Computational Analysis of Water Atomization in Spray
Desuperheaters of Steam Boilers
by
Paul M. Bovat Jr.
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Dr. Norberto Lemcoff, Thesis Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December 2013
© Copyright 2013
by
Paul M. Bovat Jr.
All Rights Reserved
i
Contents
List of Tables ..................................................................................................................... v
List of Figures ................................................................................................................... vi
Acknowledgment ............................................................................................................ viii
Nomenclature .................................................................................................................... ix
1.0 Introduction ................................................................................................................ 1
1.1 Desuperheater Components ................................................................................... 2
1.1.1 Spray Feedwater Pipe ......................................................................................... 2
1.1.2 Spray Water Nozzle (Desuperheater Nozzle) ..................................................... 2
1.1.3 Desuperheater Shell ............................................................................................ 3
1.1.4 Desuperheater Shell Liner................................................................................... 3
2.0 Atomization Basics ...................................................................................................... 5
2.1 Types of Atomizers................................................................................................ 5
2.1.1 Pressure Atomizers ............................................................................................. 6
3.0 Analysis Methods ........................................................................................................ 8
3.1 Desuperheater Nozzle Pressure Drop .................................................................... 8
3.2 Standard Operating Conditions into Air .............................................................. 10
3.3 Actual Operating Conditions ............................................................................... 10
4.0 Modeled Analyzed ..................................................................................................... 12
4.1 Desuperheater Shell ............................................................................................. 12
4.2 Desuperheater Shell Liner ................................................................................... 13
4.3 Spray Feedwater Pipe .......................................................................................... 13
4.4 Desuperheater Nozzle Assembly ......................................................................... 14
4.4.1 Nozzle Body ..................................................................................................... 14
4.4.2 Orifice Plate ...................................................................................................... 14
4.4.3 Nozzle Cap........................................................................................................ 15
5.0 Computational/Analytical Analysis of Pressure Loss ............................................... 16
ii
5.1 Analytical Pressure Loss Analysis ....................................................................... 16
5.1.1 Inlet Conditions ................................................................................................ 16
5.1.2 Area Enlargement and Contraction .................................................................. 18
5.1.3 Calculations in the Spray Nozzle ...................................................................... 20
5.1.4 Calculations in the Nozzle Cap......................................................................... 25
5.2 Computational Pressure Loss Analysis ............................................................... 25
5.2.1 The Realizable 𝒌 − 𝜺 Model............................................................................. 26
6.0 The Energy Equation ................................................................................................. 28
7.0 Discrete Phase Model (DPM) .................................................................................... 29
7.1 Particle Motion Theory ........................................................................................ 29
7.2 Turbulent Dispersion of Particles ........................................................................ 30
7.3 Laws for Drag Coefficients ................................................................................. 30
7.4 Laws for Heat and Mass Exchange ..................................................................... 30
7.4.1 Inert Heating or Cooling (Law 1/Law 6) .......................................................... 30
7.4.2 Droplet Vaporization (Law 2) and Droplet Boiling (Law 3) ........................... 31
7.5 Atomizer Model Theory ...................................................................................... 32
7.5.1 Pressure-swirl Atomizer ................................................................................... 32
7.5.2 Film Formation ................................................................................................. 32
7.5.3 Sheet Breakup and Atomization ....................................................................... 34
7.6 Secondary Breakup Model Theory ...................................................................... 34
8.0 Droplet Evaporation and Droplet Lifetime ................................................................ 36
8.1 Drop Lifetime ...................................................................................................... 37
9.0 Analysis of the Results .............................................................................................. 39
9.1 Pressure Drop Across the Spray Nozzle .............................................................. 39
9.2 Standard Operating Conditions into Air Results ................................................. 45
10. Conclusions................................................................................................................ 53
References........................................................................................................................ 54
iii
Appendix A: Types of Atomizers .................................................................................... 55
Appendix B: ANSYS Fluent Screen Shots – Turbulent Model ...................................... 57
Appendix C: ANSYS Fluent Screen Shots – DPM Model.............................................. 64
iv
List of Tables
Table 3-1: Water Flow For CFD Run Analysis ............................................................... 10
Table 9-1: Results Comparison Calculated vs. CFD ....................................................... 45
v
List of Figures
Figure 1-1: Schematic of Desuperheater [1] ...................................................................... 1
Figure 1-2: Components of a Desuperheater [1] ............................................................... 4
Figure 1-3: Desuperheater Nozzle ..................................................................................... 4
Figure 2-1: Plain Orifice Atomizer [4] .............................................................................. 6
Figure 2-2: Pressure-Swirl Atomizer [5] ........................................................................... 7
Figure 3-1: Desuperheater Water Flow Path ..................................................................... 9
Figure 4-1: Desuperheater Assembly for Analysis .......................................................... 12
Figure 4-2: Desuperheater Shell ...................................................................................... 12
Figure 4-3: Desuperheater Shell & Liner Assembly ....................................................... 13
Figure 4-4: Water Feedwater Pipe and Nozzle Assembly ............................................... 14
Figure 4-5: Desuperheater Nozzle Body ......................................................................... 14
Figure 4-10: Orifice Plate ................................................................................................ 15
Figure 4-11: Nozzle Cap .................................................................................................. 15
Figure 5-1: Pressure Drop Analysis Schematic (inches) ................................................. 16
Figure 5-2: Flow Region 1 for Calculations (inches) ...................................................... 18
Figure 5-3: Flow Region 2 and 3 for Calculations (inches) ............................................ 20
Figure 5-4: Flow Region 4, 5 and 6 for Calculations (inches) ........................................ 20
Figure 5-5: Flow Region 7A for Calculations (inches) ................................................... 21
Figure 5-6: Flow Through/Around the Orifice Plate and Nozzle Cap’s Teeth ............... 24
Figure 5-7A: Flow Region 7B for Calculations (inches)................................................. 24
Figure 5-7B: Cross Section of Flow Region 7B .............................................................. 25
Figure 5-8: Flow Region 8-10 for Calculations (inches) ................................................. 25
Figure 9-1: Mesh for Run 1 Through Run 11 .................................................................. 40
Figure 9-2: Pressure Contours for Run F1 ....................................................................... 40
Figure 9-3: Pressure Contours for Run F1 ....................................................................... 41
Figure 9-5: Velocity Vectors for Run F1 ......................................................................... 42
vi
Figure 9-6: Pressure Contours for Run F11 ..................................................................... 43
Figure 9-7: Pressure Contours for Run F11 ..................................................................... 43
Figure 9-8: Velocity Vectors for Run F11 ....................................................................... 44
Figure 9-9: Velocity Vectors for Run F11 ....................................................................... 44
Figure 9-10: Pressure Contours for Operating Conditions .............................................. 46
Figure 9-11: Pressure Contours for Operating Conditions .............................................. 46
Figure 9-12: Velocity Vectors for Operating Conditions ................................................ 47
Figure 9-13: Velocity Vectors for Operating Conditions ................................................ 47
Figure 9-14: Mesh for the Desuperheater for Operating Conditions ............................... 48
Figure 9-15: Mesh for the Desuperheater for Operating Conditions (Close up) ............. 49
Figure 9-16: Model Boundary Locations ........................................................................ 49
Figure 9-17: Spray Angle and Temperature Contours .................................................... 49
Table 9-18: Droplet Diameter vs Distance ...................................................................... 50
Figure 9-19: Cut Plan Approx. 1.5 ft from Nozzle Outlet ............................................... 51
Figure 9-20: Cut Plan Approx. 3.5 ft from Nozzle Outlet ............................................... 51
Figure 9-21: Cut Plan Approx. 5.5 ft from Nozzle Outlet ............................................... 52
Figure 9-22: Cut Plan at Outlet of Desuperheater (9ft) ................................................... 52
Figure A-1: Square Spray Atomizer [6] .......................................................................... 55
Figure A-2: Duplex Atomizer [7] .................................................................................... 55
Figure A-3: Dual Orifice Atomizer [8] ............................................................................ 55
Figure A-4: Rotary Atomizer [8] ..................................................................................... 56
Figure A-5: Air-Assist Atomizer [2] ............................................................................... 56
Figure A-8: Airblast Atomizer [8] ................................................................................... 56
Figure A-9: Effervescent Atomizer [8]............................................................................ 56
vii
Acknowledgment
I would like to thank Jerry Chase, PE and Sam Dunning for their help and guidance with
ensuring I have all the correct information about the desuperheater design.
Thank you to Dr. Norberto Lemcoff for his guidance as my thesis advisor and especially
his patience while I write my paper.
Special thanks to Dr. Shiling Zhang, PhD (United Technologies Research Center), Dr.
Lou Chiappetta, PhD (United Technologies Research Center), Dr. Yen-Ming Chen, PhD
and Dr. Luke Munholand, PhD (ANSYS, Inc.) for their help with using Fluent with CFD
Post and taking the time out of their busy schedule to review my work and ensure all of
the data is accurate and makes sense. I am forever grateful for all of your help!
Jason Phesent for his expertise in editing technical papers; without him this would have
read very poorly.
Most importantly my wife Angela and children for their patience and knowing that “it’s
almost done” is truly around the corner. Love you all and thanks!!
