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New approach relevant to floor Nusselt number in floor heating system
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Energy Conversion and Management 49 (2008) 1134–1140 www.elsevier.com/locate/enconman
New approach relevant to floor Nusselt number in floor heating system Refet Karadag˘ a b
a,*
, Ismail Teke
b
Department of Mechanical Engineering, Harran University, 63300 S ß anlıurfa, Turkey Department of Mechanical Engineering, Yıldız Technical University, Istanbul, Turkey Received 30 January 2007; accepted 9 September 2007 Available online 18 October 2007
Abstract In this study, the Nusselt number over the floor is analysed numerically for different thermal conditions in a floor heated room. A new equation related to floor Nusselt number is developed. In the literature, there have been a number of equations, different from each other and appropriate only for the conditions at which the studies were performed, have been presented. While the mentioned equations were dependent only on the floor Rayleigh number, numerical data obtained in the current study show that the floor Nusselt number depends not only on the floor Rayleigh number but also the wall and ceiling Rayleigh numbers. Therefore, the new equation developed in the current study is a function of the Rayleigh numbers over the floor, wall and ceiling surfaces. This equation is compared with those found in the literature. It is seen that while the maximum deviation from the numerical data is 35% for the equations given in the literature, it is only 10% for the new equation. Therefore, the given equation is verified to be more reliable and appropriate for the calculation of Nusselt number. 2007 Elsevier Ltd. All rights reserved. Keywords: Floor heating; Natural convection; Nusselt number; Rayleigh number
1. Introduction Floor heating systems, which have been used since ancient times, have some advantages compared to other heating systems as: A more comfortable environment can be obtained because of the homogeneous temperature distribution. As the heat is accumulated on the floor, the effect of sudden air temperature drops outside is felt less [1]. So far, there have been some studies on floor heating systems. A theoretical and experimental study was performed by Badran and Hamdan [2] for an under floor heating system using solar collectors and solar ponds. Rekstad et al. [3] presented a new approach of temperature control and energy metering in low temperature heating systems. Cho and Zaheer-uddin [4] explored a predictive control strategy as a means of improving the energy efficiency of *
Corresponding author. Tel.: +90 414 344 00 20; fax: +90 414 344 00
31. E-mail address: [email protected] (R. Karadag˘). 0196-8904/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2007.09.008
intermittently heated radiant floor heating systems. They conducted both computer simulations and experiments to assess and compare the energy performance of the predictive control strategy with an existing conventional control strategy. Athienitis and Chen [5] investigated the influence of the cover layer, thermal mass thickness and incident solar radiation on the floor temperature distribution and on the energy consumption in floor heating systems. Cho and Zaheer-uddin [6] tested and evaluated two flow control schemes in an experimental facility consisting of two identical 3 · 4.4 · 3.8 m3 rooms. They used a conventional onoff control for a valve controlled by feedback from the room thermostat and a two parameter switching control with switching intervals ranging from 30 s to 60 min. Zaheer-uddin et al. [7] made a numerical study on dynamic modelling and optimal control of an embedded piping floor heating system in order to minimize the energy input to the boiler. Khalifa [8] presented an extensive review of the studies about the natural convective heat transfer coefficient on isolated vertical and horizontal surfaces with special interest in their application to building geometries.
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
1135
Nomenclature A h H L T W Nu Ra Gr f
surface area (m2) convective heat transfer coefficient (W m2 K) height (m) length (m) temperature (C) width (m) Nusselt number Rayleigh number Grashof number coefficient defined in Eq. (7)
Kilkis and Ritter [9] gave an approximate panel surface heat output algorithm as a function of size and orientation of the heated space and outdoor exposure. Khalifa and Marshall [10] developed some correlations for the heat transfer coefficients on interior building surfaces using a real size indoor test cell, the dimensions of which were 2.95 · 2.35 · 2.08 m3 (L · W · H). Li et al. [11] investigated the heat transfer coefficient experimentally in an occupied office room with the dimensions of 3.4 · 2.6 · 4 m3 under normal working conditions. They gave a correlation related to the floor convective heat transfer coefficient that is valid only if the difference in the temperature between the floor surface and the indoor air is low. Min et al. [12] conducted experimental studies for three different floor heated rooms (3.6 · 2.4 · 7.35 m3, 3.6 · 3.6 · 7.35 m3 and 3 9 3.6 · 2.4 · 3.6 m ) and for 10 –1011 of Rayleigh number. By using ‘‘computational fluid dynamics’’ (CFD), Awbi [13] investigated the natural convective heat transfer coefficients of wall, floor and ceiling heated surfaces for 3 · 2.3 · 3 m3 dimensioned room corresponding to the ranges of 9 · 108–1 · 1011 of Grashof number. Alamdari and Hammond [14] and CIBSE [15] give a correlation for Nusselt number as a parameter of the Grashof number over the floor in a floor heating system. Awbi and Hatton [16] recommended correlations related to natural convective heat transfer coefficients for wall, ceiling and floor heated surfaces in enclosures. Table 1 presents the correlations relevant to floor Nusselt number or convective heat transfer coefficient for the floor heating systems that are given in the literature. It is seen from the table that the equations given for the floor convective heat transfer coefficient in Refs. [9,11,16] are dependent only on the temperature difference over the floor. Similarly, the floor Nusselt number is a function only of the floor Rayleigh number or Grashof number in the equations specified in Refs. [12–15]. It is seen that there are no other parameters concerned with the wall and ceiling thermal conditions in the equations. As a result, when the room thermal conditions and dimensions change, the equations will not give reasonable results. Therefore, the primary objective of this numerical study is to discover and propose a new equation for the floor Nusselt number
DT
temperature difference (C)
Subscripts c ceiling f floor w wall i indoor air wc general wall–ceiling value wcm wall–ceiling average value
Table 1 Equations given in literature for the floor surface in the floor heating system Correlations
Conditions
Reference
h ¼ ð1 2:22 105 zÞ2:627 0:08 ð4:96 2:12ðT f T i Þ0:31 L Þ
Analytical study
[9]
h = 3.08Æ(Tf Ti)0.25
3.4 · 2.6 · 4 m Office room (experimental)
[11]
3.6 · 2.4 · 7.35 m (experimental)
[8,12]
Nu = 0.269ÆGr0.308
3 · 2.3 · 3 m, DT = 5– 35 C Gr = 9.108–1.1011 (experimental)
[13]
Nu = [(0.52ÆGr1/4)6 + (0.126Gr1/3)6]1/6
0 < Gr < 1 Laminar/ turbulent
[14]
Nu = 0.132ÆGr1/3
108 < Gr < 1010
[15]
Nu ¼ 0:33Ra0:33 0:31
2:16ðT f T i Þ h¼ L0:08
h¼
2:175 :ðT f L0:076
0:308
T iÞ
8
9.10 < Gr < 7.10
10
[16]
as a function of the floor, wall and ceiling Rayleigh numbers. 2. Computational method In this study, a numerical solution method is used for two different situations (insulated and non-insulated ceiling). In both situations, the solutions are obtained at different room dimensions (L = 1–6 m, H = 1–3.25 m) and floor temperatures (Tf = 28–57 C). The wall temperature varies from 10 to 25 C for the insulated ceiling condition. In the non-insulated ceiling situation, the wall temperature is maintained at 15 C and the ceiling temperature is changed between 10 and 25 C. The Nusselt number over the surfaces and the temperature of the indoor air are calculated numerically for different thermal conditions and room dimensions that are specified in Table 2 extensively. As the room dimensions and surface temperatures are selected within extensive ranges, the results can be used in different applications of the floor heating system. A sche-
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
1136
Table 2 Thermal boundary conditions and dimensions at which this numerical study was performed First situation (insulated ceiling thermal conditions) No
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Dimensions (m · m · m)
Temperatures (C)
1·1·1 1 · 1.75 · 1 1 · 2.5 · 1 1 · 3.25 · 1 2·1·2 2 · 1.75 · 2 2 · 2.5 · 2 2 · 3.25 · 2 4·1·4 4 · 1.75 · 4 4 · 2.5 · 4 4 · 3.25 · 4 6·1·6 6 · 1.75 · 6 6 · 2.5 · 6 6 · 3.25 · 6 3 · 2.5 · 5 3 · 3.25 · 5 4 · 2.5 · 6 4 · 3.25 · 6
Tc H
Second situation (non-insulated ceiling thermal conditions) No
Wall
Floor
Ceiling
10–25
28–57
Insulated
Tw g
Ti L Tf L
Fig. 1. Schematic drawing of the room heated from floor for the cases studied.
