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Step response for free convection between parallel walls

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Abstract

This paper analyzes the heat transfer for laminar natural convection of a viscous, incompressible, Newtonian fluid between parallel vertical plates, due to a step change in the temperature of one of the two plates. The momentum conservation and energy conservation equations are analytically solved applying the Laplace transform technique. The temperature profile, which is presented as an original solution, is a fast converging series of sinusoidal and exponential functions. It is very easy to be used and highlights that the transient distribution of the temperature tends to the steady-state one when the numerical value of dimensionless time approaches the Prandtl number. The results, obtained for different fluids (liquid sodium, air, water and lubricating oil), show that the thermal transient is always faster than the dynamic transient.

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Abbreviations

c p :

Specific heat at constant pressure (J kg−1 K−1)

D :

Equivalent diameter (2 H, m)

g :

Gravitational acceleration (m s−2)

Gr :

Grashof number (g β Η 3 (T 1 −T 0 ) ν −2)

h :

Heat transfer coefficient (W m−2 K−1)

H :

Distance between two vertical walls (m)

k :

Fluid thermal conductivity (W m−1 K−1)

Nu :

Nusselt number (h D k −1)

p :

Pressure (Pa)

p s :

Hydrostatic pressure (Pa)

P :

Dimensionless pressure

Pr :

Prandtl number (ρ o ν c p k −1)

R :

Buoyancy force distribution parameter

s :

Coordinate of the Laplace transform

t :

Dimensionless time

t’ :

Time (s)

T :

Fluid temperature (K)

T 0 :

Ambient and initial (fluid and wall) temperature (K)

T 1 :

Wall temperature at x = 1, t > 0 (K)

u :

Fluid velocity (m s−1)

U :

Dimensionless fluid velocity

x, z:

Dimensionless Cartesian coordinates

Y :

Function defined in Eq. (18)

Z :

Dimensionless height of the channel

β 0 :

Thermal expansion coefficients at temperature T 0 (K−1)

μ :

Dynamic viscosity (Pa s)

ν :

Kinematic viscosity (m2 s−1)

θ :

Dimensionless temperature

ρ :

Fluid density (kg m−3)

ρ o :

Fluid density at temperature T 0 (kg m−3)

τ :

Dimensionless tangential stress

ξ, ζ :

Cartesian coordinates (m)

ψ :

Integration variable

b :

Fluid bulk

ss :

Steady-state

w :

Wall

References

  1. Aung W (1972) Fully developed laminar free convection between vertical plates heated asymmetrically. Int J Heat Mass Transfer 15(8):1577–1580

    Article  MATH  Google Scholar 

  2. Aung W, Fletcher LS, Sernas V (1972) Developed laminar free convection between vertical flat plates asymmetric heating. Int J Heat Mass Transf 15(11):2293–2308

    Article  MATH  Google Scholar 

  3. Vajravelu K, Sastri K (1977) Fully developed laminar free convection flow between two parallel vertical walls. Int J Heat Mass Transf 20(6):655–660

    Article  Google Scholar 

  4. Tanda G (1988) Natural convection in partially heated vertical channels. Heat Mass Transfer (Warme und Stoffubertragung) 23(5):307–312

    Google Scholar 

  5. Terekhov VI, Ekaid AL (2011) Laminar natural convection between vertical isothermal heated plates with different temperatures. J Eng Thermophys 20(4):416–433

    Article  Google Scholar 

  6. Kettleborough CF (1972) Transient laminar free convection between heated vertical plates including entrance effects. Int J Heat Mass Transf 15(5):883–896

    Article  MATH  Google Scholar 

  7. Joshi HM (1988) Transient effects in natural convection cooling of vertical parallel plates. Int Comm Heat Mass Transf 15(2):227–238

    Article  Google Scholar 

  8. Buonomo B, Manca O (2010) Natural convection slip flow in a vertical microchannel heated at uniform heat flux. Int J Therm Sci 49(8):1339–1344

    Article  Google Scholar 

  9. Buonomo B, Manca O (2012) Transient natural convection in a vertical microchannel heated at uniform heat flux. Int J Therm Sci 56:35–47

    Article  Google Scholar 

  10. Weng HC, Chen C-K (2008) Variable physical properties in natural convective gas microflow. J Heat Transf 130(8):082401

    Article  Google Scholar 

  11. Al-Nimr MA, El-Shaarawi MAI (1995) Analytical solutions for transient laminar fully developed free convection in vertical channels. Heat Mass Transfer 30(4):241–248

    Article  Google Scholar 

  12. Paul T, Jha BK, Singh AK (1996) Transient free convection flow in a vertical channel with constant heat flux on walls. Heat Mass Transf 32(1–2):61–63

    Article  Google Scholar 

  13. Singh AK, Gholami HR, Soundalgekar VM (1996) Transient free convection flow between two vertical parallel plates. Heat Mass Transf 31(5):329–332

    Article  Google Scholar 

  14. Singh AK, Paul T (2006) Transient natural convection between two vertical walls heated/cooled asymmetrically. Int J Appl Mech Eng 11(1):143–154

    MATH  Google Scholar 

  15. Ajibade AO, Jha BK (2009) Transient natural convection flow between vertical parallel plates with temperature dependent heat sources/sinks. Int J Heat Technol 27(2):87–94

    Google Scholar 

  16. Ajibade AO, Jha BK (2010) Transient natural convection flow between vertical parallel plates: one plate isothermally heated and the other thermally insulated. J Process Mech Eng 224(4):247–252

    Article  Google Scholar 

Download references

Acknowledgments

The research leading to these results has received the financial support of the Italian MIUR in the PRIN 2009 framework.

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Authors and Affiliations

  1. Department of Industrial Engineering, University of Parma, Parco Area delle Scienze n. 181/A, 43124, Parma, PR, Italy

    Marco Spiga & Pamela Vocale

Authors
  1. Marco Spiga
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  2. Pamela Vocale
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Correspondence to Pamela Vocale.

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Spiga, M., Vocale, P. Step response for free convection between parallel walls. Heat Mass Transfer 51, 1761–1768 (2015). https://doi.org/10.1007/s00231-015-1539-x

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  • DOI: https://doi.org/10.1007/s00231-015-1539-x

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