modeling of passive chilled beams for use in …...modeling of passive chilled beams for use in...

Modeling of Passive Chilled Beams for Use in

E�cient Control of Indoor-Air Environments

Samantha H. Erwin

Thesis submitted to the Faculty of theVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Sciencein

Mathematics

Je↵rey T. Borggaard, ChairTerry L. HerdmanLizette Zietsman

June 7, 2013Blacksburg, Virginia

Keywords: Computational Fluid Dynamics, ANSYS Fluent, Chilled BeamsCopyright 2013, Samantha H. Erwin

Modeling of Passive Chilled Beams for use inE�cient Control of Indoor-Air Environments

Samantha H. Erwin

(ABSTRACT)

This work is done as a small facet of a much larger study on e�cient control of indoor airenvironments. Halton passive chilled beams are used to cool rooms and the focus of this workis to model the beams. This work also reviews the mesh making process in Gmsh. ANSYSFluent was used throughout the entire research and this thesis describes the software and acareful description of the case study.

Acknowledgments

I would like to thank my advisor, Dr. Je↵ Borggaard for his support and guidance over thepast two years as well as his wife Dr. Lizette Zietsman. Thanks to Dr. Terry Herdman forhis support to begin my studies at Virginia Tech and Dr. Maeve McCarthy for sparking myinterest in mathematical research. Additionally, the entire ICAM family that has created asupport network that encouraged me along the way.

Also, I would like to thank all of my many friends whose unwavering support helped mestay sane. Finally, my family has been a constant source of support, encouragement, andinspiration during my entire life, academic career, and graduate school. My gratitude to myparents, Dwight and Terri Erwin, and my grandmother, Virginia Bartlett.

iii

Contents

List of Tables vii

List of Figures ix

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Energy E�cient Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Halton Chilled Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Problem Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Model Equations 6

2.1 Boussinesq Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Variables and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.4 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.5 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 GMSH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3 Exporting the Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 ANSYS Fluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

iv

2.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.4 Cell Zone Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.5 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.6 Dynamic Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.7 Reference Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.8 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.9 Monitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.10 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Numerical Example 21

3.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 ANSYS Fluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Problem Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Mass Flow Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Static Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.3 Static Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.4 Cooling Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Conclusion 51

Bibliography 52

A GMSH Source Code 53

B ANSYS Fluent Settings 64

B.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

v

B.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

vi

List of Tables

2.1 Variables Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Mesh Quality of Varying Beam widths . . . . . . . . . . . . . . . . . . . . . 26

3.2 Average Mass Flow Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Average Static Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Average Static Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Area Weighted Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Cooling Capacity of Varying Beam Widths . . . . . . . . . . . . . . . . . . . 49

B.1 General Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

B.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.3 Viscous Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.4 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

B.5 Cell Zone Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

B.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

B.7 Boundary Conditions Continued . . . . . . . . . . . . . . . . . . . . . . . . . 68

B.8 Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

B.9 Reference Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

B.10 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

B.11 Solution Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

B.12 Monitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

B.13 Solution Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

vii

B.14 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

viii

List of Figures

1.1 Depiction of a Simulated Airflow in a Room [10] . . . . . . . . . . . . . . . 3

1.2 Sketch of a Halton Passive Beam . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Sketch of a Slice of the Passive Chilled Beam . . . . . . . . . . . . . . . . . . 5

3.1 Height View of a Halton Passive Beam . . . . . . . . . . . . . . . . . . . . . 22

3.2 Geometry of Passive Chilled Beam . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Comparison View of The Halton Passive Beam Diagram and the Geometry . 23

3.4 Entire Geometry of Mesh for the Passive Chilled Beam . . . . . . . . . . . . 24

3.5 3D Mesh of the Slice of a Halton Passive Chilled Beam . . . . . . . . . . . . 24

3.6 3D Geometry of a Widened Slice of a Halton Passive Chilled Beam . . . . . 25

3.7 3D Model of a Halton Passive Chilled Beam . . . . . . . . . . . . . . . . . . 25

3.8 Velocity Vectors of Slice with Width 440mm . . . . . . . . . . . . . . . . . . 29

3.9 Velocity Vectors of Slice with Width 80mm . . . . . . . . . . . . . . . . . . . 30

3.10 Static Temperature of Slice with Width 440mm . . . . . . . . . . . . . . . . 31

3.11 Static Temperature of Slice with Width 80mm . . . . . . . . . . . . . . . . . 32

3.12 Mass Flow Rate Data for the Halton Passive Chilled Beam at 400mm . . . . 33




3.16 Mass Flow Rate Data for the Halton Passive Chilled Beam at 80mm . . . . . 37

3.17 Static Temperature Data for the Halton Passive Chilled Beam at 440mm . . 39

ix




3.21 Static Temperature Data for the Halton Passive Chilled Beam at 80mm . . . 43

3.22 Static Pressure Data for the Halton Passive Chilled Beam at 440mm . . . . 44




3.26 Static Pressure Data for the Halton Passive Chilled Beam at 80mm . . . . . 48

x

Chapter 1

Introduction

1.1 Introduction

In 2009, 39% of the energy used in the United States was for operating buildings, whileonly 28% of the energy used was for transportation. These statistics alone are motivation todecrease the energy used in buildings. However, add in that if we can reduce the energy usagein buildings by 50% it would be the same energy savings as removing every passenger vehicleand small truck from the road, and the motivation to reduce energy usage by buildings isevident [1].

A large part of the energy used in buildings is for climate control. In many other facets of lifecomputational tools have been used to improve quality; however, they have not been usedto the fullest in the design and control of buildings. As a step in this direction, we attemptto model and simulate the use of modern climate control devices, such as chilled beams,as a means to control the indoor air environment in buildings. However, direct numericalsimulation of all of the intricacies of these control mechanisms would make the simulationintractable. Thus a simplified model of the Halton passive chilled beam is developed from acombination of energy balances and computational fluid dynamics.

Chilled beams contain pipes through which temperature controlled water flows. The beamsare typically placed near the ceiling of the room where hot air accumulates. The beam chillsthe air around it causing it to become denser; hence the cooled air falls and is replaced bywarmer air that circulates through the chilled beam [14]. There are companies that produceboth active and passive chilled beams, however this research focuses on the Halton passivechilled beam.

The primary purpose is to model the Halton chilled beam in such a way that it can beincorporated into a large room model. To model the fluid dynamics forces produced bychilled beams we used ANSYS Fluent to simulate the flow and Gmsh to mesh a representative

1

flow domain.

1.2 Energy E�cient Buildings

In the past ten years the codes for new buildings have responded to the push for more energye�ciency. The cost to produce a more e�cient building is greater than a normal buildinghowever, users typically see payback within the first five years [7]. Energy can be saved byadjusting lighting, heating, cooling, and the general building process.

This research is a small part of a larger project to model and control energy usage in buildings.The overall project models the entire building as a complex system and seeks energy e�ciencyby developing control strategies that utilize more building physics. These coupled modelsmust include the building envelope, the indoor air environment, and models for controldevices. A portion of this project is to model and control the indoor air environment inthe Purdue Living Laboratory. One room in this laboratory can be seen in Figure 1.1.In this model room the cool air is falling from chilled beams in the ceiling of the room.However, in this model used constant forcing terms for the chilled beams that were takenfrom product literature. The goal of this research is to develop a better understanding ofthe cooling capacity of the chilled beams by using computational fluid dynamics to simulatethe flow through these devices. We hope that a better understanding of the fundamentalmechanisms of these devices will lead to better simulation and control of the cooling processin the room. For this study Halton passive chilled beams are used, which are designed toprovide competitive life cycle costs as well as comfort.

1.3 Halton Chilled Beams

Halton o↵ers two distinct types of chilled beams: Active and Passive. The active chilledbeams provide both heating and cooling while ventilating [2]. The ventilation applicationuniformly mixes the air to improve the air quality. Alternatively, the passive beam o↵ersno ventilation or heating capacities. Rooms are cooled by free convection that occurs whenchilled water is circulated through the pipes [10].

The passive chilled beam system is designed to fulfill requirements of sustainable energye�cient buildings. These beams provide excellent indoor conditions both acoustically andthermally. Chilled water circulates through the passive chilled beam and convection coolsthe room. The room is heated with a completely di↵erent system, unlike the active chilledbeam which was mentioned earlier. Ventilation is provided through separate wall or ceilingdi↵users, and occasionally floor di↵users [11].

The passive chilled beam is composed of a coil that doubles through the beam. Chilledwater circulates through the coils at a temperature four to 11 degrees celsius less than that

2

Figure 1.1: Depiction of a Simulated Airflow in a Room [10]

3

Figure 1.2: Sketch of a Halton Passive Beam

of the room. The temperature di↵erence between the room and the chilled water is closelymonitored to prevent condensation [10]. In Figure 1.2 the bottom side of the passive chilledbeam is shown and the coils can be clearly seen. It also contains fins that cross throughoutthe beam. These fins generate more surface area to remove thermal energy from the air,however they also create a drag on the circulation of air that e↵ects the cooling capacity ofthe beam.

For this particular project the widest passive chilled beam (615mm) will be modeled. Thecasing height is 130mm with a coil height of 100mm, and a coil diameter of 15mm. Thepassive beam is installed fully exposed and is to be hung at least 0.25 times the width ofthe beam. Thus this beam will be hung a minimum of 153.75mm below the ceiling [10]. Formodeling purpose the beam will be enclosed in an arbitrary domain to see the full convectionand flow pattern. The general idea behind the model will be discussed next.

1.4 Problem Outline

As previously mentioned the goal is to better understand the cooling capacity of the Haltonpassive chilled beam. The numerous fins that run throughout the beam cause drag and makethe model a challenge. In order to model this problem we will look at only a small slice ofthe beam as seen in Figure 1.3 outlined in red. This slice has half of a fin on either side withthe goal to be for the model to incorporate the drag as flow moves through the slice.

Simulations were run in ANSYS Fluent, a computational fluid dynamics software with broadphysical modeling capabilities. ANSYS Fluent requires a mesh of the slice an we createdthis using Gmsh, a three dimensional finite element mesh generator. A deeper explanationof Gmsh is given in Section 2.2 and then later the actual mesh generation process used

4

Figure 1.3: Sketch of a Slice of the Passive Chilled Beam

for the model will be discussed. ANSYS Fluent is further explained in Section 2.3. Adetailed explanation of the specific case study will be discussed as well. Due to the thinnessof the object being modeled, the slice was expanded in order to complete computations.Solutions were calculated with the slice expanded to varying widths. The mass flow rate,static temperature, and static pressure were computed on the inflow and outflow boundariesof the air. For each width the cooling capacity was calculated and compared to the dataprovided by Halton about their passive chilled beam.

5

Chapter 2

Model Equations

In this chapter the background behind the formulation of the model is discussed. The modelequations are derived and the software used is discussed. This includes in-depth descriptionsof Gmsh and ANSYS Fluent.

2.1 Boussinesq Equations

Navier-Stokes equations are used to describe flow of viscous fluids. These equations are acoupled system of equations that give the relationship between velocity (u), pressure (p),temperature (T ), and density (⇢). Although the equations were derived in the 19th century,the existence and smoothness of their solutions is still unproven. The Navier-Stokes equationsare derived from the equations describing conservation of mass, conservation of momentum,and conservation of energy.

