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Simplified model for a ventilated glass window under forced air flow conditions
Applied Thermal Engineering 26 (2006) 295–302 www.elsevier.com/locate/apthermeng
Simplified model for a ventilated glass window under forced air flow conditions K.A.R. Ismail
a,*
, J.R. Henrı´quez
b
a b
Depto. de Engenharia Te´rmica e de Fluidos—FEM-UNICAMP CP: 6122 CEP 13083-970 Campinas, SP, Brazil Depto. de Eng. Mecaˆnica—DEMEC, UFPE Av. Acadeˆmico He´lio Ramos, S/N CEP 50740-530, Recife, PE, Brazil Received 12 February 2004; accepted 30 April 2005 Available online 14 July 2005
Abstract This paper presents a study on a ventilated window composed of two glass sheets separated by a spacing through which air is forced to flow. The proposed model is one dimensional and unsteady based upon global energy balance over the glass sheets and the flowing fluid. The external glass sheet of the cavity is subjected to variable heat flow due to the solar radiation as well as variable external ambient temperature. The exchange of radiation energy (infrared radiation) between the glass sheets is also included in the formulation. Effects of the spacing between the glass sheets, variation of the forced mass flow rate on the total heat gain and the shading coefficients are investigated. The results show that the effect of the increase of the mass flow rate is found to reduce the mean solar heat gain and the shading coefficients while the increase of the fluid entry temperature is found to deteriorate the window thermal performance. 2005 Elsevier Ltd. All rights reserved. Keywords: Ventilated glass window; Simplified model; Thermally effective windows
1. Introduction During the last 30 years, window technology, research and development achieved a relatively high technological standard. Investments in research and development led to new generation of materials and design options which offer better thermal efficiencies and high performance. There are different advanced systems which allow better control and reduce the heat gain or loss depending on the design options. Among these options one can include windows with two and three glass sheets with and without selective properties [1–3], filled with absorbing gasses [4] and evacuated double glass windows [5], ventilated glass windows both forced and * Corresponding author. Tel.: +55 19 788 3376; fax: +55 19 788 3722. E-mail addresses: [email protected] (K.A.R. Ismail), rjorge @ufpe.br (J.R. Henrı´quez).
1359-4311/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2005.04.023
natural [6–8], windows with internal venetian, windows filled with phase change material (PCM) fixed or moving [9–11], intelligent windows sensible to incident solar radiation such as photocromatics, thermocromatics and electrocromatics [12,13]. Sparrow et al. [14] studied the interaction between radiation and natural convection in a channel formed by two vertical plates one subject to constant temperature and zero heat flux over the other. They concluded that the radiation effects are dominant for moderate and high Grashoff number and are negligible for low Grashoff number. Bar-Cohen and Rohsenow [15] presented correlations for optimizing the spacing between vertical plates for the cases isothermal plates and symmetrically or non symmetrically heated plates. Morrone et al. [16] studied numerically the problem of optimizing the spacing between symmetrically heated vertical plates cooled by naturally induced fluid flow and obtained correlation for the optimum spacing in terms of the
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296
Nomenclature A b c e Fi,k F G h Io J k m_ n Q
SC T t V w x
heat transfer area, m2 channel gap, m specific heat glass thickness, m view factor solar heat gain coefficient irradiation, W/m2 convection heat transfer coefficient solar radiation, W/m2 radiosity, W/m2 thermal conductivity mass flow in the channel per unit depth, kg/(s m) number of control volumes net radiant heat transfer between a differential element and the other elements formed by the cavity surfaces, W shading coefficient temperature time volume width coordinate along the flow direction
q s e r D
density or reflectance transmittance emissivity Stefan–Boltzmann constant, 5.57 · 108 W/ (m2 K4) increment
Superscript k level of the time in the discretized equations Subscripts c channel eg external glass ext external f fluid i center position at the control volume or i surface ig internal glass int internal k k surface M, M + 1 down and up interfaces of control volume i
Greek symbols k wavelength, lm a absorbance Grashoff number and the mass flow rate. Campo et al. [17] investigated the effect of adiabatic sections at entry and exit of a channel whose wall are heated by constant flux resulting in inducing natural fluid flow. The effects over the temperature profiles and the induced mass flow are presented and discussed. This paper present a study on a ventilated window composed of two glass sheets separated by a spacing through which air is forced to flow. The model is one dimensional, unsteady and is based upon global energy balance over the glass sheets as well as the flowing fluid. The external glass sheet is subjected to complex boundary conditions such as variable heat flow (solar radiation) and variable external ambient temperature. The radiant energy exchange of infrared radiation between the glass sheets is also included in the formulation. Effects of the variation of the forced mass flow rate on the total heat gain and the shading coefficients are presented.
2. Formulation of the problem The problem under consideration is simplified as two parallel glass sheets of height L, width w and a gap b,
e
b
e
Solar radiation Io L
Tint hint
Text hext w
Fluid flow Fig. 1. Scheme of a ventilated glass window.
and open at the top and bottom extremities as shown in Fig. 1. Circulating air forced in the channel between the glass sheets from the bottom to the top cools the glass sheets by removing the heat as it passes from the bottom to the top. The external glass sheet receives the incident solar radiation, part of which is absorbed
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302
Incident solar radiation
external glass
convection and thermal radiation
For the control volume i in the fluid one can write
internal glass thermal radiation
Solar radiation transmitted (directly)
convection
convection and thermal radiation Solar radiation absorbed
Solar radiation absorbed
conduction
conduction
Fig. 2. Mechanisms of heat transfer for a ventilated glass window.
in the external glass sheet and the rest hits the internal sheet where some of the energy is absorbed and the rest is then delivered to the internal ambient. The surfaces of the glass sheets in contact with the air flowing in the channel exchange heat by convection and also energy by radiation between the surfaces themselves. Part of the solar radiation crossed the first glass sheet is absorbed by the internal glass sheet, increases its temperature and causing exchange of heat by convection and radiation due to the temperature difference between the glass sheet surface and the internal ambient, fixed at 24 C. Fig. 2 shows the mechanisms of heat transfer for a ventilated glass window. The governing equations of the problem are deduced by performing energy balances over the control volumes in the fluid and the glass sheet. As can be seen in Fig. 3, the glass-fluid system is subdivided into n layers in the flow direction forming a grid of 3 · n control volumes (3 control volumes in each layer). Considering an arbitrary layer i, one can write the energy balance for each control volume in the layer formed by the external glass sheet, fluid and internal glass sheet.
n Point at the center of control volume Points at the interface of control volume
i+1
297
M+1 i M
2 2 1 1
Fig. 3. Control volumes in the channel and the glass sheets.
