# modelling of design of solar steam generating collector fields

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Nov 10, 2013
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Modelling and Design of Direct

Solar Steam Generating Collector

Fields

M. Eck

W.-D. Steinmann

German Aerospace Center (DLR),

Institute of Technical Thermodynamics,

Pfaffenwaldring 38-40, 70569

Stuttgart, Germany

The direct steam generation (DSG) is an attractive option regarding the economic improvement of parabolic trough technology for solar thermal electricity generation in the ¨

multi megawatt range. According to Price, H., Lupfert, E., Kearney, D., Zarza, E., Cohen, G., Gee, R. Mahoney, R., 2002, ‘‘Advances in Parabolic Trough Solar Power Technology,’’ J. Sol. Energy Eng., 124 and Zarza, E., 2002, DISS Phase II-Final Project Report, EU Project No. JOR3-CT 980277 a 10% reduction of the LEC is expected compared to conventional SEGS like parabolic trough power plants. The European DISS project has proven the feasibility of the DSG process under real solar conditions at pressures up to 100 bar and temperatures up to 400°C in more than 4000 operation hours (Eck, M., ¨

Zarza, E., Eickhoff, M., Rheinlander, J., Valenzuela, L., 2003, ‘‘Applied Research Concerning the Direct Steam Generation in Parabolic Troughs,’’ Solar Energy 74, pp. 341– 351). In a next step the detailed engineering for a precommercial DSG solar thermal power plant will be performed. This detailed engineering of the collector ﬁeld requires the consideration of the occurring thermohydraulic phenomena and their inﬂuence on the stability of the absorber tubes. ͓DOI: 10.1115/1.1849225͔

Introduction

The modelling of the DSG process in parabolic troughs is of

special interest for the detailed design of the collector ﬁeld. To identify critical process conditions that may cause e.g. an overheating of the absorber tubes a tool has been developed that considers all thermohydraulic aspects relevant for the design of a collector ﬁeld such as the ﬂow pattern in the evaporation section, the pressure loss and the heat transfer.

A design tool has been developed at DLR calculating all relevant process parameters including pressure drop, temperature ﬁeld and stress in the absorber tubes. The models implemented in this design tool have been validated in detail at the DISS test facility under real DSG conditions for pressures between 30 and 100 bar and inner diameters between 50 and 85 mm. The models have been implemented into a MATLAB® program to allow for a ﬁrst quick determination of critical process conditions. Once critical process conditions have been identiﬁed the FEM package ANSYS® is used for a detailed investigation. This article summarizes the models used and shows the design procedure for a DSG collector ﬁeld. The design program has proven to be a reliable tool for the detailed design of DSG collector ﬁelds.

heat transfer coefﬁcient the knowledge of the ﬂow pattern is essential for the calculation of the temperature distribution in the absorber tubes.

In practice various ﬂow pattern maps are used for the determination of the present ﬂow pattern. A frequently used ﬂow pattern map is that of Taitel and Dukler ͓1͔. This ﬂow pattern map is valid for horizontal and slightly inclined tubes. It has been developed for a two-phase ﬂow without heat supply. To consider the effect of heating the models of Taitel and Dukler have been extended by some empirical equations ͓2͔. At DLR a different approach was chosen. During the experiments at the PRODISS and DISS test

facility mainly wavy and annular ﬂow occur ͑see Fig. 1͒. An empirical correlation has been developed that predicts the transition from wavy to annular ﬂow. This model has been validated in a wide range of process parameters including different tube diameters ͓3͔. The mass ﬂux density where the transition occurs can be calculated according to Eq. ͑1͒.

ͩ

˙

m g,t ϭ ͑ 46.6ϩ0.595pϩ0.0119p 2 ͒ 1ϩ1.3

˙

q

56

ͪ

(1)

Determination of the Boundary Conditions

A proper design of the collector ﬁeld has to guarantee that the stress and the temperature in the absorber tubes do not exceed the limits of the absorber material used. To determine the stress and the temperature ﬁeld the most important boundary conditions such as heat transfer coefﬁcients, phase distribution, operation pressure, ﬂux distribution and ﬂux density have to be known. In the subsequent sections the models for the determination of these boundary conditions will be presented.

Flow Pattern. In the evaporator section of the collector loop a two-phase ﬂow occurs. Depending on the process parameters different ﬂow pattern and thus different phase distribution will occur. Since the present phase distribution determines the present Contributed by the Solar Energy Division and presented at the ISEC2004 Portland, Oregon, July 11–14, 2004 of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the ASME Solar Division; April 27, 2004; ﬁnal revision August 10, 2004. Associate Editor: J. Davidson.

Journal of Solar Energy Engineering

Equation ͑1͒ is an equation between quantities where the pressure p is given in bar, the heat ﬂux density q is given in kW/m2. The mass ﬂux density m t is calculated in kg/m2 s.

Wetting Angle. In case of annular ﬂow the complete inner

circumference of the absorber tube is wetted by saturated water whereas at wavy ﬂow the inner circumference is only partly wetted. In this case the boundary between the wetted and the unwetted region is of special interested. This boundary is expressed by the wetting angle wet ͑see Fig. 2͒. To determine the wetting angle of a wavy ﬂow the liquid level for a ﬁctitious stratiﬁed ﬂow is calculated ﬁrst. Afterwards this liquid level is multiplied by a so called wave factor to get the wetting level of a wavy ﬂow. At stratiﬁed ﬂow the liquid level depends on the void fraction ͑the ratio of the cross section occupied by steam to the total inner cross section of the tube͒ that is determined by the equation of Rouhani ͓4͔.

Copyright © 2005 by ASME

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Fig. 1 Main ﬂow patterns for direct steam generation

ϭ

ͫ

ͩ

˙

˙

˙

x

x

1Ϫx

˙

ϩ

͑ 1ϩ0.12͑ 1Ϫx ͒͒

g

g

l

ϩ

˙

1.18͑ 1Ϫx ͒͑ g ͑ l Ϫ g ͒͒ 0.25

˙

m l0.5

ͬ

ͪ

Ϫ1

(2)

Fig. 3 March of the speciﬁc pressure loss along the collector loop „d i Ä50 mm, MÄ1 kgÕs…

The normalized liquid level h l /d i is determined by the implicit equation ͑3͒.

