Numerical investigation on mixed convection flow in a trapezoidal cavity heated from below
Ilham Tmartnhad
a, Mustapha El Alami
b,*, Mostafa Najam
b, Abdelaziz Oubarra
aaGroupe Energétique, Laboratoire de mécanique, Faculté des Sciences, Université Hassan II Ain Chock, Km8, route d’El Jadida BP 5366, Maârif Casablanca 20100, Morocco
bGroupe de Thermique, LPMMAT, Département de Physique, Faculté des Sciences, Université Hassan II Ain Chock, Km8, route d’El Jadida BP 5366, Maârif Casablanca 20100, Morocco
a r t i c l e i n f o
Article history:
Received 13 December 2006
Received in revised form 2 October 2007 Accepted 25 May 2008
Available online 21 July 2008
Keywords:
Numerical study Trapezoidal cavity Opening site Tilted wall Mixed convection
a b s t r a c t
A numerical study of mixed convection from a trapezoidal cavity is carried out. Two openings are adjusted on the plates of the cavity. The inlet opening is horizontal or vertical, while the outlet one is placed horizontally on the bottom wall. The Navier–Stokes equations are solved using a control volume method and the SIMPLEC algorithm is used for the treatment of pressure–velocity coupling. Special emphasis is given to detail the effect of the Reynolds number on the heat transfer generated by mixed convection. The results are given for the parameters of control as, Rayleigh number (Ra= 105), Prandtl number (Pr= 0.72), the inlet and outlet opening width are respectively (C1= 0.38 andC2= 0.25), the incli- nation of the tilted wall (h= 22°) and Reynolds number (106Re61000). The results show that the flow structure and the heat transfer depends significantly on the inlet opening site. Two principal kinds of the problem solution are raised.
Ó2008 Elsevier Ltd. All rights reserved.
1. Introduction
Mixed convection usually induced in cavities or channels con- taining heating elements on one of its walls or on both walls is important from both theoretical and practical points of view. In fact, this configuration can be encountered in various engineering applications.
Numerous studies related to mixed convection in cavities have been reported in order to investigate the heat transfer and fluid flow in such geometries. This phenomenon is important in nature and in many practical transport process devices, such as furnaces, electronics cooling, solar collectors, processing equipment and oth- ers. In this paper, the interest is dictated by the encountered prob- lems in commercial refrigerating cavities. The food products exposed inside this one are cooled by a circulation of fresh air. Many studies show that the higher part of the cavity is badly cooled. Con- sequently, the presence of non-durable product at this place re- quires a rigorous control of the temperature of conservation, Chengwang and Patterson[1]. In the case of trapezoidal or triangu- lar cavities, only the natural convection (Rayleigh–Bénard type) was treated. Indeed, Boussaid et al.[2]made a study of natural heat and mass transfer in a trapezoidal cavity heated from below. The re- sults show that for weak slopes, the flow is Rayleigh–Bénard kind.
But for high values of the slope, the flow is connected rather with the case of the rectangular cavity type. Kalache[3]studied the sta-
bility of the two-dimensional numerical solutions in a trapezoidal closed cavity. It is shown that for weak slopes (h< 30°) and for Ray- leigh numberRa62104, three-dimensional instabilities appear in the narrow part of the cavity. The flow structure show that a recirculating cells appear at the higher zone of the cavity. The exis- tence of these cells is unfavourable with heat transfer.
In a trapezoidal cavity, simulating a solar water distiller, an experimental study of natural convection, was led by Tiwari et al. [4]. The Rayleigh number and the slope value constitute the principal parameters of this work. The results are presented in the form of correlations: the Nusselt number is presented as a function of Rayleigh number and slope value. The case of a closed trapezoidal cavity simulating a solar heat collector was studied experimentally and numerically by Reynolds et al.[5]. The lower wall of the cavity is transparent and receives a concentrated solar flow. On the other hand the higher wall simulates an exchanger.
