Nusselt number for the natural convection and surface thermal radiation in a square tilted open cavity

International Communications in Heat and Mass Transfer 32 (2005) 1184 – 1192 www.elsevier.com/locate/ichmt

Nusselt number for the natural convection and surface thermal radiation in a square tilted open cavityB J.F. Hinojosaa, R.E. Cabanillasb, G. Alvarez c, C.E. Estradaa,T a

b

CIE-UNAM, AP 34, Temixco, 62580, Morelos, Mexico Depto. de Ing. Quı´mica y Met., Universidad de Sonora, Hermosillo, Sonora, Mexico c CENIDET-SNIT-SEP, AP 5-164, Cuernavaca, 62490, Morelos, Mexico Available online 8 June 2005

Abstract In this communication, the numeric results of the heat transfer by natural convection and surface thermal radiation in a tilted 2D open cavity are presented. This study has importance in the thermal design of receivers for solar concentrators. The opposite wall to the aperture in the cavity holds a constant temperature of 500 K, while the temperature of the surrounding fluid interacting with the aperture is 300 K. The other walls are kept insulated. The results in the steady state are obtained for a Rayleigh range from 104 to 107 and for an inclination angles range of the cavity from 08 to 1808. The results show that the Nusselt numbers increase with the Rayleigh number except the convective Nusselt number for 1808, where it stays almost constant. The convective Nusselt number changes substantially with the inclination angle of the cavity, while the radiative Nusselt number is insensitive to the orientation change of the cavity. D 2005 Elsevier Ltd. All rights reserved. Keywords: Convection; Open cavity; Nusselt number

1. Introduction In several thermo solar concentration systems, the solar concentrator rotates maintaining its optical axis pointing directly toward the sun. During the tracking, the geometry of the concentrator B

Communicated by A. Majumdar and C. Grigoropoulos. T Corresponding author. E-mail address: [email protected] (C.E. Estrada).

0735-1933/$ - see front matter D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2005.05.007

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allows to reflect the solar rays inside the receiver (open cavity) located at the focal point of the concentrator. The receiver, when it rotates, will change the dynamic of the fluid and the heat transfer in the tilted open cavity. To have an accurate receiver’s thermal design, this process is important to study. On the other hand, the high temperatures of the thermo solar system make necessary to include the radiative heat exchange among the walls of the cavity, in the mathematical model. During the past two decades, several experimental results and numerical calculations have been presented for describing the phenomenon of natural convection in open cavities [1–16]. These studies have been focused to study the effect on flow and heat transfer of different Rayleigh numbers, aspect ratios and tilted angles. Also, these works studied the occurrence of transition and turbulence and how the boundary conditions in the aperture are considered. However, very few papers present results of the combined heat transfer by natural convection and thermal radiation in open cavities. Lage et al. [17] studied numerically the heat transfer by natural convection and surface thermal radiation in a two-dimensional open top cavity; the numerical approach used by the authors consisted of solving separately the steady state equations of natural convection and thermal radiation, assuming a temperature distribution on the vertical adiabatic wall. Balaji and Venkateshan [18] obtained steady state numerical results for the interaction of surface thermal radiation with free convection in an open top cavity, whose left wall was considered isothermal, and the right and bottom walls were adiabatic and their temperature distributions were determined by an energy balance between convection and radiation in each surface element of the walls; radiation was found to enhance overall heat transfer substantially (50–80%) depending on the radiative parameters. Deghan and Behnia [19] studied numerically and experimentally the combination between natural convection, conduction and radiation heat transfer in a discretely heated open top cavity; the comparison of the numerical results with the experimental ones showed that the accurate prediction of the flow and temperature patterns depended strongly on the consideration of the heat transfer by radiation. Ramesh and Merzkirch [20] made an experimental study of the combined natural convection and thermal radiation heat transfer, in a cavity with top aperture; they found out that the surface thermal radiation heat transfer in cavities with walls of high emissivities had a significant change in the flow and temperature patterns and therefore influence the natural convection heat transfer coefficients. The studies presented in references [17–20] have been oriented to obtain results for a specific orientation of the open cavity (top aperture). The purpose of this paper is to quantify the thermal losses by natural convection and surface thermal radiation in tilted open cavities for a complete angle range of 08 to 1808. To do that, the natural convection and the thermal radiation are simultaneously solved. 2. Physical problem and mathematical model The heat transfer in a two-dimensional square tilted open cavity of length L, is considered in the present investigation as shown in Fig. 1. The assumptions for this problem follow. The opposite wall to the aperture was kept to a constant temperature T H equal to 500 K, while the surrounding fluid interacting with the aperture was fixed to an ambient temperature T l of 300 K. The two remaining walls were insulated. The fluid was air (Pr = 0.71) and Newtonian, and the fluid flow was laminar. The fluid was radiatively nonparticipating and the walls of the cavity were gray and diffuse emitters (with e = 1). The properties of the

