CIRCUMFERENTIAL TEMPERATURE VARIATION IN
SUPERHEATER TUBES WITH MUTUAL IRRADIATION AS
APPLIED TO A SOLAR RECEIVER STEAM GENERATOR
by
Stewart John Wyatt
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
__________________________________
Norberto O. Lemcoff. Project Adviser.
Rensselaer Polytechnic Institute
Hartford, Connecticut
April, 2012
(For Graduation May, 2012)
© Copyright 2012
by
Stewart John Wyatt
All Rights Reserved
ii
CONTENTS
CIRCUMFERENTIAL TEMPERATURE VARIATION IN SUPERHEATER TUBES
WITH MUTUAL IRRADIATION AS APPLIED TO 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 .............................................................................................. 7
2. Nomenclature ............................................................................................................... 8
3. Formulation................................................................................................................ 10
3.1
Model Development ......................................................................................... 10
3.2
Model Parameters............................................................................................. 10
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 ....... 23
3.4.4
Model 4 – Temperature Distribution within a Thin Shell with Uniform
and Parallel Incident Radiation ............................................................ 27
3.4.5
Model 5 - Temperature Distribution within a Thick Shell with Uniform
and Parallel Incident Radiation and Convection .................................. 35
3.4.6
Model 6 – Heat Transfer by Mutual Irradiation of Adjacent Tubes .... 48
4. Results........................................................................................................................ 62
5. Conclusions................................................................................................................ 65
iii
6. References.................................................................................................................. 67
7. Appendix.................................................................................................................... 71
Model 1 - Heat Loss by Radiation to the Environment from a Single Half Tube. ............. I
Model 2 - Heat Loss by Convection to the Environment from a Single Half Tube. ......... II
Model 3 - Heat Loss by Forced Convection to the Internal Fluid. .................................. III
Model 4 - Temperature Distribution within a Thin Shell with Uniform and Parallel
Incident Radiation. ..................................................................................................... IV
Model 5 - Temperature Distribution within a Thick Shell with Uniform and Parallel
Incident Radiation and Convection. ........................................................................... V
Model 6 - Heat Transfer by Mutual Irradiation of Adjacent Tubes. ............................... VI
iv
LIST OF TABLES
Table 1. Operating parameters of a central solar receiver steam generator. ................... 11
Table 2. Tube metal temperature variation with conductivity ........................................ 29
Table 3. Tube metal temperature variation with outside diameter ................................. 30
Table 4. Tube metal temperature variation with tube wall thickness ............................. 38
Table 5. Tube metal temperature variation with internal convection coefficient ........... 39
Table 6. Thermal conductivity of carbon, alloy and stainless steels at varying metal
temperatures ............................................................................................................. 40
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). .... 6
Figure 3. Idealized hemispherical spectral emissivity of black nickel. .......................... 15
Figure 4. Gray body half tube of infinite length enclosed by a black body. .................. 17
Figure 5. Radiative heat loss to the environment at 300 K. ............................................ 17
Figure 6. Convective heat loss from an isothermal tube to the environment at 300 K .. 21
Figure 7. Temperature variation of superheated steam in a tube with constant thermal
irradiation ................................................................................................................. 23
Figure 8.
Long, thin walled cylinder with external collimated incident radiation,
circumferential conduction, and, internal and external surface absorptivity and
diffuse emissivity...................................................................................................... 27
Figure 9.
Circumferential temperature distribution with varying conductivity using
Heaslet and Fuller (1964) data ................................................................................. 31
Figure 10. Circumferential temperature distribution with varying thermal conductivity
.................................................................................................................................. 32
Figure 11. Circumferential temperature distribution with varying outside diameter ..... 33
Figure 12. Circumferential temperature distribution with varying internal absorptivity
and emissivity ........................................................................................................... 34
Figure 13. Long, thick walled cylinder with external collimated incident radiation, radial
and circumferential conduction, and, internal and external convection. .................. 35
Figure 14. Non-dimensional isotherms in a tube using Mackowski 2011 data. ............. 41
Figure 15. Isotherms in a tube at design conditions ....................................................... 42
Figure 16. Isotherms in tubes of varying wall thickness. ............................................... 43
Figure 17. Isotherms in tubes with varying internal convection coefficient. ................. 46
Figure 18. Thermal radiation enclosure with cylindrical surfaces ................................. 48
Figure 19. Thermal radiation enclosure consisting of the flat isothermal surfaces ........ 49
Figure 20.
Surface temperature distribution on adjacent cylinders with mutual
irradiation ................................................................................................................. 51
Figure 21. Simplified gray enclosure with the open surface as a black body whose
temperature corresponds to the irradiation ............................................................... 53
vi
Figure 22. Hottel’s crossed string method for flat isothermal surfaces of infinite length.
.................................................................................................................................. 54
Figure 23. View factor limitations of isothermal strips. ................................................. 56
Figure 24. Cavity influence on surface temperatures ..................................................... 59
Figure 25.
Directional absorptance of a blackened surface for artificial sunlight
transmitted through glass (Howell, Bannerot and Vliet 1982, figure C-9) .............. 61
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
Concentrated solar thermal technology constitutes an important alternative when considering the use of solar energy for electricity production. The radiant energy is used to
heat a working fluid to a high temperature. The objective of this report is to analyze the
circumferential temperature variations within a superheater tube of a solar receiver steam
generator. The tube was heated by concentrated, collimated solar irradiation with major
cooling by an internal steam flow. The influence on the temperature distribution of
external convection, heat loss by radiation and conduction within the tube were considered. Finally, the significance of mutual irradiation of adjacent tubes was analyzed as a
gray body enclosure composed of isothermal strips.
Practical models that may be
applied to solar receiver steam generator designs were obtained for an isolated tube.
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, that is, the irradiation is parallel and uniform as opposed to diffuse. 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 the 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 non-uniform 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
1
the flow of heat: Radiation from the outside, convection and conduction to the
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 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)). 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 dual axis tracking heliostats to concentrate solar radiation onto a tower mounted central receiver (Anderson and Kreith 1987),
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 high pressure superheated
steam for use in a turbo generator, as is typical in conventional electricity generating
plants.
3
Membrane panels of tangent tubes may be applied to the 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). Furthermore, a radiative heat flux similar to that of
a coal fired utility plant furnace, 270 kW/m2 (Stultz and Kitto 1992), 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):
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.
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
4
2006). 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.
Concentrating solar collectors are typically used to (Kreith and Kreider 1978):
a) Increase energy delivery temperatures in order to achieve a thermodynamic
match between temperature level and task.
b) Improve thermal efficiency by reducing the heat loss area relative to the receiver area. There would also be a reduction in transient effects, since the
thermal mass is usually much smaller than for a flat plate collector.
c) Reduce cost by replacing an expensive receiver by a less expensive reflecting or refracting area.
Appley and Bird (1984) 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).
6
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 one
dimensional conduction application may be described by Fourier’s law as (Modest
1993):
qx  k
T
x
Equation 1
Similarly, convection is typically described by the correlation (Modest 1993):
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):
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)
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
m2
Area
Bi
-
Biot number
cp
J/(kg.K)
Specific heat at constant pressure
D
m
Diameter
Fx-y
-
View factor for radiation from body x to y.
g
m/s2
Acceleration due to gravity
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
Re
-
Reynolds number
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
air
-
air
B
-
Blackbody
D
-
Diameter
f
-
Fluid
L
-
Length
m
-
Mean
o
-
Average, or, Outside
proj
-
Projected
sol
-
Solar
surf
-
Surface
1
-
Outside of tube
2
-
Inside of tube

