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Natural convection fluid flow and heat transfer is considered within a fluidsaturated porous enclosure with inclined diathermal partitions at 135˚. The primary objective of the paper is to attenuate the heat transfer rate across a differentially heated porous enclosure that is encountered in thermal insulations. This suppression is brought about by obstructing the flow with diathermal partitions, targeted to isolate two specific regions of the enclosure. The temperature distribution in porous enclosure with no partitions reveal that, there exists a high temperature gradient in the vicinity of lower half of hot face and upper half of cold face. Isolating these regions from the rest of enclosure is the physics behind employing multiple inclined partitions. Momentum equations are defined by Darcy model and the governing equations are discretized using Finite Difference Method (FDM) and solved by Successive Accelerated Replacement (SAR) scheme. Fluid flow and heat transfer analysis is performed for following parameters: modified Rayleigh number (Ra*) from 10^{2} to 10^{3}, partition ratio (PR) from 0 to 1, aspect ratio (AR) as 1. One of the main findings of this study is that, to suppress the natural convection effectively, the fluid flow along the lower half of the hot wall and upper half of the cold wall must be well obstructed or mechanically isolated as these regions bear high temperature gradient. Maximum reduction in average Nusselt number is obtained for PR=0.8 irrespective of the value of Ra*.
natural convection, porous medium, diathermal partition, numerical schem
Porous media, depending upon the application, may serve to enhance or suppress the heat transfer across it. Applications like cooling in electronic packages, packed bed reactors are examples of heat transfer enhancement while thermal insulation, solar energy collection involve attenuating the heat transfer. The present study concerns with suppression of natural convection heat transfer across a differentially heated porous enclosure. A detailed literature based on convection in porous media can be found in the works of Bear, Greenkorn and Neild and Bejan [13]. Natural convection fluid flow and heat transfer within a fluidsaturated porous medium having internal obstructions, which may be conducting or insulated, has been the focus of research since past decade. These researches have been driven, predominantly, by the early works like those of Bejan and Anderson, Jang and Chen and Bejan who showed that internal obstructions within a differentially heated porous enclosure, could considerably attenuate the rate of heat transfer across it [46]. Tasnim et al. showed that employing an obstruction in a cavity is comparable to reducing Da of the porous medium [7]. Jang and Chen, studied the effects of offcenter diathermal partition and inclination angle of porous enclosure on natural convection heat transfer [5]. At lower Ra values partition location is found more influential than that at higher Ra values. The least value of Nusselt number is obtained at centrally positioned partition and at roughly 30̊ inclination angle. Vasseur and Wang, studied the effect of a thermal barrier in a vertical porous layer separated by N equally spaced partitions [8]. The Nusselt number, in the boundary layer regime, has been found to vary inversely with (1+ N)^{4/5} while the greatest reduction in heat transfer, in the intermediate regime, has been obtained when the partition is located centrally. Varol and Oztop and Varol et al. analyzed a 2D solution of natural convection in solid adiabatic thin ﬁn connected to fluidsaturated porous right triangular enclosures [9, 10]. Varol et al. illustrated the effects of inclinedly inserted conductive thin plate on natural convection ﬂow in a cavity ﬁlled with a porous medium [11]. They showed that heat transfer and ﬂow ﬁeld are strongly depended on position of the plate and was found to be higher for 135̊ inclined plate.
Studies mentioned hitherto were based on Darcy model. Hsiao and Chen considered BrinkmanForchheimer extended Darcy flow model to study a steady natural convection about a heated corrugated plate inserted in an enclosed porous medium useful in casting engineering [12]. Sathiyamoorthy and Narasimman studied the effect of the length and position of the ﬁn on modiﬁcation of the recirculating cells and isotherms within the porous enclosure using the Brinkman–Forchheimerextended Darcy model [13]. Tasnim et al. considered nonDarcy model without the Forchheimer term to study free convection through a porous medium which was differentially heated from side with internal ﬂow obstructions [7]. Both the aforementioned studies revealed that average Nusselt number falls as the length of the ﬁn increases for all locations. The overall heat transfer rate can be managed with an appropriate selection of the ﬁn position and length. Fewer studies related to partitions in porous media are available in last five years. Handful among them are the works of Ziad who numerically investigated transient magnetohydrodynamic natural convection in partitioned square enclosure filled with a fluidsaturated porous medium with internal heat generation [14]. Rees et al. and Rees illustrated convection in a horizontal ﬂuidsaturated porous layer heated from below and divided it into a number of similar sublayers by impervious and inﬁnitesimally thin horizontal diathermal partitions [15, 16]. Chordiya and Sharma have reported the effects of several kinds of orientations of internal partitions within a differentially heated porous enclosure [17].
