Thermal-hydraulic analysis of the improved TOPAZ-II power system using a heat pipe radiator

Nuclear Engineering and Design 307 (2016) 218–233

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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

Thermal-hydraulic analysis of the improved TOPAZ-II power system using a heat pipe radiator Wenwen Zhang, Dalin Zhang ⇑, Wenxi Tian, Suizheng Qiu, G.H. Su School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China

h i g h l i g h t s
The system thermal-hydraulic model of the improved space thermionic reactor is developed.
The temperature reactivity feedback effects of the moderator, UO2 fuel, electrodes and reflector are considered.
The alkali metal heat pipe radiator is modeled with the two dimensional heat pipe model.
The steady state and the start-up procedure of the system are analyzed.

a r t i c l e

i n f o

Article history: Received 5 January 2016 Received in revised form 17 July 2016 Accepted 18 July 2016 Available online 2 August 2016 JEL classification: K. Thermal Hydraulics

a b s t r a c t A system analysis code coupled with the heat pipe model is developed to analyze the thermal-hydraulic characteristics of the improved TOPAZ-II reactor power system with a heat pipe radiator. The core thermal-hydraulic model, neutron physics model, and the coolant loop component models (including pump, volume accumulator, pipes and plenums) are established. The designed heat pipe radiator, which replaces the original pumped loop radiator, is also modeled, including two-dimensional heat pipe analysis model, fin model and coolant transport duct model. The system analysis code and the heat pipe model is coupled in the transport duct model. Steady state condition and start-up procedure of the improved TOPAZ-II system are calculated. The results show that the designed radiator can satisfy the waste heat rejection requirement of the improved power system. Meanwhile, the code can be used to obtained the thermal characteristics of the system transients such as the start-up process. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction A space nuclear power system converts the energy from a nuclear heat source into electricity to power a particular load or application. Nuclear power sources are attractive for using in space for its high power-to-mass ratio and efficiency, long lifetime, selfsufficiency and design flexibility. A generic space nuclear power system consists of the energy source, primary heat transport system, an energy conversion technique and a radiator for heat rejection (Diwekar and Morel, 1993). The TOPAZ-II power system developed by the former Soviet Union is the most technologically advanced and experienced thermionic system ever built (Bennett et al., 1996). It provides electricity by means of thermionic static energy conversion and releases the waste heat by a pumped loop radiator. Radiators in orbit around the Earth are exposed to an environment of micro-meteoroids and space debris particles. Although the original pumped loop radiator has a simple structure, ⇑ Corresponding author. Fax: +86 29 82664460. E-mail address: [email protected] (D. Zhang). http://dx.doi.org/10.1016/j.nucengdes.2016.07.020 0029-5493/Ó 2016 Elsevier B.V. All rights reserved.

but it has the problem of potential single point failure, which means the rupture of any radiation tube would lead to the LOCA of the whole system. Long duration space missions impose very strict reliability requirements. In order to avoid the single point failure, a heat pipe radiator comprised of a pumped fluid in a transport duct that moves across the evaporator section of a heat pipe array is designed for the TOPAZ-II system in our former work. Each heat pipe of the radiator can be considered as an independent element. Heat pipe failures due to environment hazards don’t mean the overall heat rejection system failure. The heat pipe radiators are widely used in the advanced space nuclear power system design, such as the SPACE-R thermionic reactor system (Von Arx and Alan Vincent, 1999), thermocouple reactor system SP-100 system (El-Genk and Seo, 1990), SAIRS system (El-Genk and Tournier, 2004a,b,c), SC-SCoRe system (Schriener and El-Genk, 2014), HPSTMCs power system (El-Genk and Tournier, 2004a,b,c). To analyze the heat pipe used in the space applications, Tournier and El-Genk (1992, 1995) developed a two-dimensional heat pipe transient analysis model (HPTAM) to simulate the transient operation of

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219

Nomenclature t P C c Q

P D U L h p T A H f V R Z

m X m M

time (s) power (W) delayed neutron precursor concentration (m3) specific heat (jkg1K1) heat source (Wm3) surface area (m2) diameter (m) perimeter (m), z-component of velocity (ms1) length (m) enthalpy (jkg1) pressure (Pa) temperature (K) area (m2) heat transfer coefficient (Wm2K1) view factor volume (m3), average velocity at cross-section (ms1) perfect gas constant, radial position (m) axial position (m) specific volume (m3kg1) quality of vapor mass (kg) molecular weight (mol1)

Greek symbols b delayed neutron fraction k heat conductivity (Wm1K1) decay constant of precursors (s1)

fully-thawed heat pipes and the startup of heat pipes from a frozen state. The NASA Lewis Research Center developed a heat transfer analysis code to analyze the heat pipe radiator (Hainley, 1991). Numerical simulations are also performed for the TOPAZ-II power system. A system analysis code, CENTAR (The Code for Extended Non-linear Transient Analysis of Extraterrestrial Reactors), was developed and used to analyze the TOPAZ-II system following a loss of coolant accident (Standley et al., 1992). The code TITAM (Thermionic Transient Analysis Model), developed by the El-Genk group, was used to simulate the operation of the thermionic fuel element (El-Genk et al., 1992). The extended version adding the thermal-hydraulic models of the coolant loop, EM pump, radiator and other components, was used to perform steady-state and transient analysis of the TOPAZ-II system (El-Genk et al., 1994). A system model capable of evaluating the system performance under conditions from both natural and hostile threats and system startup/shutdown transients are developed for the multicell thermionic space power reactor, S-PRIME (Von Arx and Alan Vincent, 1999). The heat pipe radiator of the S-PRIME system is modeled to allow for cases of unequal irradiant heating. The heat pipe model includes the evaporation and condensation rates, as well as the freezing and thawing. This radiator model employs much experiment data and empirical correlations, and the generality and flexibility are not good. In this work, a thermal-hydraulic system model has been developed for the improved TOPAZ-II system, which can be used to evaluate transient and steady state cases. Furthermore, coupling with the point reactor kinetic equations and the reactivity feedback model, the reactivity changes can be calculated with changes of the reactor temperatures. A two-dimensional heat pipe analysis model using the finite element method is applied to model the heat pipe radiator. The steady state condition with the nominal operation parameters and the start-up procedure are calculated and analyzed in the present study.

