Natural Convection Heat Transfer in a Saltbox Roof with by tdj18264

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									                           Proceedings of Clima 2007 WellBeing Indoors



Natural Convection Heat Transfer in a Saltbox Roof with Eave in Winter
Day Conditions

Ahmet Koca, Hakan F. Oztop and Yasin Varol

Firat University, Turkey

Corresponding email: ysnvarol@gmail.com


SUMMARY

In this study, a numerical analysis has been performed to examine the natural convection heat
transfer and flow field inside a saltbox roof with eave in winter day conditions. This analysis
is important for applications since it shows the effective parameters on natural convection
heat transfer. The governing equations of natural convection in streamfunction-vorticity form
were solved using central difference method to obtain flow and temperature fields inside the
roof. Also, the Successive Under Relaxation (SUR) technique was used to solve linear
algebraic equations. Results are presented by streamlines, isotherms and local and mean
Nusselt number by using effective parameters as Rayleigh number, aspect ratio of the roof,
and length of eave. The results indicated that both eave length and aspect ratio can be used as
control parameter for heat transfer.

Keywords: natural convection, saltbox roof, eave


INTRODUCTION

Natural convection is formed in some parts of buildings due to temperature difference
between cold and hot surfaces. The roof is an important part of the building because it
protects the building from the environmental effects. Natural convection heat transfer
occurred inside the roof due to temperature difference which depends on the climate such as
winter or summer.

Natural convection heat transfer analyzed in differentially heated square or rectangular
enclosures in earlier studies due to simplicity of numerical solution [1]. However, the number
of study about partially heated enclosures is very limited and they investigated mostly for
square cross sectional geometries [2-4].

Researchers the natural convection in roofs using different numerical techniques for triangular
cross-sectioned roofs (Gable roofs) were investigated by different researchers. Asan and
Namli [5] investigated the laminar natural convection heat transfer inside the triangular roof
with different aspect ratio in winter day boundary conditions. They observed that, both aspect
ratio and Rayleigh number affect the heat transfer inside the roof. Tzeng et al. [6], Akinsete
and Coleman [7], Holtzman et al. [8] and, recently, Ridouane et al. [9] investigated the
laminar natural convection inside the triangular cross-section enclosure which is applicable
for roof geometry of building at different constant temperature or constant heat flux boundary
conditions. On the other hand, Varol et al. [10,11] investigated the natural convection in
different roof shapes including Gambrel and Saltbox roofs. They observed that natural
                          Proceedings of Clima 2007 WellBeing Indoors



convection flow fields strongly depend on the geometrical shape and thermal boundary
conditions of the roof. Moukalled and Acharya [12] analyzed the natural convection heat
transfer in a trapezoidal roof with baffles that located at the top and bottom of the wall. They
indicated that baffles are effective parameters on thermal field inside the roof.

The main purpose of the present study is to analyze the flow and thermal fields inside the
saltbox roof with eave in winter day conditions. According to knowledge of the authors and
literature review given above, laminar natural convection heat transfer has not been
investigated yet for partially heated roofs. Thus, the present study will be the first attempt in
that area and it will help the designers and constructers from the energy saving point of view.

The photo and physical model of the saltbox roof with eave and the boundary conditions is
shown in Figure 1 with coordinate system. It is considered that the inclined surface and eave
have constant cold temperature. Also, room ceiling (bottom surface of the roof) has constant
hot temperature to simulate winter day boundary conditions. The vertical boundary has
adiabatic with height, H. Thus, total length of the bottom of roof (L) can be calculated by
adding the length of room ceiling (B) and the length of eave (E). An aspect ratio can be
defined as the ratio of the height of the roof to the length of bottom wall (AR=H/L) which can
be changed according to climatical conditions.




Figure 1a. Photo of saltbox roof from Ref. [14].
             y



                     TCold, u=0, v=0
    H

                             THot, u=0, v=0
                                                                        x
                                                                TCold
Insulated wall
   u=0
  v=0
  ∂T / ∂x = 0


                                B                        E
                                    L
Figure 1b. Physical model
                          Proceedings of Clima 2007 WellBeing Indoors



GOVERNING EQUATIONS AND NUMERICAL SOLUTIONS

The governing equations of natural convection (Eqs. 1-3) are written in streamfunction-
vorticity form for laminar regime in two-dimensional form for steady, incompressible, and
Newtonian fluid with Boussinesq approximation. It is assumed that radiation heat exchange is
negligible according to other modes of heat transfer and the gravity acts in vertical direction.

