Circumferential Temperature Variation in Superheater Tubes with
Mutual Irradiation for a Solar Receiver Steam Generator
by
Stewart John Wyatt
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING (MECHANICAL)
Approved:
Professor of Practice Norberto O. Lemcoff
Rensselaer Polytechnic Institute
Troy, New York
May, 2012
© Copyright 2012
by
Stewart John Wyatt
All Rights Reserved
ii
CONTENTS
Circumferential Temperature Variation in Superheater Tubes with Mutual Irradiation for
a Solar Receiver Steam Generator ................................................................................ i
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction.................................................................................................................. 1
1.1
Circumferential Temperature Variation ............................................................. 1
1.2
Central Solar Receiver Steam Generators .......................................................... 3
1.3
Mutual Irradiation .............................................................................................. 6
2. Nomenclature ............................................................................................................... 8
3. Formulation................................................................................................................ 11
3.1
Model Development ......................................................................................... 11
3.2
Model Parameters............................................................................................. 11
3.3
Selective Surfaces ............................................................................................ 12
3.4
Heat Transfer Models ....................................................................................... 16
3.4.1
Model 1 – Heat Loss by Radiation to the Environment from a Single
Half Tube ............................................................................................. 16
3.4.2
Model 2 – Heat Loss by Convection to the Environment from a Single
Half Tube ............................................................................................. 18
3.4.3
Model 3 – Heat Loss by Forced Convection to the Internal Fluid ....... 24
3.4.4
Model 4 – Temperature Distribution within a Thin Shell with Uniform
and Parallel Incident Radiation ............................................................ 28
3.4.5
Model 5 - Temperature Distribution within a Thick Shell with Uniform
and Parallel Incident Radiation and Convection .................................. 33
3.4.6
Model 6 – Heat Gain and Loss by Irradiation of Adjacent Tubes ....... 47
4. Solution ...................................................................................................................... 56
5. Results........................................................................................................................ 57
iii
6. Conclusions................................................................................................................ 58
7. References.................................................................................................................. 59
Appendix............................................................................................................................ 1
iv
LIST OF TABLES
Table 1. Operating parameters of a central solar receiver steam generator. ................... 11
Table 2. Tube metal temperature variation with conductivity from Figure 10 .............. 30
Table 3. Tube metal temperature variation with tube wall thickness from Figure 14 and
Figure 15 .......................................................................................................................... 37
Table 4. Tube metal temperature variation with internal convection coeffient from
Figure 14 and Figure 16 ................................................................................................... 38
Table 5. Thermal conductivity of carbon, alloy and stainless steels at varying metal
temperatures ..................................................................................................................... 39
v
LIST OF FIGURES
Figure 1. Tangent tube arrangement for use in a central receiver solar steam generator . 3
Figure 2. Central solar receiver steam generator and heliostat array (Boulden 2012, 34).
........................................................................................................................................... 6
Figure 3. Idealised hemispherical spectral emissivity of black nickel. .......................... 16
Figure 4. Gray body half tube of infinite length enclosed by a black body. .................. 18
Figure 5. Radiative heat loss to the environment at 300 K. ............................................ 18
Figure 6. Convective heat loss from an isothermal tube to the environment at 300 K. a)
L = 1 m , b) L = 10 m, c) L = 50 m. ................................................................................ 24
Figure 7. Temperature variation of superheated steam in a tube with constant thermal
irradiation ......................................................................................................................... 24
Figure 8.
Long, thin walled cylinder with external collimated incident radiation,
circumferential conduction, and, internal and external surface absorptivity and diffuse
emissivity. ........................................................................................................................ 28
Figure 9. Circumferential temperature distribution with varying conductivity. Using
Heaslet and Fuller (1964) data: qrad = 1381 W/m2, Do = 520 mm, t = 10 mm, 1=0.2, 1 =
0.5, 2 = 2 = 1. ................................................................................................................ 32
Figure 10. Circumferential temperature distribution with varying thermal conductivity.
Conductivities are 27.9 W/(m.K) for SA213 T91 at 500 C (solid line), 14.0 W/(m.K) for
50 % conductivity (o markers), 41.9 W/(m.K) for 150 % conductivity (+ markers) ...... 32
Figure 11. Circumferential temperature distribution with varying internal absorptance
and emittance. Absorptance and emittance are = = 0.001 (solid line) and = = 1
(+ markers)....................................................................................................................... 33
Figure 12. Long. thick walled cylinder with external collimated incident radiation, radial
and circumferential conduction, and, internal and external convection. ......................... 34
Figure 13. Non-dimensional isotherms in a tube using Mackowski 2011 data. ............. 40
Figure 14. Isotherms in a tube at design conditions ....................................................... 41
Figure 15. Isotherms in tubes of varying wall thickness. (a) t = 0.63 mm (10 % of
design), (b) t = 3.15 mm (50 % of design), (c) t = 9.45 mm (150 % of design) ............. 44
Figure 16. Isotherms in tubes with varying internal convection coefficient. (a) h = 2360
W/(m2.K) (50 % of design), (b) h = 7080 W/(m2.K) (150 % of design) ......................... 46
vi
Figure 17. Thermal radiation enclosure with cylindrical surfaces ................................. 47
Figure 18. Thermal radiation enclosure consisting of the flat isothermal surfaces ........ 48
Figure 19. Hottel’s crossed string method for flat isothermal surfaces of infinite length.
......................................................................................................................................... 50
Figure 20. View factor limitations of isothermal strips. ................................................. 52
Figure 21.
Directional absorptance of a blackened surface for artificial sunlight
transmitted through glass (Howell, Bannerot and Vliet 1982, figure C-9) ..................... 55
vii
ACKNOWLEDGMENT
To Gwendolyn, Emma and Juliette. For their patience.
To Doctor Marco Simiano. For the project idea.
To Professor Norberto Lemcoff. For his insights.
viii
ABSTRACT
Type the text of your abstract here.
ix
1. Introduction
1.1 Circumferential Temperature Variation
This report describes the circumferential temperature variations within a tube located in a membrane panel of tangent tubes (Figure 1) as may be used in the superheater of
a concentrated solar steam generator of the central receiver “tower” type. The aim is to
analyze the influence of the mutual irradiation of the adjacent tubes on the circumferential temperature distribution.
The solar radiation incident on the tube is collimated, so the energy absorbed by the
cylindrical surface is a function of the tubes projected area normal to the radiation
vector. The projected area is a cosine relation with the maximum at the tube crown,
declining to zero at the tube tangent position. The tube surface temperature profile is
therefore non-isothermal with a maximum at the tube crown.
The non-isothermal nature of the tube surface temperature indicates that a non-zero
net radiation interchange will occur between tube surface and those parts of the adjacent
tube surface at a different temperature. Therefore, the surface temperature of the tube
will be influenced by the incident solar radiation, mutual irradiation of the adjacent tube
and radiation exchange with the ambient. Conduction and convection modes of heat
