Steam generator structure: Continuous model and constructal design

INTERNATIONAL JOURNAL OF ENERGY RESEARCH Int. J. Energy Res. 2011; 35:336–345 Published online 26 March 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/er.1694

Steam generator structure: Continuous model and constructal design Y. Kim1, S. Lorente2 and A. Bejan1,,y 1

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, U.S.A. Universite´ de Toulouse, UPS, INSA, LMDC (Laboratoire Materiaux et Durabilite´ des Constructions), 135 avenue de Rangueil, F-31 077 Toulouse Cedex 04, France 2

SUMMARY This study shows that the main features of a steam generator can be determined based on the method of constructal design. The generator is endowed with freedom to morph, and then is optimized by putting the right components in the right places. The number of steam tubes is sufficiently large, so that the steam generator may be modeled as continuous. The total volume of the assembly and the volume of the steam tubes are fixed. The geometry is free to vary in the search for maximum heat transfer density. The steam flow in the tubes is modeled in two ways: single-phase and two-phase fully developed turbulent flow. Results of the analysis are: the location of the flow reversal (i.e. the demarcation between the tubes of the downcomer and those of the riser), the optimal spacing between adjacent tubes, and the number of tubes for the downcomer and the riser. Copyright r 2010 John Wiley & Sons, Ltd. KEY WORDS constructal; distributed energy systems; crossflow heat exchanger; steam generator; power generation Correspondence A. Bejan, Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, U.S.A. y E-mail: [email protected]

Contract/grant sponsor: Doosan Heavy Industries & Construction Co., Ltd. Received 23 July 2009; Revised 13 January 2010; Accepted 17 January 2010

1. INTRODUCTION According to constructal theory, the generation and evolution of flow configuration in nature is a universal phenomenon that is summarized by the constructal law: ‘For a finite-size flow system to persist in time (to live) its configuration must change in time such that it provides greater and greater access to its currents’ [1]. The constructal law complements the existing principles (mass, momentum and energy conservation, and the second law) and provides new insights into finding the flow configuration. There is now a broad field of research driven by the constructal law, and it covers design phenomena ranging from geophysics to biology, engineering, and social dynamics [1–18]. Steam generators are a major domain of technology development in the power generation industries [19–21]. Recently, we showed that the configuration of a steam generator can be deduced based on constructal Copyright r 2010 John Wiley & Sons, Ltd.

design [22]. This was a simple demonstration of the design method, which was based on the simplifying assumption that the steam generator consists of just one unbeated downcomer tube and the riser tubes. Here, we consider the more promising design direction where the steam generator is free to have a large number of tubes, so many that its material can be distributed through the available volume. Steam generators with many tubes are the preferred configuration in the power generation industries [23–25]. In this study, we consider large numbers of steam tubes, such that the steam generator architecture can be modeled as continuous.

2. MODEL Consider the flow of high-temperature gas across bundle of water tubes shown in Figure 1. The steam

Steam generator structure

Y. Kim, S. Lorente and A. Bejan

W m TF

Tout H z=L

z=0 m1,down

m1,up x out,up

f

g H D x in,up

x out,down

Figure 1. Steam generator with a large number of vertical two-phase tubes in crossflow with a horizontal stream of hot gas.

generator fills a parallelepipedic volume HLW, which is traversed by N equidistant steam tubes of diameter D and height H. Two volume constraints are in place: the total volume occupied by hot gas and the total volume of water tubes. _ P and inlet Hot combustion gases of capacity rate mc temperature TF flow from right to left. Steam flows in crossflow, inside the tubes, being driven in a vertical loop by the thermosyphon effect due to the fact that the tubes close to z 5 L are heated more intensely than the tubes close to z 5 0. The temperature of hot flue gas is not high enough to transform the water in the tubes into superheated steam at the outlet of each water tube; hence, the temperature of water in the tubes remains constant at the boiling temperature TB corresponding to pressure of the upper and lower plenums, which receive and separate the steam generated in the tubes. The net mass flow rate of water through a horizontal plane across all the tubes is zero. The flow rate of the circulated water rate and the flow architecture, i.e. the location of the flow reversal, are unknown. In this study, we seek to configure the continuous steam generating system such that it generates most steam.

