Effect of Time Periodic Boundary Conditions on Convective Flows in a Porous Square Enclosure with Non-Uniform Internal Heating

Transp Porous Med (2010) 85:885–903DOI 10.1007/s11242-010-9597-5

Effect of Time Periodic Boundary Conditionson Convective Flows in a Porous Square Enclosurewith Non-Uniform Internal Heating

H. Saleh · I. Hashim · N. Saeid

Received: 3 May 2010 / Accepted: 23 May 2010 / Published online: 16 June 2010© Springer Science+Business Media B.V. 2010

Abstract The problem of unsteady natural convection in a square region filled with afluid-saturated porous medium having non-uniform internal heating and heated laterally isconsidered. The heated wall surface temperature varies sinusoidally with the time about fixedmean temperature. The opposite cold wall is maintained at a constant temperature. The topand bottom horizontal walls are kept adiabatic. The flow field is modelled with the Darcymodel and is solved numerically using a finite difference method. The transient solutionsobtained are all periodic in time. The effect of Rayleigh number, internal heating parameters,heating amplitude and oscillating frequency on the flow and temperature field as well as thetotal heat generated within the convective region are presented. It was found that strong inter-nal heating can generate significant maximum fluid temperatures above the heated wall. Thelocation of the maximum fluid temperature moves with time according to the periodicallychanging heated wall temperature. The augmentation of the space-averaged temperature inthe cavity strongly depends on the heating amplitude and rather insensitive to the oscillatingfrequency.

Keywords Time-periodic boundary conditions · Natural convection · Darcy’s law ·Non-uniform heat generation

H. Saleh · I. Hashim (B)School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM, Bangi, Selangor,Malaysiae-mail: [email protected]

H. Salehe-mail: [email protected]

N. SaeidDepartment of Mechanical, Materials and Manufacturing Engineering, The University of NottinghamMalaysia Campus, 43500 Semenyih, Selangor, Malaysiae-mail: [email protected]

123

886 H. Saleh et al.

List of symbolsA, a Amplitude, dimensionless amplitudef Dimensionless oscillating frequencyg Gravitational accelerationG Measure of the internal heat generation� Width of the cavityK Permeability of the porous mediump Local heating exponentQ Space-averaged temperatureRa Rayleigh numbert ′, t Time, dimensionless timetp Dimensionless periodt Denoting a summation of time steady state and initial periodsT Fluid temperatureu, v Velocity components in the x- and y-directionsx ′, y′ & x , y Space coordinates & dimensionless space coordinates

Greek symbolsαm Effective thermal diffusivityβ Thermal expansion coefficientγ Internal heating parameterψ ′, ψ Stream function, dimensionless stream functionθ Dimensionless temperatureμ Dynamic viscosityν Kinematic viscosityω Oscillating frequency

Subscriptav Averagec Coldh Hotmax Maximumss Basic steady state

Superscript

– denoting a quantity average over time

1 Introduction

Convective flows within porous materials have occupied the central stage in manyfundamental heat transfer analyses and have received considerable attention over the lastfew decades. This interest is due to its wide range of applications, for example, high perfor-mance insulation for buildings, chemical catalytic reactors, packed sphere beds, grain storage

123

Effect of Time Periodic Boundary Conditions 887

and such geophysical problems as the frost heave. Porous media are also of interest in relationto the underground spread of pollutants, to solar power collectors, and to geothermal energysystems. Convective flows can develop within these materials if they are subjected to someform of a temperature gradient. Many applications are discussed and reviewed by Inghamand Pop (1998), Pop and Ingham (2001), Ingham et al. (2004), Ingham and Pop (2005), Vafai(2005), Nield and Bejan (2006).

In some situations of considerable practical importance, porous material provides its ownsource of heat, giving an alternative way in which a convective flow can be set up throughlocal heat generation within the porous material. Such a situation can arise through radioac-tive decay or through, in the present context, a relatively weak exothermic reaction within theporous material. This can happen, for example, in the self-induced heating of coal stockpiles(Brooks and Glasser 1986; Brooks et al. 1988) and bagasse-piles (Sisson et al. 1992; Sextonet al. 2001; Gray et al. 2002). Convective flows are governed by non-linear partial differ-ential equations, which represent conservation laws for the mass, momentum and energy.Numerical treatment usually used to solve the equations or if possible by an analytic method.Numerical investigation of natural convection in porous enclosure having uniform internalheating firstly conducted by Haajizadeh et al. (1984). Then continued by Rao and Wang(1991) and analytically by Joshi et al. (2006). Recently, Mealey and Merkin (2009) movedaway from the study of uniform internal heating to that of non-uniform internal heating andsolved the problem numerically and analytically. However, Haajizadeh et al. (1984), Rao andWang (1991), Joshi et al. (2006) and Mealey and Merkin (2009) considered the steady flowonly.

