Analysis of the Heat Transfer Effects of Tube Configuration in a Tube Bundle in Horizontal, TwoPass Condensers
by
Jennifer Lynne Tansey
A 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:
_________________________________________
Craig Wagner, Ph.D., Project Advisor
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2011
© Copyright 2011
by
ii
Jennifer Lynne Tansey
All Rights Reserved
iii
CONTENTS
LIST OF TABLES v
LIST OF FIGURES
vi
NOMENCLATURE
ix
ACKNOWLEDGMENT
xi
ABSTRACT
1.
xi
Introduction
12
1.1
Background 12
1.2
Problem Description 1
1.3
Previous Work
2.
2.1
Methodology
Theory
1
2
2
2.1.1
Equations of Conservation
2.1.2
Creating the Steam-Air Mixture Velocity Profile
2.1.3
Heat Transfer Relations and Coefficients
2.1.4
Mass Transfer Relations
2.2
2
3
4
6
Mathematical Model 6
2.2.1
Creating the Baseline Conditions
2.2.2
Calculating Initial Values Based on Assumed Temperatures
2.2.3
Calculating Heat and Mass Transfer Coefficients
2.2.4
Calculating the Initial Heat Flux and Temperature Differences 7
2.2.5
Iterating to Solve for an Adjusted Heat Flux and Temperature Differences
8
2.2.6
Recalculating Temperature Dependent Properties
2.2.7
Reiterating to Obtain Converged Values for Heat Flux and Temperatures
9
2.2.8
Calculating the Outlet Circulating Water Temperature of the First-Pass9
2.2.9
Calculating the Outlet Temperature of the Second-Pass
2.2.10
7
Performing an Energy Balance
iv
9
7
7
9
9
2.3
Assumptions 10
2.4
Initial Conditions
11
2.4.1
Condenser Dimensions
2.4.2
Operating Conditions 11
2.5
3.
3.1
11
Numerical Analysis - Modeling Using RANS Solver
12
Results 12
Analytical Results
12
3.1.1
Case 1
13
3.1.2
Case 2
22
3.1.3
Case 3
30
3.1.4
Case 4
39
3.1.5
Case 5
47
3.1.6
Case 6
57
3.2
Comparison of Results Using Mehrabian-Based Velocity Profile
3.3
Comparison of Results Using FLOW3D-Based Velocity Profile 67
3.4
Comparison of Mehrabian and FLOW3D-Based Results
4.
Conclusions
73
5.
References
75
6.
Appendix A – Steam-Air Mixture Velocity Profiles
7.
Appendix B – FLOW3D Input File for Case 1
64
70
75
76
LIST OF TABLES
Table 1. Case 1 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature ... 15
Table 2. Case 1 Calculated Temperatures ................................................................................... 16
Table 3. Case 1 Energy Balance ................................................................................................... 16
Table 4. Case 1 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data...................................................................................................................... 21
Table 5. Case 1 Comparison of Mehrabian and FLOW3D Calculated Temperatures .................. 22
Table 6. Case 2 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature ... 25
v
Table 7. Case 2 Calculated Temperature Comparison with Case 1 ............................................. 26
Table 8. Case 2 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data...................................................................................................................... 29
Table 9. Case 3 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature ... 33
Table 10. Case 3 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data...................................................................................................................... 38
Table 11. Case 4 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature . 41
Table 12. Case 4 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data...................................................................................................................... 47
Table 13. Case 5 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature . 50
Table 14. Case 5 Calculated Temperatures ................................................................................. 51
Table 15. Case 5 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data...................................................................................................................... 56
Table 16. Case 6 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature . 59
Table 17. Case 6 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data...................................................................................................................... 63
Table 18. Case Comparison of Circulating Water Temperatures ................................................ 66
Table 19. Case Comparison of Circulating Water Temperatures Using FLOW3D Data ............... 69
Table 20. Case Comparison of Mehrabian and FLOW3D Circulating Water Temperatures........ 73
LIST OF FIGURES
Figure 1. Cross-Sectional View of Tube Configurations ................................................................. 1
Figure 2. Tube Row Orientation................................................................................................... 11
Figure 3. Case 1 Tube Configuration ............................................................................................ 14
Figure 4. Case 1 First-Pass Circulating Water Temperature vs Tube Length ............................... 14
Figure 5. Case 1 Second-Pass Circulating Water Temperature vs Tube Length .......................... 15
Figure 6. Case 1 FLOW3D Velocity Magnitude Contours ............................................................ 18
Figure 7. Case 1 FLOW3D Mixture Temperature Contours .......................................................... 19
vi
Figure 8. Case 1 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
...................................................................................................................................................... 20
Figure 9. Case 1 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D
Data .............................................................................................................................................. 20
Figure 10. Case 2 Tube Configuration .......................................................................................... 23
Figure 11. Case 2 First-Pass Circulating Water Temperature vs Tube Length ............................. 24
Figure 12. Case 2 Second-Pass Circulating Water Temperature vs Tube Length ........................ 24
Figure 13. Case 2 FLOW3D Velocity Magnitude Contours .......................................................... 27
Figure 14. Case 2 FLOW3D Mixture Temperature Contours ....................................................... 27
Figure 15. Case 2 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
...................................................................................................................................................... 28
Figure 16. Case 2 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D
Data .............................................................................................................................................. 29
Figure 17. Case 3 Tube Configuration .......................................................................................... 31
Figure 18. Case 3 First-Pass Circulating Water Temperature vs Tube Length ............................. 32
Figure 19. Case 3 Second-Pass Circulating Water Temperature vs Tube Length ........................ 32
Figure 20. Case 3 FLOW3D Velocity Magnitude Contours .......................................................... 34
Figure 21. Case 3 FLOW3D Mixture Temperature Contours ....................................................... 36
Figure 22. Case 3 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
...................................................................................................................................................... 37
Figure 23. Case 3 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D
Data .............................................................................................................................................. 38
Figure 24. Case 4 Tube Configuration .......................................................................................... 39
Figure 25. Case 4 First-Pass Circulating Water Temperature vs Tube Length ............................. 40
Figure 26. Case 4 Second-Pass Circulating Water Temperature vs Tube Length ........................ 41
Figure 27. Case 4 FLOW3D Velocity Magnitude Contours .......................................................... 43
Figure 28. Case 4 FLOW3D Mixture Temperature Contours ....................................................... 44
Figure 29. Case 4 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
...................................................................................................................................................... 46
Figure 30. Case 4 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D
Data .............................................................................................................................................. 46
Figure 31. Case 5 Tube Configuration .......................................................................................... 48
vii
Figure 32. Case 5 First-Pass Circulating Water Temperature vs Tube Length ............................. 49
Figure 33. Case 5 Second-Pass Circulating Water Temperature vs Tube Length ........................ 49
Figure 34. Case 5 FLOW3D Velocity Magnitude Contours .......................................................... 52
Figure 35. Case 5 FLOW3D Mixture Temperature Contours ....................................................... 52
Figure 36. Case 5 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
...................................................................................................................................................... 53
Figure 37. Case 5 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D
Data .............................................................................................................................................. 54
Figure 38. Case 6 Tube Configuration .......................................................................................... 57
Figure 39. Case 6 First-Pass Circulating Water Temperature vs Tube Length ............................. 58
Figure 40. Case 6 Second-Pass Circulating Water Temperature vs Tube Length ........................ 58
Figure 41. Case 6 FLOW3D Velocity Magnitude Contours .......................................................... 60
Figure 42. Case 6 FLOW3D Mixture Temperature Contours ....................................................... 60
Figure 43. Case 6 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
...................................................................................................................................................... 62
Figure 44. Case 6 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D
Data .............................................................................................................................................. 62
Figure 45. Comparison of Average First-Pass Circulating Water Temperature vs Length........... 64
Figure 46. Comparison of Average Second-Pass Circulating Water Temperature vs Length ...... 65
Figure 47. Comparison of Average First and Second-Pass Outlet Circulating Water Temperature
...................................................................................................................................................... 67
Figure 48. Comparison of Average First-pass Circulating Water Temperature vs Length Using
FLOW3D Data ............................................................................................................................... 68
Figure 49. Comparison of Average Second-pass Circulating Water Temperature vs Length Using
FLOW3D Data ............................................................................................................................... 69
Figure 50. Comparison of Average First and Second-Pass Outlet Circulating Water Temperature
Using FLOW3D Data...................................................................................................................... 70
Figure 51. Comparison of Velocity Profiles on First-Pass Circulating Water Temperature ......... 71
Figure 52. Comparison of Velocity Profiles on Second-Pass Circulating Water Temperature .... 72
viii
NOMENCLATURE
Symbol
Units
Description
A
m2
Area
Aw
m2
Total tube surface area
cp,m
kJ/(kg·K) Specific heat of the mixture
cp,s
kJ/(kg·K) Specific heat of steam
cp,cw
kJ/(kg·K) Specific heat of the circulating water
di
m
Tube inner diameter
do
m
Tube outer diameter
Dsa
m2/s
Diffusivity of steam in air
ΔE
kW
Change in energy in the system
Ein
kW
Energy added to the system
Eout
kW
Energy leaving the system
fc
-
Friction factor
Fl
-
Inundation factor
Fs
-
Superheat factor
Fv
-
Vapor shear factor
g
m/s2
Gravitational constant
H
J/kg
Static enthalpy of the steam-air mixture
Ha
J/kg
Static enthalpy of air
Hs
J/kg
Static enthalpy of steam
k
W/(m·K) Thermal conductivity
kc
W/(m·K) Condensate thermal conductivity
kw
W/(m·K) Tube wall thermal conductivity
Km
-
Mass transfer coefficient
L
kJ/kg
Latent heat of condensation
𝑚̇
kg/s
Mass flow rate
𝑚′′′
kg/(m3·s) Mass condensation rate of vapor per unit volume
𝑚′′
kg/(m2·s) Mass condensation rate of vapor per unit area
ix
Ma
amu
Molecular mass of air
Mm
amu
Molecular mass of mixture
Ms
amu
Molecular mass of steam vapor
n
-
nth tube row
N
-
Total number of tube rows
Nu
-
Nusselt number (0.33Rec0.6Pr1/3)
p
Pa
Condenser pressure
ps
Pa
Bulk steam partial pressure
ps,i
Pa
Steam partial pressure at the interface
Px
m
Transverse tube pitch
Pr
-
Prandtl number (cp,m·μ/k)
q"g
W/m2
Heat flux to the interface
q"d
W/m2
Net enthalpy flux due to species diffusion
q"
W/m2
Heat flux
Q
W
Rf
Energy
(m2·K)/W Fouling factor
Re
-
Reynolds number ((ρ·Ucr·do)/μ)
Sc
-
Schmidt number (μ/(ρ·Dsa))
t
s
Time
T
K
Bulk temperature of steam-air mixture
T0
K
Reference temperature
Tf
K
Film temperature
Ti
K
Interface or condensate surface temperature
Tl
K
Circulating water temperature
Tl_out
°C
Outlet circulating water temperature
Tsat
K
Mixture saturation temperature
Two
K
Tube outer surface temperature
u
m/s
Velocity component in the x-direction
U
m/s
Velocity vector
Ucr
m/s
Resultant crossflow velocity (u2+v2)1/2
x
v
m/s
Velocity component in the y-direction
V
m3
Control volume
w
m/s
Velocity component in the z-direction
Ya
kg/kg
Mass fraction of air
Ys
kg/kg
Mass fraction of steam
Greek Symbols
Units
Description
αc
W/(m2·K) Condensate film heat transfer coefficient
αc*
W/(m2·K) Coefficient
αg
W/(m2·K) Gas-side heat transfer coefficient
αl
W/(m2·K) Coefficient
αwl
W/(m2·K) Tube outer wall-circulating water heat transfer coefficient
μ
Pa·s
Laminar dynamic viscosity
μt
Pa·s
Turbulent dynamic viscosity
μe
Pa·s
Dynamic mixture viscosity
ρ
kg/m3
Mixture density
ρc
kg/m3
Condensate density
σh
-
Laminar Prandtl number
σh,t
-
Turbulent Prandtl number
τ
Pa
φ
-
Ackerman correction factor
φH
-
Coefficient
Shear stress
ACKNOWLEDGMENT
I would like to acknowledge and thank my family for their constant encouragement and
support of my academic endeavors, especially over the course of my Graduate studies. I would
also like to thank my advisor, Dr. Wagner, for providing guidance and technical expertise on this
project.
ABSTRACT
xi
This study presents the analyses of the effect of the arrangement of tubes in a tube bundle in a
horizontal, two-pass condenser on the amount of heat transferred to the circulating water in
the tubes. The tube bundle is assumed to act as a staggered tube bank in cross-flow with
downward superheated steam flow. The saturated circulating water is assumed to be
turbulent flow. Previously defined relationships for heat transfer through tube banks, including
condensate inundation, vapor shear, and the effect of tube surface geometry are used in
analyzing six tube configurations to determine the largest change in temperature of the
circulating water. The heat flux in the system is defined as a function of the condenser and
tube material properties, tube geometry, tube spacing, condensate inundation and steam
velocity. Numerical modeling of the six tube configurations using a Reynolds-averaged NavierStokes (RANS) approach is presented to confirm the analytical results. Analyses of the
analytical and numerical results from the six configurations examined provide the optimal tube
arrangement for maximum heat transfer to the circulating water. It is found that the circulating
water temperature is dependent on the steam-air mixture velocity profile used. The most heat
transfer occurs in the tubes rows where the steam-air mixture velocities are the highest.
Furthermore, the magnitude of the velocity profile is proportional to the magnitude of the
change in circulating water temperature.
Introduction
Background
The steam cycle that occurs most commonly in thermal power plants can be ideally
represented by the Rankine Cycle. The Rankine Cycle is a thermodynamic cycle involving liquid
water, steam and a mixture of the two as the working fluids. In the Rankine Cycle, water is
adiabatically compressed and circulated by a pump to a boiler. The boiler isothermally
vaporizes the water into steam and directs the steam to a turbine, which adiabatically expands
the steam and converts heat energy into rotational energy by turning a shaft. The last step in
the Rankine Cycle involves isothermal condensation of the exhaust steam from the turbine
back into water to supply the pumps, which completes the cycle. A condenser is used to
condense the expanded steam into liquid and to maintain vacuum on the turbine outlet to
increase the available energy across the turbine.
Horizontal, two-pass condensers are commonly used in steam cycles. Two-pass
condensers have an inlet/outlet waterbox with the inlet and outlet of the circulating water on
the same side of the condenser, and a return waterbox on the opposite side of the condenser
to direct the circulating water from the first-pass into the second-pass tubes. The circulating
water moves through tubes that are arranged together to form a tube bundle. Conventional
horizontal, two-pass condensers have the top half of the tubes as the “cold” first-pass and the
bottom half of the tubes as the “warmer” second-pass. This tube configuration allows the
coldest circulating water to first come into direct contact with the condensing steam.
Condensers are typically kept under vacuum conditions to increase the pressure drop
across the turbine, thereby maximizing the power output. The vacuum system in the
xii
condenser removes non-condensibles in the condenser. The non-condensibles decrease the
heat transfer in the tubes and increase condenser pressure.
Problem Description
The objective of this project is to analyze different tube configurations in a tube bundle
to determine the best arrangement for the maximum amount of heat transferred to the
circulating water in a horizontal, two-pass condenser. The six configurations shown below will
be examined.
Figure 1. Cross-Sectional View of Tube Configurations
The dark blue and light blue portions of the cross-sectional views in Figure 1 represent
the cold first-pass and warmer second-pass in the tube bundle, respectively.
Previous Work
1
Several papers have been written providing heat transfer, vapor velocity, film
condensation and pressure drop correlations over horizontal tube banks based on experimental
results and detailed simulations using computational fluid dynamics (CFD) models.
An analysis of a two-pass condenser was performed by Malin [1] using a CFD model
simulating flow and heat transfer. In Malin’s work, a single-phase approach for the steam-air
mixture flow within the condenser was used to calculate the performance of a condenser with
a superheated steam supply. The simulated condenser employs the use of two tube bundles of
parallel staggered tubes with the first-pass entering the lower bundle and exiting the condenser
through the upper tube bundle.
Browne and Bansal [2] examined variations in experimental observations made in over
70 papers to provide an overview of condensation heat transfer on horizontal tube bundles for
downward flowing condensing vapor. The effects of surface geometry, condensate inundation,
vapor shear and gravity were studied.
Wilson and Bassiouny [3] provided results for laminar and turbulent flow of air across a
single tube row as well as staggered and in-line tube banks. The effects of flow and tube
geometry on the Nusselt number, friction factor, velocity and turbulence kinetic energy profiles
are presented therein.
Mehrabian [4] evaluated the heat transfer and pressure drop of air over a single,
circular tube and over a tube bank based on experimental results. Additionally, a relationship
between the velocity distribution of air in cross flow and pressure drop over horizontal tubes
was provided.
Methodology
Theory
The heat transferred from turbulent superheated steam-air mixture in downward cross
flow over horizontal tubes to turbulent flow within the tubes is predicted using RANS
equations. The mathematical model presented by Malin [1] is adapted for each of the six tube
configurations analyzed in this study.
Equations of Conservation
The governing conservation equations for analyzing a time independent, superheated
steam and air mixture are the conservation of mass, momentum and energy equations. All but
the conservation of energy equation are solved in a two-dimensional Cartesian coordinate
system under steady-state conditions. The energy equation is solved as a one-dimensional
system along the length of the tubes. The conservation of mass is defined as:
∇ ∙ (𝜌𝒖) = −𝑚′′′
(1)
2
Where the source term,
and is defined by Malin [1].
, is the mixture mass condensation rate per unit volume
The conservation of momentum governs the fluid dynamic behavior of the system and
relates the velocity field components, pressure, and external forces, as well as the fluid
properties including density and dynamic viscosity. Both the laminar and turbulent viscosities
are taken into consideration in the conservation of momentum.