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g

TH

y

Horizontal

φ

x

T∞

L

Fig. 1. Schematic diagram of the open cavity.

fluid were considered constant except for the density in the buoyant force term in the momentum equations, according to the Boussinesq approximation. Even though the temperatures assumed are relatively high, Vierendeels et al. [21] presented benchmark solutions for different Rayleigh numbers for a closed square cavity problem with large temperature differences between the vertical walls; they found out that for a fixed Rayleigh number, the Nusselt number decreases very slightly (1.6%) when the temperature difference increases from 10 K to 720 K. The transient state dimensionless conservation equations governing the transport of mass, momentum and energy in primitive variables are expressed as BU BV þ ¼0 BX BY

ð1Þ

BU BðU 2 Þ BðU V Þ BP þ þ ¼  þ Bs BX BY BX BV BðU V Þ BðV 2 Þ BP þ þ ¼  þ Bs BX BY BY Bh BðU hÞ BðV hÞ 1 þ þ ¼ Bs BX BY ðPrRaÞ1=2










Pr Ra

Pr Ra

1=2


1=2


B2 U B2 U þ 2 BX BY 2

B2 V B2 V þ BX 2 BY 2

B2 h B2 h þ 2 BX BY 2

 þ hcos/

ð2Þ

 þ hsin/

ð3Þ

 ð4Þ

where Ra = [gb(T H  T l)L 3] / am is the Rayleigh number, and Pr = m / a is the Prandtl number. The above equations were non-dimensionalized by defining X ¼ x=L; Y ¼ y=L; s ¼ Uo t=L; P ¼ ð p  pl Þ=qUo2 ; U ¼ u=Uo ; V ¼ v=Uo ; h ¼ ðT  Tl Þ=ðTH  Tl Þ

ð5Þ

the reference velocity U o is related to the buoyancy force term and was defined as U o = (gbL(T H  T l))1/2 [8].

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The initial and the boundary conditions for the momentum and energy equations were taken as the ones used by Chan and Tien [5] and Mohamad [10]. An important characteristic of those boundary conditions is the use of an approximation in the aperture plane, which allows us to truncate the computational domain and to reduce the computational time. The initial and boundary conditions were taken as follows: P ð X ; Y ; 0Þ ¼ U ð X ; Y ; 0Þ ¼ V ð X ; Y ; 0Þ ¼ hð X ; Y ; 0Þ ¼ 0

ð6Þ

U ð0; Y ; sÞ ¼ V ð0; Y ; sÞ ¼ U ð X ; 0; sÞ ¼ V ð X ; 0; sÞ ¼ U ð X ; 1; sÞ ¼ V ð X ; 1; sÞ ¼ 0

ð7Þ




BU BX




 ¼ X ¼1

BV BX

 ¼0

ð8Þ

X ¼1

hð0; Y ; sÞ ¼ 1

ð9Þ


hð1; Y ; sÞ ¼ 0 if U b0 or Bh BY

Bh BX

 ¼ 0 if U N0

ð10Þ

X ¼1

 ¼ Nr Qr

ð11Þ

Y ¼0;1

where N r = rT H4L / k(T H  T l), is the dimensionless parameter of conduction–radiation and Q r = q r / rT H4, is the dimensionless net radiative heat flux on the corresponding insulated wall. Eq. (11), which is not in references [5] and [10], was obtained applying an energy balance on the surface by considering the transmission of heat by radiation and convection. To obtain the net radiative heat fluxes over the walls, the radiosity-irradiance formulation was used, dividing the surface in elements according to the mesh used to solve the natural convection equations. View factors were evaluated using Hottel’s crossed string method [22]. The average convective Nusselt number was calculated integrating the temperature gradient over the heated wall as Z 1 P Bh dY ð12Þ Nuc ¼  BY 0 The average radiative Nusselt number was obtained integrating the dimensionless net radiative fluxes over the heated wall [23], by the following mathematical relationship P

Nur ¼ Nr

Z

1

ð13Þ

Qr dY 0

The total average Nusselt number was calculated by summing the average convective Nusselt number and the average radiative Nusselt number

