A 2 D Finite Element Study on the Flow Pattern and Temperature Distribution for an Isothermal Spherical Furnace with the Aperture

Calibration of radiation thermometers is one of the important research activities in the field of metrology. Many researchers in recent times have conducted numerical simulations on the calibration furnace to understand and overcome the experiment limitations. This paper presents a 2D numerical free convective study on the calibration furnace with the aperture using finite element method. The focused issues here are: aspect ratio effect on the flow pattern and temperature fields, heat transfer mechanism in the aperture zone as well as in hump regime. It is concluded that flow and temperature fields follow the same behavior in the hump regime as well as in the aperture zone. Also, it concluded that penetrative convection is more dominant for the enclosure of high aspect ratio.


Introduction
Temperature measurement of the very hot objects by means of contact thermocouple is not always handy.For this purpose, radiation thermometer is used to measure temperature in terms of radiation emitted by the hot object.Radiation thermometer is a temperature measurement instrument and measures temperature of the very hot object from a distance.For accuracy of the instrument, it must be calibrated.Calibration is an act of adjusting instrument by comparison against any standard surface.The Saturn (a hollow spherical cavity) is one of such standards used in radiation thermometer calibration [1].For details on calibration furnace and the finite element modeling to study experiment limitations, one can refer authors' previous work [2].
Many research efforts have been done finite element modeling [3,4] of calibration furnaces, industrial furnaces, heated spheres [5,6] and solar cavity receivers for quality equipment.Recently, Oluwole et al. [7] studied the flow patterns in two salt bath furnaces using finite element analysis.The implications of the heat flows on long term stability of furnace performance were evaluated.Khoei et al. [8] developed a finite element model that is employed to simulate the furnace rotation and analyze energy flows inside the furnace.Also, finite element, stream function vorticity solutions for steady state incompressible Navier Stokes equations are derived in papers [9,10].
Natural convection in fluid filled enclosures [11,12] with heating bottom and sides [13] have been studied extensively.Sarris et al. [14] studied numerically natural convection in a rectangular enclosure with heating from the top wall with all others insulated by varying Rayleigh number from 10 2 to 10 8 and the aspect ratio from 0.5 to 2. Recently, Hartlep et al. [15] performed simulations of Rayleigh-Benaurd convection in a large box of aspect ratio 10 over a range of Rayleigh and Prandtl numbers provide important insight into the choice of the aspect ratio.In another study, Lee et al. [16] stressed the need for very large aspect ratio domains by studying the natural convection in a horizontal fluid layer with a periodic array of internal square cylinders.
Finite element mesh is a part of numerical simulation study.Sensitivity analysis for parameter dependent optimization problems is an active area of research in the context of solution of partial differential equations.Becker and Vexler [17,18] investigated sensitivity of the mesh based on relative condition numbers which describe the influence of small changes in measurements on the value of interest functional.
This paper addresses a 2D numerical free convective study on the calibration furnace with the aperture.In the next section, the 2D problem, governing equations and the corresponding boundary conditions are systematically discussed.In Section 3, mathematical formulation of 2D problem and then numerical simulation schematically presented.Section 4 presents precise results on flow and temperature fields near the proximity of the furnace surface and in the aperture plane.

Statement of the Problem
The problem is addressed in the context of temperature distribution and 2D flow pattern for the furnace in three various types of enclosures, but, the method, ideas and results can be compared to enclosure of any dimension and 3D model.The qualitative behavior of flow and heat transfer is the same for a general 2D model and 3D model due to bi-axial symmetry of the sphere.For this purpose, a furnace slice, obtained by cutting along the equatorial line, kept in an enclosure of two dimensions is considered as a computational domain for the present study.The coordinates chosen for the enclosure computational domain of aspect ratio 1 are E (0, 0), F (0.3, 0), G (0.3, 0.3) and H (0, 0.3).The furnace dimensions are taken from the experiment and it is shown in Figure 1.The enclosure is filled with air to keep maintain experiment conditions.