The transient behaviour of natural convection in an enclosure has been extensively stud-ied due to the relevance to many industrial applications. For example, in the cooling ofelectronic equipment, the electrical components are periodically energized intermittentlyand, therefore, heat is generated in an unsteady manner (Kazmierczak and Chinoda, 1992).Daily or yearly period solar radiation of reservoirs where the geographical situation refersto heating amplitude (Lage and Bejan, 1993) is a good explanation of sinusoidal oscillat-ing boundary conditions. Kazmierczak and Chinoda (1992) investigated numerically naturalconvection of water in a square cavity heated from the side with a time sinusoidal periodictemperature. It was concluded that although the instantaneous heat flux through the hot wallfluctuates greatly in time, the time-averaged heat transfer across the enclosure is rather insen-sitive to the time-dependent boundary condition. Lage and Bejan (1993) studied numericallyand analytically natural convection in a configuration heated by a pulsating heat flux. Theyshowed that the amplitude of oscillation of the heat flux through a vertical surface reachesmaximum values for a given value of the angular frequency. Next, Antohe and Lage (1996)studied experimentally the amplitude effect on convection induced by the time periodic heat-ing. It is shown that the convection intensity within the enclosure increases linearly withheating amplitude while the resonance frequency was shown to be independent of the heat-ing amplitude for both clear fluid and porous medium configurations. Recently, Nithyadeviet al. (2006), Wang et al. (2007, 2008a,b) and El Ayachi et al. (2008) studied time-oscillatingboundary condition on natural convection in porous enclosure. All these studies indicate thatheat transfer in a system can be significantly augmented. However, the heat transfer wasdecreased by increasing the amplitude and the frequency for infinite domain and using thenon-equilibrium model as investigated by Saeid and Mohamad (2005).

The aim of present study is to investigate numerically the problem of unsteady natural con-vection in a square cavity filled with a fluid-saturated porous medium having non-uniforminternal heating and heated laterally with a sinusoidal time variation. Flow development,temperature distribution and the space-averaged temperature in the cavity will be presented

123

888 H. Saleh et al.

graphically. To the best of our knowledge, investigation of the effect of time periodicboundary conditions on convective flows in a porous square enclosure having internal heat-ing has not received due attention. Similar research conducted by Kim et al. (2002) for clearfluid found the secondary peak resonance for the higher internal heating. Another similarresearch conducted by Al-Taey (2009) for the chaotic natural convection found new chaoticform appears in stream function pattern and temperature distribution.

2 Mathematical Formulation

Consider unsteady, two-dimensional natural convection flow in a square region filled with afluid-saturated porous medium (Fig. 1a). The co-ordinate system employed is also depicted inthis figure. The top and bottom surfaces of the convective region are assumed to be thermallyinsulated, the face y′ = � is held at the constant cold temperature Tc. The face y′ = 0 variessinusoidally in time about a mean value hot temperature T h, with amplitude A and frequencyω. The hot wall is greater than the cold wall at all times, as graphically depicted in Fig. 1b.

Heat is assumed also to be generated internally within the porous medium at a rate pro-portional to (T − Tc)

p (p ≥ 1), where T is the local temperature. This relation, as explained

Fig. 1 Schematic representationof the model

(a)

(b)

123


by Mealey and Merkin (2009), is an approximation of the state of some exothermic process.The fluid and porous medium properties are assumed to be constant except for the varia-tion of density with temperature in the buoyancy term in Darcy’s equation for the fluid flow(Boussinesq approximation). The porous medium is taken to be homogeneous and isotropic.Under these assumptions, the equations governing the model are:

∂2ψ ′

∂x ′2 + ∂2ψ ′

∂y′2 = gKβ

ν

∂T

∂y′ , (1)

σ∂T

∂t ′+ ∂ψ ′

∂y′∂T

∂x ′ − ∂ψ ′

∂x ′∂T

∂y′ = αm

(∂2T

∂x ′2 + ∂2T

∂y′2

)+ G (T − Tc)

p, (2)

where K is the permeability of the porous medium, ν the kinematic viscosity,β the coefficientof thermal expansion, g the acceleration due to gravity, αm the effective thermal diffusivity,σ the ratio of composite material heat capacity to convective fluid heat capacity and G ameasure of local heat generation. The stream function ψ , introduced to satisfy the continuityequation, gives the velocity components as u = ∂ψ ′/∂y′, v = ∂ψ ′/∂x ′. Equations 1 and 2are to be solved subject to the following initial and boundary conditions:

at t ′ = 0, ψ ′ = 0, T = Tc,

on x ′ = 0, x ′ = �, ψ ′ = 0, ∂T∂x ′ = 0, (0 < y′ < �),

on y′ = �, ψ ′ = 0, T = Tc, (0 < x ′ < �),

on y′ = 0, ψ ′ = 0, T = T h + A sinωt ′, (0 < x ′ < �).