P

P

P

Nut ¼ Nuc þ Nur ¼

Z 0

1



Bh dY þ Nr BX

Z

1

Qr dY 0

ð14Þ

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3. Numerical method of solution Eqs. (1)–(4) were discretized using staggered uniform control volumes. The convective terms were approximated by the SMART scheme [24] and the diffusive terms with the central differencing scheme. The SIMPLEC algorithm [25] was used to couple continuity and momentum equations. The fully implicit scheme was used for the time discretization. Due to the coupling between the natural convection and the surface thermal radiation, the radiative balance was solved in every time step using an iterative method of subsequent approaches. The results were obtained with a 100  100 grid mesh and the dimensionless time step used for the calculations was 1  10 3, after an independence mesh study. 4. Results and discussion In order to validate the numerical code, the pure natural convection with Pr = 1, in a square open cavity was solved, and the results were compared with the ones reported by Chan and Tien [4] obtained with an extended computational domain. In Table 1, the comparison between the average Nusselt numbers is presented, it shows that the highest percentage difference was 4.7% for Ra = 104 and the lowest was 0.3% for Ra = 107, with an average percentage difference of 1.6%. The same problem was solved but for Pr = 0.71, varying the Rayleigh number in the range of 104 to 107 and the inclination angle between 08 and 1808. In Table 2, the calculated Nusselt numbers for pure natural convection are presented and they will be used for comparison purpose. 4 In Table 3, for the Rayleigh number range of 10P to 107 and inclination angle range of 08 to 1808, the P values of the average convective Nusselt number (Nu ), average radiative Nusselt number (Nu c r ) and total P average Nusselt number (Nut ) are presented. The comparison between the total Nusselt numbers presented in Table 3 and the convective Nusselt numbers of Table 2 shows that the inclusion of the radiative heat transfer mechanism in the mathematical model, changes the numerical prediction of the thermal losses in the open cavity. When the surface radiation is considered, the total Nusselt number increases between 62.9% (Ra = 105 and / = 458) and 3284% (Ra = 107 and 1808). The percentages differences are more important for inclination angles grater than 908. It is noted that, in some cases, for instance Ra = 106 and / = 608, the calculations of the convective Nusselt numbers did not reach the steady state. For these cases, the mean values and the corresponding standard deviations are presented. In Fig. 2a, the values of the Nusselt numbers that appear in Table 3, corresponding to Ra = 104, are graphed. It is noted that the values of the average convective Nusselt number and the total Nusselt number, varied significantly with respect to the inclination of the cavity. The angles among the range 458–908 favored the heat transfer by natural convection, while angles bigger than 908 hindered it. The average

Table 1 Comparison of the heat transfer results for the square open cavity with Pr = 1 Ra 4

10 105 106 107

P

Nu

Difference (%)

This work

Chan and Tien [4]

3.57 7.75 15.11 28.70

3.41 7.69 15.0 28.6

4.7 0.8 0.7 0.3

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Table 2 Average Nusselt numbers for the pure natural convection /

Ra

08 108 308 458 608 908 1208 1358 1508 1808

104

106

106

107

2.47 2.54 3.34 3.59 3.69 3.44 2.30 1.63 1.15 1.00

6.91 6.54 F 0.93 6.42 7.04 7.40 7.44 4.80 2.64 1.46 1.00

10.63 F 1.84 11.6 F 1.59 12.25 13.41 14.15 14.51 9.37 4.03 1.75 1.00

22.33 F 4.94 23.47 F 3.51 23.24 F 0.28 25.31 F 0.21 26.94 F 0.17 27.58 17.54 5.96 2.01 1.00

radiative Nusselt number stayed constant for all the angles. The values of the average radiative Nusselt number were always bigger than the average convective Nusselt number for the full range of inclination angles of the cavity. This fact indicates that the exchange of thermal radiation between walls is quantitatively more relevant that the convective phenomenon. Table 3 P P Values of the average convective Nusselt number (Nuc ), average radiative Nusselt number (Nur ) and total average Nusselt number /

Ra = 104 (Nr = 4.91)

Ra = 105 (Nr = 10.57)