and the volume fraction occupied by water f¼

p D2 N 4 LW

ð2Þ

The stream of hot gases is modeled as single phase with constant properties. Assuming that N is sufficiently large so that we may treat the gas temperature T as a continuous function of longitudinal portion z, we write that the hot stream loses enthalpy to match the transfer of heat to the tubular structure TB, _ P dT ¼ mc

hA ðT  TB Þ dz L

ð3Þ

We assume that the heat transfer coefficient h is a known constant, at best a function of the gas flow rate _ which is also specified. Integrating Equation (3) m, from z 5 0, where T 5 Tout, to z we obtain   hA z ð4Þ TðzÞ  TB ¼ ðTout  TB Þ exp _ PL mc or as the overall DT 5 TFTB is specified, TðzÞ  TB ¼ DT

temperature _ P LÞ expðhAz=mc _ PÞ expðhA=mc

difference

ð5Þ

The geometric constraints of this crossflow heat exchanger are the total volume

In sum, the temperature difference TTB varies continuously in the longitudinal direction, and hence does the heat transfer rate from the gas to the water in the tubes, regardless of the water direction and flow rate in the tubes. For example, the heat transfer rate into one tube located at position z is

V ¼ HLW

hpDHðT  TB Þ ¼ m_ 1 ðhout  hin Þ

3. LONGITUDINAL TEMPERATURE DISTRIBUTION

ð1Þ

Int. J. Energy Res. 2011; 35:336–345 r 2010 John Wiley & Sons, Ltd. DOI: 10.1002/er

ð6Þ 337

Y. Kim, S. Lorente and A. Bejan

Steam generator structure

where m_ 1 is the water mass flow rate through the tube, and hin ¼hf

ð7Þ

hout ¼ hf 1xout hfg

ð8Þ

In Equation (7), we are assuming that the stream received by every tube from the plenum contains saturated liquid. The steam quality at the tube exit is related to the specific volume of the steam at the exit, vout ¼ vf 1xout vfg

ð9Þ

or in terms of densities, rout 1 ¼ 11xout vfg rf rf

ð10Þ

where (rrefr1)1/2 is proportional to m_ 1 and y¼

_ P LÞ expðhAz=mc _ PÞ expðhA=mc

ð17Þ

B¼

  hfg rf 1 rf vfg rout

ð18Þ

Equations (13)–(17) show that m_ 1 varies longitudi_ P LÞ; regardless of nally in proportion with expðhAz=mc whether the flow is up or down. The change in direction occurs at the longitudinal location (zc) obtained by requiring Z L ð19Þ m_ 1 dz ¼ 0 0

Eliminating xout among Equations (6)–(10), we find   m_ 1 hfg rf 1 ð11Þ hpDHðT  TB Þ ¼ rf vfg rout

The result is zc 1 11eNtu ¼ ln L Ntu 2

where (TTB) is a known function of longitudinal position, Equation (5).

where Ntu ¼

4. FLOW REVERSAL The mass flow rate m_ 1 is driven by the weight of its liquid column relative to the weight of the neighboring columns. If the hydrostatic pressure difference between the two plenums is DPref ¼ rref gH

ð12Þ

where rref is the reference density in a column of height H, then the net pressure difference that drives m_ 1 upward through its tube is DP1 ¼ ðrref  r1 ÞgH

rf 1rout 2

Consider the hot-gas side of the crossflow heat exchanger, and ask how the geometry affects the global performance. The geometry is represented by the total volume V 5 HLW, which is fixed, and by the dimensions H, L, W, D, and S, where S is the spacing between adjacent tubes (Figure 2). If we use V1/3 as the

338

W

up

down

m H

ð14Þ

z = zc

z=0

S

z=L

W

D

L S

ð15Þ

where the factor is C 5 32 fn/p2 and f is the friction coefficient, which in the fully rough regime is practically constant. A more detailed two-phase flow model is presented in Section 7. Eliminating m_ 1 and rout/rf among Equations (11)–(15), we obtain   y C 1=2 ¼ ðrref  r1 Þ1=2 hpDHDT B gD5

ð21Þ

ð13Þ

The water flow rate m_ 1 is driven by DP1. We illustrate this qualitatively with the simplest model of flow fiction in a tube, namely fully developed turbulent (fully rough) and single-phase fluid, DP1 m_ 2 ¼ C 15 H D

hA _ P mc

5. GEOMETRY ON THE GAS SIDE

Here r1 is the average density of the two-phase mixture in the m_ 1 tube. Approximating the distribution of density of two-phase water as linear along the tube, we estimate the average density r1 ffi

ð20Þ

ð16Þ

Savg

S

The small-S limit

Figure 2. Geometric details of the steam generator, viewed from the side and from above.