matic presentation of the heated floor room is given in Fig. 1. A commercial computational fluid dynamics (CFD) package FLUENT, which uses a control volume based technique to convert the governing equations to algebraic equations is used for the solutions. Because of its accuracy, robustness and convenience, it is one of the most widely used commercial codes for simulating engineering fluid flow and heat transfer problems [17–20]. The GAMBIT 1.3-mesh generator associated with the solver has been used to plot and mesh the figure of the floor heated room. In order to ensure grid independence solutions, a series of
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Dimensions (m · m · m)
1·1·1 1 · 1.75 · 1 1 · 2.5 · 1 1 · 3.25 · 1 2·1·2 2 · 1.75 · 2 2 · 2.5 · 2 2 · 3.25 · 2 4·1·4 4 · 1.75 · 4 4 · 2.5 · 4 4 · 3.25 · 4 6·1·6 6 · 1.75 · 6 6 · 2.5 · 6 6 · 3.25 · 6 3 · 2.5 · 5 3 · 3.25 · 5 4 · 2.5 · 6 4 · 3.25 · 6
Temperatures (C) Wall
Floor
Ceiling
15
28–57
10–25
trial calculations have been conducted for two types of grid distributions (constant and variable grid size distributions). It has been found that the constant grid distribution is not suitable because the results differ depending on the grid size. So, the variable grid distribution of very small size near the surfaces but larger towards the center (the ratio of the smallest grid size to the largest grid size is 0.01) is used in the present study. Table 3 gives the Nusselt numbers at different grid sizes or numbers for the variable grid distribution (the dimension of room is 2 · 1.75 · 2 m). It is seen that the difference between the results of the grid numbers of 56,000 (average grid size is 5 · 5 · 5 cm) and 110,000 (average grid size is 4 · 4 · 4 cm) is 0.85%. Therefore, the grid number is chosen as 110,000 (4 · 4 · 4 cm). Further, the results are checked by equalizing the conduction heat transfer in the boundary layer region with the convection heat transfer in the free flow region. It has been observed that the equality is verified approximately. A constant temperature boundary condition is applied to the inner surfaces. The radiation heat transfer at the surfaces is neglected in order to calculate the convective heat transTable 3 The Nusselt numbers at different grid sizes or numbers for variable grid distribution Grid number
Mean grid size (cm · cm · cm)
Nu
7200 13,750 31,581 56,000 110,000
10 · 10 · 10 8·8·8 6·6·6 5·5·5 4·4·4
427.058 436.598 442.998 450.36 454.2142
The room dimension is 2 · 1,75 · 2 m and the ratio of the smallest grid size to the largest grid size is 0.01.
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
fer coefficient. In fact, it should be taken into account to compute the total heat transfer coefficient because a significant part of the heat transfer over the surfaces is by radiation. The relation between the radiative and convective heat transfer coefficients over the surfaces in a floor heated room was investigated by the present authors [21]. The standard k–e turbulence flow model is used. The studies done in the literature show that this model is appropriate for solution of the free convection. Omri and Galanis [22] calculated the turbulent flow of air in a 2D cavity with Ra = 1.58 · 109 using the standard and re-normalization group (RNG) k–e models and three different near wall treatments. The turbulence quantities predicted by the k– e model were in very good agreement with large eddy simulation. Peng and Davidson [23] investigated the buoyant flow in a cavity for a relatively low turbulence level (Ra = 1.58 · 109) by means of large eddy simulation. They found the numerical predictions were in good agreement with experimental values [24]. The segregated solution method has been used in the study. Corvaro and Paroncini [25] investigated, numerically and experimentally, the natural convection heat transfer in a square cavity heated from below and cooled by the sidewalls. They used the segregated method in the numerical solution and compared the results with experimental data. They found that the numerical results were suited to the experimental data. The physical properties of air have been taken from the tables given by Holman [26] and shown in Table 4.
