heat transfer in an open tilted cavityCombined natural convection and radiation
’CENIDET-SNIT-SEP,Mdxico.’Centro de hestigacidn en Energia- UNAM,Mdxico.’Depto. de k g . Quimicay Met., Universidadde Sonora, Sonora, MLxico.R. E. Cabanillas’, C.A. Estrada2and G. Alvarez3
Abstract
pattern of flow and temperature fields.great influence on the total amount of heat lost by the cavity as well as on theside at 300 K. The results indicate that the radiation and the tilted angle have(Pr=0.7) for different tilted angles, with the isothermal wall at 500 K and the opencoefficient was calculated for several values of the Grashof number for airthe radiative energy flux boundary conditions. The average heat transfersolved by finite differences using A D 1 technique. Coupled to this equations areopaque isothermal vertical wall. The momentum and energy equations wereparticipant fluid. The cavity has two opaque adiabatic horizontal walls and oneand radiation heat transfer in an open tilted square cavity containing air as a nontwo dimensional computational model of the interaction of natural convectionradiation combined with natural convection heat transfer. This study presents aheat transfer in open cavities [1,2,3], but they do not include the interaction ofEarlier investigators have solved the problem considering only natural convectionapplications ranging fi-om passive solar heating of rooms to solar concentration.Combined natural convection and radiation in open cavities have various
number.Keywords: natural convection, radiation heat transfer, open cavity, total Nusselt
cooling of electronics devices, and solar collectors among others.several engineering problems such as passive heating and cooling in buildings,The heat transfer in cavities has been widely studied due to its importance in
account for different title angles.surroundings when the convection and radiation mechanisms are taking intotransfer that occurs naturally between a heated 2D open cavity and itscavities. The present work is an effort to understand the basic phenomena of heatrevision of the reported articles on the subject for different conditions of theconvection and radiation, Balaji [6]. Cabanillas in 2001 [3] made a widelysubject and there are a fewer number that consider the combined effect ofrespect to open cavities, there are a small number of articles that deal with therecompilation work of Catton in 1978 [4] and Ostmch in 1988 [ 5 ] .However, within close cavities has been studied with great attention as can be seen in the
Several studies have been done i n cavities, in particular the natural convection
2 Description of the problem
downwards.values from 0" when the aperture of the cavity sees upwards to 180" when it seeswalls and the opening. The angle of the cavity (41) is fixed but it can take severalwall to the fluid, as well as by radiation from the heated wall to the adiabaticon with T = 1 and the heat transfer process starts by convection from the heateddimensionless temperature of T = 0. At time T = 0 the isothermal wall is turnedvary (Boussinesq approximation). The fluid is initially static and with afor the density in the buoyancy term of the energy equation which is allowed tois consider as an incompressible newtonian fluid with constant properties, exceptisothermal as it is shown in figure 1. The cavity is filled with air (Pr =0.7), whichConsider an open rectangular cavity with three walls; two adiabatic and one
equations are expressed in a dimensionless form as:automatically satisfied and the pressure is eliminated as a solution variable. Thestreamfunction-vorticity formulation is used, the continuity equation isenergy for a square cavity with radiative exchange at the boundaries. Since theThe governing equations are those of the conservation of mass, momentum and
P l y =-W (1)
d t d X dY Gr112
-++u-+v-= ~~ V 2 TdT dT dTd t d X d Y p & ? $ / 2
The velocities are related to the streamfunction by
(3)
The dimensionless variables were chosen as follow:
where
U , = ( ~ ~ A T H ) ” ~
The initial conditions in the cavity are the following,
F o r z = O a n d O < X < l , O < Y < l ,W(X,Y,O) = 0, yJ(X,Y,O)= 0, U(X,Y,O)= V(X,Y,O), ( 5 )T(X,Y,O)= 0, T (O,Y,O)= 1.0.