K d

q e h

r

neutron generation time (s) wall thickness (m) density (kgm3) emissivity angle (degree) Stefan–Boltzmann constant (Wm2K4)

Subscripts f fission, fluid U fuel G gap e electrode E emitter C collector SI inner steel pipe SO outer steel pipe M moderator R reflector v vapor l liquid wc welded copper pw pipe wall s surrounding, solid sin inner surface sout outer surface

2. General description and mathematical model description The TOPAZ-II is a single-cell thermionic reactor system which can produce 4.5–5.5 kWe for three years of autonomous and continuous operation in space. The major TOPAZ-II subsystems are: the reactor and combined thermionic converters, the primary coolant loop, secondary systems, such as the cesium supply system, the radiation shielding and power distribution system, the instrumentation and control (I&C) system. Highly enriched uranium fuel heats the thermionic emitter, enabling electron flow. The liquid metal coolant system transfers waste heat to the radiator, limiting collector temperature. An electromagnetic (EM) pump provides the motive force for coolant flow. The radiation shield composed of a stainless steel shell that contains lithium hydride attached to the lower part of the reactor limits the neutron and gamma dose rate to the rest of the spacecraft. The cesium system supplies cesium (Cs) to the interelectrode gap, improving converter efficiency. The I&C system monitors conditions, accomplishing start-up, operational control, and emergency shut-down functions. The main components of TOPAZ-II system are shown schematically in Fig. 1. The main design parameters of the core and heat rejection system are summarized in Table 1. 2.1. Thermal hydraulic model and neutron kinetics model of the core The TOPAZ-II reactor is a small, zirconium hydride moderated, epi-thermal design with high enriched U-235 fuel, which contains 37 single-cell TFEs that combine the fission heat source with the thermionic converters. The TFEs are set within axial channels in the five stacked zirconium hydride moderator blocks. The reactor core is surrounded by radial and axial beryllium (Be) reflectors. The radial reflector contains three safety drums with independent drive motors, and nine control drums attached to a single control drive motor located beneath the shield. Each drum contains a

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Fig. 1. The improved TOPAZ-II system schematic.

Table 1 Main design parameters of the improved TOPAZ-II system. Parameter

Value

Thermal power (kWt) Electrical power (kWe) Fuel type and enrichment U-235 loading (kg) Reactor coolant Coolant mass flow rate at BOL (kg/s) System pressure (MPa) Active core height (mm) Active core diameter (mm) Reactor height (mm) Reactor diameter (mm) System mass (kg) System length (m) Shield height (mm) Shield mass (kg) Reactor coolant inlet temperature at BOL/MAX (K) Reactor coolant outlet temperature at BOL/MAX (K)

115 4.5–5.5 UO2-96% 27 NaK-78 (78%Na, 22%K) 1.3 0.1 375 260 920 408 1061 3.9 580 390 743/773 843/873

116-degree section of boron–silicate–carbide neutron poison to control the nuclear reaction by drum rotation. The waste heat from the reactor is removed by the eutectic sodium–potassium (NaK) coolant. The coolant enters the reactor core through the lower plenum, then passes through the outer surface of the TFE and exits through the upper plenum. Fig. 2 presents the configuration of the reactor. 2.1.1. Reactor kinetics The heat generation in a nuclear reactor consists of two parts: fission and decay powers. The reactor point kinetics equations with six-family delayed neutrons are applied to calculate the core fission power variation following an external reactivity insertion and incorporates reactivity feedback due to Doppler and temperature effects. The point-kinetics equations may be expressed as:

P 6 X dPf q  6i¼1 bi Pf þ ki C i ¼ dt K i¼1

ð1Þ

dC i bi ¼ Pf  ki C i ; dt K

ð2Þ

i ¼ 1; 2; . . . ; 6

The total reactivity introduction to the reactor core is given as:

q ¼ qDrum þ qDf þ

X

ð3Þ

qTi

i

The first term on the right side of Eq. (3) is the external reactivity input to the nuclear reactor core, which is provided by the conP trol or safety drums. The second and third terms qDf and i qTi in Eq. (3) are the reactivity feedback due to Doppler effect in the fuel region, and the sum of temperature reactivity feedbacks in different regions of the TFE and moderator. The reactor kinetics model is coupled with the reactor thermal model in order to incorporate the various reactivity feedback effects. The Doppler reactivity feedback is expressed as (El-Genk and Paramonov, 1994):

qU ¼ 0:432  103 ðT U  T 0 Þ  0:64

sffiffiffiffiffiffi ! TU 1 T0

ð4Þ

where T f is the fuel average temperature, T 0 is the reference temperature (300 K). Moderator reactivity is the largest contributor to the overall temperature reactivity and it is positive which is quite different from the PWRs. In this study, only the temperature feedback reactivity in the moderator and electrodes are considered, which are expressed, respectively, as (Astrin, 1996):

qTM ¼ 60  U  1:04  103 ðT M  T 0 Þ; where

ð5Þ

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221

Fig. 2. Reactor configuration and the TFE schematic.