viii
Nomenclature
𝐴 = Area (𝑖𝑛2 or 𝑓𝑡 2 )
𝐴𝑎 = Air cone area (𝑖𝑛2 or 𝑓𝑡 2 )
𝐴𝑝 = Surface area of the particle (𝑖𝑛2 or 𝑓𝑡 2 )
𝐴𝑝𝑎 = Total inlet ports area (𝑖𝑛2 or 𝑓𝑡 2 )
𝐴0 = Discharge orifice area(𝑖𝑛2 or 𝑓𝑡 2 )
𝑎 = Liquid jet radius (𝑖𝑛)
𝑎1 , 𝑎2 , 𝑎3 = Constants that apply over several ranges of Re (dimensionless)
𝐵 = Transfer number (dimensionless)
𝐵1 = Break-up time constant set equal to 1.73 (dimensionless)
𝐵𝑀 = Mass transfer number (dimensionless)
𝐵𝑇 = Rate of droplet evaporation (dimensionless)
𝐶 = Flow coefficient for orifices and nozzles (dimensionless)
𝐶3𝜀 , 𝐶𝜇 = Constants (dimensionless)
𝐶𝐷 = Drag coefficient (dimensionless)
𝐶𝑑 = Discharge coefficient for orifices and nozzles (dimensionless)
𝐵𝑇𝑈
𝐶𝑝 = Heat capacity of the particle (𝑙𝑏∙°𝑅)
𝐵𝑇𝑈
𝐶𝑝,𝑗 = Heat capacity of the particle of species j (𝑙𝑏∙°𝑅)
𝐷 = Droplet diameter(𝑖𝑛 or 𝑓𝑡)
𝐷𝑒 = Droplet diameter for evaporation (𝑖𝑛 or 𝑓𝑡)
𝐷ℎ = Hydraulic diameter (𝑖𝑛 or 𝑓𝑡)
𝐷𝑖,𝑚 = Mass diffusion coefficient for species i (dimensionless)
𝐷𝑓𝑤𝑝 = Feedwater pipe diameter (𝑖𝑛 or 𝑓𝑡)
𝐷𝑇,𝑖 = Thermal (Soret) diffusion coefficient (dimensionless)
𝐷𝑡 = Turbulent diffusivity (dimensionless)
𝐷𝑠 = Swirl chamber diameter (𝑖𝑛 or 𝑓𝑡)
𝐷0 = Initial diameter(𝑖𝑛 or 𝑓𝑡)
𝑑0 = Discharge orifice diameter(𝑖𝑛 or 𝑓𝑡)
𝑑𝐿 = Ligament diameter (𝑖𝑛 or 𝑓𝑡)
ix
𝐸 = Energy transfer due to conduction (𝐵𝑇𝑈)
𝐹⃗ = Additional acceleration term (𝑙𝑏𝑓 )
𝐹𝐷 = Drag force per unit particle mass (𝑙𝑏𝑓 )
𝑓 = Friction factor (𝑖𝑛 or 𝑓𝑡)
𝐺 = Incident radiation (𝐵𝑇𝑈)
𝐺𝑏 = Generation of turbulence kinetic energy due to buoyancy (dimensionless)
𝐺𝑘 = Generation/Production of turbulence kinetic energy due to the mean velocity
Gradients (dimensionless)
𝑓𝑡
𝑔 = Gravitational constant (𝑠2 )
𝐵𝑇𝑈
ℎ = Sensible enthalpy (
𝑙𝑏
)
ℎ0 = Film height (𝑖𝑛 or 𝑓𝑡)
ℎ𝐿 = Head loss in feet of pipe due to friction loss (𝑖𝑛 or 𝑓𝑡)
𝐼 = Radiation intensity (energy per area of emitting surface per unit solid angle)
⃗J⃗j = Diffusion flux of species j ( 𝑙𝑏2𝑚 )
𝑓𝑡 ∙𝑠
⃗𝐽⃗𝑖 = Mass diffusion ( 𝑙𝑏2𝑚 )
𝑓𝑡 ∙𝑠
𝐾 = Resistant coefficient (dimensionless)
𝐾𝐴 = Atomizer constant (dimensionless)
𝐾𝑆 = Wave number corresponding to the maximum growth rate (Ω) (dimensionless)
𝐾𝑣 = Velocity coefficient (dimensionless)
𝐵𝑇𝑈
𝑘 = Fluid thermal conductivity based on translational energy only (ℎ𝑟∙𝑓𝑡∙℉)
𝐵𝑇𝑈
𝑘 = Kinetic energy ( 𝑙𝑏 )
𝑚
𝐵𝑇𝑈
𝑘𝑒𝑓𝑓 = Effective conductivity(
)
ℎ𝑟∙𝑓𝑡∙℉
𝑘𝑓 = Relative roughness (dimensionless)
𝐵𝑇𝑈
𝑘𝑡 = Turbulent thermal conductivity (ℎ𝑟∙𝑓𝑡∙℉)
𝐿 = Latent heat of fuel vaporization (
𝐵𝑇𝑈
𝑙𝑏
)
𝐿1 = The distance of the upstream tap from the upstream face of the orifice plate and the
pipe diameter (𝑖𝑛 or 𝑓𝑡)
x
𝐿′2 = The ratio of the distance of the downstream tap from the downstream face of the
orifice plate, and the valve inside diameter (𝑖𝑛 or 𝑓𝑡)
𝐿𝑏 = Length of breakup (𝑖𝑛 or 𝑓𝑡)
𝐿𝑓𝑤𝑝 = Length of feedwater pipe (𝑖𝑛 or 𝑓𝑡)
𝑙𝑏
𝑀𝐴 = Molecular weight of air (𝑚𝑜𝑙)
𝑙𝑏
𝑀𝐷 = Molecular weight of droplet (𝑚𝑜𝑙)
𝑙𝑏
𝑚̇ = Mass flow rate ( ℎ𝑟 )
𝑚𝑝 = Particle mass (𝑙𝑏)
𝑚𝑝,0 = Initial mass of the particle (𝑙𝑏)
𝑁𝑢 = Nusselt number (dimensionless)
𝑂ℎ = Ohnesorge number (dimensionless)
𝑙𝑏
𝑃 = Pressure( 𝑓𝑡𝑚2 )
𝑙𝑏
𝑃𝐷 = Droplet vapor pressure at the drop ( 𝑓𝑡𝑚2 )
𝑙𝑏
Δ𝑃𝑙 = Pressure loss across the nozzle ( 𝑓𝑡𝑚2 )
𝑔𝑎𝑙
𝑄̇ = Volumetric flow rate (𝑚𝑖𝑛)
𝑔𝑎𝑙
𝑄ℎ𝑡 = Rate of heat transfer to drop from surrounding gas (𝑚𝑖𝑛)
Re = Relative Reynolds number (dimensionless)
𝑟 = Radius of the droplet (𝑖𝑛 or 𝑓𝑡)
𝑟0 = Radial distance from center line to mid-line of sheet at the atomizer exit (𝑖𝑛 or 𝑓𝑡)
𝑆ℎ = Heat of chemical reaction (dimensionless)
𝑆𝑖𝑗 , 𝑆𝑗𝑘 , 𝑆𝑘𝑖 = Mean rate-of-strain tensor (dimensionless)
𝑆𝑘 , 𝑆𝜀 = User-defined source terms (dimensionless)
𝑆𝑐𝑡 = Turbulent Schmidt number (dimensionless)
𝑇 = Local fluid temperature (℉ or °R)
𝑇𝑎 = Taylor number (dimensionless)
𝑇𝑝 = Particle temperature (℉ or °R)
𝑇𝑟𝑒𝑓 = 536.67°𝑅
xi
𝑇𝑠 = Droplet surface temperature (℉ or °R)
𝑇∞ = Local temperature of the continuous phase (℉ or °R)
𝑡 = Time (s)
𝑡𝑒 = Drop evaporation time (𝑠)
𝑡𝐹 = Film thickness (𝑖𝑛 or 𝑓𝑡)
𝑡𝑠 = Sheet thickness (𝑖𝑛 or 𝑓𝑡)
𝑓𝑡
𝑈 = Free stream velocity ( 𝑠 )
𝑓𝑡
𝑢 = Velocity magnitude ( 𝑠 )
𝑓𝑡
𝑢′ = Fluctuating component of turbulence velocity ( 𝑠 )
𝑓𝑡
𝑢̅2 = Incompressible strained mean flow ( 𝑠 )
𝑊𝑒 = Weber number (dimensionless)
𝑋 = Ratio of the area of air cone to the area of the final discharge orifice (dimensionless)
𝑌𝐷 = Mass fraction of droplet vapor (dimensionless)
𝑌𝑖 = Mass fraction of species i (dimensionless)
𝑌𝑗 = Mass fraction of species j (dimensionless)
𝑌𝑀 = Contribution of fluctuating dilatation in turbulence to overall dissipation rate
(dimensionless)
𝑍 = Elevation (𝑖𝑛 or 𝑓𝑡)
Greeks Letters
Δ = Difference between two values, the change in (dimensionless)
Λ = Corresponding wavelength (dimensionless)
Ω = Solid angle (degrees)
Ω𝑟 = Maximum growth rate (dimensionless)
̅ ij = Rate-of-rotation tensor viewed in a moving reference frame (dimensionless)
Ω
𝛽 = Ratio of small to large diameter in orifice and nozzle, and contractions or
enlargements in pipes (dimensionless)
𝜀 = Absolute roughness coefficient (𝑖𝑛 or 𝑓𝑡)
𝑓𝑡 2
𝜀 = Dissipation rate ( 𝑠3 )
𝜀𝑝 = Particle emissivity (dimensionless)
xii
𝜃 = Spray cone half angle (degrees)
𝜃𝑅 = Radiation temperature (℉ or °R)
𝜆 = Mean free path of the fluid (dimensionless)
𝜆∗ = Wavelength for maximum growth rate (dimensionless)
𝜆𝑠𝑡 = Evaporation constant (dimensionless)
𝑙𝑏
𝑚
𝜇 = Molecular viscosity of the fluid (𝑓𝑡∙𝑠
)
𝑙𝑏
𝑚
𝜇𝑡 = Turbulent (or eddy) viscosity (𝑓𝑡∙𝑠
)
𝜈 = Kinematic viscosity (
𝑓𝑡 2
𝜈𝑡 = Kinematic viscosity (
𝑠
)
𝑓𝑡 2
𝑠
)
𝑙𝑏
𝜌 = Density of water at 70F and various pressures ( 𝑓𝑡𝑚3 )
𝑙𝑏
𝜎 = Surface tension of the liquid ( 𝑖𝑛𝑓 )
𝜎𝑆𝐵 = Stefan-Boltzmann constant (dimensionless)
𝜏 = Breakup time (𝑠)
ωk = Angular velocity (radians per second)
Subscripts
1, 2, 3 … = Flow regions
𝐷 = Droplet
𝑑𝑣 = Diverting valve
𝑓𝑤𝑝 = Feedwater pipe
𝑓𝑤𝑛 = Feedwater valve
𝑔 = Gas
𝑖𝑛 = Inlet flow
𝑙 = Liquid
𝑝 = Partical
𝑜𝑝 = Orifice plate and teeth combined
𝑠 = Surface
𝑤𝑐 = Weld ring in contraction section
𝑤𝑒 = Weld ring in enlargement section
xiii
Abstract
High temperature steam generated in boilers can damage turbines or other equipment if
the temperature is not properly controlled. Atomized water at a lower temperature is
sprayed into the steam in order to control its temperature. However, if the steam
temperature is lowered too much it can cause the steam to change phase and condense in
the turbine. In this thesis, the physics behind water atomization, droplet evaporation and
droplet lifetime is discussed. Computational Fluid Dynamics studies were carried out
using ANSYS Fluent to analyze the nozzle pressure drop and the atomization process.
The results from the computational analysis show that the pressure loss is approximately
17% greater than the value given by an industry standard. This can be due to the mesh
size limitations in the version of the code used. The results also show that the steam
temperature is reduced by only 4.5℉. This is not expected, since the design steam
temperature to enter the desuperheater is expected to be 572℉. However, the distance at
which the droplets are completely evaporated is predicted very accurately to be 1.48ft
downstream of the nozzle. Recommendations are made for better control of the
desuperheater temperature.
xiv
1.0 Introduction
In the power industry, it is essential that power plant boilers have regulated steam
temperature so that the boiler and turbine-generator can operate properly and efficiently.
One control device on such boilers is the spray-type desuperheater. These units are
primarily used to control temperature in superheater and reheater steam circuits. The
temperature of the steam tends to increase with the boiler in operation and therefore, the
spray is triggered and water taken from the feed pump discharge or the economizer inlet
(depending on the potential for thermal shock in the spray system) is released for steam
regulation. A typical desuperheating system consists of a control valve, piping, isolation
and check valves, in addition to a control temperature indicator. A replaceable liner is
installed for protection against erosion and thermal shocking of the desuperheater
pressure shell, which would otherwise occur as a result of intermittent desuperheater
spray. Spray water desuperheaters must utilize boiler quality feedwater because of the
devices location in proximity to the steam circuitry. By design, the desuperheater is
located within the steam piping system. This construction ensures that there is sufficient
time for spray water evaporation to occur before the steam actually reaches the steam
turbine (for exit-stage installations) or the superheat/reheat elements (for inter-stage
installations). Figure 1-1 illustrates a typical desuperheating installation [1].
Figure 1-1: Schematic of Desuperheater [1]
1
1.1 Desuperheater Components
There are many different designs of desuperheaters that consist of the same basic
components. These components include the spray water feed pipe, spray water nozzle,
desuperheater shell, and desuperheater shell liner. Below are the descriptions of each of
the desuperheater individual components.