terms of the streamfunction as:The hydrodynamic boundary conditions for the walls 1, 2 and 3 are given in
f o r T = z a n d O < X < l , O < Y < l ,yl (X,O,T)= 0; U(X,O:t)= V(X,O,t)=oyJ (X,l,T) = 0; U(X,l,t) = V(X,l,t)=oyl (O,Y,T)= 0; U(0,YJ) = V(0,YJ) =o
the wall.determined by the Taylor series expansion of the streamfunction i n the vicinity of
The boundary conditions on vorticity are not known explicitly, but are
The following energy balances give the temperature boundary conditions forwalls 1 and 3:
%l = 4 1 1 (7)
including the opening side.convection and qrl is the radiative heat exchange between that wall and the otherswhere qcl is the heat flux gave by the wall to the fluid through conduction-
Thus, for wall 1 (adiabatic wall),
k a [ $ g = " = % l
defining N, and Qrl as
Nr =ka(TH-q)OT;L Q,., =$
Y
and substituting, it is obtained the dimensionless form
[g] = 4 &y=o
Similarly, for wall 3 (adiabatic wall),
For wall 2 the thermal boundary condition is
For 0 < Y 5 1 and X = 0, T = 1. (10)
that the vorticity boundary conditions that are similar to the temperature onesthe cavity, it is assumed that convection dominates over conduction. It was foundzero temperature because it is the extcrior temperature. For thc fluid that leavesThe condition for the open side are as follow. The fluid entering the cavity has a
Advanc~d Computu t iodMcthods in H a t T I U ~ L ~ P I ~99
cavity’s entrance are parallel. Thus, the open side boundary conditions are:i W 3 X = 0, which causes that V = 0, this means that the streamlines at thehave a better convergence [7]. Also, for the stream function, it is assumed that
f o r O < Y < l a n d X = l ,
*=o, v=o,a xif the fluid enters to the cavity
T = 0 , W = 0 (12)
if the fluid leaves the cavity,
radiosity is given bygrid that it is used in the convective problem. Thus, for each element of a wall theTo solve the radiative balance the radiosity formulation is used with the same
J, = E , + ( I - - E ~ ) ~ I ? ~ ~ . J ~ (14)
j = I
where Fkjis the view factor from the k-element to the j-element.
will be given byFor a given wall formed by n elements, the net heat transferred by radiation (q,)
q r , = J , - 9 1 (15)
and q, is the amount of radiation received on wall 1.where qrl is the net heat for wall 1, J, is the radiative energy that leaves that wall
The average Nusselt number is given by
Nu, = Nu, + Nu, (16)
are defined byWhere Nu, is the Nusselt number for convection and Nu, is for radiation, which
given byconvection heat transfer from the same wall. Then, the total Nusselt number isWhere qrad is the heat transferred by radiation by the heated wall and qconvis the
4 Method of Solution
net heat flux equation (15).Simpson's rule. The net radiative heat fluxes were calculated from the radiativemethod of successive approximation. The integrals were evaluated by themethod of Paceman and Rachford [S]. The radiosity equations were solved by thedifference technique was used. The streamline equation (1) was solved by thedifferential equations (2)and (3). The AD1 (Alternating-Direction-Implicit) finiteand central space differences approximated the derivatives of the partialall coupled through the boundary conditions and/or the variables. Forward timeinitial and boundary conditions define the problem completely. The equations areGoverning equations (1)-(3), along with the radiative flux equations and their
4.1 Validation
present study for different Rayleigh numbers (Ra) when the cavity is 90" oriented.between those Nusselt numbers (Nu) obtained by different authors including thepublished by other authors were considered. Table 1 shows a comparisonnumerical code of the present work, for the convective case only, the resultsscheme was stable i n a wide range of Grashof numbers (Gr). To validate thetests that are detailed described by Cabanillas [3] showed that the numericala comparison with reported results in the literature. The convergence and stabilityThe numerical solution was validated with convergence and stability tests and by
primitive variables (with very little variation between them).Chan-Tien [9] and Mohamad [ I O ] used the SIMPLER numerical method inand the code are correct for the case analysed. It is important to mention that
The great similarity between these values show that the numerical procedure
agreement with the 14.5 found in the present study.Nu values of 14.39, 14.9 and 14.4 respectively. These Nu values are in
For a Gr = lo6and Pr = 1.0, Angirasa [7],Comini [ l l ]and Balaji [6] reported
Advmcmi Compututiod Methods in H a t Trumf?v 1 0 1
Table 1. Nusselt number comparison for the convective case. The cavity isoriented 90".