-
Monochromatic at wavelength 

-
Ambient conditions
Miscellaneous
Subscripts
9
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 thermal models are:
Model 1 - Heat Loss by Radiation to the Environment from a Single Half Tube.
Model 2 - Heat Loss by Convection to the Environment from a Single Half Tube.
Model 3 - Heat Loss by Forced Convection to the Internal Fluid.
Model 4 - Temperature Distribution within a Thin Shell with Uniform and Parallel Incident Radiation.
Model 5 - Temperature Distribution within a Thick Shell with Uniform and Parallel Incident Radiation and Convection.
Model 6 - Heat Transfer by Mutual Irradiation of Adjacent Tubes.
Numerical models are created in MatLab software except model 3. Model 3 uses an
Excel spreadsheet with WinSteam properties for steam. All are included as an appendix.
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.
.
10
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 velocity1
10 to 25 m/s
Tube
Material1
SA213 T91
Conductivity3
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
Absorptance4
0.95
Emittance4
0.09
Outside diameter1
50.8 mm
Wall thickness5
6.3 mm
Length
 50 m
Ambient
Air temperature
 50 C (323 K)
Wind Speed6
0 to 25 m/s
Concentrated Solar Irradiation
 300 kW/m2
Solar flux1
Notes:
1
Stultz and Kitto 1992
2
Saturation temperature at maximum operating pressure
3
ASME 2010 (b), Material group F.
4
Duffie and Beckman 2006, Black chrome on Ni-plated steel.
5
ASME 2010 (a), PG-27.2.1
6
Anderson and Kreith 1987
11
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 temperatures has limited spectral overlap for application to a solar receiver steam generator.
Therefore, separate absorptivity and emissivity values are used while obeying Kirchhoff’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).
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) 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 reemitting 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
12
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).
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). Black
chrome (chrome-oxide coating) and black nickel (nickel-oxide coating) are 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, 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, figure 1.11). This is in
agreement with Wien’s displacement law which gives the wavelength max at which the
blackbody intensity is a maximum for a given temperature (Howell, Siegel and Menguc
2011):
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). The
monochromatic hemispherical emittance and absorptance of a surface,  and , respectively, are given by (Kreith and Bohn 1986):
 