A new physics that is being brought out in the present study is the way of obstructing the convection current, precisely, by orienting the diathermal partition to isolate two specific regions on the porous enclosure. The temperature distribution in a differentially heated porous enclosure reveal that in the vicinity of lower half of hot face and upper half of cold face, there exists high temperature gradient which leads to higher convection heat transfer, particularly, in these two regions. This can also be noticed from the higher values of local Nusselt number along these parts of the wall. Thereby, in order to suppress the natural convection heat transfer within the enclosure as a whole, these two regions must be specifically dealt with. The current study introduces a novel way to isolate these regions by employing multiple inclined partitions across the lower half of hot face and upper half of cold face. Besides, to the authors’ best knowledge, such a study is yet to appear in the literature.
The objective of the present study is to suppress the natural convection fluid flow and heat transfer across a differentially heated square porous enclosure with multiple diathermal inclined partitions attached in it. The effect of position of partitions is studied and an optimum value of partition ratio has been found out that yields lowest average Nusselt number. Darcy model is considered with Boussinesq approximation to account for density variation. The whole point of such a model is sought to contribute in the redesign of a thermal insulations to effectively attenuate the convection heat transfer rate across it.
Figure 1 shows the schematic diagram of the model under study with geometric details and coordinate system. The fluidsaturated square porous enclosure has length L. Left and right walls are maintained at temperature T_{h} and T_{c} respectively, such that T_{h} > T_{c}. The top and bottom wall are insulated. Two diathermal partitions inclined at 135˚ are attached on southwest and northeast regions of the enclosure. The end points of either of the partitions are at distance D from origin and from farthest point from origin respectively. The study is confined to twodimensional flow, since in the postcritical range of Darcymodified Rayleigh number (Ra*) the convection pattern is expected to be twodimensional at first, Rees [17]. The flow is assumed to be steady, laminar and incompressible. The porous medium is assumed to be isotropic, homogenous and also that it is in local thermal equilibrium with the saturated fluid. Thermophysical properties of air are kept constant, except for density ρ. Density is assumed to change with temperature T according to Boussinesq approximation. A Darcy model is considered without the Forchheimer term, since the current study is focused primarily on analyzing a natural convection flow in a porous cavity and not on convective boundary layer flow over the surface of a body in a high porous media, as was discussed in detail by Bejan and Poulikakos [18].
Figure 1. Schematic diagram of the model
Mindful of these assumptions, the governing equations for conservation of mass, momentum and energy for solid and porous medium in dimensional form can be written as,
$∂u/∂x+∂v/∂y=0$ (1)
$u=K/μ ∂p/∂x$ (2)
$u=K/μ (∂p/∂x+ρg)$ (3)
$(ρc_p)f(u ∂T/∂x+v ∂T/∂y)=k_p(∂^2T)/(∂x^2)+(∂^2T)/(∂y^2)$ (4)
These equations are subjected to following boundary conditions,
Left wall (x=0, 0 ≤ y ≤ L): u=v=0, T=T_{h}
Right wall (x=L, 0 ≤ y ≤ L): u=v=0, T=T_{c}
Bottom and Top wall (0 ≤ x ≤ L, y=0, L): u=v=0 (5)
The matching conditions at diathermal partitions,
Lower Partition (x+y=D):u=v=0, $∂T⁄∂n^=∂T⁄∂n^+$
Upper Partition (x+y=2LD):u=v=0, $∂T⁄∂n^=∂T⁄∂n^+$ (6)
Following dimensionless variables are used to nondimensionalize above equations and boundary conditions,
$X=x/L; Y=y/L; U=u/(α_e⁄L); V=v/(α_e⁄L); θ=(TT_e)/(T_hT_c)$ (7)
Here, effective thermal conductivity is given by, $α_e=k _p/(ρc_p)_f$
