Thermal Analysis of a Hot Gas Duct for a High Temperature Gas
Cooled Nuclear Reactor
by
Viram Pandya
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:
_________________________________________________
Ernesto Gutierrez-Miravete, Engineering Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December 2010
© Copyright 2010
by
Viram Pandya
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF SYMBOLS ....................................................................................................... vii
ACKNOWLEDGMENT .................................................................................................. ix
ABSTRACT ...................................................................................................................... x
1. Introduction.................................................................................................................. 1
1.1
Evolution of Nuclear Power ............................................................................... 1
1.2
Early Helium Cooled Reactors .......................................................................... 2
1.3
General Atomics Design .................................................................................... 3
2. Methodology ................................................................................................................ 5
2.1
Assumptions ....................................................................................................... 5
2.2
Critical Radius of Insulation .............................................................................. 5
2.3
Mathematical Model .......................................................................................... 6
2.3.1
Steady State Heat Conduction................................................................ 6
2.3.2
Governing Equations .............................................................................. 6
2.4
Thermal Resistances........................................................................................... 8
2.5
Design Properties ............................................................................................... 9
2.6
Heat Exchanger Theory.................................................................................... 12
2.7
Convective Heat Transfer ................................................................................ 13
2.8
2.7.1
Entry Length......................................................................................... 13
2.7.2
Turbulence............................................................................................ 14
Model Description ............................................................................................ 16
2.8.1
Simplified Model ................................................................................. 16
2.8.2
Real Model ........................................................................................... 17
3. Results........................................................................................................................ 19
3.1
Insulation Thermal Conductivity ..................................................................... 19
iii
3.2
Hot and Cold Stream Exit Temperatures ......................................................... 19
3.3
COMSOL Modeling ........................................................................................ 19
4. Conclusion ................................................................................................................. 26
5. References.................................................................................................................. 27
Appendix A – Thermal Resistances Calculation ............................................................. 29
Appendix B – Heat Exchanger Calculation ..................................................................... 31
Appendix C – Entry-Length Calculation ......................................................................... 32
iv
LIST OF TABLES
Table 1: HGD Physical Properties ................................................................................... 10
Table 2: HGD Fluid Properties ........................................................................................ 10
Table 3: Helium Properties at 1223 K and 763 K (both at 3 MPa) ................................. 11
Table 4: Alloy 800H Properties ....................................................................................... 11
Table 5: Ceramic Insulation Properties ........................................................................... 12
Table 6: Outlet Temperature Summary ........................................................................... 13
Table 7: COMSOL Simplified Model Dimensions ......................................................... 17
Table 8: Hot and Cold Stream Outlet Temperatures ....................................................... 21
v
LIST OF FIGURES
Figure 1: Nuclear Power Evolution Timeline Showing Generation Breakdown [18]....... 2
Figure 2: General Atomics Gas-Turbine Modular Helium Reactor (GT-MHR) [16] ....... 4
Figure 3: Cross section of the HGD [6] ............................................................................. 4
Figure 4: Variation of heat transfer rate with radius [17] .................................................. 6
Figure 5: Thermal Resistance Model with Multiple Layers [3] ........................................ 9
Figure 6: Graphical Representation of Time-Averaged and Fluctuating Parameter [1] . 15
Figure 7: COMSOL Simplified Model Illustration ......................................................... 16
Figure 8: Magnified View of Simplified Model Mesh .................................................... 17
Figure 9: COMSOL Real Model Illustration ................................................................... 18
Figure 10: Magnified View of Real Model Mesh ........................................................... 18
Figure 11: Plot of Hot Stream Outlet Temperature vs. Cold Stream Inlet Velocity ....... 20
Figure 12: Real Model – Temperature Through All Layers at HGD Midpoint .............. 22
Figure 13: Simple Laminar Model Close-up at HGD Midpoint...................................... 22
Figure 14: Real Laminar Model Close-up at HGD Midpoint.......................................... 23
Figure 15: Simple Turbulent Model – Temperature Along Separator Centerline ........... 24
Figure 16: Real Turbulent Model – Temperature Along Insulation Centerline .............. 24
Figure 17: HGD Breakdown for Thermal Resistance Setup ........................................... 29
vi
LIST OF SYMBOLS
k = thermal conductivity (W/m·K)
h, hhot, hcold = convection heat transfer coefficient (W/m2·K)
rcr = critical radius of insulation (m)
r = radius (m)
qr = heat flux in radial direction (W)
A = area normal to direction of heat transfer (m2)
Rt,cond = thermal resistance for conduction (K/W)
Rt,conv = thermal resistance for convection (K/W)
L = length of cylinder (m)
P = Pressure (Pa)
T = Temperature (K)
T0 = Absolute Temperature = 273 K
Cp = specific heat, constant pressure (J/kg*K)
C, Chot, Ccold = heat capacity rate (W/K)
Cr = heat capacity ratio
U = overall heat transfer coefficient (W/m2*K)
NTU = number of transfer units
ε = effectiveness
q, qmax, Qmax = heat transfer rate (W)
ṁ = mass flow rate (kg/s)
T = temperature (K)
Re = Reynolds number
µ = dynamic viscosity (kg/m*s)
ν = kinematic viscosity (m2/s)
α = thermal expansion coefficient (/ºC)
ρ = density (kg/m3)
θ = angular position (radians)
V = velocity (m/s)
Dh = hydraulic diameter (m)
Pr = Prandtl number
vii
LIST OF SYMBOLS (continued)
Nu = Nusselt number
xh = hydrodynamic entry length (m)
xt = thermal entry length (m)
u = x-component of velocity
v = y-component of velocity
ū = time-averaged x-component of velocity
u’ = fluctuating x-component of velocity
F = body force (N)
e = internal energy (J)
i = enthalpy (J)
τ = shear stress (Pa)
x = axial direction (direction going through the HGD)
vr = velocity in the radial direction (m/s)
vθ = velocity in the θ direction (m/s)
viii
ACKNOWLEDGMENT
I would like to thank my family and friends for their support and encouragement
throughout the semester. I would also like to thank Professor Gutierrez and the entire
RPI staff for their assistance and persistence to continue challenging myself and to stay
on track.