1137
The Rayleigh numbers over the floor, wall and ceiling can be calculated by using numerical data as follows: g b ðT f T i Þ pr L3 m2 g b ðT i T w Þ Raw ¼ pr H 3 m2 g b ðT i T c Þ Rac ¼ pr L3 m2 Raf ¼
ð2Þ
For the insulated ceiling conditions (Rac = 0), Eq. (1) takes the following form n Nuf ¼ C 1 Ram f þ C 2 Raw
ð3Þ
Numerical data show that there is a relation between the Rayleigh numbers over the floor and wall. The relation is given in Figs. 2 and 3 for insulated ceiling conditions. It is seen that the relation varies depending on the dimensions. In order to simplify Eq. (1), a general wall–ceiling Rayleigh number (Rawc) is defined as, Rawc ¼
Raw Aw þ Rac Ac Aw þ Ac
ð4Þ
Figs. 4 and 5 show the variation of the general wall–ceiling Rayleigh number with the floor Rayleigh number. It is seen from the figures that the relation between the Rayleigh numbers is not identical for all dimensions. In order to efface this difference, Eq. (4) is modified as, 2
Raw Aw 1:25 ðL=H Þ þ Rac Ac Aw þ Ac
3. Results and discussion
Rawcm ¼
Numerical data obtained for insulated and non-insulated ceiling conditions at different dimensions are used in order to discover a new general equation related to floor Nusselt number as a function of the floor, wall and ceiling thermal conditions. The equations given in the literature (Table 1) are functions of the Rayleigh or Grashof numbers over the floor only. Karadag˘ and Teke [27] explored the effect of the wall and ceiling thermal conditions and dimensions on the floor Nusselt number, and they found that the floor Nusselt number was increasing with respect to the room dimensions and temperature differences over the floor, wall and ceiling surfaces. From this result, it can be understood that the floor Nusselt number is proportional to the Rayleigh numbers around the floor, wall and ceiling surfaces. Therefore, the new general equation can be given as;
The variation of the floor Rayleigh number with wall–ceiling average Rayleigh number (Rawcm) is shown in Figs. 6 and 7. It is seen from these figures that the relation between the Rayleigh numbers is identical for all dimensions. Therefore, Eq. (1) is modified as,
Nuf ¼ C 1
Ram f
þ C2
Ranw
þ C3
Rapc
ð1Þ
3
Nuf ¼ C 1 Ramf þ C 2 Ranwcm
ð6Þ
From Figs. 6 and 7, a correlation can be obtained between the Rayleigh numbers as, 2.4E+10 2.0E+10 1.6E+10
Raw 1.2E+10 8.0E+09
LxW = 2x2 LxW = 4x4
4.0E+09
Table 4 The properties of the air depending on the temperatures [22] T (K)
q (kg/m )
Cp (J/kg K)
l (kg/ms)
K (W/mK)
250 300 350 400
1.4128 1.1774 0.998 0.882
1005.3 1005.7 1009 1014
1.599 · 105 1.846 · 105 2.075 · 105 2.286 · 105
0.0222 0.0262 0.03 0.0336
ð5Þ
0.0E+00 0.0E+00
LxW = 3x5 LxW = 6x6
6.0E+10
1.2E+11
1.8E+11
2.4E+11
3.0E+11
3.6E+11
Raf
Fig. 2. The variation of the floor Rayleigh number with the wall Rayleigh number at different floor dimensions for insulated ceiling conditions (Tw = 15, H = 2.5).
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
1138 3.6E+10
1.8E+11
3.0E+10
1.5E+11
2.4E+10
1.2E+11
Raw 1.8E+10
Ra wcm
9.0E+10 6.0E+10
1.2E+10 H = 1.75
6.0E+09
3.0E+10
H = 2.5 H = 3.25
0.0E+00 0.0E+00
0.0E+00 0.0E+00
3.0E+10
6.0E+10
9.0E+10
1.2E+11
1.5E+11
1.8E+11
9.0E+10
1.8E+11
2.7E+11
Ra f
3.6E+11
4.5E+11
5.4E+11
Ra f
Fig. 3. The variation of the floor Rayleigh number with the wall Rayleigh number at different heights for insulated ceiling conditions (Tw = 15, L · W = 4 · 4).
Fig. 6. The relation between the floor Rayleigh number and wall–ceiling average Rayleigh number at different floor dimensions for non-insulated ceiling conditions (L · W = 2 · 2–6 · 6, Tc = 10–25, H = 2.5).
5.4E+10
1.2E+11
4.5E+10 1.0E+11
3.6E+10 8.0E+10
Rawcm 2.7E+10
Ra wc 6.0E+10 1.8E+10 4.0E+10
LxW=4x4(Tw=10) LxW=4x4(Tw=25) LxW=4x6(Tw=10) LxW=4x6(Tw=25) LxW=6x6(Tw=10) LxW=6x6(Tw=25)
2.0E+10 0.0E+00 0.0E+00
1.0E+11
2.0E+11
3.0E+11
4E0E+11
5.0E+11
9.0E+09 0.0E+00 0.0E+00
3.5E+10
7.0E+10
1.1E+11
6.0E+11
Ra f
Fig. 4. The variation of the general wall–ceiling Rayleigh number with the floor Rayleigh number at different floor dimensions for non-insulated ceiling conditions (H = 2.5).
3.0E+10 2.4E+10
2.1E+11
Fig. 7. The relation between the floor Rayleigh number and wall–ceiling average Rayleigh number at different heights for non-insulated ceiling conditions (L · W = 4 · 4, Tc = 10–25, H = 1.75–3.25).
Wall–ceiling temperature conditions
The ratio of height to characteristic length
Tw > Tc
0.166 < H/L < 1
c 2:14 ½ðT wTT Þ HL 0:35 f
1 6 H/L 6 3.25
c 12:97 ½ðT wTT Þ HL 0:76 f
0.166 < H/L < 1.75
0:93:ðHL Þ0:21
1.175 < H/L 6 3.25
3:8:ðHL Þ1:16
0.166 < H/L 6 0.5
1:2 ½ðT c TTf w Þ þ HL 0:04
0.5 < H/L 6 3.25
0:8 ½ðT c TTf w Þ þ HL 0:2
1.8E+10
Tw = Tc 1.2E+10
H=1.75(Tw=10) H=1.75(Tw=20) H=2.5(Tw=10) H=2.5(Tw=20) H=3.25(Tw=10) H=3.25(Tw=20)
6.0E+09 0.0E+00 0.0E+00
1.8E+11
Table 5 The coefficient (f) defined in the current study in Eq. (7)
3.6E+10
Rawc
1.4E+11
Raf
3.5E+10
7.0E+10
1.1E+11
1.4E+11
1.8E+11
Tw < Tc and Qc = 0
2.1E+11
Raf
Fig. 5. The variation of the general wall–ceiling Rayleigh number with the floor Rayleigh number at different heights for non-insulated ceiling conditions (L · L = 4 · 4).
Rawcm ¼ 0:3583 Ra0:9957 f
ð7Þ
Using correlation (7) and Eq. (6), the Eq. (8) is found. Nuf ¼ 0:15 Ra0:339 þ 0:01069 Ra0:4166 f wcm f
f
ð8Þ
The coefficient (f) defined in the current study is given in Table 5 with respect to the thermal conditions and dimensions. In Figs. 8–10, the new equation developed in the current study (Eq. (8)) and the equations given in the literature (Table 1) are compared with the numerical data. It is seen that the results of the equations given in the literature are different from each other. As they do not contain the effect of wall and ceiling thermal conditions, they match the numerical values for similar conditions only, but they devi-
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
suitable to use the new equation for the operation conditions specified in Table 2.