1.1.1 Spray Feedwater Pipe
The spray feedwater pipe (Figure 1-2 – Item 1) is designed to have a pressure of
approximately 1180𝑝𝑠𝑖 and a velocity between 8
𝑓𝑡
𝑠
𝑓𝑡
and 19 𝑠 , depending on the boiler
type. The length of the pipe is dependent on the individual design of the boiler. The
1
1
spray feedwater pipe has a pipe diameter that ranges from 1 2 𝑖𝑛 to 2 2 𝑖𝑛. The pipe is
designed using various materials such as: P-1 (carbon steel), P-4 (1Cr – 0.5Mo to 1-¼Cr
– 5Mo), and P-5 (2-¼Cr – 1Mo).
1.1.2 Spray Water Nozzle (Desuperheater Nozzle)
The spray water nozzle (Figure 1-2 – Item 2) (also known as the desuperheater nozzle) is
categorized as a pressure swirl atomizer. Section 2.0 describes the theory of atomization
and section 2.1 describes the difference between the various types of atomizers. The
desuperheater nozzle consists of three individual parts: nozzle body, nozzle tip, and
orifice plate. Figure 1-3 illustrates each individual component.
Nozzle Body: The nozzle body is made of forged steel, from P-1 (SA-106 C), P-4 (1Cr –
0.5Mo to 1-¼Cr – 5Mo), and P-5 (SA-335-P22 and SA-182-F22, CL. 3). Figure 1-3
shows an outlet spray nozzle, while other desuperheater designs may contain additional
spray outlets.
Orifice Plate: This multifunctioning plate is designed to either work as an orifice or as a
flow swirl assist. The orifice plate design has one hole centered in the plate of diameter
𝐷𝑜𝑝 . The flow swirl assist design not only has a hole centralized to the plate of diameter
𝐷𝑝𝑙1 , but smaller holes arranged around the outside of the center hole in a circular
pattern of diameter 𝐷𝑝𝑙2 . This type of design causes the flow to swirl as it exits the
nozzle tip. The material used in each of the above designs, consists of various grades of
stainless steel.
2
Nozzle Tip: The nozzle tip can be designed to assist the swirl of the flow or act as an
orifice. The swirl assist design includes systematically designed “teeth” (Figure 1-3) to
cause the flow to swirl as it exits the tip. This nozzle tip is typically used in conjunction
with the orifice plate design. The orifice tip design is used with the swirl assist orifice
plate to achieve approximately the same swirl characteristics. Various grades of stainless
steel are used in either design.
1.1.3 Desuperheater Shell
The desuperheater shell (Figure 1-2 – Item 3) is a pipe through which superheated steam
is carried from the superheater or reheater assemblies to the turbine assembly or to
another superheater/reheater assembly, depending on the individual design. The shell
diameter can range from 20𝑖𝑛 to 32𝑖𝑛. Like most other components, the actual length of
the shell varies depending on the boiler design. The material of the desuperheater shell
can be: P-1 (carbon steel), P-4 (1Cr – 0.5Mo to 1-¼Cr – 5Mo), or P-5 (2-¼Cr – 1Mo).
1.1.4 Desuperheater Shell Liner
The desuperheater shell liner (Figure 1-2 – Item 4) is intended to protect the
desuperheater shell and, unlike other components, is a replaceable part. The shell liner
can be a seamless pipe or a welded plate, while the thickness and length is dependent
upon the desuperheater shell. This is important because, if the spray does not evaporate
in time, the droplet can damage the desuperheater shell. The shell liner is fastened to the
desuperheater using screws, some of which go through the liner while others are pushed
against the liner in order to hold it into place. The liner material can be: P-1 (carbon
steel), P-4 (1Cr – 0.5Mo to 1-¼Cr – 5Mo), or P-5 (2-¼Cr – 1Mo).
3
Figure 1-2: Components of a Desuperheater [1]
Figure 1-3: Desuperheater Nozzle
4
2.0 Atomization Basics
Sprays may be produced in various ways and are used for various applications. The
atomization process consists in the development of liquid sheets that eventually become
ligaments and then break up into droplets. This process determines the shape of the
resulting spray, as well as its detailed characteristics, which include density, drop
velocity, and drop size distributions as functions of time and space. Characteristics of the
spray, determined by the internal geometry of the atomizer and the liquid properties of
the fluid, can have a noticeable effect on the droplet size and evaporation lifetime for the
spray. Lord Rayleigh1 postulated that the increase of small disturbances will ultimately
lead to the breakup of the jet, which will then form drops having a diameter nearly twice
that of the jet [2]. Ligaments vary in diameter and, when they collapse, the size of the
drops that are formed will also vary in diameter. Larger droplets that are created by this
process breakup even further (secondary breakup) transforming into even smaller
droplets. A typical spray can include a wide range of drop sizes. Knowledge of the drop
size diameter and distribution is helpful in analyzing process applications in sprays,
especially in calculations dealing with heat and/or mass transfer between the dispersed
liquid and the surrounding gas. Due to the difficulty in determining spray drop size
distributions, various mean or median droplet diameters are generally used. Mass,
volume or number median diameters are determined from the droplet size distribution
curves. In this work, the diameter that will be used is the Sauter mean diameter (SMD or
𝐷32 ), which represents the ratio of the volume to the surface area of the spray [2].
2.1 Types of Atomizers
To produce a good atomization, a high relative velocity between the liquid to be
atomized and the surrounding air or gas is required. Some atomizers accomplish this by
discharging the liquid at a high velocity into a relatively slow-moving stream of air or
gas. Some examples are the various forms of pressure atomizers and rotary atomizers,
which eject the liquid at high velocity from the periphery of a rotating cup or disk. Other
types of atomizers are twin-fluid, air-assist, or airblast atomizers [2].
1
Lord Rayleigh (John William Strutt) - The Nobel Prize in Physics 1904 was awarded to Lord Rayleigh
"for his investigations of the densities of the most important gases and for his discovery of argon in
connection with these studies" [3].
5
2.1.1 Pressure Atomizers
When a liquid is discharged through a small aperture at high pressure, the energy is
converted into kinetic energy [2]. There are various types of pressure atomizers: plain
orifice atomizers, pressure-swirl (simplex) atomizers, square spray atomizers, duplex
atomizers, and dual orifice atomizers (see Appendix A). Figures A-1 through Figure A-5
show the various pressure atomizers configurations.
Plain Orifice Atomizers: A circular orifice is used, which could be a cone, or a
cylinder, or a plate, which creates a round jet of liquid. The optimum atomization is
accomplished using orifices that are small in size; however, the difficult part is keeping
liquids free from foreign particles. The particles generally limit the minimum size to
about 0.0118𝑖𝑛 [2]
Figure 2-1: Plain Orifice Atomizer [4]
Pressure-Swirl (Simplex) Atomizers: A circular outlet orifice is preceded by a swirl
chamber into which liquid flows through a number of tangential holes or slots (Figure 22). The swirling liquid creates a core of air or gas that extends from the discharge orifice
to the rear of the swirl chamber. The liquid then emerges from the discharge orifice as an
annular sheet, which spreads radially outward, forming a hollow conical spray. Included
spray angles range from 30° to almost 180°, depending on the application. The finest
atomization occurs at high delivery pressures and wide spray angles. For some
applications a spray in the form of a solid cone is preferred. This can be achieved by
using an axial jet or with the use of other mechanical devices to inject droplets into the
center of the hollow conical spray pattern produced by the swirl chamber. These two
methods of injection create a bimodal distribution of drop sizes, with the droplets
concentrated at the center of the spray generally larger than those near the edge [2].
6
Figure 2-2: Pressure-Swirl Atomizer [5]
7
3.0 Analysis Methods
Each analysis will be run using ANSYS Fluent 14.0, a Computational Fluid Dynamics
(CFD) software. The theory of a Pressure-Swirl Atomizer is used to analyze the system,
and the following physics will be discussed: pressure drop across the nozzle, secondary
droplet break-up, droplet size compared to droplet lifetime for evaporation, and
evaporation time with distance needed for evaporation. The reason for evaluating the
discharge temperature in the desuperheater shell and liner is to ensure that the
temperature leaving the desuperheater is suitable for the downstream equipment. Droplet
lifetime will be analyzed to determine the length of the boundary required
(desuperheater shell and liner) for complete atomization evaporation. Both CFD and
analytical calculations will be carried out in order to evaluate the atomization process.
The first stage in the analysis will be to compare an industrial standard of a spray nozzle
with different pressure drop and flow rates. The second stage will be similar to the first
with the exception of the fluid temperature, which will be set to what would be seen in a
typical boiler. The third and final stage will be to analyze the desuperheater with the
spray flow injected into the steam in order to regulate the steam temperature.
3.1 Desuperheater Nozzle Pressure Drop
The first stage in the analysis will check the nozzle design against that commonly used
in the industry, while validating the CFD analysis. Water flows through the water feed
pipe at a temperature of 70℉, and at different flow rates and pressures. Figure 3-1
illustrates the flow path corresponding to this analysis. The center hole in the orifice
plate will be
5
64
𝑖𝑛 ≈ 0.391𝑖𝑛, and the nozzle cap center hole will be set at
9
64
𝑖𝑛 ≈
0.609𝑖𝑛. The intent is to evaluate the pressure drop across the nozzle, and compare it to
𝑙𝑏
values commonly used in industry. Table 3-1 shows the mass flow rate (ℎ𝑟) of the water
used in the CFD runs. Each flow is converted from a mass flow rate to a volumetric flow
rate using equation 3.1. Each flow corresponds to a different pressure, which ranges
𝑙𝑏
from 100𝑝𝑠𝑖𝑎 to 2, 000𝑝𝑠𝑖𝑎 (i.e. 44,500 ℎ𝑟 corresponds to 100𝑝𝑠𝑖𝑎).
8
The CFD analysis will be done using a Realizable 𝑘 − 𝜀 model. The inlet boundary
condition will be a “Mass Flow Inlet” whereas the outlet will be a “Pressure Outlet”,
which will be set to zero.
𝑚̇
𝑄̇ =
𝜌
(3.1)
Figure 3-1: Desuperheater Water Flow Path
9
Table 3-1: Water Flow For CFD Run Analysis
Run F1
Run F2
Run F3
Run F4
Run F5
Run F6
Flow (𝑙𝑏⁄ℎ𝑟)
44,500
77,000
99,500
117,900
126,000
140,900
Flow (𝑔𝑎𝑙⁄𝑚𝑖𝑛)
89.00
153.90
198.80
235.40
251.50
281.10
100
300
500
700
800
1,000
Run F7
Run F8
Run F9
Run F10
Run F11
Flow (𝑙𝑏⁄ℎ𝑟)
154,000
166,500
178,000
189,000
199,000
Flow (𝑔𝑎𝑙⁄𝑚𝑖𝑛)
307.00
331.70
354.40
376.10
395.80
1,200
1,400
1,600
1,800
2,000
Pressure
Drop (𝑙𝑏𝑠⁄𝑖𝑛)
Pressure
Drop (𝑙𝑏𝑠⁄𝑖𝑛)
3.2 Standard Operating Conditions into Air
This analysis will be performed using pressures and temperatures that one might see in
an actual desuperheater system. With the feedwater properties set to 370℉, and the inlet
𝑙𝑏
flow set to 41,250 ℎ𝑟, the pressure loss will be calculated. The feedwater pipe will have
the same roughness as a typical industrial pipe, and the flow will be atomized into
ambient air conditions.
