Parametric evaluation of a heat pipe-radiator assembly for nuclear space power systems

Thermal Science and Engineering Progress 13 (2019) 100368

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Parametric evaluation of a heat pipe-radiator assembly for nuclear space power systems Luis F.R. Romanoa, Guilherme B. Ribeiroa,b, a b

T

⁎

Aeronautics Institute of Technology, São José dos Campos, Brazil Institute for Advanced Studies, São José dos Campos, Brazil

A R T I C LE I N FO

A B S T R A C T

Keywords: Heat pipe Radiator Brayton Nuclear Space

The research and technological development towards compact energy conversion systems for space applications allow the emergence of new mission possibilities, especially those directed for deep space explorations. Besides the final efficiency, the most crucial factor of an energy conversion system for nuclear propulsion purposes is the total mass and size of the system. Considering the Closed Brayton Cycle (CBC) as the energy conversion system for a nuclear power system, a numerical analysis was carried out in order to predict the thermal performance of cold side of the system (i.e., heat pipes and radiator) for initial design purposes. The complete space heat piperadiator array was discretized in control volumes where a variation of geometrical parameters was included, resulting in a stepped trapezoidal-shaped radiator (RAD) as output. The heat capacity was limited by the geometry of each panel section, being its heat pipe modeled to fit the given geometry and verified against operational limits. The proposed design-based model considered a physical and thermal coupling with temperature drops along the heat pipe (HP) axial direction, the radiator panel surface, and the cold heat exchanger duct, providing reasonable global parameters to aid the design considering mass and size optimization of a heat piperadiator assembly. The number of heat pipes and the total heat pipe-radiator assembly mass and length were evaluated for different heat pipe spacing, heat transfer rate, cold heat exchanger (CHE) inlet temperatures, and radiation shield shadow angles. It was observed a point of minimum HP-RAD mass and length when the heat pipe spacing and CHE inlet temperature are varied, for a given heat transfer rate and shadow angle.

1. Introduction Energy conversion systems are a vital part of space power systems. During a space mission, the proper use of electricity is required and plays an important role on mission success. Due to this reason, highpower density and a full power availability regardless of the ambient condition are crucial aspects which make nuclear energy the chosen option for the space exploration [2,22]. When compared to ground-based power systems, space power systems present a set of novel aspects, such as low power level, lightweight and a radiant heat rejection from the energy conversion scheme. For propulsion purposes, lighter energy conversion systems for the same power output enables more available mass to be used as the payload. Furthermore, the high cost involved in sending power systems into space by current launch technologies makes their mass and size critical factors that can constrain the feasibility of nuclear power systems for space propulsion [4]. Therefore, special attention should be focused on these aspects, since they have a high influence on the viability of the

⁎

use of such power systems. As any other conversion system, an amount of heat must be extracted by the low-temperature heat sink. In space, heat rejection is performed by a set of radiators (RAD) attached to the cold side of the energy conversion system by heat pipes (HP). Thus, space radiators operate assisted by a set of heat pipes which are responsible for extracting heat from the conversion system working fluid to radiant panels, where rejection to the external ambient occurs. As pointed out by other studies [24,33], the heat pipe-radiator assembly commonly has the highest mass contribution on space power systems. Hence, it is a critical component which must be designed taking into account its mass and heat extraction capacity. Fig. 1 displays a conceptual craft presenting two full radiator sets. As can be seen, the heat rejection system, which consists of the heat pipe-radiator assembly, is the largest component of the spacecraft. For this kind of application, proper protection of the components of the energy conversion system and payload against radiation coming from nuclear fission products is required. Due to this reason, a shielding element separates physically the nuclear core from

Corresponding author at: Aeronautics Institute of Technology, São José dos Campos, Brazil. E-mail address: [email protected] (G.B. Ribeiro).

https://doi.org/10.1016/j.tsep.2019.100368 Received 30 December 2018; Received in revised form 8 May 2019; Accepted 10 June 2019 2451-9049/ © 2019 Elsevier Ltd. All rights reserved.

Thermal Science and Engineering Progress 13 (2019) 100368

L.F.R. Romano and G.B. Ribeiro

Nomenclature

A C cp D d F f G h h K k L m m ṁ MW N P Per Q̇

R r Re S T t V w v

Subscripts

Area, m2 Constant, Specific heat, J/kg.K Diameter, m Depth, m Form factor, Friction factor, Thermal conductance, W/K Specific enthalpy, J/kg Heat transfer coefficient, W/m2 K Permeability, m2 Thermal conductivity, W/m K Length, m Mass, kg Fin parameter, Mass flow rate, kg/s Molecular weight, Number, Pressure, Pa Perimeter, m Heat transfer rate, W Ideal gas constant, J/kg K Radius, m Reynolds number, Number of radiator set Temperature, K Thickness, m Volume, m3 Width, m Specific volume, m3/kg

b boil c cap che con cor e ef end ent ext fin gr h hp i inf i nt l lv max mix n rad sat sink son sp ss ti tot v va visc

Greek Symbols

Δ π μ ∊ σ σ η θ ρ ε σ

Difference, Number Pi Dynamic viscosity, Pa.s Porosity, Tensile stress, Pa Stefan-Boltzmann constant Efficiency, Angle, ° Density, kg/ m3 Emissivity, Surface tension, N/m

Base Boiling Condenser Capillary structure Cold heat exchanger Container Corrected fin Evaporator Effective Heat pipe end Entrainment External Fin Groove Hydraulic Heat pipe Interface Bulk fluid Internal Liquid phase Liquid-vapor Maximum Mixture Nucleation Radiator Saturation Heat sink Sonic Space Radiation shadow shield Titanium Total Vapor Vapor duct Viscous

Abbreviations

CBC CHE CV HP RAD

Closed Brayton Cycle Cold Heat Exchanger Control volume Heat Pipe Radiator

planar radiative heat transfer characterized the radiator. The temperature drop along the Cold Heat Exchanger (CHE) of the Brayton cycle was neglected in this analysis. Thus, a fixed CHE temperature was applied in the radiator. Vlassov et al. [36] presented an optimization of a heat pipe-radiator assembly, where the internal HP geometry and the dimensions of the saddle and radiator panel were the chosen variables under analysis. The Generalized Extremal Optimization technique was applied in this study and two different fluids were considered in the HP: acetone and ammonia. Zhang et al. [39,40] provided a thermal analysis of the space nuclear reactor TOPAZ-II. In this study, the original pumped loop radiator was replaced by liquid metal heat pipes and radiator panels as the heat removal mechanism. A steady-state model using finite difference method was used to compare the pumped loop radiator with the heat pipe-radiator assembly. As shown by the results, the heat pipe radiator satisfied the thermal requirements of the TOPAZ-II power system under

other components, creating a radiation shadow where the energy conversion system and the payload are placed. As shown in Fig. 1, the shielding element creates a trapezoidal-shaped shadow with a specific shadow angle, promoting the radiator panel footprint. This configuration was used as a reference to develop the numerical representation present in the current work in order to obtain dimensional and thermal guidelines. Regarding space power systems, the Closed Brayton Cycle (CBC) is considered as a promising energy conversion system, since it promotes higher power output to radiator area ratio when compared to other dynamic cycles [33] and [9]. Considering these aspects, several studies have focused on the performance prediction of a CBC for space power systems [3,13,31,15,23]. Fig. 2 presents a schematic layout of the space power system and the major components required for its operation. Ribeiro et al. [28] performed an endoreversible modeling and optimization of a CBC for nuclear space power systems. In this study, heat pipes were represented by thermal conductances, whereas a simple 2