Following dimensionless parameters are used while nondimensionalizing above equations and boundary conditions,
$Ra^* =(Kgβ∆TH)/(να_e); PR=D/L$ (8)
Velocities are represented in terms of stream function ($\psi$). The relation is given as,
$U=∂ψ/∂Y; V=∂ψ/∂X$ (9)
Using the above relations along with dimensionless variables and parameters, Eq. (1)(6) can be rewritten in nondimensional stream function and vorticity formulation as,
$(∂^2ψ)/(∂X^2)+(∂^2ψ)/(∂Y^2)=Ra^*(∂θ)/∂X$ (10)
$∂ψ/∂Y ∂θ/∂X∂ψ/∂X ∂θ/∂Y =∂^2θ/∂X^2+∂^2θ/∂Y^2 $ (11)
These equations are subjected to following boundary conditions,
Left wall (X=0, 0 ≤ Y ≤ 1):ψ=0,θ = 1
Right wall (X=1, 0 ≤ Y ≤ 1):ψ=0,θ = 0
Bottom and Top wall (0 ≤ X ≤ 1, Y = 0, 1):ψ = 0 (12)
The matching conditions at diathermal partitions,
Lower Partition (X+Y=PR):ψ=0 $∂θ⁄∂n^=∂θ)⁄∂n^+$
Upper Partition $(X+Y=2PR): ψ=0 $∂θ⁄∂n^=∂θ)⁄∂n^+$
The above equations are numerically solved and the results are analyzed qualitatively as well as quantitatively. Qualitative outcomes are visualized using streamlines and isotherms. On the other hand, quantitative outcome is analyzed by estimating the Nusselt number, local (Nu) as well as average (Nu_{avg}).
$Nu_h,c=∂θ/∂X_{X=0,1}$ (13)
$Nu_avg=∫_0^1Nu_h,c(Y)dY $ (14)
The coupled differential governing equations are converted into algebraic equations using Finite Difference Method (FDM) and solved by Successive Accelerated Replacement (SAR) scheme. Central differencing with secondorder accuracy is used to discretise governing equations, while second order forward and backward differencing is employed at the wall boundaries. The stream function and temperature have been solved using SAR scheme for all inner grid points. A suitable value of accelerating factor (ω) is chosen based on the accuracy obtained with regards to number of iterations required to attain convergence. Local Nusselt number along hot and cold wall are estimated and average Nusselt number is calculated by numerically integrating them using Simpson’s 1/3^{rd} rule.
3.1 SAR scheme
Sinha and Sharma, Mishra et al. and Chordiya and Sharma have illustrated the applicability of the SAR scheme for solving system of partial differential equations in the study of twodimensional natural convection fluid flow and heat transfer in porous media [1923]. The basic idea of this scheme is to guess the profile for each transport property that satisfies the boundary conditions. If λ is a transport property and the error in a typical governing equation at a grid point (i,j) at n^{th} iteration is $λ_{ij}^n$. Then, (n+1)^{th }approximation of the variable λ is obtained as,
$λ _{ij} ^{n+1}=λ_{ij}^n ω(λ_{ij}^n)/(∂λ_{ij}^n)⁄(∂λ_{ij}^n)$ (15)
The accelerating factor, ω, varies from 0 to 2. The criterion set for convergence of stream function at all inner grid points is described as below. Value of ε is the error tolerance limit with a small positive value.
$(∑_{i=2}^{imax1} ∑_{j=2}^{jmax1}λ_{ij}^{n+1}λ_{ij}^n)/(∑_{i=2}^{imax1} ∑_{j=2}^{jmax1}λ_{ij}^{n+1})<ε$ (16)
The important feature of this method is that the corrected value of the variable is immediately used upon becoming available. A numerical code based on this method is developed to solve the governing equations along with prescribed boundary conditions.
A numerical analysis has been performed to access the natural convection fluid flow and heat transfer in fluidsaturated porous enclosure with multiple inclined partitions for following parameters: 10^{2}≤Ra^{*}≤10^{3}; 0≤PR≤1. The influence of position of partition within the enclosure is evaluated by noticing streamlines and isotherms and convection heat transfer is estimated by calculating the value of average Nusselt number. An inhouse computational code is written to solve the current numerical problem. Pertaining to the present configuration, the present code has been subjected to two validation checks; by comparing it with studies of, first, natural convection in plain porous enclosure and second, natural convection in porous enclosure with single inclined partition by Varol et al. [11].