(3)

For convenience, Eqs. 1–3 are transformed into dimensionless form by introducing thefollowing variables:

ψ = gβK��T

νψ ′, θ = T − Tc

�T, a = A

�T,

t = t ′αm

σ�2 , x = x ′

�, y = y′

�, (where�T = T h − Tc > 0). (4)

This results in the following non-dimensional equations:

∂2ψ

∂x2 + ∂2ψ

∂y2 = ∂θ

∂y, (5)

∂θ

∂t+ Ra

[∂ψ

∂y

∂θ

∂x− ∂ψ

∂x

∂θ

∂y

]= ∂2θ

∂x2 + ∂2θ

∂y2 + γ θ p, (6)

subject to initial and boundary conditions:

t = 0, ψ = 0, θ = 0,x = 0, x = 1, ψ = 0, ∂θ

∂x = 0, (0 < y < 1),y = 1, ψ = 0, θ = 0, (0 < x < 1),y = 0, ψ = 0, θ = 1 + a sin f t, (0 < x < 1).

(7)

In the above equations,

Ra = gβK��T

αmν(Rayleigh number),

γ = G�2(�T )p−1

αm(internal heating parameter),

f = ωσ�2

αm(oscillating frequency).

123

890 H. Saleh et al.

The dimensionless period of the temperature oscillation, tp, can be expressed as follows,

tp = 2π

f. (8)

When internal heating exist inside the enclosure, thermal performance at hot and cold wallcompletely different. So that rather than Nusselt number, we prefer to measure the thermalperformance over the entire convective region by:

Q =1∫

0

1∫0

θ(x, y) dxdy. (9)

Q is the space-averaged temperature and the temporal average of the space-averaged tem-perature in the cavity for one period can be defined as:

Qav = 1

tp

tp∫0

Q dt. (10)

3 Numerical Method and Validation

The finite difference method is employed to solve Eqs. 5 and 6 subjected to conditions in(7). The central difference method was applied for discretizing the spatial derivatives. Analternating direction implicit (ADI) scheme is applied for discretizing the time derivatives.The resulting algebraic equations are solved by the tri-diagonal matrix algorithm (TDMA).

A grid independent test was performed using sets of grids in range 11 × 11 to 121 × 121for Ra = 100, γ = 2, p = 1, a = 0.4, f = 20π and after the basic steady state and twooscillating initial periods as shown in Fig. 2. The results showed insignificant differences forthe 81 × 81 grids to above. Therefore, for all computations in this article, a 81 × 81 uniformgrid was employed.

The time step is chosen to be uniform �t = 10−4 which has been used also by Saeidand Pop (2004). After the steady state achieved for the mean hot wall, based on our trial

Fig. 2 Grid independency study: Q and 10ψmax versus grid size

123


Table 1 Comparison of Nufor some results from theliterature at the mean hot wall ina steady state for no internalheating

References Ra = 102 Ra = 103

Walker and Homsy (1978) 3.097 12.96

Bejan (1979) 4.200 15.800

Gross et al. (1986) 3.141 13.448

Manole and Lage (1992) 3.118 13.637

Saeid and Pop (2004) 3.002 13.726

Present study 3.120 13.193

experiment, the optimal time step for one period is tp/100. For each time step during thecomputations, the convergence of computations is declared when the maximum relativechanges in the variables (ψ, θ) between two successive iterations are less than 10−5. In orderto validate the computation code, the previously published problems on natural convection ina differentially heated cavity, filled with a fluid-saturated medium without internal heating,were solved. Table 1 shows the average Nusselt number,

Nu =1∫

0

∂θ

∂y

∣∣∣y=0

dx, (11)

are in good agreement with the solutions reported by the literature. These comprehensive ver-ification efforts demonstrated the robustness and accuracy of the present numerical method.

4 Results and Discussion

The analyses in the undergoing numerical investigation are performed in the following domainof the associated dimensionless groups: the heating amplitude, 0.01 ≤ a ≤ 0.99, the oscil-lating frequency, 5π ≤ f ≤ 200π , the internal heat generated parameter, 2 ≤ γ ≤ 4, theexponent in the local-heating term, 1 ≤ p ≤ 3 and the Rayleigh number, 0 ≤ Ra ≤ 200.The value of Ra, γ and p were chosen based on studies conducted by Mealey and Merkin(2009) that the solutions exist only for a finite range of Ra and γ and p.