ͫ

ͩ

ͩ

hl

sin 2 arccos 2 Ϫ1

di

1

hl

ϭ

arccos 2 Ϫ1 Ϫ

di

2

ͩ

ͪ

ͪͪ ͬ

(3)

The wetting angle ﬁnally results from Eq. ͑4͒.

ͩ

hl

Ϫ1

wetϭarccos

d i /2

ͪ

ͩ ͪ ͩ ͪ

dp

dl

h wϭ f h l

˙

˙

RϭAϩ3.43x 0.685͑ 1Ϫx ͒ 0.24

ͩ

ϫ 1Ϫ

ͩ ͪ

1 2

w ,

ϭ

d 2

1ph

g

l

ͪ

0.89

Ϫ0.25

ϭ0.316 Re

with

˙

˙

Aϭ ͑ 1Ϫx ͒ 2 ϩx 2

Frl ϭ

(6)

According to the investigations during the European DISS

project ͓5͔ the pressure loss of the occurring two phase ﬂow is determined best with the correlations of Friedel. For a two-phase ﬂow the pressure drop is deﬁned as the product of the singlephase water ﬂow and the two-phase ﬂow multiplier R.

(7)

1ph,l

ͩ ͪ ͩ ͪ

g

l

0.8

•

g

l

0.22

FrlϪ0.47 Wel0.0334

(5)

Pressure Drop. In the turbulent single phase water or steam

ﬂow the pressure drop is determined by the Blasius equation ͑6͒. dp

dl

2ph

dp

dl

The single phase water ﬂow is calculated according to Eq. ͑6͒ assuming that the complete mass ﬂux is liquid. The two-phase ﬂow multiplier is calculated with Eq. ͑8͒:

(4)

At a wavy ﬂow the wetting angle depends on the wave height h w . The ratio of wave height to the liquid level h l is expressed by the wave factor f which is in the range of 1.7 to 2.

ϭR

ͩ ͪ

g g

l l

(8)

0.8

(9)

˙

͑ ml͒2

gd i

(10)

˙

m 2d i l

(11)

Wel ϭ

The parameters g and l are calculated for turbulent steam ﬂow (RegϾ1055) as follows:

ͫ

g ϭ 0.86859 log

ͩ

Reg

1.964 log͑ Reg ͒ Ϫ3.8215

ͪͬ

Ϫ2

(12)

For laminar steam ﬂow (RegϽ1055)

gϭ

64

Reg

For turbulent water ﬂow (RelϾ1055)

ͫ

l ϭ 0.86859 log

ͩ

Rel

1.964 log͑ Rel ͒ Ϫ3.8215

(13)

ͪͬ

Ϫ2

(14)

For laminar water ﬂow (RelϽ1055)

lϭ

Fig. 2 Schematic cross section of an evaporation tube

372 Õ Vol. 127, AUGUST 2005

64

Rel

(15)

Figure 3 displays the local speciﬁc pressure loss at different axial positions of a collector loop for different operation pressures.

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Fig. 4 Heat transfer coefﬁcient as a function of the spec. enthalpy for different heat ﬂux densities. In two-phase region only the heat transfer coefﬁcient in the wetted and heated region is ˙

displayed. „pÄ100 bar, M Ä1 kgÕs, d i Ä50 mm…

Fig. 5 Heat transfer coefﬁcient as a function of the spec. Enthalpy for different pressures. In the two-phase region only the heat transfer coefﬁcient in the wetted and heated region is dis˙ ˙

played. „q Ä40 kWÕm2 , M Ä1 kgÕs, dÄ50 mm…

The two-phase ﬂow pressure loss in the bends is determined by the equation of Chisholm ͓6͔ calculating the friction factors according to ͓7͔.

cient as a function of the spec. Enthalpy along the collector loop for different boundary conditions.

At this point all boundary conditions at the inner circumference are deﬁned as function of the known process parameters, the operation pressure, and the ﬂuid mass ﬂux. In the following sections the boundary condition at the outer circumference has to be deﬁned.

Heat Transfer Coefﬁcient. The heat transfer coefﬁcient for the single-phase water or steam ﬂow is calculated according to the equation of Dittus–Boelter ͓8͔.

␣ 1phϭ0.0235 Re0.8 Pr0.48

d

(16)

In case of the two-phase water-steam ﬂow three different heat transfer phenomena with different heat transfer coefﬁcients can be identiﬁed.

1. Heat transfer to the steam in the unwetted region;

2. Heat transfer to the water in the wetted and heated region; 3. Heat transfer to the water in the wetted and unheated region.

Collector Efﬁciency. Parabolic trough collectors only use the direct normal irradiation ͑DNI͒. The DNI is reﬂected on the absorber tube. Due to optical losses at the reﬂector, the glass envelope and the absorber surface only a certain fraction of the DNI is absorbed by the absorber tube. In addition the absorber tube has thermal losses due to convection and radiation. All these losses are determined by the collector efﬁciency. Once the collector efﬁciency is known the heat ﬂux on the outer surface of the absorber tube is known.

The heat transfer coefﬁcient for the steam phase and the water phase in the unheated region can be determined with Eq. ͑16͒. In the wetted and heated region nucleate boiling occurs. According to Gungor and Winterton ͓9͔ the heat transfer coefﬁcient ␣ 2ph,l is determined by a ﬁrst term taking into account the convective heat transfer ␣ conv and a second term for the heat transfer by the nucleate boiling ␣ NB .

␣ 2ph,l ϭE ␣ convϩS ␣ NB

(17)

The supression factor S is determined as follows:

Sϭ ͓ 1ϩ1.15•10Ϫ6 E 2 Ret1.17͔ Ϫ1

(18)

The enhancement factor E is calculated according to Eq. ͑19͒. Eϭ1ϩ24000Bo 1.16ϩ1.37X Ϫ0.86

a

(19)

With the boiling number Bo.