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L.F.R. Romano and G.B. Ribeiro

assembly geometry, the performance prediction will drastically change if any input parameter is altered. For that reason, the proposed parametric evaluation provides a good understanding of the system model behavior and can be used alongside the engineering design. The titanium-water heat pipe is physically attached to radiator panels made of aluminum, and are disposed along an aluminum duct. This heat pipe-radiator assembly is intended to extract the remaining heat coming to the cold heat exchanger duct, where a He-Xe gaseous mixture flows. In nuclear power systems a radiation shadow is created by the radiation shield angle, commonly applied to protect the payload and electronic devices that could be influenced by the nuclear fission products. This angle is considered in this study in a manner to make use of all available heat transfer area, while also avoiding the degradation of the HP working fluid, container, and heat transfer capabilities. The heat pipe modeling is based on the lumped thermal conductance method in a manner to compute the temperature drop between its evaporator and condenser. The radiator panel is based on the radiative heat transfer coefficient, whereas the fin efficiency is evaluated taking into account the temperature drop along the radiator panel. Several HP-RAD elements are computed for the assembly, creating a heat sink configuration until the required heat transfer rate is accomplished. The modeling presents temperature drops between the CBC working fluid, the HP evaporator and condenser. Variable-sized radiator panels, HP condenser lengths and external diameters, for a specific HP wick geometry. Furthermore, it also considers a temperature drop between each HP evaporator along the CHE duct, via an energy balance along the CHE, promoting a non-isothermal system. Considering the operational temperature range, titanium and water were chosen as material and working fluid of the heat pipes, respectively [38,1]. Thus, the heat pipe container and wick structure are made of titanium. Rectangular grooves were considered as the HP capillary wick, whereas for the radiator panel aluminum sheets were applied. The use of aluminum sheets may no longer be widespread for this kind of operation but this design-based model will allow the evaluation of different materials in future studies. Initially, the model is ought to generate data to evaluate which geometrical arrangement will provide the lowest total mass within a feasible length for the cold side of the CBC, therefore being of great interest to start a space radiator engineering design and guiding the development of an operational rejection system within the predicted optimal range. The model also enables to simulate chosen specifications and predict temperature drops and thermal behavior in each section, as well as to evaluate overall system dimensions if needed in future applications. With modeling that predicts the assembly thermal performance, a parametric evaluation and sensitivity analysis of geometrical parameters were performed in order to provide data to guide future experiments and designs during the conception phase. In addition, it is important to bear in mind that any aspect regarding major manufacturing processes and structural analysis of such equipment are out of the scope of this study. Also, it is relevant to point that the method used to connect the HP with the radiator panel is yet to be defined. Therefore a simple thermal connection with no resistance was considered in this case.

typical operating conditions. Wang et al. [37] proposed a model that predicts the transient behavior of a heat pipe-radiator unit for space applications. Potassium was the chosen fluid for heat pipes, and it was shown that the heat removal unit promoted satisfactory responses under transient conditions. Focusing on a radiative sink for single-phase mechanical pump fluid loops, Rai et al. [27] carried out a semi-analytical study to mitigate the final mass of the space radiator panel. Considering fixed parameters as heat transfer rate, radiator thickness, and fluid inlet temperature and mass flow rate, a configuration of radiator length and panel spacing was found for a minimum radiator mass. Using genetic algorithms, the work of Hull et al. [14] performed a mass optimization a typical radiator. The finite element analysis was used for the radiator modeling and based on an initial radiator design, evolutionary algorithms were applied to extract a radiator profile which minimizes mass and maximizes the heat transfer. Juhasz [16,17] described a mathematical model which considers variable-size radiator panels. The proposed analysis was valid for a pumped loop and for heat pipe heat exchangers. In his model, the temperature drop between the heat pipe evaporator and condenser was neglected, as well as the radiative panel efficiency. Comparison between rectangular and different trapezoidal shapes was performed. Tomboulian [34] proposed a model and then evaluated experimentally a lightweight radiator for nuclear electric propulsion vehicles. In order to decrease to total heat radiator mass, several low-density materials were tested. According to results, the radiator panel made of carbon fiber provided a substantial decrease of radiator area and mass, when compared to copper, molybdenum and carbon-carbon composite. The works of Tomboulian [34] and Juhasz [17] considered simplifications such as iso-geometric radiator panels and isothermal radiative rejection with no temperature drop along the HP length. Those have provided useful data to their needs given the experimental aid to validate and correct the models. This work, on the other hand, has a theoretical approach and it is intended to serve as a design-based tool to size lightweight heat pipe-radiators which meet the energy conversion system requirements. Therefore, considering several geometric and thermal parameters, a mathematical model coupling the radiator, the heat pipe, and the cycle CHE is proposed. This study aims the development of a numerical model to represent the heat sink configuration of a closed Brayton cycle for propulsion purposes. This model can provide aid for the heat sink development, allowing the designer to predict the thermal operation while considering each geometric change done upon the assembly during the design phase. Since the thermal operation is coupled to the HP-RAD

2. Mathematical modeling 2.1. Heat pipe The thermal operation is calculated upon a provided initial geometry and varies within a defined geometric range when needed. The final dimensions of the system can be associated with the thermal operation and heat extracting capabilities of the simulated system in order to evaluate compactness, overall sizes and masses, and temperature drop along the CHE, as calculated between assembly elements. For the prediction of the thermal behavior of heat pipes, the proposed modeling

Fig. 1. Conceptual nuclear-powered spacecraft design with two symmetrical radiator sets – Adapted [17]. 3

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L.F.R. Romano and G.B. Ribeiro

Fig. 2. Schematic layout of a nuclear space power system – Adapted [12].

is based on the lumped thermal conductance model. For this matter, the modeling considers an association of thermal conductances, one for each HP section, which also allows the computation of the temperature field along the HP, for a given heat transfer rate the pipe is subjected to. Axial symmetry was assumed, and a typical representation of the HP thermal conductances with their respective thermal resistances is shown in Fig. 3. The container and wick resistances in the adiabatic region were assumed negligible in this model. Hence, the heat pipe is modeled as a one-dimensional heat transfer case. The thermal conductances related to radial conduction of the container at the evaporating and condensing regions are defined as −1