𝜕𝑢
𝜌 𝜕𝑡 + 𝜌𝑢 ∙ ∇𝒖 = −∇𝑝 + ∇𝝉 + 𝐹 − 𝑚′′′𝑢
2
(2a)
∇𝝉 = ∇ ∙ (𝜇𝑒 ∇𝒖) − 3 (𝜇𝑒 ∇ ∙ 𝒖)
(2b)
𝜇𝑒 = 𝜇 + 𝜇𝑡
(2c)
Two general energy equations are used, Equations (3) and (4). Equation (3) describes
the energy of the mixture, and is a function of the mixture static enthalpy, diffusivity of the
steam in air, and the enthalpy transport from the bulk steam-air mixture to the vapor-liquid
interface.
𝜇
𝜇
∇ ∙ (𝜌𝐻𝒖) = ∇ ∙ ((𝜎 + 𝜎 𝑡 ) ∇𝐻) + 𝑞"𝑔 + 𝑞"𝑑
ℎ
ℎ,𝑡
(3)
The conservation of energy equation, Equation (4), relates the energy transferred to
the system through the tube wall and into the circulating water.
𝜕𝑇
𝜌𝑐𝑝 ( 𝜕𝑡 + 𝑢 ∙ ∇𝑇) + ∇ ∙ (−𝑘∇𝑇) = 𝑄
(4)
Equation (4) is simplified assuming steady-state, one-dimensional flow to calculate the
circulating water temperature in the tubes to the following equation.
𝑞" = 𝑚̇𝑐𝑝 ∆𝑇
(5)
Creating the Steam-Air Mixture Velocity Profile
A velocity profile for the steam in the condenser is established using the correlation of
pressure drop and velocity at each tube row for cross-flow of air over a tube bank created by
Mehrabian [4].
3
∆𝑃 = 𝑓𝑐 𝑁𝜌
𝑈2
2
(6)
The inlet steam velocity is calculated based on the condenser open area and steam
mass flow rate, Equation (7). Similarly, Equation (7) is used to calculate an overall velocity for
the tube region by changing the area to account for the presence of the tube bundle in the
condenser.
𝑚̇ = 𝜌𝐴𝑈
(7)
A friction factor, fc, is calculated based on the inlet velocity in the region of the
condenser without tubes, Equation (8).
𝑃
1.08
𝑓𝑐 = [0.023 + 0.11/ (𝑑𝑥 − 1)
𝑜
] 𝑅𝑒𝑐 −0.15
(8)
This friction factor is applied to the correlation established by Mehrabian [4] to
calculate the pressure drop for each row, from which the velocity at the given row is
determined. Px is the pitch, or the centerline to centerline distance between tubes and is
further defined in section 2.4.1.1. For this evaluation, the vertical pitch and horizontal pitch are
equal. The Reynolds number used in calculating the friction factor is based on the overall
velocity in the tube bundle.
Heat Transfer Relations and Coefficients
The heat transferred from the steam-air mixture to the circulating water is modeled
using three major equations: heat transfer from the steam-air mixture to the interface, heat
transfer through the condensate film and heat transfer through the tube wall to the circulating
water.
Heat Transfer from the Steam-Air Mixture to the Interface
The first part of the heat transfer occurs from the mixture to the interface. The
interface is defined as the region between the mixture and the condensate surrounding the
horizontal tubes. As described in Malin [1], this heat transfer takes into account the effect of
the heat released at the interface due to the condensing vapor, as well as the sensible heat
transferred through the diffusion layer to the interface. This heat flux is represented in the
following equation.
𝑞"=m𝐿 + 𝜑𝛼𝑔 (𝑇 − 𝑇𝑖 )
4
(9a)
𝜑 = 𝜑𝐻 /(1 − exp(𝜑𝐻 )
(9b)
𝜑𝐻 = 𝑚"𝑐𝑝,𝑠 /𝛼𝑔
(9c)
𝑘
𝛼𝑔 = (𝑑 )(0.33𝑅𝑒𝑐 0.6 𝑃𝑟1/3 )
(9d)
𝑜
The Ackerman-correction factor, φ, is used to account for the effect of mass transfer on
the gas side heat transfer coefficient, αg, for flow across staggered tube banks. The gas side
heat transfer coefficient is a function of the thermal conductivity of the steam-air mixture at
the film temperature, which is the average of the interface temperature and the inlet steam
temperature, the tube outer diameter, the Reynolds number and the Nusselt number. The
Reynolds number is calculated from the assumed velocity profile at each tube row.
Heat Transfer through the Condensate Film
Consistent with Malin [1], the second element of heat transferred from the mixture to
the circulating water is the heat flux through the condensate film, expressed below in Equation
(10a). The equation for this heat flux takes into consideration vapor-shear (Equation (10d)),
superheat (Equation (10e)) and inundation of condensate on the tubes (Equation (10f)) in the
heat transfer coefficient, αc.
𝑞" = 𝛼𝑐 (𝑇𝑖 − 𝑇𝑤𝑜 )
(10a)
𝛼𝑐 = 𝛼𝑐 ∗ 𝐹𝑣 𝐹𝑙 𝐹𝑠
(10b)
1/4
𝛼𝑐 ∗ = 0.735[𝑘𝑐 3 𝐿𝜌𝑐 2 𝑔/(𝑑𝑜 𝜇𝑐 (𝑇𝑖 − 𝑇𝑤𝑜 ))]
(10c)
11.8
𝐹𝑣 = 1 + 0.0095𝑅𝑒𝐴 √𝑁𝑢𝑐
𝐹𝑠 =
𝐹𝑙 = [
𝑚′′′ =
𝑐𝑝,𝑠 (𝑇−𝑇𝑠𝑎𝑡 )
𝐿
∑𝑛
𝑖=1 𝑚′′′𝑛
𝑚′′′𝑛
𝑚"𝐴𝑤
𝑉
(10d)
(10e)
−0.223
]
(10f)
(10g)
The Reynolds number in Equation (10d) is the approach Reynolds number of the
mixture [1], which, for this evaluation, uses the assumed inlet velocity of the mixture. Equation
(10e) factors the latent heat of condensation of the superheated steam into the condensate
film heat transfer coefficient. The inundation factor, Equation (10f), evaluates the cumulative
condensation rate per unit volume, 𝑚′′′, over the condensation rate per unit volume of a given
row. The condensation rate is further defined in the Mass Transfer Relations, section 2.1.4.
Heat Transfer through the Tube Wall
5
The third step in calculating the overall heat transfer in the system is determining the
heat transferred through the tube wall to the circulating water. This heat flux is calculated
using Equation (11a).
𝑞" = 𝛼𝑤𝑙 (𝑇𝑤𝑜 − 𝑇𝑙 )
(11a)
The heat transfer coefficient, αwl, depends on the geometry and material properties of
the tube, as well as the flow conditions inside the tube.
𝛼𝑤𝑙 =
𝛼𝑙 =
𝑑
[𝑑 𝛼𝑜
𝑖 𝑙
+
𝑑
𝑑𝑜 ln⁡( 𝑜)
𝑑𝑖
2𝑘𝑤
−1
+ 𝑅𝑓 ]
(11b)
𝑘𝑙
0.023𝑅𝑒𝑙 0.8 𝑃𝑟𝑙 0.4
𝑑𝑖
(11c)
The initial operating conditions of the circulating water are used to evaluate the
Reynolds and Prandtl numbers in calculating αl.
Mass Transfer Relations
In agreement with Malin [1], mass transfer relations are used in order to accurately
evaluate the total heat transferred in the system. The condensation mass flux, which is
necessary to calculate the heat flux from the steam-air mixture to the condensate interface, is
described by the following equation [1].
𝑀
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑚" = 𝜌𝐾𝑚 𝑀 𝑠 ln⁡(1 +
𝑝𝑠 −𝑝𝑠,𝑖
𝑚
𝑝−𝑝𝑠
𝑃𝑟 2/3
𝛼𝑔 /(𝜌𝑐𝑝,𝑚 )
𝑆𝑐
)
(12a)
𝐾𝑚 = ( )
(12b)
𝑝𝑠 = 𝑝/[1 + 𝑌𝑎 𝑀𝑠 /(𝑌𝑠 𝑀𝑎 )]
(12c)
The condensate mass flux represents the rate at which condensate is formed per unit
area along the tube. It is a function of both the mixture, steam, interface and film properties.
Km is defined as the mass transfer coefficient and is dependent upon the dimensionless ratios of
the Prandtl and Schmidt numbers, the gas side heat transfer coefficient, and the mixture
density and specific heat.
Mathematical Model
6
A mathematical model was created to analyze the outlet circulating water temperature
in the six tube configurations by solving for the heat flux in an iterative manner, first solving for
the circulating water temperature in the first-pass and then again at the exit of the secondpass.
Creating the Baseline Conditions
The initial conditions, operating parameters, condenser and tube bundle geometry,
number of tubes, areas and volumes are established. These values are the same across all six
cases, except for the number of first and second-pass tubes in each row. The steam, air,
mixture, condensate and coolant properties are calculated following the methodology
described in Malin [1]. Using these values, the mixture velocity is calculated in the regions of
the condenser with and without tubes, as described in section 2.1.2. A velocity is calculated for
each row of tubes in the tube bundle.
Calculating Initial Values Based on Assumed Temperatures
Values for the heat and mass transfer coefficients, as defined in sections 2.1.3 and
2.1.4, are calculated based on the properties taken at the assumed initial temperatures of the
interface, film, tube outer wall and circulating water. The temperature at the interface of the
mixture and condensate, the interface temperature, is assumed based on the steam and
circulating water inlet temperatures. The film temperature is then calculated by averaging the
steam inlet temperature and the assumed interface temperature. An estimated value for the
tube outer wall temperature is assumed based on the inlet circulating water temperature and
the initial assumed interface temperature.
Calculating Heat and Mass Transfer Coefficients
The heat and mass transfer coefficients, Equations (9d), (12b), (10b) and (11b), are
calculated using the values defined in section 2.2.2. The gas side heat transfer coefficient,
Equation (9d), is used to solve for the Ackerman correction coefficient, Equation (9c).
The mass transfer coefficient, Km, as defined in Equation (12b), can then be calculated
using the gas side heat transfer coefficient. Using the mass transfer coefficient, the mass
condensate rate per unit area, m”, and per unit volume, m’’’, are obtained from Equations (12a)
and (10g).
The condensate film heat transfer coefficient, which is dependent upon the mass
transfer coefficient, can then be determined once Equations (10c) through (10f) are calculated.
The last heat transfer coefficient, αwl, is calculated from the geometric properties of the
tube and the thermodynamic and transport properties of the circulating water using Equation
(11b).
Calculating the Initial Heat Flux and Temperature Differences
An initial value of the heat flux for each row is found from Equation (9a) using the
previously established values of the mass condensation rate, gas side heat transfer coefficient
and the assumed interface temperature. Equations (10a) and (11a) are rearranged, and using
7
the initial heat flux value, are then solved for the temperature differences between the
interface temperature and the outer wall temperature of the tube, and the difference between
the outer wall temperature and circulating wall temperature.
Iterating to Solve for an Adjusted Heat Flux and Temperature Differences
Equation (9a) is modified to include the temperature differences between the interface
and outer wall, and outer wall and circulating water, shown below.
𝑞"=m"𝐿 + 𝜑𝛼𝑔 (𝑇 − 𝑇𝑙 − (𝑇𝑖 − 𝑇𝑤𝑜 ) − (𝑇𝑤𝑜 − 𝑇𝑙 ))
𝑞"
𝛼𝑐
(13a)
𝑞"
))
𝛼𝑤𝑙
𝑞"=m"𝐿 + 𝜑𝛼𝑔 (𝑇 − 𝑇𝑙 − ( ) − (
(13b)
The heat flux is solved in an iterative fashion using the Newton-Raphson method by
formulating Equation (13b) as a transcendental equation. Equation (13b) is set equal to a
function, F(q”), which is then set equal to zero.
𝑞"
𝑞"
𝐹(𝑞")=q" − m"𝐿 − 𝜑𝛼𝑔 (𝑇 − 𝑇𝑙 − (𝛼 ) − (𝛼 )) = 0
𝑐
(14)
𝑤𝑙
The derivative of F(q”) is taken, Equation (15), and set equal to the definition of the
derivative of F(q”). This relation is shown in Equation (16).
1
1
𝐹 ′ (𝑞") = 1 − 𝜑𝛼𝑔 (− 𝛼 − 𝛼 )
𝑐
𝐹 ′ (𝑞") =
𝐹(𝑞")𝑑𝑒𝑠𝑖𝑟𝑒𝑑 −𝐹(𝑞")𝑐𝑢𝑟𝑟𝑒𝑛𝑡
𝑞"𝑛𝑒𝑤 −𝑞"𝑐𝑢𝑟𝑟𝑒𝑛𝑡
(15)
𝑤𝑙
1
1
= 1 − 𝜑𝛼𝑔 (− 𝛼 − 𝛼 )
𝑐
(16)
𝑤𝑙
The desired value of F(q”) is zero, which corresponds to the computed heat flux
equaling the actual heat flux. Substituting zero into Equation (16) for the desired value of F(q”),
an updated value for the heat flux value can be solved in the following.
1
1
𝑞"𝑛𝑒𝑤 = 𝑞"𝑐𝑢𝑟𝑟𝑒𝑛𝑡 + (0 − 𝐹(𝑞")𝑐𝑢𝑟𝑟𝑒𝑛𝑡 )/ (1 − 𝜑𝛼𝑔 (− 𝛼 − 𝛼 )) (17)
𝑐
𝑤𝑙
The new value of q” calculated in Equation (17) is then used to obtain new values of Ti
and Two as shown below.
8
𝑇𝑤𝑜 = 𝑇𝑙 +
𝑞" 𝑛𝑒𝑤
𝛼𝑤𝑙
𝑇𝑖 = 𝑇𝑤𝑜 +
𝑞" 𝑛𝑒𝑤
𝛼𝑐
(18)
Recalculating Temperature Dependent Properties
With the new values of Ti and Two, from section 2.2.5, the film temperature is
recalculated. An average temperature is used to calculate the thermodynamic properties
dependent upon the film temperature, including dynamic viscosity, thermal conductivity,
specific heat, and Prandtl and Schmidt numbers.
As these values are updated to reflect the new temperatures, the gas side heat transfer
coefficient (Equation (9d)), Ackerman correction factor (Equation (9b)), mass transfer
coefficient (Equation (12b)), interface partial pressure of steam, mass condensation rates
(Equations (12a) and (10g)), remaining heat transfer coefficients (Equation (10b) and (11b)) and
heat flux (Equation (9a)) are also recalculated using the new temperature values.
Reiterating to Obtain Converged Values for Heat Flux and Temperatures
Once a new set of values has been calculated in section 2.2.6, updated values of the
interface and outer tube wall temperature are calculated, following the steps outlined in
sections 2.2.4 and 2.2.5. This process is repeated until the values for the heat flux, interface
and outer tube wall temperatures converge. The iterative process is completed for each row in
the first-pass.
Calculating the Outlet Circulating Water Temperature of the First-Pass
Performing the iterative algorithm will yield a heat flux for each row of tubes in the
first-pass. The outlet circulating water temperature can then be determined for each row using
the simplified energy equation in Equation (5). An average, uniform outlet temperature for the
first-pass is calculated by multiplying the number of tubes in each row by the outlet
temperature of that row and dividing by the total number of tubes in the first-pass. This results
in a weighted average of the outlet temperatures. New circulating properties are calculated
using this first-pass outlet circulating water temperature. This outlet temperature for the firstpass becomes the inlet circulating water temperature for the second-pass tubes.
Calculating the Outlet Temperature of the Second-Pass
The outlet temperature of the second-pass tubes is calculated in the same manner as
for the first-pass, repeating steps outlined in sections 2.2.2 through 2.2.8. The averaged outlet
circulating water temperature for the second-pass is the temperature of the water exiting the
two-pass condenser. This value will be compared among all six cases to determine the tube
configuration that results in the highest circulating water temperature, thus transferring the
most heat to the system.
Performing an Energy Balance
9
A heat balance was performed on the entire system to ensure that the iterative
methodology outlined in the previous sections yielded accurate results. The change in energy
in the steam, circulating water, condensate and through the tube walls was calculated
according to Equation (19).
(𝑚̇𝑐𝑝,𝑐𝑤 𝑇𝑤𝑙 )𝑜𝑢𝑡 − (𝑚̇𝑐𝑝,𝑐𝑤 𝑇𝑤𝑙 )𝑖𝑛 + ⁡(𝑚̇𝑐𝑝,𝑠 (𝑇 − 𝑇0 )𝑌𝑠 )𝑜𝑢𝑡 − ⁡(𝑚̇𝑐𝑝,𝑠 (𝑇 − 𝑇0 )𝑌𝑠 )𝑖𝑛 + 𝑚"𝐿𝐴 +
(𝑘𝑤 𝑇𝑤𝑜 )𝑜𝑢𝑡 − (𝑘𝑤 𝑇𝑤𝑜 )𝑖𝑛 = 0
(19)
A small margin of error in the change in energy in the system is expected and
considered reasonable, given the assumptions and simplifications made throughout the
analysis.
The outlet steam and air mass flow rates were calculated from the outlet steam and air
densities, which were based on the converged outlet mixture temperature. All other variables
in the overall energy balance were either the assumed initial conditions or values obtained in
the iterations performed to calculate the heat flux.
Assumptions
To determine the maximum outlet temperature of the circulating water, the following
assumptions are made to simplify the problem in order to solve it using the mathematical
model outlined in section 2.2.
1.
A symmetrical condenser with a horizontal, two-pass tube bundle is examined,
where the steam inlet opening to the condenser is in the fore/aft and athwartship
center of the condenser.
2.
The algorithm evaluates the condenser in two dimensions, assuming that the heat
transfer is constant along the length of the tube bundle, but varies through and
around the circumference of the tube bundle.
3.
The condenser dimensions and material are constant as well as the steam inlet
velocity for all six cases.
4.