2400 2000 1600
4. Conclusions
Nuf 1200 800
In this study, in a floor heating system, a common Rayleigh number for the wall and ceiling is defined, and a correlation is obtained as,
Numerical values Equation (Current study) Kilkis (1998) Li et al. (1983) Min et al. (1956)
400 0 0.0E+00
1139
8.0E+10
1.6E+11
2.4E+11
3.2E+11
4.0E+11
4.8E+11
Ra f
Fig. 8. The comparison of the new equation developed in the current study (Eq. (7)) and the equations given in the literature (Table 1) with the numerical data at different floor dimensions (L · W = 2 · 2–6 · 6, H = 1.75, Tw = Tc = 15).
2
Raw Aw 1:25 ðL=H Þ þ Rac Ac Aw þ A c ¼ 0:3583 Ra0:9957 f
Rawcm ¼ Rawcm
Moreover, a new equation related to the floor Nusselt number is developed as a function of the Rayleigh numbers as, þ 0:01069 Ra0:4166 Nuf ¼ 0:15 Ra0:339 f wcm f
2400
A table (Table 5) is obtained for the coefficient (f) as depending on the thermal conditions and dimensions. The new equation and the equations given in the literature are compared with the numerical data. Results show that the maximum deviation is 10% for the new equation while it is 35% for those in the literature. Therefore, it will be more appropriate to use the equation of the present study.
2000 1600
Nuf 1200 800 Numerical values equation (current study) Kilkis (1998) Li et al. (1983) Min et al. (1956)
400 0 0.0E+00
9.0E+10
1.8E+11
2.7E+11
3.6E+11
4.5E+11
References 5.4E+11
Ra f
Fig. 9. The comparison of the new equation developed in the current study (Eq. (7)) and the equations given in the literature (Table 1) with the numerical data at different floor dimensions (L · W = 2 · 2–6 · 6, H = 2.5, Tw = Tc = 15).
2400 2000 1600
Nuf 1200 800 Numerical values Equation (current study) Kilkis (1998) Li et al. (1983) Min et al. (1956)
400 0 0.0E+00
8.0E+10
1.6E+11
2.4E+11
3.2E+11
4.0E+11
4.8E+11
Ra f Fig. 10. The comparison of the new equation developed in the current study (Eq. (7)) and the equations given in the literature (Table 1) with the numerical data at different floor dimensions for insulated ceiling conditions (L · W = 2 · 2–6 · 6, H = 1.75, Tw = 15).
ate from them up to 35% depending on the thermal conditions and dimensions. However, the new equation matches the numerical values for all conditions (maximum deviation 10%) because it includes the influence of all the surfaces thermal conditions. For this reason, it will be more
[1] Bozkır O, Canbazog˘lu S. Unsteady thermal performance analysis of a room with serial and parallel duct radiant floor heating system using hot airflow. Energy Buildings 2004;36:579–86. [2] Badran AA, Hamdan MA. Comparative study for under-floor heating using solar collectors or solar ponds. Appl Energy 2004;77:107–17. [3] Rekstad J, Meir M, Kristoffersen AR. Control and energy metering in low temperature heating systems. Energy Buildings 2003;35:281–91. [4] Cho SH, Zaheer-uddin M. Predictive control of intermittently operated radiant floor heating systems. Energy Convers Manage 2003;44:1333–42. [5] Athienitis AK, Chen Y. The effect of solar radiation on dynamic thermal performance of floor heating systems. Sol Energy 2000;69(3):229–37. [6] Cho SH, Zaheer-uddin M. An experimental study of multiple parameter switching control for radiant floor heating systems. Energy 1999;24:433–44. [7] Zaheer-uddin M, Zheng GR, Cho SH. Optimal operation of an embedded piping floor heating system with control input constraints. Energy Convers Manage 1997;38(7):713–25. [8] Khalifa AJN. Natural convective heat transfer coefficient a review I. Isolated vertical and horizontal surfaces. Energy Convers Manage 2001;42:491–504. [9] Kilkis B, Ritter LT. An analytical model for the design of in-slab electric heating panels. ASHRAE Trans 1998;SF-98-9-5. [10] KHalifa AJN, Marshall RH. Validation of heat transfer coefficients on interior building surfaces using a real – sized indoor test cell. Int J Heat Mass Transfer 1990;33(10):2219–36. [11] Li LD, Beckman WA, Mitchell JW. An experimental study of natural convection in an office room, large time results. Unpublished Report, Solar Energy Laboratory, University of Wisconsin, Madison, 1983, Ref. [9]. [12] Min TC, Schutrum LF, Parmelee GV, Vouris JD. Natural convection and radiation in a panel-heated room. ASHRAE Trans 1956;62: 337–58.
1140
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
[13] Awbi HB. Calculation of convective heat transfer coefficients of room surfaces for natural convection. Energy Buildings 1998;28:219–27. [14] Alamdari F, Hammond GP. Building Services Res Technol 1983;4:106–12. [15] CIBSE. CIBSE Guide, Vols. A–C. London: CIBSE; 1986. [16] Awbi HB, Hatton A. Mixed convection from heated room surfaces. Energy Buildings 2000;32:153–66. [17] Gan G, Riffat SB. CFD modelling of air flow and thermal performance of an atrium integrated with photovoltaics. Build Environ 2004;39:735–48. [18] Male P, Croon MHJM, Tiggelaar RM, Berg A, Schouten JC. Heat and mass transfer in a square microchannel with asymmetric heating. Int J Heat Mass Transfer 2004;47:87–99. [19] Zhang Z, Li Y. CFD simulation on inlet configuration of plate-fin heat exchangers. Cryogenics 2003;43:673–8. [20] Dubovsky V, Ziskind G, Druckman S, Moshka E, Weiss Y, Letan R. Natural convection inside ventilated enclosure heated by downwardfacing plate: experiments and numerical simulations. Int J Heat Mass Transfer 2001;44:3155–68.
[21] Karadag˘ R, Teke I. The relation between radiative and convective heat transfer coefficients over the surfaces in a floor heated room. J Eng Mach 2005;548:21–9 [in Turkish]. [22] Omri M, Galanis N. Numerical analysis of turbulent natural convection in a cavity. In: 13th int heat transfer conf Sydney Australia; 2006. [23] Peng S-H, Davidson L. Large eddy simulation for turbulent buoyant flow in a confined cavity. Int J Heat Fluid Flow 2001;22(3):323–31. [24] Omri M, Galanis N. Numerical analysis of turbulent buoyant flows in enclosures: influence of grid and boundary conditions. Int J Therm Sci; doi:10.1016/j.ijthermalsci.2006.10.006. [25] Corvaro F, Paroncini M. Experimental analysis of natural convection in square cavities heated from below with 2D – PIV and holographic interferometry techniques. Exp Therm Fluid Sci; doi:10.1016/ j.expthermflusci.2006.07.006. [26] Holman JP. Heat transfer. 7th ed. London: McGraw Hill; 1992. [27] Karadag˘ R, Teke I. The effect of wall thermal conditions on the floor Nusselt number in the floor heating system. J Eng Natural Sci 2004;3:84–93 [in Turkish].