ðqcV Þf
oT f;i _ ¼ mcðT f;M T f;Mþ1 Þ þ hc AðT eg;i T f;i Þ ot þ hc AðT ig;i T f;i Þ
ð1Þ
In a similar manner, one can write the energy balance for the control volume i situated in the external glass sheet. In this case the external glass is exposed to the solar radiation and absorbs part of this radiation. The glass surfaces in contact with the external ambient exchanges heat by both convection and radiation with the respective ambient, while the glass surface in contact with the flow in the channel is subject to forced convection and heat exchange by infrared radiation with the other differential elements situated in the external glass sheet. The energy balance can be written as ðqcV Þeg
oT eg;i ¼ hext AðT ext T eg;i Þ þ hc AðT f;i T eg;i Þ ot þ reeg AðT 4ext T 4eg;i Þ þ aeg I o þ Qeg;i
ð2Þ
The energy balance on the control volume situated in the internal glass sheet, where one of the glass surfaces is in contact with the fluid flow in the channel subject to forced convection, heat exchange by radiation with the other differential elements of the glass sheets and absorption of some solar radiation transmitted through the external glass. The glass surface in contact with the internal ambient exchanges heat by natural convection and infrared radiation. The energy balance can be written in the form ðqcV Þig;i
oT ig ¼ hint AðT int T ig;i Þ þ hc AðT f;i T ig;i Þ ot þ reig AðT 4int T 4ig;i Þ þ aig seg I o þ Qig;i
ð3Þ
In Eq. (1), the temperature at the frontiers of the control volume and its center appear in the equation. Considering that the control volume is small, one can consider that the temperature at the center as the mean of the frontier temperature T f;Mþ1 þ T f;M 2 which when substituted in Eq. (1) one can write T f;i ¼
ð4Þ
ðqcV Þf oðT f;Mþ1 þ T f;M Þ _ ¼ mcðT f;M T f;Mþ1 Þ ot 2 T f;Mþ1 þ T f;M þ hc A T eg;i 2 T f;Mþ1 þ T f;M þ hc A T ig;i 2 ð5Þ
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302
298
Eqs. (2), (3) and (5) can be solved numerically by finite difference. The discretized equations for the fluid, external glass and internal glass can be written successively as _ ÞDt 2ðm=w hc Dt k kþ1 kþ1 2 T f;Mþ1 ¼ T f;M þ 1 þ T qbDx qcb f;M _ 2ðm=wÞDt hc Dt k 2 þ 1 T qbDx qcb f;Mþ1 hc Dt þ 2 T keg;i þ T kig;i ð6Þ qcb k T kþ1 eg;i ¼ T eg;i þ
hext Dt hc Dt ðT ext T keg;i Þ þ ðqcÞeg eeg ðqcÞeg eeg
Qeg;i Dt aeg DtI o þ ðqcÞeg eeg ðqcÞeg eeg Dxw
k T kþ1 ig;i ¼ T ig;i þ
þ
ð7Þ
k¼1
where Fk,i is the view factor between the surfaces k and i. Therefore, making use of the reciprocity relation between the view factors AiFi,k = AkFk,i, one can rewrite Eq. (11) in the form n X F i;k J k ð12Þ Gi ¼ From Eqs. (10) and (12), considering that qi = 1 ei, one can write n X F i;k J k ð13Þ J i ¼ ei rT 4i þ ð1 ei Þ
Qi ei ðrT 4i J i Þ ¼ Ai ð1 ei Þ
ð8Þ
Eqs. (6)–(8) are applied for each control volume from i = 1 to i = n. Initially the temperature field in the channel and in the glass sheets is known and considered uniform. From the boundary condition at entry to the channel, the temperature and the fluid mass flow are known at this point, permitting calculating the exit temperature from the first control volume in the fluid. Knowing this temperature and the temperature at entry, can be obtain an average temperature in the first fluid control volume and hence one can calculate the temperature of the first control volume of the internal and external glass. This procedure is repeated for each control volume by marching in the vertical direction until the last control volume in the channel. This procedure is repeated for each time interval until the simulation time is terminated. The net radiant heat transfer between a surface and the rest of the surfaces forming the channel Qeg,i and Qig,i, is determined by the radiosity method. The radiant energy balance over the generic i surface can written as Qi ¼ J i Gi Ai
Ai Gi ¼ A1 F 1;i J 1 þ A2 F 2;i J 2 þ A3 F 3;i J 3 þ þ An F n;i J n n X ¼ Ak F k;i J k ð11Þ
Combining Eqs. (9) and (10) one can also write
reig Dt ðT 4 ðT kig;i Þ4 Þ ðqcÞig eig int
Qig;i Dt aig seg DtI o þ ðqcÞig eig ðqcÞig eig Dxw
Considering that G is the irradiation received from all the surfaces one can write
k¼1
hint Dt hc Dt ðT int T kig;i Þ þ ðqcÞig eig ðqcÞig eig
ðT kf;i T kig;i Þ þ
ð10Þ
k¼1
reeg Dt ðT kf;i T keg;i Þ þ ðT 4 ðT keg;i Þ4 Þ ðqcÞeg eeg ext þ
J i ¼ ei rT 4i þ qi Gi
ð14Þ
Thus the problem of calculating the radiant heat transfer in a cavity of n isothermal surfaces is reduced to applying Eq. (13) to each of the n surfaces forming a set of n equations to be solved simultaneously. Now consider the case of a cavity formed by two vertical parallel plates with a temperature gradient along the plate surface. Dividing each plate into n equal segments and assuming that each segment is an isothermal surface, a cavity of 2n surfaces is formed as shown in Fig. 4. Fig. 4 illustrates the radiant heat exchange between the surface i of plate 1 and the surface n + 1 to
ð9Þ
where Ji is the rate of radiant energy leaving the surface per unit area and Gi is the rate of irradiation received, per unit area, from all the surfaces of the cavity.
Fig. 4. Radiant heat exchange between i element of plate 1 and the n + 1 to 2n element of plate 2.
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302
2n of plate 2. One can observe that the surface i does not exchange radiant heat with the surface in the same plane because the view factor in this case is zero. Also, the fictitious bottom and upper surfaces are ignored to simulate the physical situation corresponding to infinite parallel plates. Under these conditions one can apply Eq. (13) for each of the surfaces as below
k normal
φi
r
z y
J i ¼ ei rT 4i þ ð1 ei Þ½F i;nþ1 J nþ1 þ F i;nþ2 J nþ2 þ for i ¼ 1 at n
dAk
φk normal
dAi
þ F i;2n J 2n ;
299
x i
ð15Þ Fig. 5. Parameters used to calculate the view factor.
and, J i ¼ ei rT 4i þ ð1 ei Þ½F i;1 J 1 þ F i;2 J 2 þ þ F i;n J n ; for i ¼ n þ 1 at 2n
ð16Þ
Thus one has a set of 2n equations in 2n unknowns which can be written as a1;1 J 1 þ a1;2 J 2 þ a1;3 J 3 þ þ a1;n J n þ a1;nþ1 J nþ1 þ þ a1;2n J 2n ¼ C 1 a2;1 J 1 þ a2;2 J 2 þ a2;3 J 3 þ þ a2;n J n þ a2;nþ1 J nþ1 þ þ a2;2n J 2n ¼ C 2 an;1 J 1 þ an;2 J 2 þ an;3 J 3 þ þ an;n J n þ an;nþ1 J nþ1 þ þ an;2n J 2n ¼ C n anþ1;1 J 1 þ anþ1;2 J 2 þ anþ1;3 J 3 þ þ anþ1;n J n þ anþ1;nþ1 J nþ1 þ þ anþ1;2n J 2n ¼ C nþ1 anþ2;1 J 1 þ anþ2;2 J 2 þ anþ2;3 J 3 þ þ anþ2;n J n þ anþ2;nþ1 J nþ1
Fig. 6. Coordinate system for the calculation of the view factor between two elements of the parallel plates (element i and element k).