ix
ABSTRACT
Current designs of high temperature helium cooled nuclear reactors call for two different
pressure vessels: one which contains the nuclear core to produce heat and one to convert
the heat to electricity or some other useful form. Connecting these two vessels is a coaxial tube, commonly called the hot gas duct. The inner tube of the hot gas duct
transports hot helium from the core to the power conversion vessel while the outer tube
takes helium cooled from the various heat exchangers back to the reactor. This project
describes a model developed to carry out a thermal analysis on this duct using both
advanced heat transfer concepts and finite element computer software to model the hot
gas duct of a currently operating experimental reactor. It examines the thermal insulation
necessary to minimize heat losses and the various layers within the duct to compensate
for large temperature gradients and thermal expansion. In addition, the thermal
conductivity of the insulation is calculated along with exit temperatures of both the hot
and cold streams.
x
1. Introduction
Although dormant for over 30 years, reactor design and development has been
ramping up recently in anticipation of the nuclear renaissance: the revival of the nuclear
power industry to meet today’s demand for green energy. Many people are hesitant
about nuclear power due to the radioactive waste that is produced and required to be
stored underground. However, many of these people agree that supplying the world’s
energy demand without carbon emissions is impossible with renewable energy sources
alone. Recent advances in nuclear reactor design may have the answer to a cleaner and
safer commercial nuclear power industry.
1.1 Evolution of Nuclear Power
The timeline of commercial nuclear power is broken up into numerous
generations as shown in Figure 1. The earliest research reactors and prototypes consisted
of Generation I in the 50’s and 60’s followed by Generation II reactors built up to the
end of the 90’s. Much of the nuclear reactors we are familiar with today such as the
Pressurized Water Reactor (PWR) and the Boiling Water Reactor (BWR) are Generation
II. Generation III and III+ have already been designed and many are currently being built
or have been approved to be built in countries such as Japan, France and China. These
reactors incorporate advances in nuclear technology such as passive safety systems,
more fuel burn up resulting in less waste at shutdown, and increased thermal efficiency.
All these improvements result in a reactor with a longer lifetime and smaller waste
profile. Even as Generation III reactors are being built, Generation IV reactors are
underway in different phases from research and development to conceptual design,
incorporating advances such as the ability to use nuclear waste as fuel, greater energy
extraction from a given amount of fuel, and improved safety systems. Advances in the
construction of commercial plants are also being incorporated such as the use of steel
plate construction and modularization techniques.
A popular Generation IV reactor design is the High Temperature Gas-cooled
Reactor (HTGR) which is graphite moderated and helium cooled. One of the biggest
advantages of the HTGR is the ability to obtain core outlet temperatures as high as
1
1000°C. This high temperature justifies uses other than electricity production such as
process heat for hydrogen production, desalination and oil refining. Additionally, the
reactor is designed using only passive safety systems. Even in the event of a loss of
coolant casualty, which in today’s plants is a severe problem causing all sorts of safety
systems to trip online, the HTGR would be able to remove heat from the core simply
using natural convection and conduction.
Figure 1: Nuclear Power Evolution Timeline Showing Generation Breakdown [18]
1.2 Early Helium Cooled Reactors
The first helium cooled nuclear reactor to produce electricity was built at the
Peach Bottom Atomic Power Station in Pennsylvania and operated from 1967 to 1974
[19]. Two helium reactors were built a few years prior in England and Germany and
operated as experimental reactors. The Peach Bottom unit was very successful,
producing a maximum of 40 MW to the grid at a thermal efficiency of approximately
39% [11]. In 1976 the Fort St. Vrain Generating Station came online in northern
Colorado. This was also a high temperature helium cooled reactor which operated until
1989. Although generally successful near the end of its commercial life, this design was
plagued by problems initially. The biggest problem originated in the complex steam
turbine helium circulators. The water lubricated bearing design of the circulators allowed
2
water to enter the inert environment and create devastating corrosion issues [20].
However, both the Peach Bottom and Fort St. Vrain reactors proved that a helium
cooled, graphite moderated nuclear reactor was not only viable, but inherently safe as
well. Although they were generally successful, the pressurized water reactor was chosen
over its helium cooled counterpart mainly due to Captain Hyman Rickover’s decision to
use the pressurized water reactor as a prototype for naval submarines, claiming they
were simpler and closer to maturity. The helium cooled reactor was put on the shelf until
its recent emergence to lay the groundwork for the next generation of reactors.
1.3 General Atomics Design
A General Atomics proposed design for an HTGR is shown below in Figure 2.
The single tube connecting the two vessels is the hot gas duct (HGD), which is a coaxial
pipe. A typical cross section of the HGD is shown in Figure 3. Hot helium from the core
runs through the liner tube while cooler helium from the power production vessel runs
through the pressure tube. The elevated temperatures and moderate pressures up to about
4 MPa that the HGD experiences also create significant stresses and thermal gradients.
In addition, insulation must be added between the hot and cold tubes to minimize heat
losses. Flow characteristics must also be determined to account for convective effects.
Insulation is also used to ensure the pressure tube is kept below its maximum allowable
temperature. The proper materials must be purchased or engineered, as in the case of the
Japan Atomic Energy Agency (JAEA) to build the High Temperature Test Reactor
(HTTR).
3
Hot gas duct
Figure 2: General Atomics Gas-Turbine Modular Helium Reactor (GT-MHR) [16]
Figure 3: Cross section of the HGD [6]
4
2. Methodology
2.1 Assumptions
Since a thermal analysis of a coaxial tube with turbulent flow conditions is
complex, numerous assumptions were made to help simplify the problem.
Assume:

Both hot and cold helium streams are smooth pipes with a negligible friction
factor.

When performing hand calculations, the case of constant heat rate per unit of
tube length was used. This is in contrast to the constant surface temperature case.

There is no natural convection.

Steady state conditions for both hot and cold streams.

All material and fluid properties are constant.
2.2 Critical Radius of Insulation
For radial systems there exists an optimal insulation thickness. This can be
proven by the fact that there are competing effects when increasing insulation. The first
effect is that the conduction resistance increases when adding insulation. The competing
effect is that as insulation is added, the total surface area of the system increases, causing
the convection resistance to decrease. Thus there must be an optimal insulation thickness
that minimizes heat loss. This is called the critical radius of insulation and is given by
the following equation:
𝑟𝑐𝑟 =
Where
𝑘
ℎ
(1)
k = thermal conductivity (W/m·K)
h = convection heat transfer coefficient (W/m2·K)
Above this critical radius, the heat flux decreases with increasing thickness of insulation
while below it, the heat flux increases up to the critical radius as Figure 4 below
illustrates.