The CFD analysis will be the same as in section 3.1. The boundary conditions will be set
𝑙𝑏
to a “Mass Flow Inlet” of 687.5𝑚𝑖𝑛 and a pressure of 1,180𝑝𝑠𝑖. The outlet will have a
boundary condition of a “Pressure Outlet” and will be set to zero. The energy equation
will be activated and the temperatures at the boundary conditions will be set to 370℉.
3.3 Actual Operating Conditions
This analysis will evaluate using actual operating conditions inside a boiler
desuperheater. This is similar to section 3.2 in regards to the flow through the feedwater
pipe. However, the discharge will not be into air but into superheated steam. The flow,
temperature, and pressure entering the desuperheater will be a superheated steam at
𝑙𝑏
1,760,000 ℎ𝑟 at 650℉ and 682𝑝𝑠𝑖𝑔, respectively.
10
The CFD analysis will be carried out using a Realizable 𝑘 − 𝜀 model, coupled with the
energy equation and species models active. The boundary conditions for the energy
equation will have an inlet temperature of 650℉. The species model will only consist of
water in the form of steam, at 650℉ and 682𝑝𝑠𝑖. The inlet will be a “Mass Flow Inlet”
𝑙𝑏
with a mass flow of 29,333.3 𝑚𝑖𝑛, and a pressure of 682𝑝𝑠𝑖. The outlet will be a
“Pressure Outlet” set to zero and the wall effects set to “Escape”. The Discrete Phase
model is activated and the “Pressure Swirl Atomizer” is selected. The properties for the
water injection will be the same as stated in section 3.2.
11
4.0 Modeled Analyzed
The desuperheater was modeled using Autodesk Inventor 2014, and was taken from a
plant design from 1970 [9]. Figure 4-1 shows the design of the system. The basic
dimensions of the desuperheater assembly are presented in the next section.
Figure 4-1: Desuperheater Assembly for Analysis
4.1 Desuperheater Shell
The desuperheater shell has an outside diameter of 25𝑖𝑛 with a wall thickness of 0.75𝑖𝑛.
This gives the inside diameter a dimension of 23.5𝑖𝑛, with a total overall length of
245𝑖𝑛. The designed length, according to the boiler design drawings, is 120𝑖𝑛.
However, 125𝑖𝑛 (5𝐷) was added to the inlet of the shell to ensure that the flow entering
is developed throughout the pipe (see Figure 4-2). The material is SA-515 Grade 70
(Silicon-Killed steel plate for boilers and other pressure vessels) with an absolute
roughness coefficient, 𝜀 of 225 × 10−6 𝑓𝑡. The relative roughness
𝑘𝑓 =
𝜀
𝐷ℎ
(4.1)
is calculated to be k f  115 106 .
Figure 4-2: Desuperheater Shell
12
4.2 Desuperheater Shell Liner
The desuperheater shell liner has an outside diameter of 22𝑖𝑛 and a wall thickness of
0.75𝑖𝑛. Therefore, the inside diameter is 20.5𝑖𝑛 with a liner length of 106𝑖𝑛. Fasteners
hold the liner centered in the desuperheater which are attached to the shell in order to
hold the liner in place. The fasteners protrude into the secondary flow area but will have
a negligible effect on the flow (see Figure 4-3), and therefore will be omitted from the
flow analysis. The material is comprised of P-1 carbon steel, using the same absolute
roughness coefficient of 225 × 10−6 𝑓𝑡. The relative roughness is calculated using
Equation 4.1 which yields 𝑘𝑓 = 132 × 10−6 . The desuperheater shell liner is installed
2𝑖𝑛 from the centerline of the feedwater pipe location, and 2𝑖𝑛 from the end of the
desuperheater shell (see Figure 4-3).
Figure 4-3: Desuperheater Shell & Liner Assembly
4.3 Spray Feedwater Pipe
The spray feedwater pipe is made of 1.5𝑖𝑛 schedule XXH (double extra strong), with an
inside diameter of 1.1𝑖𝑛. The pipe is assembled 135𝑖𝑛 from the inlet of the
desuperheater shell; however, the actual design is 10𝑖𝑛 from the inlet of the
desuperheater. The material is SA-335-P11 (boiler pipe) with the same absolute
roughness coefficient as above, and a relative roughness 𝑘𝑓 = 2.46 × 10−3. At the other
end of the nozzle there is another 1.5𝑖𝑛 schedule XXH pipe which protrudes through the
other side of the shell, which is used as support during operation. However, for ease of
calculation the pipe was modeled as a solid piece (see Figure 4-4).
13
Figure 4-4: Water Feedwater Pipe and Nozzle Assembly
4.4 Desuperheater Nozzle Assembly
The desuperheater nozzle consists of three separate components: the nozzle body, orifice
plate and the nozzle tip.
4.4.1 Nozzle Body
The nozzle body has an inlet inside diameter of 1.159𝑖𝑛 (see Figure 4-5). The material is
SA-182 Grade F11, Class 2 (forged steel) and has a relative roughness 𝑘𝑓 = 2.33 ×
10−3.
Figure 4-5: Desuperheater Nozzle Body
4.4.2 Orifice Plate
At the center of the orifice plate (Figure 4-10), there is a hole with the diameter of 25
𝑖𝑛 ≈
64
0.391𝑖𝑛. The plate is inserted into the nozzle body and centered with respect to the
14
centerline of the nozzle outlet. The material of the orifice plate is AISI Grade 420,
stainless steel, and has a relative roughness 𝑘𝑓 = 1.536 × 10−3 .
Figure 4-10: Orifice Plate
4.4.3 Nozzle Cap
The nozzle cap is shown in Figure 4-11, and it represents a pressure swirl atomizer. The
material is AISI Grade 420, stainless steel, and has a relative roughness 𝑘𝑓 = 985 ×
10−6, with an inside diameter of
39
64
𝑖𝑛 ≈ 0.609𝑖𝑛. As shown, the nozzle cap has “teeth”
which cause the liquid to swirl as it leaves the nozzle assembly. The teeth are designed
in such a way to ensure an evenly distributive swirl. Without the swirling teeth, the
atomizer will only be categorized as a plane orifice atomizer.
Figure 4-11: Nozzle Cap
15
5.0 Computational/Analytical Analysis of Pressure Loss
In this section the rigorous calculations to evaluate the pressure loss across the nozzle
will be discussed. The analysis in section 5.1 will be based on the equations from Crane
[10]. The computational analysis, based on the Fluent Theory Guide [14], will be
discussed in section 5.2.
5.1 Analytical Pressure Loss Analysis
It is important to understand that the pressure drop across the nozzle is designed to
ensure that there is enough pressure in the system to get proper atomization. Figure 5-1
shows the system to be analyzed. Water is used at 70℉ and various input pressures
(depending on the flow). The density is based on the properties at the inlet pressure and
temperature. In order to calculate the pressure drop, equations from Crane [10] are used
and described in the following sections.
Figure 5-1: Pressure Drop Analysis Schematic (inches)
5.1.1 Inlet Conditions
Minor pressure losses are common in long stretches of pipe. As shown is Figure 5-1,
there is just over 21𝑖𝑛 of pipe before there is a change in flow area. The flow rate can be
calculated from equation 5.1, while the velocity in the pipe is obtained from equation
5.2.
16
𝑚̇𝑓𝑤𝑝 = 𝐴𝑓𝑤𝑝,1 𝜌𝑙 𝑢𝑓𝑤𝑝,𝑖𝑛
𝑢𝑓𝑤𝑝,𝑖𝑛 =
𝑚̇𝑓𝑤𝑝
(5.1)
(5.2)
𝐴𝑓𝑤𝑝,1 𝜌𝑙
Once the velocity is known, the Reynolds number can be calculated from
𝑅𝑒𝑓𝑤𝑝 =
𝑢𝑓𝑤𝑝,𝑖𝑛 𝐷𝑓𝑤𝑝,1
(5.3)
𝜈𝑙
The friction factor, 𝑓𝑓𝑤𝑝 , in the pipe is calculated using the wall roughness, 𝜀 (see
Equation 4.1), and is evaluated from
𝑓𝑓𝑤𝑝,1 =
0.25
𝑘𝑓,1
5.74
[log(
+
)]
3.7𝐷𝑓𝑤𝑝,1 𝑅𝑒0.9
𝑓𝑤𝑝,1
2
(5.4)
The minor head loss due to friction is calculated by
ℎ𝐿,1 = 𝑓𝑓𝑤𝑝,1
𝐿𝑓𝑤𝑝,1
𝐷𝑓𝑤𝑝,1
(
2
𝑢𝑓𝑤𝑝,𝑖𝑛
2𝑔
)
(5.5)
The pressure drop in the pipe due to friction, and based on the head loss, is then
calculated
∆𝑃𝑙,1 = ℎ𝐿,1
𝜌𝑙
144
(5.6)
The new velocity, based on velocity head and change in pressure, is calculated using
Bernoulli’s Theorem2
𝑃𝑓𝑤𝑝,𝑖𝑛
𝑢𝑓𝑤𝑝,2 = √2𝑔∆𝑍 + 2𝑔 (
2
𝜌𝑙
−
𝑃𝑓𝑤𝑝,1
𝜌𝑙
2
− 2𝑔ℎ𝐿,1
) + 𝑢𝑓𝑤𝑝,𝑖𝑛
(5.7)
Daniel Bernoulli (1700-1782) was a Swiss mathematician and is particularly remembered for his
applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in
probability and statistics.
17
A schematic of the flow region is shown in Figure 5-2.
Figure 5-2: Flow Region 1 for Calculations (inches)
5.1.2 Area Enlargement and Contraction
As shown in Figure 5-1, the flow area changes several times, thus creating a pressure
drop. First, the flow enters the swaged region which has an increase in diameter, creating
a minor loss in pressure, characterized by a resistant coefficient as stated by Crane [10].
For a small pipe and a sudden enlargement, the following equation is used
2
𝐾𝑓𝑤𝑝,2 = (1 − 𝛽𝑓𝑤𝑝,1,2
)
2
(5.8)
where,
𝛽𝑓𝑤𝑝,1,2 =
𝐷𝑓𝑤𝑝,1
𝐷𝑓𝑤𝑝,2
(5.9)
The flow then enters the weld ring, initially decreasing the flow area and then increasing
it. This type of geometry is similar to a venturi; however, according to Crane [10], this
geometry does not fit the proper criteria for a venturi effect. Therefore, the assumption is
to treat the geometry like a contraction in diameter, followed by an enlargement in
diameter. This will create two different resistant coefficients as shown below
2
𝐾𝑓𝑤𝑝,𝑤𝑐 = 0.5(1 − 𝛽𝑓𝑤𝑝,3,4
)
18
(5.10)
2
𝐾𝑓𝑤𝑝,𝑤𝑒 = (1 − 𝛽𝑓𝑤𝑝,5,6
)
2
(5.11)
where,
𝛽𝑓𝑤𝑝,3,4 =
𝛽𝑓𝑤𝑝,5,6 =
𝐷𝑓𝑤𝑝,4
(5.12)
𝐷𝑓𝑤𝑝,3
𝐷𝑓𝑤𝑝,5
(5.13)
𝐷𝑓𝑤𝑝,6
Equations 5.10 and 5.12 are used for a contraction in diameter while Equations 5.11 and
5.13 are used for an enlargement in diameter. Calculating ℎ𝐿,𝑗 (subscript j represents the
appropriate flow region) for each location and solving for ∆𝑃𝑙,𝑗 in that region, yields a
velocity change. So, just as the area changes, so does the velocity. Setting the area ratios
equal to the ratio of the velocities yields
𝐴𝑓𝑤𝑝,𝑗
𝐴𝑓𝑤𝑝,𝑘
=
𝑢𝑓𝑤𝑝,𝑗
𝑢𝑓𝑤𝑝,𝑘
(5.14)
Rearranging Equation 5.14 and solving for 𝑢𝑓𝑤𝑝,𝑘
𝑢𝑓𝑤𝑝,𝑘 =
𝐴𝑓𝑤𝑝,𝑗 𝑢𝑓𝑤𝑝,𝑗
𝐴𝑓𝑤𝑝,𝑘
(5.15)
Introducing the velocity calculated in Equation 5.7, a final velocity entering the spray
nozzle is determined from Equation 5.15. The head loss is calculated using the following
equation
ℎ𝐿,𝑗 = 𝐾𝑓𝑤𝑝,𝑗
2
̅𝑓𝑤𝑝
𝑢
2𝑔
(5.16)
while the pressure differential is evaluated from Equation 5.6. See Figures 5-3 and 5-4
for the locations of the different regions.