þ þ anþ2;2n J 2n ¼ C nþ2 F i;k ¼
a2n;1 J 1 þ a2n;2 J 2 þ a2n;3 J 3 þ þ a2n;n J n þ a2n;nþ1 J nþ1
1 2pAi
I I ck
þ þ a2n;2n J 2n ¼ C 2n
ci
þ lnðrÞ dzi dzk
or in matrix form as ½a fJ g ¼ fCg
½lnðrÞ dxi dxk þ lnðrÞ dy i dy k
ð17Þ
ð20Þ
Choosing two arbitrary elements and adopting the coordinate system shown in Fig. 6, one can write 2
The solution of the system given by Eq. (18) can be written as 1
fJ g ¼ ½a fCg
ð18Þ
One can use the general expression given by Eq. (19) to calculate the view factor between finite surfaces. The parameters of Eq. (19) are defined in Fig. 5. Z Z 1 cosð/i Þ cosð/k Þ F i;k ¼ dAk dAi ð19Þ Ai Ai Ak pr2 After some mathematical manipulations and by applying Stokes theorem, the surface integrals of Eq. (19) are transformed into line integrals written as
2
b ¼ ðy k y i Þ and r2 ¼ ðxk xi Þ þ ðzk zi Þ þ b2 which when substituted into Eq. (20) one obtains 2 3 I I qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pAi F i;k ¼ 4 ln ðxk xi Þ2 þ ðzk zi Þ2 þ b2 dxi 5dxk ck
þ
ci
I ck
2 3 I qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 4 ln ðxk xi Þ þ ðzk zi Þ þ b dzi 5dzk ci
ð21Þ When the equation is applied to the control elements Ci and Ck, one can write
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302
300 Z
Z
w
xk ¼0
w
600
xi ¼0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ðxk xi Þ2 þ ððk 1ÞDz iDzÞ2 þ b2 dxi dxk Z 0 Z w qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ln ðxk xi Þ2 þ ðkDz iDzÞ2 þ b2 dxi dxk þ þ þ þ þ þ
Z Z Z Z Z
xk ¼w w xk ¼0 0
Z
xi ¼0
xk ¼w
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ðxk xi Þ2 þ ððk 1ÞDz ði 1ÞDzÞ2 þ b2 dxi dxk
0
xi ¼w
Z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ðxk xi Þ2 þ ðkDz ði 1ÞDzÞ2 þ b2 dxi dxk
0
xi ¼w
ðk1ÞDz
Z
zi ¼ði1ÞDz
kDz
iDz
Z
zk ¼ðk1ÞDz
Z
zk ¼kDz kDz
300
Solar radiation 1.0E-03 kg/s/m
200
1.8E-03 kg/s/m 3.6E-03 kg/s/m
100
5.4E-03 kg/s/m
0
simple glass 8 mm thickness -100
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln w2 þ ðzk zi Þ2 þ b2 dzi dzk
4
Z
ði1ÞDz
ln zk ¼ðk1ÞDz
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðzk zi Þ2 þ b2 dzi dzk
ð22Þ
zi ¼iDz
The above equation is used to calculate the view factor between any two elements of parallel plates shown in Fig. 6. The numerical code of the ventilated glass window is optimized to reduce the computational time while keeping good precision. The tests realized for a double glass window of 1.0 m height, 0.6 m width and 3 mm spacing between the glass sheets with air mass flow rate of 5.4 · 103 kg/s/m indicated that 20 grid points are adequate for the present calculations.
10
12
14
16
18
20
The total heat gain, the coefficient of solar heat gain and the shading coefficient are important parameters for window performance calculations and comparisons. To calculate the total heat gain one must take into account the solar energy passing directly through the transparent surface and the solar energy reaching the internal ambient after being absorbed by the glass and redirected by the heat transfer mechanisms. In the case of transient conditions, the total heat gain can be calculated by energy balance over the window surface in contact with the internal ambient n 1 X qtotal ¼ hint DxðT ig;i T int Þ þ rei DxðT 4ig;i T 4int Þ nDx i¼1 þ sig seg I o Dx
3. Results and discussion Fig. 7 shows the effect of the air mass flow on the temperature of the internal glass sheet at its mid height position. As can be seen the increase of the mass flow rate reduces the internal glass temperature and hence reduces the total heat gain in comparison with the case of a simple glass window as in shown in Fig. 8. 33 31 29
Temperature [°C]
8
Fig. 8. Variation of the total heat gain with time.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðwÞ2 þ ðzk zi Þ2 þ b2 dzi dzk
zi ¼iDz
6
Time [h]
zi ¼ði1ÞDz ði1ÞDz
ln Z
400
7.2E-03 kg/s/m
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ðzk zi Þ2 þ b2 dzi dzk
iDz
zk ¼kDz
ðk1ÞDz
500
Total heat gain [W/m²]
2pAi F i;k ¼
ð23Þ
where the subscript i refers to a given control volume of the internal glass sheet. The sum of energy balances over each element allows calculating the total heat gain (qtotal). The solar heat gain coefficient (F) is the fraction of incident solar radiation admitted through a window. The solar heat gain can be obtained indirectly by calculating the total heat gain with and without solar radiation, subtracting one from the other and dividing by the incident solar radiation. q qwithout I F ¼ with I ð24Þ I The shading coefficient (SC) is used to establish a comparison parameter relating the solar heat gain of a desired system to a reference system
27 25 mass flow rate [ kg/s/m]
23
SC ¼
1.0E-03 1.8E-03 3.6E-03 5.4E-03 7.2E-03 simple glass - 8mm thickness external ambient temperature
21 19 17 15 4
6
8
10
12
14
16
18
20
Time [h]
Fig. 7. Effect of the air mass flow rate on the internal glass surface temperature.