Int. J. Energy Res. 2011; 35:336–345 r 2010 John Wiley & Sons, Ltd. DOI: 10.1002/er

Steam generator structure

Y. Kim, S. Lorente and A. Bejan

fixed length scale of the entire architecture, then there are five dimensionless geometrical features, ~ ¼ ðH; L; W; D; SÞ=V 1=3 ~ L; ~ W; ~ D; ~ SÞ ðH;

ð22Þ

and one constant H~ L~ W~ ¼ 1

ð23Þ

In sum, the architecture design has four degrees of freedom. Equation (3) shows that the total heat transfer rate from the hot gas to all the tubes is _ P ðTF  Tout Þ q ¼mc _ P DTð1  eNtu Þ ¼mc

ð25Þ

Because of the large spacings, the free stream velocity of the hot gas is m_ Uffi ð26Þ rF HW where the gas density rF is treated as a constant. The total drag force experienced by the assembly is F ¼ F1

LW S S

ð27Þ

This force is balanced by DPFHW, and from this balance, we deduce the mass flow rate in the (a) limit,  1=2 2 H 2 W 2 S2 m_ a ¼ DPF rF ð28Þ LD CD Next, Equation (21) requires estimates for Aa and ha. The total contact area is Aa ¼pDH

L W pDV ¼ 2 S S S

1=3

ð29Þ

Int. J. Energy Res. 2011; 35:336–345 r 2010 John Wiley & Sons, Ltd. DOI: 10.1002/er

1=2

ha D 0:62 PrF ReD ¼ 0:31 kF ½11ð0:4= PrF Þ2=3 1=4 "   #4=5 ReD 5=8  11 282000

ð30Þ

where kF and PrF are properties of the hot gas. When 10oReDo105, Equation (30) is approximated adequately by

ð24Þ

where DT 5 TFTB, fixed. Fixed is also the pressure difference (DPF) that drives the gas stream horizontally through the multitube structure. When the spacing between tubes (S) is large, the thermal contact between gas and tubes is poor (Ntuo o1), and q is small. When the spacing is _ vanishes vanishingly small, the mass flow rate (m) and so does q. Between these two S extremes, there must be an optimal spacing for which q is maximum when the remaining three geometrical features are fixed ~ H, ~ and L). ~ This expectation is sup(for example, D, ported by several designs developed in constructal theory [3,4], where optimal spacings were predicted and validated experimentally [26] for parallel and staggered plates and cylinders in crossflow. Here, we pursue the optimal spacings by focusing on Equation (24) when only S varies. To calculate q with Equation (24), we need m_ and Ntu. The following analysis is based on the intersection of asymptotes method [3]. (a) The large S limit. First, we assume that S is sufficiently larger than D so that the drag force on each tube is independent of the flow around neighboring tubes. The drag force on the tube is 1 F1 ¼ CD H D rU 2 2

The heat transfer coefficient for a long cylinder is [27]:

ha D 1=2 ffi Ca ReD kF

ð31Þ

where Ca is a dimensionless factor of order 1, "   #4=5 1=3 0:62PrF ReD 5=8 11 Ca ¼ 282000 ½11ð0:4=PrF Þ2=3 1=4

ð32Þ

and ReD 5 UD/nF, with U being provided by Equation (26). In the end, Equation (21) yields Ntu;a ¼

p Ca kF 1=2

cP mF

VD1=2 1=2

S2 H 1=2 W 1=2 m_ a

ð33Þ

Finally, in the large S limit the number Ntu,a approaches zero, and this means that the group (1  eNtu;a ) approaches Ntu,a, and that Equation (24) approaches the asymptote  1=4 1=4 1=4 2 kF r F 1=4 VD qa ¼ pCa 1=2 DTDPF 3=2 1=4 ð34Þ CD S L mF This shows that the total heat transfer rate decreases monotonically as S increases. For this reason, we turn our attention to the opposite limit, in which S decreases. (b) The small S limit. When the cylinders barely touch, the flow is slow, the thermal contact between the hot gas and the tubes is excellent, Ntu is a large number, and Equation (24) reduces to _ P DT qb ¼ mc

