
T(y)?
q(y)?y
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