Heat transfer reduction between two finite concentric cylinders using radiation shields; Experimental and numerical studies

International Communications in Heat and Mass Transfer 65 (2015) 94–102

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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Heat transfer reduction between two finite concentric cylinders using radiation shields; Experimental and numerical studies☆ Mohammad Sadegh Motaghedi Barforoush ⁎, Seyfolah Saedodin Faculty of Mechanical Engineering, Semnan University, Semnan, Iran

a r t i c l e

i n f o

Available online 20 April 2015 Keywords: Natural convection Radiation Experimental measurement Radiation shields Numerical simulation

a b s t r a c t Energy consumption and its efficient utilization are two important factors of thermal systems. This work concerns with numerical and experimental studies about the surface radiation and natural convection effects on the heat transfer and flow field between two finite concentric cylinders, using one radiation shield between them. This study reveals material and geometric effects of the radiation shield on heat losses from two concentric cylinders enclosure at different temperatures and enclosure pressures. The enclosure consists of two concentric cylinders with hotter inner cylinder and colder outer one. The radiation shield with three different materials (aluminum, copper and steel) is inserted between the cylinders at two different radial positions. Validations are carried out for the temperature of the radiation shield with experimental data and numerical ones. After validation, forty eight different experiments and numerical simulations are carried out by varying the inner cylinder temperature between 373 K and 673 K at two enclosure pressures of 0.2 and 1.0 atm, corresponding to three different materials as radiation shields. The outer cylinder temperature from experiments is used in numerical simulations. The results show that the enclosure pressure and radiation shield emissivity together are responsible for reduction in the total heat loss from the inner cylinder. It was also found that among the three considered materials as radiation shields, copper is the most effective one to reduce the heat loss. In a specific case, the total heat loss with copper radiation shield was 14.99% and 57.7% lower than steel and aluminum shields, respectively. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Recently challenging issues regarding energy consumption in domestic and industrial facilities as well as their negative footprint of the nature such as CO2 emission have forced humankind to meticulously analyze and optimize thermal systems from heat transfer point of view. To this end, many studied have been conducted in various thermal systems considering conjugate convection and radiation heat transfer [1–4]. Conjugate radiation and convection heat transfer has attracted considerable attention due to its wide range of scientific and industrial applications such as nuclear reactors [5,6], process industry [7,8], thermal energy storage systems [9], cooling of electronic components [10,11], heat exchangers [12–14], thermal processing of moving plates [15], radiative cooling systems [16] and even for building heating where the radiation and natural convection have considerable effect at the room temperature [17–19]. These studies could be categorized into two types. The first method is to incorporate the convection and radiation effects of the boundary conditions into the energy equation within the ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (M.S.M. Barforoush).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.04.014 0735-1933/© 2015 Elsevier Ltd. All rights reserved.

solid material. Using this approach and assuming unidirectional heat flow within the material, one is able to model a thermal process with ordinary differential equations [12–16]. This method is mainly used to obtain the temperature distribution within solid materials. The other approach is to solve the coupled partial differential equations of continuity, momentum and energy together, by assuming the effects of convection and radiation [5,7–11,17–19]. In this approach, in most cases, computational fluid dynamics should be employed to tackle the coupled equations. Although this method is more challenging, it can be used to address temperature fields within both fluid and solid materials. In many articles, the effect of radiation on the temperature and flow fields has been neglected in enclosures [20,21]. Despite numerous studies for natural convection, few of heat transfer investigations have been concentrated on the radiation mode. A meticulous investigation through available literature about conjugate convection and radiation shows the influential effect of radiation on the thermal behavior of a system at high temperatures. Kuznetsov and Sheremet [22], and Martyushev and Sheremet [23] investigated the effect of Grashof number, transient factor, optical thickness and solid walls thermal conductivity on the local thermo-hydrodynamic characteristics and integral parameters in an enclosure having finite thickness conducting walls and local heating at the bottom of the cavity. Sharma et al. [24] studied the conjugate turbulent natural convection and surface radiation

M.S.M. Barforoush, S. Saedodin / International Communications in Heat and Mass Transfer 65 (2015) 94–102

Nomenclature a g Gr I Lc n p q″ R !0 s ! s T V

Absorption coefficient Gravitational acceleration m/s2 Grashof number Radiation intensity W/sr Characteristic length m Refractive index Pressure atm Heat flux loss Radius m Scattering direction vector m Direction vector m Temperature K Velocity m/s