For the six cases that are analyzed in the study, the inlet temperature of the
circulating water is kept constant, along with the individual tube size, spacing
and number of tubes, thereby keeping the overall tube bundle size constant.
5.
The circulating water is turbulent flow and complete mixing of the water is
assumed at the end of the first-pass to ensure a uniform circulating water inlet
temperature for the second-pass tubes.
10
6.
The circulating water temperature does not become a function of the number of
tubes in each pass, since the number of first and second-pass tubes is kept as
equal as possible.
7.
In the FLOW3D models, the steam-air mixture flow is assumed laminar in order
to accelerate the simulations so that a comparison to the Mehrabian velocity
approach may be made.
Initial Conditions
Condenser Dimensions
A two-pass condenser with a rectangular upper shell is used in the analyses. The upper
shell is 3.048 meters long, 0.6096 meters high and 0.6096 meters wide. The tube bundle is
approximately 0.3048 meters in diameter and is centered in the upper shell of the condenser.
The steam inlet opening is centered on the top of the condenser and is 0.3048 meters wide and
0.6096 meters long. The opening between the upper shell and the hotwell that collects the
condensate formed around the tubes is 0.1524 meters wide and runs the length of the
condenser. The condenser is assumed to be made of alloy 316L stainless steel.
Number of Tubes
There are 217 titanium tubes in the tube bundle, each with a 0.0143-meter outer
diameter. The tubes are 20 Birmingham Wire Gage (BWG), resulting in an inner diameter of
0.0125-meters. Two alternating rows of tubes are used to create the bundle. One row, row
“A,” is centered with a tube on the vertical centerline of the tube bundle. The alternating row,
row “B,” is offset by half of one tube diameter from the tube bundle vertical centerline. The
tube pitch, or centerline-to-centerline distance between tubes, Figure 2, is the same in the
vertical and horizontal direction.
Figure 2. Tube Row Orientation
Operating Conditions
11
Shell-Side Parameters
Superheated steam enters the condenser through the steam inlet opening on the top
of the condenser at 116.5°C. This inlet temperature, along with several other parameters,
including the steam and air mass inflow rates and condenser operating pressure, have been
kept constant in accordance with Malin [1]. The steam mass inflow rate is set to 8.63 kg/s, the
air mass inflow rate to 2.78 g/s and the condenser pressure to 34500 Pa.
Circulating Water Parameters
The circulating water is assumed to enter the tubes at 21.11°C at a velocity of 1.524
m/s. A fouling factor of 1.5·105 (m2K)/W is used to account for the cleanliness inside the tubes.
Numerical Analysis - Modeling Using RANS Solver
FLOW3D, CFD software developed by Flow Science Inc., is used to simulate the
condenser for each of the six cases. The condenser geometry, initial conditions, operating
parameters and assumptions made in the heat and mass transfer algorithm, sections 2.2
through 2.4, were used to create the FLOW3D models. Analyzing the condenser and tube
bundle using FLOW3D generated a steam-air mixture velocity profile, which was used to
confirm the velocity profile created in the heat and mass transfer algorithm.
A numerical mesh was created for each of the six cases. A large grid was generated
that included the entire cross-section of the condenser. A smaller, denser grid embedded
within the larger grid was created for the tube bundle. This nested grid permitted greater
resolution around the individual tubes. The first-pass tubes and second-pass tubes were
grouped into separate subcomponents within the nested grid. These tube regions were further
arranged into separate subcomponents for Cases 3 through 6 in order to group together the
tubes exhibiting similar heat fluxes and circulating water temperatures, which varied as a result
of the tube configurations. Since the subcomponents are treated as having the same
properties, smaller subcomponents had properties closer to the actual properties of the
individual tubes that made up each subcomponent. The average circulating water temperature
and overall heat transfer coefficient was calculated for each subcomponent. In order for
FLOW3D to treat the tubes as having a constant circulating water inlet temperature, fixed
surface heat transfer coefficients were applied to the tubes, thus assuming the tubes were
maintained at a constant temperature. This was necessary to prevent the tube inlet circulating
water temperature from converging to a higher temperature with the steam inlet temperature,
preventing any heat transfer from occurring.
Results
Analytical Results
The six tube configurations presented in section 1.2 were analyzed to determine the
outlet circulating water temperature using the mathematical model described in section 2.2.
12
Since the tube bundle contains an odd number of tubes, the number of tubes has been divided
as equally as possible in the first and second-passes to prevent the number of tubes in a
particular pass from influencing the circulating water temperature.
The mathematical model, based on the work of Malin [1], employs an iterative solution
method to solve for the heat flux, and subsequently for the outlet circulating water
temperature. Applying the algorithm to the six cases yielded values for the heat flux from the
steam-air mixture to the circulating water, the outer tube wall temperature, the interface
temperature and the circulating water temperature for every row of tubes in the tube bundle.
The heat flux distribution through the tube bundle was analyzed by graphing the change in
circulating water temperature for each row along the length of the tubes in the first and
second-passes. The six cases were compared by evaluating the average circulating water
temperatures of the first-pass and second-pass tubes.
An energy balance was performed for each case to validate the algorithm results. The
results of the energy balance for Case 1 are provided and are representative of the results
obtained from each case since the methodology presented in section 2.2.10 was followed for
all six cases.
The six tube configurations were modeled in FLOW3D, which provided the velocity of
the steam-air mixture. The FLOW3D velocity profiles were used in the algorithm to calculate
circulating water temperatures. Comparisons between the initial results from the algorithm
using velocity profiles based on Mehrabian [4] and those obtained using FLOW3D data are
presented. The velocity profiles used are provided in Appendix A – Steam-Air Mixture Velocity
Profiles. The FLOW3D input file used to simulate Case 1 is provided in Appendix B – FLOW3D
Input File for Case 1. The FLOW3D input files for Cases 2 through 6 are similar to that shown in
Appendix B.
Case 1
The Case 1 tube bundle is split horizontally, with the first-pass tubes comprised of the
top 12 rows of tubes and half of the tubes in row 13. The second-pass is comprised of half of
the tubes in row 13 and all of the tubes in the remaining bottom 12 rows. The tube
configuration is shown below in Figure 3.
13
Figure 3. Case 1 Tube Configuration
This configuration was chosen to demonstrate the effect on the circulating water
temperature of a horizontal tube configuration with the first-pass on the top half of the tube
bundle. The number of tubes in the first-pass totals 109 and the number of tubes in the
second-pass totals 108.
Algorithm Results Using Mehrabian-Based [4] Velocities
The results from Case 1 show that for both the first and second-passes, the highest
heat transfer occurs in the lower the tube rows. This results in a higher temperature for the
circulating water. This correlation is seen in Figure 4 and Figure 5, and presented quantitatively
in Table 1 and Table 2. The topmost row in each pass, row 1 and row 13, has the coolest
circulating water temperatures in the first and second-pass, respectively.
Figure 4. Case 1 First-Pass Circulating Water Temperature vs Tube Length
14
Figure 5. Case 1 Second-Pass Circulating Water Temperature vs Tube Length
Table 1. Case 1 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
1
1.274E+04
23.284
13
2.597E+04
29.302
2
1.547E+04
23.749
14
2.646E+04
29.385
3
1.729E+04
24.059
15
2.692E+04
29.463
4
1.870E+04
24.299
16
2.735E+04
29.537
5
1.985E+04
24.496
17
2.776E+04
29.607
6
2.083E+04
24.664
18
2.815E+04
29.674
7
2.169E+04
24.810
19
2.852E+04
29.737
8
2.246E+04
24.941
20
2.888E+04
29.798
9
2.316E+04
25.061
21
2.922E+04
29.857
10
2.380E+04
25.170
22
2.955E+04
29.913
11
2.439E+04
25.271
23
2.986E+04
29.966
15
12
2.494E+04
25.364
24
3.016E+04
30.016
13
2.546E+04
25.452
25
3.045E+04
30.066
There is a larger temperature variation of the outlet circulating water temperature in
the first-pass tubes (2.168°C), than in the second-pass tubes (0.764°C), showing that less
variation in heat transfer is occurring in the lower half of the tube bundle, resulting in a more
uniform temperature rise in the circulating water in that half. The averaged circulating water
outlet temperature for the Case 1 first-pass tubes is calculated to be 24.872°C and for the
second-pass tubes it is 29.657°C. The change between the inlet circulating water temperature
and the outlet circulating water temperature of the first-pass is 3.762°C. The change between
the inlet and outlet circulating water temperatures of the second-pass is 4.785°C. These values
are shown below in Table 2, as well as in the comparison of all six cases in Table 18. There is a
greater amount of heat transfer taking place in the second-pass tubes than in the first-pass.
The overall change in circulating water temperature in Case 1 is 8.547°C.
Table 2. Case 1 Calculated Temperatures
Case 1 Temperatures (°C)
Inlet
21.11
First-Pass Outlet
24.872
Second-Pass Outlet
29.657
ΔT Between Passes
4.786
Overall ΔT
8.547
The larger change in circulating water temperature in the second-pass can be
attributed to the velocity profile. The cumulative mass condensation rate is larger as more
condensate continues to form moving from the top of the tube bundle to the bottom. This
could lead the higher heat flux rates, and consequently, the higher circulating water
temperatures.
To verify the algorithm, an energy balance was performed on the entire system,
according to Equation (19). The results are provided in Table 3.
Table 3. Case 1 Energy Balance
E out (kJ/s)
16
E in (kJ/s)
ΔE (kJ/s)
Circulating Water
(mcpΔT)
242.12
236.09
6.02
Air (YaHama)
0.03
0.02
0.01
Steam (YsHsms)
0.00
1909.64
-1909.64
Tube Wall (kΔT)
6591.90
6809.32
-217.42
Condensation
(m"LA)
1907.25
Total
8741.30
1907.25
8955.06
Percent Change in Energy
-213.76
-2.45%
The energy balance shows less than 3% error between the energy introduced into the
system and the energy leaving the system. This error is considered acceptable, and can be
attributed to either simplifications made in the study or an iterative convergence error.
FLOW3D Results
The values used to create the condenser tube configuration, initial conditions and
operating parameters for Case 1 were used in a FLOW3D model to provide velocity profiles,
from which new values for the circulating water temperature were calculated. The resulting
velocity profile is shown in the Figure 6 velocity magnitude contours.
17
Figure 6. Case 1 FLOW3D Velocity Magnitude Contours
18
Temperature (K)
Condenser Height (m)
Unlike the velocity profile assumed by Mehrabian in the initial calculation of the
circulating water temperature, Figure 6 shows that the mixture velocity slows down as it moves
through the tube bundle and accelerates out through the bottom opening. There is some
localized swirling effect in the upper corners of the condenser shell as the steam enters the
condenser. The strongest velocity through the tubes is in the upper center third of the
configuration where the steam is moving downward through the condenser inlet opening
without any interference or directional change. The FLOW3D-predicted steam-air mixture
temperature contours are shown in Figure 7 for the Case 1 tube configuration.
The Case 1 FLOW3D shaded temperature contours show the cooling of the steam-air
mixture as it travels through the tube bundle. The Figure 6 velocity values are used in the
algorithm and produced the circulating water temperature distribution shown in Figure 8 and
Figure 9 for the first-pass and second-pass tubes, respectively.
19
Figure 7. Case 1 FLOW3D Mixture Temperature Contours
Figure 8. Case 1 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
Figure 9. Case 1 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
20
The circulating water temperature values for the first and second-pass tube rows are
provided in Table 4, in addition to the heat flux at each row. The heat flux and temperature
generally decrease at lower rows in the tube bundle. This corresponds to the decreasing
velocities through the tube bundle. However, unlike the assumed velocity profile based on
Mehrabian [4], FLOW3D provides an accurate depiction of the velocity profile through the tube
bundle due to the inclusion of the condenser operating parameters. The FLOW3D velocity
profile shows that as the mixture moves through the tube bundle, the velocity does not
consistently change in magnitude, as it does with Mehrabian [4]. There are local increases in
velocity in rows 2, 5, 14, 23 and 25, which cause the circulating water temperature in these
rows to be slightly higher than the row located directly above, exhibited in Figure 8 and Figure
9. The effect of this is seen in the circulating water temperatures given in Table 4.
Table 4. Case 1 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
1
8.988E+03
22.644
13
5.497E+03
23.317
2
9.016E+03
22.649
14
5.645E+03
23.342
3
8.740E+03
22.601
15
5.277E+03
23.280
4
8.269E+03
22.521
16
5.204E+03
23.267
5
8.677E+03
22.591
17
4.564E+03
23.158
6
8.471E+03
22.556
18
4.405E+03
23.131
7
8.399E+03
22.543
19
4.036E+03
23.068
8
7.424E+03
22.377
20
3.975E+03
23.058
9
7.311E+03
22.358
21
3.672E+03
23.006
10
7.260E+03
22.349
22
3.400E+03
22.960
11
6.061E+03
22.145
23
3.458E+03
22.969
12
6.072E+03
22.147
24
3.297E+03
22.942
13
5.182E+03
21.995
25
3.403E+03
22.960
The outlet circulating water temperatures for the first and second-passes calculated in
the initial algorithm using the velocity profile based on Mehrabian, and those calculated using
FLOW3D are provided in Table 5. The velocity profile using Mehrabian was much larger,
21
resulting in more heat transferred to the system and significantly higher circulating water
temperatures. The first-pass outlet temperature is 22.380°C and the second-pass outlet
temperature is 23.147°C. According to the evaluation performed using FLOW3D data, the
circulating water temperature increased by only 2.037°C, compared to 8.547°C based on the
evaluation using Mehrabian velocities.
Table 5. Case 1 Comparison of Mehrabian and FLOW3D Calculated Temperatures
Temperature (°C)
Mehrabian
FLOW3D
First-Pass Outlet
24.872
22.380
Second-Pass Outlet
29.657
23.147
ΔT Between Passes
4.786
0.767
Overall ΔT
8.547
2.037
The significantly lower velocities obtained from FLOW3D result in very little heat
transfer to the circulating water compared to Mehrabian. This shows a proportional
relationship between the magnitude of the steam-air mixture velocity and the circulating water
temperature.
Case 2
Similar to Case 1, the Case 2 tube bundle is split horizontally. However, the first and
second-passes are reversed from the Case 1 configuration, such that the first-pass is on the
bottom half of the bundle and the second-pass is on the top. The first-pass contains 108 tubes
and the second-pass contains 109 tubes. Any potential effect on circulating water temperature
due to the change in one tube from the first to second-pass between Cases 1 and 2 is
considered negligible.
22
Figure 10. Case 2 Tube Configuration
Algorithm Results Using Mehrabian-Based [4] Velocities
As was calculated in Case 1, for each pass, the lower the tube row is in the bundle, the
larger the heat flux and the higher the circulating water temperature. This correlation is shown
in Figure 11 and Figure 12. The actual temperatures and heat flux associated with each tube
row are presented in Table 6.
23
Figure 11. Case 2 First-Pass Circulating Water Temperature vs Tube Length
Figure 12. Case 2 Second-Pass Circulating Water Temperature vs Tube Length
24
Table 6. Case 2 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
13
2.650E+04
25.630
1
1.256E+04
28.138
14
2.700E+04
25.715
2
1.524E+04
28.597
15
2.747E+04
25.795
3
1.703E+04
28.902
16
2.792E+04
25.872
4
1.841E+04
29.138
17
2.834E+04
25.944
5
1.955E+04
29.331
18
2.875E+04
26.013
6
2.051E+04
29.496
19
2.913E+04
26.078
7
2.136E+04
29.640
20
2.951E+04
26.142
8
2.211E+04
29.768
21
2.986E+04
26.203
9
2.280E+04
29.885
22
3.020E+04
26.260
10
2.342E+04
29.993
23
3.052E+04
26.315
11
2.400E+04
30.091
24
3.083E+04
26.368
12
2.454E+04
30.183
25
3.112E+04
26.418
13
2.505E+04
30.269
With the first-pass tubes on the bottom half of the tube bundle, more heat is
transferred to the circulating water, resulting in an average outlet temperature in the first-pass
tubes of 25.996°C and in the second-pass tubes of 29.699°C.
Both the first and second-pass circulating water temperatures increase along the length
of the tube; however, more heat is transferred to the first-pass tubes than to the second-pass
tubes. Since the first-pass tubes are heated to a higher circulating water temperature, this
creates a higher inlet circulating water temperature for the second-pass tubes, compared to
the Case 1 configuration. Overall, the tube configuration in Case 2 allows more heat to be
transferred to the circulating water than Case 1, as shown in the temperature comparison in
Table 7.
25
Table 7. Case 2 Calculated Temperature Comparison with Case 1
Temperature (°C)
Case 1
Case 2
First-Pass Outlet
24.872
25.996
Second-Pass Outlet
29.657
29.699
ΔT Between Passes
4.786
3.703
Overall ΔT
8.547
8.589
FLOW3D Results
Velocity (m/s)
Condenser Height (m)
As was done with Case 1, the Case 2 properties and parameters were used to create a
simulation for the Case 2 tube configuration in FLOW3D. The velocity profile of the steam-air
mixture through the Case 2 tube bundle is shown in Figure 13 and steam-air mixture
temperature contours in Figure 14.
Condenser Width (m)
26
Figure 13. Case 2 FLOW3D Velocity Magnitude Contours
Temperature (K)
Condenser Height (m)
The Case 2 FLOW3D velocity contours are similar to the one from Case 1; however the
steam-air mixture temperature contours differ from Case 1 and show greater distinction
between the first and second-pass tubes. This is seen in the Case 2 condenser temperature
contours, Figure 14.
Condenser Width (m)
Figure 14. Case 2 FLOW3D Mixture Temperature Contours
In Figure 14, the location of the colder first-pass tubes is evident. The steam-air
mixture is cooled as it progresses downward through the tube bundle, and cools even more
once it reaches the first-pass tubes. Since the warmer second-pass tubes are located on the
top half of the tube bundle, there is not as much heat transfer occurring compared to the lower
half of the tube bundle.
27
The resultant velocity for each tube row from Figure 13 created the circulating water
temperature distribution using the established algorithm for the first-pass and second-pass,
Figure 15 and Figure 16, respectively.
