Effect of the position of the hot source on mixed convection in a rectangular cavity

Abstract

Keywords: Numerical Heat Transfer; Rectangular Enclosure; Mixed Convection; Position of the Hot Source; Cooling; Sinusoidal Temperature; Finite Element Methods

Introduction

Recently, the numerical analysis of mixed convection flow structure and thermal phenomena inside ventilated cavities has become the most extensive research subject for its functional importance and application as: production

oil, air heating-cooling mechanism, nuclear reactors, heat exchangers and others. In general, several scientific researchers have studied flow dynamics and heat transfer in both two and

*Corresponding author. *E-mail address: sereirtewfik@gmail.com This paper was recommended for publication in revised form by Regional Editor Hasan Köten Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

three-dimensional [1]–[4] configurations in various geometric shapes [5]–[14].Due to its high performance and limited energy demand, a study of laminar mixed convection cooling of a rectangular cavity with a constant source of heat flow was the subject of [15]to identify optimum location of inlet and outlet for better cooling efficiency. For various ventilation arrangements, free and forced convection was investigated in a rectangular enclosure with a source heated from below. S. Mekroussi and al. [16] studied by numerical simulation the mixed air convection (Pr=0.71) in a cavity with corrugated walls. The influence of several control parameters on the flow such as the Richardson number, amplitude ratio and phase difference have been considered. Taylor and al. [17] carried out an experimental study of mixed convection in a ventilated cavity subjected to a heat flux on the vertical wall. [18] For the fields of flow and temperature distributions, numerical results are recorded in a heated solid enclosed in a square cavity with several geometric configurations. [19] studied two-dimensional mixed convection inside a ventilated cavity with central solid circular heated horizontal conductor for different cylinder size (0-0.6) and also Richardson number 0 <Ri < 500 using the finite element method. As a result, the phenomenon inside the cavity with or without a cylinder depends on both the transfer diameter of the cylinder and the number Ri. [20] and [21] analyzed two-dimensional mixed convection in enclosures with adiabatic walls with two cylinders heated inside for a different diameter and a constant distance between the two cylinders. [22] Using a finite element method, a computer survey was carried out on mixed convection in a square cavity, in a ventilated square enclosure with an inlet located on the left edge of the insulated vertical wall, where the exhaust port is attached to the top of the heated vertical surface, this study examined and explained the complicated relationship between buoyancy and the flow of force. [23]Study of heat transfer and mixed convection flow patterns driven by the cover in a trapezoidal cavity with sinusoidal temperature at the bottom wall. [24]The mixed convection driven by the cover with a sinusoidal temperature in the two side walls in a square cavity was studied. [25]studied the two-dimensional ventilated rectangular cavity for various inlets and outlets using the finite element method. [26]three-dimensional mixed convection in a Ventilated cavity for different locations of the inlet and outlet ports at various Reynolds and Richardson numbers. The present work was carried out to study the heat transfer and the fluid patterns inside a rectangular enclosure in terms of the mean Nusselt number at the heated sinusoidal wall and the conductive square solid block inside the rectangular cavity for various parameters in order to achieve optimal mixed convection conditions. Finally, an empirical correlation equation is developed to calculate the mean Nusselt number.

Description Of The Problem

The studied physical system illustrated in Figure 1, consists of a rectangular cavity of width H and length L contains a square heat source of dimension d on each side. The two openings of dimension w, one of which is located in the right wall which serves for the outlet of hot air and the other for the inlet of cold air located in the upper wall. The cavity is stressed by a hot temperature of the sinusoidal shape in the right vertical wall and the other left wall is kept at a cold temperature, so that both sides of the upper and lower part of the cavity are thermally insulated (adiabatic). The simplifying assumptions are as follows: he flow is assumed to be two-dimensional and • T laminar • The fluid is Newtonian • The properties of fluid are constant • The viscous dissipation and compressibility effects are considered negligible • The heat exchange by radiation and the variations in density of the fluid are neglected except in the term of buoyancy. In the fluid zone : ∂U ∂V + =0 ∂X ∂Y