Greek letter β Volume expansion coefficient 1/K ε Emissivity Φ Scattering phase function μ Dynamic viscosity Pa-s ν Kinematic viscosity m2/s ρ Density kg/m3 σs Scattering coefficient Ω′ Solid angle sr ζ Volume viscosity coefficient Pa-s Subscripts al Aluminum s Steel cu Copper in Inner cylinder sh Radiation shield out Outer cylinder tot Total con Convection rad Radiation enc Enclosure

in rectangular enclosures with various aspect ratios applying finite volume method. The enclosure is heated from the bottom and cooled from other walls, typically encountered in liquid metal fast breeder reactor subsystems. They developed the correlation for the mean convection Nusselt number in terms of Rayleigh number and aspect ratio, which is proposed for design purposes. Rao et al. [25] paid special attention to study the effectiveness considering different values of fin heights, emissivity, and number of fins in a horizontal fin array by natural convection and radiation. Rabhi et al. [26] numerically studied the effects of surface radiation and number of partitions on the heat transfer and flow structures in a rectangular enclosure, inclined by 45° with respect to the horizontal plane. They found that the total heat transfer in the enclosure is increased under thermal radiation heat flux and reduced significantly with increasing the number of partitions. Nouanégué and Bilgen [27] have numerically investigated conjugate conduction, convection and radiation heat transfer in solar chimney systems. Sun et al. [28] studied the effects of radiation interchanges amongst surfaces on the transition from steady, symmetric flows about the cavity centerline to complex periodic flow using the control volume code ANSYS FLUENT. Premachandran and Balaji [29] studied a numerical investigation of conjugate mixed convection from vertical channels with four discrete protruding heat sources mounted on the right side wall of the channel. Special attention has been paid to

95

understand the effects of buoyancy and radiation heat transfer on flow and heat transfer characteristics of the channel. More recently, Saravanan and Sivaraj [30] considered a theoretical study to understand the interaction of surface radiation and natural convection in an air filled cavity with a centrally placed thin heated plate. The vertical walls of the cavity are cooled while the horizontal ones are insulated. Montiel-Gonzalez et al. [31] validated their experimental data regarding the conjugate natural convection with surface radiation in an open cavity. They used finite volume method and the SIMPLEC algorithm to numerically investigate the problem. They showed that for most of the probed temperatures, the deviation between the two methods is less than 10%. Apart from direct approaches, the inverse methods have been employed to find various characteristics of convective-radiative systems. Moghadassian and Kowsary [32] interestingly investigated the strength of heaters in a 2D enclosure to produce the desired temperature and heat flux distribution. The Levenberg–Marquardt algorithm was chosen to perform the iterative search procedure. De Faoite et al. [33] have opted in favor of inverse heat flux estimation of a plasma discharge tube using thermocouple data and a radiation boundary condition. When the atmosphere pressure is considerably low, the convection heat transfer can be neglected within a thermal system [34]. Therefore, the thermal radiation dominates the heat transfer within the system. If one can control heat radiation in this situation, high-temperature insulated structure can be achieved [35]. Under high-vacuum conditions and using radiation shields to reduce the heat transfer [36,37], few studies have been done. In this situation it is possible to introduce a simplifying approach for calculating the radiant energy using the concept of net radiation heat transfer, which provides an easy way for solving a variety of situations. However, less attention has been given to the situation when a low-vacuum is available between two or more radiation shields. Application of multilayer composite materials as radiation shields has been discussed in an interesting patent by Bowers et al. [38]. Miyakita et al. [39] have developed a multilayer material with non-interlayer-contact spacer for space cryogenic missions. They also modeled temperature filed within different layers of the developed blanket. Good agreement between the measured temperature and the numerical results for the outermost layer was achieved. Although it is possible to reduce heat transfer using radiation shields between two surfaces, it is also possible to increase the heat transfer rate by using further number of radiation shields [40]. Therefore, regarding the number of radiation shields, optimization may be needed in a system [40]. The main aim of this work is to present an experimental study together with numerical validation, to reduce radiation and free convection heat transfer by inserting shields between two finite concentric cylinders at different enclosure pressures. For this purpose, two different radial positions between two cylinders with three different materials (aluminum, copper, and steel) as radiation shields have been selected. Due to the direct effect of emissivity on the radiation, temperature-dependent emissivity was considered for all materials. In the numerical modeling, a conjugate analysis is carried out in which the mass, momentum and energy balance equations in the enclosure are concurrently solved. Furthermore, it was assumed that the density of the air is a function of temperature and pressure following the ideal gas law. This study helps researchers and industrial communities to have better insight regarding the heat transfer reduction when using one or two radiation shields between two concentric cylindrical geometries. 2. Experimental setup Schematic of the experimental setup is illustrated in Fig. 1, which consists of the cylindrical enclosure, temperature sensors, heaters, a vacuum pump, a vacuum gauge, and a data acquisition system. Enclosure consists of two concentric vertical cylinders filled with air