8  2   T þ175 T T T > > < 0:007413 TM0 þ 0:006795 TM0  0:014208 1 6 TM0 6 0 T 0 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi U¼  > T 0 þ100 TM TM > : 0:03027 > T 0 þ175  T0 T0 T0 T0 ð6Þ sffiffiffiffiffi sffiffiffiffiffiffi ! ! TE TC 2  1  7:44  10 1 q ¼ 0:12 T0 T0 T e

ð7Þ

The value of the drum reactivity is initially negative since the drums contain reactor poison plates. Outward rotation of the drums reduces the magnitude of this negative reactivity, thus adding positive reactivity to the core. The reactivity worth of control drums is curve fitted with the data from the reference (Paramonov et al., 1994) as a function of their angular position, h (in degrees):

qD ¼ 9:41  1011 h5  3:57  108 h4 þ 2:21  106 h3 þ 3:72  104 h2  2:5  103 h

ð8Þ

2.1.2. Heat transfer in the thermionic fuel element Fig. 3 presents the schematic dimensions of a TOPAZ-II TFE with an equivalent moderator cell. The TFEs are single-cell, cylindrical thermionic converters. A single-cell converter utilizes the entire length of the reactor core as a single device. The emitter tube is made of a monocrystal Mo–Nb alloy substrate and a thin tungsten surface layer. The polycrystal-Mo collector tube, placed coaxial with the emitter tube, is surrounded by a thin Al2O3 insulator and a stainless-steel cladding, separated by a helium (He) filled small gap. During nominal reactor operation, the inter-electrode gap of the TFEs is filled with Cs vapor at a pressure of approximately 20.0 torr. Beryllium oxide pellets on both ends of the fuel stack provide axial reflection and they have central holes that match up with the holes in the fuel. Fig. 3 presents the schematic diagram of a TOPAZ-II TFE model. The reactor thermal model for each channel is represented by a single TFE that is thermally coupled to the coolant flowing in the reactor core. Heat conduction is considered in all the solids such as fuel, electrodes, stainless steel pipes. Internal heat source is considered just

Fig. 3. The radial control volume division of the TFE.

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in the fuel as fission equivalent source term. The conversion efficiency of TOPAZ-II, for the purpose of determining the nominal thermal power of the nuclear reactor, is taken equal to 5%. Thus, 5% of the reactor power is delivered to the electric circuit from the electrodes and the coolant transports only 95% of this power. To simplify the calculation, the Al2O3 insulator is treated as part of the collector. The conduction model is based on cylindrical coordinate. The basic equations for heat conduction are as follows: Fuel:

qU cU

  @T U 1 @ @T U kU r ¼ þ QV r @r @t @r

ð9Þ

Emitter:

qE cE V E

   @T E kG ¼ PE ðT U  T E Þ þ PU eUE r T 4U  T 4E @t dG     k Cs ðT E  T C Þ  PE eEC r T 4E  T 4C þ dCs

ð10Þ

Collector:

qC cC V C

 kCs ðT E  T C Þ ð1  gTE Þ dCs     k He 4 4  PC eIS r T C  T SI þ ðT C  T SI Þ dHe

@T C ¼ PE @t







eEC r T 4E  T 4C þ

ð11Þ

2.1.3. Heat conduction in the whole reactor The basic structure of the TOPAZ-II reactor is solid except the narrow coolant channel and gas gaps, so the heat conduction at the core cannot be neglected. Meanwhile, there is radiation between the outer surface of reactor and space. As a result, the temperature of the reactor is not uniform along the radial direction. Therefore, to predict the thermal characteristics of the reactor, not only the single TFE but also the integrated reactor are modeled and calculated. Fig. 4 presents the two-dimensional control volume division of the reactor body. The moderator is divided into four layers in the radial direction, and each layer corresponds to a circle of TFE components. The cross sectional area of the control volume is the actual area of the moderator, deducting the area of the TFEs and coolant channel. The distance between the adjacent volumes equals that of the centers of TFEs located in the adjacent control volumes. The reflectors are also divided, including the lateral reflector and the upper and lower reflectors. During the calculation, the heat generated within the TFE is used as inner heat source of the moderator, and the temperature of each moderator layer is treated as the external boundary of TFEs. The thermal conduction through the moderator and reflector are depicted by the following differential equations: Moderator inside: i

i i i M cM V M

q

Stainless steel inner wall:

qSI cSI V SI

   @T SI kHe ðT C  T SI Þ þ eCS r T 4C  T 4SI ¼ PC @t dHe

 PSI H T SI  T f

þ pðri þ r iþ1 ÞLi ð12Þ

Stainless steel outer wall corresponding to moderator:

qSO cSO V SO

@T SO ¼ PSOI HðT f  T SO Þ @t    kCO2 ðT SO  T M Þ þ eSM r T 4SO  T 4M  PSOO dCO2

ð14Þ

Lateral reflector:

qiR ciR V iR

ð13Þ

Both the gas heat conduction and radiation heat transfer are considered in calculating the heat transfer of the gas gap between the adjacent solid structures. And the heat transfer boundary of the inner surface of the fuel rod and the outside surface of the equivalent moderator cell are zero flux conditions and the heat transfer boundary of the rod surface is the heat convection condition between the stainless steel walls and the sodium–potassium coolant. To obtain the detailed temperature distribution, the fuel pin is divided into three nodes each in the radial direction while the other solid structures just contain one radial control volume. In the axial direction, each layer is divided into numbers of nodes in both the fuel section and the reflector section based on the precision requirement, as shown in Fig. 3. The thermophysical properties of the working fluid and reactor materials are extracted from the references (Agnew, 1995; Xue et al., 2008; Von Arx and Alan Vincent, 1999) and shown in Table 2.