ð35Þ

The mass flow rate can be estimated by visualizing the channel of average spacing Savg inhabited by the streamlines that flow along L, through two adjacent rows of tubes. The smallest spacing (S) occurs where the stream channel is pinched by two tubes that almost touch. Because of the regular arrangement of tubes, we can write Savg ¼ sS

ð36Þ

where s is a factor of order 1, but greater than 1, for example, 2. For the elemental channel represented by the volume Savg  L  H, the mass flow rate in the Poissuille flow limit is m_ 1;F ¼ rU1 Savg H

ð37Þ 339

Y. Kim, S. Lorente and A. Bejan

Steam generator structure

in which [27] U1 ¼

2 Savg

DPF 12mF L

ð38Þ

The number of such channels is W/D. The total mass flow rate is m_ ¼

3 V DPF Savg 12nF DL2

ð39Þ

which in combination with Equations (35) and (39) yields qb ¼ cP DT

DPF s3 S3 V 12nF DL2

ð40Þ

This shows that in the small S limit the total heat transfer rate decreases steeply as S decreases. (c) The intersection of asymptotes. The maximum of q with respect to S is located sufficiently accurately by intersecting the two asymptotes, Equations (34) and (40). The result is "  #2=9 2 1=4 12pCa ~ Sopt ¼ Be1=6 D~ 5=18 L~ 7=18 ð41Þ 1=4 CD PrF s3 Be ¼

DPF V 2=3 aF mF

ð42Þ

where Equation (42) shows the dimensionless pressure difference number introduced by Bhattacharjee and Grosshander [28] (also see Reference [29]). The factor in square brackets in Equation (41) is approximately 1, and can be neglected. The mass flow rate that corresponds to opt is  3 s kF 1=3 S~3 V Be ð43Þ m_ opt ¼ 12 cP D~ L~ 2 and, if we regard the group [s3/12] as a number of order 1 and use Equation (41), we obtain m_ opt

kF  V 1=3 Be1=2 D~ 1=6 L~ 5=6 cP

qmax  kF DTV

1=3

1=2

Be

~ 1=6

D

~ 5=6

L

The modeling of the tube surfaces as isothermal allowed us to decouple the search for geometry on the gas side (Section 5) from the search for geometry on the steam side. The two sides are not uncoupled. The location of the flow reversal (Equation (20)) depends on the group Ntu, Equation (21), which according to the analysis that led to Equation (39) is Ntu;opt ¼

pCa 1=2

PrF



s3 12

"

2 CD

1=4

pCa 1=4

s3 PrF

 Be1=12 D~ 1=36 L~ 5=36

#7=9 Be1=12 D~ 1=36 L~ 5=36

ð47Þ

Equation (20) is plotted in Figure 3. The numbers of tubes with downflow and upflow are ndown ¼

zc n L

 zc  nup ¼ 1  n L

ð45Þ

ð48Þ

ð49Þ

1

ð46Þ

Seen from the gas side, the maximized total heat transfer rate depends on two of the remaining free ~ The qmax value dimensions of the assembly, D~ and L. increases as both D~ and L~ decrease: thinner tubes and a shorter gas flow path are better for increasing the global heat transfer rate. Another result of this analysis is that the scale of qmax is kFDTV1/3, which means that qmax increases in proportion with the linear scale of the steam generator, V1/3. The volumetric heat transfer density qmax/V decreases in proportional with V2/3. 340

6. GEOMETRY ON THE STEAM SIDE

ð44Þ

The corresponding number of tubes in this optimal assembly is LW 1 n¼ ¼ 2 S S S~ H~ ffiBe1=3 D~ 5=9 L~ 7=9 H~ 1