             ∂2Ψ ∂ 2Ψ
     −Ω =          +                                                                    (1)
             ∂X 2 ∂Y 2
     ∂ 2 Ω ∂ 2 Ω 1 ⎛ ∂Ψ ∂Ω ∂Ψ ∂Ω ⎞            ⎛ ∂θ ⎞
           +       = ⎜          −      ⎟ − Ra ⎜    ⎟                                    (2)
     ∂X  2
             ∂Y  2
                      Pr ⎝ ∂Y ∂X ∂X ∂Y ⎠      ⎝ ∂X ⎠
     ∂ 2 θ ∂ 2 θ ∂Ψ ∂θ ∂Ψ ∂θ
           +        =         −                                                         (3)
     ∂X 2 ∂Y 2 ∂Y ∂X ∂X ∂Y

The employed non-dimensional variables are given as

          x     y     ψ Pr      ω(L) 2 Pr                     T - Tcold
     X=     , Y= , Ψ=      , Ω=           ,             θ=                ,             (4)
          L     L      υ           υ                         Thot − Tcold
        ∂ψ       ∂ψ       ⎛ ∂v ∂u ⎞    β g(Thot − Tcold )L3 Pr       υ
     u=    , v=−    , ω = ⎜ − ⎟ , Ra =
                          ⎜ ∂x ∂y ⎟                            , Pr = .                 (5)
        ∂y       ∂x       ⎝       ⎠              υ 2
                                                                     α

Boundary conditions for the considered model are depicted on the physical model (Figure
1(b)). In this model, boundary conditions as

On all solid walls,        u=v=0
                           ∂T
On the adiabatic wall,         =0
                            ∂x
On the bottom wall,        0 < x < B, T=THot,          B ≤ x ≤ L, T= TCold
On the inclined wall,      T= TCold

Local and mean Nusselt numbers are calculated via Eq. 6 a and b, respectively.

          ⎛ ∂θ ⎞
                                    B
   Nu x = ⎜ −  ⎟ ,             Nu = ∫ Nu x dx                                         (6 a.b)
          ⎝ ∂Y ⎠ Y =0               0


Governing equations in streamline-vorticity form (Eqs. 1-3) are solved through finite
difference method. Algebraic equations are obtained via Taylor series and they solved using
Successive Under Relaxation (SUR) technique, iteratively. The central difference method is
used for discretization procedure. The detailed solution technique is well described in the
literature [13]. The convergence criterion, 10-4, is chosen for all depended variables and value
of 0.1 is taken for under-relaxation parameter. Some grid tests are made between 34x23 and
352x235 to obtain optimum grid dimension. The test results showed that 214x143 grid
dimension is enough for calculations.
                         Proceedings of Clima 2007 WellBeing Indoors



RESULTS AND DISCUSSION

A numerical simulation is performed for a saltbox attic with eave for winter season. Different
parameters such as Ra number, eave length and aspect ratio (AR) of the attic were tested to
see the effects of these parameters on natural convection.

Figure 2 shows flow field (by streamline, on the left) and temperature distribution (by
isotherms, on the right) for different Rayleigh numbers and E=6%, AR=0.6. Both AR and
eave length represent real values which is used in architecture. In the case of small Ra number
(Ra=104), conduction mode of heat transfer is dominant to convection and almost parallel
temperature distribution is formed. For both Ra=104 and 105, two circulation cells are formed
in different rotation directions. However, multiple cells are obtained for the highest Ra
number, namely Ra=106 (Figure 2c). At this value of Ra number, plumlike temperature
distribution is observed. However, for all values of Ra number, at the intersection point of
bottom wall and eave, isotherms are stored due to higher temperature difference.




a)




b)




c)
Figure 2. Streamlines (on the left) and isotherms (on the right) at E=%6, AR=0.6, a) Ra=104,
b) Ra=105, c) Ra=106
                         Proceedings of Clima 2007 WellBeing Indoors



Figure 3 is presents the flow and temperature distribution for AR=0.9 and Ra=106 to show the
effects of eave length. As can be seen from the figure, length of eave affects the isotherms and
streamlines due to length of heated and cooled part of the bottom wall are changed. Strong
plumelike distributions are observed for isotherm. Even, they show mushroom shaped
distribution near the right corner. It means that strong convection is occurred. For the highest
value of eave, two cells are formed which one of them locates to the top of the triangle like
region and the other forms almost at the middle of enclosure (Figure 3(c)).




a)




b)




c)
Figure 3. Streamlines (on the left) and isotherms (on the right) at AR=0.9, Ra=106, a) E=3%,
b) E=6%, c) E=9%
                                   Proceedings of Clima 2007 WellBeing Indoors