∂U ∂U ∂P 1  ∂ 2 U ∂ 2 U  +V =− + + ∂X ∂Y ∂X Re  ∂X 2 ∂Y 2 

∂V ∂V ∂P 1  ∂ 2 V ∂ 2 V  +V =− + + +Riθ ∂X ∂Y ∂Y Re  ∂X 2 ∂Y 2 

Dimensionless variables are defined as follows: y x u v p d , Y= , U= , V= , P= 2 , D= , H H ui ui ρu i H λs (T-Ti ) , λ= θ= (Th -Ti ) λf X=

The dimensionless numbers that appear in the previous equations are:

Figure 1. The physical model. gβL ( Th -Ti ) UH ν Gr Pr= , Re= o ; Ri= 2 , Gr= α υ Re υ2 3

At the vertical solid-fluid interfaces:  dθ s   dθ    = λ   dX fluid dX  solid At the horizontal solid-fluid interfaces:  dθ s   dθ    = λ   dY fluid dY  solid Boundary condition The boundary conditions of this analysis are: • Lower horizontal wall : U = 0; V = 0 and

U = 0, V = − 1 and θ = 0 At right vertical wall: U = V = 0, θh = ε.sin(πY+φ), • At the outlet: convective boundary condition, (CBC ) , P=0 3 H With w = 0.1H L= H and d= 2 3 The average Nusselt number in vertical right wall and in the solid block of cavity is evaluated by: • In the right vertical wall :

• Left vertical wall : θ = 0, U = V = 0 At upper horizontal wall:

Figure 2. Isotherms for Ri = 5, Lx = 0.5, Ly = 0.25, λ = 0.5 and Re = 100. According to Saha et al. [15] and in order to obtain a better value of the average Nusselt number, it is calculated from the average temperature and nom from the input temperature. The average Nusselt number at the surface of the square solid is: Ly

Figure 3. Average Nusselt Number for Lx = 0.5, Ly = 0.25, λ = 5, Re=100 between M. M. Rahman [19] and our work.

The finite element method is used to resolve the governing equations. A direct linear system solver is adopted with a stationary nonlinear solver described by [27] and [28]. T h e relative tolerance for error is assumed to be 10-6 used with denser grids clustered in regions close to the heat source and enclosure walls.

Our work has been validated with the work of Rahman et al [19]. We compared the isotherms and the average Nusselt number when the solid square was located at ly = lx = 0.5 with data from our results at Re = 100, Ri = 1.0, λ =

Numerical Method

Table 1. Grid sensitivity Check at Re=100, Ri=1.0, λ = 5, d=L/3, Lx=1.5/2 and Ly=0.5 Elements

Figure 4. The streamlines for a number of Ri from 0 to 10 and Re from 50 to 200 with an amplitude ε = 1, and a phasedeviation φ = 0 for the “A” configuration. in Table 1. The grid size 11884 elements is adopted since it provides more satisfaction results and less calculating time.

Results And Discussion

Streamlines and isotherms The current lines for a Richardson number from 0 to 10 and different Reynolds number from 50 to 200 with an amplitude of and a phase deviation of the configuration “A” are shown in Figure 4, In the case of forced convection Ri = 0, two small rotating vortices are formed near the left cold wall and the bottom wall of the cavity, respectively. As the Reynolds number increases, this vortex area becomes larger. As Ri increases, the rotating vortex becomes larger and the flow rate is strong near the horizontal cold wall and the appearance of a recirculation zone towards the side of the right hot wall. Increasing the Richardson number

increases the buoyancy effect and improves energy transport which leads to an increase in flow force. Figure 5 represents the isotherms for a Richardson number from 0 to 10 and different Reynolds number from 50 to 200 with an amplitude of and a phase deviation of the configuration “A”, when the convection is forced (Ri = 0) the isotherms are more concentrated vertically around the heat source. Since the greatest amount of heat flow is removed from the source and dissipated through the outlet opening, hence the air flow from the inlet to the outlet to influence the heat transport and isotherms with high values ​​tend to concentrate near the heat source so that the conduction mode of heat transfer dominates that of convection which is confirmed by layered isotherms. The mixed convection regime becomes more noticeable with the increase in Ri because the energy transport increases due to the flow of conductive heat emitted by

