1. Introduction
Solar Thermal Power Plants (STPP) can be used to generate electricity in a manner
similar to that of a traditional fossil fuel plant in that superheated steam is used to drive a
turbine-generator. The inherent disadvantage of solar plants is that the energy source,
solar radiation, is cyclic (daily and seasonally) and intermittent. This may be addressed
by the use thermal energy storage to allow the plants electricity generation to follow the
demand rather than the instantaneous solar flux.
The use of molten salt as the heat transfer fluid within a solar thermal power plant is
investigated. The molten salt provides the energy link between the concentrated solar
flux at the receiver and the steam generated for the turbine generator as shown in Figure
1. Molten nitrate salt is used as both the heat transfer fluid and the storage medium, with
heat gained from the concentrated solar flux at the receiver, and heat rejected at the
steam generator. The steam generator supplies steam at a temperature close to the peak
salt temperature of 565oC, this allows operation of the Rankine cycle turbine at conditions similar to a conventional fossil fuel plant. The peak salt temperature given is that
of the Solar Tres 15 MWe plant in Spain that incorporates 16 hours of full power storage
requirements [Mills 2004]. Both salt storage tanks remain above the freezing point of
the molten salt.
1
Figure 1. Schematic of the Solar Tres solar thermal power plant with molten salt energy storage
[Medrano et al 2010]
The use of molten nitrate salt (60%w NaNO3, 40%w KNO3) is investigated with respect to its heat transfer characteristics at the high Reynolds and Nusselt numbers found
in the receiver of a practical solar thermal power plant. A review of the literature has
provided data for this and other molten salts but only below the required Reynolds and
Nusselt number ranges.
The validity of the assumption of the heat transfer fluids constant properties, in particular dynamic viscosity, is reviewed with regard to convective heat transfer
calculations when applied to STPP applications. The hydraulic and thermal boundary
layers are considered for flow in both tubes and annuli. The velocity and temperature
profiles are developed to determine the sensitivity to the constant property assumption.
2
2. Nomenclature
Symbol
Unit
Description
A
m2
Area
Bi
-
Biot number
cp
J.kg-1.K-1
Specific heat capacity
D
m
Diameter
f
-
Fanning friction factor
h
W.m-2.K-1
Convection heat transfer coefficient
h
J.kg-1
Specific enthalpy
k
W.m-1.K-1
Thermal conductivity
L
m
Length
Nu
-
Nusselt number
Pr
-
Prandtl number
q
W
Heat transfer rate
q”
W.m-2
Heat flux
r
m
Radius
Re
-
Reynolds number
T
o
Temperature
V
m.s-1
Velocity
V
m3
Volume
e
mm
Absolute surface roughness
m
Pa.s
Dynamic viscosity
n
m2.s-1
Kinemetic viscosity
r
kg.m-3
Density
Roman Symbols
C or K
Greek Symbols
3
Miscellaneous
r, q, z
-
Cylindrical coordinates
x, y, z
-
Cartesian coordinates
c
-
Characteristic
D
-
Diameter
e
-
Electric
f
-
Fluid
i
-
Inside
l
-
Liquid
o
-
Outside
s
-
Smooth, solid, surface
th
-
Thermal
w
-
Weight, wall
Subscripts
Abbreviations
DNI
Direct Normal Irradiance
HTF
Heat Transfer Fluid
MS
Molten Salt
STPP
Solar Thermal Power Plant
4
3. Thermodynamic Properties of Molten Salt
The molten salt studied is composed of 60%w NaNO3/40%w KNO3 as used in the
Solar Thermal Power Plant (STPP) Solar Two. Solar Two was located in Daggett
California and operated between 1995 and 1999 by a consortium of Southern California
Edison and the US Department of Energy. The thermal storage system was designed to
supply full steam generation for three hours, a capacity of 105 MW.hth or 35 MW. This
mixture melted at 207oC, was thermally stable to about 600oC, and offered a favorable
combination of high density, low vapor pressure, moderate specific heat, low chemical
reactivity, and low cost [Medrano et al 2010].
The thermodynamic properties of molten salt are identified in order to be applicable
to a solar thermal power plant operating characteristics identified in Section 4.
Thermodynamic properties of steam and water are provided in order to provide relative values for comparison.
The thermodynamic properties of 60%w NaNO3/40%w KNO3 as established by
Sandia National Laboratories are [Zavoico 2001]:

Melted salt can be used over a temperature range of 260 oC to 621 oC.

As temperature decreases, it solidifies at 221 oC and starts to crystalize at
238 oC.

Fluid salt property formulas as a function of temperature are valid for 300 oC
< T < 600 oC.