transfer also influence the tube temperature profile.
Ungar and Mekler (1960) studied the circumferential temperature distribution in
thin walled tubes (Dm/t > 10) exposed to nonuniform radiation and noted:
If the radiant heat received by the outer surface of a tube were transmitted to the
fluid inside the tube in a purely radial direction through the tube metal and the inside film, the temperature differences between the fluid and the outer tube
surface at the various points around the tube circumference would be proportional to the flux intensities impinging at these points. However, since the tube metal
is a relatively good conductor of heat, there is also a flow of heat in the circumferential direction within the metal; heat then flows from points exposed to a
higher radiation intensity to points exposed to a lesser intensity. Thus the temperature of each circumferential tube element depends on four possible paths for
the flow of heat: Radiation from the outside, convection and conduction to the
1
fluid inside the tube, and conduction to or from the two adjoining tube elements.
Under steady-state operating conditions, the element assumes a temperature
which permits it to be at thermal equilibrium with its entire surroundings.
The tubes of the solar steam generator are heated by a concentrated collimated solar
flux to the front face from an array of heliostats to increase the temperature of the
superheated steam flowing within. As well as the intended heat transfer from the tube to
the steam, the tube is also cooled by conduction (axially and circumferentially), external
convection and thermal radiation exchange with the surroundings. The various modes of
heat loss are considered with a focus on the influence of the radiation interchange
between any tube and its surroundings consisting of the ambient and adjacent tubes.
The resulting equilibrium temperature distribution within the tube as influenced by
the various heat flows is of interest to the designer of steam generators as the maximum
mean tube metal temperature defines the maximum allowable stress to be used for the
design of cylindrical components under internal pressure (ASME 2010 (a), PG-27.4.2).
For the design of internal pressure retaining parts, an ideal tube circumferential temperature distribution is a uniform one as any variation will increase the maximum mean
temperature for a given heat transfer, resulting in an increased wall thickness or a
stronger material.
Various factors prevent a circumferentially uniform temperature
distributions for this application including a collimated solar flux onto the front of a
cylindrical surface, an insulated rear surface of the tube, finite conductivity of the tube
material, and, a thermal radiation view factor from the surface of the tube that varies
from full exposure to the ambient to full exposure to the adjacent tube at the tube crown
and tube tangent respectively.
2
Figure 1. Tangent tube arrangement for use in a central receiver solar steam generator
1.2 Central Solar Receiver Steam Generators
Central solar receiver steam generators consist of consists of dual axis tracking heliostats to concentrate solar radiation onto a tower mounted central receiver (Anderson
and Kreith 1987, 28), as shown in Figure 2. The radiant energy is used to heat the
working fluid to a high temperature. The working fluid considered for this project is
3
high pressure superheated steam for use in a turbo generator as is typical in conventional
electricity generating plants.
Membrane panels of tangent tubes may be applied to central receiver of solar power
system as the heat transfer medium between the concentrated solar irradiation and the
working fluid. The tube panels are typically flat but arranged to form an approximate
cylindrical surface.
With the aim of maximizing the use of existing technology, an ideal solar steam
generator used for electricity production utilizes the same equipment as a fossil fuel fired
electric utility including high temperature materials and steam turbines. Therefore, for
design purposes, the solar absorber considered is assumed to perform at nominally the
same outlet steam conditions as a contemporary fossil fuel fired plant. For a sub-critical
pressure steam generator in a 593 MWe electric utility plant, steam conditions are 540 C
at 16.5 MPa (Stultz and Kitto 1992, 2-18). Furthermore, a radiative heat flux similar to
that of the utility plant furnace, 270 kW/m2 (Stultz and Kitto 1992, 4-22), will be used
for the concentrated solar radiation.
As an example of an existing central solar receiver steam generator used for electricity generation, the Solar One plant located in Barstow, California, and completed in
1982 consisted of (Duffie and Beckman 2006, 634):
a) 10 MWe electric generating capacity
b) 71100 m2 of reflectors
c) 13.7 m high and 7 m diameter central receiver
d) 69 mm diameter tubes welded together to form panels
e) Average solar radiation absorptance of 0.96 by the non-selective flat black
painted panels
f) Superheated steam production of 14.14 kg/s at 516 C
g) Absorbing surface maximum operating temperature of 620 C
Each of the heliostats reflects the incident solar energy onto the cylindrical steam
generator located atop of the central tower. Each heliostat may move with two axis
control allowing the array of heliostats to act as a parabolic surface with the steam
generator at the focal point.
4
Solar collectors are a group of heat exchangers that converts incident solar energy
into heat. The simplest collectors include a flat plate without optical concentration, and
without solar tracking, utilizing up to approximately 1100 W/m2 (Duffie and Beckman
2006, 238). Flat plate collectors are utilized in applications such as the supply of domestic hot water. To increase the temperature of the energy supplied by the absorber, an
optical concentrator is located between the source of the radiation (the sun) and the
absorber surface. Concentrating collectors systems include parabolic troughs, parabolic
dish and central receivers. Parabolic troughs utilize a line focus while parabolic dish and
central receivers use a point focus.
Appley and Bird (1984, 23) noted that the higher concentration ratios of point focus
concentrators allowed higher working fluid temperatures, producing better Rankine
cycle efficiency. Furthermore, the point focus central receiver with a Rankine cycle and
thermal storage is the most applicable of the thermal systems analyzed for commercial
development in a large electric utility plant in the 50 to 200 MWe range. This project is
applied to a central receiver steam generator for utility application.
5
Figure 2. Central solar receiver steam generator and heliostat array (Boulden 2012, 34).
1.3 Mutual Irradiation
The solar receiver steam generator superheater tubes are exposed to conduction,
convection and radiation heat transfer mechanisms. A distinguishing feature of conduction and convection from radiation is their difference in temperature dependencies. A
6
one dimensional conduction application may be described by Fourier’s law as (Modest
1993, 1):
qx  k
T
x
Equation 1
Similarly, convection is typically described by the correlation (Modest 1993, 1):
q  h(T  T )
Equation 2
The thermal conductivity and convection coefficient are k and h respectively. Both
k and h may be a function of temperature, but for many applications the conduction and
convection heat transfer is treated as linearly proportional to the temperature difference.
However, radiative heat transfer rates are generally proportional to the difference of
temperatures to the fourth power (Modest 1993, 1):
q  T 4  T4
Equation 3
Therefore, radiative heat transfer becomes the dominant heat transfer mode as temperatures increase. All heat transfer modes are applicable to this project but the high
temperature of the solar irradiation source (5780 K) and the superheater tubes (~ 1000
K) are indicative of the important role of thermal radiation.
The collimated solar irradiation of the cylindrical surface results in non-isothermal
tube surface temperature. This temperature profile is further modified by the radiation
exchange with the cooler ambient as well the surface of the adjacent tube.
Earlier work with mutual irradiation of surfaces includes the radiant interaction of a
fin and isothermal tube (Chung and Zhang 1991; Sparrow and Eckert 1962; Kreith 1962,
122) with the aim of providing heating and cooling for space applications. The present
work considers the mutual irradiation of non-isothermal tubes with a tangent tube
arrangement. The non-isothermal tubes are approximated as a curved surface composed
of a large number of flat isothermal strips to model the circumferential temperature
variation.
7
2. Nomenclature
Symbol
Unit
Description
a
-
Radius ratio (a = r2/r1)
A
m2
Area
Bi
-
Biot number
cp
J/(kg.K)
Specific heat at constant pressure
D
m
Diameter
E
W/m2
Emissive power of a radiating body
Fx-y
-
View factor for radiation from body x to y.
g
m/s2
Acceleration due to gravity
G
W/m2
Incident radiative flux onto a surface
Gr
-
Grashof number
hc
W/(m2.K)
Convection coefficient
k
W/K
Thermal conductivity
L
m
Length
kg/s
Mass flow
Nu
-
Nusselt number
Pr
-
Prandtl number
q
W/m2
Heat flux
Q
W
Heat rate
r
m
Radius
r
-
Radius, non-dimensional
t
m
Wall thickness
T
K
Temperature
T
-
Temperature, non-dimensional
U
m/s
Velocity
Roman Symbols
•
m
8
Greek Symbols

