# Entropy generation in non-Newtonian fluids due to heat and mass transfer in the entrance region of ducts

- 443 Downloads
- 11 Citations

## Abstract

A new solution for the Graetz problem (hydrodynamically developed forced convection in isothermal ducts) extended to power-law fluids and mass transfer with phase change at the walls is presented. The temperature and concentration spatial distributions in the corresponding entrance regions are obtained for two geometries (parallel-plates duct and circular pipe) in terms of appropriate dimensionless parameters. They are used to illustrate the effects of the fluid nature on the velocity, temperature and concentration distributions, on the axial evolution of the sensible and latent Nusselt numbers as well as on the local entropy generation rate due to velocity, temperature and concentration gradients.

## Keywords

Nusselt Number Entropy Generation Sherwood Number Slug Flow Circular Pipe## List of symbols

*A*Duct area

*Br*Brinkman number = (ζ

*u*_{ b }^{ n+1})/(*k*(*T*_{ in }−*T*_{ w })*D*_{ h }^{ n−1})*C*Dimensionless species concentration = (

*ω*−*ω*_{ w })/(*ω*_{ in }*−**ω*_{ w })*c*_{p}Specific heat of the fluid (J kg

^{−1}K^{−1})*D*Mass diffusion coefficient (m

^{2}s^{−1})*D*_{h}Hydraulic diameter (m)

*H*Half distance between parallel-plates (m)

*h*_{fg}Specific enthalpy of vaporization (J kg

^{−1})*Ja*Jakob number = (

*c*_{ p }(*T*_{ in }−*T*_{ w })/*h*_{ fg })*K*Thermal conductivity (W m

^{−1}K^{−1})*m*Metric parameter (

*m*= 0 for a flat duct,*m*= 1 for a circular pipe)*M*_{i}Molecular weight for species

*i*(kg mol^{−1})*n*Flow behavior index (dimensionless)

*Nu*_{l}Latent Nusselt number

*Nu*_{s}Sensible Nusselt number

*p*Pressure (Pa)

*P*Dimensionless pressure = (

*p/*(*ρ**u*_{ b }^{2}))*Pe*_{M}Péclet number for mass diffusion = (

*u*_{ b }*D*_{ h })/*D**Pe*_{T}Péclet number for thermal diffusion = (

*ρ C*_{ p }*D*_{ h }*u*_{ b })/*k**R*Pipe radius

- \( \bar{R} \)
Universal gas constant (J mol

^{−1}K^{−1})*R*_{h}Hydraulic radius (m)

*Sh*Sherwood number

*S*Dimensionless entropy generation = (\( s_{g}^{*} D_{h}^{2} /k \))

*S*_{T},*S*_{V},*S*_{C}Dimensionless entropy generation due to temperature, velocity, concentration

- \( s_{g}^{*} \)
Entropy generation rate (W m

^{−3}K^{−1})*u*Axial velocity (m s

^{−1})*U*Dimensionless axial velocity = (

*u*/*u*_{ b })*x*,*y*Axial and transverse coordinates (m)

*x**= 4α

*X*/((α +*m*+ 1)*Pe*_{ T })*x***= 4α

*X*/((α +*m*+ 1)*Pe*_{ M })*X*,*Y*Dimensionless axial and transverse coordinates (

*x*/*D*_{ h },*y*/*R*_{ h })

## Greek symbols

- α
= (

*n*+ 1*)*/*n*- ζ
Coefficient of consistence (Pa s

^{n})- θ
Dimensionless temperature = (

*T*−*T*_{ w })/(*T*_{ in }−*T*_{ w })*μ*Dynamic viscosity (Pa s)

*ρ*Density (kg m

^{−3})*Φ*Viscous dissipation (W m

^{−3})*ω*Mass fraction (kg of diffusing species/kg of mixture)

## Indices

*b*Bulk or average value

*in*Inlet

*w*Wall

## Notes

### Acknowledgments

This project is part of the R&D program of the NSERC Chair in Industrial Energy Efficiency established in 2006 at “Université de Sherbrooke”. The authors acknowledge the support of the Natural Sciences & Engineering Research Council of Canada, Hydro Québec, Rio Tinto Alcan and Canmet-Energy Research Center. The second author acknowledges gratefully 1 year as invited professor at Université de Sherbrooke (Canada), during 2010–2011.