4.1 Temperature Field at Ra = 0

When Ra = 0, Eq. 6 for the temperature becomes independent of that for the flow (5). Inthis case, the problem reduces to one dimensional parabolic equation. The basic steady stateor corresponding to non-oscillating case (a = 0) has been solved analytically by Mealey andMerkin (2009). At p = 1 and γ > 0, the solution is given by:

θ(y) = sin√γ (1 − y)

sin√γ

. (12)

This steady solution is included with the transient result of temperature field at γ = 4 andp = 1 as depicted in Fig. 3. Initially, at t = 0 along the y space θ = 0, as in Eq. 7. Atvery beginning when the temperature at the left wall begin heated results the temperatureof the left wall is higher than that of the fluid inside the enclosure. The hot wall transmitsheat to the fluid and raises the temperature of fluid particles adjoining the left wall. As time

123

892 H. Saleh et al.

Fig. 3 Time history of temperature field at Ra = 0, γ = 4 and p = 1

increases, the heat transfers continuously from the left side of the enclosure to the middleand to the right adjoining the cold wall. When no internal heat source inside the enclosure,the distribution of temperature will be a linear function of y. The heat being conducted awayinto the surrounding fluid causes the fluid temperature to rise sharply. This implies that themaximum fluid temperature is greater than the temperature of the hot wall and the tempera-ture distributes nonlinearly as seen at t = 0.2. This is due to heat generated inside the porousmaterial. Still increasing time, the maximum fluid temperature more pronounced above thehot wall and located away from the heated wall. After that the same behaviour was foundfor further increasing the time. This implies the steady or permanent state were reached.Numerical simulation gave the steady state reach at t = 0.625. Figure 3 also demonstratedthe robustness and accuracy of our result compared with the result by Mealey and Merkin(2009).

After the steady state achieved, the next phase is to investigate the effect of the sinusoidaloscillating hot wall. Temperature evolution during one period for f = 20π , a = 0.8, γ = 4and p = 1 is presented in Fig. 4. In order to obtain a periodic oscillating solution indepen-dent of initial state, often in as few as two oscillating periods were needed to be calculated(Kazmierczak and Chinoda, 1992). We denoted t as a stable state which is the summationof the time steady state and the two initial periods. Figure 4 shows the temperature evolu-tion during one period. At very beginning, t ≤ t ≤ t + 2tp/8, the maximum temperatureoccurs at the heated wall. Increasing time result the maximum temperature occurs at the fluidinside the enclosure. Careful inspection of Fig. 4 discloses that the temporal maximum fluidtemperature location moves away from the heated wall appropriate with time increasing.

4.2 Effect of the Time Dependency on Flow and Temperature Field

For Ra > 0, the temperature inside the porous cavity becomes dependent on the flow field.The momentum equation (5) and the energy equation (6) are solved sequentially. The methodhas described in the numerical method and validation section. Figures 5 and 6 show the tran-sient result of the flow and temperature field for the non-oscillating case. Initially, at t = 0,left and right wall are cold and there is no fluid motion inside the enclosure. As heatingstarted, the fluid temperatures adjoining the hot wall rise. The fluid moves due to buoyancy

123


Fig. 4 Temperature evolution during one period for f = 20π , a = 0.8, γ = 4 and p = 1 at Ra = 0 fromt = t to t = t + 7tp/8

(a)

(d) (e) (f)

(b) (c)

Fig. 5 Time history of stream lines at Ra = 200, γ = 4 and p = 1

force from the left region of the cavity to the right region of the cavity. This movement createsa clockwise circulation cell inside the left enclosure. As time increases, the temperature hasbeen well distributed from the left wall to the right wall, so that the clockwise circulationcell occupies the whole enclosure. The core of the cell moves to the centre of the enclosure.This motion finally becomes permanent when the steady state achieved.

123

894 H. Saleh et al.

(a)

(d) (e) (f)

(b) (c)

Fig. 6 Time history of temperature field at Ra = 200, γ = 4 and p = 1

The time history of the temperature field is described as follows. At the very beginning,t = 0.001 to t = 0.008, the isotherms are almost parallel and vertical as shown in Fig. 6a, b,c. This implies that conduction or diffusion mode is dominant. For t = 0.03, it is observedthat the thermal boundary layer develops at the hot and cold boundaries as shown in Fig. 6d.This implies that the onset of convection occurs. Internal heat source reacts and gives themaximum temperature in the cavity more than the heated wall temperature. Increasing timeresults in the maximum temperature more pronounced and located away from the heated wallas shown in Fig. 6. Insignificant differences of flow and temperature field observed whentime takes longer. This implies the steady state has reached at the previous time, at t = 0.145.This time is much shorter than the previous case with Ra = 0, where the time required toreach the steady state was t = 0.625.