Mathematical Model
The computational domain (Ω) is assumed to be union of the two sub domains Ω 1 (gas) and Ω 2 (solid).The schematic sketch of the computational domain is shown in The flow is incompressible air and two dimensional, there is no viscous dissipation, gravity acts in vertical direction, air properties are constant and density variations are neglected except the buoyancy term.Continuity and momentum equations are defined in the air domain Ω 1 as follows: ( ) The heat energy equation is defined in the entire computational domain (Ω) to study the heat transport equation is as follows.
where everywhere except at the heat source region.0 Q =

Mathematical Formulation
The steady state incompressible Navier-Stokes equations in velcoty-vector potential formulation are presented as with no slip boundary conditions while the velocity is given by The stream function, and hence the velocity, is then calculated from ψ ω Δ = .Weak form of Equations ( 3)-( 5) is as follows: find such that ( ) ( ) where ⋅ , denotes the standard inner product on 1 and X be the standard continuous finite element space with the 2nd degree polynomial on each element of a triangulation.Let The velocity h u is obtained from the stream thro function ugh Clearly the velocity h u sat co t isfies the divergence free ndition everywhere and he normal velocity h u n ⋅ is continuous across element boundary.The absolute value of the extreme stream function is evaluated at the stagnant fluid, i.e.
The heat transfer equation defined on fin Ω 1 as follows: where is a finite element space of sec 2 h Y ia ond degree polynom l defined on 1 Ω .On Ω 2 , heat transfer equation is defined as follows: A quadratic triangular finite element m mu ic, there is a rapid deents on locally refined meshes.esh of maxim element size 0.02 is chosen for the current numerical study.Maximum element size is defined as the ratio of maximum edge length to the unit length of the element.
Sensitivity of the mesh is examined for two different meshes composed of 9989 nodes and 10,667 nodes based on relative condition numbers.From Table 1, measurements at m k (the point on the cavity surface exactly opposite to the aperture) have more influenced than b k (point at the cavity bottom) and t k (point at the cavity top).It is reasonable to say that at m k , the solution gradient is very high due to curvature.Therefore, the mesh is refined locally towards direction normal to m k .

Results and Discussion
As the heat passes through ceram crease in temperature but not zero and it is shown in Three enclosures of aspect ratio 0.5 (narrow), 1 and 1.5 (wide) are chosen for current numerical investigation.Also, it is important to notify the mensionless temperature θ is 1 at the heating coil B (0.26, 0.4) only when cavity attains the experiment temperature T s = 409.8˚Cat C (0.29, 0.4).No radiation effects are considered in the current study.Typical flow and temperature fields, mechanism of heat transfer and thermal penetration in the aperture region are thoroughly discussed by varying Rayleigh numbers from 10 4 to the high Rayleigh number achieved.