!

i T i1 M  TM r i  r i1 ! i kiM þ kiþ1 T iþ1 M M  TM þ Q iTFE 2 r iþ1  ri

dT M ki1 þ kiM ¼ pðri1 þ r i ÞLi M dt 2

! i dT R ki1 þ kiR T Ri1  T iR ¼ pðri1 þ r i ÞLi R dt 2 r i  r i1 ! i i iþ1 k þ kR T iþ1 R  TR þ pðri þ riþ1 ÞLi R 2 r iþ1  r i

ð15Þ

The radiation boundary condition is applied as the outer surface of the reactor body, as shown below:

kR

  end1 4 T end 4 R  TR ¼ eR r T end  T R s r end  r end1

ð16Þ

2.2. Radiator model The radiator removes heat from the coolant before it returns to the reactor core and rejects heat to outer space. The TOPAZ-II original radiator is a pumped loop radiator, which consists of the following elements: upper header, lower header, radiating elements and supporting frame. The headers are connected by 78 radiation elements. Each element consists of a stainless steel pipe and a copper strip welded to the pipe (Paramonov and El-Genk, 1995). Based on the design of the radiator of SPACE-R reactor power system, combined with the original TOPAZ-II radiator features, a truncated

Table 2 Thermophysical properties of NaK coolant and part of reactor materials. Parameter

Value

Thermal conductivity of NaK (Wm1K1)

kNaK ¼ 21:8 þ 7:33  103 T

Specific heat capacity of NaK (Jkg1K1) Thermal conductivity of emitter (Wm1K1) Thermal conductivity of collector (Wm1K1) Thermal conductivity of helium (Wm1K1) Thermal conductivity of stainless steel (Wm1K1)

cNaK ¼ 1:20  9:96  104 T þ 9:66  107 T 2  2:78  1010 T 3 kEmitter ¼ 120:0  1:4  102 T kCollector ¼ 110:0  1:5  102 T kHe ¼ 0:415 þ 3:12  103 T  9:71  106 T 2 þ 3:50  107 T 2:5  3:70  109 T 3 kSteel ¼ 10:0 þ 1:7  102 T

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W. Zhang et al. / Nuclear Engineering and Design 307 (2016) 218–233 Table 3 Design features of the heat pipe radiator.

Fig. 4. Control volume division of reactor.

cone radiator using heat pipe is designed in the former work, as shown in the right part of Fig. 5. An annular transport duct which has a rectangular cross section is employed. In this situation the pipes are coupled by forced convection with the duct fluid. The heat pipes actually penetrate the duct wall from both the top and bottom sides with a staggered arrangement. The transport duct carries the fluid heated by the reactor core to the evaporator section of heat pipes. The design values of the parameters is shown in Table 3. 2.2.1. The radiating element model A heat pipe consists mainly of a shell as the container, a capillary structure or wick to transport liquid by surface tension, and a vapor region to provide vapor passage. The capillary structure is saturated by the working fluid in the liquid state, and the vapor space is occupied by the working fluid in the vapor state. Heat and cooling devices are applied to the outer surface of the heat pipe

Parameter

Value

Transport duct Wall material Length of cross section/mm Width of cross section/mm Diameter of the duct ring/mm Number of heat pipes

Stainless steel 110 40 1800 200

Heat pipe Pipe wall material Porous wick material/type Working fluid Heat pipe total length/mm Evaporator length/mm Condenser length/mm Radius of vapor space/mm Wick average porosity/mm Wick thickness/mm Pipe wall thickness/mm

Inconel Stainless steel/mesh Potassium 600 100 500 7.5 0.6 0.5 1

Fin Fin material Fin Thickness/mm Effective surface emissivity

Carbon–Carbon 0.4 0.9

wall. In the longitudinal direction, the heat pipe consists of an evaporator and a condenser, also shown in Fig. 6. The liquid potassium in the wick structure flows very slow for the much larger density of the liquid potassium than that of vapor. In addition, the thermal conductivity of liquid sodium is large and the thickness of the wick is very thin. It is then assumed that influence of the liquid flow in the wick is negligible and the pure conduction model is applied in the wick. Therefore, heat transfer through the wick structure can be analyzed using heat conduction model. The governing equations for the container and wick heat conduction is expressed as follows:

Ci

    @T i 1 @ @T i @ @T i þ i ¼ 1; 2 Rki ki ¼ R @R @Z @t @R @Z

ð17Þ

where, the subscripts i ¼ 1 and i ¼ 2 denote the pipe container and the capillary structure, respectively. The effective thermal conductivity of the wick can be expressed as follows:

Fig. 5. Schematic of the heat rejection radiator.