The experimental and numerical results show the existence of recirculating cells which generate isothermal zones inside the cav- ity. Sieres et al.[6]conducted a numerical investigation of lamina natural convection with and without surface-to-surface radiation in a class of right-angled triangular cavities filled with air. Their re- sults presented for extreme Rayleigh numbers (103and 106) show that the mean convective Nusselt number at the hot wall increases when the height Rayleigh number increases and when the aperture angle, located at the lower vertex of the cavity, decreases. The sur- face radiation alters the air velocity and temperature fields inside the cavity, this leads to an elevation in the convective Nusselt number. A numerical investigation of two-dimensional turbulent 0196-8904/$ - see front matterÓ2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.enconman.2008.05.017
* Corresponding author. Tel.: +212 22 23 06 80; fax: +212 22 23 06 74.
E-mail address:elalami_m@hotmail.com(M. El Alami).
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natural convection in an air filled isosceles triangular cavity was conducted by Ridouane et al.[7]for two values of Rayleigh num- bers: 1.58109 and 51010. The results of this investigation show that the fluid is turbulent and flows in a narrow strip along the walls where the velocity and temperature change sharply.
For a fixed Rayleigh number, this geometry holds high turbulence levels when compared to the square cavity but stays stationary in the cavity core. A finite element analysis is performed by Natarajan et al.[8]to investigate the influence of uniform and non-uniform heating of bottom wall on natural convection flows in a trapezoidal cavity. It shows that the cavity is filled with two clockwise and anti-clockwise convective cells. The local Nnusslet number is found to be increased for both uniform and non-uniform heating cases.
The average Nusselt number is found to follow power law variation with Rayleigh number for convection dominant regimes. In the case of a rectangular cavity provided with two openings on high part of the vertical walls and subjected to a horizontal jet, we find the study of Papanicolaou and Jaluria[9]. Contrary to the preceding case, their results (presented forRe= 100) show that the fresh air blast passes in the higher zone of the cavity without providing to refresh its low part. The latter is occupied by a convective cell hot- ter than the forced flow. Raji and Hasnaoui [10] conducted a numerical study in rectangular open cavity. Their results show, for the configuration HB, that heat transfer passes by a maximum for a critical Reynolds number (Rem= 100) corresponding to the disappearance of the cells of recirculation.
To our knowledge, there is no study of mixed convection in trapezoidal or triangular cavities. However, the majority of the re- sults presented in natural convection show that the higher part of the cavity is badly cooled.
The non-durable products placed in the higher zone of the cav- ity will not be well preserved and thus ventilation using a fresh air blast is essential. We think that horizontal or vertical ventilation of the trapezoidal cavity can make a substantial improvement of the heat transfer to the level of the hot wall and solves this problem.
So, we applied a fresh air jet through vertical or horizontal opening arranged respectively on the vertical right wall or on the horizontal one. The second opening, placed to the horizontal wall, will allow to evacuate the heated air.
In the present paper, a numerical study of mixed convection in two-dimensional trapezoidal cavity with two openings and heated from below is carried out. The forced flow is assumed to be steady.
The main objective of this study is to analyse the effect of the forced flow on the natural convection cells and also on the steady and unsteady nature of the flow for various values of Reynolds number. It is demonstrated that the natural convection cells disap- pear for critical Reynolds numbers which will be presented as functions ofRa.
2. Physical problem and governing equations
The geometry of the problem herein investigated is depicted in Fig. 1. The system is made of a trapezoidal cavity. Openings are ad- justed on the plates as shown in this figure and two configurations are studied in this work. In the first case, the inlet opening is placed on the vertical wall in the right of the cavity (hereafter called case 1). For the second configuration, the inlet opening is adjusted on the lower plate of the cavity (case 2). This wall is heated with a constant temperatureT0H. The upper one is cold at a temperature T0C, while the other sides of the cavity are insulated.
The flow is considered laminar, incompressible and the Bous- sinesq approximation has been applied. The dimensionless govern- ing equations can be written as
oU oxþoV
oy¼0 ð1Þ
oU ot þUoU
oxþVoU oy¼ oP
oxþ1 Re
o2U ox2þo2U
oy2
!
ð2Þ
oV otþUoV
oxþVoV oy¼ oP
oyþ 1 Re
o2V ox2þo2V
oy2
! þ Ra
PrRe2T ð3Þ oT
otþUoT oxþVoT
oy¼ 1 PrRe
o2T ox2þo2T
oy2
!