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Fig. 1. Schematic of experimental setup.

and another thin hollow cylinder as a radiation shield which is placed between them. Fig. 2 shows the main cylindrical enclosure and the used radiation shields. Inner and outer cylinders' material is steel. Also aluminum, copper and steel plates with thickness of 2 mm are used as radiation shield for different experiments. Experiments are carried out by changing the inner cylinder temperature, radiation shield materials and/or diameters, and the enclosure air pressure. Using different inputs for power sources and fixing the temperature of the inner cylinder, the temperature of the radiation shields and the outer cylinder can be measured with K-type thermocouples. It has been stated that the thermocouples provide accuracy of about 2 K. The temperature of the inner cylinder is controlled in the range of 373–673 K during different experiments. Thermocouples are mounted at the central location of each cylinder. The upper and lower base sides of the structure were insolated, although in high temperatures they were not totally effective. The insolation of the base sides and the high aspect ratio of height to radius of the cylindrical system would allow us to assume negligible heat transfer

from the bottom and top surfaces of the structure. The system operates in a room with ambient temperature of 303 K. 3. Numerical modeling To construct the numerical analysis, a three-dimensional model of two finite concentric cylinders with two different positions for radiation shields is prepared. For solving the continuity, momentum and energy equations, FLUENT unsteady solver has been used and SIMPLE algorithm is imposed for coupling pressure and velocity fields. A time step of 0.5 s has been used. The simulation is continued to reach a steady value for the temperature of the radiation shield and its neighbor fluid flow. Obtaining this condition shows that the steady state is achieved. It is worthy to note that by using steady solver directly, the convergence of the velocity field cannot be ensured, and therefore we are urged to employ the unsteady solver. As mentioned in Section 2, the upper and lower surfaces are assumed isolated. The temperature of the outermost

Fig. 2. Experimental facilities, (a) cylindrical enclosure, (b) radiation shields.

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cylinder is read from the experiments and included into the modeling. The differential equation of continuity for a compressible fluid can be written as follows:
! ∂ρ þ ∇: ρ V ¼ 0: ∂t

ð1Þ

Also momentum equations are presented in their deferential form as: ! !
∂V ! ! μ 
! 2! ! ρ þ V :∇ V ¼ ρ g −∇p þ μ∇ V þ ζ þ ∇ ∇: V : 3 ∂t

ð2Þ

! where V is the velocity vector in (r, θ, z) directions, p indicates the pressure, ρ refers to the density, μ is the dynamic viscosity, and ζ stands for the volume viscosity coefficient. Discrete ordinate model (DO) is selected for radiation modeling, which solves the radiation transfer equation (RTE) for a finite number of discrete solid angles, each associated with a ! vector direction s fixed in the global system. The DO model considers ! ! RTE at position r in the direction s as follows: 4
  

2 σT !! !! ! ∇: I r ; s s ¼ −ða þ σ s ÞI r ; s þ an π Z4π
 σ 0 ! !0 ! !0 þ s Ið r ; s ÞΦ s ; s dΩ 4π

ð3Þ

! !0 where I is radiation intensity, s is direction vector, s is scattering direction vector, a is absorption coefficient, σs is scattering coefficient, Φ is scattering phase function, Ω′ is solid angle and the n expresses refractive index. The flow regime in natural convection is governed by the Grashof number in an enclosure, which represents the ratio of the buoyancy force to the viscous force acting on the fluid [41]: gβðT h −T c ÞL3c ν2

