WOLFRAM NOTEBOOK

Effect of Viscous Dissipation on Heat Transfer in Laminar Flow

​
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
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.

External Links

Permanent Citation

Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.