Energy Conversion and Management 49 (2008) 1134–1140 www.elsevier.com/locate/enconman
New approach relevant to floor Nusselt number in floor heating system Refet Karadag˘ a b
a,*
, Ismail Teke
b
Department of Mechanical Engineering, Harran University, 63300 S ß anlıurfa, Turkey Department of Mechanical Engineering, Yıldız Technical University, Istanbul, Turkey Received 30 January 2007; accepted 9 September 2007 Available online 18 October 2007
Abstract In this study, the Nusselt number over the floor is analysed numerically for different thermal conditions in a floor heated room. A new equation related to floor Nusselt number is developed. In the literature, there have been a number of equations, different from each other and appropriate only for the conditions at which the studies were performed, have been presented. While the mentioned equations were dependent only on the floor Rayleigh number, numerical data obtained in the current study show that the floor Nusselt number depends not only on the floor Rayleigh number but also the wall and ceiling Rayleigh numbers. Therefore, the new equation developed in the current study is a function of the Rayleigh numbers over the floor, wall and ceiling surfaces. This equation is compared with those found in the literature. It is seen that while the maximum deviation from the numerical data is 35% for the equations given in the literature, it is only 10% for the new equation. Therefore, the given equation is verified to be more reliable and appropriate for the calculation of Nusselt number. 2007 Elsevier Ltd. All rights reserved. Keywords: Floor heating; Natural convection; Nusselt number; Rayleigh number
1. Introduction Floor heating systems, which have been used since ancient times, have some advantages compared to other heating systems as: A more comfortable environment can be obtained because of the homogeneous temperature distribution. As the heat is accumulated on the floor, the effect of sudden air temperature drops outside is felt less [1]. So far, there have been some studies on floor heating systems. A theoretical and experimental study was performed by Badran and Hamdan [2] for an under floor heating system using solar collectors and solar ponds. Rekstad et al. [3] presented a new approach of temperature control and energy metering in low temperature heating systems. Cho and Zaheer-uddin [4] explored a predictive control strategy as a means of improving the energy efficiency of *
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31. E-mail address: [email protected] (R. Karadag˘). 0196-8904/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2007.09.008
intermittently heated radiant floor heating systems. They conducted both computer simulations and experiments to assess and compare the energy performance of the predictive control strategy with an existing conventional control strategy. Athienitis and Chen [5] investigated the influence of the cover layer, thermal mass thickness and incident solar radiation on the floor temperature distribution and on the energy consumption in floor heating systems. Cho and Zaheer-uddin [6] tested and evaluated two flow control schemes in an experimental facility consisting of two identical 3 · 4.4 · 3.8 m3 rooms. They used a conventional onoff control for a valve controlled by feedback from the room thermostat and a two parameter switching control with switching intervals ranging from 30 s to 60 min. Zaheer-uddin et al. [7] made a numerical study on dynamic modelling and optimal control of an embedded piping floor heating system in order to minimize the energy input to the boiler. Khalifa [8] presented an extensive review of the studies about the natural convective heat transfer coefficient on isolated vertical and horizontal surfaces with special interest in their application to building geometries.
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Nomenclature A h H L T W Nu Ra Gr f
surface area (m2) convective heat transfer coefficient (W m2 K) height (m) length (m) temperature (C) width (m) Nusselt number Rayleigh number Grashof number coefficient defined in Eq. (7)
Kilkis and Ritter [9] gave an approximate panel surface heat output algorithm as a function of size and orientation of the heated space and outdoor exposure. Khalifa and Marshall [10] developed some correlations for the heat transfer coefficients on interior building surfaces using a real size indoor test cell, the dimensions of which were 2.95 · 2.35 · 2.08 m3 (L · W · H). Li et al. [11] investigated the heat transfer coefficient experimentally in an occupied office room with the dimensions of 3.4 · 2.6 · 4 m3 under normal working conditions. They gave a correlation related to the floor convective heat transfer coefficient that is valid only if the difference in the temperature between the floor surface and the indoor air is low. Min et al. [12] conducted experimental studies for three different floor heated rooms (3.6 · 2.4 · 7.35 m3, 3.6 · 3.6 · 7.35 m3 and 3 9 3.6 · 2.4 · 3.6 m ) and for 10 –1011 of Rayleigh number. By using ‘‘computational fluid dynamics’’ (CFD), Awbi [13] investigated the natural convective heat transfer coefficients of wall, floor and ceiling heated surfaces for 3 · 2.3 · 3 m3 dimensioned room corresponding to the ranges of 9 · 108–1 · 1011 of Grashof number. Alamdari and Hammond [14] and CIBSE [15] give a correlation for Nusselt number as a parameter of the Grashof number over the floor in a floor heating system. Awbi and Hatton [16] recommended correlations related to natural convective heat transfer coefficients for wall, ceiling and floor heated surfaces in enclosures. Table 1 presents the correlations relevant to floor Nusselt number or convective heat transfer coefficient for the floor heating systems that are given in the literature. It is seen from the table that the equations given for the floor convective heat transfer coefficient in Refs. [9,11,16] are dependent only on the temperature difference over the floor. Similarly, the floor Nusselt number is a function only of the floor Rayleigh number or Grashof number in the equations specified in Refs. [12–15]. It is seen that there are no other parameters concerned with the wall and ceiling thermal conditions in the equations. As a result, when the room thermal conditions and dimensions change, the equations will not give reasonable results. Therefore, the primary objective of this numerical study is to discover and propose a new equation for the floor Nusselt number
DT
temperature difference (C)
Subscripts c ceiling f floor w wall i indoor air wc general wall–ceiling value wcm wall–ceiling average value
Table 1 Equations given in literature for the floor surface in the floor heating system Correlations
Conditions
Reference
h ¼ ð1 2:22 105 zÞ2:627 0:08 ð4:96 2:12ðT f T i Þ0:31 L Þ
Analytical study
[9]
h = 3.08Æ(Tf Ti)0.25
3.4 · 2.6 · 4 m Office room (experimental)
[11]
3.6 · 2.4 · 7.35 m (experimental)
[8,12]
Nu = 0.269ÆGr0.308
3 · 2.3 · 3 m, DT = 5– 35 C Gr = 9.108–1.1011 (experimental)
[13]
Nu = [(0.52ÆGr1/4)6 + (0.126Gr1/3)6]1/6
0 < Gr < 1 Laminar/ turbulent
[14]
Nu = 0.132ÆGr1/3
108 < Gr < 1010
[15]
Nu ¼ 0:33Ra0:33 0:31
2:16ðT f T i Þ h¼ L0:08
h¼
2:175 :ðT f L0:076
0:308
T iÞ
8
9.10 < Gr < 7.10
10
[16]
as a function of the floor, wall and ceiling Rayleigh numbers. 2. Computational method In this study, a numerical solution method is used for two different situations (insulated and non-insulated ceiling). In both situations, the solutions are obtained at different room dimensions (L = 1–6 m, H = 1–3.25 m) and floor temperatures (Tf = 28–57 C). The wall temperature varies from 10 to 25 C for the insulated ceiling condition. In the non-insulated ceiling situation, the wall temperature is maintained at 15 C and the ceiling temperature is changed between 10 and 25 C. The Nusselt number over the surfaces and the temperature of the indoor air are calculated numerically for different thermal conditions and room dimensions that are specified in Table 2 extensively. As the room dimensions and surface temperatures are selected within extensive ranges, the results can be used in different applications of the floor heating system. A sche-
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1136
Table 2 Thermal boundary conditions and dimensions at which this numerical study was performed First situation (insulated ceiling thermal conditions) No
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Dimensions (m · m · m)
Temperatures (C)
1·1·1 1 · 1.75 · 1 1 · 2.5 · 1 1 · 3.25 · 1 2·1·2 2 · 1.75 · 2 2 · 2.5 · 2 2 · 3.25 · 2 4·1·4 4 · 1.75 · 4 4 · 2.5 · 4 4 · 3.25 · 4 6·1·6 6 · 1.75 · 6 6 · 2.5 · 6 6 · 3.25 · 6 3 · 2.5 · 5 3 · 3.25 · 5 4 · 2.5 · 6 4 · 3.25 · 6
Tc H
Second situation (non-insulated ceiling thermal conditions) No
Wall
Floor
Ceiling
10–25
28–57
Insulated
Tw g
Ti L Tf L
Fig. 1. Schematic drawing of the room heated from floor for the cases studied.