Boϭ

˙

q

˙

m tot͑ h Љ Ϫh Ј ͒

(20)

˙

where m tot is the total mass ﬂux density. The Martinelli parameter ˙

X a is a function of the ﬂuid properties and the steam quality x . The convective heat transfer coefﬁcient again is calculated according to Eq. ͑16͒. The heat transfer coefﬁcient due to nucleate boiling is determined according to Cooper ͓10͔.

Figures 4 and 5 display the course of the heat transfer coefﬁJournal of Solar Energy Engineering

Fig. 6 Collector efﬁciency as a function of the difference between operation and ambient temperature „TÀT a … for different values of the DNI „Ä0 deg, Cermet with Vacuum, LS-2 collector…

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not be higher than 50 K above the ﬂuid temperature ͓16͔. To allow for a very fast and reliable determination of the highest absorber temperature a simpliﬁed analytical solution for the calculation of the circumferential temperature distribution on the medium center line has been developed and implemented into the design tool. Based on this result the maximum and minimum temperature at

the outer surface will be calculated.

Temperature on the Medium Center Line. The analytical

calculation of the temperature is based on a division of the cross section in four segments ͑see Fig. 8͒.

1.

2.

3.

4.

Fig. 7 Typical heat ﬂux distribution along the outer surface of an absorber tube and its approximation by a gaussian and a

rectangular distribution with Ä60 deg

Empirical correlations describing the collector efﬁciency for several collector models exist. For LS-2 collectors the efﬁciency has been investigated in ͓11͔ where different equations have been derived for different absorber conﬁgurations. Formulations for the EuroTrough collector are presented in ͓12͔.

The typical course of the efﬁciency of a parabolic trough collector is displayed in Fig. 6 for an LS-2 collector. The efﬁciency of the collector decreases with an increasing operation temperature and a decreasing DNI. The maximum efﬁciency at TϪT a ϭ0, where no thermal losses occur, is called optical efﬁciency. These functions are valid for an incidence angle of 0 deg ͑the sun is perpendicular to the aperture area of the collector͒. The effect of different incidence angles is considered by the incidence angle modiﬁer ͑IAM͒. Again empirical correlations have been developed for different collectors. The IAM for an LS-2 collector is given in ͓11͔. Functions for an LS-3 collector are given in ͓13͔ and ͓14͔.

Every parabolic trough collector mentioned above can be used for direct steam generation. For the determination of the heat input into the absorber tube the designer of the collector ﬁeld has to choose the efﬁciency and IAM curve for the collector used.

Wetted and heated;

Unwetted and heated;

Unwetted and unheated;

Wetted and unheated.

In case of the preheating and the superheating section with the according single-phase ﬂow only two different sections can be identiﬁed, which are particular cases of the common case.

It is assumed that the boundary conditions in the different sections are known and constant. For the exact segmentation into the four sections the wetting angle and the position of the rectangular heat ﬂux distribution must be known. The position of the heat ﬂux distribution depends on the position of the sun that can be determined according to e.g. ͓17͔. Figure 9 is used for the derivation of the accounting equations. At the outer surface ⌬Q sol represents the heat input by the solar energy, ⌬Q conv represents the heat transfer at the inner surface and Q resp. ⌬Q the conductive heat transfer to adjacent segments. It can be written that

˙

˙

˙

dQ ϩdQ convϭdQ sol

(21)

with

2

d T

˙

dQ ϭϪs⌬z 2 dy

dy

(22)

˙

dQ convϭa⌬z ͑ TϪT f ͒ dy

(23)

˙

˙

dQ solϭ⌬zq soldy

(24)

In the heated region the heat ﬂux density is:

˙

q sol,h ϭ col DNI d ap

360

d m͑ eϪ b ͒

(25)

Flux Distribution on the Outer Surface. Once the heat ﬂux

on the outer surface is known its distribution has to be determined. A typical ﬂux distribution at the outer circumference of an absorber tube for parabolic trough collectors is displayed in Fig. 7 ͑taken from ͓15͔͒. This distribution shows a local minimum at an angle of 180 deg caused by the shadow of the absorber tube on the mirror facets. This distribution is similar to a Gaussian distribution with a standard deviation of 60 deg as displayed in Fig. 7. Additionally the approximation by a rectangular distribution with its edges at 100 deg and 260 deg is presented. In all cases the integral of the curve is the same.

FEM analyses of the resulting temperature distribution in the absorber cross section have shown that there is no signiﬁcant difference between the temperature distribution calculated with a ﬂux distribution according to the Gaussian distribution and the one calculated with the typical parabolic trough distribution. Accordingly the Gaussian distribution is chosen for the accurate FEM analysis whereas the rectangular distribution is used for a simpliﬁed analytical solution presented in the next section.

Temperature Distribution

The maximum temperature of the absorber tube is the most

critical parameter of a DSG collector loop. This temperature may 374 Õ Vol. 127, AUGUST 2005

Fig. 8 Schematic illustration of an absorber cross section with the four different sections

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Fig. 9 Developed view of an absorber tube segment for the

derivation of the analytical solution. The depth of the Segment is ⌬z.

It is assumed that the thermal radiation at the outer surface is constant and equivalent to:

˙

q sol,unhϭϪ

͑ optϪ ͒ DNId ap

d m

(26)

The differential equation for the determination of the temperature distribution is given by: d 2T

dy

2

ϪK 2 TϭϪ

1

˙

q ϪK 2 T f

s sol

(27)

with

K 2ϭ

␣

s

(28)

And the solution:

T ͑ y ͒ ϭC 1 cosh͑ Ky ͒ ϩC 2 sinh͑ Ky ͒ ϩ

˙

q sol

ϩT f

␣

(29)

To determine the two coefﬁcients C 1 and C 2 two boundary

conditions per segment have to be known. Since the four sections are connected to each other, the boundary conditions at the end of one section is the same as at the beginning of the next section. At this location the according temperatures as well as their gradient have to be the same. For example:

T 1 ͑ yϭl 1 ͒ ϭT 2 ͑ yϭ0 ͒

(30)

dT 1 ͑ yϭl 1 ͒ T 2 ͑ yϭ0 ͒

ϭ

dy

dy

(31)

and

a temperature of 310°C whereas the steam is slightly superheated to a temperature of 320°C. The circumferential angle is counted clockwise from the crest.