1 D ⎡ ⎤ ln ⎛⎜ con ⎞⎟ Gcon, e = ⎢ 2πkcon, e Le ⎝ Dcap ⎠ ⎥ ⎣ ⎦

(1)

and −1

1 D ⎡ ⎤ ln ⎜⎛ con ⎟⎞ Gcon, c = ⎢ 2πkcon, c Lc ⎝ Dcap ⎠ ⎥ ⎣ ⎦

(2)

Fig. 4. Heat pipe profile with rectangular grooves as the wick.

where kcon denotes the container thermal conductance. Variables Dcon and Dcap represent the container and capillary wick external diameter, respectively, as shown in Fig. 4. The terms Le and Lc are the evaporator and condenser length, respectively. The container thickness tcon is based upon its external diameter and calculated inwards using on the Barlow’s formula [26]. It correlates the tensile stress of titanium σti and the HP internal pressure, which is the saturation pressure Psat at the temperature Tmax = Thp, e + 0.1Thp, e [K], being Thp, e the HP evaporator temperature. Thus,

tcon =

Dcon − Dcap 2

Moreover, a lower thickness limit of 0,25 mm was introduced on the code to avoid extremely thin tubes obtained at lower saturation pressures. Above this threshold, the value was considered as calculated by the presented Eq. (3). Similarly to Eqs. (1) and (2), the wick structure is also characterized by a radial conduction heat transfer. Thus, the thermal conductance present in the evaporator and condenser is represented as follows −1

=

Psat [Tmax ] Dcon 2σti

Dcap ⎞ ⎤ 1 ln ⎛ Gcap, e = ⎡ ⎢ 2πkcap, e Le ⎥ ⎝ Dva ⎠ ⎦ ⎣ ⎜

(3)

Fig. 3. Axisymmetric description of HP thermal conductances. 4

⎟

(4)

Thermal Science and Engineering Progress 13 (2019) 100368

L.F.R. Romano and G.B. Ribeiro

and

ΔPcap = −1

Gcap, c

Dcap ⎞ ⎤ 1 ln ⎛ =⎡ ⎢ 2πkcap, c Lc ⎥ ⎝ Dva ⎠ ⎦ ⎣ ⎜

(5)

⎡ C (f Re v ) μ v v ΔPv = ⎢ ⎢ 2Ava ρ hlv, e D va v 2 ⎣

( )

(wfin kl kti dgr ) + wgr kl (0.185wfin kti + dgr kl ) (wgr + wfin )(0.185wfin kti + dgr kl )

Lef =

−1

(8)

Ngr =

where R is the ideal gas constant, MW is the water molecular weight and hlv is the enthalpy of vaporization at the vapor duct temperature Tva , for each heat pipe section. Moreover, the thermal conductance related to the vapor duct is detailed as

(

T

+ Tva, e

−1

K= (9)

rh =

∊=

(10)

2 ∊ rh 2 fl Rel

(18)

2wgr dgr wgr + 2dgr

(19)

wgr wgr + wfin

(20)

(11)

⎡ 2πL k ef cap, e Tva, e ̇ =⎢ Qboil ⎢ ρ h ln Dcap lv, e D va ⎣ v, e

In order to keep the fluid circulation which is promoted by the capillary pump, the maximum capillary pressure drop ΔPcap must be equal or greater than the sum of the vapor pressure drop ΔPv , along the vapor duct, and the liquid pressure drop ΔPl , along the wick. As this system is to be operated in a low gravity environment, the current model considers only liquid and vapor pressure drops. Thus,

ΔPcap ≥ ΔPv + ΔPl

(17)

Eqs. (12)–(15) characterize a linear system that is to be solved simultaneously in order to compute the HP maximum heat transfer rate ̇ , defined within ΔPl and ΔPv . To avoid due the capillary operation Qcap liquid dryout in the evaporating region, it is crucial that the HP operates under the boiling limit. For that matter, the maximum heat transfer ̇ is determined according to Chi [6] as rate due to boiling limitation Qboil follows

and

1 1 1 1 1 1 1 1 = + + + + + + Gtot Gcon, e Gcap, e Gi, e Gva Gi, c Gcap, e Gcon, c

πDva (wgr + wfin )

and

̇ [n] Qhp Gtot [n]

(16)

The friction factor-Reynolds number product ( fl Rel ) is calculated based on a 3rd degree polynomial equation as a function of the groove width and depth ratio (wgr / dgr ), based on data displayed by Chi [6]. Likewise (flRel ), the hydraulic radius rh and porosity ∊ are also a function of groove dimensions as

In Eq. (9), variable Pva, e denotes the vapor saturation pressure for the temperature Tva, e , at the evaporator zone in the vapor duct, whereas Pva, c is the vapor saturation pressure, for the condenser temperature Tva, c . ̇ and the enthalpy of vaporization hlv, va The HP heat transport rate is Qhp and density ρv, va were computed based on the average vapor temperature at the evaporator and condenser (Tva, e and Tva, c ). During this entire analysis, material and fluid properties were always obtained using the library provided by the REFPROP software [21], as a function of the average temperature of the heat pipe section (i.e., evaporator and condenser). The temperature drop ΔThp along a given heat pipe n was computed based on the overall thermal coṅ as follows: ductance and considering the HP heat transfer rate Qhp

ΔThp [n] =

Le + Lc 2

Furthermore, the rectangular grooves can be characterized as a porous media with a defined permeability K and porosity ∊ as follows

) ⎤⎥ ⎥ ⎦

(15)

where μ , is the dynamic viscosity. Variables Ava and Acap are the crosssectional area of the vapor duct and capillary wick, respectively, and La is the length of the HP adiabatic region. For laminar and incompressible flow, ( fv Re v ) and C assumed values of 16 and 1, respectively. The term Acap is a function of the number of grooves (Ngr ) found in the capillary wick, which is obtained by rounding up the value obtained by Eq. (17), presented ahead

and

⎡ (Pva, e − Pva, c ) va, c 2 Gva = ⎢ ̇ hlv, va ρv, va Qhp ⎢ ⎣

(14)

and

(7)

2 ⎡ RTva, c 2πRTva, c ⎤ Gi, c = ⎢ MW 2πLc hlv, c 2 ⎥ ⎣ ⎦

⎤ ̇ ⎥ Lef Qcap ⎥ ⎦

μl ⎤ L Q̇ ΔPl = ⎡ ⎢ KAcap ρ hlv, e ⎥ ef cap l ⎣ ⎦

−1

⎡ RTva, e 2πRTva, e ⎤ Gi, e = ⎢ MW 2πLe hlv, e 2 ⎥ ⎣ ⎦

2

and

(6)

where wfin denotes the fin width, whereas wgr and dgr denote the groove width and depth, respectively. These variables characterize the groove profile, as shown in Fig. 3. The thermal conductivity of water and titanium are represented as kl, and kti , respectively. Faghri [11] defines the thermal conductance related to the liquid-vapor interface at the evaporator and condenser based on the following equations: 2