19
Figure 5-3: Flow Region 2 and 3 for Calculations (inches)
Figure 5-4: Flow Region 4, 5 and 6 for Calculations (inches)
5.1.3 Calculations in the Spray Nozzle
As the flow enters the spray nozzle, the flow becomes more complex to calculate. The
nozzle is broken up into three separate sections, where the velocities are calculated and
averaged. Each pressure loss is calculated and added for a total pressure loss. The nozzle
is broken up as follows: nozzle entrance, orifice plate, and flow through the nozzle cap’s
swirl teeth. Assumptions have been made in order to calculate the pressure drop across
the different sections. For example, flow through an orifice, flow through a diverting
valve, and flow through the cap’s teeth using the total open area as an orifice. Figures 55 to 5-7 show the flow regions in which the calculations are to be computed.
20
Nozzle Entrance. The geometry is similar to a diverting valve, and therefore it is used
as a best estimate for the resistance coefficient. In order to calculate the pressure loss in
the nozzle, a flow coefficient, 𝐶𝑣,𝑑𝑣 , is used based on a 1.5𝑖𝑛 diverting valve open 50%
[13]. The resistance coefficient is obtained from Crane [10]
4
𝐾𝑑𝑣 = 890.3
(𝐷𝑓𝑤𝑛,7𝐴 )
2
(𝐶𝑣,𝑑𝑣)
(5.17)
This region is labeled 7A (see Figure 5-5). The head loss in the nozzle is calculated
using Equation 5.16, and the pressure drop using Equation 5.6.
Figure 5-5: Flow Region 7A for Calculations (inches)
Orifice Plate and Nozzle Cap’s Teeth. Due to the difficulty of this flow analysis,
assumptions are made in order to calculate a pressure drop across the nozzle. As shown
in Figure 5-6, the flow is assumed to be evenly distributed throughout the orifice plate
and nozzle cap’s teeth. With that assumption, analyzing the flow and using the
calculations as an orifice seems to be most appropriate. Calculating the flow through an
orifice is described very well by Crane [10]. In order to calculate an orifice diameter, the
total flow area of the geometry is calculated. First, the area in the orifice plate is
calculated by simply using a typical area equation. However, calculating the flow area
21
through the nozzle cap teeth is not as simple due to the complexity of the geometry.
Therefore, a cross-sectional area was taken from a CAD software. Then, both areas are
added together to get a total area. This area was used to calculate the equivalent orifice
diameter. To calculate the pressure drop, several equations are required. Such equations
are the diameter ratio, 𝛽𝑓𝑤𝑛,6,7𝐵 , discharge coefficient, 𝐶𝑑,𝑜𝑝 , and the flow resistant
coefficient, 𝐾𝑜𝑝 . The flow resistant coefficient has been derived below using the
following flow relationships
𝑄̇𝑜𝑝 = 𝐴𝑜𝑝 𝑢𝑜𝑝
(5.18)
2
𝑄̇𝑜𝑝 = 19.64𝐷𝑜𝑝
𝐶 √ℎ𝐿,𝑜𝑝
𝐶=
𝐶𝑑,𝑜𝑝
4
√1−𝛽𝑓𝑤𝑛,6,7𝐵
ℎ𝐿,𝑜𝑝 = 𝐾𝑜𝑝
2
𝑢𝑜𝑝
2𝑔
(5.19)
(5.20)
(5.21)
Therefore,
𝐾𝑜𝑝 =
4
3.198×10−3 𝑔(1−𝛽𝑓𝑤𝑛,6,7𝐵
)
𝐶𝑑,𝑜𝑝
(5.22)
Equations 5.19, 5.20, and 5.21 were formulated by Crane [10] and used for flow through
an orifice. Crane [10] states that the orifice flow coefficient, 𝐶, is a dimensionless value
directly related to the discharge coefficient. Variables 𝑄, 𝐴, 𝑢, 𝐷, and ℎ𝐿 are the flow of
the fluid in the pipe, area of the pipe, velocity of the fluid, diameter of the pipe, and the
head loss in feet of pipe, respectively. The discharge coefficient is a dimensionless value
and relates the actual flow rate to the theoretical flow rate through a primary device. It is
given by
22
2
𝐶𝑑,𝑜𝑝 = 0.561 + 0.0261(𝛽𝑓𝑤𝑛,6,7𝐵 ) −
0.7
2
8
0.216(𝛽𝑓𝑤𝑛,6,7𝐵 ) + 0.000521 (
(0.0188 + 0.0063𝐽)(𝛽𝑓𝑤𝑣,6,7𝐵 )
106 (𝛽𝑓𝑤𝑛,6,7𝐵 )
3.5
106
+
)
𝑅𝑒7𝐴
0.3
(𝑅𝑒 )
7𝐴
+
4
(𝛽𝑓𝑤𝑛,6,7𝐵 )
(0.043 + 0.08𝑒 −10𝐿1 − 0.123𝑒 −7𝐿1 )(1 − 0.11𝐽)
4
1−(𝛽𝑓𝑤𝑛,6,7𝐵)
1.3
0.031(𝑀2′ − 0.8(𝑀2′ )1.1 )(𝛽𝑓𝑤𝑛,6,7𝐵 )
+
[0.011(0.75 − 𝛽𝑓𝑤𝑛,6,7𝐵 )(2.8 − 𝐷𝑓𝑤𝑛,7𝐴 )]
𝛽7 =
𝐷7𝐵
𝑀2′ =
)
𝑅𝑒7𝐴
2𝐿′2
1−𝛽𝑓𝑤𝑛,6,7𝐵
ℎ𝐿,7 = 𝐾𝑜𝑝
2
𝑢𝑓𝑤𝑛,7
∆𝑃𝑙,7 = ℎ𝐿,7
(5.23)
(5.24)
𝐷6
19000𝛽𝑓𝑤𝑛,6,7𝐵 0.8
𝐽=(
−
2𝑔
𝜌𝑙
144
(5.25)
(5.26)
(5.27)
(5.28)
The last term of Equation 5.23 was added because the inside diameter of the nozzle is
less than 2.8𝑖𝑛. After solving for the head loss (Equation 5.27) and pressure drop
(Equation 5.28), it is necessary to calculate the non-recoverable pressure drop (NRPD).
The NRPD is the difference in static pressure between the pressure measured on the
upstream side of the primary device before the influence of the approach impact
pressure, and that measured on the downstream side of the primary device where the
static pressure recovery can be considered completed [10]. The NRPD is given by
23
𝑁𝑅𝑃𝐷7 = ∆𝑃𝑙,7 [
4
2
2
4
2
2
√1−(𝛽𝑓𝑤𝑛,6,7𝐵 ) (1−(𝐶𝑑,𝑜𝑝 ) )−𝐶𝑑,𝑜𝑝 (𝛽𝑓𝑤𝑛,6,7𝐵 )
]
(5.29)
√1−(𝛽𝑓𝑤𝑛,6,7𝐵 ) (1−(𝐶𝑑,𝑜𝑝 ) )+𝐶𝑑,𝑜𝑝 (𝛽𝑓𝑤𝑛,6,7𝐵 )
Figure 5-6: Flow Through/Around the Orifice Plate and Nozzle Cap’s Teeth
Figure 5-7A: Flow Region 7B for Calculations (inches)
24
Figure 5-7B: Cross Section of Flow Region 7B
5.1.4 Calculations in the Nozzle Cap
The calculations in the nozzle cap will be similar to the calculations in section 5.1.2
(regions 4 through 6). The flow enters region 8, is constricted (region 9), and then exits
the nozzle through region 10 (see Figure 5-8), and discharges into the atmosphere.
Equations 5.10 and 5.12 are used to find the resistant coefficient and diameter ratios,
respectively. Equation 5.14 will be factored in to calculate the exiting velocity.
Figure 5-8: Flow Region 8-10 for Calculations (inches)
5.2 Computational Pressure Loss Analysis
Fluent uses different types of models for the analysis of turbulent flow. One common
model is the 𝑘 − 𝜀 model, which has two equations. This allows a turbulent length and
time scale to be estimated by solving two different turbulent equations (see Equations
5.30 and 5.31). The dissipation rate (𝜀) and kinetic energy (𝑘) are based on the transport
equations for the standard 𝑘 − 𝜀 models. There are three variants of the 𝑘 − 𝜀 model:
25
Standard 𝑘 − 𝜀 model, RNG 𝑘 − 𝜀 model and the realizable 𝑘 − 𝜀 model. The realizable
𝑘 − 𝜀 model will be discussed in the following section [14].