E (T )
Eb (T )
Equation 5
 
G (T )
Gb (T )
Equation 6
Kirchhoff’s radiation law states in essence that the monochromatic emittance is
equal to the monochromatic absorptance for any surface (Kreith and Bohn 1986).
13
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.
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):
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):

   (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
Equation 10
  0.9*0.93848  0.2*(1  0.93848)
  0.86
Equation 11
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 respec14
tively. This characteristic will be employed in the gray surface enclosure model and was
also used by Heaslet and Lomax (1962):
“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 lowtemperature 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. Idealized hemispherical spectral emissivity of black nickel.
15
3.4 Heat Transfer Models
Model 1 – Heat Loss by Radiation to the Environment from a Single Half
Tube
3.4.1
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
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), 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
δ kj
( ε
j=1
j
- Fk-j
1- ε j Q j N
)
=  Fk-jσ(Tk4 - Tj4 )
ε j A j j=1
Equation 12
Q1
=F1-2 .ε1.σ(T14 - T24 )
A1
Equation 13
q1-2 =F1-2 .ε1.σ(T14 - T24 )
Equation 14
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 in-
16
creased 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.
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.
17
Model 2 – Heat Loss by Convection to the Environment from a Single Half
Tube
3.4.2
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) for the turbulent region boundary layer of a vertical cylinder:
NuL 
hL
 0.13(GrL Pr)1/3
kf
Equation 15
The convection heat transfer coefficient h is noted to be independent of length L as
(Kreith and Bohn 1986):
GrL 
g . (Tsurf  Tair ) L3
Equation 16
2
and:
Pr 
c p .
k



Equation 17
The resulting convection coefficient and heat loss are:
  g  (Tsurf - Tair ) 
h  0.13k f 

.


1
3
Equation 18
qsurf -air  h(Tsurf - Tair )
Equation 19
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
18
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 20
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 exists, and this will be
applied to this study. While the McAdams relationship previously considered was not a
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):
1
qcyl  k (Tsurf
 g  (Tsurf - Tair )  4
- Tair ) 
 (- '( , 0))cyl
4 L 2


Equation 21
where:

2( L / ro )
1
4
( g  (Tsurf - Tair )ro3 / (4 2 ))
1
Equation 22
4
( - θ'(ξ,0))cyl is tabulated (Minkowycz and Sparrow 1974).
Similarly, Le Fevre and Ede (Ede 1967) 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 23
The McAdams, Minkowycz and Sparrow and Le Fevre and Ede equations, are
shown in Figure 6 a, b, c for varying tube length. The additional complexity of the nonisothermal nature of the superheater tube is yet to be addressed, however, it may already
19
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) 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,
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,
Table 1) model for n=1 (linear) and Pr = 0.7 (air) provides:
qnon-isothermal  1.49qisothermal
Equation 24
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) and
Yeh and Wiener (1984) 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 num-
20
bers 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 radiation 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.
Figure 6. Convective heat loss from an isothermal tube to the environment at 300 K
(a) L = 1 m
21
Figure 6. Convective heat loss from an isothermal tube to the environment at 300 K
(b) L = 10 m
Figure 6. Convective heat loss from an isothermal tube to the environment at 300 K
(c) L = 50 m.
22
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 caused by 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.
23
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)
Kreith and Bohn 1986, (3) Dittus-Boelter equation from Incropera and De Witt 1990).
•
U m D2
4m
Re D 