Figure 15. Case 2 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
28
Figure 16. Case 2 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
Similar to the results obtained from the Case 1 FLOW3D data, the Case 2 circulating
water temperature generally decreases with each lower row of tubes, although certain rows in
each first and second-passes do not follow this trend, because of the resultant velocity
associated with that row. A localized increase in velocity increases the circulating water
temperature to a value higher than that of the row located immediately above in the tube
bundle. This is seen in rows 18 and 25 in the first-pass and rows 2, 5, 10, and 14 in the secondpass. The heat flux and outlet circulating water temperatures for each row in the two passes
are provided in Table 8.
Table 8. Case 2 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data
First-Pass
Second-Pass
29
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
13
5.424E+03
22.036
1
9.288E+03
23.468
14
5.565E+03
22.060
2
9.292E+03
23.468
15
5.218E+03
22.001
3
9.085E+03
23.433
16
5.135E+03
21.987
4
8.609E+03
23.352
17
4.535E+03
21.884
5
9.913E+03
23.574
18
4.579E+03
21.892
6
8.761E+03
23.378
19
4.173E+03
21.823
7
8.684E+03
23.365
20
3.947E+03
21.784
8
7.719E+03
23.200
21
3.832E+03
21.764
9
7.205E+03
23.113
22
3.640E+03
21.732
10
7.515E+03
23.165
23
3.541E+03
21.715
11
6.321E+03
22.962
24
3.375E+03
21.687
12
6.273E+03
22.954
25
3.522E+03
21.712
13
5.383E+03
22.802
Based on these results, the circulating water is heated by 2.096°C to an average outlet
temperature of 23.206°C.
Case 3
The Case 3 tube configuration has the first and second-pass tubes divided along a 45degree angle, where the first-pass is above the rotated axis and the second-pass is below. This
configuration was chosen to determine the impact of exposing more of the second-pass tubes
directly to the steam entering the condenser, while the majority of the first-pass tubes remain
in the upper portion of the tube bundle. In the first-pass there are 109 tubes and in the
second-pass there are 108 tubes. The results of Case 3 can be compared to those of Case 1 to
evaluate the effect of moving some of the first-pass tubes down to lower rows in the tube
bundle.
30
Figure 17. Case 3 Tube Configuration
Algorithm Results Using Mehrabian-Based [4] Velocities
The results for Case 4 using a Mehrabian-based velocity profile through the tube
bundle shows the lower tube bundles have the greatest change in circulating water
temperature. The bottom-most row of tubes in the first-pass, row 20, and in the second-pass,
row 25, has the hottest circulating water temperatures. These results are presented in Figure
18, Figure 19 and Table 9.
31
Figure 18. Case 3 First-Pass Circulating Water Temperature vs Tube Length
Figure 19. Case 3 Second-Pass Circulating Water Temperature vs Tube Length
32
Table 9. Case 3 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
1
1.319E+04
23.360
6
2.120E+04
28.713
2
1.601E+04
23.841
7
2.205E+04
28.859
3
1.788E+04
24.161
8
2.282E+04
28.990
4
1.934E+04
24.408
9
2.352E+04
29.109
5
2.053E+04
24.611
10
2.416E+04
29.219
6
2.154E+04
24.784
11
2.475E+04
29.319
7
2.242E+04
24.935
12
2.530E+04
29.413
8
2.322E+04
25.070
13
2.582E+04
29.501
9
2.394E+04
25.193
14
2.630E+04
29.584
10
2.460E+04
25.306
15
2.676E+04
29.661
11
2.520E+04
25.409
16
2.719E+04
29.735
12
2.577E+04
25.505
17
2.760E+04
29.805
13
2.630E+04
25.595
18
2.799E+04
29.871
14
2.679E+04
25.680
19
2.836E+04
29.934
15
2.726E+04
25.759
20
2.872E+04
29.995
16
2.770E+04
25.835
21
2.906E+04
30.053
17
2.812E+04
25.906
22
2.938E+04
30.109
18
2.852E+04
25.974
23
2.969E+04
30.162
19
2.889E+04
26.038
24
2.999E+04
30.212
20
2.926E+04
26.101
25
3.027E+04
30.261
21
2.961E+04
26.161
The average outlet circulating water temperature for the first-pass rows is 25.097°C
and for the second-pass rows is 29.758°C. The Case 3 configuration increases the circulating
water temperature by 8.648°C from the initial inlet temperature in the first-pass rows.
33
FLOW3D Results
Velocity (m/s)
Condenser Height (m)
The Case 3 velocity contours are shown below in Figure 20 and the steam-air mixture
temperature contours are shown in Figure 21.
Condenser Width (m)
Figure 20. Case 3 FLOW3D Velocity Magnitude Contours
34
35
Temperature (K)
Condenser Height (m)
Figure 21. Case 3 FLOW3D Mixture Temperature Contours
36
The first-pass tubes can be seen above the 45-degree axis dividing the two passes. The
steam-air mixture is cooling as it moves downward through the tube bundle, particularly in the
region where the first-pass tubes are in the lower potion of the tube bundle. The FLOW3D data
is used in the algorithm to create the circulating water temperature distribution. As seen in
Figure 22, Figure 23 and Table 10, there is an increase in velocity at row 5, which caused the
circulating water temperature to rise above the temperature of the higher elevation rows. The
velocity decreased through the tube bundle, except at rows 2, 5, 10, 14, 18 and 25 where there
were local increases.
Figure 22. Case 3 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
37
Figure 23. Case 3 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
Table 10. Case 3 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
1
9.361E+03
22.707
6
8.702E+03
23.859
2
9.357E+03
22.707
7
8.629E+03
23.846
3
9.152E+03
22.672
8
7.671E+03
23.683
4
8.674E+03
22.590
9
7.139E+03
23.592
5
9.978E+03
22.813
10
7.481E+03
23.650
6
8.812E+03
22.614
11
6.292E+03
23.448
7
8.741E+03
22.602
12
6.239E+03
23.439
8
7.774E+03
22.437
13
5.312E+03
23.280
9
7.236E+03
22.345
14
5.457E+03
23.305
10
7.583E+03
22.404
15
5.087E+03
23.242
38
11
6.380E+03
22.199
16
5.027E+03
23.232
12
6.326E+03
22.190
17
4.405E+03
23.126
13
5.387E+03
22.030
18
4.478E+03
23.138
14
5.534E+03
22.055
19
4.032E+03
23.062
15
5.160E+03
21.991
20
3.837E+03
23.029
16
5.099E+03
21.981
21
3.654E+03
22.998
17
4.468E+03
21.873
22
3.508E+03
22.973
18
4.542E+03
21.886
23
3.328E+03
22.942
19
4.090E+03
21.809
24
3.179E+03
22.917
20
3.892E+03
21.775
25
3.228E+03
22.925
21
3.707E+03
21.743
The Case 3 FLOW3D analysis results in an average first-pass and second-pass circulating
water outlet temperature of 22.374°C and 23.192°C, respectively.
Case 4
Case 4 is the opposite configuration from Case 3, where the first-pass is on the bottom,
Figure 24. This provides a clear comparison between Case 4 and Case 3, showing the impact of
reversing the passes, as well as a comparison to Case 2, where all of the first-pass tubes were
on the bottom half of the tube bundle.
Figure 24. Case 4 Tube Configuration
39
Algorithm Results Using Mehrabian-Based [4] Velocities
The results obtained from the algorithm for the Case 4 tube configuration follow the
same correlation established in the previous cases. Similar to that shown in Case 2, where the
first-pass is on the bottom of the bundle, the circulating water through the first-pass tubes
heats up more than in the Case 3 configuration, which results in a greater increase in the
second-pass circulating water temperature. The circulating water temperature for the first and
second-passes along the length of the tubes is seen in Figure 25 and Figure 26, respectively.
The calculated values of the circulating water temperature at each row, and the corresponding
heat flux, are provided in Table 11.
Figure 25. Case 4 First-Pass Circulating Water Temperature vs Tube Length
40
Figure 26. Case 4 Second-Pass Circulating Water Temperature vs Tube Length
Table 11. Case 4 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
6
2.158E+04
24.791
1
1.254E+04
28.002
7
2.246E+04
24.941
2
1.523E+04
28.460
8
2.325E+04
25.075
3
1.701E+04
28.764
9
2.397E+04
25.198
4
1.839E+04
29.000
10
2.463E+04
25.310
5
1.952E+04
29.193
11
2.523E+04
25.414
6
2.049E+04
29.357
12
2.580E+04
25.510
7
2.133E+04
29.501
13
2.633E+04
25.600
8
2.208E+04
29.629
14
2.682E+04
25.684
9
2.277E+04
29.746
15
2.728E+04
25.764
10
2.339E+04
29.853
41
16
2.773E+04
25.839
11
2.397E+04
29.951
17
2.815E+04
25.911
12
2.451E+04
30.043
18
2.854E+04
25.978
13
2.501E+04
30.129
19
2.892E+04
26.043
14
2.548E+04
30.209
20
2.929E+04
26.106
15
2.592E+04
30.285
21
2.964E+04
26.165
16
2.634E+04
30.356
22
2.997E+04
26.222
17
2.674E+04
30.424
23
3.029E+04
26.276
18
2.712E+04
30.489
24
3.060E+04
26.328
19
2.748E+04
30.550
25
3.089E+04
26.378
20
2.783E+04
30.609
21
2.816E+04
30.666
The average outlet circulating water temperature for the first-pass is 25.862°C and for
the second pass is 29.654°C, which is an increase from the initial temperature of 8.544°C.
FLOW3D Results
42
Velocity (m/s)
Condenser Height (m)
The Case 4 FLOW3D velocity profile exhibits a local increase in velocity at the second
row and then decreases in magnitude until another spike in row 5. The decreasing velocity
trend continues through the tube bundle, except for rows 10, 14, 18 and 25. The overall
velocity profile, Figure 27, is similar to those obtained in the previous cases.
Condenser Width (m)
Figure 27. Case 4 FLOW3D Velocity Magnitude Contours
The FLOW3D steam-air mixture temperature distribution for Case 4 is shown in Figure
28.
43
Temperature (K)
Condenser Height (m)
Figure 28. Case 4 FLOW3D Mixture Temperature Contours
44
The configuration of the colder first-pass tubes in Case 4 is apparent in the Figure 28
contour. These tubes are readily seen below the 45-degree axis that divides the first and
second-passes.
The sudden increase in velocity at rows 2, 5, 10, 14, 18 and 25 cause the circulating
water temperatures in these particular rows to be higher than the surrounding rows, as
demonstrated in Figure 29, Figure 30 and quantitatively shown in Table 12.
45
Figure 29. Case 4 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
Figure 30. Case 4 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
46
Table 12. Case 4 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
6
8.813E+03
22.614
1
9.266E+03
23.519
7
8.740E+03
22.601
2
9.275E+03
23.521
8
7.779E+03
22.438
3
9.068E+03
23.485
9
7.245E+03
22.347
4
8.593E+03
23.405
10
7.567E+03
22.401
5
9.884E+03
23.625
11
6.362E+03
22.196
6
8.731E+03
23.428
12
6.303E+03
22.186
7
8.659E+03
23.416
13
5.394E+03
22.031
8
7.706E+03
23.253
14
5.533E+03
22.055
9
7.177E+03
23.163
15
5.166E+03
21.992
10
7.497E+03
23.218
16
5.093E+03
21.980
11
6.302E+03
23.014
17
4.469E+03
21.873
12
6.244E+03
23.004
18
4.534E+03
21.884
13
5.343E+03
22.850
19
4.092E+03
21.809
14
5.481E+03
22.874
20
3.899E+03
21.776
15
5.117E+03
22.812
21
3.689E+03
21.740
16
5.045E+03
22.799
22
3.548E+03
21.716
17
4.426E+03
22.694
23
3.354E+03
21.683
18
4.491E+03
22.705
24
3.209E+03
21.658
19
4.053E+03
22.630
25
3.272E+03
21.669
20
3.861E+03
22.598
21
3.654E+03
22.562
The average outlet circulating water temperature from the first-pass tubes is 21.939°C.
The exit circulating water temperature from the second-pass tubes is 23.190°C.
Case 5
47
The Case 5 tube configuration evaluates the outlet circulating water temperature with
the first-pass tubes on the outer edges of the bundle and the second-pass tubes centered in the
middle of the tube bundle, Figure 31. The first-pass consists of 108 tubes and the second-pass
consists of 109 tubes.
Figure 31. Case 5 Tube Configuration
Algorithm Results Using Mehrabian-Based [4] Velocities
The results from Case 4 show that for both the first and second-passes, the lower the
tube row is in the tube bundle, the larger the amount of heat transferred to the system and the
higher the temperature of the circulating water. This correlation is seen in Figure 32 and Figure
33, and presented quantitatively in Table 13. The topmost row in each pass, row 1 and row 5,
has the coolest circulating water temperatures in the first and second-pass, respectively.
48
Case 5 - First-Pass Circulating Water Temperature vs Tube Length
27.00
Row 1
Circulating Water Temperature (°C)
26.00
25.00
24.00
23.00
22.00
21.00
0
0.5
1
1.5
2
2.5
3
3.5
Row 2
Row 3
Row 4
Row 5
Row 6
Row 7
Row 8
Row 9
Row 10
Row 11
Row 12
Row 13
Row 14
Row 15
Row 16
Row 17
Row 18
Row 19
Row 20
Row 21
Row 22
Row 23
Row 24
Row 25
Average
Tube Length (m)
Figure 32. Case 5 First-Pass Circulating Water Temperature vs Tube Length
Figure 33. Case 5 Second-Pass Circulating Water Temperature vs Tube Length
Case 5 - Second-Pass Circulating Water Temperature vs Tube Length
31.00
Row 5
30.00
Row 6
Row 7
Circulating Water Temperature (°C)
Row 8
29.00
Row 9
Row 10
Row 11
Row 12
28.00
Row 13
Row 14
Row 15
27.00
Row 16
Row 17
Row 18
Row 19
26.00
Row 20
Row 21
Average
25.00
0
0.5
1
1.5
2
Tube Length (m)
49
2.5
3
3.5
Table 13. Case 5 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
1
1.319E+04
23.361
5
1.961E+04
28.755
2
1.602E+04
23.842
6
2.059E+04
28.921
3
1.789E+04
24.162
7
2.144E+04
29.066
4
1.934E+04
24.409
8
2.220E+04
29.195
5
2.053E+04
24.612
9
2.289E+04
29.313
6
2.155E+04
24.785
10
2.352E+04
29.421
7
2.243E+04
24.936
11
2.410E+04
29.520
8
2.322E+04
25.071
12
2.464E+04
29.612
9
2.394E+04
25.194
13
2.515E+04
29.698
10
2.461E+04
25.307
14
2.562E+04
29.779
11
2.521E+04
25.410
15
2.606E+04
29.855
12
2.578E+04
25.506
16
2.649E+04
29.927
13
2.631E+04
25.597
17
2.689E+04
29.996
14
2.680E+04
25.681
18
2.727E+04
30.061
15
2.727E+04
25.761
19
2.763E+04
30.123
16
2.771E+04
25.836
20
2.798E+04
30.182
17
2.813E+04
25.907
21
2.831E+04
30.239
18
2.853E+04
25.975
19
2.890E+04
26.040
20
2.927E+04
26.103
21
2.962E+04
26.162
22
2.996E+04
26.219
23
3.027E+04
26.273
24
3.058E+04
26.325
25
3.087E+04
26.375
50
The average outlet circulating water temperatures for the first and second-pass tubes
are presented in Table 14.
Table 14. Case 5 Calculated Temperatures
Case 5 Temperatures (°C)
First-Pass Outlet
25.409
Second-Pass Outlet
29.648
ΔT Between Passes
4.239
Overall ΔT
8.538
FLOW3D Results
Velocity (m/s)
Condenser Height (m)
A generally decreasing velocity profile is constructed for Case 5 in FLOW3D, which is
shown in Figure 34.
51
Figure 34. Case 5 FLOW3D Velocity Magnitude Contours
Condenser Width (m)
Figure 35. Case 5 FLOW3D Mixture Temperature Contours
52
Temperature (K)
Condenser Height (m)
The FLOW3D simulation also provided the steam-air mixture temperature contours
shown in Figure 35. The first-pass tubes are more clearly seen on the outer edges in the lower
half of the tube bundle.
The local velocities at rows 5, 10, 14, 18 and 25 increase to a magnitude greater than
that of the velocity at the row located above it in the tube bundle. The effect of the increase in
velocity causes a local increase in circulating water temperature, thereby correlating the
velocity proportionally to the circulating water temperature. For this reason, the circulating
Case 5 - First-Pass Circulating Water Temperature vs Tube Length
Using FLOW3D Data
Row 1
Row 2
Row 3
Row 4
Row 5
Row 6
Row 7
Row 8
Row 9
Row 10
Row 11
Row 12
Row 13
Row 14
Row 15
Row 16
Row 17
Row 18
Row 19
Row 20
Row 21
Row 22
Row 23
Row 24
Row 25
Average
23.00
22.80
Circulating Water Temperature (°C)
22.60
22.40
22.20
22.00
21.80
21.60
21.40
21.20
21.00
0
0.5
1
1.5
2
2.5
3
3.5
Tube Length (m)
water temperatures in rows 5, 10, 14, 18 and 25 are slightly higher than the temperature of the
circulating water in the row directly above these five rows, shown in Figure 36 and Figure 37,
and quantitatively in Table 15.