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W. Zhang et al. / Nuclear Engineering and Design 307 (2016) 218–233

The axial gradient for vapor density:

   

dX q tg X q Rv T dq 1 dp ¼  2 tg  tl þ 1 hfg M dZ dZ t dZ p The axial gradient for vapor quality:

dX q t2 ¼ dZ ðtg  tl ÞV 2 

("

1 V 2Xq  þ Mf p



tg Rv T 1 hfg M t2

ð25Þ

#

dp FV 2  dZ 8DtM f

)

_0 2m ðtg  tl Þ DV

t2

ð26Þ The axial gradient for vapor velocity:

Fig. 6. Schematic diagram of the potassium heat pipe.

k2 ¼

kl ½ðkl þ ks Þ  ð1  eÞðkl  ks Þ ½ðkl þ ks Þ þ ð1  eÞðkl  ks Þ

_ 0 Vðtg  tl Þ dX q VX q dV tm ¼ þ þ dZ D t dZ p ð18Þ

During the startup of the heat pipe, the vapor zone may be free molecular, choked, and continuum flow at various times. Even though the Reynolds number depends on the geometry of the heat pipe and the actual heat transfer rate, the results cited from the reference (Bowman, 1987) show that the vapor flow in heat pipe can be assumed laminar. The vapor with small density has a very low pressure. Even for relatively small heat flux, the vapor will flow very fast, accelerating towards sonic velocity because viscous action causes pressure and density to decrease. As a result, the compressibility should be considered. The vapor pressure drop due to friction cannot be recovered completely in the condenser section. Therefore, the temperature distribution along the length of the heat pipe are not isothermal. Studies of the distributions of temperature, pressure, and velocity in the vapor passage are essential to predict correct heat pipe performance. Thus, compressible, one-dimensional laminar flow in the vapor passage is considered. For the purpose of formulating the mass, momentum, and energy equations in one-dimensional form, the average velocity is taken. In addition, friction at the liquid–vapor interface, and momentum and energy factors are similarly calculated. Mass, momentum and energy conservation equations are given as follows:

D

d _0 ðqVÞ ¼ m dZ



ð27Þ

The axial gradient for vapor pressure: Mf m _0 VD

dp  Ef ¼ dZ

h

2hfg þ

ðtg tl Þ

t

þ E1 f

 i E V2 h V2 ðt  t Þ F h0  h þ f 2 þ 20  E1 tfg 8D  gt2 l f   2 hfg M h X t M ðt  t Þ c R T  E f fgp q t2g 1  hRv M  E f gt2 l hp Mv Tp V2

ðtg tl Þ

t

f

fg

f

ð28Þ The axial gradient for vapor temperature:

dT Rv T 2 dp ¼ dZ hfg Mp dZ

! ð21Þ

A friction factor for the surface is written as

F¼

8sv qV 2

ð22Þ

Momentum and energy factors are expressed as

Mf ¼

Ef ¼

Z

1 DV

2

1 DV 3

D

U 2 dy

ð23Þ

0

Z

D

U 3 dy

ð24Þ

0

Based on Eqs. (19–24), the gas state equation and other constitutive equations, the following equations can be derived. These equations are solved to obtained the vapor flow characteristics in the code.

ð29Þ

The basic mechanism of heat transfer through the fin is conduction combined with radiation. A radiating element consists of a heat pipe, the carbon–carbon strip welded to it and two half part of the flat fin. For each radiating element, the radiation can also be divided into four parts: heat dissipated from the welded strip (Qi2), naked part of the heat pipe wall (Qi3) and two surfaces of the flat carbon–carbon fin (Qi1 and Qi4), as shown in Fig. 7. The radiation of the welded fin and naked part of the heat pipe wall can be estimated by

ð20Þ V2 h0 þ 0 2

FV 2 8D

fg

ð19Þ

dp d F qV 2 þ ðM f qV 2 Þ ¼  dZ dZ 8D " !# Ef V 2 d _0 ¼m qV h þ D dZ 2



tg Rv T dp 1 dZ t hfg M

Fig. 7. Schematics of the C–C fin and radiation heat transfer.

W. Zhang et al. / Nuclear Engineering and Design 307 (2016) 218–233

Q i1 ¼ f wc ewc r

Q i4 ¼ f pw epw r

  phðD þ dfin Þ  4 Li T wci  T 4s 360

225



 ð360  hÞpD  4 Li T pwi  T 4s 360

ð30Þ

ð31Þ

where, f wc and f pw are the geometric view factors of the welded strip and the naked pipe, respectively; T wci and T pwi are the temperature of the welded strip and the naked pipe wall associated to the control volume i of the pipe. The T wci is obtained by one-dimension conduction calculation of the welded strip with a radiation boundary of the outer surface. Since the fin is very thin, the temperature distribution is assumed to be two-dimensional. In order to simplify the calculation, a quasi steady-state thermal model is developed to analyze the heat transfer of the fin. The energy balance equation of the flat strip is given as: 2

d T fin 2

dx

  ðf sout esout þ f sin esin Þr T 4fin  T 4s Q i2 þ Q i3 ¼ ¼ kdfin dx  k  dfin  Li

ð32Þ

2.2.2. Mathematical model of the transport duct The transport duct in the middle of the radiator is designed with two entrances and two exits. Just a quarter of the radiator is taken to be analyzed for its symmetry. One-dimensional flow and heat transfer model is established. The control volume division is shown in Fig. 8. To simplified the calculation, each five adjacent heat pipes are considered to have the same flow and heat transfer performance and only one is calculated for each pipe group. Therefore, ten heat pipes are modeled, and each one represents five adjacent heat pipes of the actual radiator. The coolant temperature and the convective heat transfer coefficient are obtained by the transport duct calculation, and they are used as the evaporator convection boundary of the heat pipe. The heat pipe and fin calculation will return dissipation heat to the transport duct model as its inner heat source.

Fig. 9. A line diagram of the models developed for the improved TOPAZ-II system.

2.3. Numerical scheme TASTIN-HP is developed to simulate the dynamic behavior of steady-state, start-up and credible accidents of the improved TOPAZ-II system by a heat pipe radiator. Modular modeling technique has been applied to develop the code, which is convenient to expand, upgrade and transplant. The models developed for TASTIN-HP are shown in Fig. 9. The thermal-hydraulics model and neutron kinetics model of the core and heat pipe radiator model are illustrated in the previous sections in detail.