The marches of the temperature are very similar in Fig. 10. The maximum deviation between the two different calculation procedures is approx. 5 K. For a ﬁrst rough estimation of the temperature ﬁeld this deviation is acceptable. Maximum Temperature. To assess the thermal load of the

absorber tube not the temperature distribution on the medium center line is of importance but the maximum temperature at the outer surface of the absorber tube, which is the hottest point, that can be determined from the known temperature on the medium center

line. Therefore it is assumed that at the location with the maximum temperature the temperature gradient in circumferential direction is zero. Thus the heat ﬂow in circumferential direction is zero. Accordingly only a heat ﬂux in radial direction occurs. The maximum temperature is determined with the equation for the

steady-state heat conduction in radial direction. Accordingly the maximum temperature at the outer surface is calculated by:

ro

rm

T max,o ϭT max,m ϩ

͑ T max,m ϪT max,i ͒

rm

ln

ri

ln

The gradient of the temperature is deﬁned by:

dT 1 ͑ y ͒

ϭC 1 K sinh͑ Ky ͒ ϩC 2 K cosh͑ Ky ͒

dy

Fig. 10 Comparison of the temperature along the medium center line of the absorber cross section calculated with the FEM package ANSYS® and the analytical solution. „two-phase-ﬂow, heated from the side…

(32)

Accordingly there are eight equations for the determination of the eight coefﬁcients C i . It is possible to simplify the equations by considering the special characteristics of the hyperbola

function:

sinh͑ 0 ͒ ϭ0

(34)

With the help of this simpliﬁed procedure a ﬁrst rough assessment of a DSG collector loop is possible and critical process situations can be identiﬁed. In a subsequent step critical situations will be investigated in more detail using the FEM package

ANSYS® .

(33)

cosh͑ 0 ͒ ϭ1

(35)

The values for the eight coefﬁcients C i are found by solving the set of eight linear equations. Knowing the coefﬁcients C i for the different segments, it is possible to calculate the temperature on the medium center line. By connecting the four segments the

march of the temperature along the medium center line of the investigated cross section is determined.

Figure 10 displays the march of the absorber temperature calculated with the FEM package ANSYS® and the analytical solution. For the calculation it was assumed that the liquid phase has Journal of Solar Energy Engineering

Design Tool

The models presented have been implemented in a simulation

tool using the programming environment MATLAB® allowing for a fast investigation of different collector loop conﬁgurations. Before starting the calculation run the boundary conditions such as pressure and temperature at the loop outlet, the direct normal irradiation, the position of the sun and the geometry of the collector loop have to be deﬁned. In case of the recirculation mode the recirculation rate and the use of an injection cooler in the superheating section has to be deﬁned too. The once-through mode AUGUST 2005, Vol. 127 Õ 375

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Table 1

Parameters of the ET-II collector

Parabola Width ͓m͔

Overall Length of a single Collector ͓m͔

Outer Diameter ͓mm͔

Inner Diameter ͓mm͔

Length of connecting pipe between two

adjacent collectors ͓m͔

Number of 90° elbows in U-bends ͓°C͔

Peak Optical Efﬁciency ͓%͔

Roughness Factor of the Absorber Pipes ͓m͔

5,76

98,5

70

55

4

8

76,5

4EϪ05

Table 2 Main Parameters of the Solar Field

Length of Evaporator ͓m͔

Length of Superheater ͓m͔

Outlet Pressure ͓bar͔

Outlet Temperature ͓°C͔

Electric Power ͓MW͔

Number of parallel rows

Fig. 11 Pressure along the collector loop for the different operation modes „pÄ100 bar, T out Ä400°C, d i Ä50 mm, DNIÄ800 WÕm2…

requires the deﬁnition of the inlet temperature and the injection mode the deﬁnition of the number and distribution of injection cooler along the collector loop.

For the calculation the collector loop is divided into a number of segments with a step size of usually 5 m. Starting from the end of the collector loop, the collector efﬁciency, the speciﬁc enthalpy, the pressure loss and the ﬂuid temperature for all investigated collector positions are calculated. Based on the calculated ﬂuid properties and ﬂow conditions the distribution of the heat transfer coefﬁcient and the temperature of the absorber tube along the collector loop is determined. In addition it is possible to display the evaporation path in the Taitel–Dukler diagram.

As an example Fig. 11 displays the pressure and Fig. 12 the

maximum temperature difference along the collector loop for the three different operation modes. According to Fig. 11 the total pressure drop is less than 2 bar for a length of the collector loop of 500 m and an outlet pressure of 100 bar.

As shown in Fig. 12 in all cases the highest temperatures do not occur in the evaporation but in the superheating section. This is

800

200

70

410

5

7

caused by the occurrence of an annular ﬂow in the evaporation section with the according good cooling of the absorber wall. Lay-Out of a DSG Collector Loop. In the previous sections

the fundamentals of DLR’s design tool are presented. This tool is used during the design phase of DSG collector loops. ͑Since parabolic trough collectors operated with a heat transfer ﬂuid such as oil are similar to the preheating section of a DSG collector loop, DLR’s design tool is also applicable to oil collector loops.͒ The detailed design of collector ﬁelds also requires the investigation of the absorber tube and of the interconnecting piping and the necessary ﬁttings in more detail. For the detailed investigation of the absorber tube the FEM package ANSYS® and for the piping and ﬁttings and the overall collector ﬁeld performance the heat balance calculation program IPSEpro® is used. The use of these three programs for the investigation of DSG collector ﬁelds is presented in the following section.

