# Thermodynamic optimization of conjugate convection from a finned channel using genetic algorithms

- 253 Downloads
- 16 Citations

## Abstract

For the first time, this study reports the results of numerical investigation of conjugate convection from a finned channel. The computational domain of investigation consists of a horizontal channel with vertical rectangular fins being mounted on outside of the channel. The equations governing two-dimensional, steady, incompressible, constant property laminar flow have been solved for the fluid flowing outside the channel. In doing this, Boussinesq assumption is assumed to be valid for the fluid flowing outside the channel along the fins. For the fluid flowing inside the channel, flow is assumed to be turbulent with forced convection as the mode of heat transfer. From a large volume of numerically generated data correlations have been proposed for (1) Nusselt number and (2) Entropy generated by the system. These correlations are finally used to obtain thermodynamic optimum where in we seek a solution with minimum total entropy generation rate for varying heat duties, by using the state-of-the art Genetic algorithms.

## Keywords

Heat Transfer Nusselt Number Natural Convection Heat Transfer Rate Entropy Generation## List of symbols

*C*_{p}isobaric specific heat capacity (J kg

^{−1}K^{−1})- \(\bar C_{\text{f}} \)
mean friction coefficient

*D*channel spacing (m)

*D*_{h}hydraulic diameter (m)

*g*acceleration due to gravity (m s

^{−2}, 9.81 m s^{−2})*G*_{ν}turbulence production (kg m

^{−1}s^{−2})*H*height of fins

*h*heat transfer coefficient of the fluid (Wm

^{−2}K^{−1})*k*thermal conductivity of fin material (Wm

^{−1}K^{−1})*L*length of the channel (m)

*L*/*D*aspect ratio

*m*_{f}mass flow rate of the inside fluid (kg s

^{−1})*n*number of fins

- Nu
_{L} Nusselt number on the basis of length of channel

*P*pressure (N m

^{−2})- Pr
Prandtl number of fluid outside

*Q*total heat transfer (W)

- Ra
_{L} Rayleigh number, \((g\beta /\nu _o^2 )(T_{{\text{in}}} - T_\infty )L^3 \Pr \)

- Re
Reynolds number of the internal flow (ρ

*u*_{in}*D*_{h}/μ_{in})*S*fin spacing (m)

*S*_{fluid flow}entropy generation rate due to fluid flow (W K

^{−1})*S*_{heat}entropy generation rate due to heat transfer (W K

^{−1})*S*_{gen}total entropy generated by the system (W K

^{−1})*S*^{*}_{gen}non dimensional entropy of the system (

*S*_{gen}/*S*_{heat})*T*temperature (K)

*u*horizontal velocity (ms

^{−1})*v*vertical velocity (ms

^{−1})*W*pumping power due to fluid friction (W)

*x*co-ordinate along the length of the channel (m)

*x*^{*}non dimensional distance (4

*x*ν/*D*^{2}*u*_{in})*y*co-ordinate along the spacing of the channel (m)

*Y*_{v}turbulent destruction (kg m

^{−1}s^{2})*Z*composite objective function

## Greek symbols

- α
_{i}′ turbulent thermal diffusivity of the inside fluid (m

^{2}s^{−1})- β
coefficient of thermal expansion (K

^{−1})- γ
heat transfer per unit mass of the system

- Δ
*T* temperature difference (K)

- λ
penalty parameter

- μ
dynamic viscosity (kg m

^{−1}s^{−1})- μ
_{t} turbulent viscosity (kg m

^{−1}s^{−1})- ν
_{i} kinematic viscosity (m

^{2}s^{−1})- ν
_{i}′ turbulent kinematic viscosity of the inside fluid (m

^{2}s^{−1})- ρ
density (kg m

^{−3})- τ
wall shear stress (N m

^{−2})- \({\bar \nu }\)
transported variable for the inside fluid (m

^{2}s^{−1})

## Subscripts

- atm
atmosphere

- al
aluminum

- b
base temperature

- c
critical value of properties

- cs
cross sectional

- d
design

- e
outside

- f
fluid

- fin
finned

- i
interface

- in
inlet

- s
surface

- unfin
unfinned

- ∞
ambient

## References

- 1.Sasikumar M, Balaji C (2002) Optimization of Convective fin systems: a holistic approach. Heat Mass Transfer 39:57–68CrossRefGoogle Scholar
- 2.Bejan A (1979) A study of entropy generation in fundamental convective heat transfers. J Heat Transfer 101:718–725Google Scholar
- 3.Bejan A (1980) Second law analysis in heat transfer. Energy 5:721–732CrossRefGoogle Scholar
- 4.Nag PK, Mukherjee P (1987) Thermodynamic optimization of convective heat transfer through a duct with constant wall temperature. Int J Heat Mass Transfer 30:401–405Google Scholar
- 5.Nag PK, Kumar N (1989) Second law optimization of convective heat transfer through a duct with constant heat Flux. Int J Energy Res 13:537–543Google Scholar
- 6.Lin WW, Lee DJ (1996) Second law analysis on a pin fin array under cross flow. Int J Heat Mass Transfer 40:1937–1945CrossRefGoogle Scholar
- 7.Balaji C, Sri Jayaram K, Venkateshan SP (1996) Thermodynamic optimization of Tubular space Radiators. J Thermophys Heat Transfer 10:705–707Google Scholar
- 8.Sahin AZ (2000) Entropy generation in turbulent liquid flow through a smooth duct subjected to constant wall temperature. Int J Heat Mass Transfer 43:1469–1478CrossRefGoogle Scholar
- 9.Welling JR, Wooldridge CB (1965) Free convection heat transfer coefficients from rectangular vertical fins. J Heat Transfer 87:439Google Scholar
- 10.Harahap F, McManus HN (1967) Natural convection heat transfer from horizontal Rectangular Fin Arrays. J Heat Transfer 89:32–38Google Scholar
- 11.Jones CD, Smith LF (1970) Optimum arrangement of Rectangular Fins on horizontal surfaces for free convection heat transfer. J Heat Transfer 92:6–10Google Scholar
- 12.Sparrow EM, Lee L (1975) Effects of fin base temperature in a multifin array. J Heat Transfer 97:463–465Google Scholar
- 13.Sobhan CB, Venkateshan SP, Seetharamu KN (1990) Experimental studies on steady free convection heat transfer from fins and fin arrays. Warme- und Stoffubertragung 25:345–352Google Scholar
- 14.Sunil Kumar S (1993) A numerical study of optimized space radiators. Ph.D Thesis Department of Mechanical Engineering, Indian Institute of Technology MadrasGoogle Scholar
- 15.Sasikumar M, Balaji C (2002) A holistic optimization of convecting radiating fin systems. J Heat Transfer 124:1110–1116CrossRefGoogle Scholar
- 16.Prakash V, Balaji C (2004) Turbulent forced convection in a parallel plate channel with natural convection on the outside. Int Commun Heat Mass Transfer 31:1027–1036CrossRefGoogle Scholar
- 17.Bejan A (1984) Convection heat transfer. Wiley, New YorkGoogle Scholar
- 18.Kim SH, Anand NK (1994) Outflow boundary condition for the temperature field in channels with periodically positioned heat sources in the presence of wall conduction. Numer Heat Transfer B 25:163–176Google Scholar
- 19.Bejan A (1980) Entropy generation through heat and fluid flow. Wiley, New YorkGoogle Scholar
- 20.Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley Longman Inc., New YorkGoogle Scholar
- 21.Jaluria Y (1998) Design and optimization of thermal systems. McGraw-Hill International EditionGoogle Scholar