-
Absorptance, or,

m2/s
Thermal diffusivity

1/K
Temperature coefficient of thermal expansion

-
Emittance

m
Wavelength

m2/s
Kinematic viscosity

kg/m3
Density

W/(m2.K4)
Stefan-Boltzmann constant

rad
Angle from normal incident radiation
x, y, z
-
Rectangular coordinates
r, , z
-
Cylindrical coordinates
B
-
Blackbody
cond
-
Conduction
conv
-
Convection
cyl
-
Cylinder
D
-
Diameter
f
-
Fluid
L
-
Length
m
-
Mean
o
-
Average, or, Outside
proj
-
Projected
rad
-
Radiation
sol
-
Solar
surf
-
Surface
Miscellaneous
Subscripts
9
air
-
air
1
-
Outside of tube
2
-
Inside of tube

-
Monochromatic at wavelength 

-
Ambient conditions

10
3. Formulation
3.1 Model Development
The thermal modeling of the tubes that collectively form the solar receiver steam
generator will consist of a staged approach beginning with simple, order of magnitude
calculation and by removing simplifying assumptions results in a realistic model.
Comparison of model results will determine the degree of accuracy gained as complexity
is added and allow the user to choose a practical level.
The model begins with a single tube with only one mode of heat transfer between
itself and the environment or the fluid flowing within.
3.2 Model Parameters
The analysis is bounded by parameters that may be considered a practical design but
potentially extending the limits of current practice for the superheater of a central
receiver solar steam generator. The values considered are given in Table 1.
Table 1. Operating parameters of a central solar receiver steam generator.
Fluid
Superheated steam
Operating outlet temperature1
 600 C (873 K)
Operating Inlet temperature2
355 C (628 K)
Operating pressure1
 17.5 MPa(g)
Tube bulk velocity3
10 to 25 m/s
Tube
Material4
SA213 T91
Conductivity5
27.4 W/(m.K) at 300 C
27.9 W/(m.K) at 400 C
27.9 W/(m.K) at 500 C
27.6 W/(m.K) at 600 C
27.0 W/(m.K) at 700 C
Absorptance6
0.95
11
Emittance6
0.09
Outside diameter7
50.8 mm
Wall thickness8
6.3 mm
Length
 50 m
Ambient
Air temperature
 50 C (323 K)
Sky temperature9
Air temperature minus 10 C (10 K)
Wind Speed10
0 to 25 m/s
Concentrated Solar Radiation
Solar flux11
 300 kW/m2
Notes:
1
Stultz and Kitto 1992, 2-18
2
Saturation temperature at maximum operating pressure
3
Stultz and Kitto 1992, 3-8
4
Stultz and Kitto 1992, 6-21
5
ASME 2010 (b), 727. Material group F.
6
Duffie and Beckman 2006, 196. Black chrome on Ni-plated steel.
7
Stultz and Kitto 1992, 18-10
8
ASME 2010 (a), PG-27.2.1
9
Rabl 1985, 218
10
Anderson and Kreith 1987, 30
11
Stultz and Kitto 1992, 4-22
3.3 Selective Surfaces
The tube properties of Table 1 are typical of those encountered in conventional utility steam generator applications. A fossil fuel fired steam generator consists of regions
with heat transfer by radiation and convection to the heat exchanger tubes. Heat transfer
within a concentrated solar collector is dominated by thermal radiation. For the fossil
fuel and solar steam generator applications, the source of the thermal radiation is a flame
and the sun respectively. In both cases the source is at a much higher temperature than
the absorbing surface. It will be shown that the difference in source and surface temper-
12
atures has limited spectral overlap. Therefore, separate absorptivity and emissivity
values are used while obeying Kirchoff’s law of equivalent spectral absorptivity and
spectral emissivity,  = .
The net radiative heat gain of a solar collector is the difference between absorbed
solar energy and radiation losses due to emission by the collector surface (Modest 1993,
123). Therefore, an ideal solar collector surface has a high absorptance for those wavelengths and directions of the incident radiation with a low emittance at the surface
conditions, referred to as a selective surface.
For solar thermal applications, an ideal collector surface as described by Howell,
Bannerot and Vliet (1982, 10) is:
To reduce the radiative losses from an absorber while at the same time maintaining a high solar absorptance, a selective surface is often applied to the absorber.
These surface coatings are composed of specially formulated paints, chemical
dips, or electroplated films that have the useful radiative property of high absorptance at important solar wavelengths (0.3-1.8 m), but low emittance in the
longer wavelengths where most of the radiant energy is emitted from the absorber. Hence, they act as a radiant heat trap, selectively absorbing solar energy but
not reemtting significant infrared radiation.
The Rankine cycle, when considered for a solar or fossil fuel fired steam generator,
has led to increasing steam temperatures for improved cycle efficiency. However, a
solar steam generator will also increase heat losses as the operating temperature rises due
to the exposure of the heating surface to the cooler ambient. The increasing temperature
differential between the surface and the ambient promotes heat loss, a reduction in
efficiency. To maximize the work output of a given heat engine, it is desirable to
operate at or near the temperature that maximizes the product of collector efficiency
(which decreases with increasing temperature) and heat engine efficiency (which increases with increasing temperature) (Howell, Bannerot and Vliet 1982, 36).
The performance of a selective surface is usually measured by the “/ ratio” where
 is the total, directional absorptivity of the material for solar irradiation, while  is the
total, hemispherical emissivity for the infrared surface emission (Modest 1993, 123).
Black chrome (chrome-oxide coating) and black nickel (nickel-oxide coating) are
13
effective solar collector coatings as they have exhibit hemispherical spectral emissivity
greater than 0.8 for wavelength less than approximately 2 m; conversely, the emissivity
is less than 0.2 and diminishing for wavelengths greater than 6 m, as indicated by
Modest (1993, 124 figure 3-32). The suitability for a solar absorber is associated with
the fact that the maximum spectral emissive power for blackbodies at 5780 K (representing the sun) and 1000 K (representing the solar absorber surface) are wavelengths of
approximately 0.5 m and 3 m respectively (Howell, Siegel and Menguc 2011, 19
figure 1.11). This is in agreement with Wien’s displacement law which gives the
wavelength max at which the blackbody intensity is maximum for a given temperature
(Howell, Siegel and Menguc 2011, 21):
max 
2897.7686
T
Equation 4
For comparison, assume the solar absorber surface is a selective material with idealized properties as shown in Figure 3, nominally that of black nickel (Modest 1993, 123).
The monochromatic hemispherical emittance and absorptance of a surface,  and 
respectively, are given by (Kreith and Bohn 1986, 442):
 
E (T )
Eb (T )
Equation 5
 
G (T )
Gb (T )
Equation 6
Kirchoff’s radiation law states in essence that the monochromatic emittance is equal
to the monochromatic absorptance for any surface (Kreith and Bohn 1986, 443). However, the total absorptance of a surface, , depends upon the temperature and spectral
characteristics of the incident radiation. The difference between the absorptance and
emittance used for the tube surface, 0.95 and 0.09 respectively, is indicated in the
following typical example and is due to the difference between the temperatures of the
source of the irradiation (5780 K) and the tube itself (~1000 K). The gray body enclosure model employed later with the assumption of  =  is justified since the mutual
irradiation by the adjacent tubes has nominally the same temperature for the emitter and
absorber.
14
The product T and the respective blackbody radiation fractions at the cut off wavelength for the source and surface temperatures are (Howell, Siegel and Menguc 2011,
871):
TSun  2*5780  11560 m.K
F0TSun  0.93848
Equation 7
TSurface  2*1000  2000  m.K
F0TSurface  0.06673
Equation 8
The hemispherical total emissivity in terms of hemispherical spectral emissivity is
(Howell, Siegel and Menguc 2011, 55):

   (T ) E (T )d 
b
 (T ) 
0
T 4
 (T )    (T ) F T  T
1
Equation 9
2
The hemispherical total emissivity and absorptivity of the surface is:
  0.9*0.06673  0.2*(1  0.06673)
  0.25
  0.9*0.93848  0.2*(1  0.93848)
  0.86
Thus the absorptance and emittance of the idealized surface are shown to be highly
dependent upon the temperature of the irradiation source and irradiated surface respectively. This characteristic will be employed in the gray surface enclosure model and was
also used by Heaslet and Lomax (1962, 458):
Both emission and reflection will be assumed diffuse and the material of the
shell opaque. A grey-body type of analysis will be used, i.e. the coefficient of
emission, absorption, and reflection are to be independent of temperature and
frequency except that two extreme temperature and frequency ranges with separate coefficients will be admitted. In this way we shall account for possible
differences between the emissivity or absorptivity in the relatively low-
15
temperature regime of the walls and the absorptivity of the incident external energy which may come from a source of much increased temperature, for
example, in the case of solar radiation.
The project specific absorptance is applied to the solar irradiation while the emittance applies to the both the diffuse emittance and re-absorptance at the surface
temperature.
Figure 3. Idealised hemispherical spectral emissivity of black nickel.
3.4 Heat Transfer Models
3.4.1
Model 1 – Heat Loss by Radiation to the Environment from a Single Half
Tube
A simple radiant heat loss model consists of a single half tube of infinite length surrounded by a black body as shown in Figure 4. Only the convex surface of the tube is
considered.
The assumptions include:
1. Heat loss occurs only by radiation to the environment.
2. Diffuse gray tube and black body environment.
3. Temperature of the half tube surface is constant around the circumference.
4. Non participating medium
16
Identifying the half tube and environment as bodies 1 and 2 respectively, the view
factors are:
F1-2 = 1
F2-1 = 0
Using the net radiation method for diffuse gray enclosures (Howell et al 2011, 211),
if the environment is a black body enclosing the gray convex surface of the tube, the heat
loss from body 1 to 2 is:
N
(
j=1
δ kj
εj
- Fk-j
1- ε j Q j N
)
=  Fk-jσ(Tk4 - Tj4 )
ε j A j j=1
Equation 10
Q1
=F1-2 .ε1.σ(T14 - T24 )
A1
Equation 11
q1-2 =F1-2 .ε1.σ(T14 - T24 )
Equation 12
A radiative heat loss of approximately 5000 W/m2 occurs assuming a surface temperature of 1000 K and emittance of 0.09 as shown in Figure 5. The solar irradiation for
the project is 300 kW/m2. However, it is noted that the solar irradiation is for the
projected area of the tube, while the heat loss is due to the circular surface of the half
cylinder. Therefore, the 5000 W/m2 heat loss corrected for the projected area is increased by a factor of /2, to 7854 W/m2 relative to the projected area. The heat loss is
equivalent to 2.6 % of the incoming irradiation.
17
Figure 4. Gray body half tube of infinite length enclosed by a black body.
Figure 5. Radiative heat loss to the environment at 300 K.
3.4.2
Model 2 – Heat Loss by Convection to the Environment from a Single Half
Tube
A simple natural convection heat loss model consists of a single half tube of infinite
length oriented vertically. Only the convex surface of the tube is considered.
The assumptions include:
1. Turbulent boundary layer with no leading edge effect.
2. Temperature of the half tube surface is constant around the circumference.
3. Temperature of the half tube surface is constant over its length.
Using the McAdams model (Kreith and Bohn 1986, 257) for the turbulent region
boundary layer of a vertical cylinder:
18
NuL 
hL
 0.13(GrL Pr)1/3
kf
Equation 13
The convection heat transfer coefficient h is noted to be independent of length L as
(Kreith and Bohn 1986, 210):
g . (Tsurf  Tair ) L3
GrL 
Equation 14
2
and:
Pr 
c p .
k