Nuc

P

Nur

P

Nut

P

Nuc

P

Nur

P

Nut

08 158 308 458 608 908 1208 1358 1508 1808

2.72 1.65 2.32 2.65 2.83 2.77 2.01 1.46 1.02 0.95

3.28 3.28 3.28 3.28 3.28 3.28 3.28 3.28 3.28 3.28

5.99 4.93 5.60 5.93 6.11 6.05 5.29 4.74 4.30 4.23

6.75 6.58 5.178 F 0.69 5.36 F 0.59 6.33 6.00 4.44 2.70 1.34 0.96

7.09 7.06 7.08 7.09 7.06 7.06 7.06 7.06 7.06 7.06

13.83 13.65 12.26 F 0.69 12.44 F 0.59 13.39 13.06 11.50 9.77 8.40 8.02

/

Ra = 106 (Nr = 22.77) P Nuc

P Nur

P Nut

Nuc

P

Nur

P

Nut

9.94 F 0.35 11.40 12.50 12.80 11.97 F 0.22 11.91 9.09 5.05 1.84 0.94

15.26 15.22 15.22 15.27 15.27 15.22 15.22 15.22 15.22 15.21

25.20 F 0.35 26.62 27.72 28.07 27.24 F 0.22 27.13 24.31 20.27 17.06 16.15

21.41 F 0.76 24.68 F 1.03 24.31 26.39 F 0.21 27.79 27.15 F 0.9 18.33 9.46 2.80 0.97

32.89 32.9 32.89 32.89 32.9 32.89 32.79 32.89 32.89 32.87

54.3 F 0.76 57.58 F 1.03 57.2 59.28 F 0.21 60.69 60.04 F 0.9 51.12 42.35 35.69 33.84

08 158 308 458 608 908 1208 1358 1508 1808

P

Ra = 107 (Nr = 49.06) P

1190

J.F. Hinojosa et al. / International Communications in Heat and Mass Transfer 32 (2005) 1184–1192 7

16

(a)

6

12

5

10

4

Nu

Nu

(b)

14

3

8 6

2

convective radiative total

1

convective radiative total

4 2

0

0 0

15 30 45 60 75 90 105 120 135 150 165 180

0

φ

30

15 30 45 60 75 90 105 120 135 150 165 180

φ

70

(c)

25

(d)

60 50

20

Nu

Nu

40

15

30

10 5

convective radiative total

0

20

convective radiative total

10 0

0 15 30 45 60 75 90 105 120 135 150 165 180

φ

0

15 30 45 60 75 90 105 120 135 150 165 180

φ

Fig. 2. Variation of the average Nusselt number with the inclination angle: (a) Ra = 104. (b) Ra = 105, (c) Ra = 106 and (d) Ra = 107.

Also, the variation of the average Nusselt numbers of Table 3, for Rayleigh number among 105 to 107 is shown in Fig. 2b–d. Notice that to elaborate these graphs, the corresponding mean values were used, when the convective Nusselt number and the total Nusselt number did not reach the steady state. For all graphs of Fig. 2, the absolute maximums occurred at / = 608, 08, 458 and 608 for Ra = 104, 105, 106 and 107, respectively. The absolute minimums occurred at / = 1808 for all the studied Rayleigh numbers. The previous described behavior of the average radiative Nusselt numbers in Ra = 104, is also observed for all the remaining Rayleigh numbers.

5. Conclusions In this paper, the numerical calculations of Nusselt numbers for the natural convection and surface thermal radiation in a tilted open cavity were presented. The main conclusions of this work are the following: 1. In this heat transfer problem, the incorporation of the radiative heat exchange modified the prediction of the total average Nusselt number, increasing its value between 62.9% (Ra = 105 and / = 458) and 3284% (Ra = 107 and 1808).

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2. The average convective Nusselt number changed substantially with respect to the whole angle range of the tilted cavity, while the average radiative Nusselt number was insensitive to the cavity orientation. 3. In general, for a fixed angle, the average Nusselt numbers increase with the Rayleigh number except for an angle of 1808, where the average convective Nusselt number stays almost constant. Nomenclature g Gravitational acceleration k Thermal conductivity L Length of the cavity Conduction–radiation number Nr Nu Nusselt number P Dimensionless pressure Pr Prandtl number Dimensionless net radiative heat flux Qr Ra Rayleigh number T Absolute temperature t Time U,V Dimensionless velocity components u,v Velocity component in x and y directions x,y Coordinate system X,Y Dimensionless coordinates Greeks a b q / m q r s

symbols Thermal diffusivity Thermal expansion coefficient Surface emissivity, dimensionless Inclination angle of the cavity Cinematic viscosity Density Stefan–Boltzmann constant Dimensionless time

Subscripts c Convective H Hot wall o Reference r Radiative t Total l Ambient References [1] P. Le Quere, J.A. Humphrey, F.S. Sherman, Numerical Heat Transfer 4 (1981) 249. [2] F. Penot, Numerical Heat Transfer 5 (1982) 421.

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