The investigated collector ﬁeld is operated in recirculation mode as proposed in ͓18͔. To guarantee a sufﬁcient cooling of the superheating collectors a length of 200 m is chosen fo

Solar Steam Generating Collector

Fields

M. Eck

W.-D. Steinmann

German Aerospace Center (DLR),

Institute of Technical Thermodynamics,

Pfaffenwaldring 38-40, 70569

Stuttgart, Germany

The direct steam generation (DSG) is an attractive option regarding the economic improvement of parabolic trough technology for solar thermal electricity generation in the ¨

multi megawatt range. According to Price, H., Lupfert, E., Kearney, D., Zarza, E., Cohen, G., Gee, R. Mahoney, R., 2002, ‘‘Advances in Parabolic Trough Solar Power Technology,’’ J. Sol. Energy Eng., 124 and Zarza, E., 2002, DISS Phase II-Final Project Report, EU Project No. JOR3-CT 980277 a 10% reduction of the LEC is expected compared to conventional SEGS like parabolic trough power plants. The European DISS project has proven the feasibility of the DSG process under real solar conditions at pressures up to 100 bar and temperatures up to 400°C in more than 4000 operation hours (Eck, M., ¨

Zarza, E., Eickhoff, M., Rheinlander, J., Valenzuela, L., 2003, ‘‘Applied Research Concerning the Direct Steam Generation in Parabolic Troughs,’’ Solar Energy 74, pp. 341– 351). In a next step the detailed engineering for a precommercial DSG solar thermal power plant will be performed. This detailed engineering of the collector ﬁeld requires the consideration of the occurring thermohydraulic phenomena and their inﬂuence on the stability of the absorber tubes. ͓DOI: 10.1115/1.1849225͔

Introduction

The modelling of the DSG process in parabolic troughs is of

special interest for the detailed design of the collector ﬁeld. To identify critical process conditions that may cause e.g. an overheating of the absorber tubes a tool has been developed that considers all thermohydraulic aspects relevant for the design of a collector ﬁeld such as the ﬂow pattern in the evaporation section, the pressure loss and the heat transfer.

A design tool has been developed at DLR calculating all relevant process parameters including pressure drop, temperature ﬁeld and stress in the absorber tubes. The models implemented in this design tool have been validated in detail at the DISS test facility under real DSG conditions for pressures between 30 and 100 bar and inner diameters between 50 and 85 mm. The models have been implemented into a MATLAB® program to allow for a ﬁrst quick determination of critical process conditions. Once critical process conditions have been identiﬁed the FEM package ANSYS® is used for a detailed investigation. This article summarizes the models used and shows the design procedure for a DSG collector ﬁeld. The design program has proven to be a reliable tool for the detailed design of DSG collector ﬁelds.

heat transfer coefﬁcient the knowledge of the ﬂow pattern is essential for the calculation of the temperature distribution in the absorber tubes.

In practice various ﬂow pattern maps are used for the determination of the present ﬂow pattern. A frequently used ﬂow pattern map is that of Taitel and Dukler ͓1͔. This ﬂow pattern map is valid for horizontal and slightly inclined tubes. It has been developed for a two-phase ﬂow without heat supply. To consider the effect of heating the models of Taitel and Dukler have been extended by some empirical equations ͓2͔. At DLR a different approach was chosen. During the experiments at the PRODISS and DISS test

facility mainly wavy and annular ﬂow occur ͑see Fig. 1͒. An empirical correlation has been developed that predicts the transition from wavy to annular ﬂow. This model has been validated in a wide range of process parameters including different tube diameters ͓3͔. The mass ﬂux density where the transition occurs can be calculated according to Eq. ͑1͒.

ͩ

˙

m g,t ϭ ͑ 46.6ϩ0.595pϩ0.0119p 2 ͒ 1ϩ1.3

˙

q

56

ͪ

(1)

Determination of the Boundary Conditions

A proper design of the collector ﬁeld has to guarantee that the stress and the temperature in the absorber tubes do not exceed the limits of the absorber material used. To determine the stress and the temperature ﬁeld the most important boundary conditions such as heat transfer coefﬁcients, phase distribution, operation pressure, ﬂux distribution and ﬂux density have to be known. In the subsequent sections the models for the determination of these boundary conditions will be presented.

Flow Pattern. In the evaporator section of the collector loop a two-phase ﬂow occurs. Depending on the process parameters different ﬂow pattern and thus different phase distribution will occur. Since the present phase distribution determines the present Contributed by the Solar Energy Division and presented at the ISEC2004 Portland, Oregon, July 11–14, 2004 of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the ASME Solar Division; April 27, 2004; ﬁnal revision August 10, 2004. Associate Editor: J. Davidson.

Journal of Solar Energy Engineering

Equation ͑1͒ is an equation between quantities where the pressure p is given in bar, the heat ﬂux density q is given in kW/m2. The mass ﬂux density m t is calculated in kg/m2 s.

Wetting Angle. In case of annular ﬂow the complete inner

circumference of the absorber tube is wetted by saturated water whereas at wavy ﬂow the inner circumference is only partly wetted. In this case the boundary between the wetted and the unwetted region is of special interested. This boundary is expressed by the wetting angle wet ͑see Fig. 2͒. To determine the wetting angle of a wavy ﬂow the liquid level for a ﬁctitious stratiﬁed ﬂow is calculated ﬁrst. Afterwards this liquid level is multiplied by a so called wave factor to get the wetting level of a wavy ﬂow. At stratiﬁed ﬂow the liquid level depends on the void fraction ͑the ratio of the cross section occupied by steam to the total inner cross section of the tube͒ that is determined by the equation of Rouhani ͓4͔.

Copyright © 2005 by ASME

AUGUST 2005, Vol. 127 Õ 371

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Fig. 1 Main ﬂow patterns for direct steam generation

ϭ

ͫ

ͩ

˙

˙

˙

x

x

1Ϫx

˙

ϩ

͑ 1ϩ0.12͑ 1Ϫx ͒͒

g

g

l

ϩ

˙

1.18͑ 1Ϫx ͒͑ g ͑ l Ϫ g ͒͒ 0.25

˙

m l0.5

ͬ

ͪ

Ϫ1

(2)

Fig. 3 March of the speciﬁc pressure loss along the collector loop „d i Ä50 mm, MÄ1 kgÕs…

The normalized liquid level h l /d i is determined by the implicit equation ͑3͒.

