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Numerical study on crossflow printed circuit heat exchanger for advanced small modular reactors
International Journal of Heat and Mass Transfer 70 (2014) 250–263
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical study on crossflow printed circuit heat exchanger for advanced small modular reactors Su-Jong Yoon a,⇑, Piyush Sabharwall a, Eung-Soo Kim b a b
Idaho National Laboratory, 2525 Fremont Avenue, Idaho Falls, ID 83415-6188, USA Seoul National University, Seoul, Republic of Korea
a r t i c l e
i n f o
Article history: Received 16 July 2013 Received in revised form 30 October 2013 Accepted 31 October 2013
Keywords: Crossflow Heat exchanger PCHE Analytical model Thermal design Cost estimation Advanced SMR
a b s t r a c t Various fluids such as water, gases (helium), molten-salts (FLiNaK, FLiBe) and liquid metal (sodium) are used as a coolant of advanced small modular reactors (SMRs). The printed-circuit heat exchanger (PCHE) has been adopted as the intermediate and/or secondary heat exchanger of SMR systems because this heat exchanger is compact and effective. The size and cost of PCHE can be changed by the coolant type of each SMR. In this study, the crossflow PCHE analysis code for advanced small modular reactor has been developed for the thermal design and cost estimation of the heat exchanger. The analytical solution of singlepass, both unmixed fluids crossflow heat exchanger model was employed to calculate a two-dimensional temperature profile of a crossflow PCHE. The analytical solution of crossflow heat exchanger was simply implemented by using built-in function of the MATLAB program. The effect of fluid property uncertainty on the calculation results was evaluated. In addition, the effect of heat transfer correlations on the calculated temperature profile was analyzed by taking into account possible combinations of primary and secondary coolants in the SMR systems. Size and cost of heat exchanger were evaluated for the given temperature requirement of each SMR. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Small modular reactors (SMR) are being developed to provide heat and electricity to the remote regions where the large nuclear power plants cannot be built. In addition, recently, the SMRs have been considered as good energy sources for the process heat application. Contrary to typical water-cooled SMRs, advanced SMRs use various coolant materials such as gases (helium, supercritical carbon dioxide), molten-salts (FLiNaK, FLiBe and KFZrF4), and liquid metal (sodium). The operating temperature and pressure of an advanced SMR can be varied by the type of reactor coolant. Table 1 describes the typical temperature and pressure conditions of SMRs according to the reactor coolants. Inlet and outlet temperatures of primary and secondary fluids are based on the intermediate heat exchanger (IHX). Thus, the inlet temperature of the primary fluids shown in Table 1 is the outlet temperature of reactor coolant. Fig. 1 shows some of the potential options that the IHX and secondary heat exchanger (SHX) could be configured in the system. It could be linked to production of power system and also could be used for providing the process heat from the reactor for industrial applications. In this case, both IHX and SHX could be either con-
⇑ Corresponding author. Tel.: +1 208 526 5407/357 8858; fax: +1 208 526 2930. E-mail addresses: [email protected], [email protected] (S.-J. Yoon). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.10.079
ventional-type heat exchanger or compact heat exchanger, such as PCHE. The word ‘‘small’’ in SMRs refers not only to the reactor power, but also to the size of the reactor. In this regard, the compact heat exchanger can be a good design option because the compact heat exchanger has a large heat transfer surface area per unit volume, which can result in the reduced size, weight, and cost. Fig. 2 shows the surface compactness that is a criterion for classifying the heat exchanger. Printed circuit heat exchanger (PCHE) is a compact heat exchanger and has been considered one of the candidate heat exchangers in designs of advanced SMRs. Fine grooves in the plate of the PCHE are made by employing the technique used for making printed circuit boards. The PCHE has been commercially manufactured by Heatric™ [1]. Fig. 3 shows the typical PCHE manufactured by Heatric Ltd. The PCHE can be classified by the flow configurations. The arrangement of flow channels are parallel in the parallel and counter flow PCHEs, whereas it is orthogonal in the crossflow PCHE. Temperature distribution of the crossflow heat exchanger is twodimensional so that it is more complicated to analyze than parallel or counter flow heat exchangers. Due to this complexity, many previous studies have used computational fluid dynamics (CFD) codes such as CFX or FLUENT. However, CFD simulation is not adequate for the design process of the PCHE since CFD simulation requires vast computational time and cost. In the CFD model of PCHE, since
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Nomenclature A C CM Cop cp C⁄ CP D, d f h I0 k L M _ m N NTU Nu OP P p Pe Pr Q R Re T t U X Y
area or overall heat transfer surface area (m2) _ p;j Þ flow stream heat capacity rate ð¼ mc material cost factor (USD/kg) operating cost factor (USD/Wh) specific heat capacity (kJ/kg K) the ratio of heat capacity rate (=Cmin/Cmax) capital cost of PCHE (USD) diameter (m) friction factor convective heat transfer coefficient (W/m2 K) modified Bessel function of the first kind and zero order thermal conductivity (W/m K) length (m) total mass of heat exchanger (kg) mass flow rate (kg/s) number of flow channel number of transfer unit Nusselt number (=hD/k) operating cost of PCHE (USD) channel pitch (m) or Pressure (Pa) the order of scheme Peclet number (=Re Pr) Prandtl number (=cpl/k) heat duty (W) thermal resistance (m2 K/W) Reynolds number (=qVD/l) temperature (K) thickness (m) overall heat transfer coefficient (W/m2 K) normalized length in x-coordinate (=x/L1NTU1) normalized length in y-coordinate (=y/L2NTU2) or Operation period of heat exchanger (yr)
the structure of PCHE is complicated and the dimensions of flow channel of PCHE are small, the fine structure of mesh is required to obtain accurate results. In this study, we developed the crossflow PCHE analysis code to perform the thermal design and cost estimation of PCHE.
2. Development of crossflow PCHE analysis code In order to perform the thermal design and cost estimation of a crossflow PCHE, the analysis code was developed based on the MATLAB program [8]. The analytical solution of single-pass, both unmixed fluids crossflow heat exchanger was employed to solve the temperature profile of crossflow PCHE. Although typical PCHEs manufactured by Heatric™ have zigzag-shaped flow channel, the straight pipe is assumed in this study. The analytical modeling method, analytical solution, and thermal design procedure are described in following sections.
V
volume (m3) or Fluid velocity (m/s)
Greek and symbols D difference d plate thickness e effectiveness of PCHE (=Q/Qmax) go surface efficiency h dimensionless temperature (=(TT2,in))/(T1,inT2,in) l dynamic viscosity of fluid (Pa s) _ DP=qÞ P pumping power ð¼ m q density (kg/m3) U approximated exact solution by Richardson extrapolation method / numerical solution Subscript f h HX in j max min L out p tot w x–s
fin grid width or Hydraulic diameter heat exchanger inlet index of fluid (1: primary, 2: secondary) maximum minimum Laplace transformation operation outlet plate total wall cross-sectional
2.1. Analytical modeling and solution of crossflow heat exchanger For single-pass crossflow design, a solution to the problem of both fluids unmixed in the heat exchanger was obtained by Nusselt [9] in the form of analytical series expansions by assuming the longitudinal conduction can be neglected. Since the flow mixing is not occurred in the PCHE, this model is applicable. The analytical model of crossflow PCHE has been developed by the energy balance between the hot and cold fluid sides shown in Fig. 4. The energy balance equations in Fig. 3 are as follows: Fluid 1 (Hot fluid): @T _ 1 cp;1 T1 _ 1 cp;1 T1 þ 1 dx dm dm dQ |{z} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} @x |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Heat transfer rate ¼ 0 Fluid enthalpy rate Fluid enthalpy rate from fluid to wall into the C:V: out of the C:V: ð1Þ
Table 1 Typical temperature and pressure conditions of advanced SMRs. Coolant type
Water-cooled [2] Gas-cooled [3] Liquid metal-cooled [4] Molten salt-cooled [5]
Coolants
Operation conditions
Primary/secondary
Inlet/outlet temperature (°C)
Water/Water He/He Na/Na FLiBe/FLiNaK
Primary
Secondary
323/295 950/637 545/390 700/650
200/293 351/925 320/526 600/690
Primary/secondary pressure (MPa)
15.0/5.0 7.0/7.0 0.1/0.1 0.1/0.1
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(a) Option 1 – 1 SHX
IHX
Reactor
SHX
IHX
Power Conversion System
SHX
IHX
Reactor
SHX
(b) Option 2 – 2 SHXs Power Conversion System
1
SHX
IHX
2
2
3 4
(c) Option 3 – 3 SHXs
Power Conversion System
Reactor IHX
3
1
5
Fig. 3. Section of crossflow PCHE [7].
In above energy balance equations, the heat transfer from the hot fluid to the cold fluid can be calculated by the term dQ that can be expressed by using the rate equations for convection and conduction as follows:
6
IHX 6
SHX
g0;1 h1 ðT1 Tw;1Þdxdy
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
SHX 4
dQ ¼
5
Fig. 1. Schematic of potential heat exchanger arrangements.
Fluid 2 (Cold fluid):
@T _ 2 cp;2 T2 þ 2 dy _ 2 cp;2 T2 dm dm dQ |{z} @y |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ 0 Fluid enthalpy rate þ Heat transfer rate Fluid enthalpy rate from fluid to wall into the C:V: out of the C:V:
¼ ð2Þ
Convection
!
from hot fluid to wall Tw;1 Tw;2 dxdy kw dw |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ð Conduction within the wall Þ
Fig. 2. Heat transfer surface area density spectrum of heat exchanger surfaces [6].
g0;2 h2 ðTw;2 T2 Þdxdy
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
¼
!
Convection from wall to cold fluid
ð3Þ
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Fig. 4. Energy balance control volume for crossflow heat exchanger [10].
By employing the definition of the overall heat transfer coefficient U ¼ dAdQDT, but neglecting the fouling thermal resistances, the heat transfer rate dQ can be given by:
dQ ¼ UdAðT1 T2 Þ
ð4Þ
where dA = dxdy. By substituting Eq. (4) into Eqs. (1) and (2), and simplifying, the following differential equations for the crossflow heat exchanger are obtained:
@h1 ðX; YÞ þ h1 ðX; YÞ ¼ h2 ðX; YÞ @X
ð5Þ
Since the Laplace transform of the derivative of function is n o L dFðnÞ ¼ sLfFðnÞgn!s Fð0Þ, Eq. (14) can be expressed as dn n!s
follow:
ph2L þ h2 ðX;0Þ þ h2L ¼ h1L )
By applying Laplace transform for variable X to Eq. (13), the following expression can be obtained with the boundary condition of h1 in Eq. (9):
n n o 1 L L dh dX
o
X!s
@h2 ðX; YÞ þ h2 ðX; YÞ ¼ h1 ðX; YÞ @Y
ð6Þ
UA UA 1 ¼ ¼ _ j cp;j m Cj Cj
Z
UdA ðj ¼ 1; 2Þ
ð7Þ
ð8Þ
If the inlet temperature distributions of primary and secondary sides are assumed to be uniform, following two boundary conditions are accompanied.
h1 ð0; YÞ ¼ 1;
h2 ðX;0Þ ¼ 0
ð9Þ
The solution of systems of partial differential equations can be obtained by implementing the Laplace transform and inverse transform. Exact solutions of this problem have been reported by Nusselt [9,11]. And there are various crossflow heat exchanger models [12–19], but it is proven that those all models are alternative expressions of Nusselt’s model [20]. Thus, in this study, the Nusselt model is employed as the analytical solution of the crossflow heat exchanger. First, the Laplace transforms for each variable can be defined as follows:
hiL ¼ hiL ðX; pÞ ¼ Lfh1 ðX; YÞgY!p ;
i ¼ 1; 2
ð10Þ
~hiL ¼ ~hiL ðs; pÞ ¼ Lfh1L ðX; pÞg ; X!s
i ¼ 1; 2
ð11Þ
LfLffðX; YÞgX!s gY!p ¼ LfLffðX; YÞgY!p gX!s
ð12Þ
To obtain the solution of the partial differential equations, first, the Laplace transform for variable Y is applied to Eqs. (5) and (6).
dh1L þ h1L ¼ h2L dX dh2L þ h2L ¼ h1L dY
þ Lfh1L gX!s ¼ Lfh2L gX!s )
ð16Þ
ðs þ 1Þ~h1L ¼ ~h2L þ 1p
~h1L ¼
The number of heat transfer units, NTU, is defined as a ratio of the overall thermal conductance to the heat capacity rate. NTU is given by:
NTUj ¼
Y!p
From algebraic Eqs. (15) and (16), ~ h1L and ~ h2L are given by:
where 0 6 X (=x/Lx) 6 NTU and 0 6 Y (=y/Ly) 6 C⁄NTU. The dimensionless temperature hj is defined by:
hj ¼ ðTj T2;in Þ=ðT1;in T2;in Þ ðj ¼ 1; 2Þ
ð15Þ
ðp þ 1Þh2L h1L ¼ 0
ð13Þ ð14Þ
! pþ1 1 1 1 ¼ p p ps þ s þ p p s þ pþ1
ð17Þ
! 1 1 1 1 ¼ p p ps þ s þ p pðp þ 1Þ s þ pþ1 ! 1 1 1 ¼ p p p þ 1 s þ pþ1
~h2L ¼
ð18Þ
The following Laplace transform pairs and their relationship are used to obtain the solutions of Eqs. (17) and (18).