Density (kg.m-3) as a function of temperature (oC):
𝜌 = 2090 − 0.636 ∗ 𝑇
5
Equation (1)
1950
Density (kg/m3)
1900
1850
1800
1750
1700
1650
250
300
350
400
450
500
550
600
650
Temperature (C)
Figure 2. Density versus Temperature for liquid 60%w NaNO3/40%w KNO3

Specific heat (J.kg-1.K-1) as a function of temperature (oC):
𝑐𝑝 = 1443 + 0.172 ∗ 𝑇
Equation (2)
1550
Specific Heat (J/(kg.K))
1540
1530
1520
1510
1500
1490
250
300
350
400
450
500
550
600
650
Temperature (C)
Figure 3. Specific Heat at Constant Pressure versus Temperature for liquid 60%w NaNO3/40%w
KNO3
6

Dynamic viscosity (Pa.s) as a function of temperature (oC):
𝜇 = 2.2714 ∗ 10−2 − 1.20 ∗ 10−4 ∗ 𝑇 + 2.281 ∗ 10−7 ∗ 𝑇 2
− 1.474 ∗ 10−10 ∗ 𝑇 3
Equation (3)
0.0035
Dynamic Viscosity (Pa.s)
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
250
300
350
400
450
500
550
600
650
Temperature (C)
Figure 4. Dynamic Viscosity versus Temperature for liquid 60%w NaNO3/40%w KNO3

Thermal conductivity (W.m-1.K-1) as a function of temperature (oC):
𝑘𝑓 = 0.443 + 1.9 ∗ 10−4 ∗ 𝑇
7
Equation (4)
Thermal Conductivity (W/(m.K))
0.5600
0.5500
0.5400
0.5300
0.5200
0.5100
0.5000
0.4900
250
300
350
400
450
500
550
600
650
Temperature (C)
Figure 5. Thermal Conductivity versus Temperature for liquid 60%w NaNO3/40%w KNO3
The Prandtl number (dimensionless) is a function of dynamic viscosity (Pa.s), specific heat at constant pressure (J.kg-1.K-1), and thermal conductivity (w.m-1.K-1). It is
calculated using Zavoico [2001] data:
𝑃𝑟 =
𝜇. 𝑐𝑝
𝑘𝑓
8
Equation (5)
12.0
Prandtl Number (-)
10.0
8.0
6.0
4.0
2.0
0.0
250
300
350
400
450
500
550
600
650
Temperature (C)
Figure 6. Prandtl Number versus Temperature for liquid 60%w NaNO3/40%w KNO3
The kinematic viscosity (m2.s-1) is a function of dynamic viscosity (Pa.s) and density (kg.m-3) and calculated using Zavoico [2001] data:
𝜇
𝜈=
𝜌
9
Equation (6)
2.0E-06
Kinematic Viscosity (m2.s-1)
1.8E-06
1.6E-06
1.4E-06
1.2E-06
1.0E-06
8.0E-07
6.0E-07
4.0E-07
2.0E-07
0.0E+00
250
300
350
400
450
500
550
600
650
Temperature (C)
Figure 7. Kinematic Viscosity versus Temperature for liquid 60%w NaNO3/40%w KNO3
It is noted that the thermodynamic properties of the molten salt, r, cp, m, kf, Pr and n
all vary with temperature as shown in Figure 2 through Figure 7. The variations, as
shown in Table 1, are typical of liquids as described by Kays et al [2005]:
For most liquids the specific heat and thermal conductivity are relatively independent of temperature, but the viscosity decreases very markedly with temperature.
This is especially so for oils, but even for water the viscosity is very temperature
dependent. The density of liquids, on the other hand, varies little with temperature.
The Prandtl number of liquids varies with temperature in much the same manner as
viscosity.
Methods available to account for property variations include the reference temperature and property ratio methods Kays et al [2005]. The reference temperature method
evaluates the properties at a chosen temperature, typically the mean of the surface and
fluids mixed mean temperature. The property ratio method evaluates all properties at the
fluids mixed mean temperature, and then those variable properties are also evaluated at
the surface temperature. A function of the variable properties ratio is then used to
correct those evaluated at the mixed mean temperature. A means for correcting for the
constant property assumption becomes increasingly important when high temperature
differentials are present between, say, the tube wall of a STPP exposed to a high solar
flux and the heat transfer fluid. Both methods will be used in this project.
10
### comment on constant property assumption used to de-couple energy equation
from continuity and moment equations – find source. See page 68 of white ####
Table 1. Property changes over operating temperature range
Property
Symbol
Units
Value at
Value at
Change from 300oC
300oC
600oC
to 600oC
Density
r
kg.m-3
1899
1708
-10%
Specific heat
cp
J.kg-1.K-1
1495
1546
+3.4%
Dynamic
m
Pa.s
0.0033
0.0010
-70%
kf
W.m-1.K-1
0.500
0.557
+11%
-
9.8
2.8
-71%
m2.s-1
1.7*10-6
5.8*10-7
-66%
viscosity
Thermal
conductivity
Prandtl
number
Kinematic
viscosity
𝑃𝑟 =
𝜇. 𝑐𝑝
𝑘𝑓
𝜈=
𝜇
𝜌
The thermodynamic properties are those of Zavoico [2001] as they are specific to
liquid 60%w NaNO3/40%w KNO3. Properties at other compositions as well as other
salts are available in Janz et al [1972].
11
4. Solar Thermal Power Plant Characteristics
As with all large power plants, the location of a STPP is dependent upon many factors, one of which is the local Direct Normal Irradiance (DNI). An evaluation of plant
sites by Avila-Marin et al [2013] included Seville, Spain; Daggett, USA; and Carnarvon,
South Africa. The characteristics of the sites are provided in Table 2.
Table 2. STPP typical location characteristics [Avila-Marin et al 2013]
Site
Seville
Daggett
Carnarvon
Spain
USA
South Africa
Latitude (o)
37.42
34.87
-30.97
Longitude (o)
-5.9
-116.8
22.13
Altitude (m)
31
588
1309
Design point DNI (W.m-2)
900
950
1000
Annual energy DNI (kW.h.m-2.y-1)
2089
2791
2995
The high temperature and exergy of the source of solar radiation makes it a valuable
energy resource. However, the low flux density at the earth’s surface as shown in Table
2 makes it a poor candidate for power generation applications when used without optical
concentration. The use of optical concentrations allow the generation of HTF temperatures useful for thermodynamic cycles
A concentrated solar flux density of nominally 1 MW.m-2 was used at the Solar Two
STPP with operational variations between 240 kW.m-2 at the outlet salt temperature of
565oC, and 850 kW.m-2 at the inlet salt temperature of 290oC [Vant-Hull 2002]. The
Solar Two flux densities will be used as the basis for this report. Note that the central
receiver flux density is three orders of magnitude greater than the DNI.
The design of central receiver tubes for STPP with molten salt HTF requires, among
others, consideration of the operating temperatures, corrosive characteristics of the fluid,
and thermal cycling. The cycling of Solar Two STPP with a 30 year design life includes
1.0x104 deep temperature cycles (nightly draining of the central receiver and cooling to
ambient temperature), and 3.0x104 shallow temperature cycles (cloud transients etc.).