 D2 
Pr 
Equation 25
cp
Equation 26
k
4
NuD  0.023Re D5 Pr 0.4
h
Equation 27
k f Nu D
Equation 28
D2
The internal convection coefficient, h, 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, Le Fevre 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 29
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.
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
24
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 30
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 31
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 32
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):
Q
2 Lk (T2  T1 )
ln(r1 / r2 )
Equation 33
For a constant tube conductivity k = 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).
The surface temperature dependence upon steam flows is supported by Yeh and
Wiener (1984):
25
“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  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 non-uniform conditions 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) at Re 
20000, and Black and Sparrow (1967) at 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.
26
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 is 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). Collimated irradiation is assumed to be 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 bounding temperature
27
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 study 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.
Therefore, 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 34
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 35
where:
1
T
 (1  4u ) 4
To
Equation 36
28
 q 
To   1 rad 
 1 . 
N
1
4
Equation 37
r2 To3
kt
Equation 38
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 39
Equation 40
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 project 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
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
Note: Data from Figure 10
29
Tmax/Tmin
The Heaslet and Fuller (1964) model is used with variation in the tube outside diameter as shown in Figure 11. Variation in outside diameter (50, 100 and 150 % of the
50.8 mm outside diameter tube) give qualified support to the use of smaller diameter
tubes to obtain reduced peak tube temperatures. A strong influence exists between the
diameter and the temperature ratio, with smaller diameter tubes providing a more
uniform temperature distribution. However, further investigation is recommended as the
analysis was conducted with a constant wall thickness. As previously noted, the Heaslet
and Fuller model assumes thin walled applications so the reduced diameter at a constant
thickness further weakens the results. The wall thickness as well as other factors,
including the internal convection coefficient will vary with diameter and will be shown
to also have an influence on the temperature distribution. The results are included in
Table 3.
Table 3. Tube metal temperature variation with outside diameter
Tube Outside
Tube Max.
Tube Min.
Tmax/Tmin
Diameter
Temperature
Temperature
mm (%)
K (C)
K (C)
25.4 (50)
2088 (1815)
2024 (1751)
1.03
50.8 (100)
2262 (1989)
1832 (1559)
1.23
76.2 (150)
2416 (2143)
1569 (1296)
1.54
Note: Data from Figure 11
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 12 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.
30
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.
Figure 9. Circumferential temperature distribution with varying conductivity using Heaslet and
Fuller (1964) data
31
Figure 10. Circumferential temperature distribution with varying thermal conductivity
32
Figure 11. Circumferential temperature distribution with varying outside diameter
33
Figure 12.
Circumferential temperature distribution with varying internal absorptivity and
emissivity
34
3.4.5
Model 5 - Temperature Distribution within a Thick Shell with Uniform and
Parallel Incident Radiation and Convection
Mackowski (2011) provides an analytical model for a long, annular cylinder with
temperature variation in both r and , convection on the internal and external surfaces,
and the outside of the pipe is exposed to a collimated source of thermal radiation, as
shown in Figure 13.
Figure 13. 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) 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:
35
Equation 41
T
(T  T,2 )k
Equation 42
 rad qr1
r
r
r1
Equation 43
a
r2
r1
Equation 44
Bi1 
h1r1
k
Equation 45
Bi2 
h2 r2
k
Equation 46
T ,1 
(T,1  T,2 )k
Equation 47
 rad qr1
g n (r )  (r ) n  a 2 n
n  Bi2
(r )  n
n  Bi2
Equation 48

n  Bi2 
g n (1)  n 1  a 2 n

n  Bi2 

Equation 49
The Mackowski model differs from that of the current project in that the radiative
emissions 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 qualifying 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 cooling effect. The project uses a selective surface with a low emittance
(0.09) so the simplification is acceptable.
Figure 14 is presented to show agreement with the results of Mackowski (2011, figure 4.16(b)), while Figure 15 uses project specific data. Figure 16 and Figure 17
compare the influence of wall thickness and internal convection coefficient of the tube
temperature distribution.
36
The project specific and off design tube wall thickness temperature profiles of Figure 15 and Figure 16 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). 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):
Bi 
 