ͫ

ͩ

ͩ

hl

sin 2 arccos 2 Ϫ1

di

1

hl

ϭ

arccos 2 Ϫ1 Ϫ

di

2

ͩ

ͪ

ͪͪ ͬ

(3)

The wetting angle ﬁnally results from Eq. ͑4͒.

ͩ

hl

Ϫ1

wetϭarccos

d i /2

ͪ

ͩ ͪ ͩ ͪ

dp

dl

h wϭ f h l

˙

˙

RϭAϩ3.43x 0.685͑ 1Ϫx ͒ 0.24

ͩ

ϫ 1Ϫ

ͩ ͪ

1 2

w ,

ϭ

d 2

1ph

g

l

ͪ

0.89

Ϫ0.25

ϭ0.316 Re

with

˙

˙

Aϭ ͑ 1Ϫx ͒ 2 ϩx 2

Frl ϭ

(6)

According to the investigations during the European DISS

project ͓5͔ the pressure loss of the occurring two phase ﬂow is determined best with the correlations of Friedel. For a two-phase ﬂow the pressure drop is deﬁned as the product of the singlephase water ﬂow and the two-phase ﬂow multiplier R.

(7)

1ph,l

ͩ ͪ ͩ ͪ

g

l

0.8

•

g

l

0.22

FrlϪ0.47 Wel0.0334

(5)

Pressure Drop. In the turbulent single phase water or steam

ﬂow the pressure drop is determined by the Blasius equation ͑6͒. dp

dl

2ph

dp

dl

The single phase water ﬂow is calculated according to Eq. ͑6͒ assuming that the complete mass ﬂux is liquid. The two-phase ﬂow multiplier is calculated with Eq. ͑8͒:

(4)

At a wavy ﬂow the wetting angle depends on the wave height h w . The ratio of wave height to the liquid level h l is expressed by the wave factor f which is in the range of 1.7 to 2.

ϭR

ͩ ͪ

g g

l l

(8)

0.8

(9)

˙

͑ ml͒2

gd i

(10)

˙

m 2d i l

(11)

Wel ϭ

The parameters g and l are calculated for turbulent steam ﬂow (RegϾ1055) as follows:

ͫ

g ϭ 0.86859 log

ͩ

Reg

1.964 log͑ Reg ͒ Ϫ3.8215

ͪͬ

Ϫ2

(12)

For laminar steam ﬂow (RegϽ1055)

gϭ

64

Reg

For turbulent water ﬂow (RelϾ1055)

ͫ

l ϭ 0.86859 log

ͩ

Rel

1.964 log͑ Rel ͒ Ϫ3.8215

(13)

ͪͬ

Ϫ2

(14)

For laminar water ﬂow (RelϽ1055)

lϭ

Fig. 2 Schematic cross section of an evaporation tube

372 Õ Vol. 127, AUGUST 2005

64

Rel

(15)

Figure 3 displays the local speciﬁc pressure loss at different axial positions of a collector loop for different operation pressures.

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Fig. 4 Heat transfer coefﬁcient as a function of the spec. enthalpy for different heat ﬂux densities. In two-phase region only the heat transfer coefﬁcient in the wetted and heated region is ˙

displayed. „pÄ100 bar, M Ä1 kgÕs, d i Ä50 mm…

Fig. 5 Heat transfer coefﬁcient as a function of the spec. Enthalpy for different pressures. In the two-phase region only the heat transfer coefﬁcient in the wetted and heated region is dis˙ ˙

played. „q Ä40 kWÕm2 , M Ä1 kgÕs, dÄ50 mm…

The two-phase ﬂow pressure loss in the bends is determined by the equation of Chisholm ͓6͔ calculating the friction factors according to ͓7͔.

cient as a function of the spec. Enthalpy along the collector loop for different boundary conditions.

At this point all boundary conditions at the inner circumference are deﬁned as function of the known process parameters, the operation pressure, and the ﬂuid mass ﬂux. In the following sections the boundary condition at the outer circumference has to be deﬁned.

Heat Transfer Coefﬁcient. The heat transfer coefﬁcient for the single-phase water or steam ﬂow is calculated according to the equation of Dittus–Boelter ͓8͔.

␣ 1phϭ0.0235 Re0.8 Pr0.48

d

(16)

In case of the two-phase water-steam ﬂow three different heat transfer phenomena with different heat transfer coefﬁcients can be identiﬁed.

1. Heat transfer to the steam in the unwetted region;

2. Heat transfer to the water in the wetted and heated region; 3. Heat transfer to the water in the wetted and unheated region.

Collector Efﬁciency. Parabolic trough collectors only use the direct normal irradiation ͑DNI͒. The DNI is reﬂected on the absorber tube. Due to optical losses at the reﬂector, the glass envelope and the absorber surface only a certain fraction of the DNI is absorbed by the absorber tube. In addition the absorber tube has thermal losses due to convection and radiation. All these losses are determined by the collector efﬁciency. Once the collector efﬁciency is known the heat ﬂux on the outer surface of the absorber tube is known.

The heat transfer coefﬁcient for the steam phase and the water phase in the unheated region can be determined with Eq. ͑16͒. In the wetted and heated region nucleate boiling occurs. According to Gungor and Winterton ͓9͔ the heat transfer coefﬁcient ␣ 2ph,l is determined by a ﬁrst term taking into account the convective heat transfer ␣ conv and a second term for the heat transfer by the nucleate boiling ␣ NB .

␣ 2ph,l ϭE ␣ convϩS ␣ NB

(17)

The supression factor S is determined as follows:

Sϭ ͓ 1ϩ1.15•10Ϫ6 E 2 Ret1.17͔ Ϫ1

(18)

The enhancement factor E is calculated according to Eq. ͑19͒. Eϭ1ϩ24000Bo 1.16ϩ1.37X Ϫ0.86

a

(19)

With the boiling number Bo.