ð4Þ

where, g is gravitational acceleration, β indicates coefficient of volume expansion, Th and Tc are the temperature of hot and cold surfaces, Lc is the characteristic length of the geometry (the distance between the hot and cold surfaces), and ν is the kinematic viscosity of the fluid. In the current work, it is assumed that the surface emissivity of the used material is a function of temperature and other thermo-physical properties are independent of temperature. The fluid is viscous, heatconducting, Newtonian, and the ideal gas law is valid for correlating the pressure, temperature and density to each other. Also, it is assumed that the flow is laminar, which is conducted from comparing the Grashof number with its critical value (about 109). The temperature-dependent emissivity of the wall's material was extracted from trustworthy reference [42] (Fig. 3). From this plot the best curve has been fitted and the following relationship is achieved. The best formulas regarding emissivity versus temperature for aluminum, copper and steel are: −11 3

ε ðT Þal ¼ 4  10

−8 2

T −2  10

T −0:0004T þ 0:943

εðT Þcu ¼ 0:0002T−0:0035 −11 3

ε ðT Þs ¼ −3  10

T þ 2  10

[43] concluded that the vertical flat plate solution is acceptable for air when ðGrD PrÞ

0

Gr ¼

Fig. 3. Temperature dependent emissivity of aluminum, steel and copper [42].

1=4

D N38: L

ð8Þ

This means that the diameter of the cylinder is sufficiently large so that the curvature effects are negligible. The dimensions and operating conditions of this work are satisfied the above inequality. Hence, for calculating the convection coefficient of outer cylinder with the periphery air, Churchill and Chu's correlation [44] for vertical plate was used: 0:670Ra1=4 Nu ¼ 0:68 þ  4=9 1 þ ð0:492= PrÞ9=16

9

f or RaL b10 :

4. Results and discussions To locate radiation shield between two cylinders two different positions are examined. For radiation shield's material three different materials, namely, aluminum, copper and steel, are used. All numerical and experimental studies are carried out using two different pressures within the enclosure, namely, 1 atm and 0.2 atm. The geometrical characteristics of main cylinders by referring to Fig. 5 are reported in Table 1. Mesh independency for the numerical simulation was investigated

ð5Þ ð6Þ

−7 2

−5

T þ 8  10

T þ 0:0794

ð7Þ

Fig. 4 shows thermal boundary conditions for this problem. In addition, the no-slip boundary condition is imposed on all solid walls. To have less time consuming computations, only inside of the enclosure was modeled, and available correlation was used to estimate the convective heat coefficient on the external side of enclosure. In the laminar free convection problems around a vertical cylinder, Sparrow and Gregg

ð9Þ

Fig. 4. Boundary conditions of the problem.

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Fig. 5. Top view configurations of main cylinders and radiation shield (not to scale).

and finally unstructured mesh of 220,729 and 210,332 cells has been chosen for the radiation shield radius of 13 and 17 cm, respectively. To validate the numerical model, the first radiation shield's temperature (R = 13 cm) from the simulation is compared with the probed temperature from the experimental study. The comparison shows a good agreement between the numerical and experimental data, which are illustrated in Fig. 6. Nevertheless, a slight deviation can be observed, which may arise from the adiabatic assumption for the upper and lower annular plates within the numerical procedure, which is not compatible very well with the experiments. Other reasons that may account for this deviation could be the isothermal assumption for the inner cylinder, and differences between the thermophysical properties of the materials used in simulations and in experiments. This level of agreement between the numerical data and experiments is valid for the second position of the radiation shield (R = 17 cm), and therefore is not repeated here. Temperature, axial velocity and isothermal-surfaces at two different pressures of enclosure for aluminum radiation shield with R = 13 cm are illustrated in Fig. 7. As it can be seen from this figure, at low pressure the air temperature is high, which indicates the lower heat capacity (mcp) of enclosed air. The hotter air is located at the top of the enclosure, because of its lower density, which yields to the non-uniform temperature distribution at the radiation shield wall (Fig. 8). In this figure the effect of upper and lower walls on the temperature distribution of radiation shield wall is visible. Since the heat fluxes on these two walls are not ideally zero, the radiations shield temperature has slight slope near to the upper and lower surfaces. Also, the volume expansion coefficient β of an ideal gas (P = ρRT) at a temperature T is equivalent to the inverse of the temperature: β¼

1 : T

ð10Þ

Hence, for an ideal gas, at the same temperature by reduction of the enclosure pressure the volume expansion coefficient remains constant.