For this service the central receiver tubes are 21 mm outside diameter, 1.25 mm wall
12
thickness, and 316 stainless steel material [Vant-Hull 2002].
The Solar Two tube
characteristics will be used as the basis for this report.
The velocity of the fluid within the tube for a heat exchanger is an optimization due
to increasing velocity producing the counteracting effects of increased pressure loss and
heat transfer coefficient. The average molten salt velocity used at Solar Two is 2.92 m.s1
[Liao et al 2014]. Variation in velocity may be used by STPP to control the salt outlet
temperature and absorbed flux [Vant-Hull 2002], therefore, a range of average molten
salt velocities from 1.0 to 5.0 m.s-1 is used as the basis for this report.
The physical characteristics of the STPP used for this project are given in Table 3.
Add fluid temperature for molten salt, and flux/molten salt temperature matches.
Table 3. STPP project physical characteristics
Project Characteristic
Symbol
Units
Value
Tube material
-
-
316 stainless steel
Tube outside diameter
Do
mm
21.0
mm
1.25
Tube wall thickness
Tube inside diameter
Di
mm
18.5
Mean fluid velocity
V
m.s-1
1.0 ≤ V ≤ 5.0
Mean fluid temperature
T
o
290 ≤ T ≤ 565
Concentrated solar flux
q”
W.m-2
240 ≤ q” ≤ 850
-
240 W.m-2 at 565oC
C
Coincident solar flux and -
850 W.m-2 at 290oC
fluid temperature
The thermal conductivity (W.m-1.K-1) of the 316 stainless steel tubes is a function of
temperature (oC) and is calculated using ASME [2010] data as:
𝑘𝑠 = 0.01431 ∗ 𝑇 + 13.96
13
Equation (7)
Thermal Conductivity (W/(m.K))
30.0
25.0
20.0
15.0
10.0
5.0
0.0
0
100
200
300
400
500
600
Temperature (oC)
Figure 8. Thermal Conductivity versus Temperature for 316 stainless steel [ASME 2010]
14
700
800
5. Heat Transfer Characteristics of Molten Salt
Convective heat transfer characteristics for fully developed (hydro dynamically and
thermally) turbulent flow inside a round tube is defined by the Nusselt number (NuD).
The Nusselt number is a function of the Reynolds (ReD) and Prandtl (Pr) numbers.
The Renolds number for flow inside a tube is:
𝑅𝑒𝐷 =
𝑉. 𝐷𝑖
𝜈
Equation (8)
The Reynolds number range for this project, with the follow conditions:
1.0 ≤ 𝑉 ≤ 5.0 𝑚. 𝑠 −1
𝐷𝑖 = 18.5 ∗ 10−3 𝑚
5.8 ∗ 10−7 ≤ 𝜈 ≤ 1.7 ∗ 10−6 𝑚2 . 𝑠 −1
is:
1.1 ∗ 104 ≤ 𝑅𝑒𝐷 ≤ 1.6 ∗ 105
The upper limit of ReD = 1.6*105 occurs at the upper velocity (V = 5.0 m.s-1) and
lowest kinematic viscosity (n = 5.8*10-7) corresponding to 600oC. Considering only the
600oC conditions but at the lower velocity (V = 1.0 m.s-1), the Reynolds number is
reduced to 20% or 3.2*104.
The project Prandtl and Reynolds number ranges are shown in Table 4. The flow is
fully turbulent for all cases with ReD > 4000.
The upper range of ReD is, in part, the motivation for this project. Published data
for heat transfer applications with molten salts are limited to ReD ≤ 50000. A survey of
available experimental data for molten salts was performed by Wu et al [2012] and
summarized in Table 5. The published data includes Prandtl numbers that cover the
range of this project, however, the Reynolds number maximum is nominally 30% of that
required for this project.
An aim of this project is to investigate the available Nusselt number correlations.
However, to facilitate order of magnitude calculations a preliminary Nusselt number
range is determined. A widely used Nusselt number correlation is the Dittus-Boelter
[Winterton 1998] equation:
𝑁𝑢 = 0.0243𝑅𝑒 0.8 𝑃𝑟 0.4
15
Equation (9)
The Nusselt number range for this project is of the order of 62 (Re = 1.1*104, Pr =
2.8) to 881 (Re = 1.6*105, Pr = 9.8). It is noted that the high heat flux, high Reynolds
number cases that are of interest for this project have a Pr = 2.8 at 600 oC, therefore, the
Nusselt number of greatest interest is nominally 534 (Re = 1.6*105, Pr = 2.8). Consideration will also be given to the high heat flux, low Reynolds number case at full
temperature (600oC). The resulting internal convection coefficients are calculated by:
ℎ=
𝑁𝑢. 𝑘𝑓
𝐷𝑖
Equation (10)
The resulting internal convection coefficients range is 1866 to 26525 W.m-2.K-1 (kf
= 0.557 W.m-1.K-1, Di = 18.5 mm). The convection coefficient corresponding to the
Nusselt number of 534 is 16077 W.m-2.K-1.
The STPP designers choice of the bulk molten salt tube velocity, and hence Reynolds number, is often a consideration of pressure loss and its impact on the pump energy
consumption. However, for heat transfer applications the Nusselt number is also a
function of the Reynolds number. It will be shown that the velocity selection and its
ultimate impact on the Nusselt number and convective heat transfer coefficient strongly
influences the tube wall to bulk fluid temperature differential. For this reason, the
velocity selected impacts the validity of the constant fluid property assumption.
The nominal Nusselt and convection coefficient values for this project are given in
Table 6.