After the basic steady state reached, the next phase is to investigate the effect of theoscillating left wall temperature. Figures 7 and 8 show the evolution of flow and tempera-ture field in one period. The heating amplitude is fixed at, a = 0.8, oscillating frequency,f = 20π , internal heat source, γ = 4 and local heating exponent, p = 1 at Rayleigh number,Ra = 200. In order to obtain a stable state,t , two oscillating periods were calculated. Thesubplots a–h of Figs. 7 and 8 are corresponding to seven phases from t = t to t = t + 7tp/8.It should have made clear that this sequence repeats itself, and the very next the flow andtemperature field generated following subplot (h) are identical to the subplot (a).

At the beginning, the streamline presented in Fig. 7 show that the flow field is dominatedby a primary circulation cell occupies the whole enclosure and rotate in a clockwise direction.As the dimensionless time increases, a weak small secondary cell appeared and it is rotating

123


Fig. 7 Sequential contour plots of flow field during one period for a = 0.8, f = 20π , γ = 4 and p = 1 atRa = 200 when a t = t , b t = t + tp/8, c t = t + 2tp/8, d t = t + 3tp/8, e t = t + 4tp/8, f t = t + 5tp/8,g t = t + 6tp/8 and h t = t + 7tp/8

in the counterclockwise direction. The secondary cell initially appears at t = t + 4tp/8, thetime at which the left hot wall temperature equals the mean hot wall temperature, T h, afterdecreasing from the maximum value. The secondary cell grows in size at t = t + 5tp/8, thenit pushes the primary cell towards the constant cooled wall at t = t + 6tp/8. Further increas-ing the time, the secondary cell shrinks, then totally disappears as the hot wall temperatureequals T h. The magnitude and location of the maximum flow rates is also time dependent.It changes with the time according to the periodically changing the hot wall temperature.

For all the subplots a–h presented in Figs. 7 and 8, the strong internal heating resultsthe maximum fluid temperatures above the heated wall. The evolution of temperature fieldis quite interesting in one period. During the first half period (Th > T h), the isothermsare more distorted compared with the second half period (Th < T h). This is related to thestrength of natural convection which is depending on heating conditions. It also observed a

123

896 H. Saleh et al.

Fig. 8 Sequential contour plots of temperature field during one period for a = 0.8, f = 20π , γ = 4 andp = 1 at Ra = 200 when a t = t , b t = t + tp/8, c t = t + 2tp/8, d t = t + 3tp/8, e t = t + 4tp/8,f t = t + 5tp/8, g t = t + 6tp/8 and h t = t + 7tp/8

well-defined thermal boundary layer on both the hot and cold walls during the first half period.The thermal boundary layer on the hot wall breaks down during the second half period. Thethermal boundary layer floats to the top of the enclosure and forms a warm pocket. Thispocket contains fluid warmer than the hot wall. At final stage during the second half period,t = t + 7tp/8, a well-defined thermal boundary layer return and the sequence repeats con-tinuously. It also observed that in one period, the reaction of the strong internal heating morepronounced during the second half period (Th < T h).

4.3 Effect of the Parameters on the Space-Averaged Temperature

In order to show the effect of Ra, the periodic oscillation of Q with time at various valuesof Ra at given a, f , γ and p are presented in Fig. 9. In order to study the effect of γ and p,

123


Fig. 9 Time-dependent behaviour of the space-averaged temperature for various value of Ra at a = 0.4,f = 20π , γ = 4 and p = 1

Fig. 10 Time-dependent behaviour of the space-averaged temperature for various value of γ and p at a = 0.4,f = 20π and Ra = 100

the periodic oscillation of Q at two combinations of γ and p when Ra, a and f fixed arepresented in Fig. 10. Next, the periodic oscillation of Q at variation a when f , Ra, γ and pconstant, are presented in Fig. 11. Finally, the effect of the frequency is studied and the resultis presented as periodic oscillations of Q at several f when a, Ra, γ and p unchanging, asshown in Fig. 12.