Equation 15
The resulting convection coefficient and heat loss are:
  g  (Tsurf - Tair ) 
h  0.13k f 

.


1
3
Equation 16
qsurf -air  h(Tsurf - Tair )
Equation 17
The preceding model assumes an isothermal surface and a cylinder that is approximated
by a flat surface, neither of which is in good agreement with the actual arrangement
consisting of long, slender tubes with an external constant radiant flux but an internal
flow with sensible heat gain in the economizer and superheater sections, negating the
isothermal assumption. The evaporator section contains a fluid with latent heat gain so
these tubes alone may be considered isothermal.
Sparrow and Gregg (1956) propose that the flat plate model is sufficient for the heat
transfer from an isothermal vertical cylinder in air if:
2
5
2
1
(
GrL 4
L
)  0.15
Do
Equation 18
Assuming typical values of Do and L of 50.8 mm and  50 m, respectively, and surface and ambient temperatures of 600 K and 300 K respectively, the criterion is not met.
Minkowycz and Sparrow (1974) analyze natural convection along an isothermal
vertical cylinder where deviation from the flat plate results exist, and it this will be
applied to this study. While the McAdams relationship previously considered was not a
19
function of tube length, L, or outside diameter, Do, both are now required so the heat loss
is specific to the physical arrangement.
The isothermal cylindrical heat loss model is (Minkowycz and Sparrow 1974, 180):
1
qcyl  k (Tsurf
 g  (Tsurf - Tair )  4
- Tair ) 
 (- '( , 0))cyl
4 L 2


Equation 19
Where:

2( L / ro )
1
4
( g  (Tsurf - Tair )ro3 / (4 2 ))
1
Equation 20
4
( - θ'(ξ,0))cyl is tabulated (Minkowycz and Sparrow 1974, 181).
Similarly, Le Fevre and Ede (Ede 1967, 18) provide the following relationship for
an isothermal vertical cylinder with natural convection.
1
4  7GrL .Pr 2  4 4(272  315 Pr) L
NuL  
 
3  5(20  21Pr) 
35(64  63Pr) Do
Equation 21
The McAdams; Minkowycz, and Sparrow; and Le Fevre, and Ede, are shown in
Figure 6 for varying tube length. The additional complexity of the non-isothermal
nature of the superheater tube is yet to be addressed, however, it may already be noted in
Figure 6 that the isothermal models diverge in results but do agree in the trend of nominally linear heat loss with increase in surface temperature.
Furthermore, Ede (1967, 21) noted there is a special difficulty in carrying out experimental work on cylinders with small values of Do/L, as the slightest movement in the
bulk of the fluid is sufficient to deflect the rising column of heated fluid away from the
upper part of the cylinder. The result is, of course, an increase in the measured heat
transfer coefficient that may be very large. The assumption of a single tube immersed in
quiescent air for a solar receiver steam generator is poor.
Sparrow and Gregg (1958) addressed free convection from a non-isothermal flat
plate with air (Pr = 0.7) as the medium. To determine the surface temperature variation
with distance along the tube, the temperature of the superheated steam within the fluid is
considered. If the steam enthalpy increases due to the product of the constant solar flux,
20
absorptivity and the projected area of the tube, the relationship between steam temperature and distance along the tube can be shown to be approximately linear (see Figure 7).
Furthermore, if the surface temperature is assumed to follow the trend of the fluid
temperature, it is also considered to be linear. Therefore, the Sparrow and Gregg (1958,
382, Table 1) model for n=1 (linear) and Pr = 0.7 (air) provides:
qnon-isothermal  1.49qisothermal
Equation 22
Assuming the flat plate non-isothermal model of Sparrow and Gregg (1958) is valid
for the superheater tube, the actual local heat transfer is nominally 150 % of that previously predicted. It is concluded that the heat transfer is greatly affected by the tube wall
temperature distribution.
The convective losses from the external surface of the tube, in particular, the large
L/Do ratio, forced convection due to wind, and the non-isothermal characteristics, result
in data of poor confidence. An alternative approach by Anderson and Kreith (1987, 30)
and Yeh and Wiener (1984, 46) provides a more direct approach by considering the
convection from the entire central solar receiver steam generator rather than the sum of
individual tubes. The receiver is treated as a cylinder with a low L/Do ratio formed by
the vertical flow tubes. The receiver height, diameter, surface temperatures, and wind
velocities, the Grashof and Reynolds numbers may be calculated. The relationship for
Nusselt numbers is provided by Anderson and Kreith. This method is not applied in this
report due to the unknown size and shape of the central receiver but it is recommended
for applications where this data is known.
Convective losses are significant in the performance of a central receiver steam generator as they contribute to reduced plant efficiency. These losses vary according to the
local surface temperature and wind speeds. However, when selecting a tube metal
temperature for the design of the heat exchanger, the minimum convective loss should
be considered as this will correspond to the highest temperature and most arduous design
condition.
Of the convective heat loss models considered, and as shown in Figure 6, the greatest heat loss for a surface temperature of 1000 K is approximately 11000 W/m2.
Correcting the heat loss by a factor of /2 to allow comparison with the incident radia-
21
tion on the projected surface, the heat loss is 17278 W/m2. The convective heat loss is
equivalent to 5.7 % of the 300 kW/m2 solar irradiation.
22
23
Figure 6. Convective heat loss from an isothermal tube to the environment at 300 K. a) L = 1 m , b)
L = 10 m, c) L = 50 m.
Figure 7. Temperature variation of superheated steam in a tube with constant thermal irradiation
3.4.3
Model 3 – Heat Loss by Forced Convection to the Internal Fluid
The tube is cooled by the internal flow of superheated steam, ranging from saturated
vapor conditions at the inlet to superheated steam (  600 C) at the outlet. The discharge
pressure is 17.5 MPa and no pressure drop is assumed between the inlet and outlet. The
inlet velocity is 10 m/s and increases with distance along the tube due to the density
change due to the increasing steam temperature. The only heat addition is considered to
be the concentrated solar flux multiplied by the tubes projected external area and surface
absorptivity, hence the increase in fluid enthalpy from the inlet saturated steam through
to the superheater outlet.
24
The Reynolds1, Prandtl2 and Nusselt3 numbers are calculated as shown, with the aim
of calculating the internal convection coefficient2 (Notes: (1) Incropera and De Witt 1990,
(2)
470,
Kreith and Bohn 1986, 210,
(3)
Dittus-Boelter equation from Incropera and De
Witt 1990, 496).
•
U m D2
4m
Re D 