r
hLc h 2

k
k
Equation 50
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 % (Ozisik 1980):
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 16, 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.
37
Variations of the tube wall thickness are presented in Figure 15 (design conditions)
and Figure 16 (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 Nu constant for this analysis. The peak metal temperature was strongly dependent upon the wall thickness, as shown in Table 4. 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 4. Tube metal temperature variation with tube wall thickness
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
Note: Data from Figure 15 and Figure 16
Variations of the internal convection coefficient are presented in Figure 15 (design
conditions) and Figure 17 (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 5. A
reduced internal convection coefficient, resulting in higher peak metal temperatures, can
result from various operating conditions in the superheater. These include typical plant
38
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 5. Tube metal temperature variation with internal convection coefficient
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
Note: Data from Figure 15 and Figure 17
The application of the Mackowski model to the central receiver steam generator requires an insulated surface in the rear of the tube where the model includes convection.
The natural convection of the external surface has been shown to have a very low Bi, so
it has a relatively small influence on the cooling of the tube. However, it is noted in
Table 4 and Table 5 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 6. 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 4 and Table 5 would indicate the possible importance of variable conductivity.
A brief review of the temperature dependence of thermal conductivity
properties of steel alloys is given in Table 6, showing a nominal trend of reducing
conductivity with increasing alloy content and/or temperature.
39
Table 6. 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), Material group A.
2
ASME 2010 (b), Material group D.
3
ASME 2010 (b), Material group F.
4
ASME 2010 (b), Material group J.
5
ASME 2010 (b), 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.
40
Figure 14. Non-dimensional isotherms in a tube using Mackowski 2011 data.
41
Figure 15. Isotherms in a tube at design conditions
42
Figure 16. Isotherms in tubes of varying wall thickness.
(a) t = 0.63 mm (10 % of design)
43
Figure 16. Isotherms in tubes of varying wall thickness
(b) t = 3.15 mm (50 % of design)
44
Figure 16. Isotherms in tubes of varying wall thickness
(c) t = 9.45 mm (150 % of design)
45
Figure 17. Isotherms in tubes with varying internal convection coefficient.
(a) h = 2360 W/(m2.K) (50 % of design),
46
Figure 17. Isotherms in tubes with varying internal convection coefficient.
(b) h = 7080 W/(m2.K) (150 % of design)
47
3.4.6
Model 6 – Heat Transfer by Mutual 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 18.
Figure 18. 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 19.
48
Figure 19. 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):
  kj
1  j
 Fk  j


j
j 1   j
N
N

4
 q j    kj  Fk  j   T j  qsol  k
j 1

Equation 51
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 52
The assumption of a diffuse gray surface indicates that the directional emissivity
and directional absorptivity 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).
49
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 3.4.6.6. 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.
The incident radiation multiplied by the absorptivity and projected area is assumed
to be conducted away by the tube. Hence the solar absorptivity is only used for the nonradiation term in the gray enclosure model. All reflection of the solar irradiation is
neglected, which is a reasonable assumption for high absorptivity materials typically
used for solar collectors but less so at reduced absorptivity. The cylindrical surface is
then modeled as diffuse gray emitters in the infrared spectral region due to the surface
temperature. This procedure is ideally suited to a flat surface as any reflection would not
interact with other surfaces.
The results of the model to project specific conditions are shown in Figure 20. The
temperature range of approximately 1500 K to 1950 K do not agree with the expected or
realistic values for this application, as they are in excess of the 1497 K equivalent
ambient temperature. Furthermore, the temperature output consists of complex numbers,
only the real part is plotted in Figure 20.
The previously presented Mackowski model of section 3.4.5 provided a peak temperature of 1017 K. Furthermore, the Heaslet and Fuller model of section 3.4.4, with the
very conservative assumption of no cooling by the steam, provided a peak temperature
of 2262 K.
50
Figure 20. Surface temperature distribution on adjacent cylinders with mutual irradiation
3.4.6.2 Open Surface
The open surface through which the collimated radiation enters the enclosure is treated
as an imaginary black surface at the ambient temperature with the solar radiation passing
through it. An alternative method as proposed by Sparrow and Cess (1967) is:
“One or more of the surfaces of the enclosure may not be material surfaces – for
instance, on open window. Each of such surfaces may be assigned equivalent radiation properties of an equivalent black-body temperature that corresponds to
51
the rate at which radiant energy passes through the fictitious surface into the enclosure.”
In both methods, the collimated nature of the solar irradiation is accounted for in the
projected area distribution of energy over the cylindrical surfaces.
The equivalent black body temperature is determined by equating the absorbed solar
energy with that emitted by a black body. The surface is then acting as a body without
influence on the energy passing through. However, the black body is a diffuse emitter,
in place of the collimated irradiation. The absorbed solar energy and equivalent emitted
energy are:
Qabsorbed  qsol A
Equation 53
Qemitted   TB4 A
Equation 54
Equating the above, the equivalent blackbody temperature is:
q  
TB   sol 
  