5
Figure 4: Variation of heat transfer rate with radius [17]
2.3 Mathematical Model
2.3.1
Steady State Heat Conduction
The heat equation for steady-state conditions with no heat generation in a
cylinder is:
1𝑑
𝑑𝑇
(𝑘𝑟 ) = 0
𝑟 𝑑𝑟
𝑑𝑟
(2)
The appropriate form of Fourier’s law for a cylindrical surface is:
𝑞𝑟 = −𝑘𝐴
Where
𝑑𝑇
𝑑𝑟
(3)
r = radius (m)
qr = heat flux in radial direction (W)
A = area normal to direction of heat transfer (m2)
2.3.2
Governing Equations
The continuity, momentum and energy equations of a compressible fluid in
vector form are shown below:
6
Continuity:
𝐷𝜌
+ 𝜌(𝛻 ∙ 𝑉) = 0
𝐷𝑡
(4)
Momentum:
𝐷𝑉
= −∇𝑃 + ∇2 𝑉 + 𝜌𝐹
𝐷𝑡
(5)
𝐷
𝑉2
(𝑒 + ) + ∇ ∙ 𝑃𝑉 + ∇ ∙ (𝜏 ∙ 𝑉) − ∇ ∙ 𝑘∇𝑇 = 𝜌(𝑉 ∙ 𝐹)
𝐷𝑡
2
(6)
𝜌
Energy:
𝜌
Where V is the fluid velocity, F is a body force, e is the internal energy and τ is the shear
stress. Applying the assumption that there is steady state flow cancels the time derivative
terms. Other reasonable assumptions which can be made include no body forces, no heat
generation or additional mechanical work, and no diffusion. Using cylindrical
coordinates and keeping only velocity in the axial direction since it is the only
significant component (the derivatives of vr and vθ with respect to r are very small)
simplifies the equations to those shown below. Note that further simplification of
Equation 9 will lead to the heat equation shown in Equation 2.
Continuity:
𝜕𝑢 1 𝜕
(𝑟𝑣) = 0
+
𝜕𝑥 𝑟 𝜕𝑟
(7)
Momentum:
𝜕𝑢
𝜕𝑢
𝑑𝑃 1 𝜕
𝜕𝑢
+ 𝜌𝑣
=−
+
(𝑟𝜇 )
𝜕𝑥
𝜕𝑟
𝑑𝑥 𝑟 𝜕𝑟
𝜕𝑟
(8)
𝜕𝑖
𝜕𝑖 1 𝜕
𝜕𝑇
𝜕𝑢
𝑑𝑃
+ 𝜌𝑣 −
(𝑟𝑘 ) − 𝜇( )2 − 𝑢
=0
𝜕𝑥
𝜕𝑟 𝑟 𝜕𝑟
𝜕𝑟
𝜕𝑟
𝑑𝑥
(9)
𝜌𝑢
Energy:
𝜌𝑢
Where u and v are the fluid velocities in the x and y axes respectively, i is the enthalpy,
µ is the dynamic viscosity and P is the pressure.
7
2.4 Thermal Resistances
For the case of one dimensional heat transfer with no internal energy generation
and constant properties, thermal resistances can be used to model the conduction of heat,
similar to electrical resistances for electrical charge. The resistances can be added in
series in the case of multiple layers as shown in Figure 5. Refer to Appendix A for the
thermal resistance calculation performed to determine the thermal conductivity of the
insulation in the HGD. The thermal resistances for conduction and convection in a
cylindrical wall are as follows:
𝑅𝑡,𝑐𝑜𝑛𝑑 =
𝑟
ln(𝑟2 )
1
2𝜋𝐿𝑘
(10)
1
ℎ𝐴
(11)
𝑅𝑡,𝑐𝑜𝑛𝑣 =
Where
Rt,cond = thermal resistance for conduction (K/W)
Rt,conv = thermal resistance for convection (K/W)
L = length of cylinder (m)
8
Figure 5: Thermal Resistance Model with Multiple Layers [3]
2.5 Design Properties
The physical dimensions of the hot gas duct were extracted from [6]. These are
the dimensions of the planned very high temperature gas cooled reactor (VHTR) being
researched at the Korea Atomic Energy Research Institute (KAERI) since data on the
General Atomics design could not be obtained. A study in [6] sizes the inner diameter of
the HGD pressure vessel according to three situations where the flow velocity, flow rate,
or dynamic pressure of the hot helium is the same as the cold. We use the case of
identical flow rates. Also, although the length of the HGD is not specifically given, we
use information about the distance between the reactor and power production vessels to
make an educated assumption on its length. Tables 1 and 2 show the design parameter
values used in this study.
9
Table 1: HGD Physical Properties
Inner radius (m)
Outer radius (m)
Hot Helium
Area (m2)
Thickness (m)
0.386
0.468
Liner Tube
0.386
0.393
0.007
0.017
Insulation
0.393
0.513
0.120
0.342
Inner Tube
Annulus (Cold
Helium)
0.513
0.523
0.010
0.033
0.523
0.626
0.103
0.372
Pressure Tube
0.626
*Length of HGD = 8.0 m
0.691
0.065
0.269
Table 2: HGD Fluid Properties
Hot Helium Temp (K)
1223
Cold Helium Temp (K)
763
Pressure (both) (MPa)
3
Flow rate (both) (kg/s)
Flow Velocity Hot Helium
(m/s)
Flow Velocity Cold Helium
(m/s)
84
65
82
Helium properties vary significantly with temperature and pressure. The following
temperature and pressure dependent equations from [10] are calculated at 1223 K and
763 K and 3 MPa in Table 3. The specific heats at constant pressure and constant
volume do not vary significantly.