 D2 
Pr 
Equation 23
cp
Equation 24
k
4
NuD  0.023Re D5 Pr 0.4
h
Equation 25
k f Nu D
Equation 26
D2
The internal convection coefficient was of the order of 4720 W/(m2.K) at the superheater outlet conditions of 600 C, 17.5 MPa(g) and 1.462 kg/s in a tube of 38.2 mm
inside diameter. The mass flow was selected according to the tube bulk velocity range
of Table 1, 10 to 25 m/s. The lowest tube velocity of 10.0 m/s is selected at the saturation temperature (17.5 MPa(g)) at the superheater inlet, providing the tube mass flow.
The resulting superheater outlet tube velocity at 600 C and 17.5 MPa(g) is 26.7 m/s,
nominally equal to the upper velocity range of 25.0 m/s.
The external natural convection coefficient was determined to be < 10 W/(m2.K).
As shown in section 3.4.2, at an assumed surface temperature of T1 = 900 K (627 C), the
surface convective heat loss is 9000 W/m2 (Figure 6a, LeFevre and Ede model). The
natural convection coefficient for this case is 7.3 W/(m2.K) using Newton’s law of
cooling:
q1-air  h(T1 - Tair )
Equation 27
The large difference between the internal and external convection coefficient supports the design of the steam flow cooling the tubes to provide useful work. Any cooling
by the ambient air is a loss in plant efficiency and is ideally minimized.
25
The internal convection coefficient is a function of both the physical conditions,
such as the tube internal diameter, and fluid properties. The fluid properties can vary
over a broad range of operating conditions such as part load operation of the plant and
transient conditions. The sensitivity of the tube metal temperature to variations in the
internal convection coefficient will be considered in section 3.4.5.
With a known internal convection coefficient of 4720 W/(m2.K), the internal surface
temperature of the tube is to be calculated. The project specific external solar irradiation
and absorptivity are 300 kW/m2 and 0.95 respectively. The energy absorbed by the
steam, without any losses to the environment, is:
Q  qsol Aproj
Equation 28
Therefore, a tube of outside diameter of D1 = 50.8 mm and 1 m length absorbs Q =
14.478 kW. The heat flux between the tube surface at the inside diameter (D2 = 38.2
mm) and the steam is q = 120.641 kW/m2 from:
q
Q
Q

Asurf  D2 L
Equation 29
The internal surface temperature of the tube is then calculated as T2 = 626 C for a
steam temperature of 600 C from the relationship:
q  h(T2  T f )
Equation 30
Assuming circumferentially uniform temperature and heat flux within the tube wall,
the relationship between the internal (T2) and external (T1) surface temperatures is
(Yener and Kakac 2008, 53):
Q
2 Lk (T2  T1 )
ln(r1 / r2 )
Equation 31
For a constant tube conductivity 27.9 W/(m.K), the external surface temperature is
T1 = 650 C. The wall differential temperature is 24 C, and Tmax/Tmin = 1.04. The
assumption of circumferentially uniform temperature and heat flux within the tube is not
valid for this application but the analysis if provided for comparison when the circumferential variations are considered (sections 3.4.4, 3.4.5 and 3.4.6).
26
The surface temperature dependence upon steam flows is supported by Yeh and
Wiener (1984, 47):
The fluid temperatures and the heat transfer coefficients inside the tubes depend on the flow rates. Consequently, the average outer surface temperatures of
tubes,… are implicitly dependent on flow rates.
The application of internal convective cooling of a tube in a central receiver solar
steam generator is subjected to circumferential variations in bore heat flux and outside
surface temperature due to factors including the cosine variation of the normal incident
radiation on the tube front surface varying between relative values of 1 at the crown
( to 0 at  = /2 radians, and an insulated rear surface. The circumferential
variation is symmetrical about the plane through  = 0 and  it is considered to be
invariant in the flow direction. For the solar collector application, the solar flux is
assumed normal to the panel of tubes, but even variations from normal will result in
nominally symmetrical tube temperature and bore flux profiles if the shadowing effect of
the adjacent tubes can be ignored.
The analysis thus far has assumed circumferentially uniform thermal boundary conditions, but the influence of variation has been considered by Sparrow and Krowech
(1977); Gartner, Johannsen and Ramm (1973); Black and Sparrow (1967); and Reynolds
(1963). Sparrow and Krowech (1977) examined solar collectors with large heat flux
spikes at discrete circumferential locations on the outer surface of the tube. Non uniform
circumferential heat flux can result in circumferential variations in both the bore heat
flux and the outer surface temperature. The results showed that for practical dimensions
and thermal properties of the collector, circumferential variations in bore heat flux and
outside surface temperature can be neglected for laminar flows. Surprisingly, turbulent
flows have a greater impact. This project Reynolds number of 1.45x106 differs from that
of other authors including Sparrow and Krowech (1977, 365) Re  20000; and Black
and Sparrow (1967, 264) Re  58000. This project makes no attempt to account for the
influence of the non-uniform circumferential heat flux on the bore heat flux. It is noted
as a possible area of future work.
27
3.4.4
Model 4 – Temperature Distribution within a Thin Shell with Uniform and
Parallel Incident Radiation
Heaslet and Fuller (1964) model a thin, conducting, cylindrical shell of infinite
length with a collimated external source of radiation, as shown in Figure 8. The thermal
conductivity of the shell is known, and radiative emissions from the inner and outer
surfaces are assumed to be diffuse. A gray body is assumed so no dependence on
radiation wavelength is considered. The resulting model of a closed, circular cylinder
with a uniform and parallel field of incident radiation providing the circumferential
temperature distribution. Temperature variations through the wall are not considered.
Figure 8. Long, thin walled cylinder with external collimated incident radiation, circumferential
conduction, and, internal and external surface absorptivity and diffuse emissivity.
The Heaslet and Fuller (1964) model considers collimated irradiation, that is, external radiation that penetrates from the outside into a participating medium (as opposed to
emission from a bounding surface), with all light waves being parallel to one another
(Modest 1993, 572). Collimated irradiation is assumed to be a applicable to the solar
receiver steam generator, however, some variation is acknowledged due to the array of
heliostats to reflect and concentrate the solar radiation. The effects of the off-normal
irradiation is not quantified, but the collimated model is assumed to provide the bound28
ing temperature distribution, as any increased distribution of the heat flux will decrease
the peak temperature at the tube crown while increasing that of the cooler tube sides.
The Heaslet and Fuller (1964) model was intended for outer space applications and
thus, did not consider external convection and assumes radiative interaction only within
the cylinder. Furthermore, the assumption of the wall thickness much less than the
diameter (t << Do) simplifies the conduction analysis to consider circumferential variations only but may be of lesser validity for this studies application, where the t:Do ratio is
nominally of the order of 1:8. The significance of conductance in the radial direction
will be considered further. The Heaslet and Fuller model is employed with the understanding that the present analysis will differ due to the presence of internal and external
convection and radial conduction. The temperature results do not reflect actual values
for the project as the model does not account for the forced convection within the tubes,
results are indicative of trends only. The temperature distribution is (Heaslet and Fuller
1964, 147):
For 0  φ 
π
:
2
πNε
u(φ)  2 12
ν β
{
1 ( α2 4 - ν 2 )
cos(ν.φ)
- νcosφ 
2
ν 1- ν
2sin( ν.π 2)
(
)
1 ( α2 4  β2 )
cosh(β.φ)
+βcosφ
2
β 1 β
2sinh(β.π 2)
)}- 14
(
For
Equation 32
π
φ π:
2
u(φ) 
πNε1
ν2  β2
- φ))
{ 1ν ( α 1-4ν- ν ) ( cos(ν(π
)
2sin( ν.π 2)
2
2
2
1 ( α 2 4  β 2 ) cosh(β(π - φ))
β 1  β2
2sinh(β.π 2)
(
)}- 14
Equation 33
Where:
1
T
 (1  4u ) 4
To
Equation 34
29
 q 
To   1 rad 
 1 . 
N
1
4
Equation 35
r2 To3
kt
Equation 36
1