(13)

Considering that there is no adiabatic region between evaporator and condenser, the vapor and liquid pressure drop were computed according to Chi [6] as follows

⎟

where kcap, e and kcap, c are the effective capillary wick thermal conductivity at the evaporating and condensing zones, respectively. The term Dva denotes the vapor duct diameter. The effective thermal conductivity was calculated as proposed by Chi [6] as follows:

kcap =

2σl wgr

( )

⎤ ⎥ ⎡ 2σl − ΔPcap⎤ ⎥ ⎥⎢ ⎣ rn ⎦ ⎦

(21)

where the nucleation radius of the vapor bubbles rn is assumed as the fixed value of 5. 10−8 m to provide values to this limit that are more realistic [6]. During operation, the HP is subjected to countercurrent flows, with different phases. The interaction of the two streams results in viscous shear forces occurring at the liquid-vapor interface, which may inhibit the capillary forces to pump the liquid back to the evaporator through the wick. The maximum heat flow allowed by the ̇ , as proposed by Cotter [8], is set as entrainment limit Qent

(12)

Furthermore, the maximum capillary pressure for a rectangular groove is calculated as a function of the fluid surface tension σl and the effective radius, which in this case is the groove width (wgr ) 5

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L.F.R. Romano and G.B. Ribeiro

σρ ̇ = Ava hlv, e ⎛⎜ l v ⎞⎟ Qent 2 ⎝ wgr ⎠

wcor [n] = wrad [n] + (22)

Perrad [n] = 2(Lc [n] + trad )

Ab, rad [n] = Lc [n] trad

(23)

hrad [n] = σrad εrad Frad (Thp, c [n]2 + Tsp2). (Thp, c [n] + Tsp)

Initially, the radiator assembly is separated in symmetrical sets by dividing the value of extracted heat and determining how much is to be extracted by each set. As an example, the radiator displayed in Fig. 1 has two sets. Making use of a control volume (CV) description, each set is then subdivided into smaller sections around a single HP where the dimensions of the RAD panel are provided as input. Each panel is responsible for the extraction of a certain amount of heat, directly proportional to its surface area, the temperature of rejection and efficiency of operation while each HP has its own temperature gradient and geometry. A visual representation of several assembly elements can be seen in Fig. 6. The numerical model predicts the thermal coupling between a given heat pipe and its RAD panel. Initially, it is assumed an isothermal assembly that operates at the fluid temperature (Tinf ). In addition, an initial heat transfer rate is provided for that CV based on the input geometry given for the panel of the first section of the RAD set. Since the radiator model utilizes the HP condenser temperature as ̇ is determined the fin root temperature, the first heat transfer rate Qrad considering no temperature drop along the HP and, for this reason, a ̇ is higher radiator efficiency ηrad is achieved. The value obtained for Qrad ̇ . then set as the HP heat transfer rateQhp Additionally, the HP thermal conductances, material, and fluid properties are also initially obtained as functions of the provided HP evaporator temperature and the previously determined heat transfer ̇ . rateQhp Subsequently, the temperature drops are calculated by applying Eqs. (10) and (11). The temperature difference between Tinf and Thp, e is calculated considering the correlations proposed by Knudsen and Katz [20] for a convective heat transfer around a cylinder. The HP internal thermal conductances are updated with the respective local temperatures and the process is restarted from the radiative rejection model.

(24)

The radiator panel model makes use of the radiative heat transfer and the fin efficiency correlation in order to determine the heat radiated through the panel. Considering a simple exchange of radiant energy between two isothermal surfaces [32], the heat transfer rate extracted by each panel is calculated as follows (25)

where σrad denotes the Stefan-Boltzmann constant and εrad denotes the panel emissivity. Commonly, a high emittance coating is considered in space applications. For this study, εrad was assumed to be equal to 0.9, as suggested by Crosby [7]. The view factor of radiating surface to sink (i.e, outer space) Frad was assumed to have the value of 1. The sink temperature Tsp is a function of the RAD panel orientation with respect to celestial bodies, like the Earth, Moon or Mars. In this study, the outer space temperature Tsp of 200 K was applied. For a rectangular fin, the heat transfer available area is obtained as a function of the HP condenser length Lc and RAD width wrad as

Arad [n] = 2(Lc [n]2wrad )

(26)

It is relevant to note that the radiator surface area description represents two double-sided thin panels. Therefore, two RAD panels are attached to each HP. The fin efficiency of the radiator panel ηrad represents the temperature drop along the fin. Each panel thermally connected to the HP condenser is modeled as a fin with adiabatic tip. Furthermore, it is assumed a mechanic coupling with negligible contact resistance. Thus, by a simple energy balance around a rectangular fin with a specified temperature at the fin root, we have

ηrad [n] =

tanh (m [n] wcor [n]) m [n] wcor [n]

(27)

and

m=

⎛⎜ hrad [n] Perrad [n] ⎞⎟ ⎝ krad Ab, rad [n] ⎠

(32)

2.3. Thermal coupling and numerical procedure

2.2. Radiator panel

̇ [n] = Arad [n] ηrad [n] σrad εrad Frad (Thp, c [n]4 − Tsp 4 ) Qrad

(31)

where trad is the RAD panel thickness, fixed as 0,5mm [30]. The term krad represents the thermal conductivity of aluminum and hrad is the radiative heat transfer coefficient, represented as follows

Dva2Ava ρv, e hlv, e Pv, e 64μl, e Lef

(30)

and

Low-flow conditions are also of concern, especially when the HP is operating at a low-temperature range or when the HP is long. For that matter, the viscous limit Q̇ visc was computed in order to evaluate if the viscous forces within the vapor core are equivalent to the required pressure gradient that drives the flow from the evaporator to the condenser. For this HP heat capacity limit, laminar and fully developed flow is considered, and the vapor pressure at the condenser region was assumed to be equal to zero (Busse, 1976). This latter hypothesis provides the absolute limit for the vapor condenser pressure. Thus, we have

Q̇ visc =

(29)

and

̇ It is worth mentioning that Eq. (22) tends to overestimate the Qent [19]. Nevertheless, in order to ensure the HP operability, a safety factor ̇ by 2. is applied in the proposed model, dividing Qent ̇ allows the evaluation of the existence Moreover, the sonic limit Qson of a choked flow in the vapor duct. Commonly, this HP limit is a concern when very small values Ava and ρv, e are applied or during the HP ̇ is calculated as startup. Thus, as proposed by Busse [5], Qson

̇ = 0, 474Ava hlv, e (ρv, e Pv, e ) Qson

trad 2

(28)

wherewcor is the corrected fin width, and Perrad and Ab, rad denote the panel perimeter and cross-sectional area, respectively. According to Fig. 5, for an adiabatic fin tip, wcor , Perrad and Ab, rad of a given panel are obtained by the following geometrical relations:

Fig. 5. Radiator panel description. 6

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L.F.R. Romano and G.B. Ribeiro

Fig. 6. Representation of the heat pipe-radiator assembly elements.