4.1 Validation of code for porous enclosure without partition
Table 1. Comparison of average Nusselt number with results from literature for PR=0 (Porous enclosure)
Authors 
Ra^{*}:10 
100 
1000 
Walker and Homsy [24] 

3.09 
12.9 
Bejan [25] 

4.20 
15.80 
Manole and Lage [26] 

3.11 
13.63 
Moya et al. [27] 
1.06 
2.80 

Baytas and Pop [28] 
1.07 
3.16 
14.06 
Present Study 
1.07 
3.16 
13.09 
Firstly, the code is compared against the benchmark solutions of Walker and Homsy, Bejan, Manole and Lage, Moya et al., Baytas and Pop [2226]. It is a classical natural convection problem in a differentially heated square porous cavity. Table 1 shows the comparison with similar parameters and boundary conditions which are: Left wall is hot, right wall is cold, top and bottom are insulated. Here, value of PR in code is set to 0.
4.2 Validation of code for porous enclosure with diathermal partition
Secondly, to authenticate the code against study of Varol et al. [11] which is based on natural convection in porous enclosure with diagonal partition. It consists of a single diagonal partition inclined at 135˚. Table 2 shows the comparison of results of present code with that of work in literature. Here, value of PR in code is set to 1.
The numerical comparison between the results obtained from present computational code agrees well in accordance with the results presented in literature. Thus, the code can be endorsed to study the problem stated in the current paper with greater assurance.
Table 2. Comparison of results with Varol et al. for PR=1 [11]
Ra* 
Varol et al. [11] 
Present Study 
100 
1.601 
1.633 
250 
2.581 
2.552 
500 
3.771 
3.831 
1000 
5.449 
5.731 
Figure 2. Streamlines (up) and isotherms (down) for square porous enclosure with: (a) no partition (PR=0); (b) inclined partitions (PR=0.5); Ra*=1000
The magnitude of stream function signifies the strength of convection fluid flow. The gradient of stream function is velocity of fluid. The value of maximum absolute stream function for Figure 2(a) and Figure 2(b) is 22.296 and 10.217 respectively. The drop in ψ_{max} is 54.18 % which is very significant, to say the least. Hence, the objective of suppressing the natural convection fluid flow is thus served very effectively by employing the inclined partition. Figure 3 shows the variation of local Nusselt number along hot and cold face for fluidsaturated porous enclosure with and without partitions for Ra*=1000 and PR=0 and 0.5.
It is evident, yet again, from the Figure 3 that there is a substantial drop of about 50% in the value of Nusselt number along the lower half of hot wall and upper half of cold wall, in the case of porous enclosure with multiple inclined partitions. The spike in the curve at Y=0.5 is due to the contact of partition since PR=0.5. There is crossflow of fluid across the partition; fluid below the partition is moving downwards as it is relatively cooler while fluid above the partition is moving upwards as it is relatively warmer. This creates a temperature difference across the partition and hence generates the spike in the curve at Y=0.5.
Figure 3. Local Nu on hot & cold wall for enclosure with (PR=0.5) and without (PR=0) inclined partitions. (Ra*=1000)
The discussion hitherto has shown that convective fluid flow has been effectively obstructed by the presence of inclined partition. The influence of multiple inclined partition on convection heat transfer can be manifested by observing the values of average Nu which is illustrated in Figure 4. The variation of average Nu is shown for values of Darcymodified Rayleigh number from 10^{2} to 10^{3} for porous enclosure with (PR=0.5) and without (PR=0) multiple inclined partitions.
It is clear from Figure 4 that average Nusselt number drops to about 50% due to presence of multiple inclined partition. As value of Ra* increases, the strength of buoyancy increases which aids in the ease of convection flow, hence the value of Nusselt number increases in both the cases. It is also noted that the drop in Nu values is slightly more for higher values of Ra*.
Figure 4. Average Nu for enclosure with (PR = 0.5) and without (PR = 0) inclined partitions
The purpose of this section was to access the effect of mere presence of the multiple inclined partition on convective fluid flow and heat transfer in fluidsaturated porous enclosure, which was performed by comparing streamlines, isotherms, maximum absolute stream function, local and average Nusselt number values with that of porous enclosure without any partitions. It is clear from all the aspects that employing the multiple inclined partition very substantially suppresses the convection fluid flow as well as heat transfer. Now, it becomes obligatory to evaluate the effect of position of partitions and to estimate the value of partition ratio PR for which least Nusselt number is obtained. This is discussed in detail in the forthcoming section.