The Boussinesq equations couple the energy equation to the momentum equation by relatingtemperature fluctuations to a body force that acts in the direction of gravity. Assume thedensity variations are negligible as well as the dissipation of mechanical energy in the energyequation. Also, assume that viscosity (µ), thermal expansion (�), thermal conductivity (),and specific heat at a constant pressure (c

p

) are all themselves constant [9]. The result is theBoussinesq model, a system of three equations whose derivation is carried out at the end ofthis section:

r · u = 0,

⇢

Du

Dt

= ⇢F�rp+r · ⌧ + �g(T � T0),

⇢c

p

DT

Dt

= ⇢f � pr · u+ �+r · (rT ),

6

Table 2.1: Variables Used

Variable Description

@⌦ boundarye energyF force, with components F

i

f heat fluxm massT stress tensor, with components T

ij

R surface forces with components Ri

t time⌦ volume

where F is an additional body force per unit volume, f is a thermal source term per unitvolume, and D/Dt is the total time derivative.

Next, the variables used are defined and then the derivations of conservation of mass, momen-tum and energy. Throughout the derivations Henningson and Berggren’s, Fluid Dynamics:Theory and Computation [12] is used as a reference.

2.1.1 Variables and Theorems

For convenience, we provide a list of additional variables used in the derivation of the Boussi-nesq equations in Table 2.1.

Let ⌦ ⇢ R3 be a region in space with boundary @⌦. The volume integral of the divergence,r ·F over ⌦ is related to the surface integral of F over the boundary @⌦ by the DivergenceTheorem:

Z

⌦

(r · F)d⌦ =

Z

@⌦

F · d(@⌦).

2.1.2 Incompressible Flows

Incompressible flows are flows in which the density of the material is constant within thefluid continuum. That is, changes in fluid pressure do not a↵ect density. Although no fluidis truly incompressible, it is considered incompressible when the speed of the flow is muchless than the speed of sound. In the case of incompressible flows the mass and momentumequations can be used to solve for pressure and velocity.

7

2.1.3 Conservation of Mass

Let the mass density of fluid at some point be ⇢ = ⇢(x1, x2, x3, t) with a nearly uniformdensity throughout some arbitrary control volume, denoted ⌦. We assume that there are nosinks or sources of mass within the volume and the fluid has a fixed, oriented boundary @⌦.The total mass of the fluid volume can be represented as

Z

⌦

⇢dV = m.

The rate of change of mass over time isZ

⌦

@⇢

@t

dV =@m

@t

. (2.1)

The only way the total mass can change is by particles flowing across the boundary, thusZ

@⌦

⇢u · ndS =@m

@t

.

Where n denotes the unit vector normal to the surface, that is n points out of the volumeelement when @⌦ is a side of a volume element ⌦. Since u denotes fluid velocity, ⇢u is theflux of mass passing through the surface.

By the divergence theorem we can transform the surface integral to a volume integral toobtain

�Z

⌦

r · (⇢u)dV =@m

@t

. (2.2)

Since (2.1) and (2.2) are equal we have:

Z

⌦

@⇢

@t

dV = �Z

⌦

r · (⇢u)dV.

This can be manipulated to showZ

⌦

@

@t

⇢dV +

Z

⌦

r · (⇢u)dV = 0Z

⌦

@

@t

⇢+r · (⇢u)dV = 0 (2.3)

@

@t

⇢+r · (⇢u) = 0.

8

The last equation is obtained by noting that (2.3) holds for any arbitrary volume ⌦. Thus,the integrand must vanish, which produces the conservation of mass equation. However, in

the special case of incompressible flows the density ⇢ is a constant. Thus@⇢

@t

= 0 and we

have

r · u = 0.

2.1.4 Conservation of Momentum

The principle idea behind conservation of momentum is that the rate of change of momentumon a region of fluid equals the sum of forces on that region. Note that the common form ofNewton’s second law, F = Ma is actually a simplification of the relationship f = D

Dt

(⇢v).Let the body forces per unit mass be F

i

and the surface forces per unit volume be R

i

.

This statement in integral form is written as,

D

Dt

Z

⌦

⇢u

i

dV =

Z

⌦

⇢F

i

dV +

Z

@⌦

R

i

dS.

Which can be written asZ

⌦

⇢

D

Dt

u

i

dV =

Z

⌦

⇢F

i

dV +

Z

@⌦

R

i

dS. (2.4)

Which, more simply, mass times acceleration is equivalent to the sum of all forces. The totalsurface force, R

i

, can be written in terms of the components of the stress tensor, Tij

suchthat the right hand side becomes

Z

⌦

⇢F

i

dV +

Z

@⌦

R

i

dS =

Z

⌦

⇢F

i

+@T

@x

j

dV. (2.5)

Then substituting (2.5) into (2.4) leads toZ

⌦

⇢

D

Dt

u

i

dV =

Z

⌦

✓⇢F

i

+@T

@x

j

◆dV

Since this holds for any volume, the integrand vanishes and leads to

⇢

Du

Dt

= ⇢F+r · T.

The tensor, T is symmetric and typically decomposed as

T

ij

= �p�

ij

+ ⌧

ij

,

9

where �p�

ij

is the thermodynamic pressure and ⌧

ij

is the viscous stress tensor. This leadsto the conservation of momentum equation

⇢

Du

Dt

= ⇢F�rp+r · ⌧.

2.1.5 Conservation of Energy

The energy equation is derived from the second law of thermodynamics which states thatenergy can be neither created nor destroyed. Thus the rate of change of the total energyequals the rate energy enters and exits plus internal energy sources. First, begin withdefinitions,

⇢[e+1

2u

i

u

i

] energy of a particle per unit volume, with e the internal energy

⇢u

i

F

i

work rate per unit volume of Fi

on a particle

n

i

u

i

T

i,j

work rate per unit area of a surface T

i

on a particle

n

i

q

i

heat loss through the surface per unit area

⇢f energy generated per unit volume.

Using the definitions and the second law of thermodynamics stated above the equation iswritten as,

D

Dt

Z

⌦

⇢[e+1

2u

i

u

i

]dV =

Z

⌦

[⇢ui

F

i

+ ⇢f ]dV +

Z

@⌦

[ni

u

i

T

i,j

� n

i

q

i

]dS.

Then apply the Divergence Theorem as seen in the conservation of mass and conservationof momentum sections to obtain,

D

Dt

Z

⌦

⇢[e+1

2u

i

u

i

]dV =

Z

⌦

[⇢ui

F

i

+ ⇢fu

i

T

i,j

� q

i

]dV.

The integrals vanish as previously discussed, and this leads to,

D

Dt

[e+1

2u

i

u

i

] = ⇢u

i

F

i

+ ⇢f +@

@x

i

(Ti,j

u

i

� q

i

). (2.6)

Now as before, separate the stress tensor,

T

i,j

= �p

i,j

+ ⌧

i,j

where p is the thermodynamic pressure and ⌧

i,j

is the viscous stress tensor. Then substitutethis into (2.6) to obtain

D

Dt

[e+1

2u

i

u

i

] = ⇢u

i

F

i

+ ⇢f +@

@x

i

(pui

+ ⌧

i,j

u

i

� q

i

)

= ⇢u

i

F

i

+ ⇢f +@

@x

i

(pui

) +@

@x

i

(⌧i,j

u

i

)� @q

i

@x

i

. (2.7)

10

Thus the change in total energy is given by (2.7) and is equal to the thermal energy plusthe mechanical energy. To find the equation for mechanical energy we use the momentumequation multiplied with velocity. Hence

u · (⇢Du

i

Dt

= ⇢F

i

+@T

i,j

@x

i

),

which implies

⇢

D

Dt

(1

2u

i

u

i

) = ⇢F

i

u

i

+�u

i

@p

@x

i

+ u

i

@⌧

i,j

@x

i

.

Then using the fact that the following relationships holds,

� @

@x

i

(pui

) = �p

@u

i

@x

i

� u

i

@p

@x

i

� @

@x

i

(⌧i,j

u

i

) = �⌧

i,j

@u

i

@x

i

� u

i

@⌧

i,j

@x

i

,

we subtract the mechanical energy from the total energy to find the thermal energy equation,

D

Dt

⇢e = ⇢f � p

@u

i

@x

i

+ ⌧

i,j

@u

i

@x

i

+@q

i

@x

i

. (2.8)

If we relate the heat flux to the temperatures gradients using Fourier’s law, we have

q

i

= �

@T

@x

i

. (2.9)

Substituting (2.9) into the thermal equation, (2.8), to obtain,

⇢

De

Dt

= ⇢f � p

@u

i

@x

i

+ ⌧

i,j

@u

i

@x

i

+@

@x

i

(@T

@x

i

). (2.10)

Let the positive definite dissipation function � be defined as

� = ⌧

i,j

@u

i

@x

i

.

Then substitute the positive definite dissipation function, � into (2.10), yields

⇢

De

Dt

= ⇢f � p

@u

i

@x

i

+ �+@

@x

i

(@T

@x

i

).

Let e = c

p

T , to obtain the following equation known as the conservation of energy equation,

⇢c

p

DT

Dt

= ⇢f � pr · u+ �+r · (rT ).

11

2.2 GMSH

Gmsh was used to create a mesh of the Halton passive chilled beam. Gmsh is a free andopen-source software that is a 3D finite element grid generator. Gmsh o↵ers a friendly userinterface but also allows users to upload text files or geo files that can be read to generatethe geometry [6].

2.2.1 Geometry

The first step in mesh generation is to build the geometry which begins by defining points.For this research all points were defined in a text file. Points are denoted using the format

Point(101) = {1, 1, 1, 0.1};

where the first three numbers are the x, y, z coordinates of the point, respectively and thefourth coordinate is the prescribed mesh element size. Note that for Gmsh to read the lineit must end with a semi-colon. Next lines are connected between the points. For example:

Line(201) = {101, 102};

where the line connects the two endpoints, or

Circle(301) = {103, 104, 105};.

The arc of a circle is formed using the two endpoints, with the center point being listedsecond. Next Line Loops are defined to create surfaces. The orientation of the line loopsis extremely important. The direction the line is pointing is important. If two lines areoriented toward each other a negative sign can be used to flip the orientation. Also, thesurface is oriented outward using the right hand rule. Line loops are simply denoted as

Line Loop(401) = {201, 202, -203, 204};.

Once the line loops are created, surfaces can then be defined as either a ruled surface or aplane surface. For this problem only plane surfaces were used. Ruled surfaces are surfacesthat can be interpolated using transfinite interpolation, which is not needed when usingANSYS Fluent to process the mesh. To declare a surface simply number the line loop, i.e.:

Plane Surface(401) = {401};

Then similar to the line loops, surface loops need to be declared to form a shell to define avolume. When looping surfaces together it is important that all of the surface’s orientationsare consistent, a negative sign may be used to orient the surface in the opposite direction.Surface orientation is directly correlated with the line orientation discussed previously.

Surface Loop(501) = {401, 402, -403, 404};

12

Finally volumes are created from the surface loops. The first item listed is the surface loopthat defines the exterior of the volume and all other items create holes inside the surfaceloop, for example:

Volume(601) = {501, 502, 503};.

Thus this volume has the exterior defined by the surface loop 501 and holes occurring at502 and 503. Before generating the mesh or exporting the file it is important to first definethe physical surfaces. Physical surfaces group elementary surfaces together so that the sameboundary condition can be set throughout. Below a sample Physical Surface is declared.

Physical Surface(701) = {401, 402, 409, 410};.

2.2.2 Mesh Generation

Gmsh o↵ers several di↵erent unstructured algorithms for mesh generation. In two dimensionsthere are three unstructured algorithms for mesh generation: MeshAdapt, Deluanay, andFrontal. However, this work focuses only in 3dimensions, and there are two unstructuredalgorithms for this, Delaunay and Frontal.