−1
𝜌 (𝐷𝑒𝑛𝑠𝑖𝑡𝑦) = 0.17623 ∗
𝑃
𝑇⁄
𝑇0
[1 + 0.53 ∗ 10−3 ∗
𝑃
1.2 ]
(𝑇⁄𝑇 )
0
𝑘𝑔
𝑚3
𝑇 0.7 𝑘𝑔
𝜇 (𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦) = 1.865 ∗ 10−5 ∗ ( )
𝑇0
𝑚∗𝑠
(12)
(13)
𝑘 (𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝐶𝑜𝑛𝑑. )
−4 ∗𝑃)
= 0.144(1 + 2.7 ∗ 10
−4
𝑇 0.71∗(1−2∗10
∗ 𝑃) ( )
𝑇0
10
𝑊
𝑚∗𝐾
(14)
Where
P = Pressure (bar)
T = Temperature (K)
T0 = Absolute Temperature = 273 K
Table 3: Helium Properties at 1223 K and 763 K (both at 3 MPa)
at 1223 K
at 763 K
3
1.18
1.88
Dynamic Viscosity (kg/m*s)
5.33E-05
3.83E-05
5195
5195
0.418
0.300
Density (kg/m )
Specific Heat, const pressure
(J/kg*K)
Thermal conductivity
(W/m*K)
The tubes of the HGD are made of Alloy 800H- a high nickel content (32%) alloy that is
highly resistant to oxidation and corrosion at elevated temperatures. The required
properties of Alloy 800H from [13] are listed in Table 4. The average thermal
conductivity value is the arithmetic average between the values at 1223 K and 763 K.
Table 4: Alloy 800H Properties
Density (kg/m3)
8030
Specific Heat, const pressure
(J/kg*K)
Thermal conductivity
(W/m*K)
500
Coeff. Thermal expansion
(m/ºC)
30.8
at 1223 K
24.55
Average
18.3
at 763 K
1.78E-05
The insulation within the HGD is made of a fibrous ceramic insulation material
mainly consisting of silicon dioxide (SiO2) and alumina (Al2O3). The fibers are wrapped
in a stainless steel net to prevent it from entering the helium stream [9]. Commercial
ceramic insulation products consisting of alumina fibers were found from [14] and its
properties are listed in Table 5 below.
11
Table 5: Ceramic Insulation Properties
Density (kg/m3)
Specific Heat, const pressure
(J/kg*K)
Thermal conductivity
(W/m*K)
192
1130
0.150
at 811 K
0.235
at 1033 K
0.320
at 1255 K
2.6 Heat Exchanger Theory
Although they have opposite goals, the HGD and a counter flow coaxial heat
exchanger are almost identical. Since the HGD has insulation between its two streams,
its purpose is to hold in heat rather than exchange it. Because of this similarity, heat
exchanger theory can be used to analytically determine the exit temperatures of the hot
and cold streams by employing the effectiveness-NTU method. First, the heat capacity
rate of each stream must be calculated and the stream with the lower value is considered
Cmin.
𝐶 = 𝑚̇ ∗ 𝐶𝑝
(15)
𝐶𝑚𝑖𝑛
𝐶𝑚𝑎𝑥
(16)
𝐶𝑟 =
We then calculate the overall heat transfer coefficient, U, to determine the number
of transfer units, NTU. Table 11.3 from [2] tabulates effectiveness equations in terms of
NTU. Once the effectiveness of the heat exchanger is known, the actual heat transfer rate
can be determined along with exit temperatures using Equations 20 through 22.
𝑈=
1
1
1
+
ℎ𝐻𝑜𝑡 ℎ𝐶𝑜𝑙𝑑
𝑁𝑇𝑈 =
𝜀=
𝑈𝐴
𝐶𝑚𝑖𝑛
𝑁𝑇𝑈
1 + 𝑁𝑇𝑈
12
(17)
(18)
(19)
𝑞 = 𝜀 ∗ 𝑞𝑚𝑎𝑥 = 𝜖 ∗ 𝐶𝑚𝑖𝑛 ∗ (𝑇𝐻𝑜𝑡 − 𝑇𝐶𝑜𝑙𝑑 )
𝑇𝐻𝑜𝑡,𝑜𝑢𝑡𝑙𝑒𝑡 = 𝑇𝐻𝑜𝑡,𝑖𝑛𝑙𝑒𝑡 −
(20)
𝑞
𝑚̇ ∗ 𝐶𝑝
(21)
𝑞
𝑚̇ ∗ 𝐶𝑝
(22)
𝑇𝐶𝑜𝑙𝑑,𝑜𝑢𝑡𝑙𝑒𝑡 = 𝑇𝐶𝑜𝑙𝑑,𝑖𝑛𝑙𝑒𝑡 +
See Table 6 for a summary of results and Appendix B for detailed calculations.
Table 6: Outlet Temperature Summary
Cmin (W/K)
U (W/m2*K)
NTU
ε
qmax (W)
Hot stream outlet temp
(K)
Cold stream outlet temp
(K)
436380
720
0.032
0.031
2.00E+08
1209
777
2.7 Convective Heat Transfer
The sheer volume and velocity of helium flowing through the HGD is a
substantial amount which follows certain velocity and thermal profiles. Since both the
hot and cold flows are turbulent, there is significant “mixing” due to random motion
which tends to increase the heat transfer rate. In addition, the turbulence of the streams
creates a boundary layer with the surface of the pipes where viscous effects cannot be
neglected.
2.7.1
Entry Length
The development of the boundary layer is caused by the no-slip boundary
condition which says the velocity of the fluid at the pipe wall must be zero. In this
boundary layer, the velocity profile varies with both the x and y axes where the x-axis is
the axial coordinate along the pipe length and the y-axis is the coordinate from one inner
pipe surface to the other. Viscous effects must also be taken into account within the
13
boundary layer. The boundary layer grows in thickness until it has completely filled the
pipe. The point along the x-axis where this occurs is called the hydrodynamic entry
length. Beyond this, the flow is considered to be fully developed and the velocity profile
varies with y only.
The same situation occurs with the temperature of the fluid flowing through the
pipe. The temperature of the fluid very close to the pipe must be equal to the surface
temperature of the pipe. In this case a thermal boundary layer develops until it reaches
the thermal entry length. The hydrodynamic and thermal entry lengths are different for
laminar and turbulent flow except for flows where the Prandtl number is equal to one. In
this case the hydrodynamic boundary layer develops at the same rate as the thermal
boundary layer. The equations for turbulent flow are given below as they will represent
the HGD.
