to Electromagnetic Radiation

Transcription

1 The Characterization of Heat Transfer in a Microchannel Heat Exchanger Exposed to Electromagnetic Radiation by SAMANTHA TIA ROSA TOW B.S., University of Colorado Colorado Springs, 2015 A thesis submitted to the Graduate Faculty of the University of Colorado Colorado Springs in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Mechanical and Aerospace Engineering Department 2016

2 ii Copyright by Samantha Tia Rosa Tow 2016 All Rights Reserved

3 iii This thesis for Master of Science degree by Samantha Tia Rosa Tow has been approved for the Department of Mechanical and Aerospace Engineering by Rebecca N. Webb, Chair James W. Stevens John Adams Date

4 iv Tow, Samantha Tia Rosa (M.S., Mechanical Engineering) The Characterization of Heat Transfer in a Microchannel Heat Exchanger Exposed to Electromagnetic Radiation Thesis directed by Dr. Rebecca N. Webb Beamed energy is a potential power source for a variety of applications. One of the potential applications is using beamed energy as a power source for a heat exchanger. Analysis of such a system requires coupling the electromagnetic response with the thermal response. A microchannel heat exchanger exposed to high frequency electromagnetic waves has been modeled to determine the wall heating due to the application of electromagnetic radiation. This was accomplished by coupling ANSYS HFSS and ANSYS Fluent software programs. The exit temperatures achievable as well as the thermal variations associated with electromagnetic radiation were determined. For this work, constant and temperature-dependent material properties were analyzed as well as different boundary conditions and geometries. Results from this work suggest that material selection is essential. In order to ensure good heat exchanger design one must evaluate the frequency and magnitude of the incoming wave against the material properties of the heat exchanger rather than only considering the thermal properties.

5 v DEDICATION To my parents and brothers.

6 vi ACKNOWLEDGEMENTS The author wishes to thank the following individuals; AFRL for their financial support, which has let me partake in numerous research projects throughout my research career Dr. Rebecca Webb, for her constant support and guidance in all things Dr. Kyle Webb, for getting me this gig in the first place Tom, for being my mentor Ryan, for being the completely objective third party observer Jake, for his contributions getting this project off the ground My fellow research assistants, for their friendship and intellectual contributions

7 Contents vii Chapter 1: Introduction Motivation Applications of Beamed Energy Project Overview... 6 Chapter 2: Background HFSS Fluent Coupling HFSS and Fluent Loss Factor Skin Depth Thermal Diffusivity Chapter 3: Geometry and Modeling Materials Microchannel Geometry Numerical Setup in Workbench: HFSS Numerical Setup in Workbench: Fluent Numerical Setup in Workbench: Coupling Method Chapter 4: Validation Mesh and Convergence HFSS Fluent Chapter 5: Results Excluding Radiative Losses and Convective Cooling... 35

8 viii Study 1a: Constant Properties Study 1b: Temperature-Dependent Properties Including Radiative Losses and Convective Cooling Study 2a: Constant Properties Study 2b: Temperature-Dependent Properties Channel Array Chapter 6: Conclusion Research Conclusion Computational Conclusion Future Work References... 67

9 List of Tables ix Table 1. Constant dielectric material properties Table 2. Constant thermal material properties Table 3. Skin depth calculations Table 4. Loss factor calculations Table 5. Fluent and COMSOL comparison data Table 6. Thermal diffusivity calculations Table 7. Radiation properties in percentages Table 8. Power calculations Table 9. Electric and magnetic field calculations Table 10. Radiation properties in percentages Table 11. Power calculations Table 12. Electric and magnetic field calculations... 58

10 List of Figures x Figure 1. General heat exchanger geometries used in this work Figure 2. Temperature-dependent relative permittivity data for the materials under analysis Figure 3. Temperature-dependent loss tangent data for the materials under analysis Figure 4. Temperature-dependent thermal conductivity data for the materials under analysis Figure 5. Temperature-dependent specific heat data for the materials under analysis Figure 6. Single channel geometry with measurements labeled Figure 7. Channel array geometry with measurements labeled Figure 8. Geometry of the HFSS set-up with boundary conditions labeled Figure 9. Geometry of the Fluent set-up including radiation and convection boundary conditions Figure 10. The basic flow chart of the coupling method Figure 11. The ΔS parameter at each converged iteration Figure 12. The magnitudes of the electric and magnetic fields as well as the power loss as a function of converged passes Figure 13. Percent change of the magnitudes as a function of converged passes Figure 14. Field validation set-up with components labeled Figure 15. The decay in energy flux density as a function of position Figure 16. Channel set-up in COMSOL with boundary conditions labeled Figure 17. Temperature contours of the case evaluated in ANSYS (left) and COMSOL (right) Figure 18. Volumetric loss density contours of the materials under analysis Figure 19. Volume average (-) and exit temperatures (- -) of the channels as a function of time Figure 20. Temperature contours of the exits of the channels at the end of the transient run Figure 21. Radiative properties as a function of time Figure 22. Power calculations as a function of time Figure 23. Electric and magnetic field calculations as a function of time Figure 24. Volume average temperature of the channel as a function of time

11 xi Figure 25. Temperature contours for both the constant (left) and temperature-dependent (right) property case Figure 26. Volume average (-) and exit temperatures (- -) of the channels as a function of time Figure 27. Temperature contours of the exits of the channels at steady state Figure 28. Full length temperature contours at steady state (side view) Figure 29. Radiative properties as a function of time Figure 30. Power calculations as a function of time Figure 31. Electric and magnetic field calculations as a function of time Figure 32. Volume average (-) and exit temperatures (- -) of the channel as a function of time Figure 33. Temperature contours at the channel exit for both the constant (left) and temperaturedependent case (right) Figure 34. Full length temperature contours (side view) of the constant (top) and temperature-dependent case (bottom) Figure 35. Volume loss density of the single channel (top) and the channel array (bottom) Figure 36. Volume average (-) and exit temperatures (- -) of the single channel and channel array geometry Figure 37. Temperature contours for both the single channel (top) and channel array (bottom) geometries Figure 38. Full length temperature contours of the single channel (top) and channel array (bottom).. 62