1
4
Equation 55
For project specific data of qsol = 300 kW/m2 and  = 0.95, the equivalent black
body temperature of the open surface is TB = 1497 K.
The use of an equivalent black body as the open surface allows the gray enclosure
equation to be used without the direct use of the solar irradiation term. The gray enclosure equation becomes:
  kj
1  j
 Fk  j


j
j 1   j
N
N

4
 q j    kj  Fk  j   T j
j 1

Equation 56
The gray enclosure equation is applied to a highly simplified model of three isothermal flat surfaces as shown in Figure 21. When considering an energy balance in
which the energy entering the open surface 3 is equivalent to the sum of those leaving
surfaces 1 and 2, an imaginary temperature results. The modified approach to improve
the poor results of section 3.4.6.1 has not achieved a useful result.
52
Figure 21. Simplified gray enclosure with the open surface as a black body whose temperature
corresponds to the irradiation
3.4.6.3 View Factors
Hottel’s crossed strings method is employed to determine the view factors 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. Hottel’s crossed string method for two
arbitrary surfaces, as shown in Figure 22, is (Modest 1993, 178):
Fi  j 
( Lbc  Lad )  ( Lac  Lbd )
2 Li
Equation 57
53
Figure 22. 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):
Ai Fi  j  Aj Fj i
Equation 58
For areas of equal depth, the reciprocity relation becomes:
Li Fi  j  L j F j i
Equation 59
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 60
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):
54
N
Fk 1  Fk 2  Fk 3  ...  Fk k  ...  Fk  N   Fk  j  1
Equation 61
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 23. The view factors from the isothermal strips of cylinder
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.
3.4.6.4 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 62
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 23. 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.
55
Figure 23. View factor limitations of isothermal strips.
3.4.6.5 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):
“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.
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
56
a black surface at the cavity temperature, having the same area as the opening (Ozisik
1973), 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) 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 3.4.6.6.
The influence of the cavity on surface temperatures can be seen in Figure 24. As
the quantity of isothermal strips in the numerical model is increased, 100 are used in
Figure 24, the tangent tube position shows a rise in temperature to be in excess of the
57
tube crown temperature. Figure 20 with the same input but only 20 isothermal strips did
not show this increase. As noted in section 3.4.6.1, the calculated temperatures are
imaginary with only the real component plotted. Also, the real component is in excess
of the equivalent black body temperature of the ambient surface, so it is presented with
reservation.
The validity of the increasing isothermal strips in the cavity region is idealized but
not necessarily realistic. The circumference of the quarter tube for the project 50.8 mm
diameter tube is 39.8 mm; therefore, 100 isothermal strips are 0.398 mm in length. The
influence of surface roughness, oxide layers and foreign matter on commercial tubes
presents a realistic impairment to the calculation using finer isothermal strips within the
cavity. Furthermore, the use of a concave, smoothly blended weld in the cavity removes
the need for the cavity results.
58
Figure 24. Cavity influence on surface temperatures
3.4.6.6 Off Normal Incident Radiation Absorptance
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):
59
“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 25 (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 been assumed to be 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, as it then enters the cavity between the tubes for further
possible reflection and absorption.
The characteristics of the cavity were noted in
section 3.4.6.5. 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.
60
Figure 25.
Directional absorptance of a blackened surface for artificial sunlight transmitted
through glass (Howell, Bannerot and Vliet 1982, figure C-9)
61
4. Results
A tube with a uniform circumferential temperature was assumed for radiation and
convection heat losses to the environment, models 1 and 2 respectively. Furthermore, a
uniform bore heat flux was assumed when the tube cooling by the internal cooling flow
of superheated steam was considered in model 3.
The collimated solar irradiation onto a cylindrical surface was shown to have a maximum flux at the crown of the tube, normal to the irradiation, and diminishing to nil at
the tube tangent where the surface normal is perpendicular to the irradiation. The
circumferential variation in irradiation, and hence, tube temperature, was analyzed in
further thermal models. A thin walled tube with radiation heat transfer at internal and
external surfaces (model 4), a thick walled thermally conducting tube with internal and
external convection (model 5) and the heat transfer by mutual irradiation by adjacent
tubes (model 6) were considered.
The heat loss by radiation to the environment of model 1, assuming a surface temperature of 1000 K and emittance of 0.09, was approximately 5000 W/m2. The 5000
