Spray and Wall Film Modeling with Conjugate Heat Transfer ...561245/...with Conjugate Heat Transfer....

Spray and Wall Film Modeling withConjugate Heat Transfer in OpenFOAM

Emil SjölinderApplied Thermodynamics and Fluid Mechanics

Degree inMaster of Sciencewith a major in

Mechanical Engineering

Linköping UniversityDepartment of Management and Engineering

LIU-IEI-TEK-A–12/01495–SE

Abstract

This master thesis was provided by Scania AB. The objective of this thesis was tomodify an application in the free Computational Fluid Dynamics software OpenFOAMto be able to handle spray and wall film modeling of a Urea Water Solution togetherwith Conjugate Heat Transfer. The basic purpose is to widen the knowledge of thevaporization process of a Urea Water Solution in the exhaust gas after treatment systemfor a diesel engine by using OpenFOAM.First, urea has been modeled as a very viscous liquid at low temperature to mimicthe solidification process of urea. Second, the development of the new application hasbeen done. At last, test simulations of a simple test case are performed with the newapplication. The results are then compared with simplified hand calculations to verifya correct behavior of certain exposed source terms.The new application is working properly for the test case but to ensure the reliability,the results need to be compared with another Computational Fluid Dynamics softwareor more preferable, real experiments. For more advanced geometries, the continueddevelopment presented last in this thesis is highly recommended to follow.

This document has been typeset using LATEX and is preferably viewed in color.

Acknowledgements

First of all I would like to thank my supervisor at Scania, Niklas Nordin, for guidanceand help in the code-jungle of OpenFOAM. Niklas has also taught me about CFD, ther-modynamics and of course, about shoes, as a great art, for which I’m grateful. I’m alsovery grateful to Raymond Reinmann, for he trusted his instincts and gave me this chal-lenge despite the low odds of completing it. I would also like to thank everyone at theFluid and Combustion simulation group for giving me useful insights to my work andappreciated morning greetings. Their smiles have lifted me up from the deepest holesof source code. Of course, I also want to thank my supervisor at Linköpings University,Jonas Lantz, for his humble and clear guidance of how to write this master thesis.

Emil SjölinderVimmerby, August 2012

Contents

1 Introduction and Background 11.1 Todays Diesel Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Exhaust Gas After Treatment Systems . . . . . . . . . . . . . . . . 21.2 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Objective of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory 62.1 Urea and Water Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Reynolds-avaraged Navier-Stokes equations . . . . . . . . . . . . . 92.2.2 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Spray Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Wall Film Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Conjugate Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 The Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Method 153.1 Modeling Liquid Urea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Development of chtMultiRegionSprayFilmFoam . . . . . . . . . . . . . . 17

3.2.1 The Wall Film Region . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 The Buffer Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.3 Assumptions and Approximations . . . . . . . . . . . . . . . . . . 20

3.3 Simulation Setup for the Test Case . . . . . . . . . . . . . . . . . . . . . . 213.3.1 Computing Platform . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.2 Simulation Domain and Boundary Conditions . . . . . . . . . . . . 213.3.3 Mesh and Solution Criteria . . . . . . . . . . . . . . . . . . . . . . 25

4 Results 274.1 Urea Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 chtMultiRegionSprayFilmFoam . . . . . . . . . . . . . . . . . . . . . . . 29

5 Discussion 345.1 Urea Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 chtMultiRegionSprayFilmFoam . . . . . . . . . . . . . . . . . . . . . . . 355.3 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Conclusion 39

A Modeling Liquid Urea 43A.1 Implementation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.2 Matlab Code for Dynamic Viscosity of Urea . . . . . . . . . . . . . . . . . 44

B chtMultiRegionSprayFilmFoam 46B.1 Development of the Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.2 Implementation of Kliq and deltaMap . . . . . . . . . . . . . . . . . . . . 56B.3 Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57B.4 File Structure for the Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . 62

C Settings for the Simulation 65C.1 Solution Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65C.2 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

List of Figures

1.1 European legislation for soot and NOx emission from Euro 1 to Euro 6 . 21.2 The simplified test domain desired to calculate . . . . . . . . . . . . . . . 5

2.1 Velocity in turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 A staggered grid for the FVM . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 T − µ curve for water and desirable T − µ curve for urea . . . . . . . . . . 163.2 Numerical grid of a fluid and a wall film . . . . . . . . . . . . . . . . . . . 173.3 Numerical grid of a fluid, a wall film, a buffer layer and a solid . . . . . . 193.4 The simulation domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 A visualization of the mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1 Calculated T − µ curve for urea . . . . . . . . . . . . . . . . . . . . . . . . 274.2 The formation of the wall film with water or urea . . . . . . . . . . . . . . 284.3 Comparison of wall film thickness for water and urea . . . . . . . . . . . . 294.4 Surface temperature at the solid for all four cases . . . . . . . . . . . . . . 304.5 Temperature drop at the surface for the solid . . . . . . . . . . . . . . . . 314.6 Heat distribution in cross sections for the steel solid . . . . . . . . . . . . 32

List of Tables

2.1 Liquid properties for water and urea . . . . . . . . . . . . . . . . . . . . . 7

3.1 Fluid and solid properties for the test simulations . . . . . . . . . . . . . . 213.2 Boundary conditions for the fluid region in the test simulations . . . . . . 223.3 Boundary conditions for the wall film region in the test simulations . . . . 223.4 Boundary conditions for the buffer layer in the test simulations . . . . . . 233.5 Thermodynamic properties for the solid . . . . . . . . . . . . . . . . . . . 243.6 Spray properties for the test simulations . . . . . . . . . . . . . . . . . . . 243.7 Number of elements and size of elements for the test domain . . . . . . . 253.8 Tolerance criterias for the test simulations . . . . . . . . . . . . . . . . . . 26

4.1 Change of conserved energy for the solid and the wall film after 1.5 s . . . 31

A.1 Parameters for T − µ relationship for water and urea . . . . . . . . . . . . 43

B.1 Region and boundary names for the tutorial guide . . . . . . . . . . . . . 57

C.1 Numerical schemes for the test simulations . . . . . . . . . . . . . . . . . 71

Nomenclature

Latin symbols

u, U velocity vector [m/s]

u, v, w U, V,W velocity [m/s]

SM external forces [N ]

Si external energy sources [J ]

m mass [kg]

p pressure [Pa]

prgh pressure with buoyancy included [Pa]

T temperature [◦C, K]

k thermal conductivity [W/m2K]

Q energy [J ]

q specific energy [J/kg]

i specific internal energy [J/kg]

h enthalpy [J/kg]

Cp specific heat at constant pressure [J/kgK]

Cv specific heat at constant volume [J/kgK]

R specific gas constant [J/kgK]

Greek symbols

µ dynamic viscosity [kg/ms]

ν kinematic viscosity [m2/s]

ρ density [kg/m3]

γ surface tension [N/m]

δ thickness [m]

Subscripts

∞ surrounding conditions

atm atmospheric conditions

melt melting point

vap vaporization point

decompose decomposition point of molecular

gen internal generation

f film conditions

Chemical compositon

H2O water

CO(NH2)2 urea

NOx nitrogen oxide

H hydrogen

N2 nitrogen

O2 oxygen

CO2 carbon dioxide

NH3 ammonia

HNCO isocyanic acid

Abbreviations

CFD Computational Fluid Dynamics

CHT Conjugate Heat Transfer

UWS Urea Water Solution

HTC Heat Transfer Coefficient

FVM Finite Volume Method

SCR Selective Catalytic Reduction

RANS Reynolds-avaraged Navier-Stokes

Chapter 1

Introduction and Background

Scania is today a manufacturer of heavy trucks, buses and diesel engines with an an-nual turnover of approximately 88 billion SEK and a total of 35000 employees in 100different countries. Approximately 3000 of these work with research and development inSödertälje. Scania’s diesel engines are developed mainly for their own trucks and busesbut also for other heavy duty vehicles, marine applications, industrial applications andpower generation [1].

This thesis is carried out at the engine development department at Scania’s research anddevelopment division in Södertälje. More specific at the Fluid and Combustion simula-tion group which are specialized in simulations of different internal flows in the dieselengine.

1.1 Todays Diesel Engines

Twenty years ago the diesel engine was a very dirty engine with high emissions and theusage was concentrated to heavy duty vehicles or other applications where high powerwas needed. As the world realized that we need to take care of our environment in termsof reducing fuel consumption and harmful emissions, a lot of diesel engine manufactur-ers started to develop more fuel efficient engines together with after treatment systemsto reduce emission of particulate matter (soot) and nitrogen oxide (NOx). Today thedevelopment has come so far that the diesel engine is a powerful challenger to otherengines developed with the aim to be environmental friendly e.g. engines that runs onrenewable fuels.

The development of the engines is mainly driven by legislations of how much harmfulemissions are allowed to be released into the air. In Europe the first regulation for dieselengines in heavy duty vehicles came 1993. This legislation was called Euro 1 and todaythe engine manufacturers are heading for the Euro 6 and Euro 7 legislation [2].

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As mentioned before the main focus is to reduce soot and NOx emission. Unfortunately,it is very hard to achieve fuel-efficient combustion in the engine with both low sootand NOx emission because when the combustion is optimized for low NOx, the sootemissions will be high and vice versa. This is also known as the Diesel Dilemma andcan be visualized as the curved black line in Fig. 1.1. One way to reduce both soot andNOx is to optimize the combustion to low soot and high NOx, and then use an aftertreatment system to reduce NOx before the emissions goes into the air, see the coloreddots in Fig. 1.1.

