overview of convection heat transfer -...

Overview of Convection Heat TransferME 448/548 Notes

Gerald Recktenwald

Portland State University

Department of Mechanical Engineering

[email protected]

February 23, 2012

ME 448/548: Convection Heat Transfer

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

ME 448/548: Convection Heat Transfer page 1

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


Continuity of heat flux at the wall

Continuity of heat flux requires

−ks∂T

∂y

∣∣∣∣y=0

= −kf∂T

∂y

∣∣∣∣y=0

q+ = kfT y y = 0+

T y y = 0+

T y y = 0

fluidsolid

y

T y y = 0

q = s

T(y+)

T(y )


Definitions (1)

Average velocity in the duct: V is the velocity that gives the correct flow rate in

m = ρV A.

V =1

A

∫A

u · n dA (1)

Similarly, the bulk temperature is defined so that the energy flowing through a cross

section is mcpTb

Tb =

∫A

(u · n)T dA∫A

(u · n) dA

(2)


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 = mcp(Tb,out − Tin) (3)


Physical Boundary Conditions

1. Uniform wall temperature

2. Uniform wall heat flux

3. Convective (external) boundary

4. Radiation (external) boundary


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



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 = mcp(Tb,out − Tin)

Remember: this equation defines Tb. Also note: Tin ≡ Tb,in.



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

h =1

Aw

∫Aw

h(xw) dA. (6)

Correlations for h in heat transfer textbooks are usually obtained from experimental

measurements. A correlation is merely a summary of the experimental data, not a

definition of h. One could also use a CFD program to generate h data.



An alternative approach to computing the average heat transfer coefficient uses the

overall heat transfer rate.

h =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 =hL

k. (8)


Duct with uniform wall heat flux (1)

u(r)

T(r)

qw

x

r



Behavior of bulk temperature:

Tb,out = Tin +Q

mcpincreases 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.



The local heat transfer coefficient is

h(xw) =qw

Tw(xw) − Tin

(9)

The average or overall heat transfer coefficient is computed with

h =Q/Aw

Tw − Tin

(10)

where Tw is the average wall temperature

Tw =1

Aw

∫Aw

Tw(xw) dA (11)

The average or overall Nusselt is

Nu =hL

k. (12)


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 thermal

resistance 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.


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


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 that

is another topic.


Conjugate Heat Transfer

conduction in the board

radiationconvection

Uin, Tin


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



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)



Constant T (y) Constant q(y) Convective BCy

q(y)

T(y)

y

q(y)

T(y)

y

q(y)

T(y)


overview of convection heat transfer -...

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