Boϭ

˙

q

˙

m tot͑ h Љ Ϫh Ј ͒

(20)

˙

where m tot is the total mass ﬂux density. The Martinelli parameter ˙

X a is a function of the ﬂuid properties and the steam quality x . The convective heat transfer coefﬁcient again is calculated according to Eq. ͑16͒. The heat transfer coefﬁcient due to nucleate boiling is determined according to Cooper ͓10͔.

Figures 4 and 5 display the course of the heat transfer coefﬁJournal of Solar Energy Engineering

Fig. 6 Collector efﬁciency as a function of the difference between operation and ambient temperature „TÀT a … for different values of the DNI „Ä0 deg, Cermet with Vacuum, LS-2 collector…

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not be higher than 50 K above the ﬂuid temperature ͓16͔. To allow for a very fast and reliable determination of the highest absorber temperature a simpliﬁed analytical solution for the calculation of the circumferential temperature distribution on the medium center line has been developed and implemented into the design tool. Based on this result the maximum and minimum temperature at

the outer surface will be calculated.

Temperature on the Medium Center Line. The analytical

calculation of the temperature is based on a division of the cross section in four segments ͑see Fig. 8͒.

1.

2.

3.

4.

Fig. 7 Typical heat ﬂux distribution along the outer surface of an absorber tube and its approximation by a gaussian and a

rectangular distribution with Ä60 deg

Empirical correlations describing the collector efﬁciency for several collector models exist. For LS-2 collectors the efﬁciency has been investigated in ͓11͔ where different equations have been derived for different absorber conﬁgurations. Formulations for the EuroTrough collector are presented in ͓12͔.

The typical course of the efﬁciency of a parabolic trough collector is displayed in Fig. 6 for an LS-2 collector. The efﬁciency of the collector decreases with an increasing operation temperature and a decreasing DNI. The maximum efﬁciency at TϪT a ϭ0, where no thermal losses occur, is called optical efﬁciency. These functions are valid for an incidence angle of 0 deg ͑the sun is perpendicular to the aperture area of the collector͒. The effect of different incidence angles is considered by the incidence angle modiﬁer ͑IAM͒. Again empirical correlations have been developed for different collectors. The IAM for an LS-2 collector is given in ͓11͔. Functions for an LS-3 collector are given in ͓13͔ and ͓14͔.

Every parabolic trough collector mentioned above can be used for direct steam generation. For the determination of the heat input into the absorber tube the designer of the collector ﬁeld has to choose the efﬁciency and IAM curve for the collector used.

Wetted and heated;

Unwetted and heated;

Unwetted and unheated;

Wetted and unheated.

In case of the preheating and the superheating section with the according single-phase ﬂow only two different sections can be identiﬁed, which are particular cases of the common case.

It is assumed that the boundary conditions in the different sections are known and constant. For the exact segmentation into the four sections the wetting angle and the position of the rectangular heat ﬂux distribution must be known. The position of the heat ﬂux distribution depends on the position of the sun that can be determined according to e.g. ͓17͔. Figure 9 is used for the derivation of the accounting equations. At the outer surface ⌬Q sol represents the heat input by the solar energy, ⌬Q conv represents the heat transfer at the inner surface and Q resp. ⌬Q the conductive heat transfer to adjacent segments. It can be written that

˙

˙

˙

dQ ϩdQ convϭdQ sol

(21)

with

2

d T

˙

dQ ϭϪs⌬z 2 dy

dy

(22)

˙

dQ convϭa⌬z ͑ TϪT f ͒ dy

(23)

˙

˙

dQ solϭ⌬zq soldy

(24)

In the heated region the heat ﬂux density is:

˙

q sol,h ϭ col DNI d ap

360

d m͑ eϪ b ͒

(25)

Flux Distribution on the Outer Surface. Once the heat ﬂux

on the outer surface is known its distribution has to be determined. A typical ﬂux distribution at the outer circumference of an absorber tube for parabolic trough collectors is displayed in Fig. 7 ͑taken from ͓15͔͒. This distribution shows a local minimum at an angle of 180 deg caused by the shadow of the absorber tube on the mirror facets. This distribution is similar to a Gaussian distribution with a standard deviation of 60 deg as displayed in Fig. 7. Additionally the approximation by a rectangular distribution with its edges at 100 deg and 260 deg is presented. In all cases the integral of the curve is the same.

FEM analyses of the resulting temperature distribution in the absorber cross section have shown that there is no signiﬁcant difference between the temperature distribution calculated with a ﬂux distribution according to the Gaussian distribution and the one calculated with the typical parabolic trough distribution. Accordingly the Gaussian distribution is chosen for the accurate FEM analysis whereas the rectangular distribution is used for a simpliﬁed analytical solution presented in the next section.

Temperature Distribution

The maximum temperature of the absorber tube is the most

critical parameter of a DSG collector loop. This temperature may 374 Õ Vol. 127, AUGUST 2005

Fig. 8 Schematic illustration of an absorber cross section with the four different sections

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Fig. 9 Developed view of an absorber tube segment for the

derivation of the analytical solution. The depth of the Segment is ⌬z.

It is assumed that the thermal radiation at the outer surface is constant and equivalent to:

˙

q sol,unhϭϪ

͑ optϪ ͒ DNId ap

d m

(26)

The differential equation for the determination of the temperature distribution is given by: d 2T

dy

2

ϪK 2 TϭϪ

1

˙

q ϪK 2 T f

s sol

(27)

with

K 2ϭ

␣

s

(28)

And the solution:

T ͑ y ͒ ϭC 1 cosh͑ Ky ͒ ϩC 2 sinh͑ Ky ͒ ϩ

˙

q sol

ϩT f

␣

(29)

To determine the two coefﬁcients C 1 and C 2 two boundary

conditions per segment have to be known. Since the four sections are connected to each other, the boundary conditions at the end of one section is the same as at the beginning of the next section. At this location the according temperatures as well as their gradient have to be the same. For example:

T 1 ͑ yϭl 1 ͒ ϭT 2 ͑ yϭ0 ͒

(30)

dT 1 ͑ yϭl 1 ͒ T 2 ͑ yϭ0 ͒

ϭ

dy

dy

(31)

and

a temperature of 310°C whereas the steam is slightly superheated to a temperature of 320°C. The circumferential angle is counted clockwise from the crest.