12 Nomenclature xii A area, m 2 AFRL E H HFSS L P Air Force Research Laboratories electric field, V/m magnetic field, A/m High Frequency Structural Simulator loss factor, F/m dielectric power loss, W S energy flux density, W/m 3 S h energy generation, W/m 3 T temperature, K V volume, m 3 a thermal diffusivity, m 2 /s c p f h h s k q v α s 0 specific heat capacity, J/kgK frequency, Hz heat transfer coefficient, W/m 2 K sensible enthalpy per unit mass, J/kg thermal conductivity, W/mK heat flux, W velocity vector, m/s absorptivity skin depth, m emissivity permittivity of free space, x F/m

13 xiii r κ μ ρ relative permittivity, F/m electrical conductivity, S/m permeability, 4π x 10-7 H/m reflectivity ϱ density, kg/m 3 Stefan-Boltzmann s constant, 5.67 x 10-8 W/m 2 K 4 ω tan( ) transmissivity angular frequency, rad/s loss tangent

14 Chapter 1: Introduction Beamed energy is the transmission of energy in the form of electromagnetic waves from a source to a system a distance away. Currently, beamed energy is a power source for a variety of applications. Such applications include microwave ovens for heating food and for use in melting, drying, and joining in industrial processes. Beamed energy is also under evaluation for use as a power source in a propulsion system. In this approach, the beamed energy is collected and transferred to the propellant by means of a heat exchanger. Typically, it has been assumed that this type of electromagnetic heating can be modeled as an applied heat flux of similar power. However, in order to fully characterize the system, it is necessary to eliminate this simplification and analyze the interaction of the electromagnetic energy with the system and the resultant heat transfer as a coupled system. Evaluating a heat exchanger exposed to beamed energy is a complex process involving heat generation caused by the absorption of the electromagnetic waves, heat transfer by conduction and convection, and the fluid flow. The numerical modeling necessary to solve such a problem requires solving several different equations and coupling the results. For example, εaxwell s equations are needed to compute the electromagnetic field, the heat diffusion equation and an energy balance are needed to predict the heat transfer in the channel and the fluid, and the Navier- Stokes equation is necessary to determine the fluid flow characteristics. Modeling the beamed energy process is a challenge because of the interdependency of the involved mechanisms. The coupling of two different software packages to solve this interdependency is a relatively new technique in computational research. This work seeks to computationally evaluate a component of the beamed energy propulsion concept by analyzing the electromagnetic and thermal design and performance of a microchannel heat exchanger exposed to a 94 GHz input. A coupled ANSYS HFSS and ANSYS Fluent analysis

15 2 was performed for various microchannel set-ups and materials to examine the thermal and electromagnetic response of the heat exchanger. The general heat exchanger geometries used for this work are shown in Fig. 1. Figure 1. General heat exchanger geometries used in this work. Results from this work suggest that material selection is key in the design of a heat exchanger for this particular use and the engineer must evaluate the frequency and magnitude of the incoming wave against the material properties of the heat exchanger rather than just the thermal properties typically evaluated. This work is a complete numerical analysis of beamed energy heating of a three-dimensional, microchannel heat exchanger where the electromagnetic and thermal analysis is fully coupled. This work goes beyond current coupled electromagnetic and heat transfer studies to evaluate the effects of high frequency exposure (94 GHz), which results in large temperature increases on the magnitude of 2000 K. This work was also able to evaluate both constant and temperature-dependent material properties for several materials not usually evaluated for heat exchanger applications and the overall analysis was transient in nature. The overarching goal is to guide future research and experimentation associated with beamed energy as a heating source to a heat exchanger. 1.1 Motivation The space industry is a multi-billion dollar a year business, and with chemical rockets reaching near maximum peak efficiency, the development of more efficient propulsion systems is needed and could promise a revolution in reduced cost and increased access to space. One possible

16 3 solution, and the top-level motivation behind this research, is the beamed energy propulsion system proposed by Parkin [1] [5]. This system uses a ground facility to generate beamed energy focused on a heat exchanger located on the spacecraft. A fluid or gas is passed through the channels of the heat exchanger, the energy from the incoming electromagnetic waves is transferred to the internal fluid, and both its temperature and velocity are increased. The heat exchanger outlet is the inlet to a nozzle, allowing the system to use the high-temperature and high-velocity propellant to produce thrust. This microwave thermal rocket is a single-fuel engine because the energy source remains on the ground. This eliminates the combustion process used by chemical rockets and the associated need to carry heavy reactive gases and the related hardware. By eliminating this weight from the thrust process, the payload of the rocket can be increased, potentially reducing the cost of operation and increasing our access to space. Since its proposition, various aspects of beamed energy propulsion have been analyzed. Gimelshein analyzed the practicality of using beamed energy to increase the thrust of solid rocket motors during launch [6]. The author concluded that by applying microwave radiation to alumina particles which were added to the propellant, the increase in the kinetic energy of the gas results in a sufficient increase to the thrust. It was determined that the total thrust would be increased by 1.5%-15% depending on the input beam. Brown also analyzed the process of beamed power transmission and its applications to the space program [7]. Stepping through the process of the beamed microwave power transmission from the transmitting to the receiving end, Brown demonstrated a scenario of a LEO to GEO transportation system using beamed energy as it energy source. Cornella even took the concept far enough to develop a ground-based power generation source which would produce constant power with a relatively constant spot diameter to the launch vehicle [8]. These studies of various aspects of the overall concept provide strong evidence of the feasibility of beamed energy propulsion. However, for the electromagnetic wave thermal rocket to be a truly viable option, more work must be done to fully characterize the electromagnetic interactions and associated heat transfer of the rocket heat exchanger.