Ra Nu Nu NuMohamad [lo] Present studyChan-Tien [9]
1 x 107
1 x IO6
1 x 105
1 x 1 0 4
1 x 103 1.07 1.31 1.10
3.41 3.44 3.07
7.69 7.41 7.16
14.3615.00 14.50
28.6028.60 26.80
obtained in this study. The agreement between the values is evident.compared. Table 2 shows the reported values by Mohamad [lo] and the onesdifferent cavity orientations with Gr = 1.5 X lo4 and Pr = 0.7 (Razlxl04) were
Also, considering only natural convection heat transfer, the Nu obtained for
Table 2. Nusselt number comparison for the convective case at different cavity'stitle angles.
Tilt Nu NuPresent studyMohamad [lo]
10" 2.57 2.82
30" 3.34 3.43
60" 3.70 3.69
90" 3.44 3.10
standard procedure and the procedure which divides each wall of the cavity withWh?. This result shows a 0.2 % difference between the calculated value with athe cavity is 3,084.48 W/m*, while the code calculates a value of 3,084.4818the surrounding are at 300 "C, then it is expected that the radiative heat leavinglike a black body, is considered. If the temperatures of the walls are 500 "C andsame temperature and with an emissivity of 1 (E = l) , that is, the cavity behaveswithout convection. The typical case of a cavity with three isothermal walls at theThe radiative calculation procedure was validated for the isothermal cavity
determined by the mesh size used.work with the same number of surface subdivisions. This subdivisions are30 segments. When the radiative part is coupled with the convective part, both
5 Results
increments of 45". The results presented correspond to the steady state.the cavity was varied from 0" (upwards cavity) to 180" (downwards cavity) withthree walls of the cavity and E = l .0 for the cavity's open side. The tilted angle of= 300°K (outside air temperature), Pr = 0.7, emissivity of 1 (E = 1.0) for all theThe parameters used in the simulations were T, = 500 K (wall tempcratures), T,
90".exists a maximum for the convective component of the Nu which occurs before1.0); for smaller values of E i t is expected that Nu, must be smaller. Second, itthe extreme case considered (where the walls' emissivity was assumed 1, E =magnitude between the two components: Nu, can be greater than Nu,, at least forof the cavity can be analyzed in three different ways. The first one is about theThe values of Nuc, Nu,, and Nu, are plotted. The effect of the orientation angle
Figure 2 shows a plot of the Nu numbers vs cavity's tilt angles for a Gr = IO5.
18
16
14
4
2
0
Figure 2. Nu,, Nu,, and Nu, numbers vs cavity's tilt angles for a Gr = lo5.