The “Delaunay” algorithm first creates an initial mesh of the union of all volumes in themodel using Si’s Tetgen algorithm [6]. Si’s Tetgen algorithm is a mesh generator that par-titions 3D geometry into tetrahedrons. Then the 2D “Delaunay” algorithm triangulationis used. In the 2D Delaunay algorithm new points are inserted successively in the centerof the largest elements and then lines are reconnected using directionally dependent linesaccording to Delaunay criterion. The Delaunay criterion states that the local properties fortwo adjacent elements results in the global property for the entire triangulation [8].

J. Schoeberl’s Netgen algorithm is used in the “Frontal” algorithm in Gmsh. Netgen splitsthe task of mesh generation into four subproblems, special point calculation, edge following,surface meshing, and volume mesh generation. There are multiple ways to select specialpoints depending on the lines or curves that connect the points. For edge detection, aninitial point for every edge is choosen and then followed until the terminal point is reached.The advancing front method is then used for surface and volume mesh generation [15].

2.2.3 Exporting the Mesh

After the mesh is generated the file is saved using the I-deas Universal file format (.unv).When saving the file the option“Save groups of nodes” must be chosen. The other optionwill eliminate the physical groups created earlier in the geometry generation process. TheI-deas Universal file

When importing a .unv file to fluent the entire mesh is automatically parameterized asaluminum however this is easily changed.

13

2.3 ANSYS Fluent

The following is a brief mathematical description of all parameters worked with in ANSYSfluent. ANSYS Fluent is a fully featured fluid dynamics solution for modeling flow andother related phenomena [3]. See Appendix I for the exacting settings used in the numericalexamples.

2.3.1 General

To begin a new case file in ANSYS Fluent, the user must first import a new mesh. Usingthe import option a user can choose to import many di↵erent file types including and I-deasUniversal file. ANSYS Fluent also o↵ers their own mesh generation software, however it wasnot used in this work. In order to continue setting up the case, first check the mesh andthe mesh quality is reported. The mesh quality is analyzed by three factors: cell squish, cellskewness, and aspect ration. The cell squish is a measure of the non-orthogonally of a cellwith respect to its faces. Bad meshes are closer to one while good meshes are closer to zero.Cell skewness indicates degenerate cells by (optimal cell size - cell size)/(optimal cell size),where the optimal cell size is an equilateral cell with the same perimeter. The aspect rationis the (max distance from center)/(min distance from center).

Once the mesh is uploaded to ANSYS Fluent the user must then define the solver type,velocity, time, and gravity. There are two choices for solver type: pressure-based or density-based. The pressure based solver is used for incompressible flows and relates to the steady-state of the conservation of mass and conservation of momentum equations [4]

I⇢u · dS = 0

I⇢u · dS = �

IpI · dS +

I⌧ · dS +

Z

V

FdV.

These equations are the integral form of the equations derived in Section 2.1.3 and Section2.1.4.

The density based solver is used for high-speed compressible flows. Unlike the pressurebased solver, this solver couples the conservation of mass, momentum and energy equationstogether. Typically the density based solver requires many more iterations than the pressure-based solver [4].

The time condition must also be set as either steady-state or transient. Steady-state assumesconstant flow and inlet conditions. Transient time steps forward in time as the solution issolved [5]. The direction of gravity (if applicable) is also set in the initial part of the caseformulation.

14

2.3.2 Models

ANSYS Fluent has several di↵erent model types built into the program including energy,viscous, multiphase, radiation, heat exchanger, chemical species, discrete phase, solidificationand melting, and acoustics. The built in energy model when employed allows users to setheat transfer parameters by simply turning the energy model on.

Enabling the viscous model allows users to choose models for di↵erent viscosities. Thesemodels include flows where viscosity can be ignored, i.e. inviscid flows, laminar flows wherefluid particles flow nearly parallel to one another with no or minimal small eddies, andturbulent flows where there is chaotic behavior [4].

In our model, we consider the realizable k-epsilon model, which ensures that u2> 0. In

particular, the Boussinesq relationship (di↵erent from the Boussinesq approximation in theenergy equation) relates the Reynolds stresses to the eddy viscosity definition, µ

t

= ⇢C

µ

k

2

✏

(Cµ

is a constant),

u2 =2

3k � 2⌫

t

@U

@x

, (2.11)

where ⌫

t

⌘ µ

t

⇢, and u2> 0. Which leads to

0 <

2

3k � 2

µ

t

⇢

@U

@x

<

2

3k � 2C

µ

k

2

✏

@U

@x

<

1

3� C

µ

k

✏

@U

@x

.

Thus for the solution to be realizable,

k

✏

@U

@x

<

1

3Cµ

must occur. But when

k

✏

@U

@x

>

1

3Cµ

occurs, u2< 0 and the solution becomes unrealizable [4]. This is the principle idea behind

the realizable k-epsilon model used.

The other models, multiphase, radiation, heat exchanger, species, discrete phase, solidifica-tion and melting, and acoustics. are not directly used in the numerical example in Chapter3, however a brief description is included of each. Multiphase flows are those where di↵erent

15

materials are not chemically related and droplets or bubbles form. Radiation models areused for radiation heat transfer. The heat exchanger model is used when a pressure drop isintroduce to a fluid to transfer heat to another fluid. Species transport is used for modelingthe mixing and transport of chemicals. Discrete phase models calculate particle or droplettrajectories. Finally, solidification and melting and acoustic models in ANSYS fluent modelstate changes and sound, respectively.

2.3.3 Materials

Fluent has a material database in which the material properties are preassigned. The usermust choose the material type either fluid or solid and then can choose a FLUENT materialor a user defined material. With either choice a user may change the material properties:density, c

p

, the specific heat, the thermal conductivity, viscosity, and the thermal expansioncoe�cient.

The density can be set as a constant or a Boussinesq approximation. The Boussinesq modeltypically allows for faster convergence for natural-convection flows. In the Boussinesq model,⇢0 = constant density, T0 = operating temperature, � = thermal expansion coe�cient (setas a constant in our example) and uses:

(⇢� ⇢0)g ⇡ �⇢0�(T � T0)g.

The following is used to eliminate ⇢ in the hydrostatic force,

⇢ = ⇢0(1� ��T0).

This model is accurate as long as changes in density are small, that is, �(T � T0) < 1 [4].

The specific heat, cp

is used when the energy equation is active. Similar to the density, itcan be set as a constant, as a function of temperature or using kinetic theory. By settingthe c

p

as a constant one is considering the material to be a “perfect gas.” Although aperfect gas is simply theoretical the problem becomes computationally easier. When thec

p

is set as temperature dependent, heat capacity is only temperature dependent and notrelated to pressure. This obviously becomes computationally more challenging when c

p

is aconstant. Kinetic theory is used in correlation with the ideal gas laws for compressible andincompressible flows [4].

The thermal conductivity is defined like the specific heat when heat transfer is active. Vis-cosity, can also be set as a constant, as a function of temperature, using kinetic theory, oruser defined. The thermal expansion coe�cient is used only when there is a Boussinesqapproximation; it is the � from the Boussinesq approximation and can be adjusted to helpwith convergence [4].

16

2.3.4 Cell Zone Conditions

A cell is the control volume into which the domain is broken up and the zone is the groupingof these cells together. Each volume within the mesh is set as a separate cell zone. The usermust set the cell zone conditions on each zone individually. Cell Zone conditions are definedin ANSYS Fluent as either fluid conditions or solid conditions.

Fluid condition cell zones occur when flow is happening within the zone. Active equationsare solved within the zone. This cell zone only requires that the type of fluid material isinputted. However, there are several other parameters a user may define at this point. Forexample, rotational axis, mesh motion, and reactions may be defined here if applicable [5].

Solid condition cell zones are used when no flow equations are solved and only heat conduc-tion problems is solved. Again, like the fluid cell zone, the only required input is the materialtype. However, a user may also define heat sources, rotational axis, and zone motion at thispoint [5].

2.3.5 Boundaries

Fluent has four main types of boundary conditions: inflow/outflow, internal face boundaries,interior, and walls. Several inflow/outflow boundary conditions are implemented includinginlet (pressure, velocity or mass flow rate), vent (pressure or outlet), intake or exhaust fan,pressure far-field or outflow. All of these boundaries permit flow to enter and exit the domain.

Internal face boundaries can be set as a fan, radiator, porous jump, wall or interior. Internalface boundaries do not have a thickness they just introduce flow changes. When choosinginterior as an internal face boundary no inputs are needed as it just connects two di↵erentcell zones. Interior boundary conditions are set using the cell zone conditions previouslydescribed in Section 2.3.4.

Wall boundary conditions can be set as symmetry, periodic, axis or walls. Symmetry bound-ary conditions are used when the expected flow and geometry are the same on either sideof the boundary. The user inputs no extra conditions for symmetric boundary conditions.Mathematical symmetric boundary conditions are summarized as zero normal velocity andzero normal gradients of all variables at a symmetry plane. Periodic boundary conditionsoccur when the flow across two planes are identical and axis boundary conditions are usedat the center line of an axis of symmetry [4].

Wall boundary conditions require several di↵erent input parameters from the user. Firstthe user must define any wall motion that may be occurring. Then the user must define theshear conditions at the walls. By default all the shear conditions on walls are set with no slipconditions. However if shear stress occurs the user must choose between specified, specularityor Marangoni. Specularity stress occurs with granular flows, and Marangoni stress is usefulwhere the shear stress is known and the fluid motion is unknown. Wall roughness must be

17

defined by the user as well. The roughness height is by default zero, which implies smoothwalls. For roughness to take a↵ect this parameter must be set at non zero. The roughnessconstant has a default value of 0.5m, at this time there is no clear guideline for adjustingthis parameter [4]. Finally, the thermal boundary conditions at walls must be input by theuser. One must choose from heat flux which relates to insulation, a set temperature for thewall, convection heat transfer, external radiation if heat from the exterior is of interest, orcombined external radiation and convection heat transfer.

2.3.6 Dynamic Mesh

The dynamic mesh model is used when the mesh boundary is changing because of motionon the domain boundary. The motion can be predefined or determined by the solution atthat time step. The motion of the mesh must be described at either the face or cell zones. Ifparts of the mesh have di↵erent types of movements the mesh must be grouped in di↵erentcell zones. Thus all of the cells in one cell zone must have the same type of movement.

2.3.7 Reference Values

Reference values can be computed from a selected area or set manually. If the compute frombar is used to choose a boundary zone it is possible that not all of the reference values will beset. Thus the user must set additional reference values manually. The user may also chooseto directly input all reference values. The following is the reference values that are manuallyset and what they are used to compute.

The reference area, pressure, and length are used for computing the force and momentcoe�cients. The reference depth is used to determine cell volumes in 2D. The referencedensity and velocity are used to compute the reference dynamic pressure. The referenceenthalpy and temperature are used for computing the total enthalpy change. The referenceviscosity is used to determine the boundary Reynolds number. Finally, the ratio of specificheat is used in turbo machinery e�ciency calculations. Also, changing the reference zoneallows the user to plot velocities relative to the motion of di↵erent zones [5].