𝑥ℎ = 4.4 ∗ (𝑅𝑒)1/6 ∗ 𝐷ℎ
(23)
𝑥𝑡 = 4.4 ∗ (𝑅𝑒)1/6 ∗ 𝐷ℎ ∗ 𝑃𝑟
(24)
From Appendix C it is shown that the only stream which reaches any type of fully
developed flow is the cold stream. This stream becomes thermally fully developed after
travelling 5.7 meters through the pipe.
2.7.2
Turbulence
The transition to turbulence occurs when laminar flow becomes unstable due to
minor disturbances. When the disturbance, such as the effects of a rough pipe, occurs,
the velocity changes and the inertia forces associated with this velocity change create
instability by magnifying the disturbance [15]. The turbulent flow undergoes random
fluid particle fluctuations and contains unsteady velocity components in all three axes.
Due to this complexity, most turbulent flow problems can only be solved using software.
Due to the random nature of turbulence, fluids undergoing turbulent flow involve
fluctuating parameters, such as velocity. The parameters must be broken down into a
mean value and a fluctuating value as shown in Figure 6. Thus, the value of the
parameter, u, at some time t is equal to:
𝑢(𝑡) = 𝑢̅ + 𝑢′
14
(25)
This is called the Reynolds average of the parameter where ū is the time-averaged
(mean) value and u’ is the fluctuating value. The Reynolds average of u(t) is defined as:
𝑡0 +𝑡
1
𝑢̅(𝑡) = ̅̅̅̅̅̅̅̅
𝑢̅ + 𝑢′ = lim ∫
𝑡→∞ 𝑡 𝑡
0
1 𝑡0 +𝑡
(𝑢̅ + 𝑢′ )𝑑𝑡 = 𝑢̅ + lim ∫
𝑢′𝑑𝑡 = 𝑢̅
𝑡→∞ 𝑡 𝑡
0
Where t0 to t is a length of time that is used to integrate out the turbulent fluctuations in
the averaging process. Reynolds averaging can also be applied to the Navier-Stokes and
energy equations as shown below.
Reynolds –averaged Navier-Stokes equation:
𝑢̅
𝜕𝑢̅
𝜕𝑢̅
1 𝜕𝑃̅
𝜕
𝜕𝑢̅
+ 𝑣̅
=−
+
(𝜈
− ̅̅̅̅̅̅
𝑢′ 𝑣 ′ )
𝜕𝑥
𝜕𝑦
𝜌 𝜕𝑥 𝜕𝑦 𝜕𝑦
(26)
Reynolds-averaged energy equation:
𝑢̅
𝜕𝑇̅
𝜕𝑇̅
𝜕
𝜕𝑇̅
+ 𝑣̅
=
(𝛼
− ̅̅̅̅̅̅
𝑣 ′𝑇 ′)
𝜕𝑥
𝜕𝑦 𝜕𝑦 𝜕𝑦
(27)
In the above equations, ̅̅̅̅̅̅
𝑢′ 𝑣 ′ is a turbulent shear stress and ̅̅̅̅̅̅
𝑣 ′ 𝑇 ′ is a turbulent heat flux in
the direction normal to the main flow.
Figure 6: Graphical Representation of Time-Averaged and Fluctuating Parameter [1]
15
2.8 Model Description
The model of the HGD was built using the Finite Element Model (FEM) in
COMSOL Multiphysics. Two different orientations were used when modeling the HGD:
simplified and real. The simplified model underwent both laminar and turbulent flow
analyses while the real model only underwent turbulent analysis, and in each case the
thermal boundary condition at the end of the pressure tube was an inward heat flux of 0
W/m2.
2.8.1
Simplified Model
In the simplified model, the hot and cold streams were approximately equal in
area and were separated by a very thin piece of Alloy 800H, herein called the separator.
Another piece of Alloy 800H surrounded the entire model from the outside, acting as the
pressure tube. The length of this model was 4 meters compared to the actual length of 8.
The simplified model was used to establish the correct laminar and turbulent conditions
and served as a baseline to compare the real model. Figure 7 and Table 7 illustrate the
simplified model and its dimensions. The mesh used in COMSOL for the simplified
model consists of 5504 elements and is shown in Figure 8.
Cold stream
Axis of symmetry
Pressure tube
Hot stream
Separator
Figure 7: COMSOL Simplified Model Illustration
16
Table 7: COMSOL Simplified Model Dimensions
Thickness
(m)
0.386
0.010
0.358
Component
Hot stream
Separator
Cold stream
Pressure
tube
0.060
Figure 8: Magnified View of Simplified Model Mesh
2.8.2
Real Model
The real model is a replica of the actual HGD with identical dimensions to that of
Table 1. All the tubes were added with their proper radius as well as the layer of
insulation. Properties used for the insulation were that of alumina ceramic fiber taken
from Table 5. This is the closest material with published properties since the data for the
insulation used on the HGD is unavailable. This model, as shown in Figure 9, only
underwent turbulent analysis since this is the realistic condition of the HGD. The mesh
for the real model consists of 2300 elements and is shown in Figure 10. It has
significantly fewer elements compared to the simplified model due to the fact that the
real model has more layers, many of which are very thin, as well as the increased length
of the HGD.
17
Axis of symmetry
Cold stream
Hot stream
Pressure tube
Liner tube
Insulation
Inner tube
Figure 9: COMSOL Real Model Illustration
Figure 10: Magnified View of Real Model Mesh
18
3. Results
3.1 Insulation Thermal Conductivity
As calculated in Appendix A, the thermal conductivity of the insulation is 1.95
W/m*K using the method of thermal resistances. Reference [9] calculates the thermal
conductivity to be 0.47 W/m*ºC through thermal tests. This difference results in a large
error percentage. The most likely reason for this large error is because [9] calculates the
thermal conductivity through thermal tests that model the HGD of the HTR-10 instead of
the KAERI design. HTR-10 is an experimental helium cooled reactor based in China
whose layout and design parameters are different. Nonetheless, the aluminum oxide fiber
insulation is identical.