matic presentation of the heated floor room is given in Fig. 1. A commercial computational fluid dynamics (CFD) package FLUENT, which uses a control volume based technique to convert the governing equations to algebraic equations is used for the solutions. Because of its accuracy, robustness and convenience, it is one of the most widely used commercial codes for simulating engineering fluid flow and heat transfer problems [17–20]. The GAMBIT 1.3-mesh generator associated with the solver has been used to plot and mesh the figure of the floor heated room. In order to ensure grid independence solutions, a series of
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Dimensions (m · m · m)
1·1·1 1 · 1.75 · 1 1 · 2.5 · 1 1 · 3.25 · 1 2·1·2 2 · 1.75 · 2 2 · 2.5 · 2 2 · 3.25 · 2 4·1·4 4 · 1.75 · 4 4 · 2.5 · 4 4 · 3.25 · 4 6·1·6 6 · 1.75 · 6 6 · 2.5 · 6 6 · 3.25 · 6 3 · 2.5 · 5 3 · 3.25 · 5 4 · 2.5 · 6 4 · 3.25 · 6
Temperatures (C) Wall
Floor
Ceiling
15
28–57
10–25
trial calculations have been conducted for two types of grid distributions (constant and variable grid size distributions). It has been found that the constant grid distribution is not suitable because the results differ depending on the grid size. So, the variable grid distribution of very small size near the surfaces but larger towards the center (the ratio of the smallest grid size to the largest grid size is 0.01) is used in the present study. Table 3 gives the Nusselt numbers at different grid sizes or numbers for the variable grid distribution (the dimension of room is 2 · 1.75 · 2 m). It is seen that the difference between the results of the grid numbers of 56,000 (average grid size is 5 · 5 · 5 cm) and 110,000 (average grid size is 4 · 4 · 4 cm) is 0.85%. Therefore, the grid number is chosen as 110,000 (4 · 4 · 4 cm). Further, the results are checked by equalizing the conduction heat transfer in the boundary layer region with the convection heat transfer in the free flow region. It has been observed that the equality is verified approximately. A constant temperature boundary condition is applied to the inner surfaces. The radiation heat transfer at the surfaces is neglected in order to calculate the convective heat transTable 3 The Nusselt numbers at different grid sizes or numbers for variable grid distribution Grid number
Mean grid size (cm · cm · cm)
Nu
7200 13,750 31,581 56,000 110,000
10 · 10 · 10 8·8·8 6·6·6 5·5·5 4·4·4
427.058 436.598 442.998 450.36 454.2142
The room dimension is 2 · 1,75 · 2 m and the ratio of the smallest grid size to the largest grid size is 0.01.
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
fer coefficient. In fact, it should be taken into account to compute the total heat transfer coefficient because a significant part of the heat transfer over the surfaces is by radiation. The relation between the radiative and convective heat transfer coefficients over the surfaces in a floor heated room was investigated by the present authors [21]. The standard k–e turbulence flow model is used. The studies done in the literature show that this model is appropriate for solution of the free convection. Omri and Galanis [22] calculated the turbulent flow of air in a 2D cavity with Ra = 1.58 · 109 using the standard and re-normalization group (RNG) k–e models and three different near wall treatments. The turbulence quantities predicted by the k– e model were in very good agreement with large eddy simulation. Peng and Davidson [23] investigated the buoyant flow in a cavity for a relatively low turbulence level (Ra = 1.58 · 109) by means of large eddy simulation. They found the numerical predictions were in good agreement with experimental values [24]. The segregated solution method has been used in the study. Corvaro and Paroncini [25] investigated, numerically and experimentally, the natural convection heat transfer in a square cavity heated from below and cooled by the sidewalls. They used the segregated method in the numerical solution and compared the results with experimental data. They found that the numerical results were suited to the experimental data. The physical properties of air have been taken from the tables given by Holman [26] and shown in Table 4.
