1. Research Scholar, ISM Dhanbad
22nd National and 11th International ISHMT-ASME Heat and Mass Transfer Conference
MATHEMATICAL HEAT TRANSFER MODELLING FOR NANOFLUID CONSIDERING INTERFACIAL NANO-LAYER
Presented by:-
Ankit Kotia
Research Scholar, ISM Dhanbad

2. Outline Of Presentation
Introduction
Objectives
Mathematical models for thermal conductivity
Mathematical models for parameters used
Interfacial layer thickness
Effective Thermal conductivity
Effective Heat capacity
Effective Viscosity
Mathematical Formulation
References

3. INTRODUCTION
In recent years heat dissipation is a matter of concern as miniaturization, complex geometries and drastically increase in functional speed in equipment lead to astronomical rise in heat flux.
Initially milli and micro size particles of higher thermal conductivity was added to base fluid, but this fluid shows demerits like erosion of flow passage, clogging of narrow channels, rapid increase in viscosity leads to reasonable drop in pressure.
To eliminate this demerits nano size particle (<100nm) are added to base fluid. Argonne National laboratory has pioneered ultrahigh thermal conductivity fluid in 1995 called Nanofluid by suspending nano-particles in conventional coolants and S.U.S Choi [2] was probably the first one who called this fluid as ‘nano-fluid
Some models for thermal conductivity of nano-fluid have been proposed by considering the Brownian motion, interfacial layer and aggregation of particles
for alumina powders with an average diameter of 10 µm, the SSA (Specific Surface Area) is only 0.15 m2g‑1. Therefore, the effect of interfacial layer is negligible. However, nano-particles have larger SSA, e.g., for alumina powders with an average diameter of 10 nm, the SSA is as large as 151 m2g
In this paper, a mathematical model for determining the heat flux of the nanofluid by considering the effect of interfacial layer has been developed.
Results obtained by using this model has been compared with heat flux obtained from the Hamilton Crosser model

4. Objectives Following are the objectives of the present work:-
To develop a mathematical model for convective heat transfer performance of nanofluid
To compare convective heat transfer results with existing models
To find out the effect different parameters on convective heat transfer performance of nanofluid.

5. Mathematical models for thermal conductivity
Maxwell conducted research on conduction through heterogeneous media and proposed equation for the effective thermal conductivity (keff), expressed as:
Hamilton and Crosser [5] extended Maxwell work to cover non-spherical particles and introduce shape factor (n). Their model for a discontinuous phase dispersed in continuous phase is:

6. Contin… Later that R.H. Davis [13] gave following model:
Yu and Choi [7] developed a modified version of Maxwell model with considering effect of interfacial nano-layer, assume that the nano-layer around each particle could be combine with the particle to form an equivalent particle expressed as:
Where β= δ/r
Later Yu and Choi [15] they renovate the Hamilton Crossers model [3] for non-spherical particles with considering interfacial nano-layer. Further various models like Xie et al [16] (2005), Ren et al [17] (2005) for effective conductivity including the effect of nano-layer have been developed.

7. Mathematical models for parameters used
Interfacial layer thickness
Using the Langmuir formula of monolayer adsorption of molecules, Wang et al [4] proposed the following relation for the thickness of the interfacial layer:
Effective Thermal conductivity
Liquid around a nano-particle takes a similar structure to the solid and consequently has different thermal conductivity compared to the base fluid
Liquid molecules close to solid surface are known to form layered structures
As the layered molecule are in an intermediate physical state between a solid and a bulk liquid , the shells would be expected to lead to a higher thermal conductivity than that of the bulk liquid.

8. Variation of thickness of Thermal- Hydrodynamic boundary layer
Contin…
Schematic model of nano-particle with interfacial layer
Recently Rizvi et al developed a new model for effective thermal conductivity. In this model, it has been assumed that the variation of thermal conductivity of interfacial layer is linearly varying along the thickness of layer
Variation of thickness of Thermal- Hydrodynamic boundary layer

9. Contin…
The portion of interfacial layer just in contact with the particle has thermal conductivity (kp) which goes on decreasing along the thickness of interfacial layer towards the base liquid medium and finally approaching the vale equal to ‘km’ just before it blends and behaves as liquid.
Finally the expression of the thermal conductivity of nanofluid with interfacial layer is expressed as

10. Effective Heat capacity
Pak and Cho have given equation for specific heat of nanofluid as follows:
Effective Viscosity
In 1972, Lundgren proposed the following equation under the form of a Taylor series, expressed as:

11. Mathematical Formulation
Variation of thickness of Thermal- Hydrodynamic boundary layer
Assuming temperature of fluid at the boundary surface is equal to wall temperature (ts) and decreases gradually to surrounding fluid till thermal boundary layer, thus no temperature gradient will exist at thermal boundary. Temperature gradient is obtained by differentiating energy equation at boundary conditions for a low velocity incompressible flow is as follows

12. Contin…
The following expression has been proposed with the assumption that the boundary layer thickness is same as the thickness of interfacial layer at initial state:
From above it can be noted that the thermal conductivity is main parameter for heat transfer rate.

13. References
Maxwell, J.C., 1904, “A Treatise on Electricity and Magnetism”, second ed., Oxford University Press, Cambridge.
Hamilton, R.L and Crosser, O.K,1962, “Thermal Conductivity of Heterogeneous Two Component Systems”, Industrial and Engineering Chemistry Fundamentals, pp
Wang, B.X., Zhou, L.P and Peng, X.F, 2003, “A fractal model for predicting the effective thermal conductivity of liquid with suspention of nano-particles”, Int. J. of Heat and Mass Transfer , pp 2665–2672
Yu, C.J., Richter, A.G., Datta, A., Durbin, M.K and Dutta, P., 2000, “Molecular layering in a Liquid on a solid Substrate: an X-Ray Reflectivity Study”. Physica B, pp 27-31
Yu, W. and Choi, S.U.S, 2003, “The role of Interfacial layer in the enhanced thermal conductivity of nanofluids: A renovated Maxwell model” , J of Nanoparticle Research, pp
Rizviet Imbesat Hassan , Jain Ayush, Ghosh Subrata Kr. and Mukherjee P. S., 2013, “Mathematical modelling of thermal conductivity for nanofluid considering interfacial nano-layer”, Heat and mass transfer pp595–600
ALundgren, T.S., 1972, “Slow flow through stationary random beds and suspensions of spheres”, J. Fluid Mech. pp 273–299
VJain Ayush, Sarkar Mayukh , Ghosh Subrata Kr. and Mukherjee P S, 2011, “A Numerical Model for Predicting the Characteristics of Nanofluids, National Seminar on nanomaterial and their applications”, NANOMAT (2011), pp
Holman J.P, 1997, 8th ed, Heat transfer, McGraw-Hill.

14. THANK YOU