3.2 Hot and Cold Stream Exit Temperatures
The extremely low effectiveness calculated in Appendix B makes sense since the
presence of insulation will significantly reduce the amount of heat that will be lost by the
hot stream, making for an ineffective heat exchanger. Through the 8 meter length of the
HGD, the hot helium decreases 14 K which is gained by the cold helium. In reality, the
cold helium temperature rise will not be as much since the insulation will contain most
of the heat.
3.3 COMSOL Modeling
Figure 11 below shows the variation of the hot stream outlet temperature as the
cold stream inlet velocity is varied between 30 m/s and 300 m/s. In all cases, the inlet
temperatures for the hot and cold stream were 1223 K and 763 K respectively. The hot
stream inlet velocity was kept constant at 65 m/s. To determine the outlet temperatures,
the Boundary Integration function was used after solving the problem. This resulted in a
value of temperature integrated over the area, so the hot stream outlet area of 0.468 m2
was divided out to arrive at the temperature.
19
Hot Stream Outlet Temperature (K)
1224
1222
1220
Simplified Turbulent
1218
Simplified Laminar
1216
Real Turbulent
1214
1212
0
50
100
150
200
250
300
Cold Stream Inlet Velocity (m/s)
Figure 11: Plot of Hot Stream Outlet Temperature vs. Cold Stream Inlet Velocity
The hot stream outlet temperature is nearly independent of the cold stream inlet velocity
for the real model. The thick insulation and large mass flow rates of both streams can
account for this. The simplified models exhibit similar behavior although the laminar
model does not lose as much energy. This result reinforces the fact that turbulence
causes efficient mixing and greater heat transfer.
In addition, the outlet temperatures of the hot and cold streams were determined
for all models and listed in Table 8 along with the calculated value obtained in Appendix
B. Again, COMSOL’s Boundary Integration tool was used to determine the temperature
integrated over the area. The outlet temperatures were determined for the normal
operating conditions shown in Table 2.
20
Table 8: Hot and Cold Stream Outlet Temperatures
Model
Simple
Laminar
Simple
Turbulent
Real Turbulent
Calculated
Outlet Temperature
(K)
Cold
Hot
Stream
Stream
770.24
1214.88
769.81
763.07
777
1218.09
1222.95
1209
The real model exhibits the smallest temperature change in both streams whereas
the calculated value undergoes the largest. This result makes logical sense since the real
model is highly insulated and the calculated values included many assumptions about the
characteristics of the flow in order to use a tabulated Nusselt number.
Figure 12 was plotted to illustrate the temperature change over the cross-section
of the HGD. The data is taken at the longitudinal midpoint of the HGD at 4 meters and
begins at the center of the hot stream, going radially outward to the edge of the pressure
tube. A close-up image of the separator in the simplified model and the insulation in the
real model is shown in Figures 13 and 14 respectively.
21
Figure 12: Real Model – Temperature Through All Layers at HGD Midpoint
Figure 13: Simple Laminar Model Close-up at HGD Midpoint
22
Figure 14: Real Laminar Model Close-up at HGD Midpoint
The linear temperature gradient between the hot and cold streams shows the benefit of
insulation as it creates a gradual change in temperature, preventing large thermal
gradients from forming and causing unnecessary stresses. The images reinforce this
point since the simple model has an uneven temperature distribution in the hot stream
along the separator. This could cause uneven thermal expansion within the HGD and
cause fractures over time. The real model, on the other hand, displays an even
temperature reduction from hot to cold streams.
Since a large focus of the thermal analysis of the HGD relies on the insulation,
plots of the temperature vs. HGD length along the centerline of the insulation or
separator are provided in Figures 15 and 16 for the turbulent case. Both plots were taken
with the HGD under normal operating conditions.
23
Figure 15: Simple Turbulent Model – Temperature Along Separator Centerline
Figure 16: Real Turbulent Model – Temperature Along Insulation Centerline
24
Again one can see why the presence of insulation is beneficial as its temperature varies
linearly from inlet to outlet with a temperature difference of about 6 K. Conversely, the
separator in the simplified model varies by about 400 K. This shows that the insulation
stays at about the same temperature through the HGD while the separator undergoes
large fluctuations, reinforcing the value provided by insulation.
25
4. Conclusion
The thermal analysis of the HGD was successful in accomplishing the given
tasks: calculating the thermal conductivity of the insulation, calculating the exit
temperatures and analyzing a finite element model of the HGD. However, it highlights
the difficulty in obtaining accurate results when performing hand calculations.
Calculation of the thermal conductivity of the insulation and the outlet temperatures of
the hot and cold streams was accomplished using thermal resistances and heat exchanger
theory respectively. Unfortunately, the proprietary nature of materials used and their
properties forced the use of data from two different reactors which skewed results of the
conductivity. Additionally, many assumptions had to be made in determining the outlet
temperatures which led to greater heat transfer than would normally occur.
The COMSOL finite element models proved useful in examining the thermal
nature of the HGD. The insulation is superior in preventing heat transfer between the
two streams as well as minimizing thermal gradients. It also allows for a uniform
thermal expansion throughout the HGD which must be taken into account when
constructing the reactor as this section will expand and cannot be constrained on both
ends.
26
5. References
1. Munson, Bruce R., Donald F. Young and Theodore H. Okiishi. Fundamentals of
Fluid Mechanics. Hoboken: John Wiley & Sons, Inc, 2006.
2. Incropera, Frank P., David P. Dewitt, Theodore L. Bergman and Adrienne S.
Lavine. Fundamentals of Heat and Mass Transfer. Hoboken: John Wiley &
Sons, Inc, 2007.
3. Bird, R. Byron, Warren E. Stewart and Edwin N. Lightfoot. Transport
Phenomena. Hoboken: John Wiley & Sons, Inc, 2002.
4. Cengel, Yunus A., and Michael A. Boles. Thermodynamics: An Engineering
Approach. New York: McGraw-Hill, 2006.
5. Hishida, Makoto, Kazuhiko Kunitomi, Ikuo Ioka, et al. “Thermal Performance
Test of the Hot Gas Ducts of Hendel.” Nuclear Engineering and Design 83
(1984): 91-103.
6. Song, Kee-nam and Yong-wan Kim. “Preliminary Design Analysis of Hot Gas
Ducts for the Nuclear Hydrogen System.” Journal of Engineering for Gas
Turbines and Power 131 (Jan 2009).