The labels 0.752, 0.647, 0.589 and 0.560 on the Q axis in Fig. 9 refer to the magnitudeof the space-averaged temperature at the basic steady state or non-oscillating case (a = 0),for Ra = 0, Ra = 40, Ra = 100 and Ra = 200, respectively. It was observed from Fig. 9that higher Ra gives lower Q when it is integrated with in time. This behaviour similar tothe non-oscillating case where the self-heating effect more pronounced at low Ra as statedby Mealey and Merkin (2009). Figure 9 shows that the temporal evolution of the Q oscil-lates with periodic variation of wall temperatures for all Ra values considered in the presentstudy. The numerical results presented in Fig. 9 show that the area of Q above the line basic

123

898 H. Saleh et al.

Fig. 11 Time-dependent behaviour of the space-averaged temperature for various value of a at f = 20π ,Ra = 100 γ = 4 and p = 1

state is larger than the area of Q below the line basic state for each Ra. This indicates theaugmentation of the Q.

Smooth curve was obtained after one period for the cases of moderate heating (γ = 2) andp = 1 as shown in Fig. 10. There exist several extra twists in the curves of Q for the case ofstrong internal heating or a high value of the local heating exponent. The present results areconsistent with the results obtained by Mealey and Merkin (2009) for the basic state. Wherethe intensity of the convection is linearly proportional to the increasing γ for the γ belowthe moderate value. This condition combined with the moderate heating amplitude (a = 0.4)results in the temporal evolution of Q moving smoothly by increasing the time.

Perfectly sinusoidal law curves were found in Fig. 11 at small heating amplitude (a =0.01, 0.2). The time-dependent Q exhibits oscillatory behaviour which is almost symmet-ric about the basic state. For larger amplitude, the qualitative behaviour of oscillation of Qis generally similar to that small heating amplitude. However, quantitative differences areostensible; the fluctuating amplitude of Q becomes substantially larger as a increases asshown in Fig. 11.

The variations of Q with t show a perfectly sinusoidal law curve at low oscillating fre-quency ( f = 5π) as depicted in Fig. 12 and it deviated from sinusoidal law for higher f .The fluctuating amplitude of Q slightly decreases as f increases. This is due to when theoscillating frequency increases, the temperature oscillation of the left sidewall does not affectthe temperature of the interior fluid region in the enclosure. These results indicate that thereis not enough time for the porous cavity to respond to the hot wall temperature oscillation athigh frequencies. Therefore, the temperature distribution inside the entire enclosure changesslightly.

4.4 Effect of the Amplitude on the Temporal Average of the Space-Averaged Temperature

The temporal average of the space-averaged temperature for one period, Qav over thespace-averaged temperature at the basic state, Qss with the heating amplitude, a, forRa = 0, 40, 100, 200 at f = 20π , γ = 4 and p = 1 are presented in Fig. 13. As shown inthe Fig. 13, the heating amplitude varies from 0.01 to 0.99. It was observed from this figurethat for all Ra, the ratio Qav/Qss is always more than 1. The augmentation of Qav rather

123


Fig. 12 Time-dependent behaviour of the space-averaged temperature for various value of f at f = 20π ,Ra = 100 γ = 4 and p = 1

Fig. 13 The normalized space-averaged temperature against amplitude for various value of Ra at f = 20π ,γ = 4 and p = 1

insensitive to the tuning heating for Ra = 0. Careful investigation shows that at the Ra = 0,the value of Qav reaches a maximum value about a = 0.2. Increasing Ra gives the Qav

reaches a maximum value at larger a. At Ra = 40, the maximal Qav occurs about a = 0.45.At Ra = 100, the maximal Qav occurs about a = 0.85. Finally, at Ra = 200, the maximalQav occurs at the maximum heating amplitude that considered in this article.

Figure 14 presents the normalized space-averaged temperature, Qav/Qss within the con-vective region against heating amplitude for two combinations of γ and p at f = 20πand Ra = 100. Significant augmentation of Qav occurs at high local heating exponent. Forany internal heating conditions, the numerical results show that the maximal Qav take placeat maximum heating amplitude (a = 0.99). The response of the system to changes in theheating amplitude is in a non-linear manner as shown in Fig. 14. This condition consistentwith the presence of non-uniform internal gives temperature distribution nonlinearly even atRa = 0 as shown in Figs. 3 and 4.

The variation of Qav/Qss against a for f = 5π, 20π, 50π, 200π at Ra = 100, γ = 4and p = 1 are depicted in Fig. 15. In general, increasing a enhances Qav and a higher

123

900 H. Saleh et al.

Fig. 14 The normalized space-averaged temperature against amplitude for various value of γ and p atf = 20π and Ra = 100

Fig. 15 The normalized space-averaged temperature against amplitude for various value of f at Ra = 100γ = 4 and p = 1

oscillating frequency gives bigger enhancement of Qav as shown in Fig. 15. Neverthe-less, for each frequency, the maximal Qav takes place when the heating amplitude isabout 0.9.