Fig. 10. Calculating scheme of the system.

Fig. 8. Control volume division of the transport duct.

Fig. 10 shows the calculation scheme of the improved TOPAZ-II system in this study. Four TFE elements are modeled, and each one represents the TFEs with the same radial distance from the core center. The moderator and reflector are also modeled. Considering the symmetry of the heat pipe radiator, only a quarter is analyzed. In addition, the reactor, heat pipe radiator, the EM pump, accumulator, coolant pipes, inlet plenum and outlet plenum are modeled in this calculation. At each time step, the core model and heat rejection loop model except the heat pipe are calculated first, as shown in Fig. 11. The transient characteristics of these components are obtained

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by solving a set of ordinary differential equations. These ordinary differential equations founded for the system can be written as:

dy ; xÞ ¼ f ðt; ~ y; ~ y dt

ð33Þ

where yis the state vector for the calculated variables, such as neutron density, temperatures and pressure. These ordinary differential equations are usually called illconditioned equations because of their stiffness (such as the point kinetics equations). The Gear method (Gear, 1971) equipped with Adams predictor–corrector method, is adopted for the numerical integration of the differential equations to improve the stability and precision of calculation. Gear method is an auto-adaptive algorithm which can select step size automatically and change orders, which uses implicit multistep integration method with backward difference method. The coding of the equations and this numerical solution method in the reactor system have been verified in the former works (Ma et al., 2015). After the calculation of the reactor and NaK coolant loop components, the radiating element calculation subroutine is called. The coolant temperature in the transport duct and the heat transfer coefficient between the coolant and heat pipe evaporator are used as the thermal boundary of the heat pipe evaporator. First, the conduction of the heat pipe container and wick are calculated with the third kind of boundary condition. Then the state of the vapor region is determined. If the continuum flow is established in the whole channel, the vapor property equations (Eqs. 25–29) are solved and the thermal resistance will be obtained. If not, the melting process is calculated and the length of the continuum flow zone will be obtained. Finally, the radiating element calculation returns the quantity of heat transferred from the coolant to the heat pipe to the transport duct model. After the above processes,

one time step calculation is finished and the calculation enters the next step. The numerical method of the radiating element calculation is different from the main system components models. For the container and capillary wick of the heat pipe, the FEM (finite element method) is adopted to calculated the heat conduction. The onedimensional vapor flow model is used to analyze the central vapor passage and a series of first-order differential nonlinear equations are obtained. The Runge–Kutta method is employed to solve these equations. This numerical solution method has been verified in the reference (Wang et al., 2013). The equations of container, wick and vapor region are solved separately. In the interface between the wick-liquid and vapor, coupling is implemented. Considering the symmetry of the fin between the adjacent pipes, just half of it is modeled. The finite difference method (FDM) is used to discretize the Eq. (32) and the discrete equations obtained are calculated by Newton iterative method. The number of axial control volumes equals to that of the heat pipe. The root temperature of fin is assumed to equal the temperature of heat pipe outer surface. Fig. 12 shows the two-dimension mesh of the heat pipe and C–C fin. 3. Results and discussion 3.1. Analysis of the steady state The results of the steady state condition were obtained by the transient calculation until the values of all the variables became steady. Fig. 13 shows the temperature distribution of each layer of the central channel components along the axial direction, in which, the fuel pellets show the radial maximum temperature while others display the radial average value. From Fig. 13, it can be seen that the highest temperature of the fuel is 1725 K, which

Fig. 11. Algorithm flow chart of the TASTIN-HP code.

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Fig. 12. Schematic of the radiating element mesh.

Fig. 13. The axial temperature distribution of each layer.

is much lower than the pellet melting temperature. Mo-Nb alloy emitter has the maximum temperature of 1602 K, and the polycrystalline Mo collector has the highest temperature 852 K. All of them are far away from the safety limit value. At the steady state, the temperature difference between the coolant and the inner pipe wall is about 4 K. The outer casing for CO2 air gap and moderator has the substantially same temperature of the coolant. Fig. 14 shows the variation of average temperature of the coolant, evaporator and condenser along the coolant flow direction. After extension, the coolant temperature decreases from 832 K to 738 K which is agreement with the TOPAZ-II primal design values. So the heat pipe radiator has the same heat dissipation capability as the pumped loop radiator under the nominal condition. It also can be seen that the main heat resistance locates between the coolant in the transport duct and outer surface of evaporator. The maximum temperature difference between them is 33 K (the coolant inlet side) and the minimum temperature difference is 22 K (the coolant outlet side). The temperature difference between the evaporator and condenser is just about 4 K. Hence, the convectional heat transfer enhancement between the coolant and the heat pipe is meaningful to improve the heat dissipation capability of radiator.

Fig. 14. Average temperature variations of the coolant, evaporator and condenser in the coolant flow direction.

Fig. 15 presents the variations of average temperature, pressure, density and velocity of the vapor along the transport duct. As we can see, the average temperature of the working fluid in different heat pipes decrease from 790 K to 719 K. The pressures and densities also decrease following the temperature drop. But the velocities of the heat pipes increase along the transport duct. Although the powers of heat pipes reduce gradually along the coolant flow direction, the lower temperature causes reducing of the vapor density so that the vapor velocity becomes larger. Fig. 16 shows the axial distributions of vapor Mach number of the ten calculated heat pipes. As we can see, the vapor Mach number of the heat pipe near the coolant outlet is larger than that of the heat pipe near the inlet. In each heat pipe, the vapor velocity increases from zero to the maximum value and then decreases to zero. In the evaporator section, the vapor accelerates due to the mass adding while in the condenser section the vapor slows down with the vapor condensing. The Mach number of the colder heat pipe is larger than the hotter one for the lower vapor pressure and higher speed. Meanwhile, the maximum Mach number of these heat pipes reaches about 0.095. As a result, the power throughput of the heat pipe is still much lower than its sonic limit.

