# Investigation of heat transfer and pressure drop of turbulent flow in tubes with successive alternating wall deformation under constant wall temperature boundary conditions

## Abstract

In this study, the effects of the presence of multi-longitudinal vortex and various rotation angles between the pitches of alternating elliptical axis (AEA) tubes on heat transfer and pressure drop of turbulent flow were numerically investigated. Turbulent flow of water fluid was simulated at Reynolds numbers of 10,000–60,000. The turbulent flow and heat transfer in the tubes were discussed in terms of parameters such as static pressure, velocity magnitude, wall shear stress, turbulent intensity, performance evaluation criterion and the field synergy principle. The results demonstrate that most heat transfer occurs in the transition zone, but this also caused a high rate of pressure drop. Increasing the rotation angle between pitches from 60° to 80° increased the heat transfer, which increased the number of the multi-longitudinal vortices from four to eight and better mixed the cold fluid with the hot fluid near the tube wall on more paths. The friction factor decreased and the average Nusselt number increased as the Reynolds number increased. Both parameters increased as the angle of the pitch rotation increased. The performance evaluation criteria for all AEA tubes at a constant pumping power showed that the highest value (1.09) was achieved at a low Reynolds number (*Re* = 10,000) in the AEA 90° tube.

## Keywords

Turbulent flow Heat transfer Multi-longitudinal vortex Field synergy principle Performance evaluation criterion Alternating elliptical axis tube## Nomenclature

*A*Area, m

^{2}- A
Major axes length of elliptical cross section, mm

- B
Minor axes length of elliptical cross section, mm

- C
Transition length, mm

*C*_{p}Specific heat, J/(kg K)

*C*Perimeter of the ellipse, m

*D*_{h}Hydraulic diameter, m

*d*Circular tube diameter of AEA tube, mm

*f*Friction factor, \(f = {{(\Delta p D_{\text{h}} )} \mathord{\left/ {\vphantom {{(\Delta p D_{\text{h}} )} {\left( {\frac{1}{2}\rho u_{\text{avg}}^{2} L} \right)}}} \right. \kern-0pt} {\left( {\frac{1}{2}\rho u_{\text{avg}}^{2} L} \right)}}\)

*G*_{k}Production of turbulence kinetic energy, kg/(m s

^{3})*g*Gravitational acceleration, m/s

^{2}*h*_{loss}Irreversible head loss, m

*K*Thermal conductivity, W/(m K)

*k*Turbulence kinetic energy, m

^{2}/s^{2}*L*Length of tube, mm

*L*_{d}Length of inlet and outlet circular tube, mm

*Nu*Nusselt number, \(Nu = (q^{\prime\prime}D_{\text{h}} )/(K (T_{\text{w}} - T_{\text{b}} ))\)

*P*Pressure, kg/(m s

^{2})- P
Pitch length, mm

- \(\overline{P}\)
Mean pressure, kg/(m s

^{2})*P*_{k}Improved production of turbulence kinetic energy, kg/(m s

^{3})*Pr*Prandtl number,

*Pr*=*µC*_{p}/*K**q*″Heat flux, W/m

^{2}*Re*Reynolds number, Re =

*ρuD*_{ h }/*µ**S*Strain tensor magnitude, s

^{−1}*S*_{ij}Components of the mean strain tensor, s

^{−1}*T*Temperature, K

*T*_{b}Average bulk temperature, K

- \(\overline{T}\)
Mean temperature, K

*T*′Turbulent temperature fluctuations, K

*u*_{i}Velocity, m/s

- \(\overline{{u_{i} }}\)
Mean velocity, m/s

*u*_{i}′Turbulent velocity fluctuations, m/s

*u**Friction velocity, m/s

*V*Velocity, m/s

*x*_{i}Cartesian coordinates, m

*Y*Distance of the closest computational node from the wall, m

*y*^{+}Dimensionless wall distance

*z*Axial distance from inlet, m

## Greek symbols

*α*Kinetic energy correction factors

*δ*_{ij}Kronecker delta

*ɛ*Turbulent dissipation rate, m

^{2}/s^{3}*ɛ*_{imn}Tensor of Levi–Civita

*θ*Angle between major axes of elliptical cross-sectional tubes

*μ*Laminar dynamic viscosity, kg/(m s)

*μ*_{t}Turbulent dynamic viscosity, kg/(m s)

*μ*_{eff}Effective dynamic viscosity, kg/(m s)