5.2.1 The Realizable 𝒌 − 𝜺 Model
The realizable 𝑘 − 𝜀 model is the most accurate analysis, as well as the most current,
within the Fluent software. The term “realizable” means that the model satisfies certain
mathematical constraints on the Reynolds stresses, consistent with the physics of
turbulent flows. The following equations are used to evaluate the turbulent dissipation
and kinetic energy
𝜕
𝜕𝑡
(𝜌𝑘) +
𝜕
𝜕𝑥𝑗
𝜕
𝜇
𝜕𝑘
(𝜌𝑘𝑢𝑗 ) = 𝜕𝑥 [(𝜇 + 1𝑡) 𝜕𝑥 ] +
𝑗
𝑗
𝐺𝑘 + 𝐺𝑏 − 𝜌𝜀 − 𝑌𝑀 + 𝑆𝑘
𝜕
𝜕𝑡
(𝜌𝜀) +
𝜕
𝜕𝑥𝑗
𝜕
𝜇
(5.30)
𝜕𝜀
(𝜌𝜀𝑢𝑗 ) = 𝜕𝑥 [(𝜇 + 1.2𝑡 ) 𝜕𝑥 ] + 𝜌𝐶1 𝑆𝜀 −
𝑗
𝑗
1.9𝜌
𝜀2
𝜀
𝑘+√𝜈𝜀
+ 1.44 𝐶3𝜀 𝐺𝑏 + 𝑆𝜀
𝑘
where,
𝜇𝑡 = 𝜌𝐶𝜇
𝐶𝜇 =
𝑘2
(5.32)
𝜀
1
𝑘𝑈∗
𝜀
(5.33)
𝐴𝑠 = √6 cos 𝜙
(5.34)
4.04+𝐴𝑆
1
𝜙 = cos −1 √6 𝑊
3
𝑊=
𝑆𝑖𝑗 𝑆𝑗𝑘 𝑆𝑘𝑖
𝑆̃ 3
1 𝜕𝑢𝑗
(5.36)
𝜕𝑢
𝑆𝑖𝑗 = ( + 𝑖 )
2 𝜕𝑥
𝜕𝑥
𝑖
26
(5.35)
𝑗
(5.37)
(5.31)
𝑆̃ = 𝑆𝑖𝑗 𝑆𝑖𝑗
𝐶1 = 𝑚𝑎𝑥 [0.43,
𝜂=𝑆
𝜀
𝑘
(5.38)
𝜂
𝜂+5
]
(5.39)
(5.40)
𝑆 = √2𝑆𝑖𝑗 𝑆𝑖𝑗
(5.41)
𝑈 ∗ = √𝑆𝑖𝑗 𝑆𝑖𝑗 + 𝛺̃𝑖𝑗 𝛺̃𝑖𝑗
𝛺̃𝑖𝑗 = 𝛺𝑖𝑗 − 2𝜀𝑖𝑗𝑘 𝜔𝑘
𝛺𝑖𝑗 = 𝛺̅𝑖𝑗 − 𝜀𝑖𝑗𝑘 𝜔𝑘
(5.42)
(5.43)
(5.44)
The Boussinesq3 relationship and the eddy viscosity definition are combined to obtain
the following expression for the normal Reynolds stress in an incompressible strained
mean flow
2
𝜕𝑈
3
𝜕𝑥
𝑢̅2 = 𝑘 − 2𝜈𝑡
3
(5.45)
Joseph Valentin Boussinesq (13 March 1842 – 19 February 1929) was a French mathematician and
physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat.
27
6.0 The Energy Equation
The energy equation is given by
𝜕
𝜕𝑡
(𝜌𝐸) + ∇ ∙ (𝑣⃗(𝜌𝐸 + 𝑝)) =
𝜕
𝜕𝑥𝑖
(𝜌𝑘𝑢𝑖 ) =
∇ ∙ (𝑘𝑒𝑓𝑓 𝛻𝑇 − ∑𝑗 ℎ𝑗 ⃗⃗⃗
𝐽𝑗 + (𝜏̅𝑒𝑓𝑓 ∙ 𝑣⃗)) + 𝑆ℎ
(6.1)
Also, the mass diffusion in turbulent flows is
𝐽⃗𝑖 = − (𝜌𝐷𝑖,𝑚 +
𝜇𝑡
) ∇𝑌𝑖 − 𝐷𝑇,𝑖
𝑆𝑐
∇𝑇
𝑡
𝑇
(6.2)
where,
𝑆𝑐𝑡 =
𝜇𝑡
(6.3)
𝜌𝐷𝑡
𝑘𝑒𝑓𝑓 = (𝑘 + 𝑘𝑡 )
𝑝
𝑣2
𝜌
2
𝐸 =ℎ− +
ℎ = ∑𝑗 𝑌𝑗 ℎ𝑗
𝑇
ℎ𝑗 = ∫𝑇
𝑟𝑒𝑓
(6.4)
(6.5)
(6.6)
𝐶𝑝,𝑗 𝑑𝑇
(6.7)
The turbulent diffusion generally overwhelms laminar diffusion. Therefore, the laminar
diffusion properties are generally not necessary in turbulent flow. The ratio between the
thermal and mass diffusivities is represented by the Lewis number,
𝐿𝑒 =
𝑘
𝐶𝑝 𝐷
(6.8)
For values of Le different from 1, transport of enthalpy due to the diffusion of the
species can have a significant effect on the enthalpy field.
28
7.0 Discrete Phase Model (DPM)
This section will describe the theory behind the Lagrangian4 discrete phase capabilities
available in Fluent. The topics discussed here are: particle motion theory, laws for drag
coefficients, laws for heat and mass exchange, atomizer model theory, and secondary
breakup model theory. Multiphase flows are being better understood due to the advances
in CFD. Two main approaches are used in the numerical analysis of multiphase flow,
Euler-Euler and Euler-Lagrange. The latter is used in the present work, while the former
is beyond the scope of this work and will not be discussed. In the Euler-Lagrange
formulation, the Navier-Stokes equations are solved for the fluid phase, while the
dispersed phase is solved by tracking a large number of particles, bubbles, or droplets
through the calculated flow field. The dispersed phase can exchange momentum, mass,
and energy with the fluid phase.
7.1 Particle Motion Theory
The trajectory of a discrete phase particle (droplet, or bubble) is predicted by integrating
the force balance on the particle. The force balance equates the particle inertia with the
forces acting on the particle, and it is written in a Lagrangian reference frame. Assuming
x flow direction in Cartesian coordinates:
𝑑𝑢𝑝
𝑑𝑡
= 𝐹𝐷 (𝑢 − 𝑢𝑝 ) +
𝐹𝐷 =
𝑅𝑒 ≡
𝑔(𝜌𝑝 −𝜌)
𝜌𝑝
18𝜇 𝐶𝐷 𝑅𝑒
2 24
𝜌𝑝 𝑑𝑝
⃗⃗𝑝 −𝑢
⃗⃗|
𝜌𝑑𝑝 |𝑢
𝜇
+ 𝐹𝑥
(7.1)
(7.2)
(7.3)
Additional forces 𝐹𝑥 that can be important under special circumstances are added in
Equation 7.1. To name a few, “virtual mass” force, thermophoretic force, Brownian
force, and Saffman’s lift force.
4
Joseph-Louis Lagrange (25 January 1736 - 10 April 1813) was an Italian Enlightenment Era
mathematician and astronomer. He made significant contributions to all fields of analysis, number theory,
and both classical and celestial mechanics.
29
7.2 Turbulent Dispersion of Particles
Turbulence helps particles disperse in the fluid phase and these are tracked using either
stochastic tracking models or particle cloud models. The stochastic tracking model will
be discussed below, while cloud models are beyond the scope of this work. The
stochastic tracking (or random walk model) includes the effect of instantaneous turbulent
velocity fluctuations on the particle trajectories through the use of stochastic methods.
The trajectories of particles can be predicted using the mean fluid phase velocity, but
also the instantaneous value of the velocity can be included to predict the dispersion of
the particles due to turbulence
𝑢 = 𝑢̅ + 𝑢′
(7.4)
If a sufficient number of representative particles are analyzed, the random effects of
turbulence on the particle dispersion can be evaluated.
7.3 Laws for Drag Coefficients
There are 6 different drag laws: spherical and non-spherical drag law, StokesCunningham drag law, high-Mach-number drag law, dynamic drag model theory, and
dense discrete phase model drag laws. In this work, the spherical drag law is used. For
smooth particles the drag coefficient, 𝐶𝐷 , can be obtained from
𝐶𝐷 = 𝑎1 +
𝑎2
𝑅𝑒
+
𝑎3
𝑅𝑒
(7.5)
7.4 Laws for Heat and Mass Exchange
There are seven different laws that are used to analyze the heat and mass exchange.
These are: inert heating or cooling (Law 1/Law 6), droplet vaporization (Law 2), droplet
boiling (Law 3), devolatilization (Law 4), surface combustion (Law 5), and
multicomponent particle definition (Law 7). Below is a brief explanation of each law
utilized.
7.4.1 Inert Heating or Cooling (Law 1/Law 6)
Law 1 and Law 6 are applied when the particle temperature is less than the vaporization
temperature, 𝑇𝑣𝑎𝑝 (Law 1)
30
𝑇𝑝 < 𝑇𝑣𝑎𝑝
(7.6)
and after the volatile fraction, 𝑓𝑣,0 ,of a particle has been consumed (Law 6)
𝑚𝑝 ≤ (1 − 𝑓𝑣,0 )𝑚𝑝,0
(7.7)
These conditions may be written as shown above. Law 1 is applied until the temperature
of the particle/droplet reaches the vaporization temperature. A non-inert particle/droplet
may proceed to obey one of the mass transfer Laws (2, 3, 4, and/or 5), returning to Law
6. The vaporization temperature, 𝑇𝑣𝑎𝑝 , is an arbitrary modeling constant used to define
the onset of the particle/droplet/volatilization laws. Law 1 or Law 6 uses the following
computation
𝑚𝑝 𝐶𝑝
𝑑𝑇𝑝
𝑑𝑡
= ℎ𝐴𝑝 (𝑇∞ − 𝑇𝑝 ) + 𝜀𝑝 𝐴𝑝 𝜎𝑆𝐵 (𝜃𝑅4 − 𝑇𝑝4 )
𝜃𝑅 = (
𝐺
4𝜎𝑆𝐵
(7.8)
1⁄4
)
𝐺 = ∫Ω=4𝜋 𝐼𝑑Ω
(7.9)
(7.10)
which is a simple heat balance to relate the particle temperature, 𝑇𝑝 (𝑡), to the
absorption/emission of radiation at the particle surface.
7.4.2 Droplet Vaporization (Law 2) and Droplet Boiling (Law 3)
Law 2 (droplet vaporization) is applied when the temperature of the droplet reaches the
vaporization temperature, and continues until the droplet reaches the boiling point, 𝑇𝑏𝑝 ,
or until the droplet’s volatile fraction is completely consumed.
𝑇𝑣𝑎𝑝 ≤ 𝑇𝑝 < 𝑇𝑏𝑝
(7.11)
𝑚𝑝 (1 − 𝑓v,0 )𝑚𝑝,0
(7.12)
31
Law 3 is applied to predict convective boiling of a discrete phase droplet. It is initiated
when the temperature of the droplet reaches the boiling temperature, 𝑇𝑏𝑝 , and while the
mass of the droplet exceeds the nonvolatile fraction, 1 − 𝑓v,0,
𝑇𝑝 ≥ 𝑇𝑏𝑝
(7.13)
𝑚𝑝 > (1 − 𝑓v,0 )𝑚𝑝,0
(7.14)
7.5 Atomizer Model Theory
For most types of injections, it is necessary to provide the initial diameter, position, and
velocity of the particles. For sprays, models are available to predict the droplet size and
velocity distributions. Atomization models use numerous attributes of the nozzle and
spray fluid, such as orifice diameter and mass flow rate, to calculate initial droplet size,
velocity, and position. For atomizer simulations, the droplets must be randomly
distributed, both spatially through a dispersion angle and in their time of release. The
atomizer models use stochastic trajectory selection to achieve random distribution.
Although there are different types of spray models that can be used in Fluent, the
pressure-swirl atomizer model is the most appropriate for this application.
7.5.1 Pressure-swirl Atomizer
The pressure-swirl atomizer is used to analyze the spray characteristics of the
desuperheater spray. Three steps are used to predict the flow transition from internal
injector flow to fully-developed spray. The steps include: film formation, sheet breakup,
and atomization. Aerodynamic instability is generally accepted as the reason for this
break up.