ð4Þ
Referring toFig. 1, the dimensionless variables are x¼x0
H0; y¼y0
H0; U¼U0 U0
; V¼V0
U0
; T¼T0T0C
T0HT0C; P¼ðP0þ
q
gy0Þq
U20 Ra¼gbDTH03am
; Pr¼m
a
withðDT¼T0 HT0CÞ
The imposed boundary conditions, in terms of Temperature and velocity, are similar to those of the natural convection flow in a ver- tical channel[11,12]:
Nomenclature
C1 dimensionless outlet opening widthðl01=H0Þ C2 dimensionless inlet opening widthðl02=H0Þ H0 maximum height of the cavity (m) g gravitational acceleration (m/s2) l01 inlet opening width (m) l02 outlet opening width (m) L0 length of the cavity (m) n normal coordinate
Nu mean Nusselt number (Eq.(5)) P0 pressure of fluid (Pa)
P dimensionless pressure Pr Prandtl number (Pr=
m
/a
)Ra Rayleigh numberðRa¼gbDTH03=
am
Þ Re Reynolds numberðRe¼H0mU0Þ T0 temperature of fluid (K) T0H temperature on the hot wall (K) T0C temperature of the cold surface (K)T dimensionless temperature of fluid ½T¼ ððT0T0CÞ=
ðT0HT0CÞ
U0,V0 velocities inx0andy0directions (m/s)
U, V dimensionless velocities inxandydirections [=(U0,V0)/
U0]
U0 average jet velocity at the entrance (m/s) x0,y0 cartesian coordinates
x, y dimensionless Cartesian coordinates [(x,y) = (x0,y0)/H0]
a
thermal diffusivityb volumetric coefficient of thermal expansion k thermal conductivity of fluid
m
cinematic viscosity of fluid h inclination of the tilted wallq
fluid densityW dimensionless stream function Subscripts
C cold
H hot
max maximum
cr critical
T= 1 on the lower wall andT= 0 on the upper inclined plane of the cavity
U=V= 0 on all the rigid walls At the inlet opening:
T¼V¼0 andU¼ 1 in the first case;
T¼U¼0 andV¼1 in the second case:
The vertical planes are insulatedoToy¼0 . At the evacuation opening:
T,U andV are extrapolated by adopting similar processes as shown in Ref.[12](Their second spatial derivative terms in the ver- tical direction are equal to zero).
The mean Nusselt number over the active horizontal wall is Nu¼
Z x2 x1
oT oy
y¼0dx ð5Þ
where
x1¼0:4 and x2¼3 in the first case;
x1¼0:4 and x2¼2:68 in the second case:
3. Numerical method
The governing equations of the problem were solved, numeri- cally, using a control volume method [13]. Quick scheme [14]
was adopted for the discretization of all convective terms of the advective transport equations (Eqs.(2)–(4)). The final discretized forms of the Eqs.(1)–(4)were solved by using the SIMPLEC (simple consistent) algorithm [15]. As a result of a grid independence study, a grid size of 12050 was found to model accurately the flow fields described in the corresponding results. Time steps con- sidered are ranging between 105and 104. The accuracy of the numerical model was verified by comparing results from the pres- ent study with those obtained by De Val Davis[16]and then with those obtained by Le Queré and De Roquefort[17]for natural con- vection in trapezoidal cavity,Table 1. Also we have compared our numerical code with the results obtained by Desrayaud and Ficher- a[18]in a vertical channel with two ribs, symmetrically, placed on the channel walls,Table 2. We note that good agreement was ob- tained inWmaxand mass flow rate terms[11–19]. When a steady state is reached, all the energy furnished by the hot wall to the fluid
must leave the cavity through the cold surface and the outlet open- ing. This energy balance was verified by less than 3% in all cases considered here.
4. Results and discussion
In the following sections, heat transfer along the hot wall, flow and temperature fields are examined for the Reynolds number (106Re61000), Rayleigh number (Ra= 105) and other parame- ters of the problem (h= 22°;C1= 0.38,C2= 0.25,Pr= 0.72). We note that the Rayleigh number value is chosen in favour of natural convection.