2.3.8 Solution Methods

The solution methods allow the user to specify solutions methods to be used in calcula-tion. The available pressure-velocity coupling schemes are SIMPLE, SIMPLEC, PISO, andcoupled. Pressure-velocity coupling schemes derive an additional condition for pressure byreformatting the continuity equation, the original form is derived in Section 2.1.3. SIMPLEis the default setting, SIMPLEC is the SIMPLE algorithm with the pressure-correction.However, with more skew meshes this can inhibit convergence. The PISO algorithm has

18

neighbor correction which helps when using a large time step; however, this algorithm isextremely computationally expensive. The coupled algorithm is another scheme that helpswith poor mesh quality or large time steps [5].

The SIMPLE algorithm solves the equations in sequential steps which lets the iterativeprocess care for the non-linearity and the coupling between equations. The basic principlebehind the algorithm is first read the best available values for all properties, then solve themomentum equation and calculate source terms for pressure correction. This is followingby solving the pressure correction equation and correcting the pressure and velocity. Theenthalpy equation and turbulence equation are then solved. Finally, write out all values forthe properties and check for convergence [13]. The SIMPLE algorithm is the only one usedfor the numerical example to follow.

Next the user must choose the types of spatial discretization schemes to be used. Spatialdiscretization schemes are used for evaluation of gradients and derivatives, interpolationschemes, momentum, turbulent kinetic energy, and energy. There are three methods inANSYS Fluent that can be used for the evaluation of gradients and derivatives: the Green-Gauss Cell-Based, the Green-Gauss Node-Based and the Least Squares Cell-Based. Thereare numerous pressure interpolation schemes are available typically the standard scheme isacceptable. However, the body-force-weighted scheme is applicable for large body forces.PRESTO! (PREssure STaggering Option) is recommended for high-speed flows and curvydomains.

When choosing the spatial discretization scheme for Momentum, Turbulent Kinetic Energy,Turbulent Dissipation Rate, and Energy a user can choose between First Order Upwind, Sec-ond Order Upwind, QUICK, or Third-Order MUSCL. The upwind schemes use an adaptivefinite di↵erence method, there is a two point (First order) and three point method (Secondorder). As expected the first order method converges faster however is less accurate thanthe second order scheme. The QUICK and Third-Order MUSCL schemes are best suited forswirling flows.

For the transient formulation explicit solvers are not available for pressure based solvers.Thus the user may only choose between first or second order implicit. In most cases firstorder implicit is su�cient.

2.3.9 Monitors

Users can create surface monitors to monitor flow rates, integral reports of a field variable,and mass average. Users may choose to print the answers to the console, plot the answers,or write them in a .out file. Reports can be made every iteration, time step, or flow time.There are several di↵erent surface integral report types available in fluent.

19

In particular there is the Mass Flow Rate given by:

Z⇢u · dA =

nX

i=1

⇢

i

vi

·Ai

.

This sums the value of the velocity multiplied by velocity and dotted with the cross sectionarea. The mass flow rate can also be written as

m = lim�t!0

�m

�t

.

In other words, the mass flow rate is the flow of mass through a surface unit per time unit.

Other surface integrals in ANSYS Fluent are computed by summing the product of the areaand some field variable, � [4], Z

�dA =nX

i=1

�i

|Ai

|.

In particular the static temperature and static pressure surface integrals are studied. Statictemperature is the temperature at a particular point and static pressure is the pressure at aparticular point.

2.3.10 Calculations

Lastly when setting up a new case in fluent the user must set up the solver iterations. Firstit is important to use the check case option. This will review all of the user’s settings andmake suggested edits. Fixed or adaptive time stepping methods are available. When usingeither method a user must first define a step size. When using the adaptive method ANSYSFluent will adapt the step size. When studying transient flows, users must set a number oftime steps to be performed. Max iterations per a time step are key for studying unsteadyflows. This makes the solver move onto the next step even though convergence may not ofoccurred yet. After the calculations are set the case file is ready to run.

In the following chapter the particular example will be discussed. The step by step processused to design the mesh and set up the case file will be discussed. Chapter 3 will directlyapply many of the equations derived in Chapter 2. Lastly, results from that particular casewill be discussed.

20

Chapter 3

Numerical Example

Halton’s Passive chilled beams will be used in modeling larger rooms to study e�cient controlof indoor air. The passive chilled beam can be seen in Figure 1.2. As previously discussed,in order to accurately model the flow around the coils in the chilled beam, only a small sliceof the beam is studied as seen in Figure 1.3. In this slice, part of a fin is on either side andthe coils run from fin to fin. However, due to the aspect ratio issues of the slender slice, themesh is widened and slowly narrowed to study the trend of the cooling capacity.

First in Chapter 3, the mesh generation is discussed as well as the specifications of the beam’sdimensions. Next, the specific settings used in ANSYS Fluent are discussed and finally theresults are analyzed.

3.1 Mesh Generation

The mesh used for this problem originally followed the specific dimensions and measurementsprovided by Halton for the CPA - Passive Chilled Beam [10]. First, refer to Figure 1.2, thebeam modeled has a width, W1, of 615mm. The total height of the beam is 130mm, whichcan be seen in Figure 3.1. The length of each beam can vary depending on needs of thespecific room. Thus by modeling just a slice of the beam, as seen in Figure 1.3, the drag andresistance can be better measured, and it can be applied to all di↵erent lengths of beams.

Within the Passive Chilled Beam coils wind back and forth, similar to a radiator. Thesecoils have a 15mm diameter. Running through the beam are fins that are 8mm apart. Againrefer to Figure 1.3 to see that the slice being modeled is one section between two fins withhalf a fin on either side. A side view of the geometry of the slice can be seen in Figure 3.2

The original mesh was made with the exact dimensions of the distance between the fins.Figure 3.3 shows a comparison of the top view of the geometry and a top view of the mesh.The geometry seen in Figure 3.2 and Figure 3.3 are the same from di↵erent view points.

21

Figure 3.1: Height View of a Halton Passive Beam

Figure 3.2: Geometry of Passive Chilled Beam

22

Figure 3.3: Comparison View of The Halton Passive Beam Diagram and the Geometry

The Halton Passive beam design guide [10] requires that all beams be mounted at least 0.25x the width of the beam from the ceiling. To preform simulations of the fluid movementthrough the chilled beam, the beam was encased in a larger volume to capture the entirefluid motion. Specifically the larger volume on either side of the beam has been extendedtwice the width of the beam and four times the height of the beam. The full geometry canbe seen in Figure 3.4. The lines in the middle of the geometry are used for refinement of themesh. This was done to help obtain convergence and better capture the flow in this area ofimportance.

Once the geometry was constructed the mesh was generated using the Delaunay 3D algorithmavailable in Gmsh. As discussed in Section 2.2.2 this algorithm first uses Si’s Tetgen algo-rithm then applies the 2D Delaunay algorithm. The final mesh resulting from this processis shown in Figure 3.5.

This mesh was then saved as an I-deas Universal file and uploaded into ANSYS Fluent.However, due to the aspect ratio and slenderness of the mesh the case would not converge.It is speculated that the built-in standard wall functions in ANSYS fluent have a large enoughboundary layer such that the flow calculations on each fin were interfering with each otherand causing divergence.

In order to obtain convergence many changes were made to the mesh while still keepingthe same properties of the Halton Passive Chilled Beam. First, the mesh was significantlywidened from 10mm to 450mm. Secondly the casing ends were made significantly larger.Both of these adjustments can be seen in Figure 3.6, where A is the casing ends. Finally,the fin edges were rounded on the inflow to help with convergence. Refer to B on Figure 3.6.

As before this geometry was nested in a larger casing to model the fluid flow movement

23

Figure 3.4: Entire Geometry of Mesh for the Passive Chilled Beam

Figure 3.5: 3D Mesh of the Slice of a Halton Passive Chilled Beam

24

Figure 3.6: 3D Geometry of a Widened Slice of a Halton Passive Chilled Beam

through the passive chilled beam. The same refinement lines were used within the largercasing. As before the mesh was generated using the Delaunay 3D algorithm. After theinitial mesh was generated the optimize 3D command and optimize 3D (Netgen) commandin Gmsh were used to optimize the mesh to improve the mesh quality. The optimize 3Dcommand uses an algorithm created by the Gmsh developers to optimize the mesh while thelater uses the Netgen algorithm [6]. Refer to Figure 3.7 to see the final mesh used to modelthe passive chilled beam.

Figure 3.7: 3D Model of a Halton Passive Chilled Beam

The final mesh was adjusted to multiple widths ranging from 80mm to 440mm and the samecase in ANSYS Fluent was ran on the mesh. The .geo file used to create and adjust thismesh is in Appendix A. Data from each width was collected and trends were studied so thatan estimate could be made about the actual dimensions. When the mesh was generated withgeometry less than ten times the actual width of the beam divergence occurred.

25

Table 3.1: Mesh Quality of Varying Beam widths

Beam Width Maximum cell squish Maximum cell skewness Maximum aspect ration

80 0.737101 0.995765 25.9515150 0.730941 0.995293 25.17367200 0.791616 0.996020 26.5710400 0.748703 0.996627 27.6848440 0.738174 0.995548 24.7307

3.2 ANSYS Fluent

This section goes through the step by step process of the case study formulation in ANSYSFluent for the Halton Passive Chilled Beam. Using the mesh generated from the previoussection, a detailed description of the process to set up a method to study the flow will bediscussed. Appedinx B contains tables of all of the settings used in the case files.

3.2.1 Problem Set up

To begin a case set up in ANSYS Fluent first the FLUENT Launcher must be opened. Thedimension is 3D, using double precision, on a serial processor. The display options used wereembed graphics windows and a workbench color scheme. All cases were ran using Version12.1.4. To move to the user interface click Okay.

In initial trials the 3D mesh in Figure 3.5 was uploaded into ANSYS Fluent as an I-deasUniversal file. However, as previously discussed due to the Aspect Ratio and built in wallfunctions in ANSYS Fluent the case would not converge. Thus the 3D model of a slice ofHalton Passive Chilled Beam, seen in Figure 3.7 was uploaded into ANSYS Fluent as anI-deas Universal file as discussed in Section 2.2.3. In ANSYS Fluent under file refer to theimport command.

Once the mesh was uploaded, the first step is to scale the mesh. All Idea-s Universal filesare imported in meters, however the file was created in mm. ANSYS Fluent o↵ers a scalingoption where the user selects that the mesh was created in mm and the mesh is automaticallyscaled. Next the imported mesh needs to be checked and mesh quality reported.

Using the exact mesh dimensions the mesh had a maximum cell squish of 0.999614, maximumcell skewness of 1, and a maximum aspect ratio of 15733.0. Along with the poor meshquality and standard ANSYS Fluent wall functions convergence was never obtained. Howeverconvergence was obtained when the mesh varied in width from 80-440mm. See Table 3.1 forthe mesh quality values of the mesh at varying widths as computed by ANSYS Fluent.

26

The pressure-based solver, with absolute velocity formulation and transient time was used.Due to the orientation of the mesh gravity is in the -9.8 Y m/s

2 direction.

The energy model is turned on so that heat transfer parameters can be adjusted. Theviscous k-epsilon (2 eqn) model, realizable is used. There are standard wall functions andFull Buoyancy E↵ects. The model constants were eventually left as the standard settings.In the original model these were varied slightly in an e↵ort to obtain convergence however,after mesh changes were made this could be left in the original settings.

In the material section it is important to have air defined as a fluid with boussinesq densityand aluminum defined as a solid. The specific values set can all be found in Appendix B.According to the constraint in ANSYS Fluent the model is accurate as long as �(T�T0) < 1.For this model the change in temperate is 7 and the thermal expansion coe�cient, � is 0.0034,hence the constraint is satisfied and the model is accurate.