Table 4. Prandtl and Reynolds number ranges for project
Salt
Prandtl range
Reynolds range
60%w NaNO3/40%w KNO3
2.8 < Pr < 9.8
1.1*104 ≤ Re ≤ 1.6*105
For a fluid temperature of 600oC:
Re = 1.6*105 at 5.0 m.s-1
Re = 3.2*104 at 1.0 m.s-1
Table 5. Prandtl and Reynolds number ranges for published molten salt data [Wu et al 2012]
Salt
Prandtl range
Reynolds range
Molten Hitec salts
3.3 < Pr < 9.1
2342 < Re < 33493
16
Lithium Nitrate
10.5 < Pr < 15.3
4104 < Re < 9536 and
18182 < Re < 46130
Flinak
1.6 < Pr < 4
2428 < Re < 9536
LiF-BeF2-ThF4-UF4
6.6 < Pr < 14.2
1542 < Re < 14210
NaBF4-NaF
4.89 < Pr < 5.64
5104 < Re < 44965
Table 6. Nusselt number and convection coefficient nominal ranges for project
Characteristic
Units
Range
Primary value
Nusselt number
-
62 < Nu < 881
530 at 5.0 m.s-1
150 at 1.0 m.s-1
Convection coefficient
W.m-2.K-1
1866 < h < 26525
16000 at 5.0 m.s-1
4400 at 1.0 m.s-1
17
6. Tube Surface Roughness
As noted previously, the flows considered are fully turbulent and of the order of ReD
≤ 1.6*105. For this application, the main advantage of turbulent over laminar flow is the
increased convective heat transfer at the expense of increased pressure loss. Similarly,
the surface roughness contributes to the heat transfer coefficient in turbulent flow in that
it increases the surface area changes the turbulence pattern close to the wall [Bhatti Shah
1987]. The effect of surface roughness on the heat transfer of turbulent flow is considered.
For the 18.5 mm inside diameter 316 stainless steel tubes considered for this project,
the surface absolute roughness is assumed to be that of drawn pipe, 0.0015 mm [Bhatti
Shah 1987]. However, &&&&&&&&& change in roughness due to corrosion? Find
reference for high temp corrosion &&&&&&&&& the in service, or worst case, absolute roughness is assumed to be that of forged steel pipe, 0.045 mm [Bhatti Shah 1987].
By use of the Moody diagram shown in Figure 9 [Bhatti Shah 1987], the equivalent
Fanning friction factor ranges are 0.0040 ≤ f ≤ 0.0065 and 0.0060 ≤ f ≤ 0.0075 for new
and corroded tubes respectively.
Test data from Solar Two STPP, as reported by Kolb [2011], calculated a Fanning
friction factor of 0.0135, furthermore, the reported manufacturers value was 0.0168. The
Kolb [2011] manufacturers’ friction factor is shown in Figure 9, in the fully rough region
this is equivalent to a relative roughness, e/Di, of 0.05. For the project tubes of 18.5 mm
inside diameter, the absolute roughness is 0.9 mm. This absolute roughness is of the
order of that found in reinforced concrete [Bhatti Shah 1987]. The Kolb [2011] friction
factor data is beyond the range initially predicted for this project, however, as it has a
basis in operating plant data it will be used as a bounding value for the analysis.
Table 7. STPP project convection characteristics
Project Characteristic
Symbol
Units
Value
Absolute surface roughness
e
mm
0.0015 (new)
0.045 (corroded)
0.9 [Kolb 2011]
Relative surface roughness
e/Di
-
18
8.1*10-5 (new)
2.4*10-3 (corroded)
5*10-2 [Kolb 2011]
Fanning friction factor
f
-
0.0040 ≤ f ≤ 0.0065 (new)
0.0060 ≤ f ≤ 0.0075 (corroded)
0.0168 [Kolb 2011]
The Fanning friction factor is a function of the Reynolds number and relative surface roughness. The Moody diagram [Bhatti Shah 1987] is used with the project data
shown in Figure 9. It is seen that the new tube can be classified as hydraulically smooth
for all operating cases. However, the corroded tube operates in the transition region,
neither hydraulically smooth nor completely rough. The Kolb [2011] tube operates in
the transition and fully turbulent region.
Figure 9. Moody diagram [Bhatti Shah 1987] with new and corroded tube characteristics (UPDATE
PICTURE and ADD KOLB friction)
Bhatti and Shah [1987] describe the three flow regimes as:
19
In the hydraulically smooth regime, e is so small that the sand grains are contained within the laminar sublayer. Hence f is not affected by e; in other words, f =
f(Re). In the transition regime, the sand grains extend partly outside the laminar
sublayer, exerting an additional resistance to the flow, in the nature of a profile drag.
This causes the friction coefficient to depend on e/r as well as on Re, i.e., for the
transition regime f = f(e/r, Re). Finally, in the completely rough regime, all sand
grains reach outside the laminar sublayer, disrupting it completely. For this situation, the friction coefficient must depend on the size of the sand grain alone, i.e., f =
f(e/r).
Various Nusselt number correlations have been developed for fully developed turbulent flow in circular duct for smooth and fully rough regimes. The cases shown to
exist within the transition between smooth and fully rough regimes will be treated be as
smooth with the following empirical correction, and as shown in Figure 10, to account
for the effect of roughness on turbulent flow (20000 < Re < 200000) [Norris 1970]:
𝑁𝑢
𝑓 𝑛
𝑓
= ( ) 𝑓𝑜𝑟 ≤ 2.5
𝑁𝑢𝑠
𝑓𝑠
𝑓𝑠
Equation (11)
𝑁𝑢
𝑓
= (4)𝑛 𝑓𝑜𝑟 > 2.5
𝑁𝑢𝑠
𝑓𝑠
Equation (12)
𝑛 = 0.68 ∗ 𝑃𝑟 0.215 𝑓𝑜𝑟 1 < 𝑃𝑟 ≤ 6
Equation (13)
𝑛 = 1 𝑓𝑜𝑟 𝑃𝑟 > 6
Equation (14)
Where
Due to the exponent n varying between 0.68 (Pr = 1) and 1.0 (Pr = 6), the effect of
roughness is more pronounced at higher Prandtl number. However, as shown in Figure
10, this trend is inconsistent.