4.5 Effect of the Frequency on the Temporal Average of the Space-Averaged Temperature

The variation of normalized space-averaged temperature, Qav/Qss with the oscillating fre-quency, f , for Ra = 0, 40, 100 and 200 at a = 0.4, γ = 4 and p = 1 are presentedin Fig. 16. The oscillating frequency limited in the range 5–200π . As the previous case(the Qav/Qss against the amplitude for Ra = 0), the effect of increasing the oscillatingfrequency rather insensitive to the augmentation of Qav. The Qav increases about 1.8%in comparison to the Qss as shown in Fig. 16. At Ra = 100, 200, the results showthat two peaks exist near f = 60π and f = 180π . The peaks refer to the local maxi-mum Qav.

123


Fig. 16 The normalized space-averaged temperature against frequency for various value of Ra at a = 0.4,γ = 4 and p = 1

Fig. 17 The normalized space-averaged temperature against frequency for various value of γ and p at a = 0.4and Ra = 100

Fig. 17 presents the Qav/Qss within the convective region against oscillating frequencyfor two combinations of γ and p at f = 20π and Ra = 100. Consistent with the pre-vious results in Fig. 14, at fixed γ , higher p gives larger augmentation of Qav. Initially,increasing f enhances Qav then approaches to a constant value for further increasing fexcept for γ = 4 and p = 3. This constant response indicates that the fluid in the interioris no longer capable responding to the high oscillatory frequency heating. At the γ = 4and p = 3 two peaks exist near f = 65π and f = 180π . The peaks indicate that thesystem becomes stiffer to pulsating heat excitation by highly exponent non-uniform heatsource.

The variations of the ratio Qav/Qss against f for a = 0.01, 0.2, 0.8 and 0.99 at Ra = 100,γ = 4 and p = 1 are depicted in Fig. 18. Initially, increasing f enhances Qav significantlythen Qav changes slightly with the oscillating frequency increasing further. In general, largera results bigger augmentation of Qav. This is consistent with the previous results as presentedin Fig. 15.

123

902 H. Saleh et al.

Fig. 18 The normalized space-averaged temperature against frequency for various value of a at Ra = 100γ = 4 and p = 1

5 Conclusions

The present numerical study modelled the effects of a time periodic boundary conditions onnatural convection in a square enclosure filled with a porous medium having non-uniforminternal heat source. The dimensionless forms of the governing equations are solved numer-ically using finite difference method. The computation code is validated with the publisheddata of the steady state for a fixed mean hot wall temperature. Detailed numerical results forflow and temperature field, time-dependent behaviour and temporal average of the space-averaged temperature in the cavity have been presented in the graphical form. The mainconclusions of the present analysis are as follows:

1. The time required to reach the basic steady state is longer for low Rayleigh number thanthat for high Rayleigh number.

2. Strong internal heating can generate significant maximum fluid temperatures above theheated wall temperature. The location of the maximum fluid temperature moves withtime according to the periodically changing the heated wall temperature.

3. Time sinusoidal oscillating boundary conditions have drastically been changing the flowand temperature field. A weak secondary cell and thermal boundary layer on the hot wallbreak down due to the effect of heated wall temperature oscillation.

4. The augmentation of the space-averaged temperature in the cavity strongly depend onthe heating amplitude and rather insensitive to the oscillating frequency.

Acknowledgements The authors would like to acknowledge the financial support received from the GrantUKM-ST-07-FRGS0028-2009.

References

Al-Taey, K.A.: Natural convection in a laterally and volumetrically heated square enclosure with time-periodicboundary conditions. Int. J. Heat Technol. 27, 77–85 (2009)

Antohe, B.V., Lage, J.L.: Natural convection in an enclosure under time periodic heating: an experimentalstudy. ASME, Heat Transfer Division, (Publication) HTD 324 (1996)

123


Bejan, A.: On the boundary layer regime in a vertical enclosure filled with a porous medium. Lett. Heat MassTransf. 6, 93–102 (1979)

Brooks, K., Glasser, D.: A simplified model of spontaneous combustion in coal stockpiles. Fuel 65, 1035–1041 (1986)

Brooks, K., Balakotaiah, V., Luss, D.: Effect of natural convection on spontaneous combustion of coal stock-piles. A.I.Ch.E. J. 34, 353–365 (1988)

El Ayachi, R., Raji, A., Hasnaoui, M., Bahlaoui, A.: Combined effect of radiation and natural convection ina square cavity differentially heated with a periodic temperature. Numer. Heat Transf. Part A 53, 1339–1356 (2008)