4.4 Effect of position of partition on natural convection fluid flow and heat transfer
This section discusses the effect of position of multiple inclined partitions within a fluidsaturated square porous enclosure on fluid flow as well as heat transfer by noticing the streamlines, isotherms, maximum absolute stream function, local and average Nusselt number values for partition ratio PR from 0 to 1. Value of PR=0 indicates enclosure without any partition and PR=1 indicates a single diagonal partition. Figure 5 demonstrates streamlines and isotherms for fluidsaturated square porous enclosure with multiple inclined partitions for PR=0.5, 0.8 and 0.9 and Ra*=1000. The range of PR is chosen such that the deviations in streamlines and isotherms are clearly noticeable. For any further low values of PR, the deviations become very ambiguous to notice. However, the effect of lower values of PR is demonstrated later in discussion of its influence on Nusselt number.
Figure 5. Streamlines (up) and isotherms (down) for square porous enclosure with: (a) PR=0.5; (b) PR = 0.8; (b) PR=0.9. (Ra*=1000)
The central stagnant portion of streamlines are broken into multiple stagnant portions as was seen previously in Figure 2. In case (a), there are four small stagnant portions at four different locations of enclosure such that those in the middle block are comparatively larger in size. The major portion of enclosure is acquired by the central block. The variation of absolute stream function value in the central block is approximately 1≤ψ≤10 with maximum absolute stream function (ψ_{max}) value of 10.217. In case (b), the stagnant portions are broken into three equal sized portions at three different locations. The enclosure is acquired by all the blocks almost equally. The variation of absolute stream function value in the central and corner blocks is approximately 1≤ψ≤6 and 1≤ψ≤8 respectively with ψ_{max} value of 9.009. While that in case (c), out of three broken stagnant portions, the ones in the corner are comparatively larger in size. The major portion of enclosure is acquired by the corner blocks. The variation of absolute stream function value in the corner blocks is approximately 1≤ψ≤9 with ψ_{max} value of 10.0185. Thus, for PR=0.8 not only the variation in stream function values, but also the maximum absolute value of stream function is found to be the least of all. Temperature distribution is shown by isotherm contours. Here, the contour level of θ=0.5 is highlighted for comparison between the three cases. Let this contour be termed as contour ‘C_{0.5}’. The end points of C_{0.5 }are marked with a pentagram and a diamond shaped marker. As seen in case (a), C_{0.5}starts at (X, Y) coordinates of (0.01, 0) and ends at (0.98, 1) which is very close to hot and cold wall respectively. In the central portion, it can be seen to extend horizontally from very near of hot wall to very near of cold wall. Also, the temperature gradient is high near the hot wall (0.1≤X≤0.3 and 0.1≤Y≤0.5) and near the cold wall as well (0.7≤X≤0.9 and 0.5≤Y≤0.9). In case (b), C_{0.5}starts at (X, Y) coordinates of (0.1, 0) and ends at (0.91, 1) which is comparatively farther to hot and cold wall respectively, as compared to case (a). In the central portion, it can be seen to extend vertically by maintaining equal distance from hot and cold wall. Also, the temperature gradient is very low near the hot wall (0.1≤X≤0.5 and 0≤Y≤0.1) and near the cold wall as well (0.5≤X≤0.9 and 0.9≤Y≤1). In case (c), C_{0.5}starts at (X, Y) coordinates of (0.09, 0) and ends at (0.91, 1) which is comparatively farther to hot and cold wall respectively, as compared to case (a). In the central portion, it can be seen to extend vertically with slight inclination towards hot and cold wall. Also, the temperature gradient is low near the hot wall (0.1≤X ≤0.6 and 0≤Y≤0.2) and near the cold wall as well (0.6≤X≤0.9 and 0.8≤Y≤1) which is slightly greater than case (b) but lower than case (a). Therefore, the temperature gradient near the hot wall and cold wall is found to be the least for PR=0.8.