*ν*Kinematic viscosity, m

^{2}/s*ρ*Density, kg/m

^{3}*σ*_{k}Turbulent Prandtl number of

*k**σ*_{ɛ}Turbulent Prandtl number of

*ɛ**τ*_{ij}Stress tensor, kg/(m s

^{2})*τ*_{w}Wall shear stress, kg/(m s

^{2})*Ω*Characteristic rotation rate, s

^{−1}*Ω*_{ij}Components of the vorticity tensor, s

^{−1}*Ω*_{m}^{rot}Components of the system rotation vector, s

^{−1}*ω*Turbulence eddy frequency, s

^{−1}

## Subscripts

*a*Section A

- avg
Average

*b*Section B

- eff
Effective

- s
Smooth tube

- w
Wall

## References

- 1.Liu S, Sakr M (2013) A comprehensive review on passive heat transfer enhancements in pipe exchangers. Renew Sustain Energy Rev 19:64–81CrossRefGoogle Scholar
- 2.Maiga SEB, Palm SJ, Nguyen CT, Roy G, Galanis N (2005) Heat transfer enhancement by using nanofluids in forced convection flows. Int J Heat Fluid Flow 26(4):530–546CrossRefGoogle Scholar
- 3.Hassan H, Harmand S (2015) Effect of using nanofluids on the performance of rotating heat pipe. Appl Math Model 39(15):4445–4462MathSciNetCrossRefGoogle Scholar
- 4.Gupta M, Kumar R, Arora N, Kumar S, Dilbagi N (2015) Experimental investigation of the convective heat transfer characteristics of TiO
_{2}/distilled water nanofluids under constant heat flux boundary condition. J Braz Soc Mech Sci Eng 37(4):1347–1356CrossRefGoogle Scholar - 5.Mousavi SM, Farhadi M, Sedighi K (2016) Effect of non-uniform magnetic field on biomagnetic fluid flow in a 3D channel. Appl Math Model 40(15):7336–7348MathSciNetCrossRefGoogle Scholar
- 6.Mokhtari M, Hariri S, Gerdroodbary MB, Yeganeh R (2017) Effect of non-uniform magnetic field on heat transfer of swirling ferrofluid flow inside tube with twisted tapes. Chem Eng Process 117:70–79CrossRefGoogle Scholar
- 7.Soltanipour H, Khalilarya S, Motlagh SY, Mirzaee I (2017) The effect of position-dependent magnetic field on nanofluid forced convective heat transfer and entropy generation in a microchannel. J Braz Soc Mech Sci Eng 39(1):345–355CrossRefGoogle Scholar
- 8.Habib MA, Mobarak AM, Attya AM, Aly AZ (1993) Enhanced heat transfer in channels with staggered fins of different spacings. Int J Heat Fluid Flow 14(2):185–190CrossRefGoogle Scholar
- 9.Wang W, Bao Y, Wang Y (2015) Numerical investigation of a finned-tube heat exchanger with novel longitudinal vortex generators. Appl Therm Eng 86:27–34CrossRefGoogle Scholar
- 10.Kundu B (2015) Beneficial design of unbaffled shell-and-tube heat exchangers for attachment of longitudinal fins with trapezoidal profile. Case Stud Therm Eng 5:104–112CrossRefGoogle Scholar
- 11.Yang S, Zhang L, Xu H (2011) Experimental study on convective heat transfer and flow resistance characteristics of water flow in twisted elliptical tubes. Appl Therm Eng 31(14):2981–2991CrossRefGoogle Scholar
- 12.Pozrikidis C (2015) Stokes flow through a twisted tube with square cross-section. Eur J Mech B 51:37–43MathSciNetCrossRefGoogle Scholar
- 13.Zhao N, Yang J, Li H, Zhang Z, Li S (2016) Numerical investigations of laminar heat transfer and flow performance of Al
_{2}O_{3}–water nanofluids in a flat tube. Int J Heat Mass Transf 92:268–282CrossRefGoogle Scholar - 14.Elsebay M, Elbadawy I, Shedid MH, Fatouh M (2016) Numerical resizing study of Al
_{2}O_{3}and CuO nanofluids in the flat tubes of a radiator. Appl Math Model 40(13):6437–6450CrossRefGoogle Scholar - 15.Guo ZY, Li DY, Wang BX (1998) A novel concept for convective heat transfer enhancement. Int J Heat Mass Transf 41(14):2221–2225MathSciNetCrossRefMATHGoogle Scholar
- 16.Li B, Feng B, He Y-L, Tao W-Q (2006) Experimental study on friction factor and numerical simulation on flow and heat transfer in an alternating elliptical axis tube. Appl Therm Eng 26(17):2336–2344CrossRefGoogle Scholar
- 17.Meng J-A, Liang X-G, Chen Z-J, Li Z-X (2005) Experimental study on convective heat transfer in alternating elliptical axis tubes. Exp Thermal Fluid Sci 29(4):457–465CrossRefGoogle Scholar