Figure 36. Case 5 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
53
Case 5 - Second Pass Circulating Water Temperature vs Tube Length
Using FLOW3D Data
24.00
23.80
Row 5
Row 6
Circulating Water Temperature (°C)
23.60
Row 7
Row 8
23.40
Row 9
Row 10
23.20
Row 11
Row 12
23.00
Row 13
Row 14
22.80
Row 15
Row 16
22.60
Row 17
Row 18
22.40
Row 19
Row 20
22.20
Row 21
Average
22.00
0
0.5
1
1.5
2
2.5
3
3.5
Tube Length (m)
Figure 37. Case 5 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
54
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
1
9.370E+03
22.709
5
9.867E+03
23.869
2
9.349E+03
22.705
6
8.715E+03
23.673
3
9.141E+03
22.670
7
8.648E+03
23.661
4
8.662E+03
22.588
8
7.687E+03
23.497
5
9.985E+03
22.814
9
7.147E+03
23.405
6
8.820E+03
22.615
10
7.504E+03
23.466
7
8.752E+03
22.603
11
6.316E+03
23.263
8
7.780E+03
22.438
12
6.307E+03
23.262
9
7.234E+03
22.345
13
5.398E+03
23.107
10
7.594E+03
22.406
14
5.463E+03
23.118
11
6.393E+03
22.201
15
5.106E+03
23.057
12
6.383E+03
22.200
16
4.978E+03
23.035
13
5.465E+03
22.043
17
4.624E+03
22.975
14
5.530E+03
22.054
18
4.662E+03
22.981
15
5.168E+03
21.992
19
4.158E+03
22.895
16
5.039E+03
21.970
20
4.072E+03
22.881
17
4.681E+03
21.909
21
3.828E+03
22.839
18
4.719E+03
21.916
19
4.209E+03
21.829
20
4.122E+03
21.814
55
Table 15.
Outlet
FLOW3D
21
3.875E+03
21.772
22
3.783E+03
21.756
23
3.562E+03
21.719
24
3.509E+03
21.709
25
3.562E+03
21.718
Case 5 First and Second-Pass Heat Flux and
Circulating Water Temperature Using
Data
56
The average outlet circulating water temperature for the first-pass is 22.186°C and for
the second-pass is 23.216°C. The change in temperature of the circulating water from the inlet
to the second-pass exit is 2.106°C.
Case 6
The last tube configuration that was analyzed is the opposite configuration of Case 5.
Case 6 evaluates the outlet circulating water temperature with the second-pass tubes on the
outer edges of the bundle and the first-pass tubes centered in the middle of the tube bundle,
Figure 38. The first-pass consists of 109 tubes and the second-pass consists of 108 tubes.
Algorithm Results Using Mehrabian-Based [4] Velocities
The velocity profile established by Mehrabian increases as the steam-air mixture flows
Figure 38. Case 6 Tube Configuration
through the tube bundle as a function of the row in the bundle. The circulating water
temperature is dependent on the mixture velocity, and as shown in Figure 39 and Figure 40,
each lower row in the first and second-passes in the tube bundle exhibits a higher circulating
water temperature than the row above it. The calculated values of the heat flux and circulating
water temperature at each row for both passes are given in Table 16.
57
Figure 39. Case 6 First-Pass Circulating Water Temperature vs Tube Length
Figure 40. Case 6 Second-Pass Circulating Water Temperature vs Tube Length
58
Table 16. Case 6 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
5
2.055E+04
24.615
1
1.259E+04
27.693
6
2.156E+04
24.788
2
1.528E+04
28.153
7
2.244E+04
24.938
3
1.707E+04
28.458
8
2.323E+04
25.073
4
1.846E+04
28.695
9
2.395E+04
25.196
5
1.960E+04
28.888
10
2.461E+04
25.308
6
2.056E+04
29.054
11
2.522E+04
25.411
7
2.141E+04
29.198
12
2.578E+04
25.508
8
2.217E+04
29.327
13
2.631E+04
25.598
9
2.285E+04
29.444
14
2.681E+04
25.682
10
2.348E+04
29.551
15
2.727E+04
25.762
11
2.406E+04
29.650
16
2.772E+04
25.837
12
2.460E+04
29.742
17
2.813E+04
25.909
13
2.511E+04
29.828
18
2.853E+04
25.976
14
2.558E+04
29.909
19
2.891E+04
26.041
15
2.602E+04
29.985
20
2.928E+04
26.104
16
2.645E+04
30.057
21
2.963E+04
26.163
17
2.685E+04
30.125
18
2.723E+04
30.190
19
2.759E+04
30.252
20
2.794E+04
30.311
21
2.827E+04
30.368
22
2.859E+04
30.422
23
2.889E+04
30.474
24
2.918E+04
30.524
25
2.946E+04
30.571
59
The average outlet circulating water temperature for the first-pass is 25.545°C and for
the second-pass is 29.649°C. In this case, the circulating water temperature increased by
8.539°C.
FLOW3D Results
Velocity (m/s)
Condenser Height (m)
In Figure 41, the FLOW3D velocity magnitude contours are shown to decrease through
the tube bundle, except at rows 2, 3, 5, 10, 14, 18 and 25. The localized increase in velocity is
consistent with other FLOW3D velocity contours, showing that the velocity profile is decoupled
from the heat transfer occurring in the system due to the configuration of the first and secondpass tubes.
Condenser Width (m)
Figure 41. Case 6 FLOW3D Velocity Magnitude Contours
The mixture temperature contours predicted in FLOW3D, Figure 42, show the first-pass
tubes in the lower half of the tube bundle, as well as the second-pass tubes in the lower half of
the tube bundle contribute most to the cooling of the steam-air mixture.
Figure 42. Case 6 FLOW3D Mixture Temperature Contours
(K)
(m)
60
Condenser Width (m)
Using the velocity profile established in FLOW3D for Case 6, the circulating water
temperature is calculated from the algorithm to produce the graphs of temperature along the
length of the tubes for both the first and second-pass rows, Figure 43 and Figure 44,
respectively. As in the graphs of the other cases using FLOW3D, the rows that do not
experience a decrease in velocity have higher circulating water temperatures relative to that of
the row above.
61
Figure 43. Case 6 First-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
Figure 44. Case 6 Second-Pass Circulating Water Temperature vs Tube Length Using FLOW3D Data
62
Table 17. Case 6 First and Second-Pass Heat Flux and Outlet Circulating Water Temperature
Using FLOW3D Data
First-Pass
Second-Pass
Row
q" (W/m2K)
Tl_out (°C)
Row
q" (W/m2K)
Tl_out (°C)
5
9.999E+03
22.816
1
9.237E+03
23.725
6
8.813E+03
22.614
2
9.262E+03
23.730
7
8.760E+03
22.605
3
9.053E+03
23.694
8
7.798E+03
22.441
4
8.578E+03
23.613
9
7.173E+03
22.334
5
9.884E+03
23.836
10
7.588E+03
22.405
6
8.711E+03
23.636
11
6.381E+03
22.199
7
8.659E+03
23.627
12
6.368E+03
22.197
8
7.707E+03
23.464
13
5.457E+03
22.042
9
7.090E+03
23.359
14
5.509E+03
22.051
10
7.501E+03
23.429
15
5.127E+03
21.985
11
6.307E+03
23.226
16
5.018E+03
21.967
12
6.293E+03
23.223
17
4.678E+03
21.909
13
5.392E+03
23.070
18
4.715E+03
21.915
14
5.445E+03
23.079
19
4.187E+03
21.825
15
5.067E+03
23.014
20
3.987E+03
21.791
16
4.959E+03
22.996
21
3.860E+03
21.769
17
4.623E+03
22.938
18
4.659E+03
22.945
19
4.138E+03
22.856
20
3.940E+03
22.822
21
3.814E+03
22.800
22
3.598E+03
22.764
23
3.497E+03
22.746
24
3.306E+03
22.714
25
3.406E+03
22.731
63
The average outlet circulating water temperature for the first and second-pass tubes is
22.150°C and 23.208°C, respectively. The overall temperature rise of the circulating water for
Case 6 is 2.098°C.
Comparison of Results Using Mehrabian-Based Velocity Profile
The results of the six cases using the Mehrabian-based velocity profile were compared
using the average circulating water temperature values. The Mehrabian-based velocity profile
increases through the tube bundle with each row, such that each successive row of tubes
experiences a higher velocity than the row above it. This velocity profile is constant throughout
all of the six cases. The comparison of the average first-pass outlet circulating water
temperature is shown below in Figure 45.
Figure 45. Comparison of Average First-Pass Circulating Water Temperature vs Length
Of the six cases, Case 2 had the most heat transferred to the first-pass tubes. The firstpass tubes in Case 2 were located on the bottom half of the tubes bundle. Case 4, which also
had the first-pass on the lower half of the tube bundle, below the 45-degree axis separating the
first and second-passes, had the second-highest outlet circulating water temperature. Cases 2
and 4 are most similar in tube configuration, thereby validating the results as these two cases
should exhibit similar amounts of heat transfer to the circulating water. Case 6, which centered
64
the first-pass tubes inside of the second-pass tubes, had the next hottest outlet circulating
water temperature, followed by the reverse tube configuration, Case 5. The tube configuration
in Cases 5 and 6 provide for an averaged amount of heat transferred relative to the cases
where the first-pass tubes are concentrated in either the upper or lower half of the bundle.
The case where all of the first-pass tubes were on the upper half of the tube bundle, Case 1,
resulted in the least amount of heat transferred to the circulating water. Case 3, which is most
similar to Case 1, has the second coldest outlet circulating water temperature. Case 3 does not
have as many first-pass tubes on the upper half of the tube bundle as Case 1, which is why the
average outlet circulating water temperature is slightly higher than that of Case 1. The results
presented in Figure 45 show that the most effective orientation of the tubes to maximize heat
transfer in the first-pass is Case 2, where the most first-pass tubes are located in the lowest
rows in the tube bundle. The stratification of the first-pass outlet circulating water
temperatures across the six cases is decreased significantly over the second-pass. The
temperatures in the second-pass appear to converge, as shown in Figure 46.
Figure 46. Comparison of Average Second-Pass Circulating Water Temperature vs Length
Unlike the first-pass, where the hottest average circulating water temperature occurred
in the case with the first-pass tubes located as low in the tube bundle as possible, there is not a
clear correlation between the average second-pass outlet circulating water temperature and
the second-pass tube orientation. Case 3, which orients the first-pass tubes on the upper half
of the bundle separated by a 45-degree axis, has the hottest outlet temperature. The largest
change in temperature from the outlet of the first-pass to the outlet of the second-pass is Case
1. These values are presented below in Table 18.
65
Table 18. Case Comparison of Circulating Water Temperatures
Temperature (°C)
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
First-Pass Outlet
24.872
25.996
25.097
25.862
25.409
25.545
Second-Pass Outlet
29.657
29.699
29.758
29.654
29.648
29.649
ΔT Between Passes
4.786
3.703
4.661
3.792
4.239
4.103
Overall ΔT
8.547
8.589
8.648
8.544
8.538
8.539
Based on the average values calculated from the six cases, Case 3 appears to be the
most favorable tube configuration for maximizing increase in temperature of the circulating
water. A comparison of the outlet circulating water temperatures for both the first and
second-passes is shown in Figure 47.
66
Figure 47. Comparison of Average First and Second-Pass Outlet Circulating Water Temperature
Again, the relative disparity in the first-pass outlet circulating water temperatures is
largely erased in the second-pass. However, Case 3 has a slightly hotter outlet temperature.
Comparison of Results Using FLOW3D-Based Velocity Profile
Similar to that discussed in section 3.2, the results predicted using FLOW3D were
compared for all six tube configurations. Unlike the velocity profile produced according to
Mehrabian [4], the velocity profile established from FLOW3D simulations resulted in a decrease
in steam-air mixture velocity as the mixture passed downward through the tube bundle. The
FLOW3D velocity profile for each case was fairly similar, which shows that for the analysis of
the FLOW3D results, the analysis of the heat transfer occurring in the six cases is independent
of the velocity profile. Therefore, the average circulating water temperatures for the first and
second-pass tubes were plotted for each case to determine the most favorable tube
configuration for heat transfer from the mixture to the circulating water. A comparison of the
first-pass circulating water temperature is shown in Figure 48 and a comparison of the secondpass circulating water temperature is shown in Figure 49.
67
Figure 48. Comparison of Average First-pass Circulating Water Temperature vs Length Using
FLOW3D Data
The range of outlet circulating water temperatures from the first-pass is relatively
small; however, Cases 1 and 3 exhibit the largest increase in temperature. These two cases are
the most similar in tube configuration and should yield similar results. Case 1 contains more
first-pass tubes on the lower half of the tube bundle than Case 3, and according to Figure 48,
Case 1 has a slightly higher outlet temperature than Case 3. Correspondingly, Cases 2 and 4
have the most similar tube configurations where the concentration of first-pass tubes is on the
lower half of the tube bundle, showing that the lower the first-pass tubes are, the less heat
transfer to the tubes occurs. Case 2, which has all of the first-pass tubes on the lower half of
the tube bundle, has the coldest outlet circulating water temperature. Cases 5 and 6 have
either the second-pass or first-pass tubes centered in the tube bundle, respectively. These
cases demonstrate that when the tubes are distributed equally in both the upper and lower
halves of the tube bundle, the resultant circulating water temperature is approximately
averaged between the calculated temperature when either pass is concentrated on one half of
the bundle.
68
The case with the hottest final exit temperature of the circulating water, or the average
of the outlet of the second-pass tubes, is not as distinct as with the first-pass tubes. The outlet
temperatures in the second-pass tubes tend to converge toward the same temperature, with
Case 1 being the coldest. This trend is shown in Figure 49.
Figure 49. Comparison of Average Second-pass Circulating Water Temperature vs Length Using
FLOW3D Data
As was seen with the first-pass, the circulating water temperatures in the second-pass
are similar for the similar tube configurations. Cases 1 and 3, Cases 2 and 4, and Cases 5 and 6
show comparable trends in increases circulating water temperature. Table 19 provides the
calculated circulating water temperatures, showing that Case 1 experiences the most heat
transfer in the first-pass, Case 2 experiences the most heat transfer from the first-pass to the
second-pass and Case 5 has the most overall heat transfer to the circulating water.
Table 19. Case Comparison of Circulating Water Temperatures Using FLOW3D Data
Temperature (°C)
Case 1
Case 2
69
Case 3
Case 4
Case 5
Case 6
First-Pass Outlet
22.380
21.884
22.374
21.939
22.186
22.150
Second-Pass Outlet
23.147
23.206
23.192
23.190
23.216
23.208
ΔT Between Passes
0.767
1.322
0.817
1.251
1.030
1.058
Overall ΔT
2.037
2.096
2.082
2.080
2.106
2.098
The average first and second-pass outlet circulating water temperatures for the six
cases are plotted in Figure 50, showing the relative differences in the first-pass outlet
temperatures and the convergence to approximately 23.2°C in the second-pass.
Figure 50. Comparison of Average First and Second-Pass Outlet Circulating Water Temperature
Using FLOW3D Data
Comparison of Mehrabian and FLOW3D-Based Results
The results from the algorithm using the Mehrabian [4] and FLOW3D-based velocity
profiles are compared to evaluate the sensitivity of the circulating water temperature to
velocity. The velocity profile based on Mehrabian [4] is row dependent and increases as the
steam-air mixture moves downward through the tube bundle. It is constant for all six cases and
has very high velocities. The velocity profile obtained from simulating the condenser in
FLOW3D is slightly different for the six cases, but all decrease as the steam-air mixture moves
downward through the tube bundle. There are localized velocity increases at the same rows
for most of the cases, which shows that the tube bundle geometry has an effect on the velocity
profile through the tubes. Compared to FLOW3D, the Mehrabian-based velocity profile is a
much larger order of magnitude. Therefore, the temperature changes seen in the cases using
70
FLOW3D data are much smaller than those from the Mehrabian-based velocities. A
comparison of the first and second-pass circulating water temperatures for the Mehrabian and
FLOW3D-based velocity profiles are shown in Figure 51 and Figure 52.
Figure 51. Comparison of Velocity Profiles on First-Pass Circulating Water Temperature
Figure 51 demonstrates the effect of the magnitude of velocity on the circulating water
temperature in the first-pass tubes. Increasing the magnitude of the velocity increases the
circulating water temperature. The change in circulating water temperature in the first-pass for
Mehrabian-based steam-air mixture velocity is substantially larger than for the FLOW3D-based
steam-air mixture velocity. The hottest first-pass cases for a Mehrabian-based mixture velocity
are with the first-pass tubes located on the lower half. Whereas a FLOW3D-based mixture
velocity shows that the hottest first-pass cases are with the first-pass tubes located on the
upper half. This difference can be attributed to the difference in velocity profiles. The firstpass tubes that experience the highest velocity, which are the lowest tube rows with
Mehrabian (Cases 2 and 4) and the highest tube rows with FLOW3D (Cases 1 and 3), have the
most heat transfer to the tubes, resulting in the highest outlet circulating water temperature.
Therefore, the velocity is proportional to the circulating water temperature. As the velocity
increases, the outlet circulating water temperature increases.
For both steam-air mixture velocity profiles, the second-pass outlet circulating water
temperatures of all six cases appear to converge. As the velocity profile based on Mehrabian is
71
larger than that of FLOW3D, the second-pass circulating water temperature calculated is hotter
for the Mehrabian-based profile than the FLOW3D-based profile. This is evident in Figure 52.
Figure 52. Comparison of Velocity Profiles on Second-Pass Circulating Water Temperature
Analogous to the first-pass circulating water temperature differences seen in Figure 51,
the second-pass tubes that are subjected to higher steam-air mixture velocities (the lower
tubes in a Mehrabian velocity profile and the higher tubes in a FLOW3D velocity profile), result
in the hottest outlet circulating water temperature. Based on a Mehrabian velocity profile, the
cases with the second-pass tubes located in the lower half of the tube bundle, Cases 1 and 3,
have the largest change in temperature between the two passes. Based on a FLOW3D velocity
profile, the cases with the second-pass tubes located in the upper half of the tube bundle,
Cases 2 and 4, have the largest change in temperature between the two passes. The calculated
temperatures of the first and second-passes, the change in temperature between the two
passes and the overall change in circulating water temperature are provided in Table 20. Also
provided in Table 20 is the change in temperature between the results using two types of
steam-air mixture velocity profiles.