Figure 5. Isotherms for a number of Ri from 0 to 10 and Re from 50 to 200 with an amplitude of ε = 1 and a phasedeviation of ϕ = 0 for the configuration “A”. the source. Since the isotherms change as Ri and Re vary, these are the concentration parameters in the analysis for all cases. Isotherms are more concentrated horizontally around the heat source and at the top of the cavity. When Re increases the isotherms are clustered near the heat source, because the induced fluid flows faster over the heat source, the heat conducted in the induced flow cannot propagate to the upper layers of the fluid. The current lines of configuration “B” for a Richardson number from 0 to 10 and different Reynolds number from 50 to 200 with an amplitude of and a phase deviation are shown in Figure 6. hot source is in the center of the cavity the flow is symmetrical for Ri> 0 and becomes larger with the increase in the number of Re. In the case of forced convection Ri = 0 the appearance of two recirculation zones one is around the hot source and the second occurs from the inlet opening to the outlet port since the flow is faster than the flow dissipated by the hot source. Figure 7 shows the isotherms for a Richardson number from 0 to 10 and different Reynolds number from 50 to 200 with an amplitude of and a phase deviation of the configuration “B”, in the case of forced convection the isotherms are

more concentrated. vertically near the hot source and as the number of Re increases, the isotherms near the cold wall and the fluid inlet become weaker than that of the source and the hot wall. On the other hand, the more Ri increases the isotherms are horizontal and always concentrated around the source since the heat flux is dominant than the flow flux. The current lines of the configuration “C” for a Richardson number from 0 to 10 and different Reynolds number from 50 to 200 with an amplitude of and a phase deviation are illustrated in Figure 8 For Ri = 0 the appearance of two recirculation zone one of the air entering through the opening which circulates around the hot source to the outlet opening and the other vortex is located in the corner between the left hot wall and the part upper cavity. When the mixed convection Ri> 0 starts, two counter-rotating vortices, one of small size located above the hot source and near the cold air inlet opening and the other large size vortex occupies the rest of the cavity and as Re increases the flow is important. The isotherms of the configuration “C” for a Richardson number from 0 to 10 and different Reynolds number from 50 to 200 with an amplitude of and a phase deviation are

Figure 6. The streamlines for Ri from 0 to 10 and Re from 50 to 200 with an amplitude of ε = 1 and a phase-deviation of ϕ = 0 for the configuration “B”.

Figure 7. Isotherms for Ri from 0 to 10 and Re from 50 to 200 with an amplitude of ε = 1 and a phase-deviation of ϕ = 0 configuration “B”.

Figure 8. The current lines for Ri from 0 to 10 and Re from 50 to 200 with an amplitude of ε = 1 and a phase deviation of ϕ = 0 for the configuration “C”.

Figure 9. Isotherms for Ri from 0 to 10 and Re from 50 to 200 with an amplitude of ε = 1 and a phase deviation of ϕ = 0 for the configuration “C”.

Figure 10. The average Nusselt number as a function of the Richardson number for different Reynolds numbers with amplitude of and phase-deviation of the three configurations. presented in Figure 9 in the case of forced convection the isotherms are concentrated vertically near the hot source and as the number of Re increases, a stratification of the isotherms at the top of the cavity for low values of Re the transfer by conduction is dominant than the convection. On the other hand, the more Ri increases, the isotherms are narrowed and they are concentrated around the source since the transfer by convection is dominant than the conduction and the more Re increases which induces a rapid flow speed of the fluid. Heat transfer rate A variation of the mean Nusselt number as a function of the Richardson number for different Reynolds numbers on