17 4 1.2 Applications of Beamed Energy Since the invention of the microwave in 1946 [9], beamed energy has been under evaluation as a potential power source for a variety of applications but food engineering has been the primary focus. There is a large amount of work related to the modeling of microwave heat transfer to foods and liquids [10] [16]. Much of this research used the finite-element method to determine the absorbed microwave power density within the food as a function of both the dielectric properties and geometry of the material. The power densities are then used to determine temperature profiles within the material. Foods under analysis included potatoes, meat products, salted gels, and bread. The results concluded that microwave heating is significantly dependent on variables such as frequency, dielectric properties, and sample size or geometry. As an interesting side note, many concluded that by adding salt the energy dissipation was increased. Hill steps outside the realm of using microwave energy to merely heat food, and examines its applications in other industrial processes such as melting, sintering, drying, and joining [17], [18]. The author formulates a mathematical model capable of modeling one-dimensional electromagnetic heating of a solid at steady state by solving εaxwell s Equations and the heat equation where all thermal, electrical, and magnetic properties of the material are nonlinearly dependent on temperature. The author notes the occurrence of hot spots in the material and concludes that in many industrial applications of microwave heating, it can be observed that heating is not uniform but rather forms regions of higher temperatures. While Hill was able to model onedimensional dielectric heating in a solid that takes into account the effects of conduction, he neglected to include other modes of heat transfer. Clark notes some advantages to using microwave energy as a tool for high-temperature processing of materials [19]. These advantages include rapid heating, decreased sintering temperatures, improved physical and mechanical properties, and some other unique properties that are not observed by other methods. However, these processes are poorly understood and a fundamental understanding of how microwave energy interacts with materials is necessary. Clark

18 5 strives to increase understanding by evaluating fundamentals such as how to include wave interactions with material and resultant power absorption, as well as, the importance of dielectric properties. The author notes that one of the limitations of modeling the microwave energy process is the lack of dielectric data in the frequency ranges as a function of temperature. The semiempirical Austin method was used to calculate certain material characteristics such as molecular structures, heats of formation, transition state energies, and infrared spectra for a variety of materials with constant dielectric properties. Alpert and Jerby were able to use the finite-difference time-domain method to derive a onedimensional heat-transfer equation to describe media irradiated by microwaves [20]. This was accomplished by coupling the electromagnetic and thermal effects together through the temperature-dependence of the material dielectric properties. The results show the evaluation of the temperature and power-dissipation profiles in the material as a function of time and space for one-dimensional applications. This approach was completely numerical. Coupling two different software packages to solve a complex interdependent problem is a relatively new technique, but the coupling between ANSYS modules for this particular problem is a valid method. Yakovlev showed that modern electromagnetic software is suitable for modeling microwave applications outside of communications, such as electromagnetic heating [21]. The current three-dimensional electromagnetic simulators were designed to model and evaluate communication and high-speed electronics applications; however, the author shows that using the software outside their design scope is possible. Yakovlev evaluated several different electromagnetic software and their ability to predict electromagnetic heating, ANSYS among them. Most of the computational models use either the finite-element method or finite-difference timedomain method. These methods require a great deal of memory but have the ability to accurately approximate electromagnetic heating in complex structures. Yakovlev outlines several conceptual and practical issues associated with the efficient use of the simulator to accurately model microwave heating.

19 6 Sabbagh presents an example of a fully coupled thermal analysis for packaged radio frequency/microwave components using Ansoft s High Frequency Electromagnetic εodule (HFSS) and ANSYS [22]. The author used HFSS to calculate the losses in the component under analysis. This loss was reported as a surface loss density and a volume loss density. These losses were then imported into ANSYS which gives a temperature distribution corresponding to the losses. This process was continued until a steady state was reached. From this information, Sabbagh was able to conclude that the process provided accurate thermal mapping which could be beneficial to the early stages of electrical component design to avoid potential overheating and failure. Sabliov provides one of first examples of coupling ANSYS programs to numerically predict the temperature of a liquid in a continuous flow in a microwave system at steady state [23]. HFSS was used to solve the electromagnetic aspect of the model and the FLOTRAN CFD Module was used to solve the steady state fluid flow and heat transfer problem. However, by assuming the dielectric properties are temperature independent the coupling was only one way. With this model, Sabliov was able to compute the temperature, velocity, electric field distribution, and power intensity in water flow within a PTFE pipe exposed to a microwave system operating at 915 MHz. By comparing analytical results to experimental, Sabliov noted noticeable changes in fluid flow rates and temperature distributions within the pipe and that they were accurately predicted. This steady state work provides the foundation for the application of beamed energy to a heat exchanger. 1.3 Project Overview This work seeks to analyze the thermal response of a three-dimensional, microchannel heat exchanger subject to an input of beamed energy in the form of electromagnetic waves. ANSYS HFSS and ANSYS Fluent software programs were coupled in order to examine numerically the electromagnetic response and the resultant thermal behavior of a channel of a heat exchanger. This model was evaluated at a high frequency with a large power input because these are the desired operating conditions for beamed energy propulsion use. These operating conditions will lead to large temperature changes across the channel. As a way to characterize these temperature

20 7 variations, constant and temperature-dependent material properties were analyzed to understand the overall effect of the system as a function of time. The single channel evaluation was then expanded to provide insight into the behavior of a channel array.

21 Chapter 2: Background This study evaluated the dielectric heating of a single channel exposed to electromagnetic waves. The channel is representative (hydraulic diameter, wall thickness, etc.) of a flow passage in a heat exchanger used for beamed energy propulsion. Dielectric heating, also known as electronic heating or high-frequency heating, is electromagnetic energy dissipated due to molecular rotation caused by electromagnetic wave propagation through the material [24]. This dissipation is referred to as a dielectric loss and affects the entire volume. The loss of electromagnetic energy correlates to a gain in thermal energy. The absorption of the electromagnetic energy by the heat exchanger material was modeled using ANSYS HFSS (High Frequency Structural Simulator). The volumetric loss density equals the thermal volumetric generation and can be used to couple the electromagnetic results to a thermal analysis in ANSYS Fluent. This chapter provides the mathematical and physical background necessary for this work. It also sets out to verify the ability of the HFSS-Fluent coupled model technique to adequately predict the behavior of the system. 2.1 HFSS ANSYS HFSS is an industry standard finite-element method solver for electromagnetic structures. HFSS has the ability to simulate a three-dimensional full-wave electromagnetic field, and is typically used for antenna design and to design complex RF electronic circuit elements [25]. Though heat exchanger heating is outside the scope of what the software was originally designed for, dielectric heating due to applied electromagnetic waves can be analyzed by HFSS. In order to generate an electromagnetic field solution, HFSS employs the finite-element method using εaxwell s equations. In general, the finite element method divides the full problem into smaller regions know as elements and represents the field in each element with a local function. In order to produce the optimal number of elements, HFSS uses an iterative process in which the