72
Table 20. Case Comparison of Mehrabian and FLOW3D Circulating Water Temperatures
Temperature (°C)
Case 1
Case 2
Case 3
Mehrabian
FLOW3D
ΔT
Mehrabian
FLOW3D
ΔT
Mehrabian
FLOW3D
ΔT
First-Pass Outlet
24.872
22.380
2.492
25.996
21.884
4.112
25.097
22.374
2.722
Second-Pass Outlet
29.657
23.147
6.511
29.699
23.206
6.494
29.758
23.192
6.566
ΔT Between Passes
4.786
0.767
3.703
1.322
4.661
0.817
Overall ΔT
8.547
2.037
8.589
2.096
8.648
2.082
Case 4
Case 5
Case 6
Mehrabian
FLOW3D
ΔT
Mehrabian
FLOW3D
ΔT
Mehrabian
FLOW3D
ΔT
First-Pass Outlet
25.862
21.939
3.923
25.409
22.186
3.222
25.545
22.150
3.395
Second-Pass Outlet
29.654
23.190
6.464
29.648
23.216
6.432
29.649
23.208
6.441
ΔT Between Passes
3.792
1.251
4.239
1.030
4.103
1.058
Overall ΔT
8.544
2.080
8.538
2.106
8.539
2.098
Overall, using a smaller magnitude velocity profile, such as the one calculated from
FLOW3D, produces a smaller change in circulating water temperature. The temperature
differences between the Mehrabian-based and FLOW3D-based velocity profiles are consistent
throughout the analyses of the six tube configurations. The analyses using a Mehrabian-based
velocity profile result in Case 3 having the hottest circulating water temperature. The analyses
using a FLOW3D-based velocity profile results in Case 5 having the hottest circulating water
temperature.
Conclusions
Six unique tube configurations in a horizontal, two-pass condenser were analyzed in an
iterative heat and mass transfer algorithm to determine the outlet circulating water
temperature through the tubes. The algorithm considers the heat transferred from the steamair mixture to the interface between the mixture and condensate, through the condensate,
through the tube wall and into the circulating water. The algorithm also takes into account the
latent heat produced by the condensate forming around the tubes. A steam-air mixture
velocity profile was established using the approach taken by Mehrabian [4]. The outlet
73
circulating water temperature for each first-pass row was calculated and using a weighted
average, a new inlet circulating water temperature was created for the second-pass tubes, from
which an exit circulating water temperature was calculated.
The results using the assumed Mehrabian-based steam-air mixture velocity profile in
the heat and mass transfer algorithm show that all six cases have higher outlet circulating
water temperatures for the second-pass tubes than for the first-pass tubes. More heat is
transferred to the first-pass tubes when these tubes are located on the bottom of the tube
bundle, such as in Cases 2 and 4. Case 2 resulted in the most heat transferred to the first-pass
tubes, resulting in the warmest circulating water at the outlet of the first-pass, with a
temperature of 25.996°C. Case 1 resulted in the largest change in circulating water
temperature from the first-pass to the second-pass with a change of 4.786°C. Overall, Case 3
resulted in the most heat transferred to the circulating water, with an average second-pass
outlet temperature of 29.758°C.
The Mehrabian steam-air mixture velocity profile that was used in the algorithm could
be more accurate and closer to the actual velocity in the condenser, similar to the velocity
profile seen in the FLOW3D simulations, by using a different correction factor. The Mehrabian
[4] approach directly increases the pressure drop, and consequently, the row velocity with each
successive row, which significantly increases the velocity at higher rows.
The steam-air mixture velocity profiles obtained from FLOW3D simulations of the six
cases decreases as the steam-air mixture moves downward through the tube bundle, due to
the tubes obstructing the mixture flow path. This velocity profile is opposite from the assumed
profile based on Mehrabian. The FLOW3D velocity profiles obtained for each of the six cases
are relatively similar and exhibit symmetry. Comparable to the results using the Mehrabianbased velocity profile, the results using FLOW3D data shows an increase in circulating water
temperature in both the first and second-passes. Case 1 has the hottest circulating water
temperature of 22.380°C at the outlet of the first-pass tubes. Case 2 has the largest change in
temperature between the first and second-passes of 1.322°C. Case 5 has the hottest secondpass circulating water temperature of 23.216°C.
In comparing results calculated from the Mehrabian and FLOW3D steam-air mixture
velocity profiles in the algorithm, the heat flux and circulating water temperature are found to
be proportional to the velocity. As the velocity increases, the heat flux and circulating water
temperature increases, consistent with thermodynamic principles. The first-pass tubes that
experience the highest velocity, which are the lowest tube rows with a Mehrabian-based
velocity profile (Cases 2 and 4) and the highest tube rows with a FLOW3D-based velocity profile
(Cases 1 and 3), have the most heat transfer to the tubes. The tubes where the highest
velocities result in the highest outlet circulating water temperature. Therefore, the velocity is
proportional to the circulating water temperature.
The FLOW3D models may be refined to more accurately compare the results of the
algorithm with those employing the Mehrabian approach to the steam-air mixture velocity.
The FLOW3D grid that was generated was relatively coarse and the flow was assumed laminar
in order to expedite simulating all six cases. A higher grid resolution and assuming a turbulent
steam-air mixture flow through the bundles would each increase the predicted maximum
74
velocity through the tubes. A grid sensitivity and/or closure model sensitivity analysis could be
performed to further validate the results obtained herein.
References
[1]
M.R. Malin, “Modeling Flow in an Experimental Marine Condenser.”
International Communications in Heat and Mass Transfer 24 (5) (1997) 597-608
[2] M.W. Browne, P.K. Bansal, “An overview of condensation heat transfer on
horizontal tube bundles.” Applied Thermal Engineering 19 (1999) 565-594
[3]
Wilson, A. Safwat, and M. Khalil Bassiouny, “Modeling of heat transfer for flow
across tube banks.” Chemical Engineering and Processing 39 (2000) 1-14
[4]
M.A. Mehrabian, “Heat Transfer and Pressure Drop Characteristics of Cross
Flow of Air Over a Circular Tube in Isolation and/or in a Tube Bank.” The Arabian
Journal for Science and Engineering 32, Number 2B (2007) 365-376
Appendix A – Steam-Air Mixture Velocity Profiles
Steam-AIr Mixture Velocity Profiles (m/s)
Mehrabian
FLOW3D
Row
Cases 1-6
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
1
48.20
26.74
26.74
26.77
26.72
26.82
26.68
2
68.16
26.93
26.80
26.79
26.81
26.76
26.85
3
83.48
25.58
25.84
25.84
25.84
25.79
25.87
4
96.40
23.26
23.57
23.58
23.57
23.52
23.59
5
107.78
25.33
30.17
30.14
30.12
30.18
30.24
6
118.06
24.33
24.37
24.31
24.32
24.36
24.31
7
127.52
24.01
24.04
24.01
24.01
24.07
24.10
8
136.33
19.41
19.62
19.61
19.64
19.64
19.72
9
144.60
18.92
17.43
17.34
17.38
17.33
17.08
10
152.42
18.71
18.76
18.82
18.75
18.87
18.84
75
11
159.86
13.71
13.93
13.98
13.91
14.03
13.98
12
166.97
13.77
13.75
13.78
13.70
14.00
13.94
13
173.78
10.49
10.57
10.46
10.48
10.72
10.69
14
180.34
10.98
11.05
10.96
10.96
10.94
10.87
15
186.67
9.79
9.90
9.72
9.74
9.75
9.62
16
192.80
9.56
9.64
9.53
9.51
9.34
9.27
17
198.73
7.64
7.80
7.60
7.61
8.23
8.23
18
204.49
7.19
7.93
7.82
7.80
8.35
8.34
19
210.09
6.19
6.77
6.54
6.55
6.87
6.81
20
215.55
6.04
6.16
6.01
6.03
6.63
6.27
21
220.87
5.28
5.85
5.53
5.49
5.97
5.93
22
226.07
4.63
5.37
5.16
5.14
5.73
5.37
23
231.15
4.77
5.12
4.72
4.67
5.18
5.12
24
236.13
4.39
4.72
4.37
4.33
5.05
4.65
25
240.99
4.64
5.08
4.49
4.48
5.18
4.90
Appendix B – FLOW3D Input File for Case 1
Water-steam-air with evaporation/condensation
&xput
remark='!! Remarks beginning with "!! " are automatically added and removed by FLOW3D.',
remark='!! Do not begin any user added remarks with with "!! ". They will be
removed',
remark='units are SI',
twfin=30.0,
remark='Final simulation time is 5 minutes',
itb=1,
remark='Sharp interface tracking on',
nmat=2,
icmprs=1,
remark='Two fluid, compressible flow',
ifenrg=3,
remark='Energy transport ON, 2nd order advection',
ihtc=3,
remark='Heat transfer ON -- comp. of conduction in solids',
ifvisc=1,
remark='Viscosity ON',
imphtc=1,
remark='Implicit heat transfer',
igmres=1,
remark='GMRES pressure solver ON',
ifdynconv=1,
ifrest=0,
resfile='flsgrf.condenser1.txt',
trest=0.500003,
iphchg=2,
remark='Two-fluid phase change model ON',
incg=1,
remark='Non-condensable gas model ON',
ifrho=3,
remark='Density transport ON, 2nd order advection',
gy=-9.817,
remark='gravity',
ifmu=1,
delt=1.0e-5,
remark='Initial time is 10 microseconds',
iorder=3,
remark='3rd order upwind advection',
tpltd(1)=0.02,
76
thpltd(1)=0.01,
tsprtd(1)=0.0001,
/
&limits
itmax=1000,
/
&props
units='si',
tunits='k',
rfnc=286.69,
remark='Gas constant of pure air (J/Kg.K)',
cvnc=1005.0,
remark='Heat capacity of pure air (J/Kg.K)',
mu1=0.00102,
remark='Viscosity of liquid water (Pa.s)',
fluid1='liquid water',
fluid2='air-water vapor',
thexf1=0.0018,
remark='Thermal expansion coeff. of liq. water',
rf2=461.5,
remark='Gas constant pure steam (J/kg.K)',
cv1=4181.44,
remark='Heat capacity of liquid water (J/kg.K)',
cv2=1900.0,
remark='Heat capacity of pure steam (J/kg.K)',
tstar=333.15,
pv1=1.013e5,
remark='Constant in psat-T equation (Pa)',
tv1=373.,
remark='Reference temperature in psat-T equation (K)',
clhv1=2.212e6,
remark='Heat of vaporization of water (J/kg)',
rsize=0.05,
remark='Accomodation coefficient for phase change',
tvexp=0.000453982, remark='Ceofficient in exponent of psat-T eq. (1/K)',
tv0=389.65,
remark='Superheat temperature (K)',
rhof=1024.39,
remark='Density of liquid water (kg/m^3)',
mu2=0.0000228,
remark='Viscosity of air/steam mixture (Pa.s)',
rhof2=0.1927,
remark='Initial density of air/steam mixture (kg/m^3)',
thc1=0.5865,
remark='Thermal conductivity of liquid water (W/m.K)',
thc2=0.02642,
remark='Thermal conductivity of air/steam mixture (W/m.K)',
/
&scalar
/
&PCAP
/
!------------------------------ super block ------------------------ Block 01
&bcdata
remark='!! Boundary condition Y Min',
ibct(3)=5,
ipbctp(3)=0,
pbct(1, 3)=34500.0,
remark='outlet pressure (Pa)',
remark='!! Boundary condition Y Max',
remark='y minimum boundary - specified velocity',
ibct(4)=6,
ipbctp(4)=0,
pbct(1, 4)=34490,
remark='inlet pressure (Pa)',
vbct(1, 4)=0,
remark='inlet velocity (m/s)',
vbct(2, 4)=-24.101,
fbct(1, 4)=0,
remark='inlet fluid fraction 0 - vapour, 1 - liquid',
remark='y minimum boundary - specified pressure',
cncbct(1, 4)=0.0002,
remark='inlet mass fraction',
tbct(1, 4)=300,
remark='inlet temperature',
tbct(2, 4)=389.65,
remark='!! Boundary condition common parameters',
timbct(1)=0,
timbct(2)=0.1,
/
&mesh
size=0.006,
px(1)=-0.3068,
px(2)=0.3068,
py(1)=-0.002,
py(2)=0.6116,
nzcelt=1,
pz(1)=0,
77
pz(2)=1.0,
/
!-------------------------------- nested block -------------------------- Block 02
&bcdata
remark='!! Boundary condition common parameters',
timbct(1)=0.0,
/
&mesh
size=0.002,
px(1)=-0.162,
px(2)=0.162,
py(1)=0.1440,
py(2)=0.4660,
nzcelt=1,
pz(1)=0,
pz(2)=1.0,
/
!------------------------------------------------------------------------------&obs
nobs=3,
remark='!! Component 1',
obsid(1)='condenser shell properties',
remark='!! Subcomponent 1',
iob(1)=1,
ioh(1)=1,
xh(1)=-0.3048,
remark='condenser shell',
remark='!! Subcomponent 2',
iob(2)=1,
ioh(2)=1,
xl(2)=0.3048,
remark='condenser shell',
remark='!! Subcomponent 3',
iob(3)=1,
ioh(3)=1,
xh(3)=-0.1524,
yl(3)=0.6096,
remark='!! Subcomponent 4',
iob(4)=1,
ioh(4)=1,
xl(4)=0.1524,
yl(4)=0.6096,
remark='!! Subcomponent 5',
iob(5)=1,
ioh(5)=1,
xh(5)=-0.0762,
yh(5)=0.,
remark='!! Subcomponent 6',
iob(6)=1,
ioh(6)=1,
xl(6)=0.0762,
yh(6)=0.,
remark='!! Component 1 properties',
rcobs(1)=3612150.00,
remark='density x specific heat',
kobs(1)=14.60,
remark='thermal conductivity (W/m/K)',
hobs1(1)=-1.0,
hobs2(1)=-1.0,
remark='allow heat transfer to all fluids',
itpobs(1)=0,
twobs(1, 1)=291.48,
remark='initial shell wall temp',
iaqsrb(1)=0,
remark='!! Component 2',
obsid(2)='lower tube bundles',
ccolor(2)=4294967040,
remark='!! Subcomponent 7',
iob(7)=2,
ioh(7)=1,
rah(7)=0.007145,
trnx(7)=-0.0381,
78
trny(7)=0.165101,
remark='!! Subcomponent
iob(8)=2,
ioh(8)=1,
rah(8)=0.007145,
trnx(8)=0.00000,
trny(8)=0.152401,
remark='!! Subcomponent
iob(9)=2,
ioh(9)=1,
rah(9)=0.007145,
trnx(9)=-0.0127,