W/m2 heat loss corrected for the projected area is increased by a factor of /2, to 7854
W/m2. For a solar irradiation of 300 kW/m2 onto the project area, the heat loss is
equivalent to 2.6 % of the incoming irradiation.
The heat loss by convection to the environment of model 2 was shown to be sensitive to the parameters used. In particular, the long length of the tube to its relatively
small diameter made it unsuitable for a flat plate assumption. Furthermore, the aspect
ratio meant that a small deflection of the bulk fluid resulted in the heated, rising fluid
moving away from the upper part of the cylinder. It is recommended that future work
consider the convective loss from the entire solar receiver rather than the sum of the
individual tubes. The aspect ratio of the solar receiver is more applicable to the available convection models than individual tubes.
Of the convective heat loss models
considered, 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 radiation 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.
62
The heat loss by forced convection to the internal fluid of model 3 is the intended
heat transfer mechanism when applied to a steam generator. The heat “loss” to the
working fluid is the means of providing the bulk of the tube cooling and a heat gain by
the working fluid. The previously discussed losses due to thermal radiation to the
environment and convection cooling by the ambient air are ideally minimized in order to
maximize the plant output. The internal convection coefficient, with the assumption of a
uniform heat flux at the bore of the tube, was calculated to be 4720 W/(m2.K) at the
superheater outlet. This assumption is not accurate for this application, as heating occurs
only on the face of the tube exposed to solar irradiation. Furthermore, as will be discussed in the following models, the irradiation of the exposed face is not uniform due to
the collimated radiation onto a cylindrical surface. Further investigation of the influence
of a non-uniform heat flux at the bore of the tube upon the convection coefficient is
recommended.
The temperature distribution within a thin shell exposed to collimated irradiation of
model 4 used the work of Heaslet and Fuller (1964). The original application was for
outer space so convection, internal and external, is not considered. Therefore, the model
cannot be directly applied to the current application but the characteristics of the results
are applicable. It was shown that the peak metal temperature that occurs at the crown of
the tube is strongly influenced by the tube conductivity and diameter. In practice, the
designer has limited influence on the conductivity of the material selected; however,
tube diameter is an important step in the design of a solar receiver steam generator. The
results indicate that a smaller diameter tube will have a reduced peak temperature, hence,
an improved design. Further work on the investigation of the relationship between tube
diameters and temperature distribution is recommended.
The temperature distribution within a thick shell exposed to collimated irradiation of
model 5 used the work of Mackowski (2011). Model 5 is considered the most reliable of
those presented thus far as it considers the combined effects of solar irradiation, internal
convection and the two dimensional conductivity of the thick walled tube. The temperature variation within the tube wall was shown to be influenced by the material
conductivity, wall thickness and internal convection coefficient. The designers greatest
influence may be with the internal convection coefficient. It is important to consider off-
63
design cases where the coefficient is likely to vary considerably from the design point
due to factors such as plant load and transient conditions.
The heat transfer by mutual irradiation of adjacent tubes was considered in model 6.
The previous models did not consider the thermal radiation exchange between the
surfaces of the adjacent tubes and the influence of it upon the temperature distribution.
A gray enclosure model was used by approximating the cylindrical surfaces as flat
isothermal strips and an imaginary surface for the ambient forming the complete enclosure. The view factors between the enclosure surfaces were calculated by Hottel’s
method. The surface representing the ambient was assumed to be a black body with a
temperature equivalent to that of the incident solar irradiation as a means of defining a
single temperature within the enclosure. The temperatures of the surfaces forming the
cylinders were to be determined. The results obtained were inconclusive as the calculated temperatures were imaginary numbers. The reason for the poor results is unknown as
no fault with the methodology could be identified. Additional work is recommended.
64
5. Conclusions
The Mackowski model (section 3.4.5) provided the most complete temperature distribution with consideration of external collimated radiation, internal and external
convection, and thick walled tube conduction. The results did not account for other
influences such as a more diffuse solar irradiation due to the array of heliostats, mutual
irradiation of the adjacent tubes and non-uniform circumferential heat flux at the tube