Next, the cell zone conditions are set. This step is extremely important in order to changethe mesh from complete aluminum to being recognized as air. Each cell zone must be setas a fluid, because there is only one fluid defined in the case set up just choosing fluid issu�cient.

Next the boundary conditions for each surface are defined. These conditions are specificallyoutlined in Table B.6 and Table B.7. The beam interior and outer walls casings are all setas interiors. Also the inflow and outflow planes of the chilled beam model are also set asinterior. These particular surfaces exists only to set up surface monitors.

The outer casing of the mesh is set as a wall with no slip conditions. These surfaces are setat a temperature 7K above the chilled beam. According to the Halton literature the passivechilled beam stays about 7 degrees cooler than the room, otherwise condensation will form onthe beams causing the ceiling to drip on the rooms inhabitants. Thus the outer walls of themodel are set at exactly 300 K. The outer casing of the beam was also set to a temperatureof 300 K with no slip boundary conditions and the material was set as aluminum.

The coils and fins that make up the side of the slice are both set as walls with a temperature of293 K with no slip boundary conditions. The material is set as aluminum. These structurescombined make up the one part of the model that is set as a cooling mechanisms. The insideof the casing around the chilled beam is set as a completely insulated boundary, that is theheat flux is set at 0 w/m2. The boundaries that represents the sliced edges of the fin are setas symmetry, since the same flow is occurring on the other side of the boundary.

The operating conditions and reference values are automatically set by ANSYS Fluent inthis case. The reference zone is the area outside of the slice of the chilled beam.

27

3.2.2 Solution

The problem is now completely set up and the user must now choose the solution methodsand monitors before the calculation run. First the solution method is set up. The pressure-velocity coupling scheme used is SIMPLE. For the spatial discretization: the gradient isthe least squares cell based, pressure is PRESTO!, momentum, turbulent kinetic energy,turbulent dissipation rate, and energy are all second order upwind. PRESTO! is used becausethe flow is considered high speed and second order upwind is used to increase accuracy. Thetransient formulation used is first order implicit. The under-relaxation factors of the solutioncontrols are pre-set by ANSYS Fluent and left in the usual way.

Next, the monitors are set. The residuals monitor is left as the standard setting of “Printand Plot”. The statistic and force monitors are left o↵. Six surface monitors are createdto monitor the movement on the inflow and outflow of the chilled beam. For each surfacemonitor the data is retrieved at every time step. Each of the monitors is written as a .outfile. A report type of Mass Flow Rate was made for both the inflow and outflow surfaces.The mass flow rate is the mass of a substance which passes through a surface with respectto time. Mathematically that is

m = lim�t!0

�m

�t

.

Also two integral report types were computed on each surface: static pressure, and statictemperature.

After all of the problem is set up and the solution methods and monitors are in place the caseis then initialized. The pre-selected initial values are su�cient to use for the case. Beforerunning the calculation the user should check the case. Occasionally ANSYS Fluent willsuggest changes to make if the case set up will cause divergence. The time stepping methodused was fixed with a time step size of 2 seconds and 3000 time steps. The first case ranfor 7200 time steps however, a steady state was achieved much earlier, thus 3000 time stepswere used in subsequent runs. Also, a max of 20 iterations per a time step limit is set.

3.2.3 Results

ANSYS Fluent o↵ers several post-processing options. After the case studies were ran usingthe graphics and animations option, graphics of the contours were produced. The velocityvectors of the slice of the beam with width 440mm is pictured in Figure 3.8 and velocityvectors for a beam width of 80mm is shown in Figure 3.9. The vectors are colored by magni-tude in meters/second and depicted the fluid movement at the final step of the calculations.In the beam of width 440mm the fluid is moving much slower to return to the beam. Onthe other hand in the narrower model, the fluid moves quicker through the beam and thevelocity continues onto the other side of the beam. The fluid quickly returns to the chilledbeam in Figure 3.9 and does not move outward throughout the room as seen in Figure 3.8.

28

Figure 3.8: Velocity Vectors of Slice with Width 440mm

29

Figure 3.9: Velocity Vectors of Slice with Width 80mm

30

ANSYS Fluent can also provide images of the static temperature. In Figure 3.10 and Figure3.11 the static temperature contours of a slice of the Passive Halton Chilled beam at di↵erentwidths are depicted. In the wider slice, Figure 3.10 the room is much warmer at the endof the simulation. The coils are the only part widened in the simulation thus there is muchless cooling surface in comparison to the entire room’s size. In Figure 3.11 the area becomesmuch cooler because there is much less room to cool. In this simulation the slice is exactlyten times the width of an actual fin slice.

Figure 3.10: Static Temperature of Slice with Width 440mm

3.3 Results

Using the monitors set in the ANSYS Fluent case file, data was collected for the beam ateach length over the time interval. This section will discuss the results and analyze the datacollected for varying beam widths starting at 440mm and narrowed to 80mm.

31

Figure 3.11: Static Temperature of Slice with Width 80mm

32

3.3.1 Mass Flow Rates

As discussed earlier, the mass flow rate is the mass of a substance which passes througha surface with respect to time. At each flow time step the mass flow rate is computed onthe inflow and outflow surfaces on the slice of the Halton Chilled beam. Throughout thesimulations the mass flow rate on the inflow and outflow are the same except the inflow hasa negative flow rate. In other words, let m

ij

denote the general mass flow rate on the inflowfor any width and m

oj

denote the general mass flow rate on the outflow for any width, wherej denotes each time step such that 1 j 6000. Then

m

ij

+ m

oj

= 0 8 j.

Plots of the mass flow rate over each slice width are in figure 3.12 to 3.16, refer to eachcaption.

Figure 3.12: Mass Flow Rate Data for the Halton Passive Chilled Beam at 400mm

33


34


35


36


37

Table 3.2: Average Mass Flow Rates

Beam Width Average Inflow Mass Flow Rate Average Outflow Mass Flow Rate

80 -0.001989437kg/s 0.001989438 kg/s150 -0.002464653kg/s 0.002464681 kg/s200 -0.002551171kg/s 0.002551166 kg/s400 -0.004760634 kg/s 0.004760609kg/s440 -0.005190752 kg/s 0.005190724 kg/s

Table 3.3: Average Static Temperature

Beam Width Average Inflow Static Temperature Average Outflow Static Temperature

80 14.61222278 K 14.55653574 K150 27.46772753 K 27.39641433K200 36.65711339 K 36.57601531K400 73.41096988K 73.28356036K440 80.7697961K 80.63059656 K

After the first 1000 seconds the mass flow rate reaches a steady state for the rest of thecalculations. An average of each mass flow rate was taken from the 1000 second to the 6000second. Refer to table 3.4 for these values. As the beam is narrowed the mass flow ratedecreases because the area for the mass to flow through decreases.

3.3.2 Static Temperature

The case files described on the previous section were ran and .out files were made containingall of the data from the surface integrals representing the static temperature. The data wasthen plotted over the flow time. In Figures 3.17 to 3.21 the static temperature of varyingbeam lengths is plotted against the flow time of the solution, refer to each figure caption.

The static temperature in ANSYS Fluent is computed by summing the product of thetemperature at each point and the area. The average static temperate hence decreasesas the area decreases as seen in Table 3.4.

38

Figure 3.17: Static Temperature Data for the Halton Passive Chilled Beam at 440mm

39


40


41


42


43

3.3.3 Static Pressure

The case files described on the previous section were ran and .out files were made containingall of the data from the surface integrals representing the static pressure. The data was thenplotted over the flow time. In Figures 3.22 to 3.26 the static temperature of varying beamlengths is plotted against the flow time of the solution, refer to each figure caption.

Figure 3.22: Static Pressure Data for the Halton Passive Chilled Beam at 440mm

Again, as seen before, the static pressure decreases as the beam width is decreased becausethe area is decreasing.

44


Table 3.4: Average Static Pressure

Beam Width Average Inflow Static Pressure Average Outflow Static Pressure

80 0.001709895 Pa -6.26E-05 Pa150 0.003627342 Pa -6.17E-05 Pa200 0.00506393Pa -8.84E-07 Pa400 0.010650184 Pa 6.41E-05 Pa440 0.011806473 Pa 1.11E-04 Pa

45


46


47


48

Table 3.5: Area Weighted Temperature

Beam Width Inflow Temperature Outflow Temperature

80 296.996398K 295.8645476K150 297.7531439K 296.9801011K200 298.0253121K 297.3659781K400 298.4185768K 297.9006519K440 298.484095K 297.9696843 K

Table 3.6: Cooling Capacity of Varying Beam Widths

Beam Width m �T Q

80 0.001989438 kg/s 1.131850399K 2.266225172150 0.002464681 kg/s 0.773042781K 1.917555003200 0.002551166 kg/s 0.659334008K 1.69288635400 0.004760609kg/s 0.517924888 K 2.481491909440 0.005190724 kg/s 0.51441069K 2.68733291

3.3.4 Cooling Capacity

To calculate the cooling capacity of each beam, the formula for cooling capacity

Q = mc

p

�T

was used. Where Q is the cooling capacity, m is the mass flow rate, �T is the changein temperature, and c

p

is the specific heat used in the case set up, 1006.43, kJ/kg.K. Thechange in temperature is related to the change in the area weighted temperature, not thestatic temperature. Thus, first the area weighted temperature is computed for each beamlength. Refer to Table 3.5 for these values.

The cooling capacity for each beam width was calculated and compared to the given Haltondata. The mass flow rate for each length was used respectively and the change betweenthe inflow and outflow area weighted temperature was computed. The calculated coolingcapacity for each length and the values used to compute it are in Table 3.6.

According to the given Halton data, the Halton Passive Chilled beam has a cooling capacityof 250watt/m. However, the calculated data models a slice of the passive halton chilledthat is originally 8mm. Thus converting the cooing capacity given by Halton, they estimatetheir chilled beams to have a cooling capacity of 2watts per an 8mm slice. According to ourcomputed data the passive chilled beams have a cooling capacity near 2 watts.

49

There are many factors that could contribute to the di↵erence between our computed dataand the Halton data. Firstly, it is unknown what assumptions the company has made whencomputing this data. Also, our model simply widens the slice. The coils and fins did notincrease in size and thus some of the cooling capacity is lost in the model.

50

Chapter 4

Conclusion

In this study, we developed a computational fluid dynamics model for a Halton passive chilledbeam. This was done by studying the flow between two cooling fins in the passive chilledbeam and embedding the problem in a warmer environment to calculate the cooling capacityof the device and the pressure drop between the fins (which can be used to estimate the dragimposed on the flow). Modeling the velocity, pressure, and temperature involved detailedgeometric modeling in Gmsh to build the finite volumes for approximating the solution tothe Boussinesq equations in ANSYS Fluent. In attempting to model the flow between aset of cooling fins, we ran into a number of aspect ratio challenges due to the thin spacingbetween consecutive fins. We constructed a sequence of meshes of progressively thinner finwidths and showed that the predicted cooling capacities were very close to those appearingin the Halton literature.

One of the limitations of the commercial software was the inability to study the precise causeof the solution divergence at the thinnest volumes. Future work will entail developing ourown software to solve the Boussinesq system to solve for the flow in this region. The nextsteps will be to couple the momentum and energy equation forcing terms studied here withfeedback control methods to develop energy e�cient control strategies for the Purdue LivingLaboratory room.

51

Bibliography

[1] Buildings Overview. PEW CENTER: Global Climate Change, May 2009.