In conclusion, if compactness is the goal, smaller steam generators have greater heat transfer density than larger steam generators. More specifically, if one large steam generator has the total heat transfer rate q 5 CqV1/3 ~ and L~ are fixed (cf. Equation (45)), then two when Be, D, steam generators (each of half size V/2) will have the total heat transfer rate 2(q/2) 5 2Cq(V/2)1/3, which exceeds by the factor 22/3 5 1.59, the q value of the single large steam generator. Alternatively, if we fix D and L of the V-size design, and divide V into two V/2 parts in which the tube diameters are D and the length scale of each V/2 part is (1/2)1/3L so that each part has the volume V/2, then the corresponding analysis shows that the combined heat transfer rate of the two parts exceeds by 21%, the original heat transfer rate, q 5 CqV1/3.

zc 0.5 L

0

0

0.5

1

Ntu Figure 3. The longitudinal location of the flow reversal, for which Ntu is given by Equation (47).

Int. J. Energy Res. 2011; 35:336–345 r 2010 John Wiley & Sons, Ltd. DOI: 10.1002/er

Steam generator structure

Y. Kim, S. Lorente and A. Bejan

According to Equations (46) and (47), these num~ L, ~ H, ~ and Be; however, the bers are functions of D, ratio ndown/nup is comparable with 1 because zc/L is order of 0.5 (Figure 3). Note that because of the ~ and L~ in Equation (47), the small exponents of Be, D, expected order of magnitude of Ntu is 1. Furthermore, because the total downward mass flow rate must equal the total upward mass flow rate, the scaling ndownnup means that the mass flow rate in one tube (m_ 1 ) has the same scale in both the parts, downcomer and riser. At this stage, we conclude that the constructal design of the gas side (Section 5) determines the flow reversal position (Figure 3) and several new relationships between the geometric features of the design, namely Equations (41)–(49).

7. TWO-PHASE FLOW MODEL

First, the assumption that all the tube surfaces are isothermal at TB had the effect of decoupling the heat transfer analysis of the gas side from the analysis of the water side. The second assumption was the singlephase modeling of the friction in one vertical tube (Equation (15)), which allowed us to develop analytically the variation of the one-tube flow rate with the longitudinal position z (Section 4). In the following analysis, we relax this second assumption and treat the flow as that of a liquid–vapor mixture [30]. Figure 4 shows the temperature distribution in the hot gas and the main features of water circulation. The ndown tubes represent the downcomer, and the nup tubes represent the riser. The sum of ndown and nup is the number of tubes in the assembly, n (Equation (46)), which is obtained by constructal design of the continuous steam generator for the gas side. The pressure drop in two-phase flow is function of pressure and quality. The pressure difference DP between two plenums is (Figure 4)

The scaling relationships developed so far are based on two assumptions that greatly simplified the analysis.

DP ¼ DPa 1DPf 1DPg

ð50Þ

T TF

q up q down

Tout TB

z zC

Steam Water

Upper plenum

.

m Tout

ΔP

TF Downcomer

Riser

Lower plenum Figure 4. The natural circulation (self-pumping effect) in the continuous steam generator model.

Int. J. Energy Res. 2011; 35:336–345 r 2010 John Wiley & Sons, Ltd. DOI: 10.1002/er

341

Y. Kim, S. Lorente and A. Bejan

Steam generator structure

where DPa and DPf are the acceleration and friction pressure drops, and DPg is the hydrostatic pressure difference for the two-phase flow [30]  2 m_ w r2 ð51Þ DPa ¼ vf A

DPf ¼ 4f

H vf D 2

DPg ¼



m_ w A

2 r3

gH r4 vf

ð52Þ

ð53Þ

The pressure difference along each tube is the same as DP, because the pressure in the plenums is uniform, and each tube is installed between the two plenums. For simplicity, we divide the tubes into two groups, downcomer and riser, in order to find the ratio of numbers of downcomer and riser tubes. Considering first the downward flow in the downcomer,  2 H vf m_ w DPdown ¼ vf r2;1;out  4f1 D 2 A1  2 1 m_ w r3;1;out 1gH r4;1;out ð54Þ  vf A1 where subscript 1 represents the downcomer, A1 ¼ ndown

pD2 4

ð55Þ

The quality values at the inlet and outlet of the riser are greater than zero due to the heat transferred to the downcomer. The two-phase pressure drop is evaluated by subtracting the inlet pressure drop from the outlet pressure drop, which yields the pressure difference equation for the riser,  2 m_ w DPup ¼ vf ðr2;2;out  r2;2;in Þ A2   H vf m_ w 2 4f2 ðr3;2;out  r3;2;in Þ D 2 A2 gH