Gray, B.F., Sexton, M.J., Halliburton, B., Macaskill, C.: Wetting-induced ignition in cellulosic materials. FireSaf. J. 37, 465–479 (2002)

Gross, R.J., Bear, M.R., Hickox, C.E.: The application of flux-corrected transport (FCT) to high Rayleigh num-ber natural convection in a porous medium. In: Proceedings of the Eighth International Heat TransferConference, San Francisco, CA (1986)

Haajizadeh, M., Ozguc, A.F., Tien, C.L.: Natural convection in a vertical porous enclosure with internal heatgeneration. Int. J. Heat Mass Transf. 27, 1893–1902 (1984)

Ingham, D.B., Pop, I.: Transport Phenomena in Porous Media. Elsevier Science, Oxford (1998)Ingham, D.B., Bejan, A., Mamut, E., Pop, I.: Emerging Technologies and Techniques in Porous Media. Klu-

wer, Dordrecht (2004)Ingham, D.B., Pop., I.: Transport Phenomena in Porous Media III. Elsevier Science, Oxford (2005)Joshi, M.V., Gaitonde, U.N., Mitra, S.K.: Analytical study of natural convection in a cavity with volumetric

heat generation. J. Heat Transf. 128, 176–182 (2006)Kazmierczak, M., Chinoda, Z.: Buoyancy-driven flow in an enclosure with time periodic boundary condi-

tions. Int. J. Heat Mass Transf. 35, 1507–1518 (1992)Kim, G.B., Hyun, J.M., Kwak, H.S.: Enclosed buoyant convection with internal heat generation under oscil-

lating sidewall temperature. J. Heat Transf. 124, 577–580 (2002)Lage, J.L., Bejan, A.: The resonance of natural convection in an enclosure heated periodically from the side. Int.

J. Heat Mass Transf. 36, 2027–2038 (1993)Manole, D.M., Lage, J.L.: Numerical benchmark results for natural convection in a porous medium cavity. In:

Heat and Mass Transfer in Porous Media. ASME Conference, HTD-vol. 216, pp. 55–60 (1992)Mealey, L.R., Merkin, J.H.: Steady finite Rayleigh number convective flows in a porous medium with internal

heat generation. Int. J. Therm. Sci. 48, 1068–1080 (2009)Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)Nithyadevi, N., Kandaswamy, P., Sivasankaran, S.: Natural convection in a square cavity with partially active

vertical walls: time-periodic boundary condition. Math. Prob. Eng. 23425, 1–16 (2006)Pop, I., Ingham, D.B.: Convective Heat Transfer: Mathematical and Computational Modelling of Viscous

Fluids and Porous Media. Pergamon, Oxford (2001)Rao, Y.F., Wang, B.X.: Natural convection in vertical porous enclosures with internal heat generation. Int. J.

Heat Mass Transf. 34, 247–252 (1991)Saeid, N.H., Pop, I.: Transient free convection in a square cavity filled with a porous medium. Int. J. Heat

Mass Transf. 47, 1917–1924 (2004)Saeid, N.H., Mohamad, A.A.: Periodic free convection from a vertical plate in a saturated porous medium,

non-equilibrium model. Int. J. Heat Mass Transf. 48, 3855–3863 (2005)Sexton, M.J., Macaskill, C., Gray, B.F.: Self-heating and drying in two-dimensional bagasse piles. Combust.

Theory Model. 5, 517–536 (2001)Sisson, R.A., Swift, A., Wake, G.C., Gray, B.F.: The self-heating of damp cellulosic materials: I. High thermal

conductivity and diffusivity. IMA J. Appl. Math. 49, 273–291 (1992)Vafai, K. (ed.): Handbook of Porous Media, 2nd edn. Taylor and Francis, New York (2005)Walker, K.L., Homsy, G.M.: Convection in porous cavity. J. Fluid Mech. 87, 449–474 (1978)Wang, G., Zeng, M., Huang, Z., Wang, Q.: Numerical study of natural convection heat transfer in a porous

cavity under time-periodic boundary conditions. Jisuan Wuli (Chin. J. Comput. Phys.) 24, 282–286 (2007)Wang, G., Wang, Q., Zeng, M., Ozoe, H.: Natural convection heat transfer in an inclined porous cavity under

time-periodic boundary conditions with positive/negative inclined angles. J. Porous Media 11, 541–555 (2008)

Wang, G., Wang, Q., Zeng, M., Ozoe, H.: Numerical study of natural convection heat transfer in an inclinedporous cavity with time-periodic boundary conditions. Transp. Porous Media 74, 293–309 (2008)

123

Effect of Time Periodic Boundary Conditions on Convective Flows in a Porous Square Enclosure with Non-Uniform Internal Heating

Documents