2
2




1




 2   4 N (1   2 )  2    4 N (1   2 ) - 2   41 2 N  


2
4 
4
 

 2   2  4(1   2 ) N 
2
Equation 37
Equation 38
4
The Heaslet and Fuller (1964) results quantify the intuitive outcomes including the
maximum temperature occurring where the radiative flux is normal ( = 0). Similarly
the minimum cylinder temperature occurs diametrically opposite the maximum, and,
higher material conductivity results in a lower temperature differential between the
maximum and minimum values. A material of infinite conductivity results in an isothermal temperature.
Confirmation of this projects application of the Heaslet and Fuller (1964) result is
provided in Figure 9, showing agreement with Heaslet and Fuller (1964) Figure 3.
The Heaslet and Fuller (1964) model is applied to parameters typical for this report
in Figure 10. Variations in thermal conductivity (50, 100 and 150 % of the conductivity
for SA213 T91 at 500 C) give qualified support to the influence of increase conductivity
in the reduction of peak tube temperatures. The results are included in Table 2.
Table 2. Tube metal temperature variation with conductivity from Figure 10
Tube Conductivity
Tube Max.
Tube Min.
W/(m.K) (%)
Temperature
Temperature
K (C)
K (C)
14.0 (50)
2364 (2091)
1670 (1397)
1.42
27.9 (100)
2262 (1989)
1832 (1559)
1.23
41.9 (150)
2211 (1938)
1897 (1624)
1.17
30
Tmax/Tmin
The dominant heat transfer mechanism within the tube of this project is forced convection from the hot tube to the internal steam flow. No attempt is made to relate the
internal surface absorptivity and emissivity used by Heaslet and Fuller (1964) to an
equivalent convective coefficient, but the importance of the internal conditions to the
tube temperature distribution is shown.
Figure 11 compares the internal radiative
properties of ==0.001 and ==1, the average tube metal temperature is unchanged
while the ratio Tmax/Tmin varies from 1.23 to 1.04 respectively. Maximizing the tubes
internal heat transfer mechanism will reduce the maximum metal temperature. This may
be applied for radiation or convection.
The designer of a steam generator is typically restrained from choosing materials
with optimum conductivity, as the dominant concern is the materials allowable stress
value at the given temperature.
However, some flexibility exists to maximize the
internal convection coefficient such as fluid velocities and tube diameters. The internal
convection coefficient was previously described in section 3.4.3.
31
Figure 9. Circumferential temperature distribution with varying conductivity. Using Heaslet and
Fuller (1964) data: qrad = 1381 W/m2, Do = 520 mm, t = 10 mm, 1=0.2, 1 = 0.5, 2 = 2 = 1.
Figure 10. Circumferential temperature distribution with varying thermal conductivity. Conductivities are 27.9 W/(m.K) for SA213 T91 at 500 C (solid line), 14.0 W/(m.K) for 50 % conductivity (o
markers), 41.9 W/(m.K) for 150 % conductivity (+ markers)
32
Figure 11.
Circumferential temperature distribution with varying internal absorptance and
emittance. Absorptance and emittance are = = 0.001 (solid line) and = = 1 (+ markers).
3.4.5
Model 5 - Temperature Distribution within a Thick Shell with Uniform and
Parallel Incident Radiation and Convection
Mackowski (2011, 119) provides an analytical model for a long, annular cylinder with
temperature variation in both r and , convection on the internal and external surfaces,
33
and the outside of the pipe is exposed to a collimated source of thermal radiation, as
shown in Figure 12.
Figure 12. Long. thick walled cylinder with external collimated incident radiation, radial and
circumferential conduction, and, internal and external convection.
The non-dimensional temperature distribution of Mackowski (2011, 124) is given
by:
T(r,φ) 

 r 
1  π.Bi1 T  ,1
1  Bi 2 ln    
π.(Bi 2  Bi1 (1  Bi 2 ln(a))) 
 a 
1
2(g1 (1)  Bi1g1 (1))
g1 (r) cos(φ) 
(1) n g 2n (r) cos(2n.φ)
2 

π n 1 (1  4n 2 )(g 2n (1)  Bi1g 2n (1))
where:
34
Equation 39
T
(T  T,2 )k
Equation 40
 rad qr1
r
r
r1
Equation 41
a
r2
r1
Equation 42
Bi1 
h1r1
k
Equation 43
Bi2 
h2 r2
k
Equation 44
T ,1 
(T,1  T,2 )k
Equation 45
 rad qr1
g n (r )  (r ) n  a 2 n
n  Bi2
(r )  n
n  Bi2
Equation 46

n  Bi2 
g n (1)  n 1  a 2 n

n  Bi2 

Equation 47
The Mackowski model differs from that of the current project in that the radiative
emmissions from the tube external surface are not considered, only the radiative absorption. Furthermore, the entire external surface is exposed to convection rather than an
insulated rear surface. However, it is presented as a means of qualifiying the influence
of tube thickness and internal convection coefficient on the tube temperature distribution. The lack of radiative emissions from the external surface is expected to provide a
conservative design with over prediction of the tube metal temperature due to the
unaccounted for cooling effect. The project uses a selective surface with a low emittance (0.09) so the simplification is acceptable.
Figure 13 is presented to show agreement with the results of Mackowski (2011, 125,
figure 4.16(b)), while Figure 14 uses project specific data. Figure 15 and Figure 16
compare the influence of wall thickness and internal convection coefficient of the tube
temperature distribution.
35
The project specific and off design tube wall thickness temperature profiles of Figure 14 and Figure 15 indicate small radial variation, and less so for thinner walls. This
result is indicative of the small Biot number (Bi) for the internal and external surfaces,
since as Bi approaches zero, the total surface resistance is very large compared to the
total internal resistance (Yener and Kakac, 2008, 62). Therefore, the temperature drop
through the wall in the radial direction will be small. Circumferential variation in
temperature is shown to be nominally Tmax/Tmin  1.1 to 1.2 with a reduction of the
variation as the tube wall thickness decreases.
For a cylinder of height much greater than the radius, the Bi is (Yener and Kakac,
2008, 184):
Bi 
 