The marches of the temperature are very similar in Fig. 10. The maximum deviation between the two different calculation procedures is approx. 5 K. For a ﬁrst rough estimation of the temperature ﬁeld this deviation is acceptable. Maximum Temperature. To assess the thermal load of the

absorber tube not the temperature distribution on the medium center line is of importance but the maximum temperature at the outer surface of the absorber tube, which is the hottest point, that can be determined from the known temperature on the medium center

line. Therefore it is assumed that at the location with the maximum temperature the temperature gradient in circumferential direction is zero. Thus the heat ﬂow in circumferential direction is zero. Accordingly only a heat ﬂux in radial direction occurs. The maximum temperature is determined with the equation for the

steady-state heat conduction in radial direction. Accordingly the maximum temperature at the outer surface is calculated by:

ro

rm

T max,o ϭT max,m ϩ

͑ T max,m ϪT max,i ͒

rm

ln

ri

ln

The gradient of the temperature is deﬁned by:

dT 1 ͑ y ͒

ϭC 1 K sinh͑ Ky ͒ ϩC 2 K cosh͑ Ky ͒

dy

Fig. 10 Comparison of the temperature along the medium center line of the absorber cross section calculated with the FEM package ANSYS® and the analytical solution. „two-phase-ﬂow, heated from the side…

(32)

Accordingly there are eight equations for the determination of the eight coefﬁcients C i . It is possible to simplify the equations by considering the special characteristics of the hyperbola

function:

sinh͑ 0 ͒ ϭ0

(34)

With the help of this simpliﬁed procedure a ﬁrst rough assessment of a DSG collector loop is possible and critical process situations can be identiﬁed. In a subsequent step critical situations will be investigated in more detail using the FEM package

ANSYS® .

(33)

cosh͑ 0 ͒ ϭ1

(35)

The values for the eight coefﬁcients C i are found by solving the set of eight linear equations. Knowing the coefﬁcients C i for the different segments, it is possible to calculate the temperature on the medium center line. By connecting the four segments the

march of the temperature along the medium center line of the investigated cross section is determined.

Figure 10 displays the march of the absorber temperature calculated with the FEM package ANSYS® and the analytical solution. For the calculation it was assumed that the liquid phase has Journal of Solar Energy Engineering

Design Tool

The models presented have been implemented in a simulation

tool using the programming environment MATLAB® allowing for a fast investigation of different collector loop conﬁgurations. Before starting the calculation run the boundary conditions such as pressure and temperature at the loop outlet, the direct normal irradiation, the position of the sun and the geometry of the collector loop have to be deﬁned. In case of the recirculation mode the recirculation rate and the use of an injection cooler in the superheating section has to be deﬁned too. The once-through mode AUGUST 2005, Vol. 127 Õ 375

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Table 1

Parameters of the ET-II collector

Parabola Width ͓m͔

Overall Length of a single Collector ͓m͔

Outer Diameter ͓mm͔

Inner Diameter ͓mm͔

Length of connecting pipe between two

adjacent collectors ͓m͔

Number of 90° elbows in U-bends ͓°C͔

Peak Optical Efﬁciency ͓%͔

Roughness Factor of the Absorber Pipes ͓m͔

5,76

98,5

70

55

4

8

76,5

4EϪ05

Table 2 Main Parameters of the Solar Field

Length of Evaporator ͓m͔

Length of Superheater ͓m͔

Outlet Pressure ͓bar͔

Outlet Temperature ͓°C͔

Electric Power ͓MW͔

Number of parallel rows

Fig. 11 Pressure along the collector loop for the different operation modes „pÄ100 bar, T out Ä400°C, d i Ä50 mm, DNIÄ800 WÕm2…

requires the deﬁnition of the inlet temperature and the injection mode the deﬁnition of the number and distribution of injection cooler along the collector loop.

For the calculation the collector loop is divided into a number of segments with a step size of usually 5 m. Starting from the end of the collector loop, the collector efﬁciency, the speciﬁc enthalpy, the pressure loss and the ﬂuid temperature for all investigated collector positions are calculated. Based on the calculated ﬂuid properties and ﬂow conditions the distribution of the heat transfer coefﬁcient and the temperature of the absorber tube along the collector loop is determined. In addition it is possible to display the evaporation path in the Taitel–Dukler diagram.

As an example Fig. 11 displays the pressure and Fig. 12 the

maximum temperature difference along the collector loop for the three different operation modes. According to Fig. 11 the total pressure drop is less than 2 bar for a length of the collector loop of 500 m and an outlet pressure of 100 bar.

As shown in Fig. 12 in all cases the highest temperatures do not occur in the evaporation but in the superheating section. This is

800

200

70

410

5

7

caused by the occurrence of an annular ﬂow in the evaporation section with the according good cooling of the absorber wall. Lay-Out of a DSG Collector Loop. In the previous sections

the fundamentals of DLR’s design tool are presented. This tool is used during the design phase of DSG collector loops. ͑Since parabolic trough collectors operated with a heat transfer ﬂuid such as oil are similar to the preheating section of a DSG collector loop, DLR’s design tool is also applicable to oil collector loops.͒ The detailed design of collector ﬁelds also requires the investigation of the absorber tube and of the interconnecting piping and the necessary ﬁttings in more detail. For the detailed investigation of the absorber tube the FEM package ANSYS® and for the piping and ﬁttings and the overall collector ﬁeld performance the heat balance calculation program IPSEpro® is used. The use of these three programs for the investigation of DSG collector ﬁelds is presented in the following section.

The investigated collector ﬁeld is operated in recirculation mode as proposed in ͓18͔. To guarantee a sufﬁcient cooling of the superheating collectors a length of 200 m is chosen fo