F desired window F reference
ð25Þ
The reference system adopted in this study is a simple glass sheet of 3.0 mm thickness, type DSA, having an extinction coefficient of a 0.0078 (1/mm) and refraction index of 1.526. Fig. 9 shows the total heat gain, solar heat gain and the heat gain due to the difference between the internal and external ambient temperatures. As can be observed,
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302 31
600 (*)Total heat gain (**) Heat exchange due to the temperature difference between the internal and external ambient
500
(*)
temperature at mid length of the external glass
29
(**) Solar radiation
27
Solar heat gain
Temperature [ºC]
400
Heat [W/m²]
301
300 200
25 23 without radiation
21
100
19
0
17
with radiation
15
-100 4
6
8
10
12
14
16
18
4
20
6
8
10
Fig. 9. Comparison between the heat gain with and without including the solar radiation exchange for the case of 1.0 · 103 [kg/s/m].
Simple glass windows 8 mm
1.0 · 103 1.8 · 103 3.6 · 103 5.4 · 103 7.2 · 103
Shading coefficient (SC)
0.779 0.776 0.769 0.765 0.762 0.89
0.855 0.852 0.844 0.840 0.836 0.978
16
18
20
33 31
mass flow rate: 7.2 E-03 kg/s/m gap between plates: 3 mm thickness of glass plates: 8mm
29
Temperature [°C]
Mass flow rate [kg/s/m]
Solar heat gain coefficient (F)
14
Fig. 10. Variation of the temperature of the external glass with and without radiation exchange between the glass sheets.
Table 1 Values of F and SC for forced convection ventilated window Ventilated window
12
Time [h]
Time [h]
27 25 23 (*)
(**)
Tamb
21
fluid entry temperature [˚C]
19
(*) 20 [˚C] (**) external ambient temperature
17 15 4
the incident solar radiation represents a dominant contribution to the total heat gain. If the instantaneous solar heat gain coefficient (F) and shading coefficient (SC), are integrated over the incidence period of solar radiation, one can obtain the average values of there coefficients. Table 1 shows the calculated average values of solar heat gain coefficient and shading coefficient for different values of mass flow rate. As can be seen both the values of SC and F decrease with the increase of the mass flow rate and are well below the case of a simple glass window. In order to verify the relative importance of the heat exchange by infrared radiation between the surfaces of the glass channel, simulations were realized with and without infrared radiation exchange and this results are shown in Fig. 10. Fig. 10 shows the temperature at mid height of the external glass with and without including the long wave radiation exchange. Similar results are obtained for the working fluid and the internal glass surface at the same position. As can be seen the effects due to long wave radiation exchange between the external and internal surfaces are very small causing a temperature variation of about 0.5 C. In order to investigate the effect of the fluid entry temperature on the performance of ventilated windows two situations were simulated; one where the fluid entry temperature is maintained constant as 20 C and the
6
8
10
12
14
16
18
20
Time [h]
Fig. 11. Effect of the fluid entry temperature on the temperature of the internal glass.
other situation where the fluid entry temperature varies in a similar manner to the variation of the external ambient temperature. The results are shown in Fig. 11. One can observe that when the entry temperature of the fluid is increased to the external ambient temperature, that is, the circulating fluid is admitted from the external ambient, the internal glass temperature increases in a corresponding manner.
4. Conclusions A mathematical numerical model to simulate a double glass ventilated window with forced air flow is formulated and implemented. The model takes into account the long wave radiation exchange between the glass sheets. Detailed analysis of the results showed that effect of including this radiation exchange is relatively small at least for the investigated flow conditions. Possibly for natural flow and stagnant conditions the effect of this thermal exchange may become important. The results show that the effect of the increase of the mass flow rate over the thermal behavior of the
302
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ventilated glass window causes a reduction of the mean solar heat gain coefficient and the shading coefficient. Consequently, this improves the performance of the double glass ventilated window in relation to a simple glass window. The increase of the fluid entry temperature of a ventilated window deteriorates the window thermal performance. Acknowledgement The authors wish to thank the CNPq for the scholarships and the financial support.
References [1] C.M. Lampert, Heat mirror coatings for energy conserving windows, Solar Energy Materials and Solar Cell 6 (1981) 1–41. [2] P.J. Sebastian, V. Sivaramakrishnan, CdSe thin films as solar control coatings, Solar Energy Materials and Solar Cell 27 (1992) 321–326. [3] P.J. Sebastian, M. Pattabi, Solar control characteristics of Cu2Se coating, Journal of Physics D: Applied Physics 25 (1992) 981–985. [4] S. Reilly, D. Arasteh, M. Rubin, The effects of infrared absorbing gases on window heat transfer: a comparison of theory and experiments, Solar Energy Materials 20 (1990) 277–288. [5] R. Sullivan, D.K. Arasteh, F.A. Beck, S.E. Selkowitz, Energy performance of evacuated glazing in residential buildings, ASHRAE Transactions 102 (part 2) (1996) 220–227. [6] J. Tanimoto, K. Kimura, Simulation study on an air flow window system with an integrated roll screen, Energy and Buildings 26 (1997) 317–325.
[7] K.H. Haddad, A.H. Elmahdy, Comparison of the monthly thermal performance of a conventional window and a supply-air window, ASHRAE Transactions 104 (part 1B) (1998) 1261–1270. [8] J.R. Henrı´quez, Modelagem e ana´lise de janelas te´rmicas, Doctorate thesis, Faculdade de Engenharia Mecaˆnica, Universidade Estadual de Campinas, Campinas-SP, Brasil, 2002. [9] J.R. Henrı´quez, Estudo nume´rico e experimental sobre vidros te´rmicos, Master thesis, Faculdade de Engenharia Mecaˆnica, Universidade Estadual de Campinas, Campinas-SP, Brasil, 1996. [10] K.A.R. Ismail, J.R. Henrı´quez, U-values, optical and thermal coefficients of composite glass systems, Solar Energy Materials and Solar Cell 52 (1998) 155–182. [11] K.A.R. Ismail, J.R. Henrı´quez, Thermally effective windows with moving phase change material curtains, Applied Thermal Engineering 21 (2001) 1909–1923. [12] K. Rottkay, M. Rubin, J. Kerr, Optical modeling of a complete electrochromic device, in: The 2nd International Conference on Electrochromics, San Diego, USA, October 1996. [13] A. Seeboth, J. Schneider, A. Parzak, Material for intelligent sun protecting glazing, Solar Energy Materials and Solar Cells 60 (2000) 263–277. [14] E.M. Sparrow, S. Shah, C. Prakash, Natural convection in a vertical channel: I. Interacting convection and radiation. II. The vertical plate with and without shrouding, Numerical Heat Transfer 3 (1980) 297–314. [15] A. Bar-Cohen, W.M. Rohsenow, Thermally optimum spacing of vertical, natural convection cooled parallel plates, Journal of Heat Transfer 106 (1984) 116–123. [16] B. Morrone, A. Campo, O. Manca, Optimum plate separation in vertical parallel-plate channel for natural convective flows: incorporation of large spaces at the channel extremes, International Journal of Heat Mass Transfer 40 (5) (1997) 993–1000. [17] A. Campo, O. Manca, B. Morrone, Numerical analysis of partially heated vertical parallel plates in natural convective cooling, Numerical Heat Transfer—Part A (Applications) 36 (1999) 129–151.