1137
The Rayleigh numbers over the floor, wall and ceiling can be calculated by using numerical data as follows: g b ðT f T i Þ pr L3 m2 g b ðT i T w Þ Raw ¼ pr H 3 m2 g b ðT i T c Þ Rac ¼ pr L3 m2 Raf ¼
ð2Þ
For the insulated ceiling conditions (Rac = 0), Eq. (1) takes the following form n Nuf ¼ C 1 Ram f þ C 2 Raw
ð3Þ
Numerical data show that there is a relation between the Rayleigh numbers over the floor and wall. The relation is given in Figs. 2 and 3 for insulated ceiling conditions. It is seen that the relation varies depending on the dimensions. In order to simplify Eq. (1), a general wall–ceiling Rayleigh number (Rawc) is defined as, Rawc ¼
Raw Aw þ Rac Ac Aw þ Ac
ð4Þ
Figs. 4 and 5 show the variation of the general wall–ceiling Rayleigh number with the floor Rayleigh number. It is seen from the figures that the relation between the Rayleigh numbers is not identical for all dimensions. In order to efface this difference, Eq. (4) is modified as, 2
Raw Aw 1:25 ðL=H Þ þ Rac Ac Aw þ Ac
3. Results and discussion
Rawcm ¼
Numerical data obtained for insulated and non-insulated ceiling conditions at different dimensions are used in order to discover a new general equation related to floor Nusselt number as a function of the floor, wall and ceiling thermal conditions. The equations given in the literature (Table 1) are functions of the Rayleigh or Grashof numbers over the floor only. Karadag˘ and Teke [27] explored the effect of the wall and ceiling thermal conditions and dimensions on the floor Nusselt number, and they found that the floor Nusselt number was increasing with respect to the room dimensions and temperature differences over the floor, wall and ceiling surfaces. From this result, it can be understood that the floor Nusselt number is proportional to the Rayleigh numbers around the floor, wall and ceiling surfaces. Therefore, the new general equation can be given as;
The variation of the floor Rayleigh number with wall–ceiling average Rayleigh number (Rawcm) is shown in Figs. 6 and 7. It is seen from these figures that the relation between the Rayleigh numbers is identical for all dimensions. Therefore, Eq. (1) is modified as,
Nuf ¼ C 1
Ram f
þ C2
Ranw
þ C3
Rapc
ð1Þ
3
Nuf ¼ C 1 Ramf þ C 2 Ranwcm
ð6Þ
From Figs. 6 and 7, a correlation can be obtained between the Rayleigh numbers as, 2.4E+10 2.0E+10 1.6E+10
Raw 1.2E+10 8.0E+09
LxW = 2x2 LxW = 4x4
4.0E+09
Table 4 The properties of the air depending on the temperatures [22] T (K)
q (kg/m )
Cp (J/kg K)
l (kg/ms)
K (W/mK)
250 300 350 400
1.4128 1.1774 0.998 0.882
1005.3 1005.7 1009 1014
1.599 · 105 1.846 · 105 2.075 · 105 2.286 · 105
0.0222 0.0262 0.03 0.0336
ð5Þ
0.0E+00 0.0E+00
LxW = 3x5 LxW = 6x6
6.0E+10
1.2E+11
1.8E+11
2.4E+11
3.0E+11
3.6E+11
Raf
Fig. 2. The variation of the floor Rayleigh number with the wall Rayleigh number at different floor dimensions for insulated ceiling conditions (Tw = 15, H = 2.5).
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
1138 3.6E+10
1.8E+11
3.0E+10
1.5E+11
2.4E+10
1.2E+11
Raw 1.8E+10
Ra wcm
9.0E+10 6.0E+10
1.2E+10 H = 1.75
6.0E+09
3.0E+10
H = 2.5 H = 3.25
0.0E+00 0.0E+00
0.0E+00 0.0E+00
3.0E+10
6.0E+10
9.0E+10
1.2E+11
1.5E+11
1.8E+11
9.0E+10
1.8E+11
2.7E+11
Ra f
3.6E+11
4.5E+11
5.4E+11
Ra f
Fig. 3. The variation of the floor Rayleigh number with the wall Rayleigh number at different heights for insulated ceiling conditions (Tw = 15, L · W = 4 · 4).
Fig. 6. The relation between the floor Rayleigh number and wall–ceiling average Rayleigh number at different floor dimensions for non-insulated ceiling conditions (L · W = 2 · 2–6 · 6, Tc = 10–25, H = 2.5).
5.4E+10
1.2E+11
4.5E+10 1.0E+11
3.6E+10 8.0E+10
Rawcm 2.7E+10
Ra wc 6.0E+10 1.8E+10 4.0E+10
LxW=4x4(Tw=10) LxW=4x4(Tw=25) LxW=4x6(Tw=10) LxW=4x6(Tw=25) LxW=6x6(Tw=10) LxW=6x6(Tw=25)
2.0E+10 0.0E+00 0.0E+00
1.0E+11
2.0E+11
3.0E+11
4E0E+11
5.0E+11
9.0E+09 0.0E+00 0.0E+00
3.5E+10
7.0E+10
1.1E+11
6.0E+11
Ra f
Fig. 4. The variation of the general wall–ceiling Rayleigh number with the floor Rayleigh number at different floor dimensions for non-insulated ceiling conditions (H = 2.5).
3.0E+10 2.4E+10
2.1E+11
Fig. 7. The relation between the floor Rayleigh number and wall–ceiling average Rayleigh number at different heights for non-insulated ceiling conditions (L · W = 4 · 4, Tc = 10–25, H = 1.75–3.25).
Wall–ceiling temperature conditions
The ratio of height to characteristic length
Tw > Tc
0.166 < H/L < 1
c 2:14 ½ðT wTT Þ HL 0:35 f
1 6 H/L 6 3.25
c 12:97 ½ðT wTT Þ HL 0:76 f
0.166 < H/L < 1.75
0:93:ðHL Þ0:21
1.175 < H/L 6 3.25
3:8:ðHL Þ1:16
0.166 < H/L 6 0.5
1:2 ½ðT c TTf w Þ þ HL 0:04
0.5 < H/L 6 3.25
0:8 ½ðT c TTf w Þ þ HL 0:2
1.8E+10
Tw = Tc 1.2E+10
H=1.75(Tw=10) H=1.75(Tw=20) H=2.5(Tw=10) H=2.5(Tw=20) H=3.25(Tw=10) H=3.25(Tw=20)
6.0E+09 0.0E+00 0.0E+00
1.8E+11
Table 5 The coefficient (f) defined in the current study in Eq. (7)
3.6E+10
Rawc
1.4E+11
Raf
3.5E+10
7.0E+10
1.1E+11
1.4E+11
1.8E+11
Tw < Tc and Qc = 0
2.1E+11
Raf
Fig. 5. The variation of the general wall–ceiling Rayleigh number with the floor Rayleigh number at different heights for non-insulated ceiling conditions (L · L = 4 · 4).
Rawcm ¼ 0:3583 Ra0:9957 f
ð7Þ
Using correlation (7) and Eq. (6), the Eq. (8) is found. Nuf ¼ 0:15 Ra0:339 þ 0:01069 Ra0:4166 f wcm f
f
ð8Þ
The coefficient (f) defined in the current study is given in Table 5 with respect to the thermal conditions and dimensions. In Figs. 8–10, the new equation developed in the current study (Eq. (8)) and the equations given in the literature (Table 1) are compared with the numerical data. It is seen that the results of the equations given in the literature are different from each other. As they do not contain the effect of wall and ceiling thermal conditions, they match the numerical values for similar conditions only, but they devi-
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
suitable to use the new equation for the operation conditions specified in Table 2.
2400 2000 1600
4. Conclusions
Nuf 1200 800
In this study, in a floor heating system, a common Rayleigh number for the wall and ceiling is defined, and a correlation is obtained as,
Numerical values Equation (Current study) Kilkis (1998) Li et al. (1983) Min et al. (1956)
400 0 0.0E+00
1139
8.0E+10
1.6E+11
2.4E+11
3.2E+11
4.0E+11
4.8E+11
Ra f
Fig. 8. The comparison of the new equation developed in the current study (Eq. (7)) and the equations given in the literature (Table 1) with the numerical data at different floor dimensions (L · W = 2 · 2–6 · 6, H = 1.75, Tw = Tc = 15).