Flow and Temperature Fields
For the aspect ratio 1, flow pattern on vorticity values in a stagnant fluid number 10 4 to the high Rayleigh number ac At the Rayleigh number 10 4 , the flow is not rotational at the region where the boundary layer entrains into the main stream.Therefore, stream function satisfies La uation.Hence the strength of circulation at the low Rayleigh number is 0.000064 (x = 0.4, y = 0.604).At the Rayleigh number 10 5 , strength of the circulation is 0.00068 (x = 0.485, y = 0.585).At this stage, convective flow cell is formed rotating in an anti-clock wise direction.At the high Rayleigh number 8.2 × 10 5 , strength of the circulation is 0.00188 (x = 0.43, y = 0.601) and therefore both clockwise and anti-clockwise flow pattern predicted with high velocity.Thus, as expected circulation strength increase with the Rayleigh number and the coordinate shifts from close to the hot cell towards inside, (in an anti-clockwise sense) i.e. the flow penetrates into the enclosure more and more with an increasing Rayleigh number.
In the aperture plane, strength of circulation is 0.0000145 (x = 0.51, y = 0.4) at the Rayleigh number 10 4 .
x in m g to e boundary layer on the circumference of ou ls are p 0.51 and x = 0.6.As the Rayleigh number increases to 10 5 , the corresponding circulatio 0.51, y = 0.4).Thus, circulation strength increases with the Rayleigh number and the cold fluid penetrates into the aperture more and more with an increasing Rayleigh number.Also, there is one more point (x = 0.58, y = 0.4) at which fluid is stagnant.The corresponding circulation strengths are 0.01 (Ra = 10 4 ), 0.064 (Ra = 10 5 ) and 0.42 (Ra = 8.2 × 10 5 ).It indicates there is a strong convective cell around that point.
For the aspect ratio 0.5, circulation strengths near the hump regime are 0.00282 (0.435, 0.6), 0.0069 (0.49, 0.582) and 0.05775 (0 4 , 10 5 and 4.9 × 10 5 respectively.At the Rayleigh number 10 4 , penetration of the flow is not vertical and hence the low flow velocity is predicted in the regime.As the Rayleigh number increases, the flow penetration gradually comes to vertical and therefore, horizontal and vertical penetration of the flow almost the same.At the low Rayleigh number, the movement of the flow at the right corner of the enclosure is very high when compared to the movement of the flow at the left corner.As the Rayleigh number increases, movement of the flow at the left corner and at the right corner almost the same and therefore, flow penetration gradually increases with the Rayleigh number.Another observation is that the absolute values of vorticity increases as the Rayleigh number increases and hence the rotation of convective cell increases with the Rayleigh number.One vortex is observed for the Rayleigh number 10 5 while the two counter rotating large vortices are predicted at the high Rayleigh number. In the aperture plane, strength of circulation is 0.0098 (x = 0.58, y = 0.4) at the Rayleigh number 10 4 .As the Rayleigh number in circulation increases from 0.05925 (x = 0.58, y = 0.4) to 0.271 (x = 0.5835, y = 0.4) correspondingly.
For the aspect ratio 1.5, at the Rayleigh number 10 4 , circulation strength is weak with |ψ ext | = 0.00085 at x = 0.401, y = 0.603.As the Rayleigh number incr 5 , correspondingly the location changes to x = 0.401, y = 0.604 with strength 0.0091 and penetration into the hump region increases.Further increase in the Rayleigh number results gradually full penetration into the hump with circulation strength 0.1068 (Ra = 6.6 × 10 5 ) at the location x = 0.05 and y = 0.603.Also, it is observed that the temperature gradient is a decreasing function of the Rayleigh number along the symmetry plane.The reason being (1) formation of the vortex a little far away from the surface involved in resisting heat transfer performance (2) penetrations from both horizontal and vertical directions are almost the same.Therefore, length of the symmetry plane participating in heat transfer to the right cold cell decreases.
In the aperture plane, circulation strength 0.0097 (x = 0.579, y = 0.4) at the Rayleigh number 10 4 .As the Rayleigh number increases to 10 5 , location is movin wards inside with strength 0.055 and level of penetration of the fluid increases.At the high Rayleigh number achieved, circulation with maximum strength 0.289 at the coordinate x = 0.58, y = 0.4.By comparison of isotherms at the high Rayleigh number in terms of aspect ratio indicates that the heat transfer rate is more at the high aspect ratio.
Figure 3(b) shows temperature contours at the high Rayleigh number achieved for aspect ratios 0.5, 1 and 2. Temperatur ter layer surface is studied.The surface temperature moves along the boundary layer in clockwise direction and entrained into the main stream at the position where in it encounters the temperature moves along the boundary layer in an anti-clockwise direction.The mechanism of heat transfer is by a single cell rotating in clockwise direction in the left half driven by hot surface in between 0.36 and 0.43.The heat is transferred to the right half along the symmetry line at x = 0.435, as a result of which a counter-clockwise rotating cell is formed.Inner cell to the left of the symmetry plane receives heat from the outer convective cell and therefore records highest temperature among hot convective cells while innermost cell to the right of symmetry plane losses heat to outer convective cells and hence records lowest temperature among cold convective cells.
In case of the aspect ratio 1, temperature gradient along the symmetry plane x = 0.435 increases as the Rayleigh number increases.
Therefore, thermal penetration is more intensive with higher temperature gradient along the symmetry plane, i.e., increasing heat transfer from the hot convective cell to the cold convective cell.From the plot, as the Rayleigh number increases, thermal penetration shift towards right while symmetry line shift towards left.Though weak cells are generated beyond penetration limit at low Rayleigh numbers, disappears as the Rayleigh number increases.
In the aperture zone, the fluid flow is stagnant in between x = 0.5 and x = 0.51 while the fluid flow penetrating through the aperture.Therefore, high temperature field is predicted in between x = 0.5 and x = 0.51.As the Rayleigh number increases, the size of stagnant flow is slightly reduced.From the temperature contours plot, isotherms bend like inverted parabolas leave the aperture toward inner surface of the cavity.Therefore, heat transfer from surface to ambient increases.Also, heat transfer behavior is same qualitatively for the aspect ratio 0.5 and 1.
Figure 4 shows the local Nusselt number along the symmetry line and the line passes through hot convective cells at the high Rayleigh number 8 ratio 1.From the plot, it clears that Nusselt numbe to zero at two instances along the symmetry plane it is almost zero at five positions along the line passes through the hot cells.One important observation is that number of positions at which Nusselt number proximity to zero represent the number of cells the line touches.