1 ðr4;2;out  r4;2;in Þ vf

ð56Þ

Here the subscript 2 represents the riser, A2 ¼ nup

pD2 4

ð57Þ

Next, for the closed loop formed by the downcomer and the riser, we write DPdown 1DPup ¼ 0

ð58Þ

Substituting Equations (54)–(57) into Equation (58), and using ndown 5 nnup and r4;1;out ¼ r4;2;in , we obtain the global flow resistance equation for the two-phase 342

flow system: gp2 r2f HD4 r2;1;out 2f1 r3;1;out H R¼ ð2r4;1;out  r4;2;out Þ ¼ 1 16m_ 2w ðn  nup Þ2 ðn  nup Þ2 D 1

ðr2;2;out  r2;2;in Þ 2f2 ðr3;2;out  r3;2;in ÞH 1 n2up n2up D

ð59Þ

If the friction factors (f1, f2) are assumed to be independent of nup, then the global flow resistance can be minimized analytically by solving qR/qnup 5 0, which yields 

n nup



 ¼ 11 opt

1=3 r2;1;out 12f1 r3;1;out H=D ½ðr2;2;out  r2;2;in Þ12f2 ðr3;2;out  r3;2;in ÞH=D

ð60Þ Consider two limiting conditions: r2442fr3H/D and r2oo2fr3H/D, which correspond to HD and H44D, respectively:    1=3 n 1 ¼ 11 ð61Þ nup opt;HD r2;2;out =r2;1;out  1 

n nup



 ¼ 11 opt;H44D

1=3 1 r3;2;out =r3;1;out  1

ð62Þ

In Equations (61) and (62), we used (f1/f2)1/31, r2;2;in ¼ r2;1;out , and r3;2;in ¼ r3;1;out . Equations (61) and (62) show that in taller continuous steam generators, the number of riser tubes is determined by the friction effect, and in shorter continuous generators, the number of riser tubes is determined by the acceleration effect. Furthermore, to determine the ratio of riser tubes, one of Equations (61) and (62) can be used because the acceleration factor r2 and frictional factor r3 for two-phase water show similar trends (they are both proportion to the quality x [30]), and the result is  1=3   n 1 ¼ 11 ð63Þ nup opt r3;2;out =r3;1;out  1 Equation (63) indicates the location of flow reversal in tubes. For example, if (n/nup)opt 5 2, then ndown 5 nup, and this means that the flow reversal occurs in the mid plane of continuous steam generator. This corresponds to r3,2,out/r3,1,out 5 2. If the number of downcomer tubes is greater than the number of riser tubes, then (n/nup)opt42 and r3,2,out/r3,1,outo2. The number of riser tubes is greater than the number of downcomer tubes when (n/nup)opto2 and r3,2,out/ r3,1,out42. Although the friction factor r3 for twophase water is a function of pressure and quality, the effect of pressure is negligible because the pressure difference in the continuous steam generator is not significant; hence, r3 is essentially a function of quality. Therefore, r3 can be evaluated from the temperature drops of hot gas though downcomer and riser, because the quality depends on the heat absorbed by the tubes. Int. J. Energy Res. 2011; 35:336–345 r 2010 John Wiley & Sons, Ltd. DOI: 10.1002/er

Steam generator structure

8. CONCLUSIONS In this study, we developed analytically the constructal design of steam generators with large numbers of tubes so many that the temperature distribution in the horizontal direction may be modeled as continuous. This feature of the model and the approximation that the tubes are essentially isothermal at the boiling temperature, led to the analytical derivation of the location of flow reversal, i.e. the intermediate location that separates the down flowing steam tubes (the downcomer) from the upflowing tubes (the riser). The flow reversal location depends only on the number of heat transfer units. On the gas side of the crossflow heat exchanger, we used the method of intersecting the asymptotes and found that the spacing between tubes can be selected such that the global heat transfer density is maximal. The optimal spacing depends on three dimensionless groups accounting for the imposed pressure difference, the tube diameter, and the horizontal flow length, Equation (41). Corresponding scaling relations were determined for the total number of steam tubes and the maximal heat transfer density of the heat exchanger. This analysis and its results are useful with regard to scaling up and scaling down the designs of steam generators. For example, the analysis showed that the total heat transfer rate increases in proportion with the length scale of the entire heat exchanger, Equation (45). A consequence of this is the fact that the heat-transfer density decreases as the length scale increases.