7. Bröckerhoff, P., J. Singh, H. Schmitt, J. Knaul, et al. “Status of Design and
Testing of Hot Gas Ducts.” Nuclear Engineering and Design 78 (1984): 215-221.
8. Inagaki, Yoshiyuki, Kazuhiko Kunitomi, Masatoshi Futakawa, Ioka Ikuo and
Yoshiyuki Kaji. “R&D on high temperature components.” Nuclear Engineering
and Design 233 (2004): 211-223.
9. Huang, Z.Y., Z.M. Zhang, M.S. Yao and S.Y. He. “Design and experiment of hot
gas duct for the HTR-10.” Nuclear Engineering and Design 218 (2002): 137145.
10. Petersen, Helge. “The Properties of Helium.” Danish Atomic Energy
Commission (1970).
11. Kingrey, K.I. “Fuel Summary for Peach Bottom Unit 1 High-Temperature GasCooled Reactor Cores 1 and 2.” Idaho National Engineering and Environmental
Laboratory (April 2003): 5-9.
12. http://www.cdeep.iitb.ac.in/nptel/Mechanical/Heat%20and%20Mass%20Transfe
r/Conduction/Module%202/main/2.6.4.html
13. Sandmeyer Steel Company. http://www.sandmeyersteel.com/A800-A800HA800AT.html
14. MatWeb. “Thermal Ceramics Kaowool Blanket.”
http://www.matweb.com/search/datasheet.aspx?MatGUID=cb830e74bc69422aa5
60a7b57494955a
15. Kays, William, Michael Crawford and Bernhard Weigand. Convective Heat and
Mass Transfer. New York: McGraw-Hill, 2005.
16. General Atomics website. http://gt-mhr.ga.com/description.php
17. Steady State Conduction Module.
http://www.cdeep.iitb.ac.in/nptel/Mechanical/Heat%20and%20Mass%20Transfe
r/Conduction/Module%202/main/2.6.4.html
18. U.S. Department of Energy. Generation IV Nuclear Energy Systems.
http://www.ne.doe.gov/geniv/neGenIV1.html
27
19. Wikipedia contributors. "Peach Bottom Nuclear Generating Station." Wikipedia,
The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 18 Nov. 2010. Web.
4 Dec. 2010.
20. Wikipedia contributors. "Fort St. Vrain Generating Station." Wikipedia, The Free
Encyclopedia. Wikipedia, The Free Encyclopedia, 13 Jul. 2010. Web. 4 Dec.
2010.
28
Appendix A – Thermal Resistances Calculation
The thermal resistance method was used to determine the thermal conductivity of
the insulation and compare it to the value given in [9]. Figure 17 below illustrates the
problem setup.
Figure 17: HGD Breakdown for Thermal Resistance Setup
𝑞𝑟𝑎𝑑𝑖𝑎𝑙 =
𝑅𝑇𝑜𝑡𝑎𝑙
𝑇∞,𝐻 − 𝑇∞,𝐶
𝑅𝑇𝑜𝑡𝑎𝑙
𝑟
𝑟
𝑟
𝑙𝑛 𝑟2
𝑙𝑛 𝑟3
𝑙𝑛 𝑟4
1
1
1
2
3
=
+
+
+
+
2𝜋𝑟1 𝐿ℎ𝐻𝑜𝑡 2𝜋𝑘𝐴 𝐿 2𝜋𝑘𝑖 𝐿 2𝜋𝑘𝐴 𝐿 2𝜋𝑟5 𝐿ℎ𝐶𝑜𝑙𝑑
(28)
(29)
Where kA is the thermal conductivity of Alloy 800H and ki is the thermal
conductivity of the insulation. In order to determine the convection coefficients (h1 and
h5) of the hot and cold helium, the Reynolds number is first calculated to determine if the
flow is laminar, turbulent, or in transition.
𝑅𝑒𝐻𝑜𝑡
𝑅𝑒𝐶𝑜𝑙𝑑
𝑘𝑔
𝑚
𝜌𝑉𝐷ℎ 1.18 𝑚3 ∗ 65 𝑠 ∗ 2 ∗ 0.386 𝑚
=
=
= 1.11 ∗ 106
𝑘𝑔
𝜇
−5
5.33 ∗ 10 𝑚 ∗ 𝑠
𝑘𝑔
𝑚
𝜌𝑉𝐷ℎ 1.88 𝑚3 ∗ 82 𝑠 ∗ (2 ∗ 0.626 − 2 ∗ 0.523)𝑚
=
=
= 8.29 ∗ 105
𝑘𝑔
𝜇
3.83 ∗ 10−5 𝑚 ∗ 𝑠
29
Therefore, both flows are turbulent. The Prandtl number is also calculated to
determine how the velocity and thermal diffusivity are developing.
𝐽
−5 𝑘𝑔
𝜇𝐶𝑝 5.33 ∗ 10 𝑚 ∗ 𝑠 ∗ 5195 𝑘𝑔 ∗ 𝐾
𝑃𝑟𝐻𝑜𝑡 =
=
= 0.662
𝑊
𝑘
0.418 𝑚 ∗ 𝐾
𝐽
−5 𝑘𝑔
𝜇𝐶𝑝 3.83 ∗ 10 𝑚 ∗ 𝑠 ∗ 5195 𝑘𝑔 ∗ 𝐾
𝑃𝑟𝐶𝑜𝑙𝑑 =
=
= 0.663
𝑊
𝑘
0.300 𝑚 ∗ 𝐾
Since the Prandtl numbers are approximately equal to 1, the heat and momentum are
diffused through the fluid at the same rates and certain simplifications can be made. The
Nusselt number for the hot and cold streams is calculated from the Dittus-Boelter
equation where n=0.4 for heating and n=0.3 for cooling.