22 9 mesh is automatically refined in critical regions. First, it generates a solution based on a coarse initial mesh and then refines the mesh in areas of high error density and generates a new mesh. This will continue until the difference in the S-parameter (to be defined in a moment) from one iteration to the next is less than the ΔS criteria and the solution has completed a minimum number of converged passes which are both set by the user [26]. It should be noted that the electromagnetic fields takes longer to converge than the S-parameter so a mesh independence study is necessary to make sure all the fields are converged. HFSS produces an S-matrix to describe what fraction of power associated with a given field excitation is transmitted or reflected at each port. These quantities are complex numbers. In a two port system, the S-matrix takes on the form shown in Eqn = [ ] Eqn. 2-1 Though it does produce a matrix of values, S 11 and S 21 are the only parameters of interest in this thesis. The S 11 parameter provides the reflectivity (ρ) of the system where ρ= S 11 2 and the S 21 parameter provides the transmissivity (τ) of the system where τ= S For this work, the electromagnetic wave input is un-polarized so two zero-modes were analyzed in the excitation. This causes the S-matrix to have 2x2 submatrices. The overall reflectivity and transmissivity can be calculated using Eqn. 2-2 and 2-3. = = Eqn. 2-2 Eqn. 2-3 The reflectivity and the transmissivity can be used to find the absorption (α) using Equation 2-4 [27]. When analyzing a heat exchanger, it is desired to have a high absorptivity and minimal reflectivity and low transmissivity. + + = Eqn. 2-4

23 10 HFSS uses waveguides to determine the wave propagation in the system. For this analysis, Floquet ports were used as the excitation for the waveguides. The field patterns of a traveling wave, dictated inside the waveguides by its excitation, is determined by solving εaxwell s equations. Equations 2-5 and 2-6 are the exact equations HFSS uses to solve the electric and magnetic fields within the system [26]. When the wave module solves these equations, it obtains an electric field pattern, E(x,y,z). It also independently uses the electric field solution to solve for the magnetic field H(x,y,z). The electric field and magnetic field is solved for at each elemental node. (,, ),, = Eqn. 2-5,, =,, Eqn. 2-6 HFSS uses the field calculations to determine the volumetric loss density at every elemental node. The volume loss density correlates to the dielectric losses in the material. The total volumetric loss density is the change in power with respect to position integrated over the volume, Eqn. 2-7, which will provide the power dissipation in Watts [22]. This value is reported as the total power loss of the entire system, or for the purpose of this thesis, the total power that was absorbed into the channel. = Eqn. 2-7 In this equation, P is the dielectric power loss, f is the frequency, 0 is the permittivity of free space equal to x F/m, r is the relative permittivity, E is the electric field, and V is the volume of the material. With this information, HFSS determines how the heat exchanger will react to the incoming electromagnetic waves, mainly how electromagnetic energy is absorbed by the heat exchanger. The dielectric properties of the material will ultimately determine the reactions.

24 Fluent ANSYS Fluent is an industry standard for computational fluid dynamics, but is not limited to fluid flow evaluations. Other applications it is commonly used for include heat transfer and reactions for industrial applications [28]. The beamed energy problem with its volumetric heat source is well within the modeling capabilities of Fluent. In order to fully characterize the heat transfer in a material, Fluent needs to solve the energy equation, using the appropriate boundary condition equations. In order for Fluent to solve a thermal problem, the energy equation must be enabled. This enables Fluent to calculate any heat transfer within the system. The energy equation Fluent uses for a solid body is provided in Eqn. 2-8 [28]. h + h = + h Eqn. 2-8 In the above equation, ϱ is the density of the material, h s is the sensible enthalpy, k is the thermal conductivity, T is the temperature, and S h is the volumetric heat source. The second term on the left-hand side of Eqn. 2-8 represents convective energy transfer due to rotational or translational motion of the solids. The velocity field, v, is computed from the motion specified for the solid zone. The terms on the right-hand side of Eqn. 2-8 are the heat flux due to conduction and volumetric heat sources within the solid. In the energy equation, the S h term also takes into account the dielectric heating within the channel in the form of generation that is entirely position dependent. Fluent also has to take into account the boundary conditions stipulated by the problem. Fluent has separate equations that it must solve for various boundary conditions. If an external radiation boundary condition is applied, Eqn. 2-9 is used to solve the heat flux to the wall. = 4 4 Eqn. 2-9 In this equation, ε is the emissivity of the material, σ is Stefan-Boltzmann s constant equal to 5.670e-8 W/m 2 K 4, A is the area, T is the surrounding temperature, and T w is the temperature of

25 12 the wall. When a convective heat transfer boundary condition is set to a boundary, Fluent uses Eqn to compute the heat flux at the wall. = h Eqn In this equation, h is the heat transfer coefficient, A is the area, T w is the wall temperature, and T is the freestream temperature. If an insulation boundary condition is necessary, the heat flux at the boundary is set to zero. The boundary conditions described here are important for the evaluation of this thesis. By implementing the energy equation to include boundary conditions and volumetric generation, Fluent can solve for the temperature profiles within the heat exchanger. 2.3 Coupling HFSS and Fluent ANSYS has the ability to connect various aspect of a design together through the Workbench interface. For the application of this thesis, the electromagnetic wave radiation results in HFSS were coupled to the thermal analysis in Fluent through Workbench. This was accomplished through EM mapping. The power loss density value determined in HFSS is defined as a position-dependent generation term in Fluent. Fluent uses this generation term to solve the energy equation and computes the temperature profiles within the material. If the model has constant properties, the results calculated in HFSS are constant, and Fluent calculates the temperature profile with the constant power loss input from HFSS for every moment in time. If the model has temperature-dependent properties, the temperature determined by Fluent is used as the reference for both the dielectric and thermal properties. After the first iteration, HFSS evaluates the model with dielectric properties at the temperature just predicted by Fluent. These new results are given to Fluent which will determine the profiles and the new temperature at which to evaluate the next iteration. This process will continue in an iterative method coupled in Workbench.