r
hLc h 2

k
k
Equation 48
Considering project specific data:
h1 = 10 W/(m2.K)
h2 = 4720 W/(m2.K)
r1 = 25.4 mm
r2 = 19.1 mm
k = 27.9 W/(m.K)
The resulting Biot numbers are given below and it is noted that, for Bi < 0.1, the one
dimensional transient temperature within a solid can be considered uniform with an error
less than approximately 5 % (Orisik 1980, 16):
Bi1 = 0.0045 (External)
Bi2 = 1.6 (Internal)
The combination of a low external and relatively high internal Biot number may be
qualified with Figure 15, as the temperature variation through the wall is negligible at
the rear of the tube ( π  φ  π ), where only convection and conduction are present.
2
However, the front face of the tube ( 0  φ  π ), especially at the tube crown ( = 0)
2
where the incident radiation is normal, external radiation is the dominant mechanism
resulting in significant through wall and circumferential temperature variations.
36
Variations of the tube wall thickness are presented in Figure 14 (design conditions)
and Figure 15 (off design conditions). All properties remained constant while the wall
thickness varied from the project specific thickness of 6.3 mm to 0.63 mm (10 %), 3.15
mm (50 %) and 9.45 mm (150 %). It is noted that the Nusselt number and hence convection coefficient will change with the difference in internal diameter, but practical
considerations such as changes to the mass flow led to the simplified approach of
keeping the Nu constant for the exercise. The peak metal temperature was strongly
dependent upon the wall thickness, as shown in Table 3. In practice, for a given design
temperature, a thinner wall requires a higher allowable stress value, typically associated
with steels of increasing alloy content. This supports the use of high alloy steels in areas
of high solar flux and high steam temperatures, as minimizing the wall thickness is
critical to limiting the peak metal temperature. Conversely, the lower alloy material in
the same application will be self-defeating, as the peak metal temperature will increase,
likely requiring an upgrade of material.
Table 3. Tube metal temperature variation with tube wall thickness from Figure 14 and Figure 15
Tube Wall
Tube Max.
Tube Min.
Tmax/Tmin
Thickness
Temperature
Temperature
mm (%)
K (C)
K (C)
0.63 (10)
946 (673)
847 (574)
1.12
3.15 (50)
977 (704)
866 (593)
1.13
6.30 (100)
1017 (744)
867 (594)
1.17
9.45 (150)
1057 (784)
870 (597)
1.21
*** see Ungar and Mekler page 274 to relate biot and radial heat flow****
Variations of the internal convection coefficient are presented in Figure 14 (design
conditions) and Figure 16 (off design conditions). All properties remained constant
while the internal convection coefficient varied from the project specific value of 4720
W/(m2.K) to 2360 W/(m2.K) (50 %) and 7080 W/(m2.K) (150 %). The peak metal
temperature is a function of the internal convection coefficient, as shown in Table 4. A
reduced internal convection coefficient, resulting in higher peak metal temperatures, can
37
result from various operating conditions in the superheater. These include typical plant
start up and low load conditions including a reduced mass flow and pressure. Therefore,
the selection of a tube metal design temperature requires consideration of the off design
conditions.
Table 4. Tube metal temperature variation with internal convection coeffient from Figure 14 and
Figure 16
Internal Convection
Tube Max.
Tube Min.
Tmax/Tmin
Coefficient
Temperature
Temperature
W/(m2.K) (%)
K (C)
K (C)
2360 (50)
1075 (802)
869 (596)
1.24
4720 (100)
1017 (744)
867 (594)
1.17
7080 (150)
995 (722)
867 (594)
1.15
The application of the Mackowski model to the central receiver steam generator requires an insulated surface in the rear of the tube where model has convection. The
natural convection of the external surface has been shown to have a very low Bi, so has a
relatively small influence on the cooling of the tube. However, it is noted in Table 3 that
the minimum temperature is typically less than the 600 C steam temperature. The
practical application of insulation to the rear section is expected to increase the tube
minimum temperature to that of the fluid or greater. The impact on the maximum tube
temperature is yet to be established.
Future work should include a review of materials other than SA213 T91 as the
thermal conductivities of the steel alloys vary as shown in Table 5. The tube temperature distribution is a function of thermal conductivity, with Heaslet and Fuller (1964)
showing that increasing tube conductivity can be used to reduce the temperature extremes.
Furthermore, constant conductivity was assumed in the Mackowski (2011) analysis.
However, the variation between maximum and minimum tube metal temperatures as
shown in Table 3 and Table 4 would indicate the possible importance of variable conductivity.
A brief review of the temperature dependences of thermal conductivity
38
properties of steel alloys is given in Table 5 showing a nominal trend of reducing
conductivity with increasing alloy content and/or temperature.
Table 5. Thermal conductivity of carbon, alloy and stainless steels at varying metal temperatures
Conductivity W/(m.K)
Carbon
SA213 T222
SA213 T913
Steel1
SA213
SA213
TP304N4
TP316N5
300 C
49.2
36.7
27.4
19.4
18.3
400 C
44.9
35.4
27.9
20.8
19.7
500 C
40.5
33.7
27.9
22.2
21.2
600 C
35.8
32.0
27.6
23.6
22.6
700 C
31.2
30.1
27.0
25.0
23.9
Notes:
1
ASME 2010 (b), 726. Material group A.
2
ASME 2010 (b), 726. Material group D.
3
ASME 2010 (b), 727. Material group F.
4
ASME 2010 (b), 727. Material group J.
5
ASME 2010 (b), 728. Material group K.
The influence on heat transfer of oxides and other scales that may form on the inner
and outer surface of tubes has not been considered. Oxides formed on the inside of heat
absorbing tubes may insulate the tube from the cooling by the steam, leading to increased tube metal temperatures. Similarly, the radiation emission and absorption by the
outer surface may be influenced by changes to the surface characteristics.
39
Figure 13. Non-dimensional isotherms in a tube using Mackowski 2011 data.
40
Figure 14. Isotherms in a tube at design conditions
41
42
43
Figure 15. Isotherms in tubes of varying wall thickness. (a) t = 0.63 mm (10 % of design), (b) t =
3.15 mm (50 % of design), (c) t = 9.45 mm (150 % of design)
44
45
Figure 16. Isotherms in tubes with varying internal convection coefficient. (a) h = 2360 W/(m2.K)
(50 % of design), (b) h = 7080 W/(m2.K) (150 % of design)
46
3.4.6
Model 6 – Heat Gain and Loss by Irradiation of Adjacent Tubes
3.4.6.1 Gray Surface Enclosure
The walls of adjacent tubes are not at a uniform temperature so heat is transferred within
and between the tubes. Conduction and convection are present but due to the high
temperatures, radiation is considered. The temperature is known to vary circumferentially and is assumed invariant along the length.
The heat exchanger surface consists of long tubes arranged tangentially to form a
flat panel. The convex external surface of each tube allows a thermal radiation exchange
with the surface of the adjacent tube and the ambient; conversely, thermal radiation is
not transferred from a tube to itself except through re-radiation from the adjacent tube.
The system is considered to be an enclosure consisting of the ambient and the /2 radian
sectors of two adjacent tubes as shown in Figure 17.
Figure 17. Thermal radiation enclosure with cylindrical surfaces
For numerical analysis, the surface of each tube is idealized as being composed of a
large quantity of equally spaced circumferential nodes connected by flat strips of infinite
length in the axial direction and isothermal characteristics. The idealized enclosure is in
Figure 18.
47
Figure 18. Thermal radiation enclosure consisting of the flat isothermal surfaces
These surfaces are assumed to be diffuse gray. The radiative exchange between
gray, diffuse surfaces within an enclosure of N surfaces is given by a system of N
equations. The general form for the kth surface is (Modest 1993, 208):
  kj
1  j
 Fk  j