Figure 1.1: European legislation for soot (PM) and NOx emission from Euro 1 to Euro6 [3]. The black curved line shows the ratio between soot and NOx emission which ispossible to achieve from the combustion process. The green dot shows the emission whenthe combustion is optimized for low soot and the blue dot shows the emission after aNOx reducing after treatment system.

1.1.1 Exhaust Gas After Treatment Systems

The basic principle for an exhaust gas after treatment system developed to reduce NOx,is to transform the NOx into other less harmful emissions. One method for this is de-veloped from the basic idea to let the NOx interact with a large amount of hydrogen(H), and then simplified it will transform into air (N2), water (H2O) and a by-product

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of carbon dioxide (CO2) [4].

Today, Scania among other manufacturers are using Selective Catalytic Reduction (SCR)together with the Urea Water Solution (UWS) AdBlue as a hydrogen carrier to achievethis interaction. In practice the UWS will vaporize in the exhaust system and then byusing the SCR application the reaction between the NOx and the vaporized UWS willstart.

The vaporization of UWS is a technical challenge due to the different thermodynamicproperties for urea and water. In atmospheric pressure water vaporizes at 100◦C andurea starts to melt and decompose at 133◦C, therefore in the temperature range of100◦C−133◦C the water in the UWS will vaporize leaving urea alone in a solid form [4].If this process continues, more and more urea will get stuck in the exhaust systemand prevent exhaust gases to come out from the engine, which will give pressure lossesand higher fuel consumption. Therefore, it is imperative that the temperature in theexhaust system is above the decomposition and vaporization temperature of urea around133◦C − 192◦C [5] to make the after treatment work correctly.

1.2 Computational Fluid Dynamics

The basic purpose of Computational Fluid Dynamics (CFD) is to predict the physics ofreal flow phenomena with the help of computers. These flow phenomena could be e. g.how the air moves inside a room or how the air moves along the wing of an airplane. Insome cases there are other fluids then air involved e. g. in a water boiler where watervaporizes into the air and raises the humidity.

When CFD first was developed it was a tool to complement expensive hardware test-ings in the industry. These hardware testings could also be hard to analyze due to itsdifficulty to measure complex flow phenomena inside a hardware, e.g. an engine. CFDis therefore commonly used to predict flow-patterns in or outside hardwares, where it ishard to do measurements. As mentioned before, CFD is a tool to predict the flow andtherefore the user should always be critical to the accuracy of the results from a CFDsimulation with respect to a real flow-pattern.

Technically described, CFD is an analyzing method from the basic idea of computingequations in a numerical grid. These equations describe a three-dimensional flow, dueto change in velocity and pressure, around or inside a given geometry and in some casesheat transfer is also included. The accuracy in the CFD analysis depends on several fac-tors such as the resolution of the grid and also which equations are used to describe theflow. A high resolution grid and complex equations, demand a lot of computer power.Therefore it is always a tradeoff between the time it takes to do the computation andthe accuracy wanted in the results.

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There are different ways to solve the motion of the flow in the CFD applications usedtoday. One common way is to use Reynolds-averaged Navier-Stokes (RANS) equationstogether with the energy equation to describe the flow and heat transfer in a given fluiddomain. To be able to close the mathematical system of above equations a turbulencemodel is used with the basic idea to predict the turbulence in the flow.

In some analysis it is desired to know how the heat transfer proceeds inside a solid, lo-cated nearby a fluid with high temperature gradients. Conjugate Heat Transfer (CHT)is commonly used together with CFD to analyze a three-dimensional heat transfer in asolid and the heat transfer between a solid and a fluid. In the NOx reducing after treat-ment system mentioned above, there are high temperature gradients between differentplaces at the wall. This is because of the large energy transfer from the wall to theUWS in order to vaporize the UWS. In this case the coupled CFD and CHT computingmethod is a good way to predict the flow and heat transfer during the vaporization ofUWS.

The most common CFD softwares used today are commercial and expensive. Open-FOAM however, is a free CFD software that has begun to challenge the commercialsoftwares. OpenFOAM [6] is a CFD solver without a Graphical User Interface (GUI)i.e. every parameter that need to be set for a particular application, are set in a text file.Pre-processing can be done in the same way but with complex geometries it is preferableto use a software with a GUI. Post-processing is preferably done in another softwaresuch as ParaView which is a free scientific visualization software with a GUI.

OpenFOAM is a software with open source which makes it possible to develop newcode or modify existing code. It is made by the concept to be object oriented in theprogramming language C++, i.e. each application consists of several classes. In thisway it is easy to change an existing code to fit a particular purpose by modifying anapplication to handle different classes.

1.3 Objective of the Thesis

The basic objective is to modify an application in OpenFOAM to handle spray and wallfilm modeling of a Urea Water Solution together with Conjugate Heat Transfer in theexhaust gas after treatment system for a diesel engine. Because the geometry in theexhaust system is very complex, the simplified test domain in Fig. 1.2 will be used toallow more time for developing and testing the code. The workflow during this degreeproject can be divided into three different parts:

• Implement the viscosity for urea into OpenFOAM’s library of liquid properties andverify that it behaves in the expected way for different temperatures.

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• Develop the new application chtMultiRegionSprayFilmFoam where the classessprayCloud and surfaceFilm will be implemented into the original applicationchtMultiRegionFoam.

• Test the developed application chtMultiRegionSprayFilmFoam with the simplifiedtest domain in Fig. 1.2 and verify the heat transfer with basic hand calculationsof heat transfer and thermodynamic.

Figure 1.2: The simplified test domain, containing a fluid and a solid region. The spraycreates a wall film which cools the upper surface of the solid and enable a heat transferfrom the solid to the wall film.

1.4 Structure of the Thesis

Chapter 2 deals with the theory of the substance urea and the SCR process in the ex-haust gas after treatment system. The Chapter continues with a theory part about thoseelements in the field of CFD which are concerned in this thesis. Chapter 3 describeshow the viscosity of urea will be determined and the approach to develop the originalapplication chtMultiRegionFoam to fit this particular purpose. This Chapter ends witha description of how the test case simulations were set up. Chapter 4 presents resultsfrom the test case simulations and Chapter 5 is a discussion of the results from the de-veloped application chtMultiRegionSprayFilmFoam and the test case simulations. Thelast Chapter contains general conclusions about this thesis and also some suggestionsfor future work in the field of exhaust gas after treatment systems together with Open-FOAM.

For a more intuitive explanation through the code development, the application namesin this thesis are denoted with blue color and the class names are denoted with greencolor.

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Chapter 2

Theory

In this chapter the basic theory of the vaporization of a Urea Water Solution (UWS)will be presented together with some CFD theory which correlates to the given problemformulation. This includes the governing equations for the flow, the wall film and thespray in the fluid region and also the governing equations for the heat transfer in thesolid region.

The temperature unit used in this thesis is Kelvin (K) and 273.15K is approximatelyequal to 0◦C and there is no difference in the scale.

2.1 Urea and Water Solution

AdBlue is a registered trademark of a UWS which contains 32.5 wt % urea and the restis water. For the basic chemical and thermodynamic properties for water and urea seeTable 2.1.

The vaporization of urea is a complicated process because the chemical decompositionof urea is approximately in the same temperature range as for the vaporization of urea.This leads to the formation of different by-products, with other vaporization propertiesthan pure urea. There is also studies of chemical decomposition of urea slightly abovethe melting temperature of urea [7]. The desired chemical decomposition and vaporiza-tion of a UWS for the Selective Catalytic Reduction (SCR) process is as follows:

The water (H2O) in the UWS will first vaporize, leaving solid urea (CO(NH2)2) behind,

CO(NH2)2(l) + H2O(l) → CO(NH2)2(s) + H2O(g) (2.1)

The solid urea will then vaporize to ammonia (NH3), which will act as the hydrogencarrier, and isocyanic acid (HNCO),

CO(NH2)2(s) → NH3(g) + HNCO(g) (2.2)

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The icocyanic acid will then transform to ammonia leaving a by-product of carbondioxide (CO2),

HNCO(g) + H2O(g) → NH3(g) + CO2(g) (2.3)

At last, ammonia will interact with nitrogen oxide (NOx) in the exhaust gases andoxygen (O2) leaving nitrogen (N2) and water behind,

4NH3(g) + 4NO(g) + O2(g) → 4N2(g) + 6H2O(g) (2.4)

For a more detailed explanation of the chemical decomposition of a UWS, with all by-products formations see [4], and for the SCR technique with urea see [8].

In OpenFOAM the dynamic viscosity for urea is modeled to be the same as for water.This is because of the almost non-existing knowledge of the behavior of pure urea inliquid form. Therefore it is imperative to first determine a better approximation ofthe real dynamic viscosity for urea to be able to more accurately simulate the givensimulation domain.

Water Urea

Chemical composition H2O CO(NH2)2

Properties at 300K

Density ρ 996 kg/m3 1230 kg/m3*

Specific heat capacity Cp 4177 J/kgK 2006 J/kgK*

Dynamic viscosity µ 889 · 10−6 kg/ms −Surface tension γ 0.073 N/m 1 N/m*

Vapor pressure pvap 3565 kPa 0 kPa*

Thermal conductivity k 0.61 W/mK −Properties at patm

Tmelt 273 K 406 K

Tvap 373 K 466 K*

Tdecompose − > 423 KHeat of vaporization ∆Hvap 40.7 kJ/mol 185.5 kJ/mol

Table 2.1: Liquid properties for water and urea from [4] and [9]. Dynamic viscosity andthermal conductivity for urea is modeled to be the same as for water in OpenFOAMand the values with a * are from the liquid properties library in OpenFOAM [5].