[2] D. Alexander and M. O’Rourke. Design considerations for active chilled beams.ASHRAE Journal, pages 50–58, September 2008.

[3] ANSYS. Fluent. Brochure, www.ansys.com, 2011.

[4] ANSYS, Inc., Canonsburg, PA. ANSYS Fluent 12.0 Theory Guide, 12.0 edition, January2009.

[5] ANSYS, Inc., Canonsburg, PA. ANSYS Fluent 12.0 User’s Guide, April 2009.

[6] J.-F. R. Christophe Geuzaine. Gmsh Reference Manual, gmsh 2.7 edition, May 2013.

[7] EPA. Building codes for energy e�ciency, 2012.

[8] P.-L. George and S. Frey. Mesh Generation. Hermes Science Ltd., 2000.

[9] M. D. Gunzburger. Finite Element Method for Viscous Incompressible Flows. AcademicPress, Inc., 1989.

[10] Halton, Helsinki, Finland. CPA Passive Chilled Beam.

[11] Halton, Helsinki, Finland. Halton-Chilled Beam Design Guide.

[12] D. S. Henningson and M. Berggren. Fluid Dynamics: Theory and Computation. NADAat KTH Royal Institute of Technology, Stockholm, Sweden, August 2005.

[13] L. Mangani and C. Bianchini. Heat transfer applications in turbomachinery. In OFCOnference. OpenFOAM Conference, November 2007.

[14] P. Mustakallio, R. Kosonen, and R. Paavilainen. New hybrid solution for energy e�cientcooling, heating and ventilation – chilled beam integrated with radiant panels. REHVAJournal, pages 81–82, May 2011.

[15] J. Schoberl. Netgen an advancing front 2d/3d-mesh generator based on abstract rules.Computing and Visualization in Science, 1:41–52, 1997.

52

Appendix A

GMSH Source Code

In this section source code is provided for generating a three-dimensional mesh of a HaltonPassive Chilled beam.

1 m2=15;2 m4=20;3 m5=25;4 m9=30;5

6 coil casing height = 130;7 coil casing width = 615;8 coil radius = 7.5;9

10 coil casing thickness = 40;11

12 //Adjustable thickness13 fin distance=400;14 fin width=10;15

16 n=2;17 m=3;18 //where n denotes space around coil casing width19 //where m denotes space around coil casing height20

21 // Points for the casing22 Point(1001) = {0, 0, 0, m4};23 Point(1002) = {0, coil casing height, 0, m4};24 Point(1003) = {coil casing width, coil casing height, 0, m4};25 Point(1004) = {coil casing width, 0, 0, m4};26 Point(1005) = {0, 0, fin distance+fin width, m4};27 Point(1006) = {0, coil casing height, fin distance+fin width, m4};28 Point(1007) = {coil casing width, coil casing height, fin distance+fin width,29 m4};

53

30 Point(1008) = {coil casing width, 0, fin distance+fin width, m4};31

32 //Points for casing Ends33 Point(1103) = {�coil casing thickness, coil casing height�fin width/2, 0,m4};34 Point(1104) = {�coil casing thickness, coil casing height�fin width/2,35 fin distance+fin width,m4};36 Point(1105) = {�coil casing thickness, 0, 0,m4};37 Point(1106) = {�coil casing thickness, 0, fin distance+fin width,m4};38 Point(1107) = {615+coil casing thickness, coil casing height�fin width/2, 0,39 m4};40 Point(1108) = {615+coil casing thickness, coil casing height�fin width/2,41 fin distance+fin width,m4};42 Point(1109) = {615+coil casing thickness, 0, 0,m4};43 Point(1110) = {615+coil casing thickness, 0, fin distance+fin width,m4};44

45

46 //Points for the fins47 Point(1009) = {0, 0, fin width/2, m4};48 Point(1010) = {0, coil casing height�fin width/2,fin width/2, m4};49 Point(1011) = {0, coil casing height�fin width/2,0, m4};50

51 Point(1012) = {coil casing width, 0, fin width/2, m4};52 Point(1013) = {coil casing width, coil casing height�fin width/2,fin width/2,53 m4};54 Point(1014) = {coil casing width, coil casing height�fin width/2,0, m4};55

56 Point(1015) = {0, 0, fin distance+fin width/2, m4};57 Point(1016) = {0, coil casing height�fin width/2,fin distance+fin width, m4};58 Point(1017) = {0, coil casing height�fin width/2,fin distance+fin width/2,59 m4};60

61 Point(1018) = {coil casing width, 0, fin distance+fin width/2, m4};62 Point(1019) = {coil casing width, coil casing height�fin width/2,63 fin distance+fin width, m4};64 Point(1020) = {coil casing width, coil casing height�fin width/2,65 fin distance+fin width/2, m4};66

67 //Points for Coils68

69 // Points for the first coil with center point (52.5, 22.5)70 x1 = 52.5;71 y1 = 22.5;72 Point(1021) = {x1, y1, fin width/2, m2};73 Point(1022) = {x1�coil radius, y1, fin width/2, m2};74 Point(1023) = {x1+coil radius, y1, fin width/2, m2};75 Point(1024) = {x1, y1+coil radius, fin width/2, m2};76 Point(1025) = {x1, y1�coil radius, fin width/2, m2};77 Point(1026) = {x1, y1, fin distance+fin width/2, m2};78 Point(1027) = {x1�coil radius, y1, fin distance+fin width/2, m2};79 Point(1028) = {x1+coil radius, y1, fin distance+fin width/2, m2};80 Point(1029) = {x1, y1+coil radius, fin distance+fin width/2, m2};

54

81 Point(1030) = {x1, y1�coil radius, fin distance+fin width/2, m2};82

83 //Points for the second coil with center point (73.5, 102.5)84 x2=73.5;85 y2=102.5;86 Point(1031) = {x2, y2, fin width/2, m2};87 Point(1032) = {x2�coil radius, y2, fin width/2, m2};88 Point(1033) = {x2+coil radius, y2, fin width/2, m2};89 Point(1034) = {x2, y2+coil radius, fin width/2, m2};90 Point(1035) = {x2, y2�coil radius, fin width/2, m2};91 Point(1036) = {x2, y2, fin distance+fin width/2, m2};92 Point(1037) = {x2�coil radius, y2, fin distance+fin width/2, m2};93 Point(1038) = {x2+coil radius, y2, fin distance+fin width/2, m2};94 Point(1039) = {x2, y2+coil radius, fin distance+fin width/2, m2};95 Point(1040) = {x2, y2�coil radius, fin distance+fin width/2, m2};96

97 //Points for the third coil with center point (297, 22.5)98 x3=297;99 y3=22.5;

100 Point(1041) = {x3, y3, fin width/2, m2};101 Point(1042) = {x3�coil radius, y3, fin width/2, m2};102 Point(1043) = {x3+coil radius, y3, fin width/2, m2};103 Point(1044) = {x3, y3+coil radius, fin width/2, m2};104 Point(1045) = {x3, y3�coil radius, fin width/2, m2};105 Point(1046) = {x3, y3, fin distance+fin width/2, m2};106 Point(1047) = {x3�coil radius, y3, fin distance+fin width/2, m2};107 Point(1048) = {x3+coil radius, y3, fin distance+fin width/2, m2};108 Point(1049) = {x3, y3+coil radius, fin distance+fin width/2, m2};109 Point(1050) = {x3, y3�coil radius, fin distance+fin width/2, m2};110

111 //Points for the fourth coil with center point (318, 102.5)112 x4=318;113 y4=102.5;114 Point(1061) = {x4, y4, fin width/2, m2};115 Point(1062) = {x4�coil radius, y4, fin width/2, m2};116 Point(1063) = {x4+coil radius, y4, fin width/2, m2};117 Point(1064) = {x4, y4+coil radius, fin width/2, m2};118 Point(1065) = {x4, y4�coil radius, fin width/2, m2};119 Point(1066) = {x4, y4, fin distance+fin width/2, m2};120 Point(1067) = {x4�coil radius, y4, fin distance+fin width/2, m2};121 Point(1068) = {x4+coil radius, y4, fin distance+fin width/2, m2};122 Point(1069) = {x4, y4+coil radius, fin distance+fin width/2, m2};123 Point(1070) = {x4, y4�coil radius, fin distance+fin width/2, m2};124

125 //Points for the fifth coil with center point (541.5, 22.5)126 x5=541.5;127 y5=22.5;128 Point(1071) = {x5, y5, fin width/2, m2};129 Point(1072) = {x5�coil radius, y5, fin width/2, m2};130 Point(1073) = {x5+coil radius, y5, fin width/2, m2};131 Point(1074) = {x5, y5+coil radius, fin width/2, m2};

55

132 Point(1075) = {x5, y5�coil radius, fin width/2, m2};133 Point(1076) = {x5, y5, fin distance+fin width/2, m2};134 Point(1077) = {x5�coil radius, y5, fin distance+fin width/2, m2};135 Point(1078) = {x5+coil radius, y5, fin distance+fin width/2, m2};136 Point(1079) = {x5, y5+coil radius, fin distance+fin width/2, m2};137 Point(1080) = {x5, y5�coil radius, fin distance+fin width/2, m2};138

139 //Points for the sixth coil with center point (562.5, 102.5)140 x6=562.5;141 y6=102.5;142 Point(1081) = {x6, y6, fin width/2, m2};143 Point(1082) = {x6�coil radius, y6, fin width/2, m2};144 Point(1083) = {x6+coil radius, y6, fin width/2, m2};145 Point(1084) = {x6, y6+coil radius, fin width/2, m2};146 Point(1085) = {x6, y6�coil radius, fin width/2, m2};147 Point(1086) = {x6, y6, fin distance+fin width/2, m2};148 Point(1087) = {x6�coil radius, y6, fin distance+fin width/2, m2};149 Point(1088) = {x6+coil radius, y6, fin distance+fin width/2, m2};150 Point(1089) = {x6, y6+coil radius, fin distance+fin width/2, m2};151 Point(1090) = {x6, y6�coil radius, fin distance+fin width/2, m2};152

153 //Points for outside154 Point(1095) = {�coil casing width*(2ˆ(n�1)), �coil casing height*(2ˆ(m�1)),155 0, 8*m9};156 Point(1096) = {coil casing width+coil casing width*(2ˆ(n�1)),157 �coil casing height*(2ˆ(m�1)), 0, 8*m9};158 Point(1097) = {�coil casing width*(2ˆ(n�1)), coil casing height+159 coil casing height*(2ˆ(m�1)), 0, 10*m9};160 Point(1098) = {coil casing width+coil casing width*(2ˆ(n�1)),161 coil casing height+coil casing height*(2ˆ(m�1)), 0, 10*m9};162 Point(1099) = {�coil casing width*(2ˆ(n�1)), �coil casing height*(2ˆ(m�1)),163 fin distance+fin width, 8*m9};164 Point(1100) = {coil casing width+coil casing width*(2ˆ(n�1)),165 �coil casing height*(2ˆ(m�1)), fin distance+fin width, 8*m9};166 Point(1101) = {�coil casing width*(2ˆ(n�1)), coil casing height167 +coil casing height*(2ˆ(m�1)), fin distance+fin width, 10*m9};168 Point(1102) = {coil casing width+coil casing width*(2ˆ(n�1)),169 coil casing height+coil casing height*(2ˆ(m�1)), fin distance+fin width,170 10*m9};171