Simplified model for a ventilated glass window under forced air flow conditions K.A.R. Ismail
a,*
, J.R. Henrı´quez
b
a b
Depto. de Engenharia Te´rmica e de Fluidos—FEM-UNICAMP CP: 6122 CEP 13083-970 Campinas, SP, Brazil Depto. de Eng. Mecaˆnica—DEMEC, UFPE Av. Acadeˆmico He´lio Ramos, S/N CEP 50740-530, Recife, PE, Brazil Received 12 February 2004; accepted 30 April 2005 Available online 14 July 2005
Abstract This paper presents a study on a ventilated window composed of two glass sheets separated by a spacing through which air is forced to flow. The proposed model is one dimensional and unsteady based upon global energy balance over the glass sheets and the flowing fluid. The external glass sheet of the cavity is subjected to variable heat flow due to the solar radiation as well as variable external ambient temperature. The exchange of radiation energy (infrared radiation) between the glass sheets is also included in the formulation. Effects of the spacing between the glass sheets, variation of the forced mass flow rate on the total heat gain and the shading coefficients are investigated. The results show that the effect of the increase of the mass flow rate is found to reduce the mean solar heat gain and the shading coefficients while the increase of the fluid entry temperature is found to deteriorate the window thermal performance. 2005 Elsevier Ltd. All rights reserved. Keywords: Ventilated glass window; Simplified model; Thermally effective windows
1. Introduction During the last 30 years, window technology, research and development achieved a relatively high technological standard. Investments in research and development led to new generation of materials and design options which offer better thermal efficiencies and high performance. There are different advanced systems which allow better control and reduce the heat gain or loss depending on the design options. Among these options one can include windows with two and three glass sheets with and without selective properties [1–3], filled with absorbing gasses [4] and evacuated double glass windows [5], ventilated glass windows both forced and * Corresponding author. Tel.: +55 19 788 3376; fax: +55 19 788 3722. E-mail addresses: [email protected] (K.A.R. Ismail), rjorge @ufpe.br (J.R. Henrı´quez).
1359-4311/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2005.04.023
natural [6–8], windows with internal venetian, windows filled with phase change material (PCM) fixed or moving [9–11], intelligent windows sensible to incident solar radiation such as photocromatics, thermocromatics and electrocromatics [12,13]. Sparrow et al. [14] studied the interaction between radiation and natural convection in a channel formed by two vertical plates one subject to constant temperature and zero heat flux over the other. They concluded that the radiation effects are dominant for moderate and high Grashoff number and are negligible for low Grashoff number. Bar-Cohen and Rohsenow [15] presented correlations for optimizing the spacing between vertical plates for the cases isothermal plates and symmetrically or non symmetrically heated plates. Morrone et al. [16] studied numerically the problem of optimizing the spacing between symmetrically heated vertical plates cooled by naturally induced fluid flow and obtained correlation for the optimum spacing in terms of the
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302
296
Nomenclature A b c e Fi,k F G h Io J k m_ n Q
SC T t V w x
heat transfer area, m2 channel gap, m specific heat glass thickness, m view factor solar heat gain coefficient irradiation, W/m2 convection heat transfer coefficient solar radiation, W/m2 radiosity, W/m2 thermal conductivity mass flow in the channel per unit depth, kg/(s m) number of control volumes net radiant heat transfer between a differential element and the other elements formed by the cavity surfaces, W shading coefficient temperature time volume width coordinate along the flow direction
q s e r D
density or reflectance transmittance emissivity Stefan–Boltzmann constant, 5.57 · 108 W/ (m2 K4) increment
Superscript k level of the time in the discretized equations Subscripts c channel eg external glass ext external f fluid i center position at the control volume or i surface ig internal glass int internal k k surface M, M + 1 down and up interfaces of control volume i
Greek symbols k wavelength, lm a absorbance Grashoff number and the mass flow rate. Campo et al. [17] investigated the effect of adiabatic sections at entry and exit of a channel whose wall are heated by constant flux resulting in inducing natural fluid flow. The effects over the temperature profiles and the induced mass flow are presented and discussed. This paper present a study on a ventilated window composed of two glass sheets separated by a spacing through which air is forced to flow. The model is one dimensional, unsteady and is based upon global energy balance over the glass sheets as well as the flowing fluid. The external glass sheet is subjected to complex boundary conditions such as variable heat flow (solar radiation) and variable external ambient temperature. The radiant energy exchange of infrared radiation between the glass sheets is also included in the formulation. Effects of the variation of the forced mass flow rate on the total heat gain and the shading coefficients are presented.
2. Formulation of the problem The problem under consideration is simplified as two parallel glass sheets of height L, width w and a gap b,
e
b
e
Solar radiation Io L
Tint hint
Text hext w
Fluid flow Fig. 1. Scheme of a ventilated glass window.
and open at the top and bottom extremities as shown in Fig. 1. Circulating air forced in the channel between the glass sheets from the bottom to the top cools the glass sheets by removing the heat as it passes from the bottom to the top. The external glass sheet receives the incident solar radiation, part of which is absorbed
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302
Incident solar radiation
external glass
convection and thermal radiation
For the control volume i in the fluid one can write
internal glass thermal radiation
Solar radiation transmitted (directly)
convection
convection and thermal radiation Solar radiation absorbed
Solar radiation absorbed
conduction
conduction
Fig. 2. Mechanisms of heat transfer for a ventilated glass window.
in the external glass sheet and the rest hits the internal sheet where some of the energy is absorbed and the rest is then delivered to the internal ambient. The surfaces of the glass sheets in contact with the air flowing in the channel exchange heat by convection and also energy by radiation between the surfaces themselves. Part of the solar radiation crossed the first glass sheet is absorbed by the internal glass sheet, increases its temperature and causing exchange of heat by convection and radiation due to the temperature difference between the glass sheet surface and the internal ambient, fixed at 24 C. Fig. 2 shows the mechanisms of heat transfer for a ventilated glass window. The governing equations of the problem are deduced by performing energy balances over the control volumes in the fluid and the glass sheet. As can be seen in Fig. 3, the glass-fluid system is subdivided into n layers in the flow direction forming a grid of 3 · n control volumes (3 control volumes in each layer). Considering an arbitrary layer i, one can write the energy balance for each control volume in the layer formed by the external glass sheet, fluid and internal glass sheet.
n Point at the center of control volume Points at the interface of control volume
i+1
297
M+1 i M
2 2 1 1
Fig. 3. Control volumes in the channel and the glass sheets.