Table 1 Main geometrical characteristics of concentric finite cylinders, position of shields and length of cylinders. Radius of the inner cylinder (R1)

Radius of the outer cylinder (R4)

Cylinders length (L)

Radius of the first radiation shield (R2)

Radius of the second radiation shield (R3)

8 cm

25 cm

60 cm

13 cm

17 cm

By assuming the constant viscosity for the fluid, the ratio of Gr number for different pressure could be obtained as follows: Gr p1 Gr p2

¼ ¼

ν p2 ν p1 μ p2 μ p1

!2 !2

ρp1 ρp2

!2 ideal gas

¼

!2    2 P 1 2 Gr p1 P ⇒ ∝ 1 : μ p1 P2 Gr p2 P2 μ p2

ð11Þ

Hence, the Gr number ratio is proportional with the square of pressure ratio. Therefore, when the pressure is reduced from 1 atm to 0.2 atm, the Gr number is reduced by the ratio of 25, which indicates the lower buoyancy force with respect to the viscous force at lower pressure and yields to the low fluctuation of flow field at this low pressure flow field. This phenomenon is clear by comparing the isothermalsurface temperature in Fig. 7. The effects of locating a radiation shield at different radius are detailed in Tables 2–4. Table 2 reports the effect of aluminum shield on the heat transfer reduction from inner cylinder at four different temperatures. Radiation shield temperatures and heat losses from inner cylinder due to radiation and convection in the various shields positions are presented. With reference to this table, the total heat losses for each case decrease by reducing the enclosure pressure, because of weakening the convection effects which consequently lead to increase the shield's temperature. Thus, at lower pressures, radiation heat losses slightly reduce because of increasing the radiation shield temperature. Also, the ability of radiation electromagnetic waves to travel through the vacuum media makes thermal radiation the main mode of heat transfer in low pressures. As another point, Table 2 expresses the heat loss increment with increasing the diameter of radiation shield. It was found that for aluminum radiation shield by setting the inner cylinder wall temperature at 673.15 K and fixing enclosure pressure at 1 atm, and changing diameter of radiation shield from 26 to 34 cm, losses ascend from 2034.3048 to 2181.1067 (W/m2), addressing 7.21 percentage of growth. Furthermore, by reducing the enclosure pressure from 1 atm to 0.2 atm, the convection heat loss is reduced by almost 63% in all cases, which shows a significant effect of enclosure pressure on the convection heat loss. Similar behavior of reduction in heat losses at various temperature and pressure has been achieved for the case of copper and steel radiation shields. Tables 3 and 4 report these results for copper and steel radiation shields. Comparing the results of these tables with Table 2 indicates that the lower emissivity of steel and copper compared to the aluminum decreases the total heat loss by alleviating the amount

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99

Fig. 6. The comparison between numerical and experimental results for radiation shield at R = 13 cm for (a) aluminum, (b) copper and (c) steel radiation shields.

of radiation loss. Hence, using copper and steel is much better than aluminum. The amount of convection heat loss from the inner cylinder is intuitively related to the radiation shield diameter. Accordingly, at the same inner cylinder temperature, enclosure pressure, and radiation

shield diameter, changing the shield's material does not significantly affect the amount of convection heat loss and it remains almost constant (according to Tables 2, 3 and 4). Hence, the enclosure pressure affects the convection heat loss, and the shield emissivity impacts on the

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Fig. 7. Temperature distribution and axial velocity for Al radiation shield.

radiation heat loss. Therefore, both heat loss mechanisms will be responsible for reducing the total heat loss from inner cylinder. 5. Conclusions This work investigated a numerical and experimental study about the effect of a cylindrical radiation shields with temperature-dependent emissivity within cylindrical enclosures to reduce heat loss. Various parameters' effects such as different materials for the radiation shields, position of the radiation shield, and pressure of the enclosure, on the heat transfer and flow field between two finite concentric inner and outer

cylinders were discussed. According to the results radiation heat losses ascend with emissivity increment, and do not depend significantly on the enclosure pressure. Nevertheless, the total heat losses for each case decrease by reducing the enclosure pressure, because of weakening the convection effects. Obtained results for copper show that the copper is a more appropriate choice to be used as a radiation shield compared to aluminum and steel, which is caused by its lower emissivity. For inner cylinder temperature of 673.15 K, enclosure pressure of 1 atm, and the radiation shield diameter of 26 cm the total heat loss is found to be 1289.818 (w/m2) when the shield is copper which is 14.99% and 57.7% lower than when steel and aluminum shields are used, respectively.