NOMENCLATURE A B Be cP C Ca CD Cq D f F F1 g h h hf hfg H k

5 area (m2) 5 variable, Equation (18) 5 dimensionless pressure difference number, Equation (42) 5 specific heat at constant pressure of hot gas (J kg1 K1) 5 constant, Equation (15) 5 dimensionless factor, Equation (32) 5 drag coefficient 5 variable 5 column diameter (m), Figure 1 5 friction coefficient 5 total drag force (N) 5 drag force on one tube (N) 5 gravitational acceleration (m s2) 5 heat transfer coefficient (Wm2 K1) 5 water specific enthalpy (J kg1) 5 specific enthalpy of saturated liquid (J kg1) 5 change of specific enthalpy between phases (J kg1) 5 height (m), Figure 1 5 thermal conductivity (W m1K1)

Int. J. Energy Res. 2011; 35:336–345 r 2010 John Wiley & Sons, Ltd. DOI: 10.1002/er

Y. Kim, S. Lorente and A. Bejan

L m_ m_ 1 m_ 1;F m_ scale m_ w n ndown nup N Ntu Pr q r2 r3 r4 R Re S Savg T TB TF Tout U U1 v vf vfg V W x z zc

5 length (m), Figure 1 5 hot gas mass flow rate (kg s1) 5 water flow rate scale in one tube (kg s1) 5 hot gas flow rate in one channel (kg s1) 5 mass flow rate scale (kg s1) 5 water mass flow rate (kg s1) 5 optimal number of tubes in the assembly, Equation (46) 5 number of downcomer tubes 5 number of riser tubes 5 number of tubes in the assembly 5 number of heat transfer units, Equation (21) 5 Prandtl number 5 heat transfer rate (W) 5 acceleration factor, Equation (51) 5 friction factor, Equation (52) 5 gravitational factor, Equation (53) 5 global water flow resistance 5 Reynolds number 5 spacing between adjacent tubes (m), Figure 2 5 average spacing between adjacent tubes (m), Figure 2 5 temperature (K) 5 water boiling temperature (K) 5 inlet hot gas temperature (K) 5 outlet hot gas temperature (K) 5 free stream velocity (m s1) 5 free stream velocity in one channel (m s1) 5 water specific volume (m3 kg1) 5 specific volume of saturated liquid (m3 kg1) 5 change of specific volume between phases (m3 kg1) 5 total volume (m3) 5 width (m), Figure 1 5 quality of liquid–vapor mixture 5 coordinate along length 5 location of flow reserval (m)

Greek symbols a DP DP1 DPa DPf DPg DT y m n r r1 rf s

5 thermal diffusivity (m2 s1) 5 pressure drop (Pa) 5 pressure drop in one tube (Pa) 5 acceleration pressure drop (Pa) 5 friction pressure drop (Pa) 5 hydrostatic pressure difference (Pa) 5 temperature difference (K) 5 variable, Equation (17) 5 dynamic viscosity (N s m2) 5 kinematic viscosity (m2 s1) 5 density (kg m3) 5 water density in the tube (kg m3) 5 density of saturated liquid (kg m3) 5 ratio between spacing and average spacing 343

Y. Kim, S. Lorente and A. Bejan

f

5 volume fraction occupied by water, Equation (2)

Subscripts 1 2 a b down F in out ref opt up

5 downcomer 5 riser 5 the large S limit 5 the small S limit 5 downcomer 5 flue gas 5 inlet 5 outlet 5 reference 5 optimum 5 riser

Superscripts 

5 dimensionless

ACKNOWLEDGEMENTS This research was supported by Doosan Heavy Industries & Construction Co., Ltd., Changwon, South Korea.

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