trny(9)=0.165101,
remark='!! Subcomponent
iob(10)=2,
ioh(10)=1,
rah(10)=0.007145,
trnx(10)=0.01270,
trny(10)=0.165101,
remark='!! Subcomponent
iob(11)=2,
ioh(11)=1,
rah(11)=0.007145,
trnx(11)=0.03810,
trny(11)=0.165101,
remark='!! Subcomponent
iob(12)=2,
ioh(12)=1,
rah(12)=0.007145,
trnx(12)=-0.0508,
trny(12)=0.177801,
remark='!! Subcomponent
iob(13)=2,
ioh(13)=1,
rah(13)=0.007145,
trnx(13)=-0.0254,
trny(13)=0.177801,
remark='!! Subcomponent
iob(14)=2,
ioh(14)=1,
rah(14)=0.007145,
trnx(14)=0.00000,
trny(14)=0.177801,
remark='!! Subcomponent
iob(15)=2,
ioh(15)=1,
rah(15)=0.007145,
trnx(15)=0.02540,
trny(15)=0.177801,
remark='!! Subcomponent
iob(16)=2,
ioh(16)=1,
rah(16)=0.007145,
trnx(16)=0.05080,
trny(16)=0.177801,
remark='!! Subcomponent
iob(17)=2,
ioh(17)=1,
rah(17)=0.007145,
trnx(17)=-0.0889,
trny(17)=0.190501,
remark='!! Subcomponent
iob(18)=2,
ioh(18)=1,
rah(18)=0.007145,
trnx(18)=-0.0635,
8',
9',
10',
11',
12',
13',
14',
15',
16',
17',
18',
79
trny(18)=0.190501,
remark='!! Subcomponent
iob(19)=2,
ioh(19)=1,
rah(19)=0.007145,
trnx(19)=-0.0381,
trny(19)=0.190501,
remark='!! Subcomponent
iob(20)=2,
ioh(20)=1,
rah(20)=0.007145,
trnx(20)=-0.0127,
trny(20)=0.190501,
remark='!! Subcomponent
iob(21)=2,
ioh(21)=1,
rah(21)=0.007145,
trnx(21)=0.01270,
trny(21)=0.190501,
remark='!! Subcomponent
iob(22)=2,
ioh(22)=1,
rah(22)=0.007145,
trnx(22)=0.03810,
trny(22)=0.190501,
remark='!! Subcomponent
iob(23)=2,
ioh(23)=1,
rah(23)=0.007145,
trnx(23)=0.06350,
trny(23)=0.190501,
remark='!! Subcomponent
iob(24)=2,
ioh(24)=1,
rah(24)=0.007145,
trnx(24)=0.08890,
trny(24)=0.190501,
remark='!! Subcomponent
iob(25)=2,
ioh(25)=1,
rah(25)=0.007145,
trnx(25)=-0.1016,
trny(25)=0.203201,
remark='!! Subcomponent
iob(26)=2,
ioh(26)=1,
rah(26)=0.007145,
trnx(26)=-0.0762,
trny(26)=0.203201,
remark='!! Subcomponent
iob(27)=2,
ioh(27)=1,
rah(27)=0.007145,
trnx(27)=-0.0508,
trny(27)=0.203201,
remark='!! Subcomponent
iob(28)=2,
ioh(28)=1,
rah(28)=0.007145,
trnx(28)=-0.0254,
trny(28)=0.203201,
remark='!! Subcomponent
iob(29)=2,
ioh(29)=1,
rah(29)=0.007145,
trnx(29)=0.00000,
19',
20',
21',
22',
23',
24',
25',
26',
27',
28',
29',
80
trny(29)=0.203201,
remark='!! Subcomponent
iob(30)=2,
ioh(30)=1,
rah(30)=0.007145,
trnx(30)=0.02540,
trny(30)=0.203201,
remark='!! Subcomponent
iob(31)=2,
ioh(31)=1,
rah(31)=0.007145,
trnx(31)=0.05080,
trny(31)=0.203201,
remark='!! Subcomponent
iob(32)=2,
ioh(32)=1,
rah(32)=0.007145,
trnx(32)=0.07620,
trny(32)=0.203201,
remark='!! Subcomponent
iob(33)=2,
ioh(33)=1,
rah(33)=0.007145,
trnx(33)=0.10160,
trny(33)=0.203201,
remark='!! Subcomponent
iob(34)=2,
ioh(34)=1,
rah(34)=0.007145,
trnx(34)=-0.1143,
trny(34)=0.215901,
remark='!! Subcomponent
iob(35)=2,
ioh(35)=1,
rah(35)=0.007145,
trnx(35)=-0.0889,
trny(35)=0.215901,
remark='!! Subcomponent
iob(36)=2,
ioh(36)=1,
rah(36)=0.007145,
trnx(36)=-0.0635,
trny(36)=0.215901,
remark='!! Subcomponent
iob(37)=2,
ioh(37)=1,
rah(37)=0.007145,
trnx(37)=-0.0381,
trny(37)=0.215901,
remark='!! Subcomponent
iob(38)=2,
ioh(38)=1,
rah(38)=0.007145,
trnx(38)=-0.0127,
trny(38)=0.215901,
remark='!! Subcomponent
iob(39)=2,
ioh(39)=1,
rah(39)=0.007145,
trnx(39)=0.01270,
trny(39)=0.215901,
remark='!! Subcomponent
iob(40)=2,
ioh(40)=1,
rah(40)=0.007145,
trnx(40)=0.03810,
30',
31',
32',
33',
34',
35',
36',
37',
38',
39',
40',
81
trny(40)=0.215901,
remark='!! Subcomponent
iob(41)=2,
ioh(41)=1,
rah(41)=0.007145,
trnx(41)=0.06350,
trny(41)=0.215901,
remark='!! Subcomponent
iob(42)=2,
ioh(42)=1,
rah(42)=0.007145,
trnx(42)=0.08890,
trny(42)=0.215901,
remark='!! Subcomponent
iob(43)=2,
ioh(43)=1,
rah(43)=0.007145,
trnx(43)=0.11430,
trny(43)=0.215901,
remark='!! Subcomponent
iob(44)=2,
ioh(44)=1,
rah(44)=0.007145,
trnx(44)=-0.1016,
trny(44)=0.228601,
remark='!! Subcomponent
iob(45)=2,
ioh(45)=1,
rah(45)=0.007145,
trnx(45)=-0.0762,
trny(45)=0.228601,
remark='!! Subcomponent
iob(46)=2,
ioh(46)=1,
rah(46)=0.007145,
trnx(46)=-0.0508,
trny(46)=0.228601,
remark='!! Subcomponent
iob(47)=2,
ioh(47)=1,
rah(47)=0.007145,
trnx(47)=-0.0254,
trny(47)=0.228601,
remark='!! Subcomponent
iob(48)=2,
ioh(48)=1,
rah(48)=0.007145,
trnx(48)=0.00000,
trny(48)=0.228601,
remark='!! Subcomponent
iob(49)=2,
ioh(49)=1,
rah(49)=0.007145,
trnx(49)=0.02540,
trny(49)=0.228601,
remark='!! Subcomponent
iob(50)=2,
ioh(50)=1,
rah(50)=0.007145,
trnx(50)=0.05080,
trny(50)=0.228601,
remark='!! Subcomponent
iob(51)=2,
ioh(51)=1,
rah(51)=0.007145,
trnx(51)=0.07620,
41',
42',
43',
44',
45',
46',
47',
48',
49',
50',
51',
82
trny(51)=0.228601,
remark='!! Subcomponent
iob(52)=2,
ioh(52)=1,
rah(52)=0.007145,
trnx(52)=0.10160,
trny(52)=0.228601,
remark='!! Subcomponent
iob(53)=2,
ioh(53)=1,
rah(53)=0.007145,
trnx(53)=-0.1143,
trny(53)=0.241301,
remark='!! Subcomponent
iob(54)=2,
ioh(54)=1,
rah(54)=0.007145,
trnx(54)=-0.0889,
trny(54)=0.241301,
remark='!! Subcomponent
iob(55)=2,
ioh(55)=1,
rah(55)=0.007145,
trnx(55)=-0.0635,
trny(55)=0.241301,
remark='!! Subcomponent
iob(56)=2,
ioh(56)=1,
rah(56)=0.007145,
trnx(56)=-0.0381,
trny(56)=0.241301,
remark='!! Subcomponent
iob(57)=2,
ioh(57)=1,
rah(57)=0.007145,
trnx(57)=-0.0127,
trny(57)=0.241301,
remark='!! Subcomponent
iob(58)=2,
ioh(58)=1,
rah(58)=0.007145,
trnx(58)=0.01270,
trny(58)=0.241301,
remark='!! Subcomponent
iob(59)=2,
ioh(59)=1,
rah(59)=0.007145,
trnx(59)=0.03810,
trny(59)=0.241301,
remark='!! Subcomponent
iob(60)=2,
ioh(60)=1,
rah(60)=0.007145,
trnx(60)=0.06350,
trny(60)=0.241301,
remark='!! Subcomponent
iob(61)=2,
ioh(61)=1,
rah(61)=0.007145,
trnx(61)=0.08890,
trny(61)=0.241301,
remark='!! Subcomponent
iob(62)=2,
ioh(62)=1,
rah(62)=0.007145,
trnx(62)=0.11430,
52',
53',
54',
55',
56',
57',
58',
59',
60',
61',
62',
83
trny(62)=0.241301,
remark='!! Subcomponent
iob(63)=2,
ioh(63)=1,
rah(63)=0.007145,
trnx(63)=-0.1270,
trny(63)=0.254001,
remark='!! Subcomponent
iob(64)=2,
ioh(64)=1,
rah(64)=0.007145,
trnx(64)=-0.1016,
trny(64)=0.254001,
remark='!! Subcomponent
iob(65)=2,
ioh(65)=1,
rah(65)=0.007145,
trnx(65)=-0.0762,
trny(65)=0.254001,
remark='!! Subcomponent
iob(66)=2,
ioh(66)=1,
rah(66)=0.007145,
trnx(66)=-0.0508,
trny(66)=0.254001,
remark='!! Subcomponent
iob(67)=2,
ioh(67)=1,
rah(67)=0.007145,
trnx(67)=-0.0254,
trny(67)=0.254001,
remark='!! Subcomponent
iob(68)=2,
ioh(68)=1,
rah(68)=0.007145,
trnx(68)=0.00000,
trny(68)=0.254001,
remark='!! Subcomponent
iob(69)=2,
ioh(69)=1,
rah(69)=0.007145,
trnx(69)=0.02540,
trny(69)=0.254001,
remark='!! Subcomponent
iob(70)=2,
ioh(70)=1,
rah(70)=0.007145,
trnx(70)=0.05080,
trny(70)=0.254001,
remark='!! Subcomponent
iob(71)=2,
ioh(71)=1,
rah(71)=0.007145,
trnx(71)=0.07620,
trny(71)=0.254001,
remark='!! Subcomponent
iob(72)=2,
ioh(72)=1,
rah(72)=0.007145,
trnx(72)=0.10160,
trny(72)=0.254001,
remark='!! Subcomponent
iob(73)=2,
ioh(73)=1,
rah(73)=0.007145,
trnx(73)=0.12700,
63',
64',
65',
66',
67',
68',
69',
70',
71',
72',
73',
84
trny(73)=0.254001,
remark='!! Subcomponent
iob(74)=2,
ioh(74)=1,
rah(74)=0.007145,
trnx(74)=-0.1397,
trny(74)=0.266701,
remark='!! Subcomponent
iob(75)=2,
ioh(75)=1,
rah(75)=0.007145,
trnx(75)=-0.1143,
trny(75)=0.266701,
remark='!! Subcomponent
iob(76)=2,
ioh(76)=1,
rah(76)=0.007145,
trnx(76)=-0.0889,
trny(76)=0.266701,
remark='!! Subcomponent
iob(77)=2,
ioh(77)=1,
rah(77)=0.007145,
trnx(77)=-0.0635,
trny(77)=0.266701,
remark='!! Subcomponent
iob(78)=2,
ioh(78)=1,
rah(78)=0.007145,
trnx(78)=-0.0381,
trny(78)=0.266701,
remark='!! Subcomponent
iob(79)=2,
ioh(79)=1,
rah(79)=0.007145,
trnx(79)=-0.0127,
trny(79)=0.266701,
remark='!! Subcomponent
iob(80)=2,
ioh(80)=1,
rah(80)=0.007145,
trnx(80)=0.01270,
trny(80)=0.266701,
remark='!! Subcomponent
iob(81)=2,
ioh(81)=1,
rah(81)=0.007145,
trnx(81)=0.03810,
trny(81)=0.266701,
remark='!! Subcomponent
iob(82)=2,
ioh(82)=1,
rah(82)=0.007145,
trnx(82)=0.06350,
trny(82)=0.266701,
remark='!! Subcomponent
iob(83)=2,
ioh(83)=1,
rah(83)=0.007145,
trnx(83)=0.08890,
trny(83)=0.266701,
remark='!! Subcomponent
iob(84)=2,
ioh(84)=1,
rah(84)=0.007145,
trnx(84)=0.11430,
74',
75',
76',
77',
78',
79',
80',
81',
82',
83',
84',
85
trny(84)=0.266701,
remark='!! Subcomponent
iob(85)=2,
ioh(85)=1,
rah(85)=0.007145,
trnx(85)=0.13970,
trny(85)=0.266701,
remark='!! Subcomponent
iob(86)=2,
ioh(86)=1,
rah(86)=0.007145,
trnx(86)=-0.1270,
trny(86)=0.279401,
remark='!! Subcomponent
iob(87)=2,
ioh(87)=1,
rah(87)=0.007145,
trnx(87)=-0.1016,
trny(87)=0.279401,
remark='!! Subcomponent
iob(88)=2,
ioh(88)=1,
rah(88)=0.007145,
trnx(88)=-0.0762,
trny(88)=0.279401,
remark='!! Subcomponent
iob(89)=2,
ioh(89)=1,
rah(89)=0.007145,
trnx(89)=-0.0508,
trny(89)=0.279401,
remark='!! Subcomponent
iob(90)=2,
ioh(90)=1,
rah(90)=0.007145,
trnx(90)=-0.0254,
trny(90)=0.279401,
remark='!! Subcomponent
iob(91)=2,
ioh(91)=1,
rah(91)=0.007145,
trnx(91)=0.00000,
trny(91)=0.279401,
remark='!! Subcomponent
iob(92)=2,
ioh(92)=1,
rah(92)=0.007145,
trnx(92)=0.02540,
trny(92)=0.279401,
remark='!! Subcomponent
iob(93)=2,
ioh(93)=1,
rah(93)=0.007145,
trnx(93)=0.05080,
trny(93)=0.279401,
remark='!! Subcomponent
iob(94)=2,
ioh(94)=1,
rah(94)=0.007145,
trnx(94)=0.07620,
trny(94)=0.279401,
remark='!! Subcomponent
iob(95)=2,
ioh(95)=1,
rah(95)=0.007145,
trnx(95)=0.12700,
85',
86',
87',
88',
89',
90',
91',
92',
93',
94',
95',
86
trny(95)=0.279401,
remark='!! Subcomponent
iob(96)=2,
ioh(96)=1,
rah(96)=0.007145,
trnx(96)=0.10160,
trny(96)=0.279401,
remark='!! Subcomponent
iob(97)=2,
ioh(97)=1,
rah(97)=0.007145,
trnx(97)=-0.1397,
trny(97)=0.292101,
remark='!! Subcomponent
iob(98)=2,
ioh(98)=1,
rah(98)=0.007145,
trnx(98)=-0.1143,
trny(98)=0.292101,
remark='!! Subcomponent
iob(99)=2,
ioh(99)=1,
rah(99)=0.007145,
trnx(99)=-0.0889,
trny(99)=0.292101,
remark='!! Subcomponent
iob(100)=2,
ioh(100)=1,
rah(100)=0.007145,
trnx(100)=-0.0635,
trny(100)=0.292101,
remark='!! Subcomponent
iob(101)=2,
ioh(101)=1,
rah(101)=0.007145,
trnx(101)=-0.0381,
trny(101)=0.292101,
remark='!! Subcomponent
iob(102)=2,
ioh(102)=1,
rah(102)=0.007145,
trnx(102)=-0.0127,
trny(102)=0.292101,
remark='!! Subcomponent
iob(103)=2,
ioh(103)=1,
rah(103)=0.007145,
trnx(103)=0.01270,
trny(103)=0.292101,
remark='!! Subcomponent
iob(104)=2,
ioh(104)=1,
rah(104)=0.007145,
trnx(104)=0.03810,
trny(104)=0.292101,
remark='!! Subcomponent
iob(105)=2,
ioh(105)=1,
rah(105)=0.007145,
trnx(105)=0.06350,
trny(105)=0.292101,
remark='!! Subcomponent
iob(106)=2,
ioh(106)=1,
rah(106)=0.007145,
trnx(106)=0.08890,
96',
97',
98',
99',
100',
101',
102',
103',
104',
105',
106',
87
trny(106)=0.292101,
remark='!! Subcomponent 107',
iob(107)=2,
ioh(107)=1,
rah(107)=0.007145,
trnx(107)=0.11430,
trny(107)=0.292101,
remark='!! Subcomponent 108',
iob(108)=2,
ioh(108)=1,
rah(108)=0.007145,
trnx(108)=0.13970,
trny(108)=0.292101,
remark='!! Component 2 properties',
rcobs(2)=0,
remark='density x specific heat',
kobs(2)=21.90,
remark='thermal conductivity (W/m/K)',
hobs1(2)=327.35,
hobs2(2)=327.35,
remark='allow heat transfer to all fluids',
itpobs(2)=0,
twobs(1, 2)=298.02,
remark='initial tube wall temp',
iaqsrb(2)=0,
remark='!! Component 3',
obsid(3)='upper tube bundles',
ccolor(3)=4278190335,
remark='!! Subcomponent 109',
iob(109)=3,
ioh(109)=1,
rah(109)=0.007145,
trnx(109)=-0.1524,
trny(109)=0.304801,
remark='!! Subcomponent 110',
iob(110)=3,
ioh(110)=1,
rah(110)=0.007145,
trnx(110)=-0.1270,
trny(110)=0.304801,
remark='!! Subcomponent 111',
iob(111)=3,
ioh(111)=1,
rah(111)=0.007145,
trnx(111)=-0.1016,
trny(111)=0.304801,
remark='!! Subcomponent 112',
iob(112)=3,
ioh(112)=1,
rah(112)=0.007145,
trnx(112)=-0.0762,
trny(112)=0.304801,
remark='!! Subcomponent 113',
iob(113)=3,
ioh(113)=1,
rah(113)=0.007145,
trnx(113)=-0.0508,
trny(113)=0.304801,
remark='!! Subcomponent 114',
iob(114)=3,
ioh(114)=1,
rah(114)=0.007145,
trnx(114)=-0.0254,
trny(114)=0.304801,
remark='!! Subcomponent 115',
iob(115)=3,
ioh(115)=1,
rah(115)=0.007145,
trnx(115)=0.00000,
trny(115)=0.304801,
88
remark='!! Subcomponent
iob(116)=3,
ioh(116)=1,
rah(116)=0.007145,
trnx(116)=0.02540,
trny(116)=0.304801,
remark='!! Subcomponent
iob(117)=3,
ioh(117)=1,
rah(117)=0.007145,
trnx(117)=0.05080,
trny(117)=0.304801,
remark='!! Subcomponent
iob(118)=3,
ioh(118)=1,
rah(118)=0.007145,
trnx(118)=0.07620,
trny(118)=0.304801,
remark='!! Subcomponent
iob(119)=3,
ioh(119)=1,
rah(119)=0.007145,
trnx(119)=0.10160,
trny(119)=0.304801,
remark='!! Subcomponent
iob(120)=3,
ioh(120)=1,
rah(120)=0.007145,
trnx(120)=0.12700,
trny(120)=0.304801,
remark='!! Subcomponent
iob(121)=3,
ioh(121)=1,
rah(121)=0.007145,
trnx(121)=0.15240,
trny(121)=0.304801,
remark='!! Subcomponent
iob(122)=3,
ioh(122)=1,
