Effect of Viscous Dissipation on Heat Transfer in Laminar Flow

Out[]=

axial temperature

Brinkman number

0.

Peclet number

5.

distance along axis

0.2

This Demonstration shows the effect of axial conduction and viscous dissipation on heat transfer between a fluid in laminar flow and a tube at constant temperature.

Consider the fully developed laminar flow of a fluid in a tube with a wall temperature

T

w

; the fluid enters at a uniform temperature

T

0

. Assuming constant physical properties and axial symmetry, the dimensionless energy equation is:

2(1-

2

r

)

∂T

∂x

=

1

2

Pe

2

∂

T

∂

2

x

+

1

r

∂

∂r

r

∂T

∂r

+

2

Br

dU

dr

,

with boundary conditions:

T(0,r)=0

,

∂T

∂x

(L,)=0

,

T(x,R)=

T

w

,

∂T(x,0)

∂r

=0

,in which the dimensionless variables are given by:

x=

ξ

2PeR

,

r=



2R

,

T=

-



o



w

-



o

,

U=

u

avg

,

u=2

u

avg

1-

2

r

R

,

Pe=

ρ

C

p

2

u

avg

R

k

=

heatconvected

heatconducted

,

Br=

μ

2

U

avg

k(

T

w

-

T

0

)

=

heatproducedbyviscousdissop[ation

heattransportedbymolecularconduction

,

where



and

ξ

are the radial and axial coordinates, respectively,

R

is the tube radius,

L

is the tube length,

ρ

is the fluid specific gravity,

C

p

is the fluid heat capacity,

u

avg

is the average laminar velocity, and

k

is the fluid thermal conductivity.

The dimensionless equation is solved using the built-in Mathematica function NDSolve; the effect of the Péclet number and the Brinkman number on the temperature distribution is shown.