The results of mean Nusselt number are shown in Figures 4 (a) and (b) for different aspect
ratios and eave lengths, respectively. Figure 4(a) shows that heat transfer decreases with
increasing AR. However, conduction mode of heat transfer is dominant to convection at lower
Ra number (Ra<104). It enhances with increasing Ra number due to domination of convection
mode of heat transfer. Mean Nusselt number decreases with increasing eave length due to
decreasing heater surface. It means that eave makes negative effect on heat transfer inside the
roof. Variation of local Nusselt number with Rayleigh number and eave length is shown in
Figure 5 and 6, respectively. In these figures, results are presented for both inclined and
bottom surfaces. As can be seen from the Figure 5 (a) that results of local Nusselt numbers are
close to each other due to domination of conduction mode of heat transfer along the bottom
wall. However, higher values were obtained for Ra=106. At this value of Rayleigh number,
local Nu number shows wavy variation. The highest Nu number was formed at the
intersection point of eave and ceiling due to high temperature difference. Local Nusselt
number shows similar variation along the inclined surfaces except intersection point of eave
and ceiling. The variation of local Nusselt number for different eave length is shown in Figure
6 (a) (along the bottom surface) and 6(b) (along the inclined surface). As can be illustrated in
Figures, length of eave directly affects the peak point of local Nusselt number due to changing
of location of maximum temperature difference. Higher values were obtained due to
increasing of heater surfaces.


       15                                                            15


       10                                                            10
  Nu




                                                                Nu




                                              AR=0.3                                                            E=3%
       5                                                                 5                                      E=6%
                                              AR=0.6
                                                                                                                E=9%
                                              AR=0.9
       0                                                                 0
            104
             4                    105
                                  5                 106
                                                     6                       104
                                                                             4                      105
                                                                                                    5                  106
                                                                                                                        6
                                  Ra                                                                Ra

a)                                             b)
Figure 4. Variation of mean Nusselt number with Rayleigh number, a) at different ARs, b) at
different eave lengths
      100                                                       100
       90         Ra=10E4                                           90
                                                                                    Ra=10E4
       80         Ra=10E5                                           80
                                                                                    Ra=10E5
       70                                                           70
                  Ra=10E6                                                           Ra=10E6
       60                                                           60
Nux




                                                              Nuy




       50                                                           50
       40                                                           40
       30
                                                                    30
       20
                                                                    20
       10
                                                                    10
        0
                                                                     0
            0      0.2      0.4         0.6   0.8         1
                                                                         0         0.2        0.4         0.6    0.8         1
                                   x                                                                 y

a)                                               b)
Figure 5. Variation of local Nusselt number for different Rayleigh numbers a) along the
bottom wall, b) along the inclined wall
                             Proceedings of Clima 2007 WellBeing Indoors



      140                                                  100
                E=3%
                                                            90       E=3%
      120       E=6%                                        80
                                                                     E=6%
      100       E=9%                                        70
                                                                     E=9%
       80                                                   60
Nux




                                                     Nuy
                                                            50
       60                                                   40
       40                                                   30
       20                                                   20
                                                            10
       0                                                     0
            0   0.2    0.4       0.6    0.8      1               0   0.2    0.4       0.6   0.8    1
                             x                                                    y

a)                                               b)
Figure 6. Variation of local Nusselt number for different eave lengths a) along the bottom
wall, b) along the inclined wall,

CONCLUSIONS

Steady state flow field by natural convection in a saltbox roof with eave for different aspect
ratios has been numerically studied. Based on the findings in this study, we conclude that heat
transfer regime is conduction for lower Rayleigh numbers. Heat transfer increases with
increasing of Rayleigh number with domination of convection mode of heat transfer. Due to
geometrical shape of the roof at least double circulation number was formed. It is observed that
number of cells is strongly depending on eave length and Rayleigh number. Higher heat transfer
was formed for lower aspect ratios. Similarly, smaller eave length enhances the heat transfer.


NOMENCLATURE

AR    aspect ratio, AR=H/L
B     length of room ceiling (m)
E     eave length (m)
g     gravitational acceleration (ms-2)
Gr    Grashof number
H     height of roof (m)
L     length of bottom wall (m)
Nu    Nusselt number
Pr    Prandtl number
Ra    Rayleigh number
T     temperature (K)
u, v axial and radial velocities (ms-1)
x,y   cartesian coordinates (m)
X, Y non-dimensional coordinates
Greek Letters
υ     kinematic viscosity (m2s-1)
Ω     non-dimensional vorticity
θ     non-dimensional temperature
β     thermal expansion coefficient (K-1)
α     thermal diffusivity (m2s-1)
Ψ     non-dimensional streamfunction
                             Proceedings of Clima 2007 WellBeing Indoors



REFERENCES

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11.   Varol, Y, Koca, A, Oztop, H F. 2006. Laminar natural convection in saltbox roofs for both
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14.   http://tms.ecol.net/realestate/sty_salt.htm

								
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