7.5.2 Film Formation
The liquid moves through the nozzle as a thin sheet which quickly vibrates and spreads
radially outward and then turning into ligaments, before finishing as droplets. The film
thickness is directly related to the diameter of the final spray, which in turn is related to
the area of the air cone. Lefebvre [2] suggested that the film thickness be determined
from the relationship between the nozzle dimensions and the size of the air cone and
discharge coefficient,
32
2𝐾𝐴 2 𝑋 2 = (1 − 𝑋)3
(7.15)
with
𝐾𝐴 =
𝑋=
𝐴𝑝𝑎
(7.16)
𝐷𝑠 𝑑0
𝐴𝑎
(7.17)
𝐴0
Assuming nonviscous fluid
𝐴𝑝𝑎 2
𝜋2 (1−𝑋)3
(𝐷 𝑑 ) = 32
𝑋2
𝑠 0
𝑋=
(𝑑0 −2𝑡𝐹 )2
𝑑02
(7.18)
(7.19)
According to Lefebvre [2], the film thickness is independent of the liquid viscosity and
liquid injection pressure. The velocity coefficient, 𝐾𝑣 , is given by
𝐶
𝐾𝑣 = (1−𝑋)𝐷
cos 𝜃
(7.20)
where
𝐶𝐷 = [
(1−𝑋)3 0.5
1+𝑋
]
(7.21)
CD is defined as the ratio of the actual discharge velocity to the theoretical velocity
corresponding to the total pressure loss across the nozzle.
The spray cone angle, (2𝜃), is determined by the swirl chamber geometry and is a
unique function of the coefficient 𝐾𝑣 [2]. The flow in a swirl atomizer assumes a
nonviscous liquid which lets the spray cone angle to be expressed as a function of nozzle
dimensions. It leads to the following expression for the mean value of the spray cone
half-angle
33
sin 𝜃 =
(𝜋⁄2)𝐶𝐷
(7.22)
𝐾𝑣 (1+√𝑋)
7.5.3 Sheet Breakup and Atomization
There are various types of models for sheet breakup, such as conical sheets, flat sheets,
and fan sheets. The radius of curvature has a destabilizing effect on the fluctuations, so
that conical sheets tend to be shorter than flat sheets [2]. The resultant mean droplet
diameter is estimated as
𝐷 = 2.13(𝑡𝑠 𝜆∗ )0.5
(7.23)
The diameter of the ligaments formed at the point of breakup can be obtained from a
mass balance. If it is assumed that the ligaments are formed from rips in the sheet twice
per wavelength, the resulting diameter is given by
𝑑𝐿 = √
8ℎ0
𝐾𝑆
(7.24)
The ligament diameter depends on the sheet thickness, which is a function of the
breakup length. The film thickness is calculated from the breakup length and the radial
distance from the center line to the mid-line of the sheet at the atomizer exit
ℎ𝑒𝑛𝑑 =
𝑟0 ℎ0
𝜃
2
𝑟0 +𝐿𝑏 sin( )
(7.25)
7.6 Secondary Breakup Model Theory
There are several models to predict spray breakup. Since the Weber number
𝑊𝑒 =
𝜌𝑔 𝑢2 𝑟
𝜎
(7.26)
is greater than 100, the wave breakup model will be used in this work. This model
assumes that the time of breakup and the resulting droplet size are related to the fastestgrowing Kelvin-Helmholtz instability, derived from the jet stability analysis. The
wavelength and growth rate of this instability are used to predict details of the newlyformed droplets. Breakup of droplets particles is calculated by assuming that the radius
34
of the newly formed droplets is proportional to the wavelength of the fastest-growing
unstable surface wave on the parent droplet
𝑟 = 𝐵0 Λ
(7.27)
In the wave model, mass is accumulated from the parent drop at a rate given by
𝜏=
3.726𝐵1 𝑎
ΛΩ𝑟
(7.28)
until the shed mass is equal to 5% of the initial particle mass. At this time, a new particle
is created with a radius given by equation 7.27. The new particle has the same properties
as the parent particle (i.e., temperature, material, position, etc.) with the exception of
radius and velocity [14]. The variables Λ and Ω𝑟 are defined as
Λ
𝑎
= 9.02
(1.045𝑂ℎ0.5 )(1+0.4𝑇𝑎0.7 )
(1+0.87𝑊𝑒 1.67 )0.6
(7.29)
and
𝜌𝑎3
Ω𝑟 (
𝜎
0.34+0.38𝑊𝑒 1.5
) = (1+𝑂ℎ)(1+1.4𝑇𝑎0.6)
where,
𝑂ℎ =
√𝑊𝑒
𝑅𝑒
𝑇𝑎 = 𝑂ℎ√𝑊𝑒
35
(7.31)
(7.32)
(7.30)
8.0 Droplet Evaporation and Droplet Lifetime
The evaporation of drops in a spray involves heat and mass transfer processes where the
heat for evaporation is transferred onto the drop surface simultaneously by conduction
and convection through the surrounding gas and vapor. A spherically symmetric model
is assumed in order to calculate an evaporating droplet.
Heat Transfer Number. Based on considerations of conductive and convective heat
fluxes across a thin shell surrounding the evaporating drop, the heat transfer number is
𝐵𝑇 =
𝐶𝑝𝑔 (𝑇∞ −𝑇𝑠 )
(8.1)
𝐿
and it represents the driving force for the evaporation process. When heat transfer rates
are controlling the evaporation, the rate of evaporation for a Lewis number of one is
obtained as
𝑘
𝑚̇𝐷 = 2𝜋𝐷𝑒 ( ) ln(1 + 𝐵𝑇 )
𝐶
𝑝
(8.2)
𝑔
This equation can only be used for steady-state evaporation. However, this equation is
usually easier to evaluate, since the magnitudes of the various terms are either contained
within the data of the problem or readily available in the literature [2].
Calculation of Steady-State Evaporation Rates. As stated above, Equation 8.2 can
only be used in steady-state evaporation. The term steady-state is used to determine the
stage in the drop evaporation process, where the drop surface has reached the wet-bulb
temperature and all of the heat reaching the surface is used in providing the latent heat of
vaporization. When 𝑇𝑠 is known, the transfer number 𝐵𝑀 is easy to evaluate from
𝐵𝑀 =
𝑌𝐷,𝑠 =
𝑃𝐷,𝑠 𝑀𝐷
𝑃𝐷,𝑠 𝑀𝐷 +(𝑃−𝑃𝐷,𝑠 )𝑀𝐴
𝑌𝐷,𝑠
1−𝑌𝐷,𝑠
(8.3)
= [1 + (
36
𝑃
𝑃𝐷,𝑠
− 1)
𝑀𝐴
𝑀𝐷
−1
]
(8.4)
At steady-state droplet evaporation 𝐵𝑀 = 𝐵𝑇 = 𝐵, and the mass rate of droplet
evaporation is given by [2]
𝑘
𝑚̇𝐷 = 2𝜋𝐷 ( ) ln(1 + 𝐵)
𝐶
𝑝
(8.5)
𝑔
Evaporation Constant. During the steady-state period of an evaporating drop, the
diameter at any instant may be related to its initial diameter by
𝐷02 − 𝐷2 = 𝜆𝑠𝑡 𝑡
(8.6)
where,
𝜆𝑠𝑡 =
8𝑘𝑔 ln(1+𝐵)
𝐶𝑝𝑔 𝜌𝐷
(8.7)
Calculation of Heat-Up Period. Chin and Lefebvre [15] have discussed the role of the
heat-up period in droplet evaporation in some detail. A quasi-steady gas phase is
assumed, in which the boundary layer around the drop has the same characteristics as a
steady boundary layer for the same conditions of drop size, and surface and ambient
temperatures. The heat transfer coefficient is determined by
𝑁𝑢 =
ℎ𝐷
𝑘𝑔
=2
ln(1+𝐵𝑀 )
𝐵𝑀
(8.8)
with the heat transferred from the gas to the drop given by
𝑄ℎ𝑡 = 𝜋𝐷2 ℎ(𝑇∞ − 𝑇𝑠 )
(8.9)
Substitution for h from Equation 8.8 into equation 8.9 gives
𝑄ℎ𝑡 = 2𝜋𝐷𝑘𝑔 (𝑇∞ − 𝑇𝑠 )
ln(1+𝐵𝑀 )
𝐵𝑀
(8.10)
8.1 Drop Lifetime
Droplet lifetime is important in the design of desuperheaters in industrial boilers.
Equipment may be damaged by a high velocity droplet impacting the desuperheater
37
walls, which may cause pitting and eventually create holes in the walls. This is also
important because measuring the temperature in the desuperheater is vital to ensure
proper regulation of the exiting temperatures. Thermocouples are usually installed in a
region downstream from where the droplets have completely evaporated. This is why it
is important to know where the droplet evaporates. In order to calculate the time of
evaporation, assuming 𝜆𝑠𝑡 is constant and integrating, Equation 8.7 yields
𝑡𝑒 =
𝐷02
𝜆𝑠𝑡
38
(8.11)
9.0 Analysis of the Results
The results were obtained by carrying out post processing with ANSYS Fluent software,
and were compared against industrial standards (see Table 3-1). Fluent offers a variety
of licenses, one of which is an academic student license that has some limitations. One
of these limitations is the amount of elements that can be analyzed. Therefore, some
small geometry modifications were made in order to maintain the integrity of the
analysis and still stay within the license parameters.
9.1 Pressure Drop Across the Spray Nozzle
Section 3.1 discussed the analysis that is performed in order to calculate the pressure
drop across the nozzle. To ensure proper operation of the desuperheater, the pressure
drop at any spray water flow rate is calculated using the following industry standard
equation [9]
𝑄
2
𝑣
∆𝑃𝑠 = ( 𝑠 ) ( 𝑠 ) (∆𝑃𝑐 )
𝑄
0.01605
(9.1)
𝑐
where,
∆𝑃𝑠 = Spray water pressure drop across the nozzle, (𝑝𝑠𝑖)
𝑄𝑠 = Spray water quantity, (𝑙𝑏𝑠⁄ℎ𝑟)
𝑄𝑐 = Spray water quantity from Table 3-1, (𝑙𝑏𝑠⁄ℎ𝑟)
𝑣𝑠 = Specific volume of the spray water at 70℉, (𝑓𝑡 3 ⁄𝑙𝑏)
∆𝑃𝑐 = Spray water pressure drop across the nozzle from Table 3-1, (𝑝𝑠𝑖)
The pressure drop across the nozzle was calculated using the equations described in
Section 5. For Run F6 the value obtained was 832𝑝𝑠𝑖, which is 16.8% lower than the
standard value of 1,000𝑝𝑠𝑖 (𝑇𝑎𝑏𝑙𝑒 3.1).
In order to verify that Table 3-1 and Equation 9-1 are accurate, eleven Fluent runs were
carried out and post processed. A standard mesh, which contained 145,731 nodes and
469,891 elements, was used (Figure 9-1).
39
Figure 9-1: Mesh for Run 1 Through Run 11
Figure 9-2 shows the pressure contours on a cross-sectional side view of the spray
nozzle for Run F1. A front view of the spray nozzle is shown in Figure 9-3. Because the
pressure at the exit is zero, a “back pressure” is the maximum pressure loss across the
nozzle. Figure 9-4 shows that the flow has been separated into the different openings of
the internal design, indicating the nozzle assembly is operating as initially intended.
Also, shown in Figure 9-4 are numerous “dead zones” or eddies within the nozzle. This
is one of the reasons why the pressure drop across the nozzle is so high.
Figure 9-2: Pressure Contours for Run F1
40
Figure 9-3: Pressure Contours for Run F1
Figure 9-4: Velocity Vectors for Run F1
41
Figure 9-5 shows the “swirling effect” of the pressure swirl atomizer. The nozzle cap’s
teeth have been designed for this purpose, typical of a pressure-swirl atomizer.