20
Figure 10. Effect of Prandtl number on heat transfer increase ratio for 20000 < Re < 200000 [Norris
1970]
The Nusselt number for a turbulent flow is sensitive to the surface roughness. As an
estimate of the sensitivity, at Re = 104 and Pr = 0.7, the local Nusselt number varies from
200 to 400 as shown in Figure 11. The doubling of the Nusselt number by variation in
only the relative surface roughness is indicative of the sensitivity of this characteristic.
The relative surface roughness of Figure 11 is from smooth to 1*10-2 which is within the
project range of smooth to 5*10-2.
Figure 11. Local Nusselt number as a function of relative surface roughness and Prandtl number in
the turbulent region [Nellis Klein 2008]
It is noted for this project, the Prandtl number range is 9.8 to 2.8 for 300oC to 600oC
respectively, with Pr < 6.0 for fluid temperatures greater than approximately 375oC.
21
### use fanning friction factor ####
### constant property assumption ##################
### smooth versus rough tubes – sensitivity. Important due to the corrosive nature
of the molten salt at high temperature. How does the corroded surface change the
roughness of the tube, and e/d ratio for small d tubes? ##################
#### due radiation onto tube wall to work out tube and fluid temperature difference.
What is limitation for Nusselt correlations – find the references that note the delta T
limit.########
22
7. Tube Wall Temperatures
The tube wall temperature, in particular, the difference between the tube wall inner
temperature and the bulk temperature of the fluid, plays an important role in the convection analysis. As the differential temperature increases, the constant fluid properties
assumption becomes less reasonable. Authors such as Gnielinski [1976] and Incropera
and DeWitt [2001] have noted the significance of the wall to fluid temperature difference in convection correlations.
As a first approximation of the wall to fluid temperature difference, the Biot number
is considered. The Biot number is a dimensionless parameter used in conduction problems that involve surface convection and is defined as [Incropera DeWitt 2001]:
𝐵𝑖 =
ℎ. 𝐿𝑐
𝑘𝑠
Equation (15)
𝑉
𝐴𝑠
Equation (16)
Where the characteristic length is:
𝐿𝑐 =
For this application, the inside and outside convection coefficients differ by orders
of magnitude so only the inside surface is considered. Therefore, the characteristic
length for the project is:
𝑟𝑚𝑒𝑎𝑛
= 4.9 𝑚𝑚
Equation (17)
2
The tube thermal conductivity, as shown in Figure 8, is in the 15 to 25 W.m-1.K-1
𝐿𝑐 =
range. The internal convection coefficient, as yet unknown, is assumed to be in the
range of 1866 to 26525 W.m-2.K-1 (Table 6). The resulting Biot number for this project
is in the range of 0.36 to 8.6. Using the projects nominal internal convection coefficient
of 16000 W.m-2.K-1 (Table 6), and tube thermal conductivity of 22.2 W.m-1.K-1 (at
575oC), the Biot number is 3.5.
For Bi << 0.1, the resistance to conduction in the solid is much less than the resistance to convection across the solid/fluid boundary [Incropera DeWitt 2001]. That is,
a relatively uniform temperature occurs throughout the solid while a rapid temperature
differential occurs within the fluid adjacent to the wall. Conversely, as Bi → ∞, a large
temperature change occurs in the solid while the surface and fluid temperature are
23
nominally equal. The temperature distribution at a solid/fluid interface is shown in
Figure 12.
The range of Biot numbers is indicative of a moderate temperature difference between the inside tube wall and the mean fluid. The magnitude of this temperature
difference is to be investigated further as it is important in selecting a Nusselt number
correlation.
Figure 12. Temperature distribution for limiting Biot numbers [similar to figure 3.9 of Yener]
Another means of determining the temperature difference between the tube inner
wall and that of the fluid is that of Mackowski [2015]. The Mackowski [2015] model as
used by Wyatt [2012] provides an analytical model for a long, annular cylinder with
temperature variation in both r and q, convection on the internal and external surfaces,
and collimated thermal radiation incident on the outside of the cylinder, as shown in
Figure 13.
24
Figure 13. Long, thick walled cylinder with external collimated incident radiation, radial and
circumferential conduction, and, internal and external convection [Wyatt 2012]
The Mackowski [2015] model was used to determine the temperature difference between the tube inner wall and the bulk fluid temperature. The base case was performed
as well as considering the incident flux range (240 kW.m-2 to 850 kW.m-2) and internal
convection coefficient (±20%). The base case uses the Solar Two STPP characteristics
of an incident flux of 240 W.m-2 with a molten salt bulk fluid temperature of 565oC
[Vant-Hull 2002] and an internal convection coefficient of 16000 W.m-2.K-1. The results
using the Wyatt [2012] code is shown in Figure 14 and the results for all cases are in
Table 8.
A weakness of the Mackowski [2015] model for this application is the assumption
of a uniform external convection coefficient. The STPP application will insulate the rear
of the tubes to prevent convective heat loss from this surface. Similarly, as discussed by
Wyatt [2012], the STPP tubes are arranged to form a panel. The panel arrangement
allows the peak tube temperature to occur at the crown as shown in Figure 14, however,
the crevice between tubes results in a second peak tube temperature to occur at nominally q = ±p/2 radians from the tube crown. These will not be considered further in this
project.