The particularity of this problem is the appearance of different solutions when varying the parameter Re.The flow structure is, essentially, composed of the open lines, which represent the forced flow, and closed cells which are due to the recirculating movement up of the jet or to natural convection phenomena. In this later case, the cell is dawn of the forced flow.
4.1. Flow structure and isotherms
(a) Case 1: Inlet opening is placed on the vertical wall in the right of the cavity (Fig. 1a)The flow structure and the thermal field are respectively, presented by the streamlines and isotherms inFig. 2a–c. ForRe= 20,Fig. 2a, a recircu- lating cell appears up on the jet of the forced flow along the inclined plate of the cavity. This cell is clockwise rotating and unfavourable to the heat exchange through this wall. It can be seen that the path of the forced flow is deformed by the strength of the natural convection cell which is located down of the jet. This cell is essentially due to the Rayleigh–Benard convection (because of the existence of a vertical thermal gradient in the cav- ity). The tightened isotherms (Re= 20) indicate that there exists a relatively, important heat transfer from the active wall (horizontal heated wall). ForRe= 100, Fig. 2b, there is no recirculating cell while the Ray- leigh–Bénard cell size increases considerably and deforms strongly the path of the forced jet. This cell is always located close the inlet opening and is counter clockwise rotating. Compared to those of the last case, the corresponding isotherms are more deformed near the inlet opening and in the middle of the cavity. Note that heat transfer along the active wall is reduced by the existence of the natural convective cell. For high val-
)
(a) Case 1
)
(b) Case 2
'
l
1l
1''
l
2l
2'H’
L’
L’
H’
h’ h’
Adiabatic walls
T=1 T=1
T=0 T=0
x, U y, V
θ θ
Fig. 1.Studied configurations.
Table 1
Code validation with in the case of rectangular cavity De Val
Davis[16]
Le Queré and De Roquefort[17]
Our results Maximum relative error (%) Ra= 104 Wmax= 5.098 – Wmax= 5.035 1.2 Ra= 105 Wmax= 9.667 – Wmax= 9.725 0.6 Ra= 106 Wmax= 17.113 Wmax= 16.811 Wmax= 17.152 2 Ra= 107 – Wmax= 30.170 Wmax= 30.077 0.3
Table 2
Code validation in the case of vertical channel with rectangular blocks
Desryaud and Fichera[18] Our results Maximum error (%)
Wmax 151.51 152.85 0.9
M 148.27 151.72 2.2
ues ofRe(ReP500),Fig. 2c, the jet passes near the hor- izontal hot wall. A close inspection of this figure shows that the convective cell is absent and a weak recirculat- ing one exists up on the jet and localised in the right cor- ner of the cavity. The corresponding isotherms are too tight near the active wall and show that the major part of heat leaves the cavity through the outlet opening.
We notify that the convective cell disappears whenRe exceeds a critical value (Recr) which depends on Ray- leigh number value.We notify that forRe= 100, the heat exchanged by the hot wall forwards equitably through the cold wall and the outlet opening,Fig. 2b. Beyond this value, the cold wall does not take part, practically, with the evacuation of heat towards outside.
(b) Case 2: Inlet opening is placed on the horizontal wall in the right of the cavity (Fig. 1b)In this case, the forced con- vective flow is vertical at the entrance. Numerical results are given for range of Reynolds number as 1006Re61000. For this configuration, natural con- vection drawing can be developed near the inlet open- ing because of the thermal gradient which exists between the horizontal and inclined walls. This ther- mal drawing will cause a vertical aspiration of air with a non-negligible mass flow rate. For this reason, we consideredRevalues up to 100.