̇ [n] = Qsink ̇ [n − 1] − Qhp ̇ [n] Qsink

The solution is obtained by an iterative method of subsequent substitutions. This procedure is repeated until the following convergence criterion (Eq. (33)), based on the HP condenser temperature, is met. The variable [o] represents the iterative step counter.

|Thp, c [o] − Thp, c [o + 1]| Thp, c [o]

and

Tinf [n + 1] = Tinf [n] − ≤ 10−3

(33)

̇ , tot Qsink S

(34)

̇ , tot is the contribution of all element Moreover, considering that Qsink assemblies on all symmetrical sets, we have ̇ [n] ∑ Qrad n=1

ṁ mix cp, mix

Lc [n + 1] = Lc [n] + 2wrad tg (θss )

Nhp

̇ , tot = Qsink

̇ [n] S Qhp (37)

where ṁ mix and cp, mix are the mass flow rate and specific heat of the working fluid of the CBC. A mixture of He-Xe with a molecular weight of 40 g/mole is considered as the energy conversion system working fluid. Nevertheless, for this study both variables were kept constant and any influence of the temperature on cp, mix was neglected. Data from Ribeiro et al. [28] were used to define ṁ mix . As mentioned by Tournier et al. [35] and El-Genk and Tournier [10], noble gases such as He and Xe under low pressures (≤2 MPa) and high temperatures (≥400 K) present an ideal gas behavior. Thus, the He-Xe thermodynamic properties were extracted from the NASA properties library, using the ideal gas mixture approach [25]. As before, the variable n refers to the assembly element that is currently being iterated. For the next element, the HP condenser length Lc and, consequently, RAD panel dimensions are updated based on the shield shadow angle θss as follows

̇ is then checked upon its operational The HP heat transfer rate Qhp ̇ is lower than its lowest limits and determined as operational if Qhp limiting value, with a safety factor of 2. Being the HP capable of rejecting at least two times the calculated heat, the thermal coupling is considered operational and the modeling process marches to the next assembly element. This procedure happens until the generated radiator ̇ set is capable of rejecting at least the amount of heat Qsink provided to ̇ , tot that set – as determined in Eq. (34). Dividing the total heat waste Qsink by the number of symmetrical radiator sets S, set as two considering the concept presented in Fig. 1, allows to reduce the computational effort and speed up iterations. Thus, we have ̇ Qsink =

(36)

(38)

According to Eq. (38), values for shadow angle θss higher than zero result in a stepped trapezoidal-shape assembly with increasingly bigger Lc . For a null θss the HP-RAD assembly forms a rectangular shape. Moreover, by not considering the HP container diameter in the condenser length increment the next dimensioned condenser is guaranteed to fall within the radiation shield shadow while also simplifying the increment dependence with variable parameters. ̇ , tot is Also, a verification is introduced in order to check if all of Qsink already accounted for, given the already simulated assembly elements. If positive, the radiator assembly is determined, the total number of HP (Nhp ) and other global variables are generated for the system analysis. Otherwise, another HP-RAD element is iterated considering the changes in temperature, length and rejection area as demonstrated. Fig. 7 presents a flowchart that summarizes the code operation, considering the input conditions, intermediate procedures for the described models,

(35)

If the HP limit verification fails, the HP external diameter is incremented by a 10% of the minimum predetermined threshold (i.e., 0.015 m), and a new iterative procedure starts for the same assembly element. This increment is repeated until a valid thermal coupling is obtained or when the provided maximum diameter is extrapolated (i.e., 0.200 m), being the HP-RAD assembly flagged as not operational and requiring a reevaluation on the initial parameters. Having the code generated an operational HP-RAD assembly within ̇ all the required conditions, the remaining heat Qsink coming from CHE ̇ ) is updated and the next HP evaporator temperature is computed (Qsink based on a simple energy balance around the CHE control volume. Thus, 7

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Table 1 Reference input data. Variable

Value

ṁ mix cp, mix Dcon dgr

1,973 kg/s 519,6 J/(kg.K)

Lc p [1] Le S Tche, in trad Tsp wfin

0,9 m 0,25 m 2 504 K 0,5 mm 200 K

wgr

0,0005 m

0,04 m 0,005 m

0,0005 m

not taken into account since it depends on a previous evaluation of the neutron transport phenomena. Furthermore, the total heat sink length Lsink is defined as the sum of all RAD panel width wrad and HP external diameter Dhp as follows N

Lsink =

∑n =hp1 (2wrad [n] + Dcon [n]) S

(43)

Also, the last HP condenser size can be prompted to provide useful information about the radiator span, being dimensioned from the CHE duct to the HP condenser end. Table 1 summarizes the baseline input data for the proposed modeling, where the geometrical characteristics of the heat pipe, including groove dimensions and CBC conditions, are presented. As mentioned, the HP container diameter Dcon and condenser length Lc are subject to changes according to the model flowchart. At the first assembly element, the CHE inlet temperature Tche, in is the same as the CBC working fluid temperature Tinf . 3. Results and discussion In order to analyze the model capabilities and evaluate the impact of the CBC final conditions on the HP-RAD assembly profile, case simulations were run varying the HP spacing (i.e., 2wrad ) and the CHE inlet temperature Tche, in . The effect of such variables on the number of heat pipes Nhp , total assembly length Lsink and mass msink were performed, for different radiation shield angles θss and heat transfer rate ̇ , tot . Qsink

Fig. 7. General model flowchart.

internal validations and also intermediate and final outputs obtained from the iterative processes. The model was implemented in the software Engineering Equation Solver – EES [18]. The heat sink (i.e, heat pipe-radiator assembly) mass is defined as the sum of all HP and RAD panels as follows

3.1. Heat pipe spacing effect The influence of HP spacing on the number of heat pipes is depicted ̇ , tot and shadow in Fig. 8 for several heat rejection requirements Qsink angles θss for a CHE inlet temperature Tche, in of 504 K. This value was assumed for the first evaluations of the code given the good efficiency/ mass results presented in an initial analysis [29]. As shown, the decrease of HP spacing results in more assembly elements, and as a coṅ , tot also sequence, Nhp increases considerably. Moreover, higher Qsink provided larger Nhp , since more assembly elements are required to ̇ , tot . It also can be pointed in achieve the increased heat rejection Qsink Fig. 8 that Nhp becomes larger for lower shadow angles θss . When a lower θss is used, shorter condenser lengths Lc are provided, resulting in lower radiator panel area. Thus, more assembly elements and heat pipes ̇ , tot . A null θss provides are needed to extract the heat transfer rate Qsink infeasible values for Nhp , therefore should be considered only as a reference parameter on this work. Fig. 9 presents the variation of the total assembly length for different HP spacing. According to results, large HP spacing tends to promote longer Lsink , since longer radiator panels wrad are computed. For shorter HP spacing, assembly elements with lower available surface areas are provided for radiative heat transfer, which culminates on the