- 18.Chen W-L, Dung W-C (2008) Numerical study on heat transfer characteristics of double tube heat exchangers with alternating horizontal or vertical oval cross section pipes as inner tubes. Energy Convers Manag 49(6):1574–1583CrossRefGoogle Scholar
- 19.Sajadi AR, Yamani Douzi Sorkhabi S, Ashtiani D, Kowsari F (2014) Experimental and numerical study on heat transfer and flow resistance of oil flow in alternating elliptical axis tubes. Int J Heat Mass Transf 77:124–130CrossRefGoogle Scholar
- 20.White FM (1991) Viscous fluid flow, 2nd edn. McGraw-Hill, New YorkGoogle Scholar
- 21.Guo ZY (2003) A brief introduction to a novel heat-transfer enhancement heat exchanger. Internal report, Department of Engineering Mechanics, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing, ChinaGoogle Scholar
- 22.Chen W-L, Guo Z, Co-K Chen (2004) A numerical study on the flow over a novel tube for heat-transfer enhancement with a linear Eddy-viscosity model. Int J Heat Mass Transf 47(14):3431–3439CrossRefMATHGoogle Scholar
- 23.Sajadi AR, Kowsary F, Bijarchi MA, Sorkhabi SYD (2016) Experimental and numerical study on heat transfer, flow resistance, and compactness of alternating flattened tubes. Appl Therm Eng 108:740–750CrossRefGoogle Scholar
- 24.Launder BE, Spalding DB (1972) Lectures in mathematical models of turbulence. Academic Press, LondonMATHGoogle Scholar
- 25.Van Doormaal JP, Raithby GD (1984) Enhancements of the SIMPLE method for predicting incompressible fluid flows. Numer Heat Transf 7(2):147–163MATHGoogle Scholar
- 26.Chorin AJ (1968) Numerical solution of the Navier-Stokes equations. Math Comput 22(104):745–762MathSciNetCrossRefMATHGoogle Scholar
- 27.Versteeg HK, Malalasekera W (2007) An introduction to computational fluid dynamics: the finite method, 2nd edn. Pearson Education, LondonGoogle Scholar
- 28.Holmes DG, Connell SD (1989) Solution of the 2D Navier-Stokes equations on unstructured adaptive grids. In: 9th computational fluid dynamics conference. AIAA, p 1932Google Scholar
- 29.Rausch RD, Batina JT, Yang HTY (1991) Spatial adaption procedures on unstructured meshes for accurate unsteady aerodynamic flow computation. In: 32nd structures, structural dynamics, and materials conference. AIAA, p 1106Google Scholar
- 30.Warming RF, Beam RM (1976) Upwind second-order difference schemes and applications in aerodynamic flows. AIAA J 14(9):1241–1249MathSciNetCrossRefMATHGoogle Scholar
- 31.Launder BE, Spalding DB (1974) The numerical computation of turbulent flows. Comput Methods Appl Mech Eng 3(2):269–289CrossRefMATHGoogle Scholar
- 32.Petukhov BS, Irvine TF, Hartnett JP (1970) Advances in heat transfer. Academic Press, New YorkMATHGoogle Scholar
- 33.Gnielinski V (1976) New equations for heat and mass-transfer in turbulent pipe and channel flow. Int Chem Eng 16(2):359–368
**(cited in [34])**Google Scholar - 34.Bergman TL, Lavine AS, Incropera FP, Dewitt DP (2011) Fundamentals of heat and mass transfer, 7th edn. Wiley, New JerseyGoogle Scholar
- 35.Fluent A (2011) ANSYS FLUENT Theory Guide. ANSYS Inc., CanonsburgGoogle Scholar
- 36.Spalart PR, Shur M (1997) On the sensitization of turbulence models to rotation and curvature. Aerosp Sci Technol 1(5):297–302CrossRefMATHGoogle Scholar
- 37.Smirnov PE, Menter FR (2009) Sensitization of the SST turbulence model to rotation and curvature by applying the Spalart–Shur correction term. J Turbomach 131(4):041010CrossRefGoogle Scholar
- 38.Çengel YA, Cimbala JM (2014) Fluid mechanics fundamentals and applications, 3rd edn. McGraw-Hill, New YorkGoogle Scholar
- 39.Tao W-Q, Guo Z-Y, Wang B-X (2002) Field synergy principle for enhancing convective heat transfer: its extension and numerical verifications. Int J Heat Mass Transf 45(18):3849–3856CrossRefMATHGoogle Scholar
- 40.Webb RL, Kim N-H (1994) Principles of enhanced heat transfer. Taylor Francis, New YorkGoogle Scholar