26 Loss Factor The dielectric properties of a material will dictate how the material will behave when irradiated. In particular, the loss tangent, tan( ), and the relative permittivity, ε r, of a material will play a big part in how the material will interact with the incoming electromagnetic radiation. The loss tangent is defined as the ratio, or angle in a complex plane, of the real part to the imaginary part of the electric field in the curl equation. For dielectrics with small loss, this angle is 1 and tan. The loss tangent is a measure of the degree to which the medium electrically conducts. It dictates how the dissipation of electromagnetic energy occurs in a material. A medium is considered to be a good electric conductor if it has a large loss tangent. Similarly, the relative permittivity is the factor by which the electric field between the charged particles decreases relative to vacuum [29]. These properties are used to quantify how the material will behave when irradiated through the determination of the loss factor L using Eqn = tan Eqn The loss factor measures the energy dissipated by a dielectric material when in an oscillating field. Usually the loss factor is used to select materials to minimize or maximize dielectric loss [24]. The loss factor will be used in a later section to evaluate the materials under analysis against each other and to validate the results produced in HFSS. 2.5 Skin Depth Skin depth or penetration depth ( s) is the characterization of how deep electromagnetic radiation can penetrate into a conducting material. It is defined as the depth at which the intensity of the radiation attenuates to 1/e or about 37% of its initial magnitude [30]. Eqn is used to determine the skin depth in a material [29]. = Eqn. 2-12

27 In this particular equation, f is the frequency, μ is the permeability equal to 4π x 10-7 H/m, and κ is the electrical conductivity. From this equation, it is clear the dielectric properties and wave frequency will govern the depth of penetration. Skin depth dictates how the energy is dissipated through the material. As the field attenuates, this energy is absorbed into the material and is converted into heat. This is dielectric heating. Skin depth will be used in a later section to validate the electromagnetic field propagation produced in HFSS. 2.6 Thermal Diffusivity In heat transfer analysis, thermal diffusivity measures the ability of a material to conduct thermal energy relative to its ability to store thermal energy. This is a very important thermal property, especially when dealing with transient problems because the thermal diffusivity is the ratio of the time derivative of temperature to its curvature. This value describes how quickly a material reacts to a change in temperature. Eqn is used to calculate the thermal diffusivity of a material and has units of m²/s [27]. = Eqn In this equation, a is the thermal diffusivity, k is the thermal conductivity, ϱ is density, and c p is the specific heat capacity. In materials with high thermal diffusivities, heat is able to move rapidly through the material because it conducts heat quickly relative to its volumetric heat capacity so it will response faster to the change in temperature. This is a very important characteristic for a heat exchanger where it is desired that it heat quickly. Thermal diffusivity calculations will be used to evaluate the materials as potential heat exchanger materials and to validate the thermal response of the system. 14

28 Chapter 3: Geometry and Modeling The electromagnetic absorption by the walls of the heat exchanger was modeled in ANSYS HFSS and the resultant heating due to the application of electromagnetic waves was modeled in ANSYS Fluent. This section lays out the materials and their properties used for this analysis. It also details the computational set-up of both HFSS and Fluent and how the two were coupled. The set-ups provided in this section were used to analyze several different studies. Study 5.1 evaluated a single channel geometry that omitted radiative and convective losses. The simplifications made in study 5.1 were then expanded to include radiative losses and convective cooling in study 5.2. The final study, study 5.3, evaluated an array of channels to determine how neighboring channels affect the energy transfer. Each of these studies were transient in nature and evaluated the response of the system as a function of time. 3.1 Materials Several different materials were analyzed and their response to the electromagnetic waves and resultant heating predicted and compared. The materials were selected based on their dielectric and thermal properties. It was required that the material have a high melting point to ensure the heat exchanger did not melt, as temperatures on the order of 2000 K were expected. In addition, to ensure effective heat exchange the material also needed to have a high absorptivity in the frequency range of interest and a large thermal conductivity. The materials evaluated were silicon carbide, alumina, PTFE Teflon, borosilicate glass, and silicon. For the first set of studies, constant material properties were analyzed. A constant relative permittivity and dielectric loss tangent were required in HFSS. The electric conductivity is another important dielectric property not necessary for HFSS but was necessary for validation calculations. The dielectric properties used in these studies are given in Table 1 [30] [32]. For the thermal aspect

29 of the study, density, thermal conductivity, specific heat capacity, and emissivity were required in Fluent. The thermal properties used in the constant property studies are given in Table 2 [33]. Table 1. Constant dielectric material properties Material Relative Permittivity Dielectric Loss Tangent Conductivity [S/m] Silicon Carbide Alumina e-11 PTFE Teflon e-23 Borosilicate Glass e-12 Silicon e-4 16 Table 2. Constant thermal material properties Material Density Thermal Conductivity Specific Heat Emissivity [kg/m 3 ] [W/mK] Capacity [J/kgK] ( ) Silicon Carbide Alumina PTFE Teflon Borosilicate Glass Silicon For the second set of studies, temperature-dependent material properties were analyzed. As in the first set of studies, the relative permittivity and dielectric loss tangent were required for the electromagnetic investigation and density, thermal conductivity, and specific heat capacity were required for the thermal study. However, all of these properties needed to be a function of temperature. The challenge with implementing dielectric temperature-dependent material properties was that these properties are also highly dependent on the frequency. It became difficult to implement the temperature-dependent material properties for the dielectric properties because there was not

30 17 much information provided by the academic community at the high frequencies and high temperatures that were under analysis. Using the information that was available, the temperaturedependent properties for both the relative permittivity and the loss tangent are shown below in Fig. 2 and 3 [34] [37]. These properties are reported as a function of temperature in degrees Celsius as required by HFSS. The markers on the plot show the data points that were available. Figure 2. Temperature-dependent relative permittivity data for the materials under analysis. Figure 3. Temperature-dependent loss tangent data for the materials under analysis.