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2.2 Governing Equations

The text in the following section is based on the book An Introduction to ComputationalFluid Dynamics, The Finite Volume Method by H. K. Versteeg and W. Malalasekera [10].

As mostly everything in the field of mechanics the behavior of a flow could be describedby Newton’s second law which states:

The rate of change of momentum for a fluid particle=

The sum of all forces acting on the fluid particle

In other words there is a way to describe the flow when all forces are known. But whenit comes to fluid dynamics one surface force, the viscous stress τij is really hard to de-termine. The viscous stress force acts in three dimensions at each of the three differentplanes in a three dimensional coordinate system. This implies that six unique viscousstress terms (due to symmetry) need to be determined to be able to describe the move-ment for a fluid particle.

The continuity equation together with the momentum equation in three dimen-sions are used to describe the movement for a fluid particle according to this viscousstress tensor plus a sum of all other active forces (see [10] for the derivation of the mo-mentum equations). This implies six unknown viscous stress variables in four equationsand therefore it is not a mathematically closed system.

The Navier-Stokes equations are derived from the momentum equations in three di-mensions with the statement that the viscous stress is proportional to the deformationrate of a fluid particle. This statement applies only for a Newtonian fluid which meansthat the viscous stress change linearly with respect to the strain rate.

Due to the given problem formulation, this thesis only regards compressible flows forNewtonian fluids. The Navier-Stokes equations reads,

∂(ρu)

∂t+ div(ρuu) = −∂p

∂x+ div(µ grad u) + SMx (2.5)

∂(ρv)

∂t+ div(ρvu) = −∂p

∂y+ div(µ grad v) + SMy (2.6)

∂(ρw)

∂t+ div(ρwu) = −∂p

∂z+ div(µ grad w) + SMz (2.7)

and the continuity equation,

∂ρ

∂t+ div(ρu) = 0 (2.8)

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together with the equation of state after which a more explicit form of the behaviorof the flow is obtained,

p = p(ρ, T ) and i = i(ρ, T ) (2.9)

e.g. perfect gas p = ρRT and i = CvT

where ρ is the density for the fluid, p is pressure and µ is dynamic viscosity. The vari-ables u, v and w define the flow velocity in the x-, y- and z-direction as well the velocityvector u contain all these velocity components. The variable SM contains the externalforces acting on the fluid particle. For the equation of state the variable i define theinternal energy for the fluid particle, R the specific gas constant, T the temperature andCv the specific heat capacity at constant volume.

With the Navier-Stokes equations 2.5-2.7, the continuity equation 2.8, the equation ofstate 2.9 and the energy equation, it is possible to get a mathematically closed systemwith seven unknowns and seven equations. The energy equation reads,

ρ∂i

∂t+ div(ρiu) = −p div u + div(k grad T ) + Φ + Si (2.10)

where k is the thermal conductivity for the fluid, Φ the dissipation function describingthe effects due to viscous stresses and Si is the internal energy of the fluid particle exceptthe kinetic energy.

2.2.1 Reynolds-avaraged Navier-Stokes equations

The Reynolds number (Re) measure the intensity of the flow within certain placeswith respect to inertia forces and viscous forces. For Reynolds numbers above Recritical,the behavior of the flow will be random and chaotic, i.e. turbulent flow. Within a cer-tain point in a turbulent flow the velocity will fluctuate randomly over time, see Fig. 2.1.

Figure 2.1: Measuring of velocity in an arbitrary point located in a turbulent flow.

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This velocity is decomposed into u(t) = U + u′(t) where U is the mean velocity vectorand u′(t) is the fluctuating components according to the turbulent behavior of the flow.This is called the Reynolds decomposition and is the start of modeling turbulence.

By introducing the flow property φ, which represents the necessary flow terms such asvelocity, pressure, temperature etc, the flow can be declared in a more overall definitionas,

φ = Φ + φ′ (2.11)

Reynolds averaging is also known as time averaging (or ensemble averaging when theboundary conditions for the flow are time-dependent) of φ in equation 2.11 as,

φ = Φ + φ′ (2.12)

where,Φ = Φ (2.13)

φ′ = limT→∞

1T

∫ T0φ′(t)dt = 0 (2.14)

T: averaging interval

To continue, all the time averaged flow terms from equation 2.12 are substituted into theNavier-Stokes equations 2.5-2.7 and the continuity equation 2.8. Thus, the Reynolds-averaged Navier-Stokes equations (RANS) for compressible flows are obtained (see[10] for the derivation of the RANS equations),

∂(ρ̄Ũ)

∂t+div(ρ̄ŨŨ) = −∂P̄

∂x+div(µ grad Ũ)+

[−∂(ρ̄u

′2)

∂x− ∂(ρ̄u

′v′)

∂y− ∂(ρ̄u

′w′)

∂z

]+SMx

(2.15)

∂(ρ̄Ṽ )

∂t+div(ρ̄Ṽ Ũ) = −∂P̄

∂y+div(µ grad Ṽ )+

[−∂(ρ̄u

′v′)

∂x− ∂(ρ̄v

′2)

∂y− ∂(ρ̄v

′w′)

∂z

]+SMy

(2.16)

∂(ρ̄W̃ )

∂t+div(ρ̄W̃ Ũ) = −∂P̄

∂z+div(µ grad W̃ )+

[−∂(ρ̄u

′w′)

∂x− ∂(ρ̄v

′w′)

∂y− ∂(ρ̄w

′2)

∂z

]+SMz

(2.17)and the continuity equation,

∂ρ̄

∂t+ div(ρ̄Ũ) = 0 (2.18)

where ρ̄ is the mean density for the fluid and P̄ is the mean pressure. The variablesŨ , Ṽ and W̃ is the density-weighted or Favre-averaged velocity components in the x, yand z-direction as well the velocity vector Ũ contain all these velocity components. Theterms located inside the brackets are the turbulent stresses, also known as the Reynoldsstresses.

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2.2.2 Turbulence Models

To be able to solve the RANS equations, the Reynolds stresses need to be modeled whichoften refers to turbulence modeling. Over the past years, several turbulence modelshave been developed such as Spalart-Allmaras, k− �, k−ω, SST k−ω, Reynolds stressmodel etc. These turbulence models contain different amount of equations to calculatethe turbulence in the flow, e.g. Spalart-Allmaras uses one equation and the Reynoldsstress model evaluates all Reynolds stresses independently with seven equations. Thek−� model is a two equation turbulence model which have been proven accurate in flowswith flat boundary layer and in flows near smooth geometries. It is widely used becauseof its stability and fast convergence in a lot of different industrial cases [10]. Therefore,the k − � turbulence model is used in this thesis where the geometry is simple and fastconvergence is desirable for testing and verifying the developed code.

The k − � turbulence model is built up by the transport equation of turbulent kineticenergy k,

∂(ρk)

∂t+ div(ρkU) = div

[µtσk

grad k

]+ 2µtSij · Sij − ρ� (2.19)

and the transport equation for the dissipation of turbulent kinetic energy �,

∂(ρ�)

∂t+ div(ρ�U) = div

[µtσ�

grad �

]+ C1�

�

k2µtSij · Sij − C2�ρ

�2

k(2.20)

where the Sij tensor is the mean deformation rate of a fluid particle and the term µt isthe eddy viscosity defined as,

µt = ρCµk2

�(2.21)

The five constants in equation 2.19-2.21 are set to default values according to [10] as,

Cµ = 0.09 σk = 1.00 σ� = 1.30 C1� = 1.44 C2� = 1.92 (2.22)

2.3 Spray Modeling

A spray is tricky to model in CFD softwares firstly due to the random velocity directionfor all the droplets coming out of the spray. Every droplet also has a random size andmass which makes it even harder to make an accurate prediction of a spray.

In OpenFOAM the spray is modeled to be a given amount of parcels coming out ofthe spray where this amount is set in the pre-processing part. The total mass injectedin the system through the spray is also set in the pre-processing part and together itis possible to determine the mass given to each parcel. Each parcel is also given onerandom number from a stochastic variable at the outlet, which declares the radius forthe droplets in that specific parcel. Thus, the mass (or volume) given to each parcelrarely match the volume for the droplets in that parcel due to the different radius for the

11

droplets. This gives that the amount of droplets in each parcel is a statistical numberwhich most certain vary from a fraction of a droplet to thousands of droplets, e.g. aparcel can contain 0.00034− 45000 number of droplets [11].

In aspect of the flow in the simulation domain, the parcels or “droplets” will impact onthe fluid particles according to Newton’s second law. This extra force is included inthe external force SM in the Navier-Stokes equations 2.5-2.7.

Unlike the Eulerian representation of the flow, the spray is modeled with a Lagrangianrepresentation, i.e. all parcels from the spray are tracked separately inside the fixednumerical grid which is used by the flow equations. This will remove the necessity forresolving the nozzle and will therefore need less computational power [11].

2.4 Wall Film Modeling

Wall film modeling in CFD softwares is a challenge because of the complex formationof the liquid film at the wall and also the multiphase treatment required to handle bothgas and liquid. The formation of the liquid film at the wall is especially hard to pre-dict due to the many external factors that will affect the formation, such as gas flow,gravity, wall properties and wall roughness. The properties of the liquid will also affectthe formation such as viscosity, which will be discussed later in this thesis, but also sur-face tension and surface shear which are a subject that will not be discussed in this thesis.

