Governing Equations

The RANS equations govern the transport of averaged flow quantities, with the whole range of scales of turbulence being modelled. The RANS-based modelling approach therefore greatly reduces the required computational effort and resources, and is widely adopted for practical engineering applications. In the following paragraph, the Reynolds-averaged Navier-Stokes equations for single phase, multi-phase and discrete phase models are presented and described [68].

3.1.1

Single Phase Model

The dimensional steady-state governing equations of fluid flow and heat transfer for the single phase model are presented and the following assumptions are made:

i. Fluid flow is incompressible and Newtonian,

ii. The Boussinesq approximation is negligible as the pipe is placed horizontally, iii. Nanoparticles are spherical and uniform in size and shape,

iv. Radiation effects and viscous dissipation are negligible.

v. Fluid phase and nanoparticles phase are in thermal equilibrium and no-slip between them and they flow with the same local velocity,

Under the above assumptions, the dimensional steady state governing equations for the fluid flow and heat transfer in the single phase model can be expressed as (Fluent [68]):

For 2D Axisymmetric Model:

Continuity equation: + + = 0 (3.1) x-momentum equation: 1 _{( ) +}1 _{( ) = −}1 _{+ 2 −}2 3 (∇. ) + + (3.2) r-momentum equation: 1 _{( ) +}1 _{( ) = −}1 _{+ +} + 2 −2_{3 (∇. ) − 2} +2₃1 (∇. ) (3.3) where (∇. ) = + + (3.4) Energy equation: + = + (3.5) For 3D model: Continuity equation: ∇. ( ) = 0 (3.6) Momentum equation: ∇. ( ) = −∇ + ∇. (∇ + ∇ ) −2_{3 ∇. +} (3.7)

Energy equation:

∇. = ∇. ∇ (3.8)

where and are the axial and radial coordinates respectively, and are the axial and radial velocity respectively, is the velocity vector, is the temperature, is the gravitational body force, is the unit tensor, is the density, is the pressure, is the dynamic viscosity and is the thermal conductivity of nanofluid.

3.1.2

Multi-phase Mixture Model

The dimensional steady-state governing equations of continuity, momentum, energy and volume concentration for the multi-phase model are presented considering the assumptions (i) to (iv) given in the single phase model. Moreover, it is assumed that there is a strong coupling between the fluid and nanoparticles phases and these phases move at the same local velocity. Interaction between the fluid and nanoparticles is also taken into account.

It is also assumed that fluid and nanoparticles phases are in local thermal equilibrium in multi-phase mixture model. It means, mean temperature of the fluid phase and the nanoparticles phase are same.

Under the above assumptions, the governing equations for the multi-phase mixture model can be expressed as (Fluent [68]):

Continuity equation:

∇. ( ) = 0 (3.9)

Momentum equation:

The multi-phase mixture model allows the phases to move at different or same velocities using the concept of drift velocity. When the phases can also be assumed to move at same velocities then the mixture model is called the homogeneous multi- phase model. Moreover, the momentum equation for the mixture can be obtained by summing the individual momentum equations for all the phases.

Energy equation:

∇. ( + ) = ∇. ( + ) ∇ (3.11)

Volume concentration equation:

∇. = −∇. _, (3.12)

Also, , , , , , , are the mass-average velocity, mixture density, viscosity of the mixture, mixture thermal conductivity coefficient, number of phases, turbulent thermal conductivity and nanoparticles concentration respectively.

These are defined as

= (3.13)

= (3.14)

= (3.15)

= (3.16)

Here, is the sensible enthalpy for phase s. The drift velocity ( , ) for the

secondary phase s is defined as

, = − (3.17)

The relative or slip velocity is defined as the velocity of the secondary phase (s) relative to the velocity of the primary phase (f):

= − (3.18)

Then the drift velocity related to the relative velocity becomes

, = − (3.19)

Manninen et al. [69] and Naumann and Schiller [70] proposed the following respective equations for the calculation of the relative velocity, , and the drag function, .

= 1 + 0.15_0.0183 . ≤ 1000_{> 1000}  (3.21) Here, the acceleration is determined by

= −( . ∇) (3.22)

And, is the diameter of the nanoparticles of secondary phase s and is the secondary phase particle’s acceleration, is the temperature, is the pressure. Also, the buoyancy term in the momentum equations (3.7) and (3.10) is approximated (Fluent [68]) by

( − ) ≈ − ( − ) (3.23)

which is considered when Boussinesq approximation is taken into account for mixed convection case. Here is the thermal expansion coefficient of the fluid, and are the reference density and temperature respectively.

3.1.3

Discrete Phase Model

In the discrete phase model, the fluid phase is considered as a continuous phase and is governed by the time averaged continuity, momentum and energy equations. Though, the solid phase is governed by momentum and energy equations. It is assumed that the solid phase occupies a low concentration of less than 10%, Moreover, each phase moves at different velocities with the assumption that the existence of local thermal equilibrium between the fluid and solid particles is not taken into account.

The governing equations for the discrete phase model can be expressed as (Fluent [68]): ∇. ( ) = 0 (3.24) ∇. ( ) = −∇ + ∇. ( ∇ ) + (3.25) ∇. = ∇. ∇ + (3.26) = − + − (3.27) =6ℎ − (3.28)

Here, Eqs. (3.27) and (3.28) represent the Lagrangian form of particle momentum and energy equations respectively. And, and are the nanoparticles velocity and temperature respectively. Finally, is the nanoparticles density.

The drag coefficient is defined as [68]:

= 18 (3.29)

Here, the factor is known as Cunningham correction which can be defined as [68]:

= 1 +2 1.257 + 0.4 . / _(3.30)

Here, is the nanoparticles mean free path and is the dynamic viscosity of base fluid.

Also, the source terms and are defined as [68]:

= _(3.31)

= _(3.32)

For transition and turbulent flow regimes, both the terms and are replaced by their effective values and defined as (Nicholas and Markatos [71] and Bacharoudis et al. [72])

= + (3.33)

= + (3.34)

In that order, is the turbulent molecular viscosity, is the constant of turbulent Prandtl number and Pr is the Prandtl number of nanofluid.