31 18 The thermal properties needed for this work have been well-studied and documented at larger temperatures. Emissivity and density will remain constant given the constraints of Fluent while the thermal conductivity and specific heat capacity can be entered as polynomials of various degrees. The temperature-dependent properties for both the thermal conductivity and specific heat capacity are shown below in Fig. 4 and 5 [33], [38] [40]. These properties are reported as a function of temperature in Kelvin as required by Fluent. Like previously, the markers on the plots show what data points were available. Figure 4. Temperature-dependent thermal conductivity data for the materials under analysis. Figure 5. Temperature-dependent specific heat data for the materials under analysis.

32 19 From the temperature-dependent property plots provided, it is clear that not all of the materials have been well-documented at the high temperatures that were under analysis for this thesis. In particular, the dielectric properties of most of the materials are deficient after 600 K. Until there is more information on the dielectric properties at high frequencies and temperatures, accurate analysis with temperature-dependent properties is not possible, except for silicon carbide. Because of this, only silicon carbide will be used in the studies that evaluate temperature-dependent properties. 3.2 Microchannel Geometry Two simplified geometries were used for this work. From previous research performed by AFRL, a heat exchanger with square microchannels and minimal wall thickness was determined to work best for this particular application [41]. To minimize the complexity of the current study initially, a single channel was evaluated with symmetry conditions on both sides. The flow channel was 1 mm square and was surrounded by a channel wall that was roughly 1/3 mm thick. The channel was 130 mm in length. This geometry is shown in Fig. 6. Figure 6. Single channel geometry with measurements labeled. In order to fully understand how neighboring channels modify the energy transfer, a second geometry was also evaluated. The simplified single channel model was expanded to include an array of three channels. The size of the flow channels and the wall thickness between the channels remained the same; however, the length of the channel was reduced to a third its size (43.33 mm) due to computational limitations. The geometry for this model is shown in Fig. 7.

33 20 Figure 7. Channel array geometry with measurements labeled. 3.3 Numerical Setup in Workbench: HFSS In the HFSS model, the 1 mm square channel was composed of air and surrounded by a wall composed of one of the various materials under investigation. Above and below the channel was a rectangular vacuum approximately one wavelength high (3.19 mm). These vacuum columns were used as the waveguide for the system. It was decided to use a Floquet port excitation with master/slave boundary condition pairs for this analysis. This allows the electromagnetic system to be modeled as a periodic structure of infinite length. In principle, this assumption utilized the single channel geometry as the unit cell to model the effects of a heat exchanger composed of many channels that were infinite in length. This assumption only neglected the electromagnetic effects on the ends of the channel which is a valid assumption because the inlet and exit of the channels would be in direct contact with other aspects of the heat exchanger set-up which currently are not under analysis. Above the top vacuum, a Floquet port was assigned to input the electromagnetic waves. Below the bottom vacuum, a second Floquet port was used. This port had no input but merely collected reflectivity and transmissivity information. One side of the channel and vacuum was given a master boundary condition. The opposite side of the channel and vacuum was given a slave boundary condition. The slave condition must be paired with the corresponding master condition. The frequency under analysis was chosen to correspond to an ideal frequency in the W band and the power corresponds to a predetermined 1e7 W/m 2 applied heat flux to the entire heat exchanger. The channel was then exposed to

34 21 electromagnetic waves at a frequency of 94 GHz with a power input of Watts. This was done by setting each of the two modes associated with the top Floquet port by half the total power or Watts each. The geometry and boundary conditions are shown in Fig 8. Figure 8. Geometry of the HFSS set-up with boundary conditions labeled. If the analysis required constant dielectric properties, the dielectric property listed in Table 1 was applied to the channel material. If the analysis required temperature-dependent properties, the thermal modifier option in HFSS was selected and tab files of the relative permittivity and dielectric loss tangent data as a function of temperature were imported and set as the channel material properties. (It should be noted that the default units for temperature in HFSS is Celsius so any imported data must be in the default units). Finally, the initial temperature of the channel was set to 300 K. During this step, both Include Temperature Dependence and Enable Feedback were also selected. 3.4 Numerical Setup in Workbench: Fluent Evaluating the system heat transfer in ANSYS Fluent required the model used in the HFSS evaluation to be modified. The two vacuums were removed as was the fluid inside the channel. Considering that air was used as the working fluid for the heat exchanger, there is little to no energy dissipation from the electromagnetic radiation through the 1 mm fluid in the channel so any thermal effects in the fluid were neglected. However, the fluid flow in the heat exchanger was not neglected,

35 22 it was addressed through the use of a convective boundary condition in an effort to simplify this already complex model. Using these assumptions, this left the channel, in its entirety, for the Fluent analysis. In Fluent, different boundary conditions were required. The inlet of the channel was set to a constant temperature of 300 K. If the analysis neglected radiation and convection all the other walls and the outlet wall were given an insulated boundary condition. If the analysis included radiation and convection, the top of the channel was given a radiation condition where the emissivity,, was set to that of the material. In addition, a convection condition was set on the inner channel walls with a heat transfer coefficient of 1000 W/m 2 K and an ambient temperature of 300 K. Because the convection described above was being used to model the fluid flow within the heat exchanger, the large heat transfer coefficient was calculated and chosen based on previous research done for AFRL. The outlet wall remained insulated. In order to couple the electromagnetic analysis with the thermal evaluation, electromagnetic mapping of the absorbed energy was required in the form of a volumetric energy source. The volumetric energy source is a generation term for each node of the Fluent mesh. This allowed Fluent to calculate the input energy of each node and produce a temperature equivalent. The geometry and boundary conditions for Fluent are shown Fig 9. Figure 9. Geometry of the Fluent set-up including radiation and convection boundary conditions. If the analysis required constant thermal properties, the thermal properties of the wall material were set to its value given in Table 1. If the analysis required temperature-dependent thermal properties, various ordered polynomials were inputted into the material properties based