ðqcV Þf
oT f;i _ ¼ mcðT f;M T f;Mþ1 Þ þ hc AðT eg;i T f;i Þ ot þ hc AðT ig;i T f;i Þ
ð1Þ
In a similar manner, one can write the energy balance for the control volume i situated in the external glass sheet. In this case the external glass is exposed to the solar radiation and absorbs part of this radiation. The glass surfaces in contact with the external ambient exchanges heat by both convection and radiation with the respective ambient, while the glass surface in contact with the flow in the channel is subject to forced convection and heat exchange by infrared radiation with the other differential elements situated in the external glass sheet. The energy balance can be written as ðqcV Þeg
oT eg;i ¼ hext AðT ext T eg;i Þ þ hc AðT f;i T eg;i Þ ot þ reeg AðT 4ext T 4eg;i Þ þ aeg I o þ Qeg;i
ð2Þ
The energy balance on the control volume situated in the internal glass sheet, where one of the glass surfaces is in contact with the fluid flow in the channel subject to forced convection, heat exchange by radiation with the other differential elements of the glass sheets and absorption of some solar radiation transmitted through the external glass. The glass surface in contact with the internal ambient exchanges heat by natural convection and infrared radiation. The energy balance can be written in the form ðqcV Þig;i
oT ig ¼ hint AðT int T ig;i Þ þ hc AðT f;i T ig;i Þ ot þ reig AðT 4int T 4ig;i Þ þ aig seg I o þ Qig;i
ð3Þ
In Eq. (1), the temperature at the frontiers of the control volume and its center appear in the equation. Considering that the control volume is small, one can consider that the temperature at the center as the mean of the frontier temperature T f;Mþ1 þ T f;M 2 which when substituted in Eq. (1) one can write T f;i ¼
ð4Þ
ðqcV Þf oðT f;Mþ1 þ T f;M Þ _ ¼ mcðT f;M T f;Mþ1 Þ ot 2 T f;Mþ1 þ T f;M þ hc A T eg;i 2 T f;Mþ1 þ T f;M þ hc A T ig;i 2 ð5Þ
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302
298
Eqs. (2), (3) and (5) can be solved numerically by finite difference. The discretized equations for the fluid, external glass and internal glass can be written successively as _ ÞDt 2ðm=w hc Dt k kþ1 kþ1 2 T f;Mþ1 ¼ T f;M þ 1 þ T qbDx qcb f;M _ 2ðm=wÞDt hc Dt k 2 þ 1 T qbDx qcb f;Mþ1 hc Dt þ 2 T keg;i þ T kig;i ð6Þ qcb k T kþ1 eg;i ¼ T eg;i þ
hext Dt hc Dt ðT ext T keg;i Þ þ ðqcÞeg eeg ðqcÞeg eeg
Qeg;i Dt aeg DtI o þ ðqcÞeg eeg ðqcÞeg eeg Dxw
k T kþ1 ig;i ¼ T ig;i þ
þ
ð7Þ
k¼1
where Fk,i is the view factor between the surfaces k and i. Therefore, making use of the reciprocity relation between the view factors AiFi,k = AkFk,i, one can rewrite Eq. (11) in the form n X F i;k J k ð12Þ Gi ¼ From Eqs. (10) and (12), considering that qi = 1 ei, one can write n X F i;k J k ð13Þ J i ¼ ei rT 4i þ ð1 ei Þ
Qi ei ðrT 4i J i Þ ¼ Ai ð1 ei Þ
ð8Þ
Eqs. (6)–(8) are applied for each control volume from i = 1 to i = n. Initially the temperature field in the channel and in the glass sheets is known and considered uniform. From the boundary condition at entry to the channel, the temperature and the fluid mass flow are known at this point, permitting calculating the exit temperature from the first control volume in the fluid. Knowing this temperature and the temperature at entry, can be obtain an average temperature in the first fluid control volume and hence one can calculate the temperature of the first control volume of the internal and external glass. This procedure is repeated for each control volume by marching in the vertical direction until the last control volume in the channel. This procedure is repeated for each time interval until the simulation time is terminated. The net radiant heat transfer between a surface and the rest of the surfaces forming the channel Qeg,i and Qig,i, is determined by the radiosity method. The radiant energy balance over the generic i surface can written as Qi ¼ J i Gi Ai
Ai Gi ¼ A1 F 1;i J 1 þ A2 F 2;i J 2 þ A3 F 3;i J 3 þ þ An F n;i J n n X ¼ Ak F k;i J k ð11Þ
Combining Eqs. (9) and (10) one can also write
reig Dt ðT 4 ðT kig;i Þ4 Þ ðqcÞig eig int
Qig;i Dt aig seg DtI o þ ðqcÞig eig ðqcÞig eig Dxw
Considering that G is the irradiation received from all the surfaces one can write
k¼1
hint Dt hc Dt ðT int T kig;i Þ þ ðqcÞig eig ðqcÞig eig
ðT kf;i T kig;i Þ þ
ð10Þ
k¼1
reeg Dt ðT kf;i T keg;i Þ þ ðT 4 ðT keg;i Þ4 Þ ðqcÞeg eeg ext þ
J i ¼ ei rT 4i þ qi Gi
ð14Þ
Thus the problem of calculating the radiant heat transfer in a cavity of n isothermal surfaces is reduced to applying Eq. (13) to each of the n surfaces forming a set of n equations to be solved simultaneously. Now consider the case of a cavity formed by two vertical parallel plates with a temperature gradient along the plate surface. Dividing each plate into n equal segments and assuming that each segment is an isothermal surface, a cavity of 2n surfaces is formed as shown in Fig. 4. Fig. 4 illustrates the radiant heat exchange between the surface i of plate 1 and the surface n + 1 to
ð9Þ
where Ji is the rate of radiant energy leaving the surface per unit area and Gi is the rate of irradiation received, per unit area, from all the surfaces of the cavity.
Fig. 4. Radiant heat exchange between i element of plate 1 and the n + 1 to 2n element of plate 2.
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302
2n of plate 2. One can observe that the surface i does not exchange radiant heat with the surface in the same plane because the view factor in this case is zero. Also, the fictitious bottom and upper surfaces are ignored to simulate the physical situation corresponding to infinite parallel plates. Under these conditions one can apply Eq. (13) for each of the surfaces as below
k normal
φi
r
z y
J i ¼ ei rT 4i þ ð1 ei Þ½F i;nþ1 J nþ1 þ F i;nþ2 J nþ2 þ for i ¼ 1 at n
dAk
φk normal
dAi
þ F i;2n J 2n ;
299
x i
ð15Þ Fig. 5. Parameters used to calculate the view factor.
and, J i ¼ ei rT 4i þ ð1 ei Þ½F i;1 J 1 þ F i;2 J 2 þ þ F i;n J n ; for i ¼ n þ 1 at 2n
ð16Þ
Thus one has a set of 2n equations in 2n unknowns which can be written as a1;1 J 1 þ a1;2 J 2 þ a1;3 J 3 þ þ a1;n J n þ a1;nþ1 J nþ1 þ þ a1;2n J 2n ¼ C 1 a2;1 J 1 þ a2;2 J 2 þ a2;3 J 3 þ þ a2;n J n þ a2;nþ1 J nþ1 þ þ a2;2n J 2n ¼ C 2 an;1 J 1 þ an;2 J 2 þ an;3 J 3 þ þ an;n J n þ an;nþ1 J nþ1 þ þ an;2n J 2n ¼ C n anþ1;1 J 1 þ anþ1;2 J 2 þ anþ1;3 J 3 þ þ anþ1;n J n þ anþ1;nþ1 J nþ1 þ þ anþ1;2n J 2n ¼ C nþ1 anþ2;1 J 1 þ anþ2;2 J 2 þ anþ2;3 J 3 þ þ anþ2;n J n þ anþ2;nþ1 J nþ1
Fig. 6. Coordinate system for the calculation of the view factor between two elements of the parallel plates (element i and element k).