In OpenFOAM the wall film is modeled to be in an own external mesh region extrudedfrom the fluid region. This mesh region is one layer thick and the flow perpendicular tothe surface is neglected which imply a two-dimensional flow along the wall. The flow ismodeled with a continuous Eulerian phase with the momentum, continuity and theenergy equation. The thickness of the liquid film is derived from the momentum andcontinuity equation [12].

The class surfaceFilm in OpenFOAM (which is used to solve the movement of the liq-uid film) can only handle one chemical component and not a multi component mixture.First of all, this implies that the liquid for the wall film is the only component whichcan be defined in the wall film region and the gas in the fluid region is not includedin the wall film region. In reality, if the velocity of the gas flow is relatively high, thegas flow located in the frontline of the liquid film (while it is being formed) most likelywill experience some adverse pressure gradients near the wall. These adverse pressuregradients can create vortices or turbulence which can interfere with the formation of theliquid film. However, if the liquid film is relatively thin and the velocity of the gas flowis low, these effects will most likely be negligible.

Secondly, because of the non multi component mixture of the liquid in the wall filmregion, the liquid film can only be defined to handle urea or water, not a UWS as for

12

the after treatment system case. Since the objective of this thesis is formulated by theusage of a UWS, the results will be different than expected and the liquid film will onlybe urea or water independently.

2.5 Conjugate Heat Transfer

Heat transfer inside a solid can be described according to the law of heat-conduction orFourier’s law in one-dimension as reads,

qx = −kA∂T

∂x(2.23)

where qx is the heat transfer rate, k is the thermal conductivity, A is the area of thesurface perpendicluar to the direction of the heat flow and ∂T/∂x is the temperaturegradient in the direction of the heat flow. With equation 2.23 it is possible to derive athree-dimensional heat-conduction equation as reads,

∂

∂x(kx

∂T

∂x) +

∂

∂y(ky

∂T

∂y) +

∂

∂z(kz

∂T

∂z) + q̇gen = ρCp

∂T

∂t(2.24)

where q̇gen is the heat generation inside the solid, ρ is the density of the material andCp is the specific heat at constant pressure [13].

Heat transfer from a fluid to a solid is mainly caused by convection but a small partalso comes from radiation which in this thesis is neglected to give more time for codedevelopment. The convection part is normally natural driven when the density for afluid changes nearby a hot or cold solid, causing the fluid to move due to the densitygradient. If external forces makes the fluid to move around at the surface of the solid,e.g. when a fan blows on a hot radiator, it is called forced convection. The convectionheat transfer can simplified be described by Newton’s law of cooling as reads,

q = h(Twall − T∞) (2.25)

where q is the heat transfer rate, h is the Heat Transfer Coefficient (HTC), Twall is thetemperature at the wall and T∞ is the surrounding temperature [13].

In CFD softwares, heat transfer between a solid and a fluid is called Conjugate HeatTransfer (CHT). This involve variations of temperature within a solid and a fluid becauseof the thermal interaction between the solid and the fluid, i.e. the three-dimensionalheat-conduction equation 2.24 in the solid is coupled with the energy equation 2.10 inthe fluid.

13

2.6 The Finite Volume Method

The Finite Volume Method (FVM), or the control volume method, is a numerical methoddeveloped to handle the transport process of diffusion and convection (advection) for thegoverning equations. The basic idea of the FVM is to divide the domain into discretecontrol volumes where each side has the specified length ∆i. Inside each control volumeand in each corner of the control volumes, certain nodal points are located. The controlvolumes together with the nodal points construct a so-called staggered grid which isvisualized in Fig. 2.2 for a two-dimensional case.

Figure 2.2: A two-dimensional staggered grid for the FVM where P is referring to thecurrent point where the flow properties will be calculated and the N , S, W and E lettersrefers to the north, south, west and east point. The grey cells refer to the staggered cellswith central point n, s, w and e, which characterize the staggered grid in the FVM.

The next step is the integration of the governing equations over the grid to yield dis-cretized equations at the arbitrary node point P . To solve these discretized equationsin point P , the flow properties in the neighbor nodes need to be known. In the middleof the simulation domain these flow properties will be unknown, but if nodal point Pis located at the beginning of the domain, the boundary conditions will define the flowproperties for the “first” node. This process will then proceed through the domain overand over until the flow properties between two iterations are less than a certain tolerancecriterion set by the user. In a three-dimensional case as in this thesis, the approach issimilar but with one more dimension (see [10] for a more detailed description of theFVM).

The three-dimensional FVM is used in OpenFOAM for solving the flow equations in thefluid and also for the heat distribution inside solids. This implies a natural connectionbetween the fluid and the solid in the simulations with CHT.

14

Chapter 3

Method

This chapter starts with presenting how urea is modeled to fit this particular applica-tion. It continues with a description of the workflow during the spray and wall filmimplementation and also the approach to use a buffer layer region with the developedapplication chtMultiRegionSprayFilmFoam. At last, there is a presentation of howthe CFD simulations for the test case are set up with all necessary parameters andapproximations.

3.1 Modeling Liquid Urea

The natural state of urea at atmospheric conditions is to be a solid. But it is difficultto simulate a phase change from solid to liquid. This is mainly because of the numericalinstability when calculating the flow equations, i.e. when the fluid instantly goes to asolid. In other words the flow equations need to be active in the fluid region but not inthe solid region. To handle a case when ice melts to water, the solver need to be capableto solve the partially melted regions in the ice and also the huge energy transfer to theice during the melting process. To make such a case the solver and the numerical gridneed to be complex which in the end requires more computational power.

Vaporization of a Urea Water Solution (UWS) in the after treatment system is a techni-cal challenge because of the vaporization of water before urea. To do an accurate CFDsimulation for this scenario, urea needs to be modeled as a liquid below TUreaMelt butstill behave as a solid. A way to achieve this is to model urea as a liquid with a viscositythat will go very high when the temperature drops below TUreaMelt (i.e. urea will belike a very viscous syrup). The correct viscosity of melted urea above TUreaMelt is hardto determine due to the complex decomposition of urea, i.e. the liquid substance ofmelted urea will most likely be a chemical compound of different substances. Thus, thecorrect behavior of pure urea in terms of viscosity is not investigated in this thesis andis not desirable either, due to the infrequent presence of pure melted urea in the aftertreatment system. In other words, the modeling of liquid urea in this thesis is only away to predict where it is possible for this chemical compound to grow stiff in the after

15

treatment system.

In OpenFOAM the liquid properties for fluids are modeled with the functions from theNational Standard Reference Data System (NSRDS) [14]. To model dynamic viscosityOpenFOAM is using the NSRDS function no. 1 which reads,

µ(T ) = eA+BT+ClogT+DTE (3.1)

The dynamic viscosity for mostly all the fluids in OpenFOAM is described by settingdifferent values for the parameters A to E in function 3.1. Water e.g. have the param-eters A to E set in a way which gives the T − µ curve shown in Fig. 3.1 (please seeappendix A.1 for the values of each parameter A to E).

One way to achieve a desirable behavior for the dynamic viscosity of urea is to use theT−µ curve for water and basically shift the curve to the right, as can be shown in Fig. 3.1,giving that the viscosity will grow very high at TUreaMelt instead of at TWaterMelt as forwater. Thus, the desirable dynamic viscosity for urea needs to be described with thefunction 3.1 by setting parameters A to E to certain values. A method for this is to usethe Matlab function lsqcurvefit with the desirable T − µ curve for urea as an inputvalue.

Figure 3.1: T − µ curve for water and a desirable T − µ curve for urea.

16

3.2 Development of chtMultiRegionSprayFilmFoam

The original application chtMultiRegionFoam in OpenFOAM v2.1.0 is developed tohandle transient compressible flow and heat transfer calculations in an arbitrary num-bers of fluid and solid regions. The three-dimensional heat transfer in the solid is handledby heatConductionFoam and the conjugate heat transfer between the solid and the fluidregions is handled by buoyantFoam, see the User Guide [15]. To calculate heat trans-fer and vaporization of liquid urea in the after treatment system case, the applicationchtMultiRegionFoam needs to be able to handle spray and wall film calculations in eachfluid region.

For the spray calculations, the class sprayCloud is implemented in the solver forchtMultiRegionFoam. In the same way the class surfaceFilm is implemented in thesolver and the new application is called chtMultiRegionSprayFilmFoam. To handlethe extra thermodynamic which comes with the liquid spray and the liquid wall filmin the gas flow, some other classes also need to be implemented. For a more de-tailed overview of the implementation, see appendix B.1. The implementation is donewith help from the Programmer’s Guide [16]. A tutorial for the developed applicationchtMultiRegionSprayFilmFoam is presented in appendix B.3.

3.2.1 The Wall Film Region

Solving the spray with sprayCloud is carried out in the ordinary fluid region, but solvingthe wall film with surfaceFilm is carried out in an own mesh region as described inSection 2.4. This mesh region is extruded from the ordinary fluid mesh region withthe application extrudeToRegionMesh, see Fig. 3.2. The input parameters for thisapplication are basically which wall to be extruded, number of cell layers in the newextruded mesh (surfaceFilm is only compatible with one cell layer) and thickness forthe new extruded mesh.