36 23 on temperature profiles provide in the Materials Section. It should be noted that the default units for temperature in Fluent is Kelvin so any imported data must be in the default units. 3.5 Numerical Setup in Workbench: Coupling Method ANSYS Workbench was used to couple the HFSS and Fluent models. The HFSS results were used as an input to Fluent and then the Fluent results were looped back to HFSS in an iterative process. After solving the HFSS set-up in Workbench, the geometry of HFSS was directed into a new geometry module in Workbench where the vacuum and fluid geometries were disabled. The new geometry, that only includes the channel, was then connected to a new mesh module where the channel was meshed finer. The new mesh and the solution of HFSS were inputted into Fluent s set-up. By connecting the modules this way, the power loss density calculated in HFSS becomes the generation input term in Fluent. Fluent can then solve the thermal effects of the entire system. The temperature results in Fluent were then looped back into HFSS by using an iterator feedback module connected to the setup of HFSS. HFSS used the new temperature information to calculate the new electromagnetic results and the process starts over again. An iteration counter allowed the system to be continuously looped for the desired amount of time. A simple flow chart of the coupling method between HFSS and Fluent in the Workbench interface is shown in Fig. 10. Figure 10. The basic flow chart of the coupling method.

37 Chapter 4:Validation This chapter sets out to verify the ability of the HFSS-Fluent coupling model to sufficiently predict the behavior of the heat exchanger. Simple validations of various aspects of the model were performed and are described below. First, a mesh analysis was performed to confirm that the electric and magnetic field solutions converged properly. Then, aspects of the HFSS model were compared to analytical solutions to make sure the fields were solved correctly. Lastly, the thermal response computed in Fluent was compared to a similar thermal set-up in a different software package to make sure the temperature magnitudes and profiles were within reason. 4.1 Mesh and Convergence To ensure that the mesh that was generated in HFSS was adequate for this finite-element method study, a mesh analysis was performed. This was done by setting the ΔS criteria or the difference in the S-parameter from one iteration to the next to 0.005, which is the value recommended by HFSS for adequate solutions, and increasing the minimum number of converged passes from 1 to 4. By default, HFSS refines the mesh from one converged pass to the next so the model with 4 converged passes is more refined than the model with one converged pass. After each converged pass had completely solved, the magnitudes of the ΔS parameter, the power loss, the electric field, and the magnetic field were calculated within the volume of the channel. By comparing the magnitudes from one iteration to the next, it is possible to see their convergence. Figure 11 shows the change in the ΔS parameter at each converged iteration.

38 25 Figure 11. The ΔS parameter at each converged iteration. It is clear from the magnitude of this plot that the change in the S-parameter is very small after one converged pass (i.e < 13e-5). There is a slight increase in the ΔS-parameter from iteration 3 to iteration 4 because the fields are still converging but the overall change is still on the order of 2e-5 and very minimal. Based on this information, the S-parameters have sufficiently converged on an answer. While the S-parameter may have converged rather nicely, that does not mean the fields have. Figure 12 shows the average magnitudes of the electric field and the magnetic field, and the power loss after each iteration.

39 26 Figure 12. The magnitudes of the electric and magnetic fields as well as the power loss as a function of converged passes. From this figure, it appears that the fields have not completely converged on an answer. If you look at the scale provided with each plot, the change between iterations is only to two decimal places or less. This means that the magnitudes are changing slightly from one iteration to the next. As a way to quantify the change in magnitude for one iteration to the next, the percent change was calculated. Figure 13 shows the percent change in the magnitudes of the electric field, the magnetic field, and the power loss.

40 27 Figure 13. Percent change of the magnitudes as a function of converged passes. This figure shows that the change in magnitude is less than 0.004%. This value is quite insignificant and it is reasonable to assume all the magnitudes have converged on a reasonable answer. With this information, it was determined that the standard size mesh would have a ΔS parameter less than and minimum number of converged passes set to 2. These settings ensured that the results are satisfactory but did not overload the computation capabilities. 4.2 HFSS In order to ensure that the results in HFSS are correct, it was necessary to make sure the fields were being solved correctly. This was important because the fields were used to solve every other aspect of the problem. Skin depth is an effective tool to analytically determine if the fields predicted by HFSS are correct. Skin depth determines how the energy will dissipate within a material. This depth is dependent on characteristics of the incoming wave such as its frequency as well as the characteristics of the material it interacts with, such as its electric conductivity and

41 28 resistivity. Using the material properties from Table 1, the skin depth for the materials under analysis have been calculated using Eqn and are reported in Table 3. Table 3. Skin depth calculations Material Skin Depth [m] Silicon Carbide Alumina 367 Teflon 1.64e9 Borosilicate Glass 519 Silicon The materials under analysis have very small electric conductivities, which makes them poor conductors. This means that the wave will penetrate many wavelengths into the material before decaying away. Based on the calculated skin depths and compared to the overall size of the channel ( mm), some of the materials evaluated have significant skin depths while others do not. For example, silicon carbide and silicon, which are considered to be the good conductors for this study because they have the largest electric conductivities, have the smallest skin depths calculations compared to the other materials but are still relatively large when compared to the size of the channel. Because of this characteristic, these materials will have a high concentration of energy absorption near the exposed surface of the material and that absorption should exponentially decay with depth. On the other hand, Teflon, borosilicate glass, and alumina have large skin depths, much greater than the size of the channel, which means the energy absorption will be more uniform and will take on a more linear decay. In these cases, most of the energy will travel through the material and large transmissivity values are expected. Using this information, a skin depth analysis for silicon carbide was performed. This was accomplished by setting up a HFSS model that consisted of a 1mm x 1mm x 10 mm silicon carbide sample with a vacuum waveguide. An un-polarized Floquet port excitation with a frequency of 94 GHz and a power input of 1 Watt was set above the vacuum. The HFSS set-up is shown in Fig. 14.

