# Flow in a Vertical Channel with Walls at Different Temperatures

Flow in a Vertical Channel with Walls at Different Temperatures

Consider a vertical parallel-plate channel of width with walls at different temperatures and , >. Two cases are studied: (1) pure free convection (i.e., the channel is closed at both ends and there is no net flow); and (2) mixed free and forced convection (a pressure gradient is present). The velocity (solution of the momentum equation) is given by

2H

T

1

T

2

T

2

T

1

v

x

gβ(-)

T

2

T

m

2

H

6ν

3

η

2

H

3

1

μ

d

dx

2

η

where is gravitational acceleration, is the thermal expansion coefficient, == is the reference temperature taken equal to the mean temperature, is the kinematic viscosity, is the dynamic viscosity, is the pressure gradient, and is the dimensionless position.

g

β

T

0

T

m

(+)

T

1

T

2

2

ν

μ

d

dx

η=

y

H

The velocity can be expressed as =η-+U1-, where is the mean velocity and = is the maximum velocity when the pressure gradient is zero.

v

x

3

2

3

v

0

3

η

2

η

U=-

2

H

3

1

μ

d

dx

v

0

gβ(-)

T

2

T

1

2

H

18

3

νIf and ≠, one recovers pure free convection. When =, the velocity profile is the expected parabolic profile corresponding to a Poiseuille flow. In general, mixed free and forced convection is observed.

U=0

T

2

T

1

T

2

T

1