9 8 p
ð19Þ
p!Y
V1 ðX; YÞ 1
Z
X
pffiffiffiffiffiffi eðuþYÞ I0 2 Yu du ¼ L1
0
9 8 p
p
;
p!Y
ð20Þ V1 ðX; YÞ V1;0 ðX; YÞ ¼ 1 V1 ðY; XÞ
ð21Þ
Thus, by applying the inverse Laplace transform of Eqs. (17) and (18), the final solutions are given by:
(
!) ) 1 1 h1 ðX; YÞ ¼ L L p p s þ pþ1 s!X p!Y
p 1 1 exp X ¼L ¼ V1 ðX; YÞ p pþ1 p!Y ( ( !) ) 1 1 1 1 1 h2 ðX; YÞ ¼ L L p p p þ 1 s þ pþ1 s!X p!Y
1 1 p exp X ¼ L1 p pþ1 pþ1 p!Y 1
(
1
¼ V1 ðX; YÞ V1;0 ðX; YÞ ¼ 1 V1 ðY; XÞ
ð22Þ
ð23Þ
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Above solutions can be easily implemented by using built-in functions of the MATLAB program. MATLAB provides the modified Bessel function of the first kind and zero order and integration of the function. The temperature solution of the crossflow heat exchanger is obtained simply and quickly by solving Eq. (19). 2.2. Thermal design of PCHE To meet heat requirements of the inlet and outlet temperatures of fluids, the size of heat exchanger is calculated through the thermal design procedure. The outlet temperature was averaged along the perpendicular to the flow direction in each fluid side since the temperature profile of crossflow heat exchanger is two-dimensional distribution. Thermal design process of crossflow PCHE is shown in Fig. 5. In general, the size of the heat exchanger is determined by the fluid temperatures. Fig. 6 shows the schematic diagram of crossflow PCHE. Geometric parameters of PCHE were tabulated in Table 2. In order to determine the size of crossflow PCHE, at first, the sizes of crossflow PCHE (Lx, Ly, and Lz) are assumed. From the assumed geometry information, the overall heat transfer surface area A is calculated as follows:
A¼
p
p d þ d Lx N1 Nz ¼ d þ d Ly N2 Nz 2 2
Fig. 6. Schematic diagram of crossflow PCHE.
Table 2 Design parameters of crossflow PCHE.
ð24Þ
The number of flow channels in each fluid side is determined as follows:
A¼
Geometric parameters
Value
Channel diameter d, (m) Channel pitch P, (m) Plate thickness tp, (m)
0.003 0.0033 0.00317
p
p ðLy þ tf Þ ðLx þ tf Þ d þ d Lx Lz ¼ d þ d Ly Lz 2 2 2tf ðtf þ dÞ 2tf ðtf þ dÞ
ð28Þ
N1 d þ ðN1 1Þtf ¼ Ly
ð25Þ
The mass flow rate through each fluid side is calculated from the heat duty and the given temperature conditions by using the following heat balance equations:
N2 d þ ðN2 1Þtf ¼ Lx
ð26Þ
_ 1 cp;1 ðT1;in T1;out Þ ¼ m _ 2 cp;2 ðT2;out T2;in Þ Q ¼m
Nz ¼ Lz =ð2tp Þ
ð27Þ
Simplifying Eqs. (25)–(27) and substituting them to Eq. (24) yields:
Vj ¼
Calculate overall heat transfer surface area, As
Calculate mass flow rate . and fluid velocity, mf , Vf
Calculate Temperature Distribuon
Calculate heat transfer coefficient, h
Calculate average outlet − temperature Tout
Resize
_j m
−out| <ε |Tout-T Yes Determinaon of HX sizes
Fig. 5. Thermal design process of crossflow PCHE.
ð31Þ
The Reynolds number and Prandtl number of each fluid side are calculated as follows:
Rej ¼
qj Vj Dh;j lj
Prj ¼
cp;j lj kj
ðj ¼ 1; 2Þ
ðj ¼ 1; 2Þ
ð32Þ
ð33Þ
The heat transfer coefficient is calculated by empirical heat transfer correlations. The heat transfer correlations used in this study will be discussed in Section 2.4. When the Nusselt number is determined by empirical correlation, the heat transfer coefficient in each fluid side is calculated as follows:
hj ¼ Nuj No
ð30Þ
2;in Þ
ðj ¼ 1; 2Þ
qj Axs;j
Calculate Number of Transfer Unit, NTU
Calculate Non-dimensional parameters (Re, Pr, Nu)
Calculate overall heat transfer coefficient, U
_ 1 ¼ c ðT QT Þ m p;1 1;in 1;out
The fluid velocity of each fluid side is calculated as follows:
Given Requirements and condions of HX: Q, Dh , Tin , Tout , P, material properes f(P, T)
Set the inial sizes of HX
thus:
_ 2 ¼ c ðT Q T m p;2 2;out
Thermal Design Process
ð29Þ
kj Dh;j
ðj ¼ 1; 2Þ
ð34Þ
Overall heat transfer coefficient of crossflow PCHE is given by:
U¼
1 h1
1 1 ¼ þ Rw þ h12 h1 þ kdww þ h1 1 2
ð35Þ
Calculated values in Eqs. (28), (30), and (35) are used to calculate the NTU in Eq. (8). When the NTU is determined, temperature profile of the crossflow heat exchanger can be calculated by the
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analytical solutions in Eqs. (22) and (23). Dimensionless temperature profiles, hj(X, Y), of crossflow PCHE are converted to Tj(x, y) as follows: Tj ðx;yÞ ¼ ðT1;in T2;in Þhj ðx;yÞ þ T2;in sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! # " Z NTU1 x Lx NTU2 NTU2 ¼ ðT1;in T2;in Þ 1 exp y þ u Io 2 yu du þ T2;in Ly Ly 0 ð36Þ
where 0 6 x 6 Lx, 0 6 y 6 Ly. The mean outlet temperatures of primary and secondary fluids can be calculated by averaging each temperature profile function as follows:
T1;out ¼ ¼
T2;out ¼ ¼
1 Ly 1 Ly 1 Lx 1 Lx
Z Z
T1 ðLx ; yÞdy 0
ðT1;in T2;in Þhj ðLx ; yÞ þ T2;in dy
ð37Þ
0
Z Z
L1
T2 ðx;Ly Þdx 0 Lx
ðT1;in T2;in Þhj ðx;Ly Þ þ T2;in dx
ð38Þ
0
Above mean outlet temperatures should be corresponded to the required outlet temperature conditions. If the mean outlet temperatures disagree with the required outlet temperatures, then the heat exchanger has to be resized until the calculated mean outlet temperatures are good agreement with the required outlet temperatures. From the calculated outlet temperatures, log mean temperature difference (LMTD) of crossflow PCHE can be obtained by:
DTLMTD ¼
ðT1;in T2;out Þ ðT1;out T2;in Þ ðT T Þ ln ðT1;in T2;out Þ 1;out
ð39Þ
2;in
In order to estimate the operational cost of heat exchanger, the pressure loss of the heat exchanger has to be calculated. Since the channel shape of PCHE is a semi-circle, the pressure loss calculation using the friction factor correlation for a circular pipe can cause an incorrect result. The fully-developed laminar friction factor of semi-circular straight pipe [21] is given by:
f Re ¼ 15:78;
channel should be employed. However, the straight pipe is assumed in this study so that the friction factor correlation by Berbish et al. [22] is employed to calculate the turbulent friction factor of semi-circular straight pipe. The friction factor is given by:
f ¼ 0:478Re0:26 ; 8200 Re 58000
ð41Þ
The pressure loss of the cross flow PCHE can be calculated by the pressure loss equation of straight pipe:
DPj ¼ 2f j
Lj q V2 Dh;j j j
ðj ¼ 1; 2Þ
ð42Þ
2.3. Thermo-physical properties of working fluids
L2
Ly
255
Re < 2300
ð40Þ
Hesselgreaves [21] suggests the turbulent friction factor of PCHE of which the flow channel is wavy. If the flow channel is wavy or zigzag-shaped, the correlations for wavy or zigzag-shaped
The working fluids used in advanced SMRs are water, helium, sodium, and molten-salts (FLiNaK, FLiBe). In this study, IAPWS IF97 standard formulation [23] is used to calculate the water properties. The helium property is based on the ‘‘National Institute of Standards Technology (NIST) Chemistry WebBook’’ data [24]. Thermophysical properties of sodium and molten-salts are summarized in Table 3. The unit of temperature, T, in Table 3 is Kelvin. Some material properties of molten-salts are not accurate and have to be investigated in the future. In this study, the effect of uncertainty of material property was investigated. The maximum uncertainty of property is assumed to be ±30%. 2.4. Correlations for heat transfer coefficient The heat transfer correlations have been developed with various fluids under the various conditions. Dittus-Boelter [28] and Gnielinski [29] correlations are well-known and widely-used heat transfer correlations, but it is known that Dittus-Boelter has definite uncertainty. Gnielinski correlation is used as the default heat transfer correlation of the developed code. The heat transfer correlations are summarized in Table 4. Each correlation is valid for limited ranges of temperature and pressure, shape of channel, and fluid material. A heat transfer coefficient calculated by the empirical correlation has an influence on the temperature profile of the crossflow heat exchanger and consequently on the heat exchanger sizing. Therefore, the appropriate selection of a heat transfer correlation is important for accurate heat exchanger sizing. In addition, the empirical heat transfer correlations developed for circular pipe might predict the heat transfer coefficient incorrectly since the flow channel of PCHE is a semi-circle. In this study, the temperature profiles according to
Table 3 Thermophysical properties of sodium, FLiNaK and FLiBe. Sodium [25] Density (kg/m3) Heat capacity (J/(kg K)) Thermal conductivity (W/(m K)) Dynamic viscosity (Pa s)
896.60679 + 0.5161343T 1.8297218 103T2 + 2.2016247 106T3 – 1.3975634 109T4 + 0.44866894 1012T5 – 0.057963628 1015T6 (38.12 – 0.069 106T2 – 19.493 103T + 10.24 106T2)/22.99 99.5 – 39.1 103T exp(6.44060.3958lnT + 556.8/T) FLiNaK (LiF-NaF-KF) [26]
Density (kg/m3) Heat capacity (J/(kg K)) Thermal conductivity (W/(m K)) Dynamic viscosity (Pa s)
2,579.3 – 0.624T 976.78 + 1.0634T 0.36 + 5.6 104T
Density (kg/m3) Heat capacity (J/(kg K)) Thermal conductivity (W/(m K)) Dynamic viscosity (Pa s)
2,413 – 0.488T 2397.73 ( ± 20%) 0.629697 + 0.0005T
2.487 105exp(4,478.62/T) FLiBe (LiF-BeF2) [26]
0.000116exp(3,755/T)
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Table 4 Summary of single-phase heat transfer coefficient correlations. Authors
Channel shape
Working fluid
Berbish [22] Kim et al. [27] Dittus-Boelter [28]
Semi-circular Semi-circular Circular
Air He Any fluid
Gnielinski (NuG) [29]
Circular
Any fluid
Nusselt number correlation 0.0228Re 3.255 + 0.00729(Re350) 0.0243Re0.8Pr0.4 for heating 0.0265Re0.8Pr0.3 for cooling ð8f ÞðRe1000ÞPr pffi
8,200 < Re < 5.83 104 350 < Re < 800, Pr = 0.66 104 < Re < 1.2 105, 0.7 < Pr < 120, L/D > 60
1þ12:7
Lyon [30] Lubarsky and Kaufman [31] Reed [32] Taylor and Kirchgessner [33], Wieland [34] Wu and Little [35] Wang and Peng [36]
Circular Circular, Annular Circular Circular
Liquid Metal Liquid Metal Liquid Metal He
7.0 + 0.025Pe0.8 0.625Pe0.4 3.3 + 0.02Pe0.8 0.021Re0.8Pr0.4
0 6 Pr 6 0.1, 300 < Pe < 104 0 6 Pr 6 0.1, 200 < Pe < 104 100 < Pe 3.2 103 < Re < 6 104, 60 < L/D
Rectangular Rectangular
N2 gas Water, Methanol
0.00222Re1.09Pr0.4 0.00805Re0.8Pr0.33
3000 < Re 2300 < Re
2.5. Cost estimation methodology The total cost of a heat exchanger consists of capital cost and operating cost. The capital cost is related to the material cost of the heat exchanger so that it can be estimated based on its weight. Therefore, the capital cost depends on the structural material and size of the heat exchanger. The total mass of heat exchanger is given by:
MHX ¼ qHX VHX
ð43Þ
The volume of the heat exchanger can be approximated as follows:
VHX ¼ Lx Ly Lz
ð44Þ
The capital cost can be calculated by multiplying the cost factor of material ($/kg) and the total mass of heat exchanger, as follows:
CP ¼ CM MHX
ð45Þ
On the other hand, the operating cost is estimated based on the thermal-hydraulic condition of the heat exchanger. Since the operating cost is defined as the cost to operate the circulation pump of the heat exchanger, it is important to optimize the design of heat exchanger to reduce the pressure loss, in turn reducing the operating cost. To estimate the operating cost, the pumping power of the heat exchanger is used. The pumping powers of the heat exchanger in hot and cold fluid sides can be defined by [37]:
P2 ¼
2300 < Re < 5 106, 0.5 < Pr < 2000
2=3 f 1Þ 8ðPr
the heat transfer correlations are compared to evaluate the effect of the heat transfer correlation.