Nhp

msink =

∑ (mhp [n] + mrad [n] + mduct [n]) n=1

(39)

and

mhp [n] = ρs Vcon [n] + ρs Vcap, s [n] + ρl Vcap, l [n] + ρv Vv + 2ρs Vcon, end

(40)

and

mrad [n] = ρrad trad wrad Lc [n]

(41)

and

mduct [n] = ρduct π (rext , duct 2 − rint , duct 2)(2wrad + Dcon )

(42)

The masses are defined simply as the component volume times their respective densities. The duct is made of aluminum, with its wall thickness arbitrated as 5 mm and its internal diameter being 0.3 m. Variables Vcap, s and Vcap, l denote the solid and liquid volumes of the capillary wick, respectively. The volume Vcon, end characterizes HP volume at both far ends. The mass related to the nuclear core shielding is 8

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minimum for msink is presented in the analysis and, consequently, an optimized HP-RAD assembly is achieved within the determined input and contour criteria. Low values of HP spacing might provide the lowest system mass but, since the radiative heat transfer surface is smaller, it only accepts as much heat as the surface is capable of rejecting to space. As the HP thermal prediction procedure is iterated based on the amount ̇ , for a smaller radiator area, a smaller heat pipe is of extraction heat Qrad expected to fulfill the required heat capacity. Although an optimized assembly profile presents the lowest mass, the overall quantity of heat pipes may be large, as a manner to overcome the lower heat transfer rate that each assembly element has to dissipate. ̇ , tot that has to be rejected is reFurthermore, the waste heat Qsink duced along with the required radiative heat transfer area. Therefore, the HP-RAD assembly is more compact and, as a consequence, a lower msink is achieved. Additionally, as shown in Fig. 10, a θss = 5° provides lower msink values than θss = 10°. Once again, θss = 0° provides an unfeasible solution, serving only as an upper limit to the cold side mass evaluation. The reduction of the radiation shield angle θss results in shorter overall Lc . Due to this reason, the influence of the Nhp on msink gets weakened and a lower msink minimum is achieved under higher HP spacing even with higher Nhp values. ̇ , tot , HP-RAD asIt also can be pointed out that, for the same Qsink semblies with lower θss have lower overall Lc and, consequently, available radiative area. On the other hand, as a manner to compensate and reach the required heat rejection capacity, Nhp and Lsink tend to increase. As the net effect, the overall Lc presented a strong influence on reducing msink , considering the code operation as proposed. Since a large Lsink might be somewhat restrictive for a device ought to be launched from the ground, this msink and Lsink tradeoff must be taken into account during the mission feasibility evaluation. As the analysis presented so far were done at a fixed temperature Tche, in , the aforementioned conclusions only make sense in a real CBC system if components are designed to adapt to temperature and pressure constraints. Nevertheless, points of minimum msink and Lsink can be obtained varying the CHE inlet temperature, Tche, in . Therefore, a proper analysis under variable CHE inlet temperature was carried out, in order to aid the HP-RAD design. Figs. 11–13 show the influence of the HP spacing on the number of heat pipes Nhp , total assembly length Lsink and mass msink for Tche, in varying from 504 to 544 K. Fixed values for the total ̇ , tot and shadow angle θss were considered, with rewaste heat Qsink spective values of 117750 W and 5°. According to Figs. 11 and 12, the increase of Tche, in results in smaller Nhp and shorter Lsink . When a higher temperature of the CBC working fluid is applied at the CHE inlet, less heat transfer area is needed for the HP-RAD assembly, due to the temperature difference increase. Consequently, fewer assembly elements are required and the number of HP drops with the increasing HP spacing and temperature Tche, in .

Fig. 8. Heat pipe number as a function of the heat pipe spacing, for Tche, in = 504 K.

Fig. 9. Total heat pipe-radiator assembly length as a function of heat pipe spacing, for Tche, in = 504 K.

̇ , tot , increasing the length need of more assembly elements to meet Qsink as well. Thus, a minimum point of Lsink is achievable when HP spacing is varied between 0.05 and 0.5 m. ̇ , tot Furthermore, as evidenced before, higher heat transfer rate Qsink presented longer Lsink , due to the additional surface area required to ̇ , tot . Additionally, it was showed a longer waste a higher value of Qsink Lsink when a lower shadow angle θss is applied. As mentioned, an assembly element with a lower θss results in shorter overall condenser lengths Lc . As consequence, more assembly elements are necessary to meet the necessary parameters for heat rejection. Fig. 10 displays the heat sink mass msink under different HP spacing, being the CHE inlet temperature still kept constant at 504 K. As shown, there is a HP spacing that promotes the lowest msink , for each set of ̇ , tot and shield angle θss . By varying the HP spacing within the speQsink cified range of 0.05 to 0.5 m, points of minimum msink are observed ̇ , tot . around 0.11 m for a θss of 10° and for all evaluated Qsink ̇ , tot , lower HP spacing tends to provide a larger Nhp , For the same Qsink whereas the heat sink length Lsink tends to reduce. On the contrary, for higher HP spacing, RAD panels with larger width wrad are computed to fill the gaps between heat pipes, increasing Lsink . With smaller panels ̇ , the high quantity of elements needed rejecting lower values of heat Qhp to reject all the heat – observed through Nhp behavior – strongly dictates the total heat sink mass msink under low HP spacing, even though HP of smaller diameters tend to be iterated. For high HP spacing, the radiative ̇ is higher, repanel area is larger and the calculated heat capacity Qhp quiring the code to iterate HP with bigger container diameters to compensate for the excess heat being rejected at each CV, even if lesser CV are iterated and lower Nhp are observed. As the net effect, a point of

Fig. 10. Total heat pipe-radiator assembly mass as a function of the heat pipe spacing, for Tche, in = 504 K. 9

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Fig. 14. Heat pipe number as a function of the CHE inlet temperature, for HP spacing of 0.1 m and θss = 5°.

Fig.11. Heat pipe number as a function of the heat pipe spacing, for Q̇sink, tot = 117750 W and θss = 5°.