rah(122)=0.007145,
trnx(122)=-0.1397,
trny(122)=0.317501,
remark='!! Subcomponent
iob(123)=3,
ioh(123)=1,
rah(123)=0.007145,
trnx(123)=-0.1143,
trny(123)=0.317501,
remark='!! Subcomponent
iob(124)=3,
ioh(124)=1,
rah(124)=0.007145,
trnx(124)=-0.0889,
trny(124)=0.317501,
remark='!! Subcomponent
iob(125)=3,
ioh(125)=1,
rah(125)=0.007145,
trnx(125)=-0.0635,
trny(125)=0.317501,
remark='!! Subcomponent
iob(126)=3,
ioh(126)=1,
rah(126)=0.007145,
trnx(126)=-0.0381,
trny(126)=0.317501,
116',
117',
118',
119',
120',
121',
122',
123',
124',
125',
126',
89
remark='!! Subcomponent
iob(127)=3,
ioh(127)=1,
rah(127)=0.007145,
trnx(127)=-0.0127,
trny(127)=0.317501,
remark='!! Subcomponent
iob(128)=3,
ioh(128)=1,
rah(128)=0.007145,
trnx(128)=0.01270,
trny(128)=0.317501,
remark='!! Subcomponent
iob(129)=3,
ioh(129)=1,
rah(129)=0.007145,
trnx(129)=0.03810,
trny(129)=0.317501,
remark='!! Subcomponent
iob(130)=3,
ioh(130)=1,
rah(130)=0.007145,
trnx(130)=0.06350,
trny(130)=0.317501,
remark='!! Subcomponent
iob(131)=3,
ioh(131)=1,
rah(131)=0.007145,
trnx(131)=0.08890,
trny(131)=0.317501,
remark='!! Subcomponent
iob(132)=3,
ioh(132)=1,
rah(132)=0.007145,
trnx(132)=0.11430,
trny(132)=0.317501,
remark='!! Subcomponent
iob(133)=3,
ioh(133)=1,
rah(133)=0.007145,
trnx(133)=0.13970,
trny(133)=0.317501,
remark='!! Subcomponent
iob(134)=3,
ioh(134)=1,
rah(134)=0.007145,
trnx(134)=-0.1270,
trny(134)=0.330201,
remark='!! Subcomponent
iob(135)=3,
ioh(135)=1,
rah(135)=0.007145,
trnx(135)=-0.1016,
trny(135)=0.330201,
remark='!! Subcomponent
iob(136)=3,
ioh(136)=1,
rah(136)=0.007145,
trnx(136)=-0.0762,
trny(136)=0.330201,
remark='!! Subcomponent
iob(137)=3,
ioh(137)=1,
rah(137)=0.007145,
trnx(137)=-0.0508,
trny(137)=0.330201,
127',
128',
129',
130',
131',
132',
133',
134',
135',
136',
137',
90
remark='!! Subcomponent
iob(138)=3,
ioh(138)=1,
rah(138)=0.007145,
trnx(138)=-0.0254,
trny(138)=0.330201,
remark='!! Subcomponent
iob(139)=3,
ioh(139)=1,
rah(139)=0.007145,
trnx(139)=0.00000,
trny(139)=0.330201,
remark='!! Subcomponent
iob(140)=3,
ioh(140)=1,
rah(140)=0.007145,
trnx(140)=0.02540,
trny(140)=0.330201,
remark='!! Subcomponent
iob(141)=3,
ioh(141)=1,
rah(141)=0.007145,
trnx(141)=0.05080,
trny(141)=0.330201,
remark='!! Subcomponent
iob(142)=3,
ioh(142)=1,
rah(142)=0.007145,
trnx(142)=0.07620,
trny(142)=0.330201,
remark='!! Subcomponent
iob(143)=3,
ioh(143)=1,
rah(143)=0.007145,
trnx(143)=0.12700,
trny(143)=0.330201,
remark='!! Subcomponent
iob(144)=3,
ioh(144)=1,
rah(144)=0.007145,
trnx(144)=0.10160,
trny(144)=0.330201,
remark='!! Subcomponent
iob(145)=3,
ioh(145)=1,
rah(145)=0.007145,
trnx(145)=-0.1397,
trny(145)=0.342901,
remark='!! Subcomponent
iob(146)=3,
ioh(146)=1,
rah(146)=0.007145,
trnx(146)=-0.1143,
trny(146)=0.342901,
remark='!! Subcomponent
iob(147)=3,
ioh(147)=1,
rah(147)=0.007145,
trnx(147)=-0.0889,
trny(147)=0.342901,
remark='!! Subcomponent
iob(148)=3,
ioh(148)=1,
rah(148)=0.007145,
trnx(148)=-0.0635,
trny(148)=0.342901,
138',
139',
140',
141',
142',
143',
144',
145',
146',
147',
148',
91
remark='!! Subcomponent
iob(149)=3,
ioh(149)=1,
rah(149)=0.007145,
trnx(149)=-0.0381,
trny(149)=0.342901,
remark='!! Subcomponent
iob(150)=3,
ioh(150)=1,
rah(150)=0.007145,
trnx(150)=-0.0127,
trny(150)=0.342901,
remark='!! Subcomponent
iob(151)=3,
ioh(151)=1,
rah(151)=0.007145,
trnx(151)=0.01270,
trny(151)=0.342901,
remark='!! Subcomponent
iob(152)=3,
ioh(152)=1,
rah(152)=0.007145,
trnx(152)=0.03810,
trny(152)=0.342901,
remark='!! Subcomponent
iob(153)=3,
ioh(153)=1,
rah(153)=0.007145,
trnx(153)=0.06350,
trny(153)=0.342901,
remark='!! Subcomponent
iob(154)=3,
ioh(154)=1,
rah(154)=0.007145,
trnx(154)=0.08890,
trny(154)=0.342901,
remark='!! Subcomponent
iob(155)=3,
ioh(155)=1,
rah(155)=0.007145,
trnx(155)=0.11430,
trny(155)=0.342901,
remark='!! Subcomponent
iob(156)=3,
ioh(156)=1,
rah(156)=0.007145,
trnx(156)=0.13970,
trny(156)=0.342901,
remark='!! Subcomponent
iob(157)=3,
ioh(157)=1,
rah(157)=0.007145,
trnx(157)=-0.1270,
trny(157)=0.355601,
remark='!! Subcomponent
iob(158)=3,
ioh(158)=1,
rah(158)=0.007145,
trnx(158)=-0.1016,
trny(158)=0.355601,
remark='!! Subcomponent
iob(159)=3,
ioh(159)=1,
rah(159)=0.007145,
trnx(159)=-0.0762,
trny(159)=0.355601,
149',
150',
151',
152',
153',
154',
155',
156',
157',
158',
159',
92
remark='!! Subcomponent
iob(160)=3,
ioh(160)=1,
rah(160)=0.007145,
trnx(160)=-0.0508,
trny(160)=0.355601,
remark='!! Subcomponent
iob(161)=3,
ioh(161)=1,
rah(161)=0.007145,
trnx(161)=-0.0254,
trny(161)=0.355601,
remark='!! Subcomponent
iob(162)=3,
ioh(162)=1,
rah(162)=0.007145,
trnx(162)=0.00000,
trny(162)=0.355601,
remark='!! Subcomponent
iob(163)=3,
ioh(163)=1,
rah(163)=0.007145,
trnx(163)=0.02540,
trny(163)=0.355601,
remark='!! Subcomponent
iob(164)=3,
ioh(164)=1,
rah(164)=0.007145,
trnx(164)=0.05080,
trny(164)=0.355601,
remark='!! Subcomponent
iob(165)=3,
ioh(165)=1,
rah(165)=0.007145,
trnx(165)=0.07620,
trny(165)=0.355601,
remark='!! Subcomponent
iob(166)=3,
ioh(166)=1,
rah(166)=0.007145,
trnx(166)=0.10160,
trny(166)=0.355601,
remark='!! Subcomponent
iob(167)=3,
ioh(167)=1,
rah(167)=0.007145,
trnx(167)=0.12700,
trny(167)=0.355601,
remark='!! Subcomponent
iob(168)=3,
ioh(168)=1,
rah(168)=0.007145,
trnx(168)=-0.1143,
trny(168)=0.368301,
remark='!! Subcomponent
iob(169)=3,
ioh(169)=1,
rah(169)=0.007145,
trnx(169)=-0.0889,
trny(169)=0.368301,
remark='!! Subcomponent
iob(170)=3,
ioh(170)=1,
rah(170)=0.007145,
trnx(170)=-0.0635,
trny(170)=0.368301,
160',
161',
162',
163',
164',
165',
166',
167',
168',
169',
170',
93
remark='!! Subcomponent
iob(171)=3,
ioh(171)=1,
rah(171)=0.007145,
trnx(171)=-0.0381,
trny(171)=0.368301,
remark='!! Subcomponent
iob(172)=3,
ioh(172)=1,
rah(172)=0.007145,
trnx(172)=-0.0127,
trny(172)=0.368301,
remark='!! Subcomponent
iob(173)=3,
ioh(173)=1,
rah(173)=0.007145,
trnx(173)=0.01270,
trny(173)=0.368301,
remark='!! Subcomponent
iob(174)=3,
ioh(174)=1,
rah(174)=0.007145,
trnx(174)=0.03810,
trny(174)=0.368301,
remark='!! Subcomponent
iob(175)=3,
ioh(175)=1,
rah(175)=0.007145,
trnx(175)=0.06350,
trny(175)=0.368301,
remark='!! Subcomponent
iob(176)=3,
ioh(176)=1,
rah(176)=0.007145,
trnx(176)=0.08890,
trny(176)=0.368301,
remark='!! Subcomponent
iob(177)=3,
ioh(177)=1,
rah(177)=0.007145,
trnx(177)=0.11430,
trny(177)=0.368301,
remark='!! Subcomponent
iob(178)=3,
ioh(178)=1,
rah(178)=0.007145,
trnx(178)=-0.1016,
trny(178)=0.381001,
remark='!! Subcomponent
iob(179)=3,
ioh(179)=1,
rah(179)=0.007145,
trnx(179)=-0.0762,
trny(179)=0.381001,
remark='!! Subcomponent
iob(180)=3,
ioh(180)=1,
rah(180)=0.007145,
trnx(180)=-0.0508,
trny(180)=0.381001,
remark='!! Subcomponent
iob(181)=3,
ioh(181)=1,
rah(181)=0.007145,
trnx(181)=-0.0254,
trny(181)=0.381001,
171',
172',
173',
174',
175',
176',
177',
178',
179',
180',
181',
94
remark='!! Subcomponent
iob(182)=3,
ioh(182)=1,
rah(182)=0.007145,
trnx(182)=0.00000,
trny(182)=0.381001,
remark='!! Subcomponent
iob(183)=3,
ioh(183)=1,
rah(183)=0.007145,
trnx(183)=0.02540,
trny(183)=0.381001,
remark='!! Subcomponent
iob(184)=3,
ioh(184)=1,
rah(184)=0.007145,
trnx(184)=0.05080,
trny(184)=0.381001,
remark='!! Subcomponent
iob(185)=3,
ioh(185)=1,
rah(185)=0.007145,
trnx(185)=0.07620,
trny(185)=0.381001,
remark='!! Subcomponent
iob(186)=3,
ioh(186)=1,
rah(186)=0.007145,
trnx(186)=0.10160,
trny(186)=0.381001,
182',
remark='!! Subcomponent
iob(187)=3,
ioh(187)=1,
rah(187)=0.007145,
trnx(187)=-0.1143,
trny(187)=0.393701,
remark='!! Subcomponent
iob(188)=3,
ioh(188)=1,
rah(188)=0.007145,
trnx(188)=-0.0889,
trny(188)=0.393701,
remark='!! Subcomponent
iob(189)=3,
ioh(189)=1,
rah(189)=0.007145,
trnx(189)=-0.0635,
trny(189)=0.393701,
remark='!! Subcomponent
iob(190)=3,
ioh(190)=1,
rah(190)=0.007145,
trnx(190)=-0.0381,
trny(190)=0.393701,
remark='!! Subcomponent
iob(191)=3,
ioh(191)=1,
rah(191)=0.007145,
trnx(191)=-0.0127,
trny(191)=0.393701,
remark='!! Subcomponent
iob(192)=3,
ioh(192)=1,
rah(192)=0.007145,
trnx(192)=0.01270,
187',
183',
184',
185',
186',
188',
189',
190',
191',
192',
95
trny(192)=0.393701,
remark='!! Subcomponent
iob(193)=3,
ioh(193)=1,
rah(193)=0.007145,
trnx(193)=0.03810,
trny(193)=0.393701,
remark='!! Subcomponent
iob(194)=3,
ioh(194)=1,
rah(194)=0.007145,
trnx(194)=0.06350,
trny(194)=0.393701,
remark='!! Subcomponent
iob(195)=3,
ioh(195)=1,
rah(195)=0.007145,
trnx(195)=0.08890,
trny(195)=0.393701,
remark='!! Subcomponent
iob(196)=3,
ioh(196)=1,
rah(196)=0.007145,
trnx(196)=0.11430,
trny(196)=0.393701,
remark='!! Subcomponent
iob(197)=3,
ioh(197)=1,
rah(197)=0.007145,
trnx(197)=-0.1016,
trny(197)=0.406401,
remark='!! Subcomponent
iob(198)=3,
ioh(198)=1,
rah(198)=0.007145,
trnx(198)=-0.0762,
trny(198)=0.406401,
remark='!! Subcomponent
iob(199)=3,
ioh(199)=1,
rah(199)=0.007145,
trnx(199)=-0.0508,
trny(199)=0.406401,
remark='!! Subcomponent
iob(200)=3,
ioh(200)=1,
rah(200)=0.007145,
trnx(200)=-0.0254,
trny(200)=0.406401,
remark='!! Subcomponent
iob(201)=3,
ioh(201)=1,
rah(201)=0.007145,
trnx(201)=0.00000,
trny(201)=0.406401,
remark='!! Subcomponent
iob(202)=3,
ioh(202)=1,
rah(202)=0.007145,
trnx(202)=0.02540,
trny(202)=0.406401,
remark='!! Subcomponent
iob(203)=3,
ioh(203)=1,
rah(203)=0.007145,
trnx(203)=0.05080,
193',
194',
195',
196',
197',
198',
199',
200',
201',
202',
203',
96
trny(203)=0.406401,
remark='!! Subcomponent
iob(204)=3,
ioh(204)=1,
rah(204)=0.007145,
trnx(204)=0.07620,
trny(204)=0.406401,
remark='!! Subcomponent
iob(205)=3,
ioh(205)=1,
rah(205)=0.007145,
trnx(205)=0.10160,
trny(205)=0.406401,
remark='!! Subcomponent
iob(206)=3,
ioh(206)=1,
rah(206)=0.007145,
trnx(206)=-0.0889,
trny(206)=0.419101,
remark='!! Subcomponent
iob(207)=3,
ioh(207)=1,
rah(207)=0.007145,
trnx(207)=-0.0635,
trny(207)=0.419101,
remark='!! Subcomponent
iob(208)=3,
ioh(208)=1,
rah(208)=0.007145,
trnx(208)=-0.0381,
trny(208)=0.419101,
remark='!! Subcomponent
iob(209)=3,
ioh(209)=1,
rah(209)=0.007145,
trnx(209)=-0.0127,
trny(209)=0.419101,
remark='!! Subcomponent
iob(210)=3,
ioh(210)=1,
rah(210)=0.007145,
trnx(210)=0.01270,
trny(210)=0.419101,
remark='!! Subcomponent
iob(211)=3,
ioh(211)=1,
rah(211)=0.007145,
trnx(211)=0.03810,
trny(211)=0.419101,
remark='!! Subcomponent
iob(212)=3,
ioh(212)=1,
rah(212)=0.007145,
trnx(212)=0.06350,
trny(212)=0.419101,
remark='!! Subcomponent
iob(213)=3,
ioh(213)=1,
rah(213)=0.007145,
trnx(213)=0.08890,
trny(213)=0.419101,
remark='!! Subcomponent
iob(214)=3,
ioh(214)=1,
rah(214)=0.007145,
trnx(214)=-0.0508,
204',
205',
206',
207',
208',
209',
210',
211',
212',
213',
214',
97
trny(214)=0.431801,
remark='!! Subcomponent 215',
iob(215)=3,
ioh(215)=1,
rah(215)=0.007145,
trnx(215)=-0.0254,
trny(215)=0.431801,
remark='!! Subcomponent 216',
iob(216)=3,
ioh(216)=1,
rah(216)=0.007145,
trnx(216)=0.00000,
trny(216)=0.431801,
remark='!! Subcomponent 217',
iob(217)=3,
ioh(217)=1,
rah(217)=0.007145,
trnx(217)=0.02540,
trny(217)=0.431801,
remark='!! Subcomponent 218',
iob(218)=3,
ioh(218)=1,
rah(218)=0.007145,
trnx(218)=0.05080,
trny(218)=0.431801,
remark='!! Subcomponent 219',
iob(219)=3,
ioh(219)=1,
rah(219)=0.007145,
trnx(219)=-0.0381,
trny(219)=0.444501,
remark='!! Subcomponent 220',
iob(220)=3,
ioh(220)=1,
rah(220)=0.007145,
trnx(220)=-0.0127,
trny(220)=0.444501,
remark='!! Subcomponent 221',
iob(221)=3,
ioh(221)=1,
rah(221)=0.007145,
trnx(221)=0.01270,
trny(221)=0.444501,
remark='!! Subcomponent 222',
iob(222)=3,
ioh(222)=1,
rah(222)=0.007145,
trnx(222)=0.03810,
trny(222)=0.444501,
remark='!! Subcomponent 223',
iob(223)=3,
ioh(223)=1,
rah(223)=0.007145,
trnx(223)=0.00000,
trny(223)=0.457201,
remark='!! Component 3 properties',
rcobs(3)=0,
remark='density x specific heat',
kobs(3)=21.90,
remark='thermal conductivity (W/m/K)',
hobs1(3)=227.11,
hobs2(3)=227.11,
remark='allow heat transfer to all fluids',
itpobs(3)=0,
twobs(1, 3)=294.26,
remark='initial tube wall temp',
iaqsrb(3)=0,
remark='!! Component common parameters',
avrck=-3.1,
/
98
&fl
nfls=1,
remark='!! Fluid Region 1',
fioh(1)=1,
ifdis(1)=0,
freg(1)=0.0,
cncreg(1)=0.0002,
fyh(1)=1.0,
pvoid=34500.0,
presi=34500.0,
remark='Initial pressure (Pa)',
cnci=0.0002,
remark='Volume fraction of air initially is 88.47',
iflinittyp=0,
/
&bf
/
&temp
tempi=300.00,
remark='Initial temperature (K)',
tvoid=300.00,
/
&motn
/
&grafic
nvplts=1, contpv(1)='tn',
/
&renderspace
ifrs=0,
ifcomp=0,
ifcompf=-1,
/
&header
project='Case 1',
/
&parts
/
Notes:
Condenser example.
#start tables:
#component(1):
#end component(1)
#component(2):
#end component(2)
#component(3):
#end component(3)
#fluid1:
#end fluid1
#fluid2:
#end fluid2
#end start tables
99