42 29 Figure 14. Field validation set-up with components labeled. After evaluating this model, the magnitude of the Poynting vector was calculated as a function of position within the silicon carbide sample. The Poynting vector represents the energy flux density [W/m 3 ] of an electromagnetic field. The Poynting vector, S, can be solved by crossing the electric field with the magnetic field, Eqn. 4.1 [25], [26]. = Eqn. 4.1 As defined by skin depth, the intensity of the Poynting vector should exponentially decay to 37% of its initial magnitude at its calculated penetration depth. Analytically, the exponential decay can be determined using Eqn. 4.2, if the calculated skin depth ( s) is known. = Eqn. 4.2 The data taken from this study, as well as, the analytical solution are plotted in Fig. 15. The dotted line on this figure is the skin depth of silicon carbide calculated earlier and the dot-dash line is the 37% energy flux density of its initial magnitude.

43 30 Figure 15. The decay in energy flux density as a function of position. Based on this figure, the numerics match the analytical values very well and both converge on the cross hairs of the skin depth calculation. From this analysis, it is clear that the electromagnetic fields are correct for the applications used in this thesis. The loss factor can also be used to validate the results in HFSS because it numerically determines the energy dissipated by a material when in an oscillating electromagnetic field. This property is dependent on the relative permittivity and loss tangent of the material. The loss factors for the materials under analysis have been calculated using equation Eqn with the materials properties from Table 1 and are reported in Table 4. Table 4. Loss factor calculations Material Loss Factor, L Silicon Carbide Alumina Teflon Borosilicate Glass Silicon

44 31 Based on this information, the material that has the largest loss factor will have the largest energy dissipation within the material. This means silicon carbide will absorb the most energy from the electromagnetic radiation, followed by alumina, silicon, borosilicate glass, and then Teflon. These calculations will be used to lend some small degree of credence to the results produced by HFSS in the results section. 4.3 Fluent In order to ensure that the thermal results provide by Fluent were correct, it was necessary to perform a thermal analysis. This analysis set out to verify that the temperature magnitudes and profiles computed by Fluent were accurate. To accomplish this, the results of a simplified ANSYS model were compared to results of a similar thermal model computed in a different software package because the complementary experiment, being performed by AFRL, is not yet complete. The software package that was used in this comparison was COMSOL, which is another Multiphysics finite-element solver commonly used for problems of this nature. The set-up used for the ANSYS portion of the analysis included the single channel geometry consisting of constant silicon carbide material properties. The channel was exposed to a 94 GHz electromagnetic wave frequency in HFSS with a power input of 100 Watts. HFSS determined the heat transfer input; in this case, it is position-dependent generation with a total power of Watts. In Fluent, the position-dependent generation was applied and the channel was given a convective boundary condition on the inside walls and the top was allowed to radiate. This is the same set-up described in Section 3.4 and is shown in Fig. 8. Using this set-up, Fluent solved for the temperature profiles within the channel. The COMSOL comparison portion of the analysis is very similar to the set-up in Fluent. This set-up involved modeling the same single channel geometry composed of silicon carbide. Like the ANSYS set-up, this model assigned convection to the inside walls and the top was allowed to radiate. The difference with this model was the primary mode of heat transfer. A heat flux was applied to the top surface of the channel and a uniform heat source set within the body of the

45 32 channel that together was equivalent in size to the generation term in the ANSYS set-up, Watts total. In principle, the application of both boundary conditions should mimic positiondependent generation within the channel. This method was used, rather than recreating the exact boundary conditions used in Fluent, because position-dependent generation is hard to compute within the volume of the original case and it becomes hard to implement this highly positiondependent generation in COMSOL. The COMSOL set-up is shown in Fig. 16. Using this set-up, COMSOL solved the temperature distribution within the channel walls. Figure 16. Channel set-up in COMSOL with boundary conditions labeled. The results for both the ANSYS and COMSOL models were analyzed at steady-state. The volume average temperature and the exit temperature are tabulated for both cases in Table 5. Table 5. Fluent and COMSOL comparison data ANSYS COMSOL Average Temperature [K] Exit Temperature [K] This information shows that the temperature magnitudes for the Fluent case are fairly close to the results determined by the second software package. This verifies that at least the magnitudes are in range with what is to be expected with the given power input.

46 33 Now, in order to validate the temperature profiles produced in Fluent, temperature contours were evaluated at the exits of the channels. Figure 17 shows the temperature profiles of the ANSYS case on the left and of the COMSOL case on the right. Figure 17. Temperature contours of the case evaluated in ANSYS (left) and COMSOL (right). These contours are by no means identical but the solutions are relatively similar. Profile wise, the top of the channel has the hottest area while the bottom inside wall, which was exposed to convective cooling, has the coolest area. Because the outside walls have an insulated boundary condition, it is expected and seen that they are slightly hotter than the inside walls. However, it is clear that position-dependent generation created by dielectric heating is hard to replicate with basic modes of heat transfer. The temperature contour for the ANSYS case are more robust than the temperature contours produced by COMSOL. Yet, the profiles are just analogous enough to verify that the temperature profiles determined by Fluent are what is to be expected with positiondependent generation. From the results of this thermal analysis, it is clear that the thermal results produced in Fluent are what is to be expected from the applications used in this thesis. As a way to establish the thermal differences between the materials, thermal diffusivity can also be used to validate the results produced by Fluent because it numerically determines how the

47 34 material will thermally behave in this transient analysis. In fact, the thermal diffusivity measures the ability of the material to conduct thermal energy relative to its ability to store thermal energy. This property is dependent on the thermal conductivity, the density, and the specific heat capacity of the material. The thermal diffusivity for the materials under analysis have been calculated using the equation in Chapter 2 and are reported in Table 6. Table 6. Thermal diffusivity calculations Material Silicon Carbide Alumina Teflon Borosilicate Glass Silicon Thermal Diffusivity [m 2 /s] 3.115e e e e e-5 Based on this information, the materials that have the largest thermal diffusivity, silicon, silicon carbide, and alumina, will have the ability to move heat more rapidly through the material. This means these materials should respond the quickest to the power input whereas borosilicate glass and Teflon have small thermal diffusivity and should respond the slowest to the power input. These calculations will be used to lend some small degree of credibility to the results produced by Fluent in the results section.

48 Chapter 5: Results This chapter describes and analyzes the results of the various numerical models. The simplified geometry is analyzed with and without radiative and convective losses in Studies 1 and 2. These studies take into account constant material properties, as well as temperature-dependent properties. Finally, the channel array geometry is analyzed in