172 //Points for refined mesh173 Point(1111) = {0, 200, fin distance+fin width, m5};174 Point(1112) = {0, 200, 0, m5};175 Point(1113) = {coil casing width, 200, 0, m5};176 Point(1114) = {coil casing width, 200, fin distance+fin width, m5};177 Point(1115) = {0, coil casing height+coil casing height*(2ˆ(m�1)),178 fin distance+fin width, 8*m9};179 Point(1116) = {0, coil casing height+coil casing height*(2ˆ(m�1)), 0, 8*m9};180 Point(1117) = {615, coil casing height+coil casing height*(2ˆ(m�1)),181 fin distance+fin width, 8*m9};182 Point(1118) = {615, coil casing height+coil casing height*(2ˆ(m�1)), 0, 8*m9};

56

183

184 //Lines for casing185 Line(2001) = {1002, 1003};186 //extd casing Line(2002) = {1002, 1001};187 Line(2002) = {1002, 1011};188 //extd casing Line(2003) = {1006, 1005};189 Line(2003) = {1006, 1016};190 Line(2004) = {1006, 1007};191 Line(2005) = {1007, 1019};192 Line(2006) = {1005, 1008};193 Line(2007) = {1004, 1001};194 Line(2008) = {1014, 1003};195

196

197 //Lines for casing ends198 Line(2118) = {1103, 1104};199 Line(2119) = {1104, 1016};200 Line(2120) = {1103, 1011};201 Line(2121) = {1010, 1017};202 Line(2122) = {1105, 1001};203 Line(2123) = {1005, 1106};204 Line(2124) = {1015, 1009};205 Line(2125) = {1106, 1105};206 Line(2126) = {1104, 1106};207 Line(2127) = {1105, 1103};208 Line(2130) = {1107, 1014};209 Line(2131) = {1019, 1108};210 Line(2132) = {1108, 1107};211 Line(2133) = {1013, 1020};212 Line(2134) = {1004, 1109};213 Line(2135) = {1108, 1110};214 Line(2136) = {1107, 1109};215 Line(2137) = {1110, 1008};216 Line(2138) = {1110, 1109};217 Line(2139) = {1012, 1018};218

219 //Lines for fins220 Line(2009) = {1012, 1004};221 Line(2010) = {1001, 1009};222 Line(2011) = {1015, 1005};223 Line(2012) = {1018, 1008};224 Line(2013) = {1015, 1018};225 Line(2014) = {1009, 1012};226 Line(2015) = {1015, 1017};227 Line(2016) = {1010, 1009};228 Line(2017) = {1020, 1018};229 Line(2018) = {1012, 1013};230 Line(2019) = {1013, 1010};231 Line(2020) = {1017, 1020};232

233 //Rounded Edges

57

234 Circle(2021) = {1003, 1014, 1013};235 Circle(2022) = {1007, 1019, 1020};236 Circle(2023) = {1002, 1011, 1010};237 Circle(2024) = {1006, 1016, 1017};238

239 //Lines for the first coil240 Circle(2025) = {1025, 1021, 1022};241 Circle(2026) = {1022, 1021, 1024};242 Circle(2027) = {1024, 1021, 1023};243 Circle(2028) = {1023, 1021, 1025};244 Circle(2029) = {1027, 1026, 1029};245 Circle(2030) = {1029, 1026, 1028};246 Circle(2031) = {1028, 1026, 1030};247 Circle(2032) = {1030, 1026, 1027};248 Line(2033) = {1024, 1029};249 Line(2034) = {1023, 1028};250 Line(2035) = {1025, 1030};251 Line(2036) = {1022, 1027};252

253 //Line loops, Surface, and Volume for first coil254 Line loop(3013) = {�2027, �2026, �2025, �2028}; // min255 Line loop(3014) = {2029, 2030, 2031, 2032}; // max256 Line loop(3015) = {2034, �2030, �2033, 2027};257 Ruled Surface(4015) = {3015};258 Line loop(3016) = {�2034, 2028, 2035, �2031};259 Ruled Surface(4016) = {3016};260 Line loop(3017) = {2025, 2036, �2032, �2035};261 Ruled Surface(4017) = {3017};262 Line loop(3018) = {2026, 2033, �2029, �2036};263 Ruled Surface(4018) = {3018};264

265 //Lines for the second coil266 Circle(2037) = {1035, 1031, 1032};267 Circle(2038) = {1032, 1031, 1034};268 Circle(2039) = {1034, 1031, 1033};269 Circle(2040) = {1033, 1031, 1035};270 Circle(2041) = {1037, 1036, 1039};271 Circle(2042) = {1039, 1036, 1038};272 Circle(2043) = {1038, 1036, 1040};273 Circle(2044) = {1040, 1036, 1037};274 Line(2045) = {1034, 1039};275 Line(2046) = {1033, 1038};276 Line(2047) = {1035, 1040};277 Line(2048) = {1032, 1037};278

279 //Line loops, Surface, and Volume for second coil280 Line loop(3019) = {�2039, �2038, �2037, �2040};281 Line loop(3020) = {2041, 2042, 2043, 2044};282 Line loop(3021) = {2046, �2042, �2045, 2039};283 Ruled Surface(4021) = {3021};284 Line loop(3022) = {�2046, 2040, 2047, �2043};

58

285 Ruled Surface(4022) = {3022};286 Line loop(3023) = {2037, 2048, �2044, �2047};287 Ruled Surface(4023) = {3023};288 Line loop(3024) = {2038, 2045, �2041, �2048};289 Ruled Surface(4024) = {3024};290

291 //Lines for the third coil292 Circle(2049) = {1045, 1041, 1042};293 Circle(2050) = {1042, 1041, 1044};294 Circle(2051) = {1044, 1041, 1043};295 Circle(2052) = {1043, 1041, 1045};296 Circle(2053) = {1047, 1046, 1049};297 Circle(2054) = {1049, 1046, 1048};298 Circle(2055) = {1048, 1046, 1050};299 Circle(2056) = {1050, 1046, 1047};300 Line(2057) = {1044, 1049};301 Line(2058) = {1043, 1048};302 Line(2059) = {1045, 1050};303 Line(2060) = {1042, 1047};304

305 //Line loops, Surface, and Volume for third coil306 Line loop(3025) = {�2051, �2050, �2049, �2052};307 Line loop(3026) = {2053, 2054, 2055, 2056};308 Line loop(3027) = {2058, �2054, �2057, 2051};309 Ruled Surface(4027) = {3027};310 Line loop(3028) = {�2058, 2052, 2059, �2055};311 Ruled Surface(4028) = {3028};312 Line loop(3029) = {2049, 2060, �2056, �2059};313 Ruled Surface(4029) = {3029};314 Line loop(3030) = {2050, 2057, �2053, �2060};315 Ruled Surface(4030) = {3030};316

317 //Lines for the fourth coil318 Circle(2070) = {1065, 1061, 1062};319 Circle(2071) = {1062, 1061, 1064};320 Circle(2072) = {1064, 1061, 1063};321 Circle(2073) = {1063, 1061, 1065};322 Circle(2074) = {1067, 1066, 1069};323 Circle(2075) = {1069, 1066, 1068};324 Circle(2076) = {1068, 1066, 1070};325 Circle(2077) = {1070, 1066, 1067};326 Line(2078) = {1064, 1069};327 Line(2079) = {1063, 1068};328 Line(2080) = {1065, 1070};329 Line(2081) = {1062, 1067};330

331 //Line loops, Surface, and Volume for fourth coil332 Line loop(3031) = {�2072, �2071, �2070, �2073}; // min333 Line loop(3032) = {2074, 2075, 2076, 2077}; // max334 Line loop(3033) = {2079, �2075, �2078, 2072};335 Ruled Surface(4033) = {3033};

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336 Line loop(3034) = {�2079, 2073, 2080, �2076};337 Ruled Surface(4034) = {3034};338 Line loop(3035) = {2070, 2081, �2077, �2080};339 Ruled Surface(4035) = {3035};340 Line loop(3036) = {2071, 2078, �2074, �2081};341 Ruled Surface(4036) = {3036};342

343 //Lines for the fifth coil344 Circle(2082) = {1075, 1071, 1072};345 Circle(2083) = {1072, 1071, 1074};346 Circle(2084) = {1074, 1071, 1073};347 Circle(2085) = {1073, 1071, 1075};348 Circle(2086) = {1077, 1076, 1079};349 Circle(2087) = {1079, 1076, 1078};350 Circle(2088) = {1078, 1076, 1080};351 Circle(2089) = {1080, 1076, 1077};352 Line(2090) = {1074, 1079};353 Line(2091) = {1073, 1078};354 Line(2092) = {1075, 1080};355 Line(2093) = {1072, 1077};356

357 //Line loops, Surface, and Volume for fifth coil358 Line loop(3037) = {�2084, �2083, �2082, �2085};359 Line loop(3038) = {2086, 2087, 2088, 2089};360 Line loop(3039) = {2091, �2087, �2090, 2084};361 Ruled Surface(4039) = {3039};362 Line loop(3040) = {�2091, 2085, 2092, �2088};363 Ruled Surface(4040) = {3040};364 Line loop(3041) = {2082, 2093, �2089, �2092};365 Ruled Surface(4041) = {3041};366 Line loop(3042) = {2083, 2090, �2086, �2093};367 Ruled Surface(4042) = {3042};368

369 //Lines for the sixth coil370 Circle(2094) = {1085, 1081, 1082};371 Circle(2095) = {1082, 1081, 1084};372 Circle(2096) = {1084, 1081, 1083};373 Circle(2097) = {1083, 1081, 1085};374 Circle(2098) = {1087, 1086, 1089};375 Circle(2099) = {1089, 1086, 1088};376 Circle(2100) = {1088, 1086, 1090};377 Circle(2101) = {1090, 1086, 1087};378 Line(2102) = {1084, 1089};379 Line(2103) = {1083, 1088};380 Line(2104) = {1085, 1090};381 Line(2105) = {1082, 1087};382

383 //Line loops, Surface, and Volume for sixth coil384 Line loop(3043) = {�2096, �2095, �2094, �2097};385 Line loop(3044) = {2098, 2099, 2100, 2101};386 Line loop(3045) = {2103, �2099, �2102, 2096};

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387 Ruled Surface(4045) = {3045};388 Line loop(3046) = {�2103, 2097, 2104, �2100};389 Ruled Surface(4046) = {3046};390 Line loop(3047) = {2094, 2105, �2101, �2104};391 Ruled Surface(4047) = {3047};392 Line loop(3048) = {2095, 2102, �2098, �2105};393 Ruled Surface(4048) = {3048};394

395 // New lines to extend casing to symmetry plane396 Line(1120) = {1020, 1019};397 Line(1121) = {1017, 1016};398 Line(1122) = {1014, 1013};399 Line(1123) = {1011, 1010};400

401 //Line loops, Surface for fins402 Line loop(3001) = {�1123, �2002, 2023};403 Plane Surface(4001) = {3001};404 Line loop(3002) = {2003, �1121, �2024};405 Plane Surface(4002) = {3002};406 Line loop(3003) = {�2023, 2001, 2021, 2019}; // curved upper fin edge (zmin)407 Ruled Surface(4003) = {3003};408 Line loop(3004) = {�2021, �2008, 1122};409 Plane Surface(4004) = {3004};410 Line loop(3005) = {2010, 2014, 2009, 2007};411 Plane Surface(4005) = {3005};412 Line loop(3006) = {2020, �2022, �2004, 2024}; // curved upper fin edge (zmax)413 Ruled Surface(4006) = {3006};414 Line loop(3007) = {2011, 2006, �2012, �2013}; // fin bottom415 Plane Surface(4007) = {3007};416 Line loop(3008) = {�2005, 2022, 1120};417 Plane Surface(4008) = {3008};418 Line loop(3010) = {2013, �2017, �2020, �2015};419 Plane Surface(4010) = {3010, 3014, 3020, 3026, 3032, 3038, 3044};420 Line loop(3011) = {�2016, �2019, �2018, �2014};421 Plane Surface(4011) = {3011, 3013, 3019, 3025, 3031, 3037, 3043};422