Free Convection in the Aperture Zone
In the aperture zone, local Nusselt number distribution along the horizontal layer of cross section is studied for various values of the aspect ratio 0.5, 1 and 1.5.Figure 5 shows the graph of local Nusselt number against Rayleigh number in the aperture plane for the aspect ratio 1.
From the plot, it clears that convection is gradually dominated and therefore penetration of cold fluid into the aperture increases as the Rayleigh number increases to 8.2 × 10 5 .Another important observation is that many weak convective cells are observed at Rayleigh number 10 4 .As the Rayleigh number increases, weak cells vanish and converted to very few strong convective cells inside plane at the high aspect ratio 1.5.Another obincreases as the aspect ratio increases from 0.5 to 1. the plane.
Figure 6 shows thermal penetration is 100% in the aperture servation is that number of strong convective cells

Conclusions
In view of results and discussions presented, the following main conclusions are drawn.

X
with zero boundary values.For the Lagrangian finite element space, ( ) ( )

Figure 2 .
Figure 2. Temperature profile at the vertical midpoint along horizontal layer of cross section.

. 2 ×
10 5 for the aspect r close while4.2.Free Convection in the Hump RegimeLocal Nusselt number distribution along the horizontal layer of cross section near the proximity of surface top for various values of the aspect ratio 0.5, 1 and 1.5 and for various values of Rayleigh number studied.For the aspect ratio 1, maximum of local Nusselt number increases as one moves from surface top to the environment.It indicates that heat transfer from the surface to the ambient medium decreases.As the Rayleigh number increases to 8.2 × 10 5 , maximum of local Nusselt number decreases as one move away from the surface.

Figure 4 .
Figure 4. Local Nusselt number along the symmetry line and the line touches hot convective cell at x = 0.4 and x = 0.435.

Figure 5 .
Figure 5. Local Nusselt number distribution along horizontal layer of cross section in the aperture zone for Rayleigh numbers 10 4 , 10 5 and 8.2 × 10 5 .

Figure 6 .
Figure 6.Local Nusselt number distribution along horizontal layer of cross section in the aperture for the aspect ratios 0.5, 1 and 1.5.

Table 1 . Relative condition numbers for point measure- m
interval range 0.3 < x < 0.5 indicates the possibility of occurrence of Rayleigh-Benaurd convection in the region above surface top.