þ þ anþ2;2n J 2n ¼ C nþ2 F i;k ¼
a2n;1 J 1 þ a2n;2 J 2 þ a2n;3 J 3 þ þ a2n;n J n þ a2n;nþ1 J nþ1
1 2pAi
I I ck
þ þ a2n;2n J 2n ¼ C 2n
ci
þ lnðrÞ dzi dzk
or in matrix form as ½a fJ g ¼ fCg
½lnðrÞ dxi dxk þ lnðrÞ dy i dy k
ð17Þ
ð20Þ
Choosing two arbitrary elements and adopting the coordinate system shown in Fig. 6, one can write 2
The solution of the system given by Eq. (18) can be written as 1
fJ g ¼ ½a fCg
ð18Þ
One can use the general expression given by Eq. (19) to calculate the view factor between finite surfaces. The parameters of Eq. (19) are defined in Fig. 5. Z Z 1 cosð/i Þ cosð/k Þ F i;k ¼ dAk dAi ð19Þ Ai Ai Ak pr2 After some mathematical manipulations and by applying Stokes theorem, the surface integrals of Eq. (19) are transformed into line integrals written as
2
b ¼ ðy k y i Þ and r2 ¼ ðxk xi Þ þ ðzk zi Þ þ b2 which when substituted into Eq. (20) one obtains 2 3 I I qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pAi F i;k ¼ 4 ln ðxk xi Þ2 þ ðzk zi Þ2 þ b2 dxi 5dxk ck
þ
ci
I ck
2 3 I qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 4 ln ðxk xi Þ þ ðzk zi Þ þ b dzi 5dzk ci
ð21Þ When the equation is applied to the control elements Ci and Ck, one can write
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302
300 Z
Z
w
xk ¼0
w
600
xi ¼0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ðxk xi Þ2 þ ððk 1ÞDz iDzÞ2 þ b2 dxi dxk Z 0 Z w qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ln ðxk xi Þ2 þ ðkDz iDzÞ2 þ b2 dxi dxk þ þ þ þ þ þ
Z Z Z Z Z
xk ¼w w xk ¼0 0
Z
xi ¼0
xk ¼w
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ðxk xi Þ2 þ ððk 1ÞDz ði 1ÞDzÞ2 þ b2 dxi dxk
0
xi ¼w
Z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ðxk xi Þ2 þ ðkDz ði 1ÞDzÞ2 þ b2 dxi dxk
0
xi ¼w
ðk1ÞDz
Z
zi ¼ði1ÞDz
kDz
iDz
Z
zk ¼ðk1ÞDz
Z
zk ¼kDz kDz
300
Solar radiation 1.0E-03 kg/s/m
200
1.8E-03 kg/s/m 3.6E-03 kg/s/m
100
5.4E-03 kg/s/m
0
simple glass 8 mm thickness -100
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln w2 þ ðzk zi Þ2 þ b2 dzi dzk
4
Z
ði1ÞDz
ln zk ¼ðk1ÞDz
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðzk zi Þ2 þ b2 dzi dzk
ð22Þ
zi ¼iDz
The above equation is used to calculate the view factor between any two elements of parallel plates shown in Fig. 6. The numerical code of the ventilated glass window is optimized to reduce the computational time while keeping good precision. The tests realized for a double glass window of 1.0 m height, 0.6 m width and 3 mm spacing between the glass sheets with air mass flow rate of 5.4 · 103 kg/s/m indicated that 20 grid points are adequate for the present calculations.
10
12
14
16
18
20
The total heat gain, the coefficient of solar heat gain and the shading coefficient are important parameters for window performance calculations and comparisons. To calculate the total heat gain one must take into account the solar energy passing directly through the transparent surface and the solar energy reaching the internal ambient after being absorbed by the glass and redirected by the heat transfer mechanisms. In the case of transient conditions, the total heat gain can be calculated by energy balance over the window surface in contact with the internal ambient n 1 X qtotal ¼ hint DxðT ig;i T int Þ þ rei DxðT 4ig;i T 4int Þ nDx i¼1 þ sig seg I o Dx
3. Results and discussion Fig. 7 shows the effect of the air mass flow on the temperature of the internal glass sheet at its mid height position. As can be seen the increase of the mass flow rate reduces the internal glass temperature and hence reduces the total heat gain in comparison with the case of a simple glass window as in shown in Fig. 8. 33 31 29
Temperature [°C]
8
Fig. 8. Variation of the total heat gain with time.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðwÞ2 þ ðzk zi Þ2 þ b2 dzi dzk
zi ¼iDz
6
Time [h]
zi ¼ði1ÞDz ði1ÞDz
ln Z
400
7.2E-03 kg/s/m
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln ðzk zi Þ2 þ b2 dzi dzk
iDz
zk ¼kDz
ðk1ÞDz
500
Total heat gain [W/m²]
2pAi F i;k ¼
ð23Þ
where the subscript i refers to a given control volume of the internal glass sheet. The sum of energy balances over each element allows calculating the total heat gain (qtotal). The solar heat gain coefficient (F) is the fraction of incident solar radiation admitted through a window. The solar heat gain can be obtained indirectly by calculating the total heat gain with and without solar radiation, subtracting one from the other and dividing by the incident solar radiation. q qwithout I F ¼ with I ð24Þ I The shading coefficient (SC) is used to establish a comparison parameter relating the solar heat gain of a desired system to a reference system
27 25 mass flow rate [ kg/s/m]
23
SC ¼
1.0E-03 1.8E-03 3.6E-03 5.4E-03 7.2E-03 simple glass - 8mm thickness external ambient temperature
21 19 17 15 4
6
8
10
12
14
16
18
20
Time [h]
Fig. 7. Effect of the air mass flow rate on the internal glass surface temperature.