423 //Line loops, Surface, and Volume for casing ends424 Line loop(3055) = {2126, 2125, 2127, 2118};425 Plane Surface(4055) = {3055};426 Line loop(3056) = {�2118, 2120, 1123, 2121, 1121, �2119};427 Plane Surface(4056) = {3056};428 Line loop(3058) = {�2125, �2123, �2011, 2124, �2010, �2122};429 Plane Surface(4058) = {3058};430 Line loop(3060) = {2015, �2121, 2016, �2124};431 Plane Surface(4060) = {3060};432 //Surface loop(5010) = {4055, 4056, 4057, 4058, 4059, 4060};433

434 //extend casing Line loop(3061) = {�2133, �2130, �2132, �2131};435 Line loop(3061) = {�2133, �1122, �2130, �2132, �2131, �1120};436 Plane Surface(4061) = {3061};437 Line loop(3063) = {2138, �2134, �2009, 2139, 2012, �2137};

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438 Plane Surface(4063) = {3063};439 Line loop(3065) = {2133, 2017, �2139, 2018};440 Plane Surface(4065) = {3065};441 Line loop(3066) = {2132, 2136, �2138, �2135};442 Plane Surface(4066) = {3066};443

444 //Lines for outside445 Line(2106) = {1101, 1097};446 Line(2107) = {1101, 1115};447 Line(2108) = {1102, 1098};448 Line(2109) = {1116, 1097};449 Line(2110) = {1097, 1095};450 Line(2111) = {1095, 1099};451 Line(2112) = {1099, 1101};452 Line(2113) = {1095, 1096};453 Line(2114) = {1096, 1100};454 Line(2115) = {1100, 1099};455 Line(2116) = {1102, 1100};456 Line(2117) = {1096, 1098};457

458 //Refinement lines459 Line(2140) = {1115, 1117};460 Line(2141) = {1116, 1118};461 Line(2142) = {1118, 1098};462 Line(2143) = {1117, 1102};463 Line(2144) = {1115, 1111};464 Line(2145) = {1116, 1112};465 Line(2146) = {1117, 1114};466 Line(2147) = {1118, 1113};467 Line(2148) = {1111, 1006};468 Line(2149) = {1112, 1002};469 Line(2150) = {1113, 1003};470 Line(2151) = {1114, 1007};471

472 //Line loops, Surface, and Volume for outside473 Line loop(3049) = {2108, �2142, �2141, 2109, �2106, 2107, 2140, 2143};474 Plane Surface(4049) = {3049};// outer y max475 Line loop(3050) = {2112, 2106, 2110, 2111};476 Plane Surface(4050) = {3050};// outer x min477 Line loop(3051) = {2113, 2114, 2115, �2111};478 Plane Surface(4051) = {3051};// outer y min479 Line loop(3052) = {2116, �2114, 2117, �2108};480 Plane Surface(4052) = {3052};// outer x max481 Line loop(3053) = {2147, 2150, �2001, �2149, �2145, 2141};482 Plane Surface(4053) = {3053};483 Line loop(3054) = {�2117,�2113, �2110, �2109, 2145, 2149, 2002,�2120, �2127,484 2122, �2007, 2134, �2136, 2130, 2008, �2150, �2147, 2142};485 Plane Surface(4054) = {3054};486 Line loop(3067) = {2144, 2148, 2004, �2151, �2146, �2140};487 Plane Surface(4067) = {3067};488 Line loop(3068) = {�2112, �2115, �2116, �2143, 2146, 2151, 2005, 2131, 2135,

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489 2137, �2006, 2123, �2126, 2119, �2003, �2148, �2144, �2107};490 Plane Surface(4068) = {3068};491

492 //inflow and outflow493 Line Loop(3069) = {2019, 2121, 2020, �2133};494 Plane Surface(4069) = {3069};495 Line Loop(3070) = {2124, 2014, 2139, �2013};496 Plane Surface(4070) = {3070};497

498 Physical Surface(6076) = {4049, 4050, 4051, 4052};499 Physical Surface(6077) = {4053, 4054, 4067, 4068};500 Physical Surface(6079) = {4070};501 Physical Surface(6080) = {4069};502 Physical Surface(6083) = {4001, 4002, 4003, 4004, 4005, 4006, 4007, 4008,503 4010, 4011};504 Physical Surface(6084) = {4055, 4056, 4058, 4061, 4063, 4066};505 Physical Surface(6085) = {4060, 4065};506 Physical Surface(6086) = {4045, 4046, 4047, 4048, 4039, 4040, 4041, 4042,507 4015, 4016, 4017, 4018, 4021, 4022, 4023, 4024,508 4027, 4028, 4029, 4030, 4033, 4034, 4035, 4036};509

510 Surface Loop(5072) = {4070, 4010, 4065, 4011, 4060, 4016, 4015, 4018, 4017,511 4022, 4021, 4024, 4023, 4028, 4027, 4030, 4029, 4034, 4033, 4036, 4035,512 4040, 4039, 4042, 4041, 4046, 4045, 4048, 4047, 4069};513 Volume(6072) = {5072};514

515 Surface Loop(5075) = {4068, 4050, 4049, 4052, 4051, 4054, 4053, 4003, 4001,516 4056, 4055, 4058, 4007, 4063, 4066, 4061, 4004, 4008, 4006, 4067, 4002,517 4005, 4069, 4070};518 Volume(6075) = {5075};519

520 Physical Volume("chilled beam") = {6072};521 Physical Volume("room") = {6075};522

523 Color Blue{ Surface{ 4015, 4016, 4017, 4018,524 4021, 4022, 4023, 4024,525 4027, 4028, 4029, 4030,526 4033, 4034, 4035, 4036,527 4039, 4040, 4041, 4042,528 4045, 4046, 4047, 4048}; }529 Color Green{ Surface{ 4049, 4050, 4051, 4052, 4053, 4054,4067,4068,530 //extend casing 4061, 4065, 4066, 4064, 4062, 4063,531 4061, 4065, 4066, 4063,532 4058, 4055, 4060, 4056, //4057,533 4007, 4008, 4006, 4002, 4010,534 4005, 4011, 4004, 4003, 4001 }; }535 Mesh.ElementOrder = 2;536 Mesh.SecondOrderLinear = 1;537

538 Coherence;539 Coherence;

63

Appendix B

ANSYS Fluent Settings

B.1 Problem Setup

In this section all o↵ the settings selected in ANSYS Fluent for the problem setup areoutlined.

Table B.1: General Parameters

Solver Type Pressure-BasedVelocity Formulation Absolute

Time TransientGravity Y: -9.8

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Table B.2: Models

Multiphase O↵Energy OnViscous k-✏*Radiation O↵

Heat Exchanger O↵Species O↵

Discrete Phase O↵Solidification & Melting O↵

Acoustics O↵*See next table

Table B.3: Viscous Setting

k-epsilon Model RealizableNearWall Treatment Standard Wall Functions

Options Full Buoyancy E↵ectsC2 1.9TKE 1TDR 1.2

Energy Prandtl 0.85Wall Prandtl 0.85

User Defined Functions None

65

Table B.4: Materials

Fluid airFLUENT Fluid Material air

Density Boussinesq, 1.025C p Constant, 1006.43

Thermal Conductivity Constant, 0.0242Viscosity Constant, 1.7894e-05

Thermal Expansion Coe�cient Constant, 0.0034Solid Aluminum

FLUENT Fluid Material aluminumDensity Constant, 2719C p Constant, 871

Thermal Conductivity Constant, 202.4

Table B.5: Cell Zone Conditions

solid-6072 Fluid Stationary Movementsolid-6075 Fluid Stationary Movement

66

Table B.6: Boundary Conditions

def interior-6072 Interior

def interior-6075 Interior

interface-perm-grp3 Interior

interface-perm-grp4 Interior

perm-grp1 WallMomentum

Wall Motion Stationary WallShear Condition No SlipWall Roughness Height = 0; Constant = 0.5

ThermalThermal Conditions Temperature

Material Name AluminumTemperature 300 K Constant

Heat Generation Rate 0 Constant

perm-grp2 Symmetry

permt-grp5-sld-6072 WallMomentum





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Table B.7: Boundary Conditions Continued

perm-grp5-sld-6075 WallMomentum











ThermalThermal Conditions Heat Flux

Material Name AluminumHeat Flux 0 w/m2 Constant







68

Table B.8: Operating Conditions

Pressure 101325Reference Pressure Location (0,0,0)

Gravity 0, -9.8Boussinesq 288.16

Table B.9: Reference Values

Reference Values Area 1Density 1.225Enthalpy 0Length 1000Pressure 0

Temperature 288.16Velocity 1Viscosity 1.7894e-05

Ration of Specific Heats 1.4Reference Zone solid-6075

69

Table B.10: Solution Methods

Pressure-Velocity Coupling scheme SIMPLE

Spatial Discreatization Gradient Least Squares Cell BasedPressure Presto!

Momentum Second Order UpwindTurbulent Kinetic Energy Second Order UpwindTurbulent Dissipation Rate Second Order Upwind

Energy Second Order Upwind

Transient Formulation First Order Implicit

Table B.11: Solution Controls

Pressure 0.3Density 1

Body Forces 1Momentum 0.7

Turbulent Dissipation Rate 0.8Turbulent Viscosity 1

Energy 1

B.2 Solution

In this section the solution methods used are outline.

70

Table B.12: Monitors

Outflow Mass Flow RateX axis Flow Time

Get Data Every 1 Time StepReport Type Mass Flow RateSurfaces interface-permanent-group3

Inflow Mass Flow RateX axis Flow Time

Get Data Every 1 Time StepReport Type Mass Flow RateSurfaces interface-permanent-group4

Outflow PressureX axis Flow Time

Get Data Every 1 Time StepReport Type IntergralField Variable Pressure ¿ Static PressureReport Type IntergralSurfaces interface-permanent-group3

Inflow PressureX axis Flow Time

Get Data Every 1 Time StepReport Type IntergralField Variable Pressure ¿ Static PressureReport Type IntergralSurfaces interface-permanent-group4

Outflow TemperatureX axis Flow Time

Get Data Every 1 Time StepReport Type IntergralField Variable Temperature ¿ Static TemperatureReport Type IntergralSurfaces interface-permanent-group3

Inflow TemperatureX axis Flow Time

Get Data Every 1 Time StepReport Type IntergralField Variable Temperature ¿ Static TemperatureReport Type IntergralSurfaces interface-permanent-group4

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Table B.13: Solution Initialization

Compute from n/aReference Frame Relative to Cell ZoneGauge Pressure 0

X Velocity 0Y Velocity 0Z Velocity 0

Turbulent Kinetic Energy 1Temperature 299.5195

Table B.14: Calculations

Time Stepping Method FixedTime Step Size 2

Number of Time Steps 3000Options n/a

Mas Iterations/Time Step 20Reporting Interval 1

Profile Update Interval 1

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