The above discussion shows that natural convection fluid flow is certainly influenced by the position of partition and maximum obstruction is obtained for PR=0.8. The effect of partition ratio on convective heat transfer can be manifested by noticing the local and average Nusselt number. Nusselt number signifies the strength of convective heat transfer relative to conductive heat transfer. Figure 6 shows the variation of local Nusselt number along hot wall (Figure 6a) and cold wall (Figure 6b) for fluidsaturated square porous enclosure with PR=0.5, 0.8 and 0.9 and Ra*=1000.
(a)
(b)
Figure 6. Local Nusselt number on (a) hot and (b) cold wall for enclosure with multiple inclined partitions. (Ra*=1000)
The spike in the curves of Figure 6 are due to presence of edges of partition at that particular enclosure height. The spikes are exactly seen to occur at Y=0.5, 0.8 and 0.9 which are the values of PR. It is clear from the graph that lowest value of local Nusselt number is obtained for PR=0.8. The difference can be noticed over the lower half of hot face and upper half of cold face where value of Nusselt number significantly high. However, the relative values of Nu are lowest for PR=0.8. The reason can be given on the basis of distribution of isotherms previously explained. The temperature gradient near hot wall and cold wall for PR=0.8 is relatively lower than that for other two cases. Figure 7 depicts the variation for average Nusselt number against a wide range of PR* plotted for Ra=100, 250, 500 and 1000.
(a)
(b)
Figure 7. (a)Variation of Nu_{avg}; (b) Percent reduction in Nu_{avg} with varying partition ratio for different values of Ra*
A comprehensive summary of above discussions about convective fluid flow and heat transfer in fluidsaturated square porous enclosure with multiple inclined partitions can be viewed in Figure 7. It can be clearly seen from Figure 7(a) that average Nusselt number increases with increase in value of Ra* for all values of PR. Further, the least value of average Nusselt number is obtained for PR=0.8 irrespective of the value of Ra*. The percent decrease in convective heat transfer is up to 71 % when PR=0.8. This is a significantly high reduction in the value of Nu_{avg}.
The present numerical study was based on suppressing the convective fluid flow and heat transfer in a fluidsaturated square porous enclosure with multiple inclined diathermal partitions attached to it. The effect of mere presence of partitions and its position in the enclosure was studied by evaluating streamlines, isotherms, maximum absolute stream function, local and average Nusselt number values for Darcymodified Rayleigh number (Ra*) 100 to 1000 and partition ratio (PR) 0 to 1. Following conclusions may be drawn from the analysis performed above,
1. To suppress the natural convection fluid flow and heat transfer, the regions near the vicinity of lower half of hot face and upper half of cold face must be well obstructed or isolated from the rest of enclosure on account of high temperature gradient generating in these regions.
2. Mere presence of partition brings about substantial reduction in values of average Nusselt number.
3. The reduction in Nusselt number is more pronounced for higher values of DarcyModified Rayleigh number.
4. The maximum in Nusselt number is about 70% obtained for PR=0.8 for higher values of Darcymodified Rayleigh number.
AR 
Aspect ratio 
c_{p} 
specific heat, J. kg^{1}. K^{1} 
D 
Distance of partition from origin, m 
g k 
gravitational acceleration, m.s^{2} thermal conductivity, W.m^{1}. K^{1} 
K 
Permeability of porous medium (m^{2}) 
L 
Length of enclosure, m 
Nu 
Nusselt number 
PR 
Partition Ratio 
Ra* 
Darcy modified Rayleigh number 
T 
Temperature, K 
U, V 
Dimensionless x and y velocity 
u, v 
x and y direction velocity (m^{2}/s) 
x, y 
Horizontal and vertical directions (m) 
X, Y 
Dimensionless horizontal and vertical directions 
Greek symbols 

a 
thermal diffusivity, m^{2}. s^{1} 
b 
thermal expansion coefficient, K^{1} 
Ɵ 
dimensionless temperature 
µ 
dynamic viscosity, kg. m^{1}.s^{1} 
ν 
Kinematic viscosity (m^{2}/s) 
ρ 
Density (kg/m^{3}) 
ψ 
Dimensionless stream function 
ω 
Acceleration factor 
Subscripts 

avg 
average 
c 
cold 
e 
effective 
h 
hot 
max 
maximum 
p 
porous 
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