The flow structure and isotherms are presented inFig. 3a–c. Gen- erally, the flow structures induced in this case are different from those observed in the last one. The difference is mainly due to ab- sence of the natural convection cell (down of the forced flow) for weak values ofReas shown inFig. 3a forRe= 100. The correspond- ing isotherms are too tight near the hot wall. WhenReincreases, a small cell appears down of the jet,Fig. 3b forRe= 120. Its size in- creases with increasingReand constrains the jet of forced flow to
pass close the inclined plane,Fig. 3c forRe= 1000. The isotherms of this figure show that the major part of heat exit trough the outlet opening. This situation is desirable because it’s favourable to the cooling of foods placed in the upper zone of the cavity. We notify that the existence of the rotating cell up of the jet, for weak values ofRe, is undesirable because it minimises heat exchange through the inclined cold wall. It prevents the fresh air to refresh foods placed top of the cavity. Fortunately, its size decreases with increas- ingReand disappears forRe= 300 (figure not presented), while the natural convective cell size becomes more and more important. Re- mark that we have the inverse of this situation in the first case, where the convective cell is absent and a strong rotating one exists for high values ofRe. Hence, the configuration with vertical forced flow can resolve the problem of foods cooling in all the cavity.
InFig. 4, we present the critical Reynolds number (Recr) varia- tion with Rayleigh number for case 1 (Recris a value ofRefor which the convective cell disappears). It curve delimits two zones: that of bottom corresponds to the forced convection mode, and that of the top is related to the mixed convection mode. We note thatRecr
curve presents two lines of different slopes in the selected Re range. These variations are correlated with the relations:
Ra¼338:3Re0:89cr For 106Re650 Ra¼0:5Re2:60cr For 506Re61000
whenRaincreases the cell of natural convection becomes very in- tense. Consequently, it is necessary to apply a very powerful jet to eliminate it and pass to the forced convection mode. This explains the break of the curve withRecr= 50 corresponding toRa= 10950.
4.2. Heat transfer and mass flow rate
The heat transfer through the hot face is given in term of Nus- selt number,Fig. 5. It presents the variation of the quantity of
(c) Re=500 (a) Re=20
(b) Re=100
Fig. 2.Streamlines and isotherms for different Reynolds number values,Ra= 105,h= 22°Case 1: the inlet opening adjusted on the vertical right wall.
a dimensional average heatQmaccording toReforRa= 105. Gener- ally,Nuincreases withRefor the two configurations. This result is awaited because more we increase the speed of the fresh air blast, more we evacuate heat from the cavity towards outside. Note that Nucurve related the vertical jet (case 2) is up to theNuone gener-
ated by the horizontal forced flow (case 1). So, we consider that the forced vertical jet (in the entrance) is favourable to heat transfer in the trapezoidal cavity. Note that the gap betweenNuvalues gener- ated in the two cases have reached its maximum (16%) for Re110.Nuvariations withReare correlated by the relations:
(a) Re=100
(b) Re=120
(c) Re =1000
Fig. 3.Streamlines and isotherms for different Reynolds number values,Ra= 105,h= 22°Case 2: the inlet opening adjusted on the horizontal lower wall.
10 100 1000
1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 Ra
Re Mixed convection zone
Forced convection zone Ra = 0.50 Re cr2.60
Ra = 338 .3 Recr 0.89
Fig. 4.Recrvariation withRa,h= 22°.
10 100 1000
1 10 100
Nu = 1.60 Re0.28
Nu = 2.40 Re 0.23
Re Nu
Case 1 Case 2Fig. 5.Nusselt variation withRefor the two studied configurations.
Nu¼1:60Re0:28 in case 1 ð6Þ
Nu¼2:40Re0:23 in case 2 ð7Þ
with a maximum error of about 4%.
This relation is comparable with those which exist in the liter- ature, Raji and Hasnaoui[10]in the case of open cavities with rect- angular geometries.
5. Conclusion
Mixed convection in a trapezoidal cavity heated from below is studied numerically. Two configurations are studied in this work:
the applied convective jet is horizontal at the entrance (case 1) or it’s vertical (case 2). In the first case (case 1), the results show that there exists a criticalReabove which the natural convection cells disappear and forced convection mode is installed. The critical values ofRedepend on Rayleigh number and its variation withRa is given inFig. 4.
In the second configuration, the vertical jet at the entrance per- mits a good ventilation of the high part of the cavity and then it’s favourable to heat exchange from the cavity towards the exterior.
Nugenerated by this configuration is important than the one of the other configuration. Hence, we conclude that the vertical ventila- tion is more favourable to foods cooling then the horizontal one.
Power law correlations betweenNuandReare proposed for the two cases.
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