Figure 3.2: A visualization of the numerical grid of a fluid and the wall film extrusionof a single cell layer mesh from the fluid region mesh.

17

Thus, in the test domain shown in Fig. 1.2, the mesh for each region is split before themesh for the wall film is created. This implies that the mesh for the wall film regionwill be extruded into the mesh for the solid region. In visualization purpose this couldbe confusing but for the solver this is not a problem because the different meshes arecalculated separately and the connection between the boundaries for each region are notaffected by the position of the boundaries.

Due to the two-dimensional flow and the separate wall film thickness variable derivedfrom the momentum and continuity equation, as presented in Section 2.4, the thicknessof the mesh is irrelevant for the formation of the liquid film.

3.2.2 The Buffer Layer

In reality, the heat transfer will go instantly from the solid to those places at the sur-face in which a liquid film is formed. In the other places where there is gas, the heattransfer will be described by a Heat Transfer Coefficient (HTC). In the wall film regionin OpenFOAM, it is not possible to calculate a HTC due to the lack of thermodynamicproperties for the gas. This is because the wall film region only contains one chemicalcomponent, i.e. the liquid film.

The connection between a fluid region and a wall film region is carried out via the settingsfor the boundary and is handled automatically with the applicationextrudeToRegionMesh. In chtMultiRegionFoam, the connection between a solid and afluid region is carried out automatically when using the application splitMeshRegions.However, when connecting a wall film region to a solid region, there will be a direct tem-perature coupling from the wall-film-to-solid boundary to the solid-to-wall-film boundaryand not a heat transfer depending on a HTC. A way to circumvent this is to introducea buffer layer between the wall film and the solid, please see Fig. 3.3. The buffer layeris a way to solve the problem with the HTC because the buffer layer is an ordinary fluidregion where the desirable thermodynamic properties for the gas will be available.

The buffer layer will basically be a thin fluid region with the same fluid properties as forthe big fluid region. This implies a heat transfer from the solid to the buffer layer, witha correct calculated HTC. The connection between the buffer layer and the wall filmwill be an ordinary fluid-to-wall-film connection, enabling a correct heat transfer fromthe buffer layer to the liquid film. However, due to the very low thermal conductivityfor air, the buffer layer will act as an insulating layer between the liquid film and the solid.

In order to have a direct heat transfer from the solid to the liquid film and a heat transferdepending on a HTC from the solid to the gas, two new scalar fields need to be createdin the chtMultiRegionSprayFilmFoam code. The first scalar field will contain the newthermal conductivity variable Kliq which will be used in the buffer layer. This variablewill have the same thermal conductivity as the original conductivity for the gas in thefluid region, except at those faces where a liquid film is formed. In these faces, Kliq

18

shall be set to the same thermal conductivity as the liquid film.

Figure 3.3: A visualization of the position for each region and the connection between.

To know where the liquid film is formed, the other scalar field will contain a new variablecalled deltaMap, which will be mapped from the wall film thickness deltaf located in thewall film region. Thus, with the variable deltaMap it is possible to know in which cellsthe liquid film is formed and by that, which cells should have greater thermal conduc-tivity by changing the variable Kliq (see appendix B.2 for the implementation of Kliqand deltaMap). The value for the new thermal conductivity is set by the user throughthe parameter KValue in the file constant/bufferLayer/surfaceFilmProperties andshall be set to the same thermal conductivity as the liquid film.

The mesh for the buffer layer is copied to be the exact same mesh as for the wall filmregion. This makes the pre-processing when setting up a buffer layer much faster.

If the buffer layer is thin, the flow calculations will have some numerical instability,i.e. for a good convergence, the buffer layer needs to be relatively thick. The mesh forthe buffer layer shall be one cell layer thick for the new approach with the Kliq anddeltaMap variable to work.

19

3.2.3 Assumptions and Approximations

• Heat transfer from radiation is assumed to be negligible. This is because the heattransfer via radiation is normally much less than the heat transfer via convection,in cases similar to the after treatment system case. In other words, the effect ofadding radiation for the after treatment system case in this thesis, will be negligiblechanges in the result and a lot of more time will be spent on developing the code.

• The thermal conductivity variable Kliq is activated in the cells where the wall filmthickness is > 10−6 m. This is because a wall film thickness less than 10−6 m isassumed to have negligible effect on the heat transfer between the wall film andthe solid.

• The thermal conductivity variable Kliq is set to a constant value by the user in thepre-processing part. Since the temperatures within the wall film do not have hightemperature gradients, and because the temperature for a liquid is often relatedto the thermal conductivity, the change of thermal conductivity between differentplaces within the wall film is most likely negligible.

20

3.3 Simulation Setup for the Test Case

Six different transient simulations are made with two different liquids for the spray andtwo different materials for the solid according to Table 3.1. Air is used as fluid in thefluid region instead of exhaust gases which implies that no chemical reactions are takingplace, i.e. chemical reactions are deactivated in all simulations.

Case Fluid Solid Spray Time Vaporization

1 Air Copper 100 g Urea 1.2 s No

2 Air Copper 100 g Water 1.2 s No

3 Air Copper 10 g Water 2.5 s Yes

4 Air Copper 10 g Urea 2.5 s Yes

5 Air Steel, 1% C 10 g Water 2.5 s Yes

6 Air Steel, 1% C 10 g Urea 2.5 s Yes

Table 3.1: Fluid and solid properties for the test simulations made with thechtMultiRegionSprayFilmFoam application.

3.3.1 Computing Platform

The CFD simulations are made on a local workstation with six CPU cores of 2.93 GHzeach and 24 GB RAM. Operating system is Linux Red Hat distribution v2.16.0 and thecode development is made from OpenFOAM v2.1.0.

3.3.2 Simulation Domain and Boundary Conditions

The simulation domain with dimensions is located in Fig. 3.4.

Figure 3.4: A sketch of the simulation domain with dimensions. The thickness of thesolid is 0.01 m and the direction of the spray is in the same direction as for the gravity,i.e negative y-direction. The spray creates a wall film which cools the upper surface ofthe solid and enable a heat transfer from the solid to the wall film.

21

The Fluid

The air in the fluid region is a mixture of 23.4% oxygen (O2) and 76.6% nitrogen (N2).The boundary conditions are set according to Table 3.2. The boundary to wall film isset to slip condition because the non-existing “gas” in the wall film region assumedto flow with the same velocity as the fluid region. The buoyantPressure condition forthe outlet calculates the normal gradient from the local density gradient. The initialconditions for velocity and temperature are set to be the same as for the inlet and theinitial pressure is 100000 Pa.

Inlet Outlet

p calculated calculated

prgh zeroGradient buoyantPressure

T fixedValue zeroGradient

- value 300K -

U fixedValue inletOutlet

- value 0.5 m/s 0.5 m/s (inletValue = 0 m/s)

Walls Boundary to wall film

p calculated calculated

prgh zeroGradient zeroGradient

T zeroGradient mapped

- value - -

U fixedValue slip

- value 0 m/s -

Table 3.2: Boundary conditions for the fluid region.

The Wall Film

The surface film model is defined to be the OpenFOAM specific thermoSingleLayerwith the surface shear coefficient set to 0.9. The liquid in the wall film is defined tocontain either water or urea. The boundary conditions are set according to Table 3.3.Vaporization is activated with the phase change model standardPhaseChange.

Walls Boundary to fluid Boundary to buffer layer

Tf zeroGradient zeroGradient zeroGradient

- value - - -

Uf fixedValue mapped fixedValue

- value 0 m/s - 0 m/s

Table 3.3: Boundary conditions for the wall film region.

22

The Buffer Layer

Air is used as fluid in the buffer layer with the same composition as the air in the fluidregion. The boundary conditions are set according to Table 3.4. The initial conditionsfor velocity and temperature are set to be the same as for the inlet and the initialpressure is 100000 Pa. The thermal conductivity Kliq in the buffer layer is set to wateraccording to Table 2.1 in all cases.

Inlet Outlet Walls

p zeroGradient zeroGradient zeroGradient

prgh zeroGradient zeroGradient zeroGradient

T fixedValue zeroGradient zeroGradient

- value 300K - -

U fixedValue inletOutlet fixedValue

- value 0.01 m/s 0.01 m/s (inletValue = 0 m/s) 0 m/s

Boundary to wall film Boundary to solid

p zeroGradient zeroGradient

prgh zeroGradient zeroGradient

T mapped turbulentTemperatureCoupledBaffleMixed

- value - -

U mapped fixedValue

- value - 0 m/s

Table 3.4: Boundary conditions for the buffer layer.

23

The Solid

The simulations are made with two different types of material for the solid, copper andsteel with 1% carbon. The thermodynamic properties for these materials are shownin Table 3.5. The initial temperature in the solid is set to 300K and the only en-ergy transfer is between the buffer layer and the solid. The other walls are adiabaticwith the OpenFOAM specific zeroGradient boundary. The surface coupled to thebuffer layer are set with the OpenFOAM specific conjugate heat transfer boundary,turbulentTemperatureCoupledBaffleMixed.

Copper (Cu)

ρ 8954 kg/m3

Cp 384 J/kgK

k 398 W/mK

Steel with 1% C

ρ 7800 kg/m3

Cp 473 J/kgK

k 43 W/mK

Table 3.5: Thermodynamic properties for the materials representing the solid in thesimulations [9].