Figure 9-5: Velocity Vectors for Run F1
Figures 9-6 and 9-7 show the pressure contours for Run F11, while Figures 9-8 and 9-9
show the velocity vectors for the same run.
42
Figure 9-6: Pressure Contours for Run F11
Figure 9-7: Pressure Contours for Run F11
43
Figure 9-8: Velocity Vectors for Run F11
Figure 9-9: Velocity Vectors for Run F11
As shown in Table 9-1, the results from the computational analysis indicate a 17%
greater pressure drop than the industrial standard values. This larger than expected
44
difference in pressure drop may be a direct result of the license limitation that has been
discussed above. To check this, further investigation would be required, without an
element limitation and increasing the accuracy by using, for example, a “second-order
upwind” analysis.
Table 9-1: Results Comparison Calculated vs. CFD
Run F1
Run F2
Run F3
Run F4
Run F5
Run F6
𝑙𝑏
Flow (ℎ𝑟
)
44,500
77,000
99,500
117,9000
126,000
140,900
Pressure (𝑖𝑛𝑙𝑏2)
100.0
300.0
500.0
700.0
800.0
1,000.0
CFD Pressure (𝑖𝑛𝑙𝑏2 )
117.5
351.5
586.9
823.9
941.0
1176.6
Pressure Difference
17.5%
17.2%
17.4%
17.1%
17.6%
17.7%
Run F7
Run F8
Run F9
Run F10
Run F11
𝑙𝑏
Flow (ℎ𝑟
)
154,000
166,500
178,000
189,000
199,000
Pressure (𝑖𝑛𝑙𝑏2)
1,200.0
1,400.0
1,600.0
1,800.0
2,000.0
CFD Pressure (𝑖𝑛𝑙𝑏2 )
1,405.5
1,642.9
1,877.6
2,116.7
2,346.6
Pressure Difference
17.1%
17.4%
17.4%
17.6%
17.3%
9.2 Standard Operating Conditions into Air Results
The same analysis as in section 9.1 was run but at the operating conditions of the boiler.
A temperature of 370℉ is used, along with an operating pressure of 1180𝑝𝑠𝑖𝑔. The
fluid properties at the mentioned temperature were entered manually. At the inlet, the
𝑙𝑏
flow rate is set to 41,250ℎ𝑟
, with a 10% dissipation rate. For screen shots of the 𝑘 − 𝜀
turbulent model in ANSYS Fluent see Appendix B. The actual pressure drop across the
nozzle was provided by the manufacturer [9], and is listed as 498𝑝𝑠𝑖𝑔. The
computational post processing results were given at the same inputs as above, the
pressure drop across the nozzle is 584.6𝑝𝑠𝑖𝑔, which is approximately a 17.4%
difference. By using a simple analysis and limited elements may have resulted in the
larger difference between the empirical and computational results. To check this theory
one would require no element limitations and an increase in accuracy by using, for
example, a “second-order upwind” analysis. Figures 9-10 and 9-11 show the pressure
45
contours at the operating pressure and temperature, while Figures 9-12 and 9-13 show
the velocity vectors.
Figure 9-10: Pressure Contours for Operating Conditions
Figure 9-11: Pressure Contours for Operating Conditions
46
Figure 9-12: Velocity Vectors for Operating Conditions
Figure 9-13: Velocity Vectors for Operating Conditions
47
9.3 Operating Conditions for the Desuperheater Results
The desuperheater flow model has a total of 27,560 nodes and 148,935 elements. The
mesh is shown in Figure 9-14, and a close up view of the spray nozzle modeled is in
Figure 9-15. The boundary condition at the inlet is a “Mass Flow Inlet” with a flow rate
𝑙𝑏
of 29,333.3 𝑚𝑖𝑛 at 650℉, and an operating pressure of 682𝑝𝑠𝑖. The boundary condition
for the outlet is a “Pressure Outlet” which is set to zero. A droplet and pressure-swirl
atomizer was used for the injection model, for the Discrete Phase Model (DPM) with the
properties of water as discussed in Section 9.2. For screen shots of the DPM in ANSYS
Fluent see Appendix C. The analysis is solved as a transient state model, and unsteady
particle tracking is selected in the DPM. The solution was solved with 20 iterations at a
time step of 0.001 seconds. The number of time steps is set of 500, so the total time
analyzed is 0.5 seconds. This time step is sufficient for the DPM to be solved. The
velocity at the desuperheater inlet is 160.5
𝑓𝑡
𝑠
and the particle will travel a maximum
of 9𝑓𝑡, that is a minimum of 0.056 seconds for the particle to reach the outlet. Equation
7.22 is used to calculate the spray half angle and is entered into the DPM injection
properties (see Figure 9-17).
The outlet average temperature is 645.5℉, which is below the maximum temperature
of 650℉. The droplet diameter is averaged throughout the flow domain, resulting in a
SMD or 𝐷32 = 0.396 × 10−3 𝑖𝑛, and a max droplet diameter of 0.700 × 10−3 𝑖𝑛. The
droplet will evaporate in 0.011 seconds and therefore, will be completely evaporated
about 1.48ft after the spray leaves the nozzle.
Figure 9-14: Mesh for the Desuperheater for Operating Conditions
48
Figure 9-15: Mesh for the Desuperheater for Operating Conditions (Close up)
Outlet
Nozzle Location
Inlet
Figure 9-16: Model Boundary Locations
13.5
Figure 9-17: Spray Angle and Temperature Contours
49
Figure 9-18 shows the droplet diameter versus the distance traveled. It can be seen that
the diameter drops off before 2𝑓𝑡 is reached.
1.0E-05
Injection - 1
9.0E-06
Injection - 2
Injection - 3
8.0E-06
Injection - 4
Injection - 5
7.0E-06
Injection - 6
Injection - 7
6.0E-06
Diameter
Injection - 8
Injection - 9
5.0E-06
Injection - 10
Injection - 11
4.0E-06
Injection - 12
Injection - 13
3.0E-06
Injection - 14
Injection - 15
2.0E-06
Injection - 16
Injection - 17
1.0E-06
Injection - 18
Injection - 19
0.0E+00
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Injection - 20
Distance (ft)
Table 9-18: Droplet Diameter vs Distance
Figures 9-19 through 9-22 show the surface temperature at different locations
downstream of the nozzle.
50
Figure 9-19: Cut Plan Approx. 1.5 ft from Nozzle Outlet
Figure 9-20: Cut Plan Approx. 3.5 ft from Nozzle Outlet
51
Figure 9-21: Cut Plan Approx. 5.5 ft from Nozzle Outlet
Figure 9-22: Cut Plan at Outlet of Desuperheater (9ft)
52
10. Conclusions
The pressure drop across the nozzle calculated using Computational Fluid Dynamics is
about 17% higher than that of an industrial standard. Better accuracy could be achieved
if a finer mesh would be used, that is without the limitations on the number of elements
that could be used in Fluent. Also, the solver could be modified to increase accuracy of
the solution by utilizing the “second-order upwind” analysis for all values.
Now that the spray water has been analyzed in the desuperheater, equipment such as
thermowells may be installed. According to the information, for this designed
desuperheater, the thermowell can be installed approximately 2ft downstream from the
spray water nozzle. However, as shown from the temperature contours, the temperature
is not completely regulated or “steady” at 2ft. In fact, installing the thermowell between
5ft and the outlet of the desuperheater would be better because the temperature variation
is not as significant and thermal readings will be more accurate. The inlet temperature of
650℉ is a maximum temperature the desuperheater should reach before regulation is
activated. The calculated outlet temperature of 645.5℉ is quite high. Actually, it was not
expected that the temperature difference between the calculated and the maximum value
would be more than half the difference between the design temperature (572℉) and the
maximum temperature (650℉). The calculated 4.5℉ temperature difference was not
expected and, in fact, it is questionable if the geometry, and/or boundary conditions
provided was accurate enough for the analysis. Alternatively, the desuperheater was
incorrectly designed when the boiler was originally built. To improve the spray water
system and make it work efficiently, lowering the temperature on the spray water and
opening the spray half angle is one of the ways to help regulate better steam
temperatures.
53
References
1. Bozzuto, Carl (Editor) (2009). Clean Combustion Technologies, A References
Book on Steam Generation and Emission Control, 5th Edition, (pp. 3-101)
Windsor, CT, Alstom Power Inc.
2. Arthur H. Lefebvre (1989). Atomization and Sprays, Taylor & Francis Group,
LLC, CRC Press
3. "The Nobel Prize in Physics 1904". Nobelprize.org. 7 Feb 2012
http://www.nobelprize.org/nobel_prizes/physics/laureates/1904/
4. The McGraw-Hill Companies. Access Science. 7 March 2012
http://accessscience.com/content/Atomization/061200
5. Wikipedia. Spray Nozzle. 7 March 2012.
http://en.wikipedia.org/wiki/Spray_Nozzle
6. Historic Naval Ships Association. Section VI – Boilers. 7 March 2012.
http://hnsa.org/doc/destroyer/steam/sec06.htm
7. Global Security. Chapter 4 Fuel System. 7 March 2012.
http://www.globalsecurity.org/military/library/policy/army/fm/1-506/Ch4.htm
8. Thermopedia. Atomization. 7 March 2012.
http://www.thermopedia.com/content/573/
9. J. Chase, Private Communication
10. Engineering Department (2011). Flow of Fluids Through Valves, Fittings and
Pipe, Crane Co.
11. Heald, C. C. (Editor) (2010). Cameron Hydraulic Data, 9th Edition, Flowserve
Corporation
12. Wikipedia. Daniel Bernoulli, 23 March 2012.
http://en.wikipedia.org/wiki/Daniel_Bernoulli
13. Dresser Masoneilan. 8000 Series 3-Way Control Valve, 09, April 2012.
http://www.dressermasoneilan.com/documents/LiteratureLibrary/reciprocating/s
pec-data/CH80000-112910.pdf
14. ANSYS Inc, November 2011. ANSYS Fluent Theory Guide, Release 14.0
15. Chin, J. S., and Lefebvre, A. H., The Role of the Heat-up Period in Fuel Drop
Evaporation, Int. J. Turbo Jet Engines, Vol. 2, 1985, pp. 315-325.
54
Appendix A: Types of Atomizers
Figure A-1: Square Spray Atomizer [6]
Figure A-2: Duplex Atomizer [7]
Figure A-3: Dual Orifice Atomizer [8]
55
Figure A-4: Rotary Atomizer [8]
Figure A-5: Air-Assist Atomizer [2]
Figure A-8: Airblast Atomizer [8]
Figure A-9: Effervescent Atomizer [8]
56
Appendix B: ANSYS Fluent Screen Shots – Turbulent Model
General Settings
57
Mesh Check & Mesh Quality
58
Turbulent Model Settings
59
Materials Selection
Operating Conditions
60
Boundary Conditions
61
62
Solution Methods and Run Settings
63
Appendix C: ANSYS Fluent Screen Shots – DPM Model
General Settings
64
Mesh Check & Mesh Quality
Energy Equation Check Box
Species Model
65
Discrete Phase Model - Tracking
66
Discrete Phase Model – Physical Models
67
Discrete Phase Model – UDF
68
Discrete Phase Model - Numerics
69
Discrete Phase Model - Injection
70
71
Materials Selection – Saturated Water
72
73
Materials Selection – Saturated Steam
Operating Condition
74
Boundary Conditions
75
76
77
Solution Methods
78
Run Settings
79