F desired window F reference
ð25Þ
The reference system adopted in this study is a simple glass sheet of 3.0 mm thickness, type DSA, having an extinction coefficient of a 0.0078 (1/mm) and refraction index of 1.526. Fig. 9 shows the total heat gain, solar heat gain and the heat gain due to the difference between the internal and external ambient temperatures. As can be observed,
K.A.R. Ismail, J.R. Henrı´quez / Applied Thermal Engineering 26 (2006) 295–302 31
600 (*)Total heat gain (**) Heat exchange due to the temperature difference between the internal and external ambient
500
(*)
temperature at mid length of the external glass
29
(**) Solar radiation
27
Solar heat gain
Temperature [ºC]
400
Heat [W/m²]
301
300 200
25 23 without radiation
21
100
19
0
17
with radiation
15
-100 4
6
8
10
12
14
16
18
4
20
6
8
10
Fig. 9. Comparison between the heat gain with and without including the solar radiation exchange for the case of 1.0 · 103 [kg/s/m].
Simple glass windows 8 mm
1.0 · 103 1.8 · 103 3.6 · 103 5.4 · 103 7.2 · 103
Shading coefficient (SC)
0.779 0.776 0.769 0.765 0.762 0.89
0.855 0.852 0.844 0.840 0.836 0.978
16
18
20
33 31
mass flow rate: 7.2 E-03 kg/s/m gap between plates: 3 mm thickness of glass plates: 8mm
29
Temperature [°C]
Mass flow rate [kg/s/m]
Solar heat gain coefficient (F)
14
Fig. 10. Variation of the temperature of the external glass with and without radiation exchange between the glass sheets.
Table 1 Values of F and SC for forced convection ventilated window Ventilated window
12
Time [h]
Time [h]
27 25 23 (*)
(**)
Tamb
21
fluid entry temperature [˚C]
19
(*) 20 [˚C] (**) external ambient temperature
17 15 4
the incident solar radiation represents a dominant contribution to the total heat gain. If the instantaneous solar heat gain coefficient (F) and shading coefficient (SC), are integrated over the incidence period of solar radiation, one can obtain the average values of there coefficients. Table 1 shows the calculated average values of solar heat gain coefficient and shading coefficient for different values of mass flow rate. As can be seen both the values of SC and F decrease with the increase of the mass flow rate and are well below the case of a simple glass window. In order to verify the relative importance of the heat exchange by infrared radiation between the surfaces of the glass channel, simulations were realized with and without infrared radiation exchange and this results are shown in Fig. 10. Fig. 10 shows the temperature at mid height of the external glass with and without including the long wave radiation exchange. Similar results are obtained for the working fluid and the internal glass surface at the same position. As can be seen the effects due to long wave radiation exchange between the external and internal surfaces are very small causing a temperature variation of about 0.5 C. In order to investigate the effect of the fluid entry temperature on the performance of ventilated windows two situations were simulated; one where the fluid entry temperature is maintained constant as 20 C and the
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Fig. 11. Effect of the fluid entry temperature on the temperature of the internal glass.
other situation where the fluid entry temperature varies in a similar manner to the variation of the external ambient temperature. The results are shown in Fig. 11. One can observe that when the entry temperature of the fluid is increased to the external ambient temperature, that is, the circulating fluid is admitted from the external ambient, the internal glass temperature increases in a corresponding manner.
4. Conclusions A mathematical numerical model to simulate a double glass ventilated window with forced air flow is formulated and implemented. The model takes into account the long wave radiation exchange between the glass sheets. Detailed analysis of the results showed that effect of including this radiation exchange is relatively small at least for the investigated flow conditions. Possibly for natural flow and stagnant conditions the effect of this thermal exchange may become important. The results show that the effect of the increase of the mass flow rate over the thermal behavior of the
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ventilated glass window causes a reduction of the mean solar heat gain coefficient and the shading coefficient. Consequently, this improves the performance of the double glass ventilated window in relation to a simple glass window. The increase of the fluid entry temperature of a ventilated window deteriorates the window thermal performance. Acknowledgement The authors wish to thank the CNPq for the scholarships and the financial support.
References [1] C.M. Lampert, Heat mirror coatings for energy conserving windows, Solar Energy Materials and Solar Cell 6 (1981) 1–41. [2] P.J. Sebastian, V. Sivaramakrishnan, CdSe thin films as solar control coatings, Solar Energy Materials and Solar Cell 27 (1992) 321–326. [3] P.J. Sebastian, M. Pattabi, Solar control characteristics of Cu2Se coating, Journal of Physics D: Applied Physics 25 (1992) 981–985. [4] S. Reilly, D. Arasteh, M. Rubin, The effects of infrared absorbing gases on window heat transfer: a comparison of theory and experiments, Solar Energy Materials 20 (1990) 277–288. [5] R. Sullivan, D.K. Arasteh, F.A. Beck, S.E. Selkowitz, Energy performance of evacuated glazing in residential buildings, ASHRAE Transactions 102 (part 2) (1996) 220–227. [6] J. Tanimoto, K. Kimura, Simulation study on an air flow window system with an integrated roll screen, Energy and Buildings 26 (1997) 317–325.
[7] K.H. Haddad, A.H. Elmahdy, Comparison of the monthly thermal performance of a conventional window and a supply-air window, ASHRAE Transactions 104 (part 1B) (1998) 1261–1270. [8] J.R. Henrı´quez, Modelagem e ana´lise de janelas te´rmicas, Doctorate thesis, Faculdade de Engenharia Mecaˆnica, Universidade Estadual de Campinas, Campinas-SP, Brasil, 2002. [9] J.R. Henrı´quez, Estudo nume´rico e experimental sobre vidros te´rmicos, Master thesis, Faculdade de Engenharia Mecaˆnica, Universidade Estadual de Campinas, Campinas-SP, Brasil, 1996. [10] K.A.R. Ismail, J.R. Henrı´quez, U-values, optical and thermal coefficients of composite glass systems, Solar Energy Materials and Solar Cell 52 (1998) 155–182. [11] K.A.R. Ismail, J.R. Henrı´quez, Thermally effective windows with moving phase change material curtains, Applied Thermal Engineering 21 (2001) 1909–1923. [12] K. Rottkay, M. Rubin, J. Kerr, Optical modeling of a complete electrochromic device, in: The 2nd International Conference on Electrochromics, San Diego, USA, October 1996. [13] A. Seeboth, J. Schneider, A. Parzak, Material for intelligent sun protecting glazing, Solar Energy Materials and Solar Cells 60 (2000) 263–277. [14] E.M. Sparrow, S. Shah, C. Prakash, Natural convection in a vertical channel: I. Interacting convection and radiation. II. The vertical plate with and without shrouding, Numerical Heat Transfer 3 (1980) 297–314. [15] A. Bar-Cohen, W.M. Rohsenow, Thermally optimum spacing of vertical, natural convection cooled parallel plates, Journal of Heat Transfer 106 (1984) 116–123. [16] B. Morrone, A. Campo, O. Manca, Optimum plate separation in vertical parallel-plate channel for natural convective flows: incorporation of large spaces at the channel extremes, International Journal of Heat Mass Transfer 40 (5) (1997) 993–1000. [17] A. Campo, O. Manca, B. Morrone, Numerical analysis of partially heated vertical parallel plates in natural convective cooling, Numerical Heat Transfer—Part A (Applications) 36 (1999) 129–151.