Fig. 8. Non-uniform distribution of temperature at aluminum radiation shield wall with D = 26 [cm].

M.S.M. Barforoush, S. Saedodin / International Communications in Heat and Mass Transfer 65 (2015) 94–102 Table 2 Heat loss reduction at different enclosure pressures and radiation shield diameters for aluminum. Tin [K]

Penc [atm]

Rsh [cm]

q″tot [w/m2]

q″rad [w/m2]

q″con [w/m2]

Tsh [K]

373.15

1.0

26 34 26 34 26 34 26 34 26 34 26 34 26 34 26 34

216.09853 244.22089 114.06107 127.57000 601.53503 650.1479 388.9346 416.6426 1181.7235 1250.731 863.76672 924.7659 2034.3048 2181.1067 1673.3564 1780.501

64.385834 67.682869 59.522705 61.58105 274.52576 283.1432 264.53052 272.1714 746.68976 770.1997 706.79474 735.439 1590.2397 1669.536 1512.2854 1580.984

151.7127 176.538 54.53837 65.98895 327.0093 367.0047 124.4041 144.4712 435.0337 480.5313 156.972 189.3269 444.0651 511.5707 161.071 199.517

315.84003 315.4229 324.80963 323.8474 350.80865 354.808 366.66977 366.4147 394.5195 403.8417 424.5514 423.7743 461.1993 471.0356 496.9335 497.8156

0.2 473.15

1.0 0.2

573.15

1.0 0.2

673.15

1.0 0.2

Table 3 Heat loss reduction at different enclosure pressures and radiation shield diameters for copper. Tin [K]

Penc [atm]

Rsh [cm]

q″tot [w/m2]

q″rad [w/m2]

q″con [w/m2]

Tsh [K]

373.15

1.0

26 34 26 34 26 34 26 34 26 34 26 34 26 34 26 34

181.9501 217.50929 78.53975 93.63763 470.9331 536.0834 248.6193 285.7342 836.2026 938.8344 496.7453 588.2612 1289.818 1506.019 909.6204 1077.518

29.65564 36.87663 26.793882 32.6534 127.6889 153.9537 122.9068 145.7381 353.78348 420.0427 336.7225 400.9641 784.2385 931.7805 746.558 882.5562

152.2945 180.6327 51.74587 60.98423 343.2442 382.1297 125.7125 139.9961 482.4191 518.7917 160.0228 187.2971 505.5795 574.2385 163.0624 194.9618

316.2734 315.6176 327.0296 325.2626 348.951 352.4652 366.5257 366.6161 387.4615 396.2674 420.0461 419.4247 449.3167 457.0383 489.1246 491.1222

0.2 473.15

1.0 0.2

573.15

1.0 0.2

673.15

1.0 0.2

These graphs and data help many industries to have better insight regarding the effects of radiation shields in cylindrical enclosures. The provided data bring about useful information regarding the differences of these three materials as radiation shields when heat loss is an important parameter, besides expenses of choosing different materials.

Table 4 Heat loss reduction at different enclosure pressures and radiation shield diameters for steel. Tin [K]

Penc [atm]

Rsh [cm]

q″tot [w/m2]

q″rad [w/m2]

q″con [w/m2]

Tsh [K]

373.15

1.0

26 34 26 34 26 34 26 34 26 34 26 34 26 34 26 34

195.877 224.1395 88.72557 103.9361 514.6136 571.6534 286.9645 325.534 940.2717 1045.2412 601.5328 689.6979 1483.125 1709.504 1120.823 1291.417

41.40969 47.57751 37.02937 42.46309 174.4819 197.1599 165.511 186.8683 474.4388 533.1396 448.378 507.0693 1018.807 1169.526 965.8142 1101.479

154.4673 176.562 51.6962 61.47301 340.1317 374.4935 121.4535 138.6657 465.8329 512.1016 153.1548 182.6286 464.318 539.978 155.0088 189.938

315.7177 315.8299 326.9408 325.1623 349.1311 354.0012 368.542 368.0114 389.6591 396.5295 422.3587 423.494 455.0599 464.0577 495.1929 496.6849

0.2 473.15

1.0 0.2

573.15

1.0 0.2

673.15

1.0 0.2

101

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