the hot wall of the cavity of the three configurations studied above is represented in the Figure 10. We notice, through the profile of the configuration “C” where the position of the source located at the bottom and to the left of the cavity. The average Nusselt number increases from Ri = 0 to Ri = 2.5 and vice versa for the case of the “A” configuration where the position is at the top right of the cavity. Then it remains almost constant beyond this value for the set of Re values of the two cases. On the other hand, in the case of the configuration “B” where the position is in the center, the average Nusselt number is almost constant as a function of Ri for different numbers of Re. Nevertheless, by comparing quantitatively the value of the transfer rate is larger in configuration “A” this is due to the increase in

Figure 11. The variation of the mean Nusselt number as a function of the Richardson number of the three configurations for each Reynolds number. convective effects and consequently to the intensification of heat exchanges between the heat source and the flowing fluid. Note that the positioning of the heat source at the top near the right vertical wall of the cavity gives a better heat transfer rate, especially for high values of the Reynolds number. The Figure 11 shows the variation of the average Nusselt number as a function of the Richardson number of the three configurations for each Reynolds number, we notice that the Nusselt number decreases by increasing the Richardson

number whatever Re for the configuration “A “. But we see in the configuration “C” the Nu increases in the phase of forced convection Ri = 0 to 2.5 then it decreases by increasing the Re. On the other hand, the configuration “B” the heat transfer rate registers a stability except for the phase of the low numbers of Ri where it experiences a decrease, then it increases with the large values of Re. The significant increase in average Nusselt number for low Ri values corresponds to a dominant forced convection in configuration « C ».

Figure 12. The Nusselt number profile as a function of the Richardson number for different values of the sinusoidal temperature amplitude ε = 1 for the three configurations. Effect of the amplitude and phase-deviation: Figure 12 illustrates the Nusselt number profile as a function of the Richardson number for different values​​ of the amplitude of the sinusoidal temperature imposed on the right vertical wall of the three configurations studied. The Nu decreases in the case of the source is positioned at the top of the right side (configuration “A”) then it becomes constant where the position of the source is in the center (configuration “B”) and the Nu increases in the case of the source is located at the bottom left side (“C” configuration). However, the average heat transfer

rate increases with decreasing excitation temperature amplitude in the three cases of configurations studied. So conversely that we vary the frequency of the imposed temperature, we recorded that the average heat transfer rate of the air increases with the increase in the frequency of the temperature. This implies that heat transfer by convection is more dominant than transfer by conduction regardless of the magnitude or frequency of the imposed temperature. But quantitatively, increasing the amplitude and frequency respectfully decreases and increases the transfer rate.

Conclusion

A digital scan of a rectangular cavity has an inlet and an outlet of air containing a heated solid body studied at three different positions. The right vertical wall is heated by a temperature of the sinusoidal shape and the other wall subjected to a cold temperature. The top and bottom of the cavity remain without heat exchange with the external environment. The study leads to the following conclusions: - As Ri increases, convection strengthens with primary vortices, in half of the cavity and conduction becomes prominent in the vicinity of the solid body and the wall in the “A” configuration. while the “B” configuration, exhibits strong convective movement adjoining the two zones located in the vicinity of the hot source which in turn stretches the primary vortices outward through the orifice in the right-side wall. In the “C” configuration, the appearance of two counter-rotating recirculation zones with strong convection in the cell located between the solid body and the right-side wall. - Heat transfer by convection is dominant over conduction, increasing the Richardson number with increasing Reynolds number. - The transfer rate increases with the decrease in the amplitude of the excitatory temperature. On the other hand, it decreases with the increase in the frequency of the sinusoidal temperature. The mean temperature of the fluid and the Nusselt number on the right wall and the square solid depend strongly on: - Entry and exit positions; - Dimension and position of the solid block.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Ethics

There are no ethical issues with the publication of this manuscript.

Dimensionless velocity components along X and Y, respectively

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

1. The article references can be accessed from the .pdf file.

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