facilitated the convective movement of the air caused by the floating forces. ForFor ti l t angles between 0" and 90 the cavity is looking upward and this
Advancc.d Computut iod Mcthods in H a t Trumfc.r 1 0 3
mechanism in the cavity.and it present a small variation, this indicates the coupling of the to heat transferwith Nu, close to unit. Third, the radiative component of the Nu is not constantgreater angles the convection is strongly diminished getting a minimum at 180"
observed:of lo5and @ = 0" , 45" , 90°, 135", 180". From this figure, the following can be
Figure 3 shows a comparison of isotherms (right) and streamlines (left) for Gr
opposite directions, allowing an incipient convective heat transfer to the exterior.effect is observed in the streamlines where appears four cells which turn inNu, presents a minimum value, as mentioned before. However, an interestingcauses the stratification of the thermal boundary layer that fulfill the cavity. Thelow density can not move upwards in the vertical direction (stagnant fluid), this3. When $ = 180" (aperture downwards), the fluid that has been heated and hasthe buoyancy as the driving force.fluid cells. These orientations allow the fluid to circulate inside the cavity having2. For tilt angles between 45" and 135", streamlines presents the formation ofcenter of the cavity and leaving at the walls.two cells turning in opposite ways, showing that the cold fluid is entering at the1. When @ = 0" (aperture upwards), the stream function presents the formation of
corresponding standard dcviation.near future this issue. In this table the Nu, presented are the average with theirsolution getting an oscillated behavior of the Nu, .It is intended to study in themention that for some tilt angles it was not possible to converge the numericalIt is notorious the increase of the Nu, with the increase of the Cr. It is important toangle o n Nu, mentioned before for Gr = 10' is repeated for the other Gr studied.varying the tilt angle of the cavity. This table shows that the effect of the tiltTable 3 shows Nu, with its components Nu, and Nu,for different values of Gr and
Table 3. Nut, Nu, and Nu, for different Gr at different tilt angles of the cavity.
Advmcmi Compututiod Mcthods in H a t Trumf?v 1 0 5
6 Conclusions
was validated by comparison with the results of other authors.other opposite to the cavity opening was isothermal. The convective part of codenon participating fluid has been presented. Two opposite walls were adiabatic andconvection and radiation in an open square cavity containing air as a laminar andA transient two-dimensional mathematical model of a combined natural
most cases, when the tilt angle is 45".the Nut occur when the cavity is downwards, while the higher values occur, inreduces the natural convection process is 180" . Therefore, the lowest values ofnatural convection process is 45". In complementary form, the angle that bestrespect to the tilted angle of the cavity. In most cases, the angle that best help thenumber. Also, there is a high dependence of the convective Nusselt number withdependence of the convective Nusselt number with respect to the Grashof
From the results, and according with previous studies, there is a high
coupling.radiative and convective heat transfer. More research is needed to investigate thisis important to mention that the Nur is not constant, showing the coupling of thehowever, it is demonstrated how high the radiative heat transfer can have. Also, itis reasonable to assume that for other values of E this number will change,the radiative transport in the heat transfer from the cavity can not be neglected. It
In all cases analyzed, with E = 1.0, the Nu, was grater than Nu, indicating that
Nomenclature
PrNrNutNurNu,NuLJiHGrGLatin
T a
T w
Ra9 1 ,
q c
91
Outside air temperature, KHeated wall temperature, KRayleigh number, PrGrNet radiation of wall iConvective heat transferIncident energy on wall iPrandtl number, d aRadiative number, oTH3LkaTotal Nusselt number,Radiative Nusselt number,Convetive Nusselt number,Nusselt number, hL/kaWidth of cavity, mRadiosity of wall yHeight of cavity, mGrashof number, gpATL3h2Gravity, d s 2
T time, S
Uo Rcfercnce velocity, d su,v Velocity components, d sU,V Dimcnsionkss elocity,u/uoW Vorticity& Y J ' Co-ordinate axisX,Y Dimensionless coordinate
Axis
Y Dimensionless streamlinesW/m2K4Stefan Boltzm. const.,Dimensionless time, tuo/LEmissivitycoefficient, UKThermal expansionAir diffusivity, m2/s
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MCxico, Septiembre de 2001, Cd. de Mexico, MCxico.Bidimensional Abierta, Tesis doctoral Universidad Nacional Authoma de
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Publishing Corp., 1978.International Heat Transfer Conference, Toronto, Vol. 6, Hemisphere
[4] Catton, I., Natural Convection in Eclosures, Proceedings of the Sixth
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110, November 1988.[5] Ostrach, S.,Natural Convection in Eclosures, Journal of Heat Transfer, Vol.
Open a Side", Numerical Heat Transfer, Pan A, Vol. 28, pp. 755-768.Simulation of Transient Natural Convection from an Isothermal Cavity
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