## References

- 1.Shah RK, London AL (1978) Laminar flow forced convection in ducts. Academic Press, New YorkGoogle Scholar
- 2.Chakraborty S, Som SK, Rahul (2008) A boundary layer analysis for entrance region heat transfer in vertical microchannels within the slip flow regime. Int J Heat Mass Transf 51:3245–3250MATHCrossRefGoogle Scholar
- 3.Niazmand H, Renksizbulut M, Saeedi E (2008) Developing slip-flow and heat transfer in trapezoidal microchannels. Int J Heat Mass Transf 51:6126–6135MATHCrossRefGoogle Scholar
- 4.Hooman K (2008) A perturbation solution for forced convection in a porous-saturated duct. J Comp Appl Math 211:57–66MathSciNetMATHCrossRefGoogle Scholar
- 5.Minkowycz WJ, Haji-Sheikh A (2009) Asymptotic behaviors of heat transfer in porous passages with axial conduction. Int J Heat Mass Transf 52:3101–3108MATHCrossRefGoogle Scholar
- 6.Vera Marcos, Liñán Amable (2011) Exact solution for the conjugate fluid–fluid problem in the thermal entrance region of laminar counterflow heat exchangers. Int J Heat Mass Transf 54:490–499MATHCrossRefGoogle Scholar
- 7.Cheng CH, Weng CJ, Aung W (2000) Buoyancy-assisted flow reversal and convective heat transfer in entrance region of a vertical rectangular duct. Int J Heat Fluid Flow 21:403–411CrossRefGoogle Scholar
- 8.Maré T, Galanis N, Voicu I, Miriel J, Sow O (2008) Experimental and numerical study of mixed convection with flow reversal in coaxial double-duct heat exchangers. Exp Therm Fluid Sci 32:1096–1104CrossRefGoogle Scholar
- 9.Chong DT, Liu J, Yan J (2008) Effects of duct inclination angle on thermal entrance region of laminar and transition mixed convection. Int J Heat Mass Transf 51:3953–3962MATHCrossRefGoogle Scholar
- 10.Haji-Sheikh A, Beck JV (2008) Entrance heat transfer in rectangular ducts with constant axial energy input. Int J Heat Mass Transf 51:434–444MATHCrossRefGoogle Scholar
- 11.Behzadmehr A, Galanis N, Laneville A (2003) Low Reynolds number mixed convection in vertical tubes with uniform wall heat flux. Int J Heat Mass Transf 46:4823–4833MATHCrossRefGoogle Scholar
- 12.Bejan A (1982) Second-law analysis in heat transfer and thermal design. Adv Heat Transf 15:1–58CrossRefGoogle Scholar
- 13.Bejan A (1996) Entropy generation minimization. CRC Press, Boca RatonMATHGoogle Scholar
- 14.Demirel Y (2000) Thermodynamic analysis of thermomechanical coupling in Couette flow. Int J Heat Mass Transf 43:4205–4212MATHCrossRefGoogle Scholar
- 15.Mahmud S, Fraser RA (2003) The second-law analysis in fundamental heat transfer problems. Int J Therm Sci 42:177–186CrossRefGoogle Scholar
- 16.Sahin AZ (1998) A second law comparison for optimum shape of duct subjected to constant wall temperature and laminar flow. Heat Mass Transf 33:425–430CrossRefGoogle Scholar
- 17.Haddad OM, Alkam MK, Khasawneh MT (2004) Entropy generation due to laminar forced convection in the entrance region of a concentric annulus. Energy 29:35–55CrossRefGoogle Scholar
- 18.Ben Mansour R, Galanis N, Nguyen CT (2006) Dissipation and entropy generation in fully developed forced and mixed laminar convection. Int J Therm Sci 45:998–1007CrossRefGoogle Scholar
- 19.Yan WM, Lin TF, Tsay YL (1991) Evaporative cooling of liquid film through interfacial heat and mass transfer in a vertical channel-I. Experimental study. Int J Heat Mass Transf 34:1105–1111CrossRefGoogle Scholar