25
Figure 14. Tube temperature profile with an incident flux 240000 W/m2, molten salt bulk temperature 565oC, internal convection coefficient 16000 W.m-2.K-1
26
Table 8. Tube and molten salt temperatures with internal convection coefficient 16000W.m-2.K-1 +/20%
Case
1H
2H
3H
4H
5H
6H
7H
8H
9H
Base
Incident flux
kW.m-2
240
545
850
240
545
850
240
545
850
Internal convection
kW.m-2.K-1
16.0
16.0
16.0
19.2
19.2
19.2
12.8
12.8
12.8
o
596
636
676
593
630
667
600
645
690
o
581
603
624
579
597
615
585
612
638
o
15
33
52
14
33
52
15
33
52
o
16
38
59
14
32
50
20
47
73
coefficient
Tube outside maximum
C
temperature
Tube inside maximum
C
temperature
Through wall temperature
C
differential (max)
Inside wall to fluid tempera-
C
ture differential (max)
For all cases:
Tube outside diameter: 21 mm
Tube wall thickness: 1.25 mm
Tube conductivity: 22.2 W.m-1.K-1
Absorptivity of tube outer surface: 0.95
External convection coefficient: 10 W.m-2.K-1
External bulk fluid (air) temperature: 27 oC
Internal bulk fluid (molten salt) temperature: 565 oC
The Mackowski [2015] model supports the previous Biot number analysis. That is,
the temperature difference from the outside of the tube through to the bulk fluid temperature is not dominated by either the tube wall temperature differential (Bi→∞) nor
the inner tube wall to bulk fluid temperature differential (Bi→0). The results show the
overall temperature differential is nominally evenly split between the tube wall and
molten salt fluid.
The Mackowski [2015] model shows the through wall temperature, and the tube
outside maximum temperature are dependent upon the incident flux. Hence the significance of controlling the flux as a function of the molten salt temperature in order to
remain within the temperature limits of the tube material. As previously noted, this was
27
a characteristic of the Solar Two STPP with flux to molten salt operational variations
between 240 kW.m-2 at the outlet salt temperature of 565oC, and 850 kW.m-2 at the inlet
salt temperature of 290oC [Vant-Hull 2002]. However, it is the tube inside wall to bulk
fluid temperature differential that is of significance to this project due to its influence on
the molten salt fluid properties.
For a given incident flux, the Mackowski [2015] model shows the greatest tube inside wall to bulk fluid temperature differential corresponding to the lowest internal
convection coefficient. This is as expected as the internal convection coefficient is
indicative of the fluids ability to remove the heat from the tube wall, however, as this
differential temperature increases, the constant property assumption is weakened.
For the incident flux of 240 kw.m-2, and the convection coefficient varying between
12.8 kW.m-2.K-1 (-20%) through 16.0 kW.m-2.K-1 to 19.2 kW.m-2.K-1 (+20%), the
internal tube wall to bulk fluid differential temperatures are 20, 16, 14oC respectively.
This project initially assumed that the high Reynolds number may be deviating from
the Nusselt number correlations. The results thus far are indicative of the reverse, that
is, low Reynolds numbers are prone to higher temperature differentials within the fluid
and hence greater fluid property variation. This projects other initial assumption that
high heat flux may also deviate from the Nusselt number correlations is supported by the
Mackowski [2015] model results.
The Mackowski [2015] model was again used to determine the temperature difference between the tube inner wall and the bulk fluid temperature. The cases were
repeated with the same incident flux range (240 kW.m-2 to 850 kW.m-2), however, the
internal convection coefficient was changed to 4400 W.m-2.K-1 ±20%. The reduced
convection coefficient base case again uses the Solar Two STPP characteristics of an
incident flux of 240 W.m-2 with a molten salt bulk fluid temperature of 565oC [VantHull 2002] but with an internal convection coefficient of 4400 W.m-2.K-1. The results
using the Wyatt [2012] code is shown in Figure 15 and the results for all cases are in
Table 9.
28
Figure 15. Tube temperature profile with an incident flux 240000 W/m2, molten salt bulk temperature 565oC, internal convection coefficient 4400 W.m-2.K-1
Table 9. Tube and molten salt temperatures with internal convection coefficient 4400W.m-2.K-1 +/20%
Case
1L
2L
3L
29
4L
5L
6L
7L
8L
9L
Base
-2
Incident flux
kW.m
240
545
850
240
545
850
240
545
850
Internal convection
kW.m-2.K-1
4.40
4.40
4.40
5.28
5.28
5.28
3.52
3.52
3.52
o
634
725
815
626
705
785
647
753
860
o
620
693
765
612
673
734
633
722
810
o
14
32
50
14
32
51
14
31
50
o
55
128
200
47
108
169
68
157
245
coefficient
Tube outside maximum
C
temperature
Tube inside maximum
C
temperature
Through wall temperature
C
differential (max)
Inside wall to fluid tempera-
C
ture differential (max)
For all cases:
Tube outside diameter: 21 mm
Tube wall thickness: 1.25 mm
Tube conductivity: 22.2 W.m-1.K-1
Absorptivity of tube outer surface: 0.95
External convection coefficient: 10 W.m-2.K-1
External bulk fluid (air) temperature: 27 oC
Internal bulk fluid (molten salt) temperature: 565 oC
The Mackowski [2015] model results with the internal convection coefficient of
4400 W.m-2.K-1 was an academic exercise only. The tube outside and inside temperatures ranges are 626oC to 860oC and 612oC to 810oC respectively, both of which are
impractical. The oxidation temperature, or minimum continuous service temperature
without excessive scaling in air, for AISI 316 stainless steel is 870oC [Siebert et al
2008]. Similarly, the maximum temperature limit for SA-213 TP316 seamless tubing in
ASME section I application is 816oC [ASME 2010]. However, the reported maximum
temperature limit for molten nitrate salt (60%w NaNO3, 40%w KNO3) is approximately
621oC [Zavoico 2001].
Although impractical, the results of Table 9 (hi = 4400 W.m-2.K-1 ± 20%) are informative when compared to Table 8 (hi = 16000 W.m-2.K-1 ± 20%). By changing the
molten salt velocity from nominally 1.0 m.s-1 to 5.0 m.s-1 to increase the internal convec-
30
tion coefficient, the outside tube wall temperature decreased from 634oC (Case 1L) to
596oC (Case 1H). While the tube material temperature is clearly an important design
consideration, the inside wall to fluid temperature differential is of importance to this
project due to its role in the fluid properties. Similarly, the inside wall to fluid temperature differential decreased from 55oC (Case 1L) to 16oC (Case 1H).