3.2. Cold heat exchanger inlet temperature effect It is also of interest of the authors of this study to evaluate the model limitations on providing operational assembly elements, allowing the designer to pinpoint possible HP limitations that may hamper the goal to achieve small and lightweight heat sinks. For that reason, a wider range of temperature was also considered, varying the CHE inlets temperature from 450 to 570 K. As already mentioned, a smaller θss culminates on a longer Lsink , and it also contributes on the msink reduction. Since the ultimate goal is to develop a feasible heat rejection system, for this analysis θss was considered as 5° and the HP Spacing was fixed at 0.1 m. This particular profile of the HP-RAD assembly can be considered close to the optimum solution, for the mass and length minimization, when considering the evaluated range presented on this work. Fig. 14 presents the variation of the heat pipe number Nhp for the Tche, in range. As can be seen, the decrease of Nhp is close to linear but tends to be slightly reduced for higher values of Tche, in . Similarly, as shown in Fig. 15, an optimal Tche, in is achievable considering the Lsink minimization. Since the HP wick geometry is fixed, it is expected that a higher Tche, in results in an overall higher temperature difference along the HP at the heat sink, reducing the radiative heat transfer area needed for the same amount of heat rejection. On the other hand, for extremely ̇ in the first assembly high inlet temperatures, the HP heat transport Qhp elements are more hampered by their operational limits, namely the boiling limit, which restricts the heat transfer rate to the panels. Since the panel dimensions are what provides the heat transfer rate for the assembly element calculations, the HP container diameter is

Fig.12. Total heat pipe-radiator assembly length as a function of the heat pipe spacing, for Q̇sink, tot = 117750 W and θss = 5°.

Fig.13. Total heat pipe-radiator assembly mass as a function of the heat pipe spacing, for Q̇sink, tot = 117750 W and θss = 5°.

Additionally, for higher Tche, in , the point of minimum msink stays close to 0.1 m for 504 and 524 K, but shifts to a lower HP spacing for 544 K. For the higher temperature, the HP container thickness tcon is increased due to the increasing vapor pressure of the working fluid, as detailed in Eq. (3). As displayed, the effect of increasing tcon overcomes the effects of decreasing Nhp and Lsink . As net effect, a heavier msink is obtained when a higher Tche, in is used as a boundary condition for a titanium-water HP. Fig. 15. Total heat pipe-radiator assembly length as a function of the CHE inlet temperature, for HP spacing of 0.1 m and θss = 5°. 10

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incremented further to overcome the boiling limitation that decreases exponentially with the increase in temperature. To support this affirmation, the limiting heat capacity curves for a single HP of the radiator assembly (here, consideringLe = 0.25 m and Lc = 0.9 m as fixed parameters) with container diameters of 0.015, 0.04, 0.08 and 0.2 m are presented in Fig. 16 as a function of their evaporator temperature. Note that the overall values for the limits will vary for each iterated HP since the HP condenser length and container diameter are constantly being altered within the model in order to fit the required specifications, therefore the presented values serve only as a reference for the limiting behavior within this present model by considering the diameter variation. ̇ , tot , more Also, in order to accomplish the required extraction of Qsink assembly elements are introduced along the heat sink and HP with bigger diameters are generated for a small increase on the heat capacity allowance, which increases drastically the assembly element mass as well. The slight reduction of the Nhp decreasing rate as well as the increase of Lsink for Tche, in higher than 550 K are observed due to this ̇ , tot resulted in reason. Alike Figs. 8 and 9, higher heat rejection Qsink larger Nhp and Lsink . The influence of the CHE inlet temperature Tche, in on the assembly ̇ , tot . mass msink is shown in Fig. 17, for different total waste heat Qsink According to these results, there is an optimum Tche, in which results in a ̇ , tot . Again, HP operation under minimum heat sink mass, for a fixed Qsink high temperatures promoted a reduction of the HP heat capacity and an increase of HP geometrical parameters. Moreover, the Nhp decrease (and assembly elements) starts to be reduced. Considering these aspects, the final HP-RAD assembly mass rises rapidly when the CHE inlet temperature gets close to the triple point, given the considerable reduction on allowable heat capacities caused by the rapid decrease of the boiling limit of the grooved titanium-water HP of each element. By running the numerical routine at the minimum mass and waste heat configurations, the program provides each HP-RAD assembly property for analysis. A graph of the temperature drop along the CHE can then be drawn (Fig. 18), together with the heat extracted by each

Fig. 17. Total heat pipe-radiator assembly mass as a function of the CHE inlet temperature, for HP spacing of 0.1 m and θss = 5°.

Fig. 18. Temperature profile and heat transfer rate along the CHE.

̇ , max ) for (a) Dcon = 0.015 m, (b) Dcon = 0.04 m, (c) Dcon = 0.08 m and (d) Dcon = 0.2 m. Fig. 16. Operational limits and maximum allowed heat capacity (Qhp 11

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length of 0,21 m, the upper and lower minima obtained at around 520 K were approximately 1200 and 1000 kg, respectively. Low mass variances were observed between 500 and 530 K, which validates the assumption that a low mass system could be achievable within the temperature range of operation [34,23,1,9]. Results also presented a total HP number ranging between 350 and 425 for lower inlet temperatures and 200–250 for higher inlet temperatures. The total HP-RAD length was observed within a 22–35 m range. Declaration of Competing Interest None. Acknowledgements Fig. 19. Calculated HP diameter, thickness and assembly masses along the CHE for the optimal configuration.

This work is part of the TERRA program (from Portuguese, an acronym for Advanced Fast Reactor Technology), sponsored by the Brazilian Air Force – FAB. Authors are indebted to the National Council for Scientific and Technological Development – CNPq (427209/2018-8) for the financial support. Valuable suggestions and comments of Roberto D. Garcia are also acknowledged.

HP at those positions. Bear in mind that this analysis is considering a symmetrical radiator, as a simplification. Therefore, two HP are to exist and operate similarly at each point of the CHE. Additionally, Fig. 19 illustrates that the overall diameter of the HP ranges around 0.03 m, and thickness around 1.5 mm, even with the routine calculating different values on each HP-RAD set. The overall masses of each component can also be seen in Fig. 19, being the duct mass fairly constant at each section whereas the HP shows a great mass change give its condenser length increase along the CHE.

Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.tsep.2019.100368. References

4. Conclusions [1] W. Anderson, P. Dussinger, R. Bonner, D. Sarraf, High temperature titanium/water and monel/water heat pipes, 4th Int. Energy Convers. Eng. Conf. Exhib. (2006), https://doi.org/10.2514/6.2006-4113. [2] T.L. Ashe, W.G. Baggenstoss, R. Bons, Nuclear reactor closed brayton cycle power conversion system optimization trends for extra-terrestrial applications, Intersociety Energy Conversion Engineering Conference, (1990). [3] W.G. Baggenstoss, T.L. Ashe, Mission design drivers for closed brayton cycle space power conversion configuration, ASME J. Eng. Gas Turbines Power (1992) 721–726. [4] M.J. Barret, B.M. Reid, System mass variation and entropy generation in 100-kwe closed-brayton-cycle space power systems, Space Technol. Appl. Int. Forum, Am. Inst. Phys. (2004). [5] C.A. Busse, Theory of the ultimate heat transfer of cylindrical heat pipes, Int. J. Heat Mass Transfer 16 (1973) 169–186. [6] S.W. Chi, Heat Pipe Theory and Practice: A Sourcebook, Hemisphere, Washington, D.C., 1976. [7] J.R. Crosby, The development and qualification of thermal controls coatings for SNAP systems, NAA-SR-9908 (1965). [8] T.P. Cotter, Heat Pipe Startup Dynamics, Proc. SAE Thermionic Conversion Specialist Conference, (1967). [9] M.S. El-Genk, Space nuclear reactor power system concepts with static and dynamic energy conversion, Energy Convers. Manag. 49 (2008) 402–411. [10] M.S. El-Genk, J.M. Tournier, Noble gas binary mixtures for gas-cooled reactor power plants, Nucl. Eng. Des. 238 (2008) 1353–1372. [11] A. Faghri, Heat Pipe Science and Technology, Taylor and Francis, New York, 1995. [12] B.M. Gallo, M.S. El-Genk, Brayton rotating units for space reactor power systems, Energy Convers. Manag. 50 (2009) 2210–2232, https://doi.org/10.1016/j. enconman.2009.04.035. [13] R.B. Harty, W.D. Otting, C.T. Kudija, Applications of Brayton Cycle Technology to Space Power, Intersociety Energy Conversion Engineering Conference, (1993). [14] P.V. Hull, K. Kittredge, M. Tinker, M. SanSoucie, Thermal analysis and shape optimization of an in-space radiator using genetic algorithms, IP Conf. Proc. 813 (2006) 81, https://doi.org/10.1063/1.2169183. [15] A.K. Hyder, R.L. Wiley, G. Halpert, D.J. Flood, S. Sabripour, Spacecraft Power Technologies, Imperial College Press, London, 2000, pp. 332–340. [16] A.J. Juhasz, Analysis and Numerical Optimization of Gas turbine Space Power Systems with Nuclear Fission Reactor Heat Sources, Ph.D. Thesis Cleveland State University, 2005. [17] A. Juhasz, Heat Transfer Analysis of a Closed Brayton Cycle Space Radiator, 5th Int. Energy Convers. Eng. Conf. Exhib. 2007, pp. 25–27, , https://doi.org/10.2514/6. 2007-4841. [18] S.A. Klein, J.L. Alvarado, Engineering Equation Solver, F-Chart, 1993. [19] B.H. Kim, G.P. Peterson, Analysis of the critical Weber number at the onset of liquid entrainment in capillary-driven heat pipes, Int. J. Heat Mass Transfer 38 (1995) 1427–1442. [20] J.D. Knudsen, D.L. Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958. [21] E.W. Lemmon, M.O. McLinden, M. Huber, REFPROP: Thermodynamic and

Results have shown that with the reduction of the HP quantity the overall system mass should diminish as well. However this reduction is attenuated by the need of larger HP used to extract higher heat transfer rates. Furthermore, for the simulated range of HP spacing, the total HP number tends to diminish when a larger HP spacing is applied. For the same HP spacing, HP-RAD assemblies subjected to greater CHE inlet temperature provided a smaller number of HP and a shorter assembly length, up to the point where the reduction of the transport properties of the selected fluid started to reduce the boiling limit and induce an assembly length increase. This behavior would also be expected to appear at the Nhp evaluations, given the slight reduction observed with the increase in temperature, but was not present within the determined temperature ranges as demonstrated given the wide range of diameters the HP containers could be generated within. The total assembly mass also increased with the increasing CHE inlet temperature due to the ticker HP container wall prompted by the Barlow’s law correlation, that which is directly affected by the exponentially higher vapor pressures of the water at temperatures close to its triple point. By studying the presented data, it became evident that an optimum HP spacing where the total assembly mass and length are minimized can be pointed with the applied methodology. The optimized assembly profile is not strongly dependent on the energy conversion cycle efficiency – which directly dictates the amount of waste heat that the radiator has to reject. Even though lower values are prompted for a lower heat transfer rate, the overall curve behavior is the same. The same can be said for the shadow angle impact, which is responsible to limit the available area for positioning the components. Moreover, optimal values of CHE inlet temperatures were also achieved, providing the assembly length and mass minimization. As evidenced by the results, for extremely high CHE inlet temperatures, the HP geometrical parameters are updated in the numerical routine as a manner to overcome the HP operational limits and satisfy the requested heat extraction. This tradeoff is to be properly studied to determine the required engineering specifications during the design phase. For a fixed HP spacing of 0.1 m, θss of 5° and an initial condenser 12

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Transport Properties of Refrigerants and Refrigerant Mixtures, User’s Guide, US DOE, NIST, USA, 2002. B. Lubarsky, Nuclear power systems for space applications, Adv. Nucl. Sci. Tech. 5 (1969). L.S. Mason, A power conversion concept for the jupiter icy moons orbiter, International Energy Conversion Engineering Conference, (2003). L.S. Mason, R.K. Shaltens, J.L. Dolce, R.L. Cataldo, Status of Brayton Cycle Power Conversion Development at NASA GRC, in: M.S. El-Genk (Ed.), Proceedings of Space Technology and Applications International Forum, Melville, 2002. B.J. McBride, M.J. Zehe, S. Gordon, ic,ic, (Ed.), NASA Glenn Coefficients for Calculating Thermodynam Properties of Individual Especies, NASA/TP – 2002 – 211556, 2002. E.S. Menon, Liquid Pipeline Hydraulics, CRC Press, 2004. P.K. Rai, S.R. Chikkala, A.A. Adoni, D. Kumar, Space radiator optimization for single-phase mechanical pumped fluid loop, J. Therm. Sci. Eng. Appl. 7 (2015) 7. G.B. Ribeiro, F.A. Braz Filho, L.N.F. Guimarães, Thermodynamic analysis and optimization of a Closed Regenerative Brayton Cycle for nuclear space power systems, Appl. Therm. Eng. 90 (2015) 250–257, https://doi.org/10.1016/j.applthermaleng. 2015.06.093. L.F.R. Romano, G.B. Ribeiro, “Optimal temperature of operation of the cold side of a closed Brayton cycle for space nuclear propulsion, International Nuclear Atlantic Conference, (2017). V.A. Saule, R.P. Krebs, B.M. Auer, Design Analysis and General Characteristics of Flat-Plate Central-Fin-Tube Sensible-Heat Space Radiators, NASA Technical Note, Lewis Research Center, Cleveland, 1965. R.K. Shaltens, L.S. Mason, Early results from solar dynamic space power system testing, AIAA J. Propul. Power (1996) 852–858.

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