- 20.Yan WM, Lin TF (1991) Evaporative cooling of liquid film through interfacial heat and mass transfer in a vertical channel-II. Numerical study. Int J Heat Mass Transfer 34:1113–1124CrossRefGoogle Scholar
- 21.Debissi C, Orfi J, Ben Nasrallah S (2001) Evaporation of water by free convection in vertical channel including effects of wall radiative properties. Int J Heat Mass Transf 44:811–826CrossRefGoogle Scholar
- 22.Salah El-Din MM (1992) Fully developed forced convection in vertical channel with combined buoyancy forces. Int Commun Heat Mass Transf 19:239–248CrossRefGoogle Scholar
- 23.Boulama K, Galanis N (2004) Analytical solution for fully developed mixed convection between parallel vertical plates with heat and mass transfer. ASME J Heat Transf 126:381–388CrossRefGoogle Scholar
- 24.Azzizi Y, Benhamou B, Galanis N, El Ganaoui M (2007) Buoyancy effects on upward and downward laminar mixed convection heat and mass transfer in a vertical channel. Int J Numer Methods Heat Fluid Flow 17:333–353CrossRefGoogle Scholar
- 25.Laaroussi N, Lauriat G, Desrayaud G (2009) Effects of variable density for film evaporation on laminar mixed convection in a vertical channel. Int J Heat Mass Transf 52:151–164MATHCrossRefGoogle Scholar
- 26.Oulaid O, Benhamou B, Galanis N (2010) Flow reversal in combined laminar mixed convection heat and mass transfer with phase change in a vertical channel. Int J Heat Fluid Flow 31:711–721CrossRefGoogle Scholar
- 27.San JY, Worek WM, Lavan Z (1987) Entropy generation in combined heat and mass transfer. Int J Heat Mass Transf 30:1359–1369CrossRefGoogle Scholar
- 28.Carrington CG, Sun ZF (1991) Second law analysis of combined heat and mass transfer phenomena. Int J Heat Mass Transf 34:2767–2773MATHCrossRefGoogle Scholar
- 29.Boulama KG, Galanis N, Orfi J (2006) Entropy generation in a binary gas mixture in the presence of thermal and solutal mixed convection. Int J Therm Sci 45:51–59CrossRefGoogle Scholar
- 30.Hashemabadi SH, Etemad SGh, Thibault J (2004) Forced convection heat transfer of Couette–Poiseuille flow of nonlinear viscoelastic fluids between parallel plates. Int J Heat Mass Transf 47:3985–3991MATHCrossRefGoogle Scholar
- 31.Mirzazadeh M, Shafaei A, Rashidi F (2008) Entropy analysis for non-linear viscoelastic fluid in concentric rotating cylinders. Int J Therm Sci 47:1701–1711CrossRefGoogle Scholar
- 32.Bouzid N, Saouli S, Aiboud-Saouli S (2008) Entropy generation in ice slurry pipe flow. Int J Refrig 31:1453–1457CrossRefGoogle Scholar
- 33.Shah RK, Bhatti MS (1987) Laminar convective heat transfer in ducts. In: Kakaç S, Shah RK, Aung W (eds) Handbook of single-phase convective heat transfer, (Chap. 3). Wiley, New YorkGoogle Scholar
- 34.Abramowitz M, Stegun IA (eds) (1968) Handbook of Mathematical Functions, U.S. Dept. of Commerce, National Bureau of Standards, Washington D.C., May 1968Google Scholar
- 35.Khellaf K, Lauriat G (1997) A new analytical solution for heat transfer in the entrance region of ducts: hydrodynamically developed flows of power-law fluids with constant wall temperature. Int J Heat Mass Transf 40(14):3443–3447MATHCrossRefGoogle Scholar