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Fig. 15. Average temperature, pressure, density and velocity variations of vapor in the coolant flow direction.

Fig. 16. Axial variations of vapor velocity of different heat pipes.

The temperature distributions of the reactor, a chosen heat pipe and its fin are displayed in Fig. 17. From the temperature contour of moderator and radial reflector in the left part of the following figure, we can see the hottest zone is located in the upper center

Fig. 18. Thermal powers and electric power.

of the reactor. The highest temperature of the moderator is about 840 K. The maximum temperature gradient is located near the interface of the moderator and radial reflector. It would lead to the maximum thermal stress of moderator. Form the temperature

Fig. 17. Temperature distribution of the heat pipe wall and wick.

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contour of heat pipe wall and wick, we can see that the temperature difference between the outer surface of the shell and inner surface of the wick is about 5 K. In the condensing section, the

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temperature of the inner surface is higher than that of the outer part and the temperature difference is about 0.7 K. This difference is caused by the length difference between the condenser and evaporator. Meanwhile, as shown in Fig. 17, the condensing section has a good isothermal performance. As a result, the fin temperature reduces in the direction vertical to the heat pipe and has a good isothermal performance in the pipe axial direction. 3.2. Analysis of the start-up procedures

Fig. 19. Temperature reactivity feedback effects during start-up.

Start-up sequence of the TOPAZ-II system is very complex, including the rotation of the safety drums and control drums, electromagnetic pump, interelectrode gap gas changing and other device actions. The start-up procedures simulated herein are assumed only for the purpose of demonstrating the capabilities of the model, so only the features that the thermal-hydraulic analysis needs are considered. Meanwhile, to compared with the startup of the original TOPAZ-II system, the same start-up sequence is used for the improved system analysis. These procedures call for the reactor star-up to begin by rotating the safety drums outward. Subsequently, the control drums are also rotated outward to make the reactor become critical. The rotation speed and direction of the control drums are adjusted to increase the reactor power at a rate of 600 W/s until it reaches 35 kW, then at 80 W/s until it reaches

Fig. 20. Temperature distributions of the reactor at different times.

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Fig. 21. Temperature of each layer during start-up (core central channel).

Fig. 24. Temperature distribution of the hottest fin at different times.

Fig. 22. Coolant temperature of core inlet and outlets.

Fig. 25. Fin temperature distribution.

Fig. 23. Startup of heat pipe.

115 kW. Approximately 1500 s after the reactor becomes critical, the TFEs are turned on by gradually replacing the helium gas in the interelectrode gap with cesium vapor. Startup batteries power

the EM pump intermittently prior to reactor startup. During startup, the battery current limits the NaK flow rate to 25% of full flow. When the coolant temperature in the lower collector reaches 500 K, the battery current switches to the maximum. Flow rate increases from 50% to 100% as the reactor power increases and the TFEs can provide useful voltage. At startup, the entire system including the reactor and the heat rejection system is assumed to be at 300 K. Fig. 18 shows the thermal power of the core, electric power added to the emitter and the radiator heat rejection power of the original system and the improved system. The code in this paper doesn’t integrate the thermionic model and the electric power is added to the electrode heat transfer model as an inner heat sink.

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As we can see, the radiator power is very low before 500 s. It is because that the working fluid of the heat pipe is still solid, and the reactor power is also very low. But after about 500 s, the radiator power increases quickly, for the melting of the potassium. After about 750 s, the heat pipe radiator is totally start-up. Meanwhile, we can see that these two types of radiator almost have the same thermal characteristics at the high level of power. Fig. 19 presents the temperature reactivity feedback of each material. Shortly after the reactor becomes critical, the temperatures of the fuel and emitter increase very rapidly, introducing a negative feedback reactivity, causing the total temperature reactivity feedback of the reactor to decrease. The ZrH moderator begins to heat up after a long delay time, approximately 300 s after the reactor becomes critical. The total reactivity feedback reaches the minimum of negative $0.26 at 260 s. At such time the rate of temperature positive reactivity feedback, due mostly to the ZrH moderator, equals the sum of the rates of the temperature reactivity feedbacks of the other core components. Beyond this point, the former continues to increase causing the total temperature reactivity feedback to turn to positive value. However, the rise rates of the ZrH moderator temperature and temperature positive reactivity feedback decreases as the ramp rate of the reactor thermal power decreases from 600 W/s to 80 W/s. The total temperature reactivity feedback reaches $0.88 at 2500 s. The temperature distribution of the reactor at different times is shown in Fig. 20. As we can see, the highest temperature of moderator increases from 300 K at initial time to 828 K at 3000 s. Meanwhile, during the core heating up progress, the radial and axial temperature differences increase gradually. The maximum radial temperature gradient locates near the interface of the moderator and lateral reflector. Fig. 21 shows the calculated highest temperatures of each layer of the central channel. After 1700 s, the fuel and emitter temperatures rise rapidly due to the charge of cesium vapor with a lower

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conductivity coefficient. As a result, the negative reactivity feedback of the fuel becomes larger, shown in Fig. 19. In contrast, the temperature of the collector, the coolant and the moderator has a slight temporary drop. Following the startup of the TFEs the rapid increase in the fuel and emitter temperatures increases the energy stored in the TFEs, but at the expense of lowering the heat transfer to the coolant. In Fig. 21, the largest temperature difference between emitter and collector is just 440 K before the start-up of TFEs, but reaches about 1050 K at 3000 s. The highest temperatures of the fuel, emitter, collector approach 2003 K, 1892 K and 852 K, respectively. The coolant inlets of the four channels modeled have the same temperature, but the coolant outlets have different temperatures for nonuniform thermal features of these channels. In Fig. 22, the temperature changes of the core inlet and four channel outlets are presented. As we can see, the central channel has the highest coolant outlet temperature, approximately 843 K. As mentioned in the prior part, the gas change in the interelectrode gap leads to a temperature drop of the components outside the interelectrode gap. Such a drop causes the energy transport from the TFEs to the primary coolant to reduce, and in turn to the radiator radiating power, to temporarily decrease, as shown in Fig. 18. Subsequently, the decrease of the coolant inlet temperature occurs after a short delay time for the thermal inertia of the coolant loop. At 3000 s, the coolant inlet temperature approaches 748 K. Fig. 23 shows the axial temperature distribution at the liquid– vapor interface during the start-up procedure. Since the NaK coolant heats the evaporator of the heat pipe, the temperature of evaporator increases and a large temperature gradient is established in the entire heat pipe as clearly shown in Fig. 23. It illustrates that the heat pipe is not effective, due to extremely small vapor density at initial stage. Temperature at the rest of the heat pipe is almost the same as the initial condition (300 K). Owing that the heat pipe evaporator is fully immersed in the NaK coolant and it has a large

Fig. 26. Key parameters variation at vapor region.

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effective thermal conductivity, the temperature drop of evaporator is very small and it can be considered to be isothermal. After the evaporator working fluid melting, a continuum flow region is established in the vapor space of the evaporator so that heat is mainly transferred through latent heat transport. However, most of the working fluid at the condenser is in the solid state, and free molecule flow prevails in most of the condenser. As time proceeds, the continuum flow region is expanded. This phenomenon can also be illustrated in Fig. 24, the temperature contour of the C–C fin. A large temperature gradient exists, where the continuum flow front is located. This front boundary moves toward the end of heat pipe slowly. At about 620 s, the continuum flow develops in the entire vapor space. Until this moment, the sonic limitation is encountered owing to the large temperature gradient in the vapor space. After about 700 s, the heat pipe becomes nearly isothermal, due to the continuous heat input, as we can see from Fig. 24. But the heat pipe average temperature still increases with the coolant temperature of the transport duct. Fig. 25 shows the average fin temperature distribution of the selected five heat pipes along the direction vertical to the heat pipe after the continuum vapor flow establishing in all heat pipes. As we can see, the total temperature difference of the fin increases following the coolant temperature. For each fin, the temperature gradient is larger near the root than that of the outer part. The

maximum temperature difference of the first heat pipe HP1 is 48 K and that of the HP9 is 34 K at 3000 s. Fig. 26 shows the variation of the vapor parameters in the pipe axial direction of the first pipe at different times. It can be seen that the vapor pressure, temperature and density all reduce from evaporator to condenser. At 620 s, the temperature drop along the pipe is 22.3 K, but it is 0.5 K at 3000 s. Meanwhile, the pressure drop is 130 Pa at 620 s but 15 Pa at 3000 s. The reason is that at 620 s, the velocity of the heat pipe is very large than that at 3000 s, as the Mach number curves in Fig. 26. The maximum Mach number is 0.33 at 620 s but only 0.05 at 3000 s. The higher speed leads to the larger pressure drop of vapor. The temperature of vapor equals to the saturation temperature, as a result, the temperature drop is also larger under colder condition. The sonic, capillary and entrainment limits of the selected heat pipes are shown in Fig. 27. As we can see, the capillary limit and the sonic limit change a lot with time. At the initial cold stage, the values of these variables are very low, but increase very fast with the increasing temperature. As a result, these two limits may be encountered firstly during the start-up procedure. Because the heat pipe power increases slowly, the heat pipe can work in a large margin of safety during the nominal condition. Different from the sonic limit and entrainment limit, the capillary limit is relatively stable. For the four heat pipes selected, the maximum value

Fig. 27. Heat transfer limits of the heat pipe during startup.

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of the capillary limit is about 1500 W. During the accident conditions, the coolant temperature would increase to a large value which may lead to the capillary limit of the heat pipe. But during the normal operation conditions, the heat pipe radiator designed has enough heat rejection capability. 4. Summary and conclusions In this work, a system analysis code is developed for the improved space thermionic reactor TOPAZ-II system using a heat pipe radiator. The heat pipe radiator applies the two-dimensional heat pipe analysis model. The reactor neutron physical and thermal-hydraulics models, the pump model and other components are established in this code. Some important conclusions can be summarized as follows: (1) The steady state calculation results are in good agreement with that of the original TOPAZ-II system using a pumped loop radiator. The designed radiator can satisfied the heat rejection requirement of the TOPAZ-II system under normal conditions and has a good isothermal performance; (2) during the startup process, the reactivity temperature feedback of the moderator is the main part of the total reactivity feedback. Furthermore, the moderator has a positive feedback coefficient, which brings many difficulties to the reactivity control; (3) the sonic limit and entrainment limit of the heat pipe rise fast with the temperature increasing and are the threats for the start-up procedure during the initial stage. The capillary limit changes a little with the temperature and keeps at a low level about 1600 W and may be encountered under the higher working temperature.

Acknowledgements The present study is support by the National Natural Science Foundation of China (Grant No. 91326201).

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