Skip to main content
SpringerLink
Log in

Natural convection in attic-shaped spaces subject to sudden and ramp heating boundary conditions

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

In this study, a discussion of the fluid dynamics in the attic space is reported, focusing on its transient response to sudden and linear changes of temperature along the two inclined walls. The transient behaviour of an attic space is relevant to our daily life. The instantaneous and non-instantaneous (ramp) heating boundary condition is applied on the sloping walls of the attic space. A theoretical understanding of the transient behaviour of the flow in the enclosure is performed through scaling analysis. A proper identification of the timescales, the velocity and the thickness relevant to the flow that develops inside the cavity makes it possible to predict theoretically the basic flow features that will survive once the thermal flow in the enclosure reaches a steady state. A time scale for the heating-up of the whole cavity together with the heat transfer scales through the inclined walls has also been obtained through scaling analysis. All scales are verified by the numerical simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

Abbreviations

A :

Slope of the attic

g :

Acceleration due to gravity

h :

Height of the attic

k :

Thermal conductivity

L :

Length of one inclined side of the roof

l :

Horizontal half length of the attic

Nu :

Nusselt number

Nu s :

Steady state Nusselt number

Nu sr :

Nusselt number at quasi-steady state

Nu h :

Nusselt number at quasi-steady mode

p :

Pressure

\( \hat{p} \) :

Dimensionless pressure

Pr :

Prandtl number

Ra :

Rayleigh number

T :

Temperature

T 0 :

Reference temperature

T c :

Cooling temperature

T h :

Heating temperature

t :

Time

t f :

Heating up time for sudden heating

t s :

Steady state time

t sr :

Quasi-steady time

t p :

Ramp time

u, v :

Velocity components

\( \hat{u},\hat{v} \) :

Dimensionless velocity components

u r :

Unsteady velocity scale

u q :

Velocity scale at quasi-steady state

u s :

Steady state velocity

\( \hat{u}_{r} \) :

Dimensionless unsteady velocity scale

\( \hat{u}_{s} \) :

Dimensionless steady-state velocity

u sr :

Quasi-steady velocity

\( \hat{u}_{sr} \) :

Dimensionless quasi-steady velocity

V :

Volume

x, y :

Coordinates

\( \hat{x},\hat{y} \) :

Dimensionless coordinates

β:

Thermal expansion coefficient

ΔT :

Temperature difference between hot surface and the ambient

δ T :

Thickness of the thermal boundary layer

δ Ts :

Steady state thickness of the thermal boundary layer

δ Tr :

Thickness of the thermal boundary layer at quasi-steady time

δ Tq :

Thickness of the thermal boundary layer at the quasi-steady stage

δ p :

Thickness of the thermal boundary layer when ramp is finished

\( \delta_{T}^{*} \) :

Dimensionless thickness of the thermal boundary layer

\( \delta_{Tr}^{*} \) :

Dimensionless thickness of the thermal boundary layer at quasi-steady time

\( \delta_{Tq}^{*} \) :

Dimensionless thickness of the thermal boundary layer at quasi-steady mode

κ :

Thermal diffusivity

ϕ:

Angle

ρ :

Density

ν :

Kinematic viscosity

θ :

Dimensionless temperature

τ :

Dimensionless time

τ r :

Dimensionless heating-up time (τ r  > τ p )

τ r′ :

Dimensionless heating-up time (τ r  < τ p )

τ s :

Dimensionless steady-state time

τ f :

Dimensionless heating-up time

τ p :

Dimensionless ramp time

τ sr :

Dimensionless quasi-steady time

References

  1. Flack RD (1980) The experimental measurement of natural convection heat transfer in triangular enclosures heated or cooled from bellow. ASME J Heat Transf 102:770–772

    Article  Google Scholar 

  2. Akinsete VA, Coleman TA (1982) Heat transfer by steady laminar free convection in triangular enclosures. Int J Heat Mass Transf 25:991–998

    Article  MATH  Google Scholar 

  3. Asan H, Namli L (2000) Laminar natural convection in a pitched roof of triangular cross-section: summer day boundary conditions. Energy Buildings 33:69–73

    Article  Google Scholar 

  4. Poulikakos D, Bejan A (1983) The fluid dynamics of an attic space. J Fluid Mech 131:251–269

    Article  MATH  MathSciNet  Google Scholar 

  5. Lin W, Armfield SW, Patterson JC (2008) Unsteady natural convection boundary-layer flow of a linearly stratified fluid with Pr < 1 on an evenly heated semi-infinite vertical plate. Int J Heat Mass Transf 51:327–343

    Article  MATH  Google Scholar 

  6. Lin W, Armfield SW, Patterson JC (2007) Cooling of a Pr < 1 fluid in a rectangular container. J Fluid Mech 574:85–108

    Article  MATH  MathSciNet  Google Scholar 

  7. Lin W, Armfield SW (2005) Scaling laws for unsteady natural convection cooling of fluid with Pr < 1 in a vertical cylinder. Phys Rev E 72:016306

    Article  Google Scholar 

  8. Lin W, Armfield SW (2005) Unsteady natural convection on an evenly heated vertical plate for Pr < 1. Phys Rev E 72:066309

    Article  Google Scholar 

  9. Lei C, Patterson JC (2002) Unsteady natural convection in a triangular enclosure induced by absorption of radiation. J Fluid Mech 460:181–209

    Article  MATH  Google Scholar 

  10. Lei C, Patterson JC (2005) Unsteady natural convection in a triangular enclosure induced by surface cooling. Int J Heat Fluid Flow 26:307–321

    Article  Google Scholar 

  11. Patterson JC, Lei C, Armfield SW, Lin W (2009) Scaling of unsteady natural convection boundary layers with a non-instantaneous initiation. Int J Therm Sci 48:1843–1852

    Article  Google Scholar 

  12. Saha SC, Patterson JC, Lei C (2010) Natural convection boundary layer adjacent to an inclined flat plate subject to sudden and ramp heating. Int J Therm Sci. doi:10.1016/j.ijthermalsci.2010.03.017

  13. Saha SC, Patterson JC, Lei C (2010) Natural convection in attics subject to instantaneous and ramp cooling boundary conditions. Energy Building. doi:10.1016/j.enbuild.2010.02.010

  14. Leonard BP, Mokhtari S (1990) ULTRA-SHARP nonoscillatory convection schemes for high-speed steady multidimensional flow. NASA TM 1-2568 (ICOMP-90-12), NASA Lewis Research Centre

  15. Patterson JC, Armfield SW (1990) Transient features of natural convections in a cavity. J Fluid Mech 219:469–497

    Article  Google Scholar 

Download references

Acknowledgments

This work was funded by the Australian Research Council (ARC).

Author information

Authors and Affiliations

  1. School of Engineering and Physical Sciences, James Cook University, Townsville, QLD, 4811, Australia

    Suvash C. Saha

  2. School of Civil Engineering, The University of Sydney, Sydney, NSW, 2006, Australia

    John C. Patterson & Chengwang Lei

Authors
  1. Suvash C. Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John C. Patterson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chengwang Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suvash C. Saha.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Saha, S.C., Patterson, J.C. & Lei, C. Natural convection in attic-shaped spaces subject to sudden and ramp heating boundary conditions. Heat Mass Transfer 46, 621–638 (2010). https://doi.org/10.1007/s00231-010-0607-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-010-0607-5

Keywords

Search

Navigation