# Numerical study of turbulent flow and heat transfer of Al_{2}O_{3}–water mixture in a square duct with uniform heat flux

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## Abstract

Turbulent flow and convective heat transfer of a nanofluid made of Al_{2}O_{3} (1–4 vol.%) and water through a square duct is numerically studied. Single-phase model, volumetric concentration, temperature-dependent physical properties, uniform wall heat flux boundary condition and Renormalization Group Theory k-ε turbulent model are used in the computational analysis. A comparison of the results with the previous experimental and numerical data revealed 8.3 and 10.2 % mean deviations, respectively. Numerical results illustrated that *Nu* number is directly proportional with *Re* number and volumetric concentration. For a given *Re* number, increasing the volumetric concentration of nanoparticles does not have significant effect on the dimensionless velocity contours. At a constant dimensionless temperature, increasing the particle volume concentration increases the size of the temperature profile. Maximum value of dimensionless temperature increases with increasing x/D_{h} value for a given *Re* number and volumetric concentration.

## Keywords

Heat Transfer Heat Transfer Coefficient Wall Shear Stress Base Fluid Skin Friction Coefficient## List of symbols

- a
_{P} Coefficient of P cell

- C
_{1ε} Turbulent model constant = 1.42

- C
_{2ε} Turbulent model constant = 1.68

- C
_{p} Specific heat of nanofluid under constant pressure (J/kg K)

- Cp
_{np} Specific heat of nanoparticle under constant pressure (J/kg K)

- C
_{μ} Turbulent model constant = 0.0845

- C
Constant part of the source term

- d
_{h} Hydraulic diameter (m)

- g
Gravitational acceleration (m/s

^{2})- I
Turbulent intensity = \({\text{u}}^{\prime } /\overline{\text{u}}\)

- k
Turbulent kinetic energy (m

^{2}/s)- l
Turbulent mixing length (m)

- l
_{ε} Length scale of turbulent kinetic energy dissipation (m)

- l
_{μ} Length scale of viscosity (m)

*Nu*Nusselt number

- P
Pressure (Pa)

*Pr*Prandtl number of nanofluid

- q″
Wall heat flux (W/m

^{2})- R′
Effect of strain in ε equation (kg/ms

^{4})*Re**Re*number*Re*_{y}*Re*number for a cell having distance y from the nearest wall- S
Modulus of mean rate of strain tensor (1/s)

- T
Temperature (K)

- T
_{b} Cross-sectional weighted average of the local fluid temperature (K) \(\left( { = \frac{1}{UA}\int {uTdA} } \right)\)

- T
_{∞} Mixed mean temperature (K)

- \(\overline{\text{u}}\)
Time averaged mean velocity (m/s)

- u′
Instantaneous velocity component (m/s)

- u
Velocity (m/s)

- \({\text{y}}^{*}\)
Non-dimensional viscous sublayer thickness

## Greek symbols

- α
Inverse effective

*Pr*number \(\left( { = \frac{1}{Pr}} \right)\)- α
_{ε} Inverse effective

*Pr*number for dissipation rate of turbulent kinetic energy \(\left( { = \frac{1}{{Pr_{\varepsilon } }}} \right)\)- α
_{k} Inverse effective

*Pr*number for turbulent kinetic energy \(\left( { = \frac{1}{{Pr_{k} }}} \right)\)- α
_{t} Inverse effective

*Pr*number for turbulent flow \(\left( { = \frac{1}{{Pr_{t} }}} \right)\)- ε
Turbulent kinetic energy dissipation rate (m

^{2}/s^{3})- κ
Von Karman constant = 0.42

- η
Rate of strain in turbulent flow = Sk/ε

- λ
Thermal conductivity of nanofluid (W/mK)

- μ
Molecular viscosity of nanofluid (kg/ms)

- ν
Kinematic viscosity of nanofluid (m

^{2}/s)- ϕ
Conservation of mass, momentum and energy equations

- Φ
_{v} Volume concentration of nanoparticles

- ρ
Density of nanofluid (kg/m

^{3})

## Subscripts

- bf
Base fluid

- eff
Effective

- i
Inlet

- m
Mean

- nb
Neighbor cell

- np
Nanoparticle

- t
Turbulent

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