The Spray

The external forces which are affecting the spray are gravity and drag. The heat trans-fer model for the interaction between the parcels and the fluid is RanzMarshall andvaporization is activated with the phase change model liquidEvaporationBoil. Theproperties for the spray are presented in Table 3.6. For liquid properties of water andurea see Table 2.1.

Nozzle type coneNozzleInjection

tstart 0 s

tend 1 s

DsprayNozzle 2 mm

θsprayCone 5◦

Uspray 1 m/s

Tspray 293K

mtotal 10 g / 100 g

Nr. of parcels 100000

Dparcels (uniform distribution) 1 mm

Table 3.6: Spray properties for the water and urea spray.

24

3.3.3 Mesh and Solution Criteria

The mesh is structured and the total number of elements and element size for each regionare shown in Table 3.7. A visualization of the mesh is shown in Fig. 3.5. An analysisof a results independent mesh has not been done since this thesis only regard spray andwall film modeling with CHT in OpenFOAM. Tolerance criterias for the most commonflow properties are listed in Table 3.8.

Fluid elements 162000

- size 1.6 mm− 3.3 mmSolid elements 54000

- size 1 mm− 3.3 mmElements in wall film and buffer layer 5400

- size 2.2 mm− 3.3 mm

Table 3.7: Number of elements and size of elements. The wall film and the buffer layerhave a thickness of 10 mm for a single cell layer beyond the listed element size in thetable.

Figure 3.5: A visualization of the mesh. The blue region is the fluid region and the greyregion is the solid region. The mesh for the wall film region and the buffer layer is notshown because these are inside the solid region.

25

Fluid Region Wall Film Buffer Layer

U 10−7 10−10 10−7

prgh 10−7 - 10−9

ρ 10−7 10−10 10−7

hs 10−7 10−10 10−7

Table 3.8: Tolerance criterias for all test case simulations for the most common flowproperties where prgh is the pressure with the buoyancy from the air and hs is thesensible enthalpy.

The tolerance criteria for the temperature in the solid region is 10−6. For a more detailedoverview of all tolerance criterias and the relaxation factors for the buffer layer (the onlyregion that use relaxation factors), see the solution setting files in appendix C.1. Thefiles with all numerical schemes used in the test simulations are located in appendix C.2and a description for each scheme is found in Table C.1.

The turbulence model used is the k − � model with default values for all modelingconstants. The initial time step is set to 0.00001 s and the maximum time step ismanually adjusted between 0.0001 s− 0.002 s where the shorter time step is set duringthe spray injection.

26

Chapter 4

Results

This chapter starts by presenting the new viscosity for urea in OpenFOAM and theresults from the behavior of liquid urea with respect to water. The results from thetest case simulations with the developed application chtMultiRegionSprayFilmFoamare then presented with a brief description of the most interesting parts.

4.1 Urea Viscosity

After several iterations with lsqcurvefit the parameters A to E are set in a way whichenables the function 3.1 to fit the desirable T −µ curve for urea shown in Fig. 4.1 (pleasesee appendix A.2 for the Matlab code and appendix A.1 for the calculated values of eachparameter A to E).

Figure 4.1: The desirable T − µ curve for urea together with the T − µ curve for ureacalculated from function 3.1.

27

Two simulations are done according to case 1-2 in Table 3.1. The simulations are donewith water respectively urea as the spray liquid to verify the new dynamic viscosity forurea. The total injected spray mass is 100 g in both cases and vaporization is not active.The temperature for the spray is 293K and the rest of the domain has a surroundingtemperature of 300K. The dynamic viscosity for water at 300K is 0.889 · 10−3 kg/msand for urea 85.2 · 10−3 kg/ms which implies that urea appear as a liquid, approxi-mately 100 times more viscous than water in these test simulations. A comparison forthe wall film thickness of water and urea is shown in Fig. 4.3 and the formation of thewall film can be seen in Fig. 4.2.

Figure 4.2: The formation of the wall film with water or urea at the upper surface of thesolid (the xz-plane in Fig. 3.4). The left picture couple show wall film thickness deltaf[m] after t = 0.2 s and the right show after t = 0.8 s. The lines show the extractionpoints which are used in Fig. 4.3 and the spray is located at z = 0.03 m.

28

In Fig. 4.2 at t = 0.8 s the wall film for the water case is formed to the end of the domainand in the case with urea it is only formed to approximately 80% of the length of thedomain. From the results of wall film thickness in Fig. 4.3 it is found that the formedwall film at the location of the spray impact at t = 0.2 s, is more extended in the casewith the water spray than in the case with the urea spray.

Figure 4.3: Comparision of wall film thickness deltaf [m] for water and urea. The leftfigure show wall film thickness at the location for the spray impact after t = 0.2 s andthe right show in the middle of the test domain after t = 0.8 s. The extraction lines arevisualized in Fig. 4.2.

4.2 chtMultiRegionSprayFilmFoam

Four simulations are done according to case 3-6 in Table 3.1. The total injected spraymass is 10 g in all cases and vaporization is active in all regions. Fig. 4.4 show thesurface temperature at the solid for all four cases and when compared to the formationof the wall film in Fig. 4.2, it is possible to verify the heat transfer from the solid to thewall film (through the buffer layer) since the heat distribution at the solid is similar tothe wall film formation.

With Table 3.5 and the dimension for the solid from Fig. 3.4, it is calculated that thecopper solid has an energy conservation of 1375 J/K and the steel solid has 1476 J/K.Simplified, this implies in a steady state condition that a water spray with the totalmass of 10 g and with the temperature of 293K, will cool the total temperature for thecopper solid 0.21K and the steel solid 0.20K, if there only occur heat transfer from thesolid to the wall film. In the same way a urea spray will cool both the copper and thesteel solid 0.10K.

29

Figure 4.4: Surface temperature T [K] at the upper surface of the solid (the xz-plane inFig. 3.4) after t = 1.5 s. The left picture couple show surface temperature with a waterspray and the right with a urea spray. The lines show the extraction points which areused in Fig. 4.5 and the spray is located at z = 0.03 m.

Since there is heat transfer from the air region to the spray, from the air region to thewall film and since the fluid in the buffer layer also will be cooled by the wall film, thetemperature drop for the solid will be much less than 0.20−0.21K and 0.10K. Table 4.1show the total temperature and change of conserved energy for the solid and the wallfilm after 1.5 s.

30

Case Ttotal,solid [K] ∆Qsolid [J ] Ttotal,film [K] ∆Qfilm [J ]

Urea, Cu 299.9931 −9.5 294.8948 38.0Urea, Steel 299.9933 −9.9 294.8949 38.0Water, Cu 299.9928 −9.6 295.1637 90.4Water, Steel 299.9930 −10.3 295.1662 90.5

Table 4.1: Total temperature and change of conserved energy (with respect to initialtemperature, 300K for the solid and 293K for the spray) for the solid and the wall filmafter 1.5 s.

Figure 4.5: Temperature drop in different places at the surface of the solid for t =1.5 s. The temperature is normalized with 0.10K and the black line refer to the initialtemperature (300K). The extraction lines are visualized in Fig. 4.4.

The temperature in Fig. 4.5 is normalized with the total temperature drop of 0.10K forthe solid if a urea spray cools the solid, as presented in the beginning of this section.This implies that the normalized temperature at the surface of the solid in the case with

31

a urea spray in Fig. 4.5, should be −1 if there is an adiabatic process and steady statecondition has been reached (if it is 0 it means that there is no heat transfer at all).The thermal conductivity for copper is almost 10 times higher than it is for steel (seeTable 3.5). It is found from Fig. 4.5 that the surface temperature at the solid differthe most between the cases with a copper solid and a steel solid which most likely arebecause of the difference in thermal conductivity.

The wall film cools the solid more widely in the case with water than urea, e.g. see crosssection z = 0.03 m for 1.5 s and z = 0.06 m for 2.5 s in Fig. 4.6. In the same figure atz = 0.20 m for 2.5 s it is possible to see the effect of re-heating the solid from the fluidregion (i.e. buffer layer) when the wall film has flowed past the z = 0.20 m location.

Figure 4.6: Cross section of the solid at four different locations. The comparison areonly for the steel solid where the top picture group show for t = 1.5 s and the bottomgroup for t = 2.5 s. The location for all cross sections are visualized in Fig. 4.4 and thecross section at z=0.03 is the spray location.

The energy loss via the outlet for the buffer layer is most uncertain in these results butcan be approximated in a simplified way. If it is assumed that all the air that goes outthrough the outlet, have been instant cooled from the inital temperature of the bufferlayer (300K) to the mean temperature of the wall film after 1.5 s (295K, see Table 4.1),it is found that during 1.5 s there is a total energy loss of −0.09 J . This is calculated

32

with the approximation that all the air which goes out through the outlet, always hasthe temperature of 295K between 0 s − 1.5 s. For this calculation, the density ρ andthe specific heat Cp for air at 295K has been used which can be found in [9].

Due to the fact that it is impossible to cool the air in the buffer layer to 295K, before thewall film has started to advance in the wall film region, this approximation is generous.If this approximation is assumed to be correct, the energy loss via the outlet for thebuffer layer will be less than 1% of the total energy transfer from the solid, which canbe found in Table 4.1. The energy loss via the buffer layer in these test simulations cantherefore be neglected if a source of error by 1% is accepted.

33

Chapter 5

Discussion

In this chapter, the behavior of the new viscosity of urea in OpenFOAM is discussedwith respect to the desirable behavior of urea in a virtual environment. The mostconcerned thoughts of the developed application chtMultiRegionSprayFilmFoam arethen discussed together with the accuracy of the simulations. In the end there are somegeneral comments of the thermodynamics of urea and water in a UWS.