interface with the steam. The diffuse solar irradiation and the influence of the radiation
exchange with the adjacent tubes is expected to reduce the Tmax/Tmin ratio, thus providing
a conservative model for the calculation of tube metal temperatures. The non-uniform
circumferential heat flux to the steam requires further investigation before its influence
on the tube temperature distribution can be determined.
The ideal tube temperature distribution for the design of a solar receiver steam generator is uniform. The variation from the ideal results in a localized maximum wall
temperature that defines the tube materials allowable stress, thus dictating material
selection and tube wall thickness.
Factors that tend to increase the Tmax/Tmin ratio include the directional properties of
radiation absorptance favoring those normal to the incoming flux and the relatively poor
conducting, thick walled tube material for this application.
Conversely, the idealized Tmax/Tmin = 1 ratio is approached by the use of a concave,
smoothly blended weld joining the tangent tubes acting as a heat sink fin nominally
normal to the irradiation. This increases the temperature at the tube tangent that is
otherwise in the minimum temperature region. Mutual irradiation of the adjacent tubes
is also predicted to cool the tube crown while maintaining or increasing that in the tube
tangent region. The supply of diffuse radiation from the array of heliostats rather than a
collimated source will also provide a better flux distribution to the tube surfaces away
from the tube crown.
The gray enclosure model (section 3.4.6) included the thermal radiation exchange
between the surfaces of adjacent tubes. However, it generated unexplained high tube
surface temperatures. Further work is required to validate the model, and then to expand
it to include the second tube at ambient conditions, thus providing a comparison of
temperatures with and without mutual irradiation of the adjacent tubes.
65
The current project identified several areas for future work:
a) The influence of circumferential variation of tube temperature on the internal
convection coefficient.
Previous work included Sparrow and Krowech
(1977); however, it could not be extrapolated to suit the current application.
b) The effect of diffuse radiation on the temperature distribution. Instead of
considering collimated solar irradiation, future work may include modeling a
more diffuse pattern due to the array of heliostats providing the incident
flux. Diffuse irradiation was assumed by Yeh and Wiener (1984) for a different solar receiver design.
c) Use of evaporator screen tubes in front of the superheater panels. Yeh and
Weiner (1984) solar receiver design used evaporator screen tubes in front of
the superheater panels as a means to control the incident flux and hence the
superheater metal temperatures. However, they did not consider the mutual
irradiation between tubes.
The correct estimation of circumferential temperature distribution within tubes of a
solar receiver steam generator is an important design step. Furthermore, an understanding of all the factors that influence the maximum to minimum temperature ratio within
the tube wall will allow its minimization and hence the optimization of the system
design.
66
6. References
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Thermal Systems.” Advances in Heat Transfer, Vol. 18, 1-86.
Apley, W. J., and S. P. Bird, 1984, “Assessment of Generic Solar Thermal Systems for
Large Power Applications.” ASME Journal of Solar Energy Engineering, Vol. 106,
22-28.
ASME, 2010 (a), ASME Boiler and Pressure Vessel Code, I, Rules for Construction of
Power Boilers, ASME, New York.
ASME, 2010 (b), ASME Boiler and Pressure Vessel Code, II, Part D, Properties (Metric): Materials, ASME, New York.
Attaway, S., 2009, Matlab: A Practical Introduction to Programming and Problem
Solving, Butterworth-Heinemann, Boston.
Black, A. W., E. M. Sparrow, 1967, “Experiments on Turbulent Heat Transfer in a Tube
with Circumferentially Varying Thermal Boundary Conditions.” ASME Journal of
Heat Transfer, Vol. 89, 258-268.
Boulden, L. L., 2012, “A Guide to Successful CSP.” Solar Power World, February, 3437.
Chung, B. T. F., and B. X. Zhang, 1991, “Optimization of Radiating Fin Array Including
Mutual Irradiations Between Radiator Elements.” ASME Journal of Heat Transfer,
Vol. 113, 814-822.
Duffie, J. A., and W. A. Beckman, 2006, Solar Engineering of Thermal Processes, 3rd
ed., John Wiley & Sons, Hoboken.
Ede, A. J., 1967, “Advances in Free Convection.” Advances in Heat Transfer, Vol. 4, 164.
Gartner, D., K. Johannsen, H. Ramm, 1974, “Turbulent Heat Transfer in a Circular Tube
with Circumferentially Varying Thermal Boundary Conditions.” International
Journal of Heat and Mass Transfer, Vol. 17, 1003-1018.
67
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7. Appendix
71
Model 1 - Heat Loss by Radiation to the Environment from a Single
Half Tube.
I
Model 2 - Heat Loss by Convection to the Environment from a Single
Half Tube.
II
Model 3 - Heat Loss by Forced Convection to the Internal Fluid.
III
Model 4 - Temperature Distribution within a Thin Shell with Uniform
and Parallel Incident Radiation.
IV
Model 5 - Temperature Distribution within a Thick Shell with Uniform
and Parallel Incident Radiation and Convection.
V
Model 6 - Heat Transfer by Mutual Irradiation of Adjacent Tubes.
VI