𝑁𝑢𝐻𝑜𝑡 = 0.023 ∗ 𝑅𝑒 4/5 ∗ 𝑃𝑟 𝑛 = 0.023 ∗ (1.11 ∗ 106 )4/5 ∗ (0.662)0.3 = 1394
𝑁𝑢𝐶𝑜𝑙𝑑 = 0.023 ∗ 𝑅𝑒 4/5 ∗ 𝑃𝑟 𝑛 = 0.023 ∗ (8.29 ∗ 105 )4/5 ∗ (0.663)0.4 = 1060
ℎ𝐻𝑜𝑡
ℎ𝐶𝑜𝑙𝑑
𝑊
𝑁𝑢𝐻𝑜𝑡 ∗ 𝑘 1394 ∗ 0.418 𝑚 ∗ 𝐾
𝑊
=
=
= 755 2
𝐷ℎ
2 ∗ 0.386 𝑚
𝑚 ∗𝐾
𝑊
1060 ∗ 0.300 𝑚 ∗ 𝐾
𝑁𝑢𝐶𝑜𝑙𝑑 ∗ 𝑘
𝑊
=
=
= 1543 2
𝐷ℎ
(2 ∗ 0.626 − 2 ∗ 0.523) 𝑚
𝑚 ∗𝐾
𝑞𝑟𝑎𝑑𝑖𝑎𝑙 = ℎ𝐻𝑜𝑡 ∗ 𝐴 ∗ (𝑇𝐻𝑜𝑡 − 𝑇𝐶𝑜𝑙𝑑 ) = 755
𝑊
∗ 0.468 𝑚2 ∗ (1223 − 763)𝐾
∗𝐾
𝑚2
= 162536 𝑊
All the constants are now known except for the thermal conductivity of the
insulation (ki).
𝑞𝑟𝑎𝑑𝑖𝑎𝑙 =
𝑇∞,𝐻 − 𝑇∞,𝐶
𝑟2
𝑟3
𝑟4
𝑙𝑛
𝑙𝑛
𝑙𝑛
1
𝑟1
𝑟2
𝑟3
1
+
+
+
+
2𝜋𝑟1 𝐿ℎ𝐻𝑜𝑡 2𝜋𝑘𝐴 𝐿 2𝜋𝑘𝑖 𝐿 2𝜋𝑘𝐴 𝐿 2𝜋𝑟5 𝐿ℎ𝐶𝑜𝑙𝑑
𝑘𝑖 = 1.95
30
𝑊
𝑚∗𝐾
Appendix B – Heat Exchanger Calculation
𝐶𝐻𝑜𝑡 = 𝐶𝐶𝑜𝑙𝑑 = 𝑚 ∗̇ 𝐶𝑝 = 84
𝑘𝑔
𝐽
𝑊
∗ 5195
= 436380
𝑠
𝑘𝑔 ∗ 𝐾
𝐾
We have calculated hhot and hcold in Appendix A and use these values to calculate U:
𝑈=
𝑞𝑚𝑎𝑥
1
1
1
= 720
𝑊
𝑚2 ∗ 𝐾
1
+
𝑊
𝑊
ℎ𝐻𝑜𝑡 ℎ𝐶𝑜𝑙𝑑
755 2
1543 2
𝑚 ∗𝐾
𝑚 ∗𝐾
𝑊
720 2
∗ 𝜋 ∗ 2 ∗ 0.386 𝑚 ∗ 8 𝑚
𝑈𝐴
𝑚 ∗𝐾
𝑁𝑇𝑈 =
=
= 0.032
𝑊
𝐶𝑚𝑖𝑛
436380 𝐾
𝑁𝑇𝑈
0.032
𝜀=
=
= 0.031
1 + 𝑁𝑇𝑈 1 + 0.032
𝑊
= 𝐶𝑚𝑖𝑛 (𝑇𝐻𝑜𝑡,𝑖𝑛𝑙𝑒𝑡 − 𝑇𝐶𝑜𝑙𝑑,𝑖𝑛𝑙𝑒𝑡 ) = 436380 ∗ (1223 − 763) 𝐾 = 2.00 ∗ 108 𝑊
𝐾
+
1
1
=
𝑞𝑎𝑐𝑡𝑢𝑎𝑙 = 𝜖 ∗ 𝑞𝑚𝑎𝑥 = 0.031 ∗ 2.00 ∗ 108 𝑊 = 6.23 ∗ 106 𝑊
𝑇𝐻𝑜𝑡,𝑜𝑢𝑡𝑙𝑒𝑡 = 𝑇𝐻𝑜𝑡,𝑖𝑛𝑙𝑒𝑡 −
𝑞
6.23 ∗ 106 𝑊
= 1223 𝐾 −
= 1209 𝐾
𝑊
𝑚̇ ∗ 𝐶𝑝
436380 𝐾
𝑇𝐶𝑜𝑙𝑑,𝑜𝑢𝑡𝑙𝑒𝑡 = 𝑇𝐶𝑜𝑙𝑑,𝑖𝑛𝑙𝑒𝑡 +
𝑞
6.23 ∗ 106 𝑊
= 763 +
= 777 𝐾
𝑊
𝑚̇ ∗ 𝐶𝑝
436380 𝐾
31
Appendix C – Entry-Length Calculation
From Appendix A we know the values of the Reynolds number, Prandtl number and
hydraulic diameter for the hot and cold streams.
Hydrodynamic Entry-Length:
𝑥ℎ,ℎ𝑜𝑡 = 4.4 ∗ (𝑅𝑒)1/6 ∗ 𝐷ℎ = 4.4 ∗ (1.11 ∗ 106 )1/6 ∗ 2 ∗ 0.386 𝑚 = 34.6 𝑚
𝑥ℎ,𝑐𝑜𝑙𝑑 = 4.4 ∗ (𝑅𝑒)1/6 ∗ 𝐷ℎ = 4.4 ∗ (8.29 ∗ 105 )1/6 ∗ 2 ∗ (0.626 − 0.523) 𝑚 = 8.5 𝑚
Thermal Entry-Length:
𝑥𝑡,ℎ𝑜𝑡 = 4.4 ∗ (𝑅𝑒)1/6 ∗ 𝐷ℎ ∗ 𝑃𝑟 = 4.4 ∗ (1.11 ∗ 106 )1/6 ∗ 2 ∗ 0.386 𝑚 ∗ 0.662
= 22.9 𝑚
𝑥𝑡,𝑐𝑜𝑙𝑑 = 4.4 ∗ (𝑅𝑒)1/6 ∗ 𝐷ℎ ∗ 𝑃𝑟
= 4.4 ∗ (8.29 ∗ 105 )1/6 ∗ 2 ∗ (0.626 − 0.523)𝑚 ∗ 0.663 = 5.7 𝑚
32