a detailed discussion of the results in terms of the streamline profiles, isotherm contours, distribution of local Nusselt number, variation of velocity components, etc., is also presented. Finally, from an application standpoint, a simple correlation for the average Nusselt number is presented, which can be used for the interpolation of the present results for the intermediate values of the governing parameters in a new application.

1. Natural Convection Heat Transfer of Viscoelastic Fluids
in a Horizontal Annulus
by
Pyari Mohan Sahu
Under the supervision of
Dr. Chandi Sasmal
Department of Chemical Engineering, IIT Ropar
MTech Thesis Presentation

2. Contents
Introduction
Background
Problem statement
Governing equations and boundary conditions
Numerical solution procedure
Results and discussion
Conclusions and future perspectives

3. Introduction
What is natural convection heat transfer?
• There is no external agency like pump or blower present.
• The driving force in this mode of heat transfer is the buoyancy-induced convection current present in the system,
originated due to the density difference of the fluid.

4. Introduction…
Why it is important to study?
Thermal sterilization of canned foods
Food processing industries Design of Heat exchangers Design of solar dish concentrator

5. Background
• Most of the studies on natural convection heat transfer have been carried out for simple Newtonian fluids like
water or air
Ketchup Paint Shaving cream Sunscreen
Polymer solutions
Pharmaceutical suspensions
However, there are many fluids present which do not obey the simple Newton’s laws of viscosity, for example

6. Background
https://link.springer.com/chapter/10.1007/978-1-4615-6907-7_5
Significant amount of studies have been carried out for these GNF fluids on natural convection heat transfer
Generalized non-Newtonian fluids (GNF) or inelastic fluids

7. Background
Newton’s law of (perfectly) viscous fluid Hooke’s law of (perfectly) elastic solid
Complex fluids are mixtures that have a coexistence between two phases: solid–liquid (suspensions or solutions of macromolecules such as
polymers), solid–gas (granular), liquid–gas (foams) or liquid–liquid (emulsions). source: Wikipedia
Robert Hooke
(1635-1703)
Isaac Newton
(1643-1727)
There is almost no study available on natural convection heat transfer in these viscoelastic fluids !!

8. Problem statement

9. Governing equations and dimensionless numbers
0
i
j
u
x



2
1 ij
i i i
j i
j i j j j
C
u u u
p
u
t x x x x x
Ra Pr Wi Ra Pr
 

  
  
  
     
 
 
     
 
1
j
j j j
u
t x x x
RaPr
  
 
   
   
 
   
 
Momentum equation:
Energy equation:
Continuity equation:
 
1
ref T ref
T T
  
 
  
 
Boussinesq approximation:
1
T
P
T




 

Thermal expansion coefficient

10. Governing equations and dimensionless numbers
*
* * * *
2 2
0
, , , , , ij
i i c C
i i ij
c ref c H C
C
x u t u T T
p
x u t p C
R u R u T T L



     

Non-dimensionalization
c T
u Rg T

 
Where the characteristic velocity
( )
ij ij j ij ij
i
k ik kj
k k k
C C u f R C
u
u C C
t x x x Wi

   

   
   
FENE-P viscoelastic constitutive equation
In the above equation, is the Kroneker delta and f (R) is the Peterlin’s approximation of the finite extensibility of the
FENE-P model defined as where and L are the extension length and maximum possible extension
length of a polymer molecule, respectively. The relation between the conformation tensor and viscoelastic stress tensor is
given by
ij
 2
2
3
( )
L
f R
L R



( )
ij
R tr C

( )
p
ij ij ij
f R C
 
 

11. Governing equations and dimensionless numbers
• Rayleigh number
3
T i
g TR
Ra




0
ref



 

 
 
  ref
k
Cp


 

 
 
 
Where and are the kinematic viscosity and
thermal diffusivity, respectively.
• Prandtl number
Pr



• Weissenberg number
c
i
u
Wi
R


• Viscosity ratio
0
s




• Polymer extensibility parameter
2
L

12. Results and discussion
Validation

13. Streamlines and velocity magnitude plots
Results and discussion
Figure 4: Streamlines and velocity magnitude plots at
(a) 3 2
10 , 0.9, 1, 10
Ra Wi L

    (b) 6 2
10 , 0.9, 1, 10
Ra Wi L

   
(c) 6 2
10 , 0.5, 1, 500
Ra Wi L

    (d) 6 2
10 , 0.5, 100, 500
Ra Wi L

   
Figure 4: Streamlines and velocity magnitude plots at
(e) 6 2
10 , 0.9, 100, 500
Ra Wi L

    (f) 6 2
10 , 0.5, 100, 500
Ra Wi L

   
(g) 6 2
10 , 0.5, 100, 10
Ra Wi L

    (h) 6 2
10 , 0.5, 100, 500
Ra Wi L

   

14. Isotherm contours
Results and discussion
Figure 5: Surface distribution of isotherm contours
(a) 3 2
10 , 0.9, 1, 10
Ra Wi L

    (b) 6 2
10 , 0.9, 1, 10
Ra Wi L

   
(c) 6 2
10 , 0.5, 1, 500
Ra Wi L

    (d) 6 2
10 , 0.5, 100, 500
Ra Wi L

   
Figure 5: Surface distribution of isotherm contours
(e) 6 2
10 , 0.9, 100, 500
Ra Wi L

    (f) 6 2
10 , 0.5, 100, 500
Ra Wi L

   
(g) 6 2
10 , 0.5, 100, 10
Ra Wi L

    (h) 6 2
10 , 0.5, 100, 500
Ra Wi L

   