5.1 Urea Viscosity

For the new T − µ curve for urea, which can be found in Fig. 4.1 the slope at low tem-perature is slightly under the desirable curve which is an effect from the behavior of theexponential function 3.1. If the slope for the calculated curve was as steep or steeperthan the desirable curve, the CFD simulations can have some numerical instability. Theopposite, as in this case, can give more stable CFD simulations when the temperaturedrops for urea, because the dynamic viscosity will not increase as fast as for the desirablecurve.

The formation of the wall film is depending on a lot of different factors such as gravity,wall properties, wall roughness etc. Viscosity is also an important factor basically de-scribing the inertia of the fluid. The only difference between the case with a water sprayand a urea spray is the liquid for the spray. This implies that the more extended wallfilm in the water case in Fig. 4.2-4.3 is most likely because of the higher inertia in urea,i.e. the viscosity for urea is modeled to be higher than it is for water.

The behavior of the urea liquid in the test simulations at ordinary room temperature, isstill similar to an ordinary fluid but should behave more as a solid or a very viscous liquidsuch as pitch. This can imply that the T − µ curve for urea in Fig. 4.1 can be steeperthan the calculated curve to make a more desirable behavior of urea at low temperature,even though it can give some numerical instability.

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5.2 chtMultiRegionSprayFilmFoam

Code Development

The development of the application chtMultiRegionSprayFilmFoam seemed to be aneasy task from the beginning but the extent of the task was slightly underestimated. Thebasic idea was to implement the already existing classes sprayCloud and surfaceFilminto the application chtMultiRegionFoam. However, since these classes are dependingon other classes and accompanying libraries which in turn are depending on some otherclasses, the extent of the implementation of sprayCloud and surfaceFilm became muchbigger.

The class sprayCloud is an extensive class with severel abilities. The capability to han-dle multi component treatment is one of those, which give the ability to handle severalchemical components such as water (H2O) and a mass fraction such as air (N2 + O2)in the parcels. Unfortunately the application chtMultiRegionFoam cannot handle thisin the ordinary fluid region. An additional solver for mass fraction, with accompa-nying classes and libraries, were therefore imperative to implement into the originalchtMultiRegionFoam application together with the handling of the spray. Thus, a lotof time was spent to make the code compile with these classes and also to troubleshootsimple spray simulations performed with this new application.

The class surfaceFilm had some surprising delimitations, i.e. the class is not able tohandle the multi component treatment either. A liquid film by a UWS (CO(NH2)2 +H2O) and a gas such as air (N2+O2) in the wall film region, need to have this multi com-ponent treatment. Unfortunately the class surfaceFilm is developed from separatelyderived transport equations with the purpose to handle two-dimensional flow with awall film thickness variable δf , as described in Section 2.4. The additional solver formass fraction, which was implemented in the chtMultiRegionFoam application, cannotwork with the surfaceFilm class in its original design. In other words, the surfaceFilmclass together with accompanying classes and libraries, need to be modified to implementmulti component treatment in the wall film. Due to the extent of this implementationand modification, this is not concerned in this thesis and is left to another OpenFOAMdeveloper.

Because of this delimitation in the surfaceFilm class the approach for this degree projecttook a different path than expected in the objective of the thesis in Section 1.3. Theresults in Chapter 4 are only for a wall film containing urea or water instead of a UWSand the buffer layer is imperative to use to calculate a HTC.

Another way to model a wall film in chtMultiRegionFoam could be by skipping thewall film region and model the wall film directly inside the fluid region. It could proba-bly be done by using an already existing multiphase code in OpenFOAM together withchtMultiRegionFoam. This approach is probably more advanced and will require a lot

35

of knowledge about OpenFOAM’s object-oriented code structure. Another way couldbe by developing the surfaceFilm class to use the faces at the boundary for the fluidregion (which are abutting the solid region), instead of a separate wall film region. Sincethe surfaceFilm class calculate a two-dimensional flow with an own wall film thicknessvariable δf , this approach may work by using the faces and not the cells in the fluidregion. However, the extent of this approach is uncertain and will probably also requirea lot of knowledge about OpenFOAM.

Thus, the simplest way to continue this degree project is to continue the code develop-ment by adding multi component treatment to the surfaceFilm class and accompanyingclasses. This means that the wall film can handle a UWS, and the HTC can be calcu-lated without the buffer layer. After this implementation it is possible to extend thisresearch about NOx reducing after treatment system by adding the chemical reactionsand associated algorithms to the simulations by the CHEMKIN package in OpenFOAM.

The assumption in Section 3.2.3 regarding that the liquid thermal conductivity Kliq isactivated in the cells where the wall film thickness is less than 10−6 m, is not deter-mined from previous research and should be investigated more before further usage ofchtMultiRegionSprayFilmFoam. This can easily be changed in the code for the imple-mentation of Kliq and deltaMap in appendix B.2. The assumption regarding that thethermal conductivity Kliq is set to a constant value by the user, should also be consid-ered with caution. A big factor for change in thermal conductivity is temperature andsince the wall film do have big changes in temperature, the thermal conductivity for theliquid should be changed as well during the simulation. A way to circumvent the usageof a constant value for the Kliq variable, is to link Kliq to the thermal conductivityfunction K for the specific liquid in the liquid properties library.

The Buffer Layer

The optimal approach to use the buffer layer is to make the buffer layer region closedwith no inlet and outlet. This makes the heat transfer between the formed liquid filmand the solid go directly trough the fluid in the buffer region. If there is a flow in thebuffer layer, this flow will make a heat exchange with the solid or liquid film and goout through the outlet causing energy losses, which imply an incorrect heat transfer tothe liquid film. Unfortunately if the buffer region is closed, the pressure will increaseunhindered due to the raise of temperature, which will result in numerical instabilityand divergence in the simulations.

In the other hand, the HTC between the buffer layer and the solid is calculated fromthe turbulent boundary layer created by the flow inside the buffer layer. Because of thepartially formed liquid film, there will be places where there is no liquid film, and theHTC in the buffer layer will be used to calculate the heat transfer between the solid andthe faces in the wall film region which do not have a formed liquid film. If there is adifferent flow in the buffer layer than in the air region, the HTC will be calculated to a

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different value than if both the buffer layer and the fluid region had the same flow. Thisimplies that the buffer layer need to have the same flow properties as the fluid regionto calculate the correct HTC between the air and the solid. In other words, there is atradeoff between the amount of energy losses from the outlet in the buffer layer that canbe accepted and the accuracy of the calculated HTC between the buffer layer and thesolid.

Time step

In a three-dimensional heat transfer the time step could be long due to the slow heattransfer inside the solid. Thus, in cases when spray and CHT are coupled, the simula-tions need to be calculated over a long time interval to be able to see the change in heatdistribution in the solid and with a small time step to be able to solve the spray, whichrequires a lot of calculation time just to calculate a simple case.

The automatic adjustment for the time step in OpenFOAM is changed with respect tothe courant number which itself is changed by the velocity of the flow. Thus, the spraywill not affect the time step until the spray parcels affecting the fluid, i.e. the adjust-ment of the time step has a small delay with respect to the spray. To compensate this,OpenFOAM uses several input parameters to make a guess for the next time step. Butthis guess could sometimes be large with respect to the spray which makes the sprayparcels travel far in the simulation domain within one time step.

It was hard to get convergence with the automatic time step during the simulationswith a urea spray. This was because the temperature in the fluid domain had hugefluctuations when the time step was relatively large. When a spray parcel travel in thedomain within a large time step it may travel through several cells. If a heat transferthen occurs to the parcels from the fluid, the energy will be taken from the nearest cells.In this case if the parcels has a large mass of a liquid with high specific heat Cp (e.g.water or urea), and the environmental fluid has a small specific heat Cp (e.g. air), it ispossible that the temperature in the nearest cells make a huge drop to compensate forthe instant energy transfer of a big amount of energy to the parcels, because of the largetime step. This could possibly be the cause of the numeric instability and divergencein the simulations. In other words, the time step needs to be small to be able to geta stable behavior in the simulations during the vaporization of the spray, because thevaporization will demand a lot of energy.

Results

The boundary against the wall film in the fluid region is set to slip condition in thesimulations as can be seen in Table 3.2. Before the liquid film is formed, this boundaryshould have a zero velocity because it is against a wall. After the liquid has been injectedinto the wall film region (by the spray) the boundary should have the same velocity as forthe liquid film. In other words, this boundary condition should be mapped to the wall

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film region by the mapped condition instead since both of these scenarios will be takeninto account by this boundary condition. Since the results in Chapter 4 are focusedon the formation of the liquid film and the heat distribution inside the solid, the slipcondition will not have a significant effect on these results. However, the heat transferfrom the fluid region to the wall film region could have a small difference due to thechange in velocity in the boundary layer for the fluid region.

It is found that the water spray form a more extended wall film at the surface of thesolid because of the less viscosity. In Fig. 4.6 at z = 0.03 m for 1.5 s it is possible tosee the more spread cooling of the wall film with water than in the case with urea. Thisindicates a relationship between the heat distribution inside the solid and the extent ofthe wall film. The heat distribution and gain of energy in the wall film is most likelyalso depending on the extent of the wall film. In Table 4.1 it is possible to see this effectbecause the gain of energy for the wall film with water is more than twice as much as forthe case with urea, which probably is because of the wider surface of the wall film andcan imply a faster energy transfer to the wall film. The change of energy conservationfor the solid is almost the same in all cases according to Table 4.1. This is most likelydue to the fact that the thermal conductivity Kliq in the buffer layer is set to the samevalue in all cases, i.e. the thermal conductivity for water.