The use of the molten salt velocity is significant to control the tube wall temperature
as shown by the above results and Vant-Hull [2002]. However, it is also significant
when selecting the convection correlation model due to the influence on the range of
fluid properties. These convection correlation models will be investigated in the next
section.
31
8. Convection Correlations for Turbulent Fully Developed Flow in
Tubes
As White [2005] noted for turbulent flow in pipes and channels, all of the available
data is in the form of correlations. As such, many correlations are available and Bhatti
and Shah [1987] listed twenty three Nusselt number correlations for smooth circular
ducts and Prandtl number of greater than 0.5, and an additional nine for the fully rough
regime of a circular duct. Of the many convection heat transfer correlations for turbulent
fully developed flow in tubes, this project is limited to those found to be in wide spread
use for this application.
The Nusselt number is the dimensionless temperature gradient occurring at the solid
fluid interface and is a measure of the convection heat transfer [Incropera DeWitt 2001].
The Nusselt number for flow in a circular tube is:
𝑁𝑢 =
ℎ. 𝐷𝑖
𝑘𝑓
Equation (18)
The determination of the Nusselt number from a correlation allows the convection
heat transfer coefficient to be calculated, and hence the heat transfer:
𝑞 = ℎ. 𝐴(𝑇1 − 𝑇2 )
Equation (19)
Applicable Nusselt number correlations are investigated.
8.1 Dittus-Boelter Correlation
One of the earliest equations for the turbulent heat transfer is a smooth tube is that
referred to as the Dittus-Boelter [Dittus Boelter 1930] correlation. As noted by Winterton [1998], the original equation (after conversion to SI units) for heating of the fluid is:
𝑁𝑢 = 0.0241𝑅𝑒 0.8 𝑃𝑟 0.4
Equation (20)
The widely presented form of the correlation for heating [Incropera DeWitt 2001,
Kreith Bohn 1986, Burmeister 1983] is:
𝑁𝑢 = 0.023𝑅𝑒 0.8 𝑃𝑟 0.4
Equation (21)
𝑁𝑢 = 0.024𝑅𝑒 0.8 𝑃𝑟 0.4
Equation (22)
1.0 ∗ 104 < 𝑅𝑒 < 1.2 ∗ 105
Equation (23)
Or [Bhatti Shah 1987]:
For:
32
0.7 < 𝑃𝑟 < 120
Equation (24)
Burmeister [1983] noted that the correlation is “reasonably accurate” when the wall
temperature does not exceed the fluid mixing-cup temperature by more than 5oC for
liquids or 55oC for gases. The properties are evaluated at the average local film temperature according to Burmeister [1983] while Winterton [1998] recommends using the
bulk fluid temperature which is more convenient. The point is moot assuming the
difference between the fluid and wall temperature is limited to 5oC.
Burmeister [1983] claims the Dittus-Boilter correlation results can be 20% high for
gases and up to 40% low for water at high Reynolds number.
8.2 Sieder-Tate Correlation
Similar in form to that of the Dittus-Boelter [1930] correlation, the Sieder-Tate
[1936] correlation differs by taking into account the viscosity gradient within the fluid
by means of a viscosity ratio. The viscosity ratio consists of the viscosity at the bulk
fluid temperature and that at the temperature of the tube wall. The original publication
by Sieder and Tate [1936] provided the results graphically but the correlation provided
by others [Incropera DeWitt 2001, Kreith Bohn 1986]:
1 𝜇
𝑁𝑢 = 0.027𝑅𝑒 0.8 𝑃𝑟 3 ( )
𝜇𝑠
Equation (25)
104 < 𝑅𝑒 < 105
Equation (26)
0.6 < 𝑃𝑟 < 100
Equation (27)
𝑓 = 𝑠𝑚𝑜𝑜𝑡ℎ 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑑𝑢𝑐𝑡
Equation (28)
For:
It is noted that the published coefficient varies from the 0.027 including 0.023
[Burmeister 1983] and 0.026 [Bird et al 2007].
All properties are evaluated at the bulk fluid temperature except for ms which is to
be evaluated at the pipe surface temperature.
The significance of the variations in fluid properties, as noted by Sieder and Tate
[1936], is:
The theoretical formulas,… for heat transfer to fluids in viscous flow in tubes,
do not take into consideration the effect of a radial temperature gradient on the distribution of the axial and radial components of the velocity. Because of the
33
magnitude of the temperature coefficient of viscosity of many liquids, a large radial
viscosity gradient results, which effects a distribution of velocity considerably different from that occurring in isothermal viscous flow. The viscosity gradient has
opposite signs for heating and cooling.
When a liquid is being heated by a hot wall, the viscosity of the fluid adjacent to the
wall will have a lower viscosity that of the bulk fluid. The result is an increase in the
velocity gradient at the wall as well as the associated improved heat transfer when
compared to the fluid at a uniform bulk temperature. Note that gases and liquids show
different characteristics as for low density gases the viscosity increases with increasing
temperature, for liquids the viscosity usually decreases with increasing temperature [Bird
et al 2007]. The distortion of a laminar velocity profile for a liquid in heated and cooled
tubes with temperature dependent viscosity is shown in Figure 16 [Kays Perkins 1985].
Figure 16. Distortion of laminar velocity profile in a heated or cooled tube when the viscosity of the
fluid depend on temperature (a) parabolic profile (b) heating of liquid (c) cooling of liquid [Kays
Perkins 1985]
8.3 Petukhov Correlation
Petukhov [1970] noted the reality of fluid properties being a function of temperature. The constant properties assumption can only be used with small temperature
differences or physical properties that only change slightly within the temperature range
34
considered. Petukhov [1970] also noted the practical difficulties in obtaining experimental data at high temperatures, large heat fluxes, and high pressures.
The Petukhov [1970] correlation is given in heat transfer texts including Incropera
and DeWitt [2001] and Kreith and Bohn [1986]. It is noted that Incropera and DeWitt
[2001] utilized the simplified correlation (Equation aa) while Kreith and Bohn [1986]
use the more accurate relationship (Equation bb).