j
j 1   j
N
N

4
 q j    kj  Fk  j   T j  qsol  k
j 1

Equation 49
The solar irradiation onto the kth surface is defined as qsol-k. Corresponding to each
surface, k takes on the values 1,2,…, N, and the Kronecker delta is defined as:
1 when k  j
0 when k  j
 kj  
Equation 50
The assumption of a diffuse gray surface indicates that the directional emissivity
and directional absoptivity do not depend on direction.
Furthermore, the spectral
emissivity and spectral absorptivity do not depend on wavelength. They can, however,
depend on temperature. Thus, at each surface temperature, for any wavelength, the
emitted spectral radiation is a fixed fraction of the blackbody spectral radiation (Howell
et al 2011, 65).
48
The incident radiation is characterized as collimated rather than diffuse so is not independent of angle. The directional dependence of the absorptivity is not considered
here but is proposed as an area of further investigation. For this application, the collimated incident radiation is normal at the crown of the tube ( = 0 radians) and
diminishes to nil at the tangent position ( = /2 radians). The grazing angles ( > /3
radians) compose a significant portion of the heat transfer surface yet, as shown by
Modest (2011, 77 figure 3-1), the surface radiation properties are heavily dependent
upon the incident angle. This is discussed further in section 3.4.6.5. Furthermore,
additional factors such as surface roughness and temperature dependence may be significant. The diffuse gray assumption provides great simplification but the significance of
this has not been quantified.
3.4.6.2 View Factors
Hottel’s crossed strings method is employed to determine the view factor between
each isothermal strip and the other surfaces of the enclosure. The diffuse view factor Fi-j
represents the ratio of the radiative energy leaving surface i that strikes surface j directly
to the radiative energy leaving surface i in the entire hemispherical space (Ozisik 1973,
122). A view factor not equal to zero will only exist where an unobstructed line of sight
exists between the two surfaces considered. Hotell’s crossed string method for two
arbitrary surfaces, as shown in Figure 19, is (Modest 1993, 178):
Fi  j 
( Lbc  Lad )  ( Lac  Lbd )
2 Li
Equation 51
49
Figure 19. Hottel’s crossed string method for flat isothermal surfaces of infinite length.
Upon calculating the view factor for the surfaces of cylinder 1 to those of cylinder 2,
the reciprocity relation may be used to determine that for the surfaces of cylinder 2 to
those of cylinder 1. However, symmetry is used. The reciprocity relation is used to
determine the view factor between the ambient surface and each isothermal strip after
the reverse is found. The reciprocity relation for two surfaces is (Howell, Siegel and
Menguc 2011, 162):
Ai Fi  j  Aj Fj i
Equation 52
For areas of equal depth, the reciprocity relation becomes:
Li Fi  j  L j F j i
Equation 53
The crossed strings method is applied by calculating the distance between nodes,
equivalent to a straight line between points (xi, yi) and (xj, yj).
The enclosure consists of the polygons representing the two cylindrical quadrants
and is completed by the ambient section. The ambient section of the enclosure is equal
to a straight line:
y  r for 0  x  2r
Equation 54
The sum of all of the fractions of energy leaving a surface and reaching the surfaces
of the enclosure must equal 1. For the kth surface in an enclosure of N surfaces (Howell,
Siegel, Menguc 2011, 170):
N
Fk 1  Fk 2  Fk 3  ...  Fk k  ...  Fk  N   Fk  j  1
Equation 55
j 1
Due to the convex external surfaces of the tubes, the view factor for the isothermal
strip to itself and all other isothermal strips on the same cylinder are zero. The only nonzero view factors will be to the ambient surface and that of the adjacent cylinder.
Additional limitations will be imposed by the inability of the isothermal strips to view
any surfaces beyond the hemispherical emittance or hidden by the horizon of the adjacent tube as shown in Figure 20. The view factors from the isothermal strips of cylinder
50
1 to those of cylinder 2 are calculated by Hottel’s crossed strings method. The view
factors between the isothermal strips and the ambient surface are then calculated by the
summation of enclosure view factors as described previously.
3.4.6.3 Surface Position and View Factors
The collimated solar irradiation on to the convex tube surface results in a surface
heat flux that is a product of the cosine of the angle between the surface normal and the
solar vector:
qsurf  qsol cos 
Equation 56
The peak surface flux occurs at the tube crown ( = 0 radians) and decreases to nil
at the tube tangent ( = /2 radians) thereby providing a non-isothermal tube surface.
However, the tube surface temperature is influenced by the radiation interchange between the tube surface and that of the adjacent tube as per section 3.4.6.1. The view
factors of those elements nearest the crown has a cooling effect as their view is dominated by the relatively cool ambient. Conversely, those surface elements near the tube
tangent are in a crevice with a view factor dominated by the adjacent tube so minimal
cooling by radiation occurs. The importance of surface position for the view factor of
surface elements is shown in Figure 20. This is a good result when considering the
reduction in the peak temperature that occurs at the tube crown and a reduction in the
tube temperature ratio, Tmax/Tmin.
51
Figure 20. View factor limitations of isothermal strips.
3.4.6.4 Cavity Absorption
The region adjacent to the tube tangent is in the form of a long valley or cavity. A
cavity has special properties for thermal radiation as noted by Incropera and De Witt
(1990, 710):
Although closely approximated by some surfaces, it is important to note that no
surface has precisely the properties of a blackbody. The closest approximation is
achieved by a cavity whose inner surface is at a uniform temperature. If radiation
enters the cavity through a small aperture, it is likely to experience many reflections before reemergence. Hence it is almost entirely absorbed by the cavity, and
blackbody behavior is approximated.
Cavities of various shapes have been studied including cylindrical cavities with one
end open (Sparrow, Albers and Eckert 1962), parallel walled grooves (Sparrow and
Gregg 1962) and V-grooves (Sparrow and Lin 1962). Studies directly applicable to the
cavity formed by two convex surfaces with non-isothermal and selective characteristics,
was not found. However, general results are considered for this application.
52
The apparent emissivity of a cavity, defined as the ratio of the actual radiative energy streaming out of the opening to the radiative energy that would have been emitted by
a black surface at the cavity temperature, having the same area as the opening (Ozisik
1973, 198), is of interest. The trend is for the apparent emissivity to show the greatest
increase for deep cavities with small opening areas to enclosed area ratios and small
emissivity. This indicates the greatest influence of the cavity adjacent to the tube
tangent.
The influence of V-groove cavities on the radiative properties of a surface was noted by Duffie and Beckman (2006, 193) as:
Surfaces of deep V-grooves, large relative to all wavelengths of radiation concerned, can be arranged so that radiation from near-normal directions to the
overall surface will be reflected several times in the grooves, each time absorbing
a fraction of the beam. This multiple absorption gives an increase in the solar
absorptance but at the same time increases the long-wavelength emittance.
However, as shown by Hollands (1963), a moderately selective surface can have
its effective properties substantially improved by proper configuration. For example, a surface having nominal properties of  = 0.60 and  = 0.05, used in a
fixed optimally oriented flat-plate collector over a year, with 55o grooves, will
have an average effective  of 0.9 0 and an equivalent  of 0.10.
The influence of the cavity on solar absorption characteristics warrants further investigation but it is noted that the practical configuration may differ from that considered
thus far. The construction of a membrane wall with tangent tube construction is likely to
require a weld within the cavity for structural purposes. Assuming the weld is of concave finish and blended smoothly into the tubes, any cavity effect is significantly
reduced.
The use of a selective surface was not found in the available literature on cavity absorption. This may be significant for solar applications due to the high / ratio.
Furthermore, the off normal absorptivity of the surface is typically poor so leads to an
increase in reflectivity at the glancing angle present within the cavity, this is discussed
further in section 3.4.6.5.
53
3.4.6.5 Off Normal Incident Radiation Absoptance
Collimated solar irradiation onto a cylindrical surface results in and angle of incidence of zero at the tube crown ( = 0 radians) and increases to /2 radians at the tube
tangent ( = /2 radians). The significance of this characteristic is noted by Duffie and
Beckman (2006, 197):
The angular dependence of solar absorptance of most surfaces used for solar collectors is not available.
The directional absorptance for solar radiation of
ordinary blackened surfaces (such as used for solar collectors) is a function of the
angle of incidence of the radiation on the surface.
The directional absorptance of a typical surface is provided in Figure 21 (Howell,
Bannerot, Vliet 1982, Figure C-9). The directional absorptance is nominally two regions, an angle of incidence 0     3 and  3     2 , the absorption is greater
than 90 % in the former, and rapidly falls to zero for the latter. A similar characteristic
is in Duffie and Beckman (2006, Figure 4.11.1).
The surface absorptance of this project has assumed it is only a function of the projected area. Considering the two regions, the projected areas are 86.6 % and 13.4 % for
the angle of incidence 0     3 and  3     2 respectively. The bias of the
large projected area in the region of high absorption supports the assumption.
However, additional uncertainty is created by the radiation in the grazing angle, if
not absorbed by the tube surface, it then enters the cavity between the tubes for further
possible reflection and absorption.
The characteristics of the cavity were noted in
section 3.4.6.4. As with the cavity, if in practice a concave weld with smooth blending
into the tubes is located at the tube tangent, the grazing angle region is likely to be
removed and replaced by a surface nominally normal to the incident radiation.
54
Figure 21.
Directional absorptance of a blackened surface for artificial sunlight transmitted
through glass (Howell, Bannerot and Vliet 1982, figure C-9)
55
4. Solution
***Surface emissivity of carbon and alloy steel tube
***how does difference in tube conductivity due to surface deposits (internal and
external) change the temp distribution????
56
5. Results
****Comment on the radiation versus convection losses. Radiation dominates at the
high temperatures so becomes important for power generation applications as the steam
temperature increase (max temp ~ 600 C or 873 K). Radiation loss is less applicable at
the economizer section (give operating temp range). Is this the reason for the different
water wall design with and without finning between the econ and superheater?
*** suitability of grade 91 for panel design. Traditionally used CS for panels.
Poorer conductivity, thermal stress, cycling, thermal expansion????
*** will differential T between front and back of panel cause bowing???? Check
thermal expansion coefficient of 91 and temperature max and min. Does the bowing
tendency vary between the inlet and outlet of the SH?
*** NOT USED **
***Yeh and Wiener (1984) use a superheater with evaporator tubes as a shield to
get ~ 300 kW/m2 instead of 1 MW/m2 (fig 2)?????? Relate this to figure 31 page 4-20
of Steam for different effectiveness of tangent tubes versus spaced or shielded tubes.
***Yeh and Wiener (1984) treat solar flux as diffuse instead of collimated due to
array of mirrors page 46. I used collimated? What is the significance?? Is collimated a
conservative method as it will give a higher peak temperature at the crown as it is the
only location with normal incident radiation.
57
6. Conclusions
***mutual irradiation tends to reduce the Tmax/Tmin ratio so a conservative design
considers a tube in isolation to determine the max tube wall temperature.
***cavity weld with a concave finish smoothly blended into the tubes on the heated
side is important to prevent any possible hot spot in the tube tangent. The effect of the
cavity is not fully understood.
***many factors, some tend to increase Tmax/Tmin eg
1.
Directional absorptance is likely > 0.90 from 0<<60 degree, but drops off to
zero between 60<<90 degree. Means crown of tube does most of the heat absoption.
2. Weld between the tangent tubes (concave and smoothly blended into tubes)
makes it more like the model by ?? with the fin between the tubes causing a heat
concentration at =90 degrees due to conduction from the fin. Many articles
about this.
3. T91 is a relatively poor conductor so does not spread the temperature out.
Many factors, some tend to decrease Tmax/Tmin eg
1. Mutual irradiation lets hot crown see the cool ambient but the relatively cool
f=90 region does not see the sky, only the other hot areas of adjacent tube.
2. Modelled as collimated radiation but likely has some diffuse characteristic due
to array of mirror providing a much wider source of irradiation tending to add
more heat to the sides and less to the crown.
3. Cavity effect may make =90 degree region a better absorber.
58
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62
Appendix