5.3 General Comments

The Thermodynamics of Urea and Water in a UWS

Due to the different specific heat capacity Cp and density ρ for water and urea in Ta-ble 2.1 the energy conservation for water is almost twice as high than for urea. Thus,in order to vaporize a UWS with 32.5 wt % urea there will for that reason be a higherenergy transfer to the water part than to the urea.

With reflection to the above conclusions of the heat distribution inside the solid andthe thermodynamics of urea and water, it is possible to conclude some things of thevaporization of UWS in the after treatment system:

The viscosity for a UWS is similar to water which implies a wall film formation similar towater. Because of the different thermodynamic properties for water and urea, the waterpart in the UWS will first vaporize leaving urea behind. The remaining urea will be aviscous liquid which means a slow advancement of the wall film. With respect to thisslow advancement and the thermodynamic properties of urea, the energy transfer fromthe solid is small implying an energy transfer direct from the exhaust gases to the wallfilm. For that reason, this is an energy source that should not be neglected in furtherresearch. However, if there is a heat generation inside the solid or at the other side of thesolid (which enable a heat transfer through the solid), this will still be the main energysource for the vaporization of the remaining urea.

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Chapter 6

Conclusion

There he goes. One of God’s own prototypes. Some kind of high-powered mutant nevereven considered for mass production. Too weird to live, and too rare to die. . .

- Hunter S. Thompson, Fear And Loathing in Las Vegas

This chapter present the general conclusion for the development of the applicationchtMultiRegionSprayFilmFoam and for the results from the test case simulations. Inthe end there will be some suggestions for future work which will be divided into twoseparate parts, one for further research of urea viscosity and the developed applicationchtMultiRegionSprayFilmFoam and one for continued development of the application.

General Conclusions

• The developed application chtMultiRegionSprayFilmFoam is working properlybut to have its fully potential it is recommended to follow the continued develop-ment on the next page.

• Before the application is used for real industrial cases, such as a real after treatmentsystem, it is most imperative to verify the accuracy of the given results withexperimental results or with another CFD software.

• The viscosity for urea presented in this thesis still behaves as a fluid at low tem-perature and should behave more as a solid. A further investigation of a moreviscous urea will therefore be recommended. However, the approach of modelingurea as a liquid with relatively high viscosity is a good method to simulate theenergy transfer from a solid to a liquid film of urea.

• The energy loss via the buffer layer can be neglected, but will still be an additionalsource of error. In cases with higher temperature gradients or if the transientsimulation will be calculated for a longer time interval, the energy loss from thebuffer layer should be investigated further to ensure that it can be neglected.

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• The time step during spray simulations with chtMultiRegionSprayFilmFoam shouldbe small to ensure numerical stability during the spray injection and the vaporiza-tion of the spray.

Future Work

Further research:

• Produce the real viscosity for the melted chemical compound obtained when ureahas molten slightly above TUreaMelt. This value can then be used to attach theT − µ curve to one correct value of viscosity at the temperature TUreaMelt.

• Modify the test case to have the same dimensions and properties for the solid andthe same conditions for the fluid and the spray, as for a real after treatment sys-tem. Then, implement a new fluid representing the UWS presented in Section 2.1,into the liquid properties library of OpenFOAM. This UWS fluid can have thesame properties as for water except for the specific heat capacity Cp and the vaporpressure pvap. These properties shall be recalculated with respect to the thermo-dynamic properties of water and urea. The new test case with the UWS liquidshould more accurately calculate the energy distribution inside the solid and thewall film, with respect to a real after treatment system.

Continued development:

• Develop the class surfaceFilm to handle a multi component mixture. This willmake it possible to specify a liquid film with a UWS (CO(NH2)2 + H2O), as inthe real after treatment system.

• Continue the development of the class surfaceFilm to handle a gas as an initialfluid in the wall film region, such as air (N2 + O2) and all its thermodynamicproperties. This makes it possible to calculate a HTC for the solid, before the wallfilm liquid has entered the wall film region and also meanwhile the liquid film isadvancing along the surface at the solid.

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Bibliography

[1] Scania AB official website, May 2012. scania.com/.

[2] European legislation, May 2012. europa.eu/legislation−summaries/.

[3] European motor news, May 2012. europeanmotornews.com/2010/03/01/euro-6-standards-pose-a-tough-challenge-for-truck-manufacturers/.

[4] A. Lundström. Urea Decomposition for Urea-SCR Applications. PhD thesis,Chalmers University of Technology, Göteborg, Sweden, 2010.

[5] OpenFOAM. Liquid Properties library, 2.1.0 edition.

[6] OpenFOAM, July 2012. openfoam.com/.

[7] P. M. Schaber, J. Colson, S. Higgins, D. Thielen, B. Anspach, and J. Brauer. Ther-mal decomposition (pyrolysis) of urea in an open reaction vessel. ThermochimicaActa, 2004.

[8] M. Koebel, M. Elsener, and M. Kleemann. Urea-scr: a promising technique toreduce nox emissions from automotive diesel engines. Catalysis Today, 2000.

[9] M. Karlsson, K. Storck, I. Andersson, and D. Loyd. Formelsamling i termo- och flu-iddynamik. Mekaniksystem, Instutionen för konstruktions- och produktionsteknik,2007. LiTH-IKP-S-508.

[10] H. K. Versteeg and W. Malalasekera. An Introduction to Computational Fluid Dy-namics, The Finite Volume Method. Pearson Education Limited, Harlow, England,second edition, 2007. ISBN 978-0-13-127498-3.

[11] N. Nordin. Complex Chemistry Modeling of Diesel Spray Combustion. PhD thesis,Chalmers University of Technology, Göteborg, Sweden, 2001.

[12] OpenFOAM Foundation. Wall film modelling: Implementation notes, January 2010.Not for redistribution.

[13] J. P. Holman. Heat Transfer. McGraw-Hill, New York, USA, tenth edition, 2010.ISBN 978-007-126769-4.

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http://scania.com/http://europa.eu/legislation_summaries/http://europeanmotornews.com/2010/03/01/euro-6-standards-pose-a-tough-challenge-for-truck-manufacturers/http://europeanmotornews.com/2010/03/01/euro-6-standards-pose-a-tough-challenge-for-truck-manufacturers/http://openfoam.com/

[14] National institute of standards and technology, May 2012. nist.gov/srd/.

[15] OpenFOAM Foundation. OpenFOAM, The Open Source CFD Toolbox - UserGuide, 2.1.0 edition, December 2011.

[16] OpenFOAM Foundation. OpenFOAM, The Open Source CFD Toolbox - Program-mer’s Guide, 2.1.0 edition, December 2011.

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http://nist.gov/srd/

Appendix A

Modeling Liquid Urea

A.1 Implementation Parameters

Implementation parameters to NSRDS function no. 1.

A B C D E

water -51.964 3670.6 5.7331 -5.3495e-29 10

urea 58.750501872 -0.962447508 -10.7315227328 -1.995911999 -2.045859082

Table A.1: Implementation parameters to function 3.1 for T-µ relationship. Parametersfor water are from the liquid properties library in OpenFOAM [5] and parameters forurea are determined with appendix A.2.

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A.2 Matlab Code for Dynamic Viscosity of Urea

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%% Program to approximate the dynamic v i s c o s i t y f o r molten urea with help% from the dynamic v i s c o s i t y f o r water . Urea s t a r t s to melt /decompose at% 406K and th e r e f o r e the v i s c o s i t y w i l l be very high at t h i s temperature .% Emil S j o l i n d e r% 2012−02−20%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%

% Melting temperature f o r ureaT urea melt =133;% Overshoot va lue to get the d e s i r a b l e dynamic v i s c o s i t y f o r ureaovershoot =23;% Temperature range f o r water and ureaTw=(250:444) ;Tu=(250+T urea melt+overshoot ) :(444+ T urea melt+overshoot ) ;

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%% Dynamic v i s c o s i t y f o r water , parameters from l i q u i d p r op e r t i e s l i b r a r y% f o r OpenFOAM v2 . 1 . 0aw = [−51.964 3 .6706 e3 5 .7331 −5.3495e−29 1 0 ] ;mu w = nsrds func101 (aw ,Tw) ;

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−%% Dynamic v i s c o s i t y f o r water and d e s i r a b l e dynamic v i s c o s i t y f o r ureaf i g u r e (1 )hold onp l o t (Tw−273 ,mu w) ;p l o t (Tu−273 ,mu w, ’m’ ) ;t ex t (−3 ,3.7 e−3 ,0 , ’ T W a t e r M e l t ’ ) ;t ex t ( 130 , 3 . 7 e−3 ,0 , ’ T U r e a M e l t ’ ) ;l egend ( ’Water ’ , ’ Urea , d e s i r a b l e ’ ) ;p l o t ( 0 , 0 : 0 . 0 0 0 0 1 : 3 . 6 e−3) ;p l o t (406 −273 ,0 :0 .00001 :3 .6 e−3) ;x l ab e l ( ’T [ ˆoC ] ’ ) ;y l ab e l ( ’ \mu [ kg/ms ] ’ ) ;

%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−�

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