2
Raw Aw 1:25 ðL=H Þ þ Rac Ac Aw þ A c ¼ 0:3583 Ra0:9957 f
Rawcm ¼ Rawcm
Moreover, a new equation related to the floor Nusselt number is developed as a function of the Rayleigh numbers as, þ 0:01069 Ra0:4166 Nuf ¼ 0:15 Ra0:339 f wcm f
2400
A table (Table 5) is obtained for the coefficient (f) as depending on the thermal conditions and dimensions. The new equation and the equations given in the literature are compared with the numerical data. Results show that the maximum deviation is 10% for the new equation while it is 35% for those in the literature. Therefore, it will be more appropriate to use the equation of the present study.
2000 1600
Nuf 1200 800 Numerical values equation (current study) Kilkis (1998) Li et al. (1983) Min et al. (1956)
400 0 0.0E+00
9.0E+10
1.8E+11
2.7E+11
3.6E+11
4.5E+11
References 5.4E+11
Ra f
Fig. 9. The comparison of the new equation developed in the current study (Eq. (7)) and the equations given in the literature (Table 1) with the numerical data at different floor dimensions (L · W = 2 · 2–6 · 6, H = 2.5, Tw = Tc = 15).
2400 2000 1600
Nuf 1200 800 Numerical values Equation (current study) Kilkis (1998) Li et al. (1983) Min et al. (1956)
400 0 0.0E+00
8.0E+10
1.6E+11
2.4E+11
3.2E+11
4.0E+11
4.8E+11
Ra f Fig. 10. The comparison of the new equation developed in the current study (Eq. (7)) and the equations given in the literature (Table 1) with the numerical data at different floor dimensions for insulated ceiling conditions (L · W = 2 · 2–6 · 6, H = 1.75, Tw = 15).
ate from them up to 35% depending on the thermal conditions and dimensions. However, the new equation matches the numerical values for all conditions (maximum deviation 10%) because it includes the influence of all the surfaces thermal conditions. For this reason, it will be more
[1] Bozkır O, Canbazog˘lu S. Unsteady thermal performance analysis of a room with serial and parallel duct radiant floor heating system using hot airflow. Energy Buildings 2004;36:579–86. [2] Badran AA, Hamdan MA. Comparative study for under-floor heating using solar collectors or solar ponds. Appl Energy 2004;77:107–17. [3] Rekstad J, Meir M, Kristoffersen AR. Control and energy metering in low temperature heating systems. Energy Buildings 2003;35:281–91. [4] Cho SH, Zaheer-uddin M. Predictive control of intermittently operated radiant floor heating systems. Energy Convers Manage 2003;44:1333–42. [5] Athienitis AK, Chen Y. The effect of solar radiation on dynamic thermal performance of floor heating systems. Sol Energy 2000;69(3):229–37. [6] Cho SH, Zaheer-uddin M. An experimental study of multiple parameter switching control for radiant floor heating systems. Energy 1999;24:433–44. [7] Zaheer-uddin M, Zheng GR, Cho SH. Optimal operation of an embedded piping floor heating system with control input constraints. Energy Convers Manage 1997;38(7):713–25. [8] Khalifa AJN. Natural convective heat transfer coefficient a review I. Isolated vertical and horizontal surfaces. Energy Convers Manage 2001;42:491–504. [9] Kilkis B, Ritter LT. An analytical model for the design of in-slab electric heating panels. ASHRAE Trans 1998;SF-98-9-5. [10] KHalifa AJN, Marshall RH. Validation of heat transfer coefficients on interior building surfaces using a real – sized indoor test cell. Int J Heat Mass Transfer 1990;33(10):2219–36. [11] Li LD, Beckman WA, Mitchell JW. An experimental study of natural convection in an office room, large time results. Unpublished Report, Solar Energy Laboratory, University of Wisconsin, Madison, 1983, Ref. [9]. [12] Min TC, Schutrum LF, Parmelee GV, Vouris JD. Natural convection and radiation in a panel-heated room. ASHRAE Trans 1956;62: 337–58.
1140
R. Karadag˘, I. Teke / Energy Conversion and Management 49 (2008) 1134–1140
[13] Awbi HB. Calculation of convective heat transfer coefficients of room surfaces for natural convection. Energy Buildings 1998;28:219–27. [14] Alamdari F, Hammond GP. Building Services Res Technol 1983;4:106–12. [15] CIBSE. CIBSE Guide, Vols. A–C. London: CIBSE; 1986. [16] Awbi HB, Hatton A. Mixed convection from heated room surfaces. Energy Buildings 2000;32:153–66. [17] Gan G, Riffat SB. CFD modelling of air flow and thermal performance of an atrium integrated with photovoltaics. Build Environ 2004;39:735–48. [18] Male P, Croon MHJM, Tiggelaar RM, Berg A, Schouten JC. Heat and mass transfer in a square microchannel with asymmetric heating. Int J Heat Mass Transfer 2004;47:87–99. [19] Zhang Z, Li Y. CFD simulation on inlet configuration of plate-fin heat exchangers. Cryogenics 2003;43:673–8. [20] Dubovsky V, Ziskind G, Druckman S, Moshka E, Weiss Y, Letan R. Natural convection inside ventilated enclosure heated by downwardfacing plate: experiments and numerical simulations. Int J Heat Mass Transfer 2001;44:3155–68.
[21] Karadag˘ R, Teke I. The relation between radiative and convective heat transfer coefficients over the surfaces in a floor heated room. J Eng Mach 2005;548:21–9 [in Turkish]. [22] Omri M, Galanis N. Numerical analysis of turbulent natural convection in a cavity. In: 13th int heat transfer conf Sydney Australia; 2006. [23] Peng S-H, Davidson L. Large eddy simulation for turbulent buoyant flow in a confined cavity. Int J Heat Fluid Flow 2001;22(3):323–31. [24] Omri M, Galanis N. Numerical analysis of turbulent buoyant flows in enclosures: influence of grid and boundary conditions. Int J Therm Sci; doi:10.1016/j.ijthermalsci.2006.10.006. [25] Corvaro F, Paroncini M. Experimental analysis of natural convection in square cavities heated from below with 2D – PIV and holographic interferometry techniques. Exp Therm Fluid Sci; doi:10.1016/ j.expthermflusci.2006.07.006. [26] Holman JP. Heat transfer. 7th ed. London: McGraw Hill; 1992. [27] Karadag˘ R, Teke I. The effect of wall thermal conditions on the floor Nusselt number in the floor heating system. J Eng Natural Sci 2004;3:84–93 [in Turkish].