𝑓
2 𝑅𝑒. 𝑃𝑟
𝑁𝑢 =
1
2
1
2
𝑓
1 + 3.4 ∗ 4𝑓 + (11.7 + 1.8𝑃𝑟 −3 ) ( ) (𝑃𝑟 3 − 1)
2
Equation
(bb29)
Where:
1
(1.82𝑙𝑜𝑔10 𝑅𝑒 − 1.64)−2
4
With a claimed error of 1% for:
𝑓=
Equation (30)
104 < 𝑅𝑒 < 5 ∗ 105
Equation (31)
0.5 < 𝑃𝑟 < 200
Equation (32)
A simplified calculation is presented by Petukhov [1970] but with a claimed error of
5-6%:
𝑓
2 𝑅𝑒. 𝑃𝑟
𝑁𝑢 =
1
2
2
𝑓
1.07 + 12.7 (2) (𝑃𝑟 3 − 1)
Equation
(aa33)
The latter formed the basis for the correlations of Gnielinski [1976].
8.4 Sleicher-Rouse Correlation
[Sleicher Rouse 1975]
8.5 Gnielinski Correlation
Of the correlations for fully developed turbulent flow in a circular tube and Pr > 0.5,
that of Gnielinski [1976] is the most commonly recommended in the general heat
transfer literature [Kays et al 2005; Incropera DeWitt 2001; Bhatti Shah 1987]. It is also
used in molten salt heat transfer literature [Liao et al 2014; Wu et al 2012; Bin et al
2009].
35
The correlations attributed to Gnielinski as presented by Bhatti and Shah [1987] are
a modification of Petukhov [1970] in order to extend the Reynolds number range from
fully developed turbulent flow range 104 ≤ Re ≤ 5*106 to include the transition range
2300 < Re ≤ 104:
𝑓
(𝑅𝑒 − 1000)𝑃𝑟
2
𝑁𝑢 =
1
2
2
𝑓
1 + 12.7 (2) (𝑃𝑟 3 − 1)
EquationA
(34)
With approximations for gases (0.5 < Pr < 1.5):
𝑁𝑢 = 0.0214(𝑅𝑒 0.8 − 100)𝑃𝑟 0.4
Equation (35)
And liquids (1.5 < Pr < 500):
𝑁𝑢 = 0.012(𝑅𝑒 0.87 − 280)𝑃𝑟 0.4
EquationB
(36)
Although not included by Kays et al [2005], Incropera and DeWitt [2001], or Bhatti
and Shah [1987], Gnielinski [1976] also published another equation to account for the
temperature dependence of the properties:
𝑁𝑢 =
𝑓
(𝑅𝑒 − 1000)𝑃𝑟
2
1
2
2
𝑓
1 + 12.7 (2) (𝑃𝑟 3 − 1)
𝑃𝑟 0.11
(
)
𝑃𝑟𝑤
EquationC
(37)
For:
2300 < 𝑅𝑒 < 106
Equation (38)
0.5 < 𝑃𝑟 < 724
Equation (39)
𝑓 = 𝑠𝑚𝑜𝑜𝑡ℎ 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑑𝑢𝑐𝑡
Equation (40)
Gnielinski [1976] validates the correlation with gases flowing through pipes with
“small temperature differences between the average gas temperature and the wall
temperature…” Furthermore, for liquids:
For heat transfer between liquids and solid wall it is possible to consider the
dependence of the properties on temperature by the ratio of Prandtl numbers in
Equation C21 at the average liquid temperature and the wall temperature, because
only the viscosity of the liquid depends very much on temperature, and large temperature differences are not usual for convective heat transfer.
Of the Gnielinski [1976] correlations available, those used within the molten salt literature are varied. Liao et al [2014] used the correlation without properties correction
36
(Equation 18), Wu et al [2012] and Bin et al [2009] used the liquid approximation but
with the Prandtl ratio correction (a combination of Equation C20 and Equation C21).
The author is not aware of the use of Equation C21 with the temperature dependent
properties correction.
The application of the Gnielinski [1976] correlation to a liquid with a large temperature difference between the wall and mean fluid will be examined as this was not
considered usual.
## mention who used the equation with Pr ratio correction in STPP work.
## talk about temperature range and Pr and Prwall range for this project
## includes property ratio ##
## need to determine wall temperature to use Pr_wall. Will it be outside the properties range???????
## used by Jianfeng due to large property variations of molten salt – has viscosity
correction ###
## see Petukhov page 531 below equation 56 for change of constant n=0.14 to
n=0.11. ##
### how large are the temperature difference (and constant property assumption)
####
37
9. Things to Do
Things to do include:
1. Determine properties of HTF (Janz, use critical properties to determine mu in
BSL – does it agree with Janz, other published data, see Apurba’s references). Is
published data at 1 atm?
2. Determine range of operating parameters to be studied (Re, Pr, Twall, Tfluid, P,
velocity, tube diameter, tube metal, heat flux,…)
3. Why do published studies stop at Re=50000?
4. Are properties a function of T only. How close to boiling point is the Twall?
How sensitive are the properties to temperature? Is constant properties assumption valid?
5. Review published Nu correlations for turbulent flow in a circular tube. Limitations on temperature differential, Re, etc.
See heat transfer handbook for
sources.
6. Use previous project to determine typical wall and fluid temperatures?
7. Parametric study of velocity profile inside annular and circular sections. Variation with Pr, Re, Di/Do, etc. to come up with correlation for the maximum
velocity location.
8. COMSOL model?
9. Boundary layer description
38
10. Conclusion
Should different velocities be used at different molten salt velocities?
Use higher velocity at higher molten salt velocity in order to increase flux. Higher
velocities will…
Or instead of controlling flux over panel (high flux at low molten salt temps and low
flux at high molten salt temps) use high flux everywhere but the velocity has to increase
as the molten salt temperature increases. 10 tubes in parallel, then 8 tubes in parallel,…
forming a single panel with constant heat flux over it?
39
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42