P1 ¼
Valid ranges
0.8
_ 1 DP1 m
ð46Þ
q1 _ 2 DP2 m
ð47Þ
q2
Thus, the operating cost can be approximated as follows [37]:
OP ¼ COP YðP 1 þ P 2 Þ
ð48Þ
Therefore, the total cost of heat exchanger becomes:
Ctotal ¼ CP þ OP ¼ CM MHX þ COP YðP 1 þ P 2 Þ
ð49Þ
As a structural material of PCHE, the nickel-based alloys such as Alloy 800H, Alloy 617, and Hastelloy N are under development. The thermo-physical properties of structural material also have an influence on the sizing and cost estimation of heat exchanger. According to the previous studies on the cost of heat exchangers [37,38], the material costs of Alloy 800H and Alloy 617 are assumed to be 120 USD/kg conservatively. The material cost of Hastelloy N is approximately 124 USD/kg [39]. The operating cost is 0.0000612 USD/Wh, which is based on the consumer price index average price data [37]. For high temperature condition of 700 °C, the thermo-physical properties and material costs of PCHE structural materials are summarized in Table 5.
3. Crossflow PCHE analysis The fluid combinations of primary and secondary sides are different for each reactor coolant of SMR. In this study, four fluid combinations of primary and secondary side are selected by taking into account the reactor coolant types of SMR. The heat duty of heat exchanger is assumed to be 300 MW. The Reynolds numbers of primary and secondary sides are varied according to the fluid property and heat duty. Test conditions are summarized in Table 6.
3.1. Grid sensitivity test In this study, the temperature profile of crossflow PCHE was obtained by solving the system of differential equations. Twodimensional grid is used to calculate the temperature profile. In developed code, the mean outlet temperature of each fluid side is calculated by averaging the temperature at the outlet. Thus, the number of grids can influence the mean outlet temperature. In order to investigate the grid effect, the grid sensitivity test was carried out. Tested numbers of grid and results are summarized in Table 7. In grid sensitivity test, x and y-axial dimensions of heat exchanger, NTUs of primary and secondary sides are assumed to be a unity. The numbers of reference grid was 40 40.
Table 5 Thermo-physical properties and cost of structural materials of PCHE. Structural material
Density (kg/m3)
Thermal conductivity (W/(m K))
Heat capacity (J/(kg K))
Cost (USD/kg)
Alloy 617 [37] Alloy 800H [38] Hastelloy N [39]
8,360 7,940 8,860
23.9 22.8 23.6
586 460 523
120 120 124
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S.-J. Yoon et al. / International Journal of Heat and Mass Transfer 70 (2014) 250–263 Table 6 Test matrix for crossflow PCHE analysis. Coolants (primary/secondary)
Inlet temperature (°C) (primary/secondary)
Pressure (MPa) (primary/secondary)
Heat duty (MW)
Re
Pr
Water/Water Helium/Helium Sodium/Sodium FLiBe/FLiNaK
320/200 950/350 545/320 700/600
15/15 7.0/7.0 0.1/0.1 0.1/0.1
300
104–105 104–105 104–105 103–104
1 0.66 0.005 10
Table 7 Grid sensitivity test result. Dimensionless mean outlet temperature Numbers of grids Primary side Secondary side
55 0.52124 0.47876
10 10 0.52251 0.47749
20 20 0.52314 0.47686
The grid sensitivity analysis by the Richardson extrapolation [40] was performed to estimate the numerical error due to the grid number. The approximation of exact solution U is given by:
U ¼ /h þ
/h /2h 2p 1
ð50Þ
where / is a numerical solution and the order of the scheme p is defined by:
log p¼
/2h /4h /h /2h
ð51Þ
log2
Fig. 7 shows the dimensionless mean outlet temperatures of the primary and secondary sides. The dimensionless temperature in the grid sensitivity test was defined in Eq. (7). The Richardson solutions of primary and secondary fluids were 0.5238 and 0.4762, respectively. The maximum relative errors between the reference grid (40 40) and Richardson solution of primary and secondary fluids were 0.48% and 0.53%, respectively. Consequently, the grid sensitivity test results show that the effect of grid number is negligible. 3.2. Effectiveness-NTU (e-NTU) method analysis The effectiveness of heat exchanger, e, is the ratio of the actual heat transfer rate to the maximum possible heat transfer rate of the exchanger and is defined by:
e¼
Q ðT1;in T1;out Þ ¼ Q max ðT1;in T2;in Þ
ð52Þ
40 40 0.52346 0.47654
80 80 0.52362 0.47638
160 160 0.52370 0.47630
320 320 0.52374 0.47626
The e-NTU method [10] is one of general methods applied for the thermal design of heat exchanger. In this method, the effectiveness of crossflow heat exchanger is a function of NTU. The relationship between the effectiveness and NTU of both unmixed fluids crossflow heat exchanger [41] is given by:
e ¼ 1 exp
1 0:22 0:78 ðNTUÞ ðexp½C ðNTUÞ 1Þ C
ð53Þ
where C⁄( = Cmin/Cmax) is the ratio of minimum to maximum heat capacity rate of fluid. The effectiveness of crossflow PCHE calculated by the developed code is compared with that of the e-NTU method. Fig. 8 shows the effectiveness-NTU plots by the developed code and e-NTU method. The effectiveness of a heat exchanger increases as the NTU increases. The effectiveness increases rapidly as the ratio of heat capacity rate decreases. The maximum relative error of effectiveness between the code and e-NTU method was 3.95%. 3.3. Effect of fluid property uncertainty The uncertainty of fluid property can be propagated into the calculated temperature profile of the heat exchanger. To evaluate the effect of uncertainty of fluid property, the uncertainty of fluid property was assumed to be ±30%. In this calculation, the heat duty, the temperatures and pressures in Table 1 were employed. Hastelloy N is employed as the structural material. Gnielinski heat transfer correlation was used to calculate the heat transfer coefficient. The
Φprimary=0.5238
0.523
0.482
0.522
0.481
0.521
Primary Secondary
0.520 0.519
0.480 0.479 0.478
0.518
0.477
Φsecondary=0.4762
0.517
0.476 0
40
80
120
160
200
240
280
320
Number of grids Fig. 7. Effect of grid number on dimensionless mean temperature profile of crossflow heat exchanger.
1.0 0.9 0.8
Effectiveness
0.483
Dimensionless mean outlet temperature of secondary fluid
Dimensionless mean outlet temperature of primary fluid
1.1 0.524
0.7 0.6 0.5
ε-NTU method PCHE analysis code C*=0.1 C*=0.1 C*=0.3 C*=0.3 C*=0.5 C*=0.5 C*=0.7 C*=0.7 C*=1 C*=1
0.4 0.3 0.2 0.1 0.0 0
1
2
3
4
5
6
7
8
9
10
NTU Fig. 8. Effectiveness-NTU comparison between e-NTU method and crossflow PCHE analysis code.
S.-J. Yoon et al. / International Journal of Heat and Mass Transfer 70 (2014) 250–263
Averaged temperature (oC)
320
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
310
300
290
800
+1.05% 750 o
dimensions in x, y, and z axial directions of heat exchanger were assumed to be 0.9897 m, 0.9897 m, and 0.634 m, respectively. Corresponding numbers of flow channels are 300, 300, and 100, respectively. Figs. 9–16 show the average temperature profiles according to the uncertainty of thermo-physical property for each coolant. The
Averaged temperature ( C)
258
700
-1.65%
650 600
Reference k (-30%) cp(-30%) μ (-30%) k (+30%) cp(+30%) μ (+30%)
550 500 450 400 350 0.0
0.2
0.4
0.6
0.8
1.0
Distance from the inlet (m) 280
+0.75%
270
Fig. 12. Average temperature profile of secondary fluid according to uncertainty of material property (Helium).
-0.47% 260 0.0
560 0.2
0.4
0.6
0.8
540
250
-0.77%
500 480 460 440 420
+0.7%
400 380
230
360 0.0
-0.47% k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
220
210
200 0.0
0.2
0.4
0.6
0.8
Fig. 10. Average temperature profile of secondary fluid according to uncertainty of material property (Water).
950
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
900 850 800 750
0.4
0.6
0.8
1.0
Fig. 13. Average temperature profile in primary side according to uncertainty of material property (Sodium).
500
+0.37% 480
-0.54%
460 440 420
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
400 380 360 340 320 0.0
700
0.2
0.4
0.6
0.8
1.0
Distance from the inlet (m)
650
+2.38%
600 550
-1.52% 500 0.0
0.2
Distance from the inlet (m)
1.0
Distance from the inlet (m)
Averaged temperature (oC)
520
240
Averaged temperature (oC)
Averaged temperature (oC)
+0.48%
o
Fig. 9. Average temperature profile of primary fluid according to uncertainty of material property (Water).
Averaged temperature ( C)
Distance from the inlet (m)
260
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
1.0
0.2
0.4
0.6
0.8
1.0
Distance from the inlet (m) Fig. 11. Average temperature profile of primary fluid according to uncertainty of material property (Helium).
Fig. 14. Average temperature profile in secondary side according to uncertainty of material property (Sodium).
density change did not have an effect on the temperature profile. In the developed code, when the heat duty is given, the mass flow rate is determined from Eq. (30). Then, the density is used to calculate the fluid velocity and Reynolds number. Since the product of density and fluid velocity is constant by Eq. (31), the Reynolds number is not changed by the density. Consequently,
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660
650 0.0
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%) 0.2
+0.62%
-0.51% 0.4
0.6
0.8
1.0
Distance from the inlet (m) Fig. 15. Average temperature profile in primary side according to uncertainty of material property (FLiBe).
650
Averaged temperature (oC)
+0.53% 640
630
-0.64% Reference k (-30%) cp(-30%) μ (-30%) k (+30%) cp(+30%) μ (+30%)
620
610
600 0.0
0.2
0.4
0.6
0.8
1.0
Distance from the inlet (m) Fig. 16. Average temperature profile in secondary side according to uncertainty of material property (FLiNaK).
the uncertainty of fluid density does not have an effect on the temperature. Uncertainties of heat capacity and dynamic viscosity resulted in the same temperature profile. The temperature profile of the crossflow analytical model is determined by the size of heat exchanger and NTU value. In this crossflow model, NTU can be approximated as a function of the product of heat capacity and dynamic viscosity so that the changes in these two parameters resulted in the same degree of change in NTU. Uncertainty of thermal conductivity led to the maximum deviation of temperature profile, except the sodium case. The effect of thermal conductivity was relatively smaller than heat capacity and dynamic viscosity for the fluid of low Prandtl number. Overall, the relative deviation of the temperature profile according to the uncertainty of fluid property was very small. Maximum deviation was 2.38% that occurred by the thermal conductivity in the primary side of helium flow. In conclusion, the uncertainty of fluid property was negligible in the thermal design of crossflow PCHE. 3.4. Effect of heat transfer coefficient correlations Comparative analyses were carried out to evaluate the effect of heat transfer correlations on the temperature profile. In this analysis, the heat duty, the temperatures, and pressures in Table 1 were employed. Hastelloy N was employed as the structural material. The dimensions of the x, y, and z axial directions of the heat exchanger were assumed to be 0.9897 m, 0.9897 m, and 0.634 m, respectively.
360
o
670
Average temperatre ( C)
680
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
330
Lubarsky and Kaufman
300 Wang and Peng
ΔT=39 K
270 Wu and Little, Kim et al.
240 0.0
0.2
0.4
0.6
0.8
1.0
Distrance from the Inlet (m) Fig. 17. Temperature profile in primary side by the heat transfer correlations (Water).
280
o
690
Figs. 17 and 18 show the temperature profiles in the primary and secondary side of the water flows, respectively. The Lubarsky and Kaufman’s correlation showed the highest temperature of primary side and lowest temperature of secondary side. Wu and Little’s and Kim et al. correlations showed the lowest temperature of primary side and highest temperature of secondary side. These two correlations predicted the heat transfer coefficients similarly for Prandtl number in the range of 0.5–1.0. Thus, the same temperature profile trend was observed in the water and helium flows of which Prandtl numbers were approximately 1.0 and 0.66, respectively. The maximum deviation among the averaged outlet temperature at the outlet was 39.45 K, which was caused by the Lubarsky and Kaufman’s correlation and Kim et al. correlation. Figs. 19 and 20 show the temperature profiles of helium flows. In the helium flow, the similar trends of temperature profiles were observed as the water flow. The maximum deviation among the averaged outlet temperature was approximately 196.01 K which was caused by the Lubarsky and Kaufman’s correlation and Kim et al. correlation. Figs. 21 and 22 show the temperature profiles of sodium flows. In this case, the effect of heat transfer correlation was smaller than the other cases. For the fluids with very low Prandtl numbers, most of empirical correlations predicted low heat transfer coefficient and the difference among them was also small. Maximum deviation of averaged outlet temperature was 15.34 K, which was caused by Gnielinski’s correlation and Kim et al. correlation. Figs. 23 and 24 shows the temperature profiles of molten salt flows. The Prandtl
Average temperatre ( C)
Averaged temperature (oC)
700
260
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
Wu and Little, Kim et al.
ΔT=39 K
240 Wang and Peng
220 Lubarsky and Kaufman
200 0.0
0.2
0.4
0.6
0.8
1.0
Distrance from the Inlet (m) Fig. 18. Temperature profile in secondary side by the heat transfer correlations (Water).
260
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550
800 Wang and Peng
700
500
400 0.0
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
0.2
0.4
ΔT=196 K Wu and Little Kim et al.
0.6
0.8
500
o
Lubarsky and Kaufman
600
Average temperatre ( C)
o
Average temperatre ( C)
900
450
350
0.2
Fig. 19. Temperature profile in primary side by the heat transfer correlations (Helium).
700
700 Wang and Peng
o
ΔT=196 K
Lubarsky and Kaufman
680 ΔT=31 K
660
640
400 0.2
0.4
0.6
0.8
620 0.0
1.0
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
0.2
Distrance from the Inlet (m)
Gnielinski
400
0.8
ΔT=31 K
620 Wang and Peng
600
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
580
ΔT=15 K
Berbish, Wu and Little, and Kim et al.
0.6
1.0
Lyon
640
o
Average temperatre ( C)
o
Average temperatre ( C)
450
0.4
0.8
660
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
0.2
0.6
Fig. 23. Temperature profile in primary side by the heat transfer correlations (Molten Salt).
550
350 0.0
0.4
Lyon
Distrance from the Inlet (m)
Fig. 20. Temperature profile in secondary side by the heat transfer correlations (Helium).
500
1.0
Kim et al. Wu and Little
Wang and Peng
0.0
0.8
720
600
500
0.6
Fig. 22. Temperature profile in secondary side by the heat transfer correlations (Sodium).
Average temperatre ( C)
o
Average temperatre ( C)
800
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
0.4
Distrance from the Inlet (m)
Distrance from the Inlet (m)
900
ΔT=15 K
400
300 0.0
1.0
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Berbish, Wu and Little, and Kim et al. Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng Gnielinski
1.0
Distrance from the Inlet (m) Fig. 21. Temperature profile in primary side by the heat transfer correlations (Sodium).
number of molten salt flows was approximately 10.0. Contrary to the sodium flow, the molten salt is a fluid with a high Prandtl number. Not only the empirical correlations developed for liquid metals but also Dittus-Boelter correlation and Gnielinski’s correlations predicted high-heat transfer coefficient. Consequently, these
560 0.0
0.2
0.4
0.6
0.8
1.0
Distrance from the Inlet (m) Fig. 24. Temperature profile in secondary side by the heat transfer correlations (Molten Salt).
correlations predicted similar temperature profiles. Maximum deviation of averaged outlet temperature was 31.04 K, which was caused by Wang and Peng’s correlation and Lyon’s correlation. By comparing heat transfer correlations for each fluid, we found that the effect of heat transfer correlations was largest in the
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S.-J. Yoon et al. / International Journal of Heat and Mass Transfer 70 (2014) 250–263 Table 8 Sizing and cost estimation results of crossflow PCHEs. Coolants
Heat exchanger size (m)
Average outlet temperature (°C)
Cost (USD) (Operation period: 20 year)
Lx
Ly
Lz
Primary
Secondary
Structural material
Capital
Operation
Water
0.396
1.518
0.634
295.3
293.6
Alloy 617 Alloy 800H Hastelloy N
3.8197 105 3.6278 105 4.1831 105
1.4049 109 1.4420 109 1.4146 109
Helium
1.128
2.095
637.8
925.3
Alloy 617 Alloy 800H Hastelloy N
1.5036 106 1.4280 106 1.6466 106
9.4799 109 9.4901 109 9.4826 109
Sodium
1.214
1.613
390.2
526.0
Alloy 617 Alloy 800H Hastelloy N
1.2459 106 1.1833 106 1.3644 106
1.2703 108 1.2770 108 1.2720 108
Molten Salt
6.352
11.880
651.0
690.2
Alloy 617 Alloy 800H Hastelloy N
4.7996 107 4.5585 107 5.2562 107
9.1158 106 9.1174 106 9.1163 106
In this crossflow PCHE analysis code, the size of heat exchanger is determined by iterative calculation to meet the given heat duty and temperature requirements. The heat duty and temperature requirements in Table 1 and three nickel-based alloys in Table 5 were used in the economic analysis. The maximum operating period of heat exchanger is assumed to be 20 years. In this analysis, the height of heat exchanger, Lz, was fixed to 0.634 m to reduce the number of cases. If the height of the heat exchanger can be varied, the length and width of the heat exchanger are determined for each height so that various heat exchanger sizes can be generated. Economic analysis results are summarized in Table 8. The sizes of heat exchangers using water, helium, and sodium coolants were similar to each other, whereas large heat exchanger size was required to meet the temperature requirements for molten salt coolants. The predicted size of crossflow PCHE for molten salt is deemed inappropriate for molten salt cooled SMR. To reduce the size of the heat exchanger, the heat duty per heat exchanger has to be reduced by increasing the number of heat exchangers. 7.4E6
Alloy 617 Alloy 800H Hastelloy N
Total cost per year (USD/yr)
7.35E6 7.3E6
Total cost per year (USD/yr)
3.5. Cost estimation results
4.755E8
Alloy 617 Alloy 800H Hastelloy N
4.752E8 4.749E8 4.746E8 4.743E8 4.74E8 4.737E8
3
6
9
12
15
18
21
Operation period (yr) Fig. 26. Total cost per year of crossflow PCHE (Helium/Helium).
6.8E6
Total cost per year (USD/yr)
helium flow. The deviation of temperature profiles by the heat transfer correlation decreased for very low or very high Prandtl number fluids. It was not easy to determine a heat transfer correlation that can be applied generally irrespective of coolant types. Consequently, the experimental and/or computational validations are deemed to be required to improve the validity and accuracy of crossflow PCHE analysis code. In addition, the appropriate heat transfer correlation for each fluid type should be applied.
Alloy 617 Alloy 800H Hastelloy N
6.7E6
6.6E6
6.5E6
6.4E6
6.3E6 3
6
9
12
15
18
21
Operation period (yr) 7.25E6
Fig. 27. Total cost per year of crossflow PCHE (Sodium/Sodium). 7.2E6 7.15E6 7.1E6 7.05E6 7E6 3
6
9
12
15
18
21
Operation period (yr) Fig. 25. Total cost per year of crossflow PCHE (Water/Water).
The capital cost of crossflow PCHE was relatively smaller than the operation cost except for molten salt heat exchanger. Due to the large size of the molten salt heat exchanger, its capital cost was estimated to be larger than operation cost. The capital cost of the molten salt heat exchanger was higher than the others, but the operation cost was much lower. The operation cost of the helium heat exchanger was the most expensive. Figs. 25–28 show the total cost per year of the heat exchangers according to the operation period. Operation cost per year decreased as the
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(Grant code: 2012-052255), under DOE Idaho Operations Office Contract DE-AC07-05ID14517.
1.2E7
Alloy 617 Alloy 800H Hastelloy N
Total cost per year (USD/yr)
1E7 8E6
References 6E6
4E6
2E6 3
6
9
12
15
18
21
Operation period (yr) Fig. 28. Total cost per year of crossflow PCHE (FLiBe/FLiNaK).
operation period increased. Total cost of the heat exchanger using Alloy 617 was cheaper than the other structural materials except for the molten salt heat exchanger. In the molten salt heat exchanger, the total cost was minimized by using Alloy 800H. The total cost per year of molten salt heat exchanger decreased rapidly as the operation period increased. 4. Conclusion In this study, the crossflow PCHE analysis code has been developed to evaluate the size and cost of heat exchangers by implementing the analytical solution of single pass, both unmixed fluids crossflow heat exchanger model. Two-dimensional temperature distribution of crossflow PHCE was calculated by the analytical solution. General methods for thermal design and cost estimation of heat exchangers were employed to determine the size and cost of heat exchanger, respectively. The information provided by this code can be used for the optimal design of advanced SMR system. The grid sensitivity test of the code showed that the effect of grid on the temperature profile was negligible. The effectiveness calculated by the code showed a good agreement with that by well-known e-NTU method. The uncertainty of fluid property was propagated into the temperature distribution, but its effect was small enough to be inconsequential. The heat transfer correlations had a considerable influence on the temperature profile. The temperature profile of crossflow PCHE using helium gases showed the largest deviation by the heat transfer correlations. The deviation of temperature profiles by the heat transfer correlation decreased for very low or very high Prandtl number fluids. However, it was not easy to determine the most accurate heat transfer correlation for the heat exchangers using various combinations of coolants. Consequently, experimental and/or computational validations are required to determine the best heat transfer correlation for each SMR. Costs of crossflow PCHE for the advanced SMR designs were investigated. Capital costs of PCHE for water, helium, and sodium flows were lower than their operation costs, whereas capital cost of molten salt heat exchanger was higher than operation cost. Total cost of heat exchanger using Alloy 617 was cheaper than the other structural materials except for the molten salt heat exchanger. In the molten salt heat exchanger, the total cost was minimized by using Alloy 800H. Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) and grant funded by the Korean government (MSIP)
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[38] Kim, I.H., 2012. Experimental and numerical investigations of thermalhydraulic characteristics for the design of a Printed Circuit Heat Exchanger (PCHE) in HTGRs, Ph.D. Thesis, KAIST. [39] Sabharwall, P., Kim, E.S., Siahpush, A., Anderson, N., Glazoff, M., Phoenix, B., Mizia, R., Clark, D., McKellar, M., Patterson, M.W., 2011. Feasibility Study of Secondary Heat Exchanger Concepts for the Advanced High Temperature Reactor, INL/EXT-11-23076. [40] L.F. Richardson, The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam, Philos. Trans. Roy. Soc. London Ser. A 210 (1910) 07–357. [41] A. Triboix, Exact and approximate formulas for cross flow heat exchangers with unmixed fluids, Int. Commun. Heat Mass Transfer 36 (2009) 121–124.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerical study on crossflow printed circuit heat exchanger for advanced small modular reactors Su-Jong Yoon a,⇑, Piyush Sabharwall a, Eung-Soo Kim b a b
Idaho National Laboratory, 2525 Fremont Avenue, Idaho Falls, ID 83415-6188, USA Seoul National University, Seoul, Republic of Korea
a r t i c l e
i n f o
Article history: Received 16 July 2013 Received in revised form 30 October 2013 Accepted 31 October 2013
Keywords: Crossflow Heat exchanger PCHE Analytical model Thermal design Cost estimation Advanced SMR
a b s t r a c t Various fluids such as water, gases (helium), molten-salts (FLiNaK, FLiBe) and liquid metal (sodium) are used as a coolant of advanced small modular reactors (SMRs). The printed-circuit heat exchanger (PCHE) has been adopted as the intermediate and/or secondary heat exchanger of SMR systems because this heat exchanger is compact and effective. The size and cost of PCHE can be changed by the coolant type of each SMR. In this study, the crossflow PCHE analysis code for advanced small modular reactor has been developed for the thermal design and cost estimation of the heat exchanger. The analytical solution of singlepass, both unmixed fluids crossflow heat exchanger model was employed to calculate a two-dimensional temperature profile of a crossflow PCHE. The analytical solution of crossflow heat exchanger was simply implemented by using built-in function of the MATLAB program. The effect of fluid property uncertainty on the calculation results was evaluated. In addition, the effect of heat transfer correlations on the calculated temperature profile was analyzed by taking into account possible combinations of primary and secondary coolants in the SMR systems. Size and cost of heat exchanger were evaluated for the given temperature requirement of each SMR. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Small modular reactors (SMR) are being developed to provide heat and electricity to the remote regions where the large nuclear power plants cannot be built. In addition, recently, the SMRs have been considered as good energy sources for the process heat application. Contrary to typical water-cooled SMRs, advanced SMRs use various coolant materials such as gases (helium, supercritical carbon dioxide), molten-salts (FLiNaK, FLiBe and KFZrF4), and liquid metal (sodium). The operating temperature and pressure of an advanced SMR can be varied by the type of reactor coolant. Table 1 describes the typical temperature and pressure conditions of SMRs according to the reactor coolants. Inlet and outlet temperatures of primary and secondary fluids are based on the intermediate heat exchanger (IHX). Thus, the inlet temperature of the primary fluids shown in Table 1 is the outlet temperature of reactor coolant. Fig. 1 shows some of the potential options that the IHX and secondary heat exchanger (SHX) could be configured in the system. It could be linked to production of power system and also could be used for providing the process heat from the reactor for industrial applications. In this case, both IHX and SHX could be either con-
⇑ Corresponding author. Tel.: +1 208 526 5407/357 8858; fax: +1 208 526 2930. E-mail addresses: [email protected], [email protected] (S.-J. Yoon). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.10.079
ventional-type heat exchanger or compact heat exchanger, such as PCHE. The word ‘‘small’’ in SMRs refers not only to the reactor power, but also to the size of the reactor. In this regard, the compact heat exchanger can be a good design option because the compact heat exchanger has a large heat transfer surface area per unit volume, which can result in the reduced size, weight, and cost. Fig. 2 shows the surface compactness that is a criterion for classifying the heat exchanger. Printed circuit heat exchanger (PCHE) is a compact heat exchanger and has been considered one of the candidate heat exchangers in designs of advanced SMRs. Fine grooves in the plate of the PCHE are made by employing the technique used for making printed circuit boards. The PCHE has been commercially manufactured by Heatric™ [1]. Fig. 3 shows the typical PCHE manufactured by Heatric Ltd. The PCHE can be classified by the flow configurations. The arrangement of flow channels are parallel in the parallel and counter flow PCHEs, whereas it is orthogonal in the crossflow PCHE. Temperature distribution of the crossflow heat exchanger is twodimensional so that it is more complicated to analyze than parallel or counter flow heat exchangers. Due to this complexity, many previous studies have used computational fluid dynamics (CFD) codes such as CFX or FLUENT. However, CFD simulation is not adequate for the design process of the PCHE since CFD simulation requires vast computational time and cost. In the CFD model of PCHE, since
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Nomenclature A C CM Cop cp C⁄ CP D, d f h I0 k L M _ m N NTU Nu OP P p Pe Pr Q R Re T t U X Y
area or overall heat transfer surface area (m2) _ p;j Þ flow stream heat capacity rate ð¼ mc material cost factor (USD/kg) operating cost factor (USD/Wh) specific heat capacity (kJ/kg K) the ratio of heat capacity rate (=Cmin/Cmax) capital cost of PCHE (USD) diameter (m) friction factor convective heat transfer coefficient (W/m2 K) modified Bessel function of the first kind and zero order thermal conductivity (W/m K) length (m) total mass of heat exchanger (kg) mass flow rate (kg/s) number of flow channel number of transfer unit Nusselt number (=hD/k) operating cost of PCHE (USD) channel pitch (m) or Pressure (Pa) the order of scheme Peclet number (=Re Pr) Prandtl number (=cpl/k) heat duty (W) thermal resistance (m2 K/W) Reynolds number (=qVD/l) temperature (K) thickness (m) overall heat transfer coefficient (W/m2 K) normalized length in x-coordinate (=x/L1NTU1) normalized length in y-coordinate (=y/L2NTU2) or Operation period of heat exchanger (yr)
the structure of PCHE is complicated and the dimensions of flow channel of PCHE are small, the fine structure of mesh is required to obtain accurate results. In this study, we developed the crossflow PCHE analysis code to perform the thermal design and cost estimation of PCHE.
2. Development of crossflow PCHE analysis code In order to perform the thermal design and cost estimation of a crossflow PCHE, the analysis code was developed based on the MATLAB program [8]. The analytical solution of single-pass, both unmixed fluids crossflow heat exchanger was employed to solve the temperature profile of crossflow PCHE. Although typical PCHEs manufactured by Heatric™ have zigzag-shaped flow channel, the straight pipe is assumed in this study. The analytical modeling method, analytical solution, and thermal design procedure are described in following sections.
V
volume (m3) or Fluid velocity (m/s)
Greek and symbols D difference d plate thickness e effectiveness of PCHE (=Q/Qmax) go surface efficiency h dimensionless temperature (=(TT2,in))/(T1,inT2,in) l dynamic viscosity of fluid (Pa s) _ DP=qÞ P pumping power ð¼ m q density (kg/m3) U approximated exact solution by Richardson extrapolation method / numerical solution Subscript f h HX in j max min L out p tot w x–s
fin grid width or Hydraulic diameter heat exchanger inlet index of fluid (1: primary, 2: secondary) maximum minimum Laplace transformation operation outlet plate total wall cross-sectional
2.1. Analytical modeling and solution of crossflow heat exchanger For single-pass crossflow design, a solution to the problem of both fluids unmixed in the heat exchanger was obtained by Nusselt [9] in the form of analytical series expansions by assuming the longitudinal conduction can be neglected. Since the flow mixing is not occurred in the PCHE, this model is applicable. The analytical model of crossflow PCHE has been developed by the energy balance between the hot and cold fluid sides shown in Fig. 4. The energy balance equations in Fig. 3 are as follows: Fluid 1 (Hot fluid): @T _ 1 cp;1 T1 _ 1 cp;1 T1 þ 1 dx dm dm dQ |{z} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} @x |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Heat transfer rate ¼ 0 Fluid enthalpy rate Fluid enthalpy rate from fluid to wall into the C:V: out of the C:V: ð1Þ
Table 1 Typical temperature and pressure conditions of advanced SMRs. Coolant type
Water-cooled [2] Gas-cooled [3] Liquid metal-cooled [4] Molten salt-cooled [5]
Coolants
Operation conditions
Primary/secondary
Inlet/outlet temperature (°C)
Water/Water He/He Na/Na FLiBe/FLiNaK
Primary
Secondary
323/295 950/637 545/390 700/650
200/293 351/925 320/526 600/690
Primary/secondary pressure (MPa)
15.0/5.0 7.0/7.0 0.1/0.1 0.1/0.1
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(a) Option 1 – 1 SHX
IHX
Reactor
SHX
IHX
Power Conversion System
SHX
IHX
Reactor
SHX
(b) Option 2 – 2 SHXs Power Conversion System
1
SHX
IHX
2
2
3 4
(c) Option 3 – 3 SHXs
Power Conversion System
Reactor IHX
3
1
5
Fig. 3. Section of crossflow PCHE [7].
In above energy balance equations, the heat transfer from the hot fluid to the cold fluid can be calculated by the term dQ that can be expressed by using the rate equations for convection and conduction as follows:
6
IHX 6
SHX
g0;1 h1 ðT1 Tw;1Þdxdy
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
SHX 4
dQ ¼
5
Fig. 1. Schematic of potential heat exchanger arrangements.
Fluid 2 (Cold fluid):
@T _ 2 cp;2 T2 þ 2 dy _ 2 cp;2 T2 dm dm dQ |{z} @y |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ 0 Fluid enthalpy rate þ Heat transfer rate Fluid enthalpy rate from fluid to wall into the C:V: out of the C:V:
¼ ð2Þ
Convection
!
from hot fluid to wall Tw;1 Tw;2 dxdy kw dw |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ð Conduction within the wall Þ
Fig. 2. Heat transfer surface area density spectrum of heat exchanger surfaces [6].
g0;2 h2 ðTw;2 T2 Þdxdy
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
¼
!
Convection from wall to cold fluid
ð3Þ
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Fig. 4. Energy balance control volume for crossflow heat exchanger [10].
By employing the definition of the overall heat transfer coefficient U ¼ dAdQDT, but neglecting the fouling thermal resistances, the heat transfer rate dQ can be given by:
dQ ¼ UdAðT1 T2 Þ
ð4Þ
where dA = dxdy. By substituting Eq. (4) into Eqs. (1) and (2), and simplifying, the following differential equations for the crossflow heat exchanger are obtained:
@h1 ðX; YÞ þ h1 ðX; YÞ ¼ h2 ðX; YÞ @X
ð5Þ
Since the Laplace transform of the derivative of function is n o L dFðnÞ ¼ sLfFðnÞgn!s Fð0Þ, Eq. (14) can be expressed as dn n!s
follow:
ph2L þ h2 ðX;0Þ þ h2L ¼ h1L )
By applying Laplace transform for variable X to Eq. (13), the following expression can be obtained with the boundary condition of h1 in Eq. (9):
n n o 1 L L dh dX
o
X!s
@h2 ðX; YÞ þ h2 ðX; YÞ ¼ h1 ðX; YÞ @Y
ð6Þ
UA UA 1 ¼ ¼ _ j cp;j m Cj Cj
Z
UdA ðj ¼ 1; 2Þ
ð7Þ
ð8Þ
If the inlet temperature distributions of primary and secondary sides are assumed to be uniform, following two boundary conditions are accompanied.
h1 ð0; YÞ ¼ 1;
h2 ðX;0Þ ¼ 0
ð9Þ
The solution of systems of partial differential equations can be obtained by implementing the Laplace transform and inverse transform. Exact solutions of this problem have been reported by Nusselt [9,11]. And there are various crossflow heat exchanger models [12–19], but it is proven that those all models are alternative expressions of Nusselt’s model [20]. Thus, in this study, the Nusselt model is employed as the analytical solution of the crossflow heat exchanger. First, the Laplace transforms for each variable can be defined as follows:
hiL ¼ hiL ðX; pÞ ¼ Lfh1 ðX; YÞgY!p ;
i ¼ 1; 2
ð10Þ
~hiL ¼ ~hiL ðs; pÞ ¼ Lfh1L ðX; pÞg ; X!s
i ¼ 1; 2
ð11Þ
LfLffðX; YÞgX!s gY!p ¼ LfLffðX; YÞgY!p gX!s
ð12Þ
To obtain the solution of the partial differential equations, first, the Laplace transform for variable Y is applied to Eqs. (5) and (6).
dh1L þ h1L ¼ h2L dX dh2L þ h2L ¼ h1L dY
þ Lfh1L gX!s ¼ Lfh2L gX!s )
ð16Þ
ðs þ 1Þ~h1L ¼ ~h2L þ 1p
~h1L ¼
The number of heat transfer units, NTU, is defined as a ratio of the overall thermal conductance to the heat capacity rate. NTU is given by:
NTUj ¼
Y!p
From algebraic Eqs. (15) and (16), ~ h1L and ~ h2L are given by:
where 0 6 X (=x/Lx) 6 NTU and 0 6 Y (=y/Ly) 6 C⁄NTU. The dimensionless temperature hj is defined by:
hj ¼ ðTj T2;in Þ=ðT1;in T2;in Þ ðj ¼ 1; 2Þ
ð15Þ
ðp þ 1Þh2L h1L ¼ 0
ð13Þ ð14Þ
! pþ1 1 1 1 ¼ p p ps þ s þ p p s þ pþ1
ð17Þ
! 1 1 1 1 ¼ p p ps þ s þ p pðp þ 1Þ s þ pþ1 ! 1 1 1 ¼ p p p þ 1 s þ pþ1
~h2L ¼
ð18Þ
The following Laplace transform pairs and their relationship are used to obtain the solutions of Eqs. (17) and (18).
9 8 p
ð19Þ
p!Y
V1 ðX; YÞ 1
Z
X
pffiffiffiffiffiffi eðuþYÞ I0 2 Yu du ¼ L1
0
9 8 p
p
;
p!Y
ð20Þ V1 ðX; YÞ V1;0 ðX; YÞ ¼ 1 V1 ðY; XÞ
ð21Þ
Thus, by applying the inverse Laplace transform of Eqs. (17) and (18), the final solutions are given by:
(
!) ) 1 1 h1 ðX; YÞ ¼ L L p p s þ pþ1 s!X p!Y
p 1 1 exp X ¼L ¼ V1 ðX; YÞ p pþ1 p!Y ( ( !) ) 1 1 1 1 1 h2 ðX; YÞ ¼ L L p p p þ 1 s þ pþ1 s!X p!Y
1 1 p exp X ¼ L1 p pþ1 pþ1 p!Y 1
(
1
¼ V1 ðX; YÞ V1;0 ðX; YÞ ¼ 1 V1 ðY; XÞ
ð22Þ
ð23Þ
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Above solutions can be easily implemented by using built-in functions of the MATLAB program. MATLAB provides the modified Bessel function of the first kind and zero order and integration of the function. The temperature solution of the crossflow heat exchanger is obtained simply and quickly by solving Eq. (19). 2.2. Thermal design of PCHE To meet heat requirements of the inlet and outlet temperatures of fluids, the size of heat exchanger is calculated through the thermal design procedure. The outlet temperature was averaged along the perpendicular to the flow direction in each fluid side since the temperature profile of crossflow heat exchanger is two-dimensional distribution. Thermal design process of crossflow PCHE is shown in Fig. 5. In general, the size of the heat exchanger is determined by the fluid temperatures. Fig. 6 shows the schematic diagram of crossflow PCHE. Geometric parameters of PCHE were tabulated in Table 2. In order to determine the size of crossflow PCHE, at first, the sizes of crossflow PCHE (Lx, Ly, and Lz) are assumed. From the assumed geometry information, the overall heat transfer surface area A is calculated as follows:
A¼
p
p d þ d Lx N1 Nz ¼ d þ d Ly N2 Nz 2 2
Fig. 6. Schematic diagram of crossflow PCHE.
Table 2 Design parameters of crossflow PCHE.
ð24Þ
The number of flow channels in each fluid side is determined as follows:
A¼
Geometric parameters
Value
Channel diameter d, (m) Channel pitch P, (m) Plate thickness tp, (m)
0.003 0.0033 0.00317
p
p ðLy þ tf Þ ðLx þ tf Þ d þ d Lx Lz ¼ d þ d Ly Lz 2 2 2tf ðtf þ dÞ 2tf ðtf þ dÞ
ð28Þ
N1 d þ ðN1 1Þtf ¼ Ly
ð25Þ
The mass flow rate through each fluid side is calculated from the heat duty and the given temperature conditions by using the following heat balance equations:
N2 d þ ðN2 1Þtf ¼ Lx
ð26Þ
_ 1 cp;1 ðT1;in T1;out Þ ¼ m _ 2 cp;2 ðT2;out T2;in Þ Q ¼m
Nz ¼ Lz =ð2tp Þ
ð27Þ
Simplifying Eqs. (25)–(27) and substituting them to Eq. (24) yields:
Vj ¼
Calculate overall heat transfer surface area, As
Calculate mass flow rate . and fluid velocity, mf , Vf
Calculate Temperature Distribuon
Calculate heat transfer coefficient, h
Calculate average outlet − temperature Tout
Resize
_j m
−out| <ε |Tout-T Yes Determinaon of HX sizes
Fig. 5. Thermal design process of crossflow PCHE.
ð31Þ
The Reynolds number and Prandtl number of each fluid side are calculated as follows:
Rej ¼
qj Vj Dh;j lj
Prj ¼
cp;j lj kj
ðj ¼ 1; 2Þ
ðj ¼ 1; 2Þ
ð32Þ
ð33Þ
The heat transfer coefficient is calculated by empirical heat transfer correlations. The heat transfer correlations used in this study will be discussed in Section 2.4. When the Nusselt number is determined by empirical correlation, the heat transfer coefficient in each fluid side is calculated as follows:
hj ¼ Nuj No
ð30Þ
2;in Þ
ðj ¼ 1; 2Þ
qj Axs;j
Calculate Number of Transfer Unit, NTU
Calculate Non-dimensional parameters (Re, Pr, Nu)
Calculate overall heat transfer coefficient, U
_ 1 ¼ c ðT QT Þ m p;1 1;in 1;out
The fluid velocity of each fluid side is calculated as follows:
Given Requirements and condions of HX: Q, Dh , Tin , Tout , P, material properes f(P, T)
Set the inial sizes of HX
thus:
_ 2 ¼ c ðT Q T m p;2 2;out
Thermal Design Process
ð29Þ
kj Dh;j
ðj ¼ 1; 2Þ
ð34Þ
Overall heat transfer coefficient of crossflow PCHE is given by:
U¼
1 h1
1 1 ¼ þ Rw þ h12 h1 þ kdww þ h1 1 2
ð35Þ
Calculated values in Eqs. (28), (30), and (35) are used to calculate the NTU in Eq. (8). When the NTU is determined, temperature profile of the crossflow heat exchanger can be calculated by the
S.-J. Yoon et al. / International Journal of Heat and Mass Transfer 70 (2014) 250–263
analytical solutions in Eqs. (22) and (23). Dimensionless temperature profiles, hj(X, Y), of crossflow PCHE are converted to Tj(x, y) as follows: Tj ðx;yÞ ¼ ðT1;in T2;in Þhj ðx;yÞ þ T2;in sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! # " Z NTU1 x Lx NTU2 NTU2 ¼ ðT1;in T2;in Þ 1 exp y þ u Io 2 yu du þ T2;in Ly Ly 0 ð36Þ
where 0 6 x 6 Lx, 0 6 y 6 Ly. The mean outlet temperatures of primary and secondary fluids can be calculated by averaging each temperature profile function as follows:
T1;out ¼ ¼
T2;out ¼ ¼
1 Ly 1 Ly 1 Lx 1 Lx
Z Z
T1 ðLx ; yÞdy 0
ðT1;in T2;in Þhj ðLx ; yÞ þ T2;in dy
ð37Þ
0
Z Z
L1
T2 ðx;Ly Þdx 0 Lx
ðT1;in T2;in Þhj ðx;Ly Þ þ T2;in dx
ð38Þ
0
Above mean outlet temperatures should be corresponded to the required outlet temperature conditions. If the mean outlet temperatures disagree with the required outlet temperatures, then the heat exchanger has to be resized until the calculated mean outlet temperatures are good agreement with the required outlet temperatures. From the calculated outlet temperatures, log mean temperature difference (LMTD) of crossflow PCHE can be obtained by:
DTLMTD ¼
ðT1;in T2;out Þ ðT1;out T2;in Þ ðT T Þ ln ðT1;in T2;out Þ 1;out
ð39Þ
2;in
In order to estimate the operational cost of heat exchanger, the pressure loss of the heat exchanger has to be calculated. Since the channel shape of PCHE is a semi-circle, the pressure loss calculation using the friction factor correlation for a circular pipe can cause an incorrect result. The fully-developed laminar friction factor of semi-circular straight pipe [21] is given by:
f Re ¼ 15:78;
channel should be employed. However, the straight pipe is assumed in this study so that the friction factor correlation by Berbish et al. [22] is employed to calculate the turbulent friction factor of semi-circular straight pipe. The friction factor is given by:
f ¼ 0:478Re0:26 ; 8200 Re 58000
ð41Þ
The pressure loss of the cross flow PCHE can be calculated by the pressure loss equation of straight pipe:
DPj ¼ 2f j
Lj q V2 Dh;j j j
ðj ¼ 1; 2Þ
ð42Þ
2.3. Thermo-physical properties of working fluids
L2
Ly
255
Re < 2300
ð40Þ
Hesselgreaves [21] suggests the turbulent friction factor of PCHE of which the flow channel is wavy. If the flow channel is wavy or zigzag-shaped, the correlations for wavy or zigzag-shaped
The working fluids used in advanced SMRs are water, helium, sodium, and molten-salts (FLiNaK, FLiBe). In this study, IAPWS IF97 standard formulation [23] is used to calculate the water properties. The helium property is based on the ‘‘National Institute of Standards Technology (NIST) Chemistry WebBook’’ data [24]. Thermophysical properties of sodium and molten-salts are summarized in Table 3. The unit of temperature, T, in Table 3 is Kelvin. Some material properties of molten-salts are not accurate and have to be investigated in the future. In this study, the effect of uncertainty of material property was investigated. The maximum uncertainty of property is assumed to be ±30%. 2.4. Correlations for heat transfer coefficient The heat transfer correlations have been developed with various fluids under the various conditions. Dittus-Boelter [28] and Gnielinski [29] correlations are well-known and widely-used heat transfer correlations, but it is known that Dittus-Boelter has definite uncertainty. Gnielinski correlation is used as the default heat transfer correlation of the developed code. The heat transfer correlations are summarized in Table 4. Each correlation is valid for limited ranges of temperature and pressure, shape of channel, and fluid material. A heat transfer coefficient calculated by the empirical correlation has an influence on the temperature profile of the crossflow heat exchanger and consequently on the heat exchanger sizing. Therefore, the appropriate selection of a heat transfer correlation is important for accurate heat exchanger sizing. In addition, the empirical heat transfer correlations developed for circular pipe might predict the heat transfer coefficient incorrectly since the flow channel of PCHE is a semi-circle. In this study, the temperature profiles according to
Table 3 Thermophysical properties of sodium, FLiNaK and FLiBe. Sodium [25] Density (kg/m3) Heat capacity (J/(kg K)) Thermal conductivity (W/(m K)) Dynamic viscosity (Pa s)
896.60679 + 0.5161343T 1.8297218 103T2 + 2.2016247 106T3 – 1.3975634 109T4 + 0.44866894 1012T5 – 0.057963628 1015T6 (38.12 – 0.069 106T2 – 19.493 103T + 10.24 106T2)/22.99 99.5 – 39.1 103T exp(6.44060.3958lnT + 556.8/T) FLiNaK (LiF-NaF-KF) [26]
Density (kg/m3) Heat capacity (J/(kg K)) Thermal conductivity (W/(m K)) Dynamic viscosity (Pa s)
2,579.3 – 0.624T 976.78 + 1.0634T 0.36 + 5.6 104T
Density (kg/m3) Heat capacity (J/(kg K)) Thermal conductivity (W/(m K)) Dynamic viscosity (Pa s)
2,413 – 0.488T 2397.73 ( ± 20%) 0.629697 + 0.0005T
2.487 105exp(4,478.62/T) FLiBe (LiF-BeF2) [26]
0.000116exp(3,755/T)
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Table 4 Summary of single-phase heat transfer coefficient correlations. Authors
Channel shape
Working fluid
Berbish [22] Kim et al. [27] Dittus-Boelter [28]
Semi-circular Semi-circular Circular
Air He Any fluid
Gnielinski (NuG) [29]
Circular
Any fluid
Nusselt number correlation 0.0228Re 3.255 + 0.00729(Re350) 0.0243Re0.8Pr0.4 for heating 0.0265Re0.8Pr0.3 for cooling ð8f ÞðRe1000ÞPr pffi
8,200 < Re < 5.83 104 350 < Re < 800, Pr = 0.66 104 < Re < 1.2 105, 0.7 < Pr < 120, L/D > 60
1þ12:7
Lyon [30] Lubarsky and Kaufman [31] Reed [32] Taylor and Kirchgessner [33], Wieland [34] Wu and Little [35] Wang and Peng [36]
Circular Circular, Annular Circular Circular
Liquid Metal Liquid Metal Liquid Metal He
7.0 + 0.025Pe0.8 0.625Pe0.4 3.3 + 0.02Pe0.8 0.021Re0.8Pr0.4
0 6 Pr 6 0.1, 300 < Pe < 104 0 6 Pr 6 0.1, 200 < Pe < 104 100 < Pe 3.2 103 < Re < 6 104, 60 < L/D
Rectangular Rectangular
N2 gas Water, Methanol
0.00222Re1.09Pr0.4 0.00805Re0.8Pr0.33
3000 < Re 2300 < Re
2.5. Cost estimation methodology The total cost of a heat exchanger consists of capital cost and operating cost. The capital cost is related to the material cost of the heat exchanger so that it can be estimated based on its weight. Therefore, the capital cost depends on the structural material and size of the heat exchanger. The total mass of heat exchanger is given by:
MHX ¼ qHX VHX
ð43Þ
The volume of the heat exchanger can be approximated as follows:
VHX ¼ Lx Ly Lz
ð44Þ
The capital cost can be calculated by multiplying the cost factor of material ($/kg) and the total mass of heat exchanger, as follows:
CP ¼ CM MHX
ð45Þ
On the other hand, the operating cost is estimated based on the thermal-hydraulic condition of the heat exchanger. Since the operating cost is defined as the cost to operate the circulation pump of the heat exchanger, it is important to optimize the design of heat exchanger to reduce the pressure loss, in turn reducing the operating cost. To estimate the operating cost, the pumping power of the heat exchanger is used. The pumping powers of the heat exchanger in hot and cold fluid sides can be defined by [37]:
P2 ¼
2300 < Re < 5 106, 0.5 < Pr < 2000
2=3 f 1Þ 8ðPr
the heat transfer correlations are compared to evaluate the effect of the heat transfer correlation.
P1 ¼
Valid ranges
0.8
_ 1 DP1 m
ð46Þ
q1 _ 2 DP2 m
ð47Þ
q2
Thus, the operating cost can be approximated as follows [37]:
OP ¼ COP YðP 1 þ P 2 Þ
ð48Þ
Therefore, the total cost of heat exchanger becomes:
Ctotal ¼ CP þ OP ¼ CM MHX þ COP YðP 1 þ P 2 Þ
ð49Þ
As a structural material of PCHE, the nickel-based alloys such as Alloy 800H, Alloy 617, and Hastelloy N are under development. The thermo-physical properties of structural material also have an influence on the sizing and cost estimation of heat exchanger. According to the previous studies on the cost of heat exchangers [37,38], the material costs of Alloy 800H and Alloy 617 are assumed to be 120 USD/kg conservatively. The material cost of Hastelloy N is approximately 124 USD/kg [39]. The operating cost is 0.0000612 USD/Wh, which is based on the consumer price index average price data [37]. For high temperature condition of 700 °C, the thermo-physical properties and material costs of PCHE structural materials are summarized in Table 5.
3. Crossflow PCHE analysis The fluid combinations of primary and secondary sides are different for each reactor coolant of SMR. In this study, four fluid combinations of primary and secondary side are selected by taking into account the reactor coolant types of SMR. The heat duty of heat exchanger is assumed to be 300 MW. The Reynolds numbers of primary and secondary sides are varied according to the fluid property and heat duty. Test conditions are summarized in Table 6.
3.1. Grid sensitivity test In this study, the temperature profile of crossflow PCHE was obtained by solving the system of differential equations. Twodimensional grid is used to calculate the temperature profile. In developed code, the mean outlet temperature of each fluid side is calculated by averaging the temperature at the outlet. Thus, the number of grids can influence the mean outlet temperature. In order to investigate the grid effect, the grid sensitivity test was carried out. Tested numbers of grid and results are summarized in Table 7. In grid sensitivity test, x and y-axial dimensions of heat exchanger, NTUs of primary and secondary sides are assumed to be a unity. The numbers of reference grid was 40 40.
Table 5 Thermo-physical properties and cost of structural materials of PCHE. Structural material
Density (kg/m3)
Thermal conductivity (W/(m K))
Heat capacity (J/(kg K))
Cost (USD/kg)
Alloy 617 [37] Alloy 800H [38] Hastelloy N [39]
8,360 7,940 8,860
23.9 22.8 23.6
586 460 523
120 120 124
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S.-J. Yoon et al. / International Journal of Heat and Mass Transfer 70 (2014) 250–263 Table 6 Test matrix for crossflow PCHE analysis. Coolants (primary/secondary)
Inlet temperature (°C) (primary/secondary)
Pressure (MPa) (primary/secondary)
Heat duty (MW)
Re
Pr
Water/Water Helium/Helium Sodium/Sodium FLiBe/FLiNaK
320/200 950/350 545/320 700/600
15/15 7.0/7.0 0.1/0.1 0.1/0.1
300
104–105 104–105 104–105 103–104
1 0.66 0.005 10
Table 7 Grid sensitivity test result. Dimensionless mean outlet temperature Numbers of grids Primary side Secondary side
55 0.52124 0.47876
10 10 0.52251 0.47749
20 20 0.52314 0.47686
The grid sensitivity analysis by the Richardson extrapolation [40] was performed to estimate the numerical error due to the grid number. The approximation of exact solution U is given by:
U ¼ /h þ
/h /2h 2p 1
ð50Þ
where / is a numerical solution and the order of the scheme p is defined by:
log p¼
/2h /4h /h /2h
ð51Þ
log2
Fig. 7 shows the dimensionless mean outlet temperatures of the primary and secondary sides. The dimensionless temperature in the grid sensitivity test was defined in Eq. (7). The Richardson solutions of primary and secondary fluids were 0.5238 and 0.4762, respectively. The maximum relative errors between the reference grid (40 40) and Richardson solution of primary and secondary fluids were 0.48% and 0.53%, respectively. Consequently, the grid sensitivity test results show that the effect of grid number is negligible. 3.2. Effectiveness-NTU (e-NTU) method analysis The effectiveness of heat exchanger, e, is the ratio of the actual heat transfer rate to the maximum possible heat transfer rate of the exchanger and is defined by:
e¼
Q ðT1;in T1;out Þ ¼ Q max ðT1;in T2;in Þ
ð52Þ
40 40 0.52346 0.47654
80 80 0.52362 0.47638
160 160 0.52370 0.47630
320 320 0.52374 0.47626
The e-NTU method [10] is one of general methods applied for the thermal design of heat exchanger. In this method, the effectiveness of crossflow heat exchanger is a function of NTU. The relationship between the effectiveness and NTU of both unmixed fluids crossflow heat exchanger [41] is given by:
e ¼ 1 exp
1 0:22 0:78 ðNTUÞ ðexp½C ðNTUÞ 1Þ C
ð53Þ
where C⁄( = Cmin/Cmax) is the ratio of minimum to maximum heat capacity rate of fluid. The effectiveness of crossflow PCHE calculated by the developed code is compared with that of the e-NTU method. Fig. 8 shows the effectiveness-NTU plots by the developed code and e-NTU method. The effectiveness of a heat exchanger increases as the NTU increases. The effectiveness increases rapidly as the ratio of heat capacity rate decreases. The maximum relative error of effectiveness between the code and e-NTU method was 3.95%. 3.3. Effect of fluid property uncertainty The uncertainty of fluid property can be propagated into the calculated temperature profile of the heat exchanger. To evaluate the effect of uncertainty of fluid property, the uncertainty of fluid property was assumed to be ±30%. In this calculation, the heat duty, the temperatures and pressures in Table 1 were employed. Hastelloy N is employed as the structural material. Gnielinski heat transfer correlation was used to calculate the heat transfer coefficient. The
Φprimary=0.5238
0.523
0.482
0.522
0.481
0.521
Primary Secondary
0.520 0.519
0.480 0.479 0.478
0.518
0.477
Φsecondary=0.4762
0.517
0.476 0
40
80
120
160
200
240
280
320
Number of grids Fig. 7. Effect of grid number on dimensionless mean temperature profile of crossflow heat exchanger.
1.0 0.9 0.8
Effectiveness
0.483
Dimensionless mean outlet temperature of secondary fluid
Dimensionless mean outlet temperature of primary fluid
1.1 0.524
0.7 0.6 0.5
ε-NTU method PCHE analysis code C*=0.1 C*=0.1 C*=0.3 C*=0.3 C*=0.5 C*=0.5 C*=0.7 C*=0.7 C*=1 C*=1
0.4 0.3 0.2 0.1 0.0 0
1
2
3
4
5
6
7
8
9
10
NTU Fig. 8. Effectiveness-NTU comparison between e-NTU method and crossflow PCHE analysis code.
S.-J. Yoon et al. / International Journal of Heat and Mass Transfer 70 (2014) 250–263
Averaged temperature (oC)
320
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
310
300
290
800
+1.05% 750 o
dimensions in x, y, and z axial directions of heat exchanger were assumed to be 0.9897 m, 0.9897 m, and 0.634 m, respectively. Corresponding numbers of flow channels are 300, 300, and 100, respectively. Figs. 9–16 show the average temperature profiles according to the uncertainty of thermo-physical property for each coolant. The
Averaged temperature ( C)
258
700
-1.65%
650 600
Reference k (-30%) cp(-30%) μ (-30%) k (+30%) cp(+30%) μ (+30%)
550 500 450 400 350 0.0
0.2
0.4
0.6
0.8
1.0
Distance from the inlet (m) 280
+0.75%
270
Fig. 12. Average temperature profile of secondary fluid according to uncertainty of material property (Helium).
-0.47% 260 0.0
560 0.2
0.4
0.6
0.8
540
250
-0.77%
500 480 460 440 420
+0.7%
400 380
230
360 0.0
-0.47% k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
220
210
200 0.0
0.2
0.4
0.6
0.8
Fig. 10. Average temperature profile of secondary fluid according to uncertainty of material property (Water).
950
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
900 850 800 750
0.4
0.6
0.8
1.0
Fig. 13. Average temperature profile in primary side according to uncertainty of material property (Sodium).
500
+0.37% 480
-0.54%
460 440 420
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
400 380 360 340 320 0.0
700
0.2
0.4
0.6
0.8
1.0
Distance from the inlet (m)
650
+2.38%
600 550
-1.52% 500 0.0
0.2
Distance from the inlet (m)
1.0
Distance from the inlet (m)
Averaged temperature (oC)
520
240
Averaged temperature (oC)
Averaged temperature (oC)
+0.48%
o
Fig. 9. Average temperature profile of primary fluid according to uncertainty of material property (Water).
Averaged temperature ( C)
Distance from the inlet (m)
260
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%)
1.0
0.2
0.4
0.6
0.8
1.0
Distance from the inlet (m) Fig. 11. Average temperature profile of primary fluid according to uncertainty of material property (Helium).
Fig. 14. Average temperature profile in secondary side according to uncertainty of material property (Sodium).
density change did not have an effect on the temperature profile. In the developed code, when the heat duty is given, the mass flow rate is determined from Eq. (30). Then, the density is used to calculate the fluid velocity and Reynolds number. Since the product of density and fluid velocity is constant by Eq. (31), the Reynolds number is not changed by the density. Consequently,
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660
650 0.0
k (-30%) cp(-30%) μ (-30%) Reference k (+30%) cp(+30%) μ (+30%) 0.2
+0.62%
-0.51% 0.4
0.6
0.8
1.0
Distance from the inlet (m) Fig. 15. Average temperature profile in primary side according to uncertainty of material property (FLiBe).
650
Averaged temperature (oC)
+0.53% 640
630
-0.64% Reference k (-30%) cp(-30%) μ (-30%) k (+30%) cp(+30%) μ (+30%)
620
610
600 0.0
0.2
0.4
0.6
0.8
1.0
Distance from the inlet (m) Fig. 16. Average temperature profile in secondary side according to uncertainty of material property (FLiNaK).
the uncertainty of fluid density does not have an effect on the temperature. Uncertainties of heat capacity and dynamic viscosity resulted in the same temperature profile. The temperature profile of the crossflow analytical model is determined by the size of heat exchanger and NTU value. In this crossflow model, NTU can be approximated as a function of the product of heat capacity and dynamic viscosity so that the changes in these two parameters resulted in the same degree of change in NTU. Uncertainty of thermal conductivity led to the maximum deviation of temperature profile, except the sodium case. The effect of thermal conductivity was relatively smaller than heat capacity and dynamic viscosity for the fluid of low Prandtl number. Overall, the relative deviation of the temperature profile according to the uncertainty of fluid property was very small. Maximum deviation was 2.38% that occurred by the thermal conductivity in the primary side of helium flow. In conclusion, the uncertainty of fluid property was negligible in the thermal design of crossflow PCHE. 3.4. Effect of heat transfer coefficient correlations Comparative analyses were carried out to evaluate the effect of heat transfer correlations on the temperature profile. In this analysis, the heat duty, the temperatures, and pressures in Table 1 were employed. Hastelloy N was employed as the structural material. The dimensions of the x, y, and z axial directions of the heat exchanger were assumed to be 0.9897 m, 0.9897 m, and 0.634 m, respectively.
360
o
670
Average temperatre ( C)
680
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
330
Lubarsky and Kaufman
300 Wang and Peng
ΔT=39 K
270 Wu and Little, Kim et al.
240 0.0
0.2
0.4
0.6
0.8
1.0
Distrance from the Inlet (m) Fig. 17. Temperature profile in primary side by the heat transfer correlations (Water).
280
o
690
Figs. 17 and 18 show the temperature profiles in the primary and secondary side of the water flows, respectively. The Lubarsky and Kaufman’s correlation showed the highest temperature of primary side and lowest temperature of secondary side. Wu and Little’s and Kim et al. correlations showed the lowest temperature of primary side and highest temperature of secondary side. These two correlations predicted the heat transfer coefficients similarly for Prandtl number in the range of 0.5–1.0. Thus, the same temperature profile trend was observed in the water and helium flows of which Prandtl numbers were approximately 1.0 and 0.66, respectively. The maximum deviation among the averaged outlet temperature at the outlet was 39.45 K, which was caused by the Lubarsky and Kaufman’s correlation and Kim et al. correlation. Figs. 19 and 20 show the temperature profiles of helium flows. In the helium flow, the similar trends of temperature profiles were observed as the water flow. The maximum deviation among the averaged outlet temperature was approximately 196.01 K which was caused by the Lubarsky and Kaufman’s correlation and Kim et al. correlation. Figs. 21 and 22 show the temperature profiles of sodium flows. In this case, the effect of heat transfer correlation was smaller than the other cases. For the fluids with very low Prandtl numbers, most of empirical correlations predicted low heat transfer coefficient and the difference among them was also small. Maximum deviation of averaged outlet temperature was 15.34 K, which was caused by Gnielinski’s correlation and Kim et al. correlation. Figs. 23 and 24 shows the temperature profiles of molten salt flows. The Prandtl
Average temperatre ( C)
Averaged temperature (oC)
700
260
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
Wu and Little, Kim et al.
ΔT=39 K
240 Wang and Peng
220 Lubarsky and Kaufman
200 0.0
0.2
0.4
0.6
0.8
1.0
Distrance from the Inlet (m) Fig. 18. Temperature profile in secondary side by the heat transfer correlations (Water).
260
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550
800 Wang and Peng
700
500
400 0.0
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
0.2
0.4
ΔT=196 K Wu and Little Kim et al.
0.6
0.8
500
o
Lubarsky and Kaufman
600
Average temperatre ( C)
o
Average temperatre ( C)
900
450
350
0.2
Fig. 19. Temperature profile in primary side by the heat transfer correlations (Helium).
700
700 Wang and Peng
o
ΔT=196 K
Lubarsky and Kaufman
680 ΔT=31 K
660
640
400 0.2
0.4
0.6
0.8
620 0.0
1.0
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
0.2
Distrance from the Inlet (m)
Gnielinski
400
0.8
ΔT=31 K
620 Wang and Peng
600
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
580
ΔT=15 K
Berbish, Wu and Little, and Kim et al.
0.6
1.0
Lyon
640
o
Average temperatre ( C)
o
Average temperatre ( C)
450
0.4
0.8
660
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
0.2
0.6
Fig. 23. Temperature profile in primary side by the heat transfer correlations (Molten Salt).
550
350 0.0
0.4
Lyon
Distrance from the Inlet (m)
Fig. 20. Temperature profile in secondary side by the heat transfer correlations (Helium).
500
1.0
Kim et al. Wu and Little
Wang and Peng
0.0
0.8
720
600
500
0.6
Fig. 22. Temperature profile in secondary side by the heat transfer correlations (Sodium).
Average temperatre ( C)
o
Average temperatre ( C)
800
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng
0.4
Distrance from the Inlet (m)
Distrance from the Inlet (m)
900
ΔT=15 K
400
300 0.0
1.0
Gnielinski Berbish Dittus-Boelter Kim et al Lyon Berbish, Wu and Little, and Kim et al. Lubarsky and Kaufman Reed Taylor and Kirchgessner, Wieland Wu and Little Wang and Peng Gnielinski
1.0
Distrance from the Inlet (m) Fig. 21. Temperature profile in primary side by the heat transfer correlations (Sodium).
number of molten salt flows was approximately 10.0. Contrary to the sodium flow, the molten salt is a fluid with a high Prandtl number. Not only the empirical correlations developed for liquid metals but also Dittus-Boelter correlation and Gnielinski’s correlations predicted high-heat transfer coefficient. Consequently, these
560 0.0
0.2
0.4
0.6
0.8
1.0
Distrance from the Inlet (m) Fig. 24. Temperature profile in secondary side by the heat transfer correlations (Molten Salt).
correlations predicted similar temperature profiles. Maximum deviation of averaged outlet temperature was 31.04 K, which was caused by Wang and Peng’s correlation and Lyon’s correlation. By comparing heat transfer correlations for each fluid, we found that the effect of heat transfer correlations was largest in the
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S.-J. Yoon et al. / International Journal of Heat and Mass Transfer 70 (2014) 250–263 Table 8 Sizing and cost estimation results of crossflow PCHEs. Coolants
Heat exchanger size (m)
Average outlet temperature (°C)
Cost (USD) (Operation period: 20 year)
Lx
Ly
Lz
Primary
Secondary
Structural material
Capital
Operation
Water
0.396
1.518
0.634
295.3
293.6
Alloy 617 Alloy 800H Hastelloy N
3.8197 105 3.6278 105 4.1831 105
1.4049 109 1.4420 109 1.4146 109
Helium
1.128
2.095
637.8
925.3
Alloy 617 Alloy 800H Hastelloy N
1.5036 106 1.4280 106 1.6466 106
9.4799 109 9.4901 109 9.4826 109
Sodium
1.214
1.613
390.2
526.0
Alloy 617 Alloy 800H Hastelloy N
1.2459 106 1.1833 106 1.3644 106
1.2703 108 1.2770 108 1.2720 108
Molten Salt
6.352
11.880
651.0
690.2
Alloy 617 Alloy 800H Hastelloy N
4.7996 107 4.5585 107 5.2562 107
9.1158 106 9.1174 106 9.1163 106
In this crossflow PCHE analysis code, the size of heat exchanger is determined by iterative calculation to meet the given heat duty and temperature requirements. The heat duty and temperature requirements in Table 1 and three nickel-based alloys in Table 5 were used in the economic analysis. The maximum operating period of heat exchanger is assumed to be 20 years. In this analysis, the height of heat exchanger, Lz, was fixed to 0.634 m to reduce the number of cases. If the height of the heat exchanger can be varied, the length and width of the heat exchanger are determined for each height so that various heat exchanger sizes can be generated. Economic analysis results are summarized in Table 8. The sizes of heat exchangers using water, helium, and sodium coolants were similar to each other, whereas large heat exchanger size was required to meet the temperature requirements for molten salt coolants. The predicted size of crossflow PCHE for molten salt is deemed inappropriate for molten salt cooled SMR. To reduce the size of the heat exchanger, the heat duty per heat exchanger has to be reduced by increasing the number of heat exchangers. 7.4E6
Alloy 617 Alloy 800H Hastelloy N
Total cost per year (USD/yr)
7.35E6 7.3E6
Total cost per year (USD/yr)
3.5. Cost estimation results
4.755E8
Alloy 617 Alloy 800H Hastelloy N
4.752E8 4.749E8 4.746E8 4.743E8 4.74E8 4.737E8
3
6
9
12
15
18
21
Operation period (yr) Fig. 26. Total cost per year of crossflow PCHE (Helium/Helium).
6.8E6
Total cost per year (USD/yr)
helium flow. The deviation of temperature profiles by the heat transfer correlation decreased for very low or very high Prandtl number fluids. It was not easy to determine a heat transfer correlation that can be applied generally irrespective of coolant types. Consequently, the experimental and/or computational validations are deemed to be required to improve the validity and accuracy of crossflow PCHE analysis code. In addition, the appropriate heat transfer correlation for each fluid type should be applied.
Alloy 617 Alloy 800H Hastelloy N
6.7E6
6.6E6
6.5E6
6.4E6
6.3E6 3
6
9
12
15
18
21
Operation period (yr) 7.25E6
Fig. 27. Total cost per year of crossflow PCHE (Sodium/Sodium). 7.2E6 7.15E6 7.1E6 7.05E6 7E6 3
6
9
12
15
18
21
Operation period (yr) Fig. 25. Total cost per year of crossflow PCHE (Water/Water).
The capital cost of crossflow PCHE was relatively smaller than the operation cost except for molten salt heat exchanger. Due to the large size of the molten salt heat exchanger, its capital cost was estimated to be larger than operation cost. The capital cost of the molten salt heat exchanger was higher than the others, but the operation cost was much lower. The operation cost of the helium heat exchanger was the most expensive. Figs. 25–28 show the total cost per year of the heat exchangers according to the operation period. Operation cost per year decreased as the
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(Grant code: 2012-052255), under DOE Idaho Operations Office Contract DE-AC07-05ID14517.
1.2E7
Alloy 617 Alloy 800H Hastelloy N
Total cost per year (USD/yr)
1E7 8E6
References 6E6
4E6
2E6 3
6
9
12
15
18
21
Operation period (yr) Fig. 28. Total cost per year of crossflow PCHE (FLiBe/FLiNaK).
operation period increased. Total cost of the heat exchanger using Alloy 617 was cheaper than the other structural materials except for the molten salt heat exchanger. In the molten salt heat exchanger, the total cost was minimized by using Alloy 800H. The total cost per year of molten salt heat exchanger decreased rapidly as the operation period increased. 4. Conclusion In this study, the crossflow PCHE analysis code has been developed to evaluate the size and cost of heat exchangers by implementing the analytical solution of single pass, both unmixed fluids crossflow heat exchanger model. Two-dimensional temperature distribution of crossflow PHCE was calculated by the analytical solution. General methods for thermal design and cost estimation of heat exchangers were employed to determine the size and cost of heat exchanger, respectively. The information provided by this code can be used for the optimal design of advanced SMR system. The grid sensitivity test of the code showed that the effect of grid on the temperature profile was negligible. The effectiveness calculated by the code showed a good agreement with that by well-known e-NTU method. The uncertainty of fluid property was propagated into the temperature distribution, but its effect was small enough to be inconsequential. The heat transfer correlations had a considerable influence on the temperature profile. The temperature profile of crossflow PCHE using helium gases showed the largest deviation by the heat transfer correlations. The deviation of temperature profiles by the heat transfer correlation decreased for very low or very high Prandtl number fluids. However, it was not easy to determine the most accurate heat transfer correlation for the heat exchangers using various combinations of coolants. Consequently, experimental and/or computational validations are required to determine the best heat transfer correlation for each SMR. Costs of crossflow PCHE for the advanced SMR designs were investigated. Capital costs of PCHE for water, helium, and sodium flows were lower than their operation costs, whereas capital cost of molten salt heat exchanger was higher than operation cost. Total cost of heat exchanger using Alloy 617 was cheaper than the other structural materials except for the molten salt heat exchanger. In the molten salt heat exchanger, the total cost was minimized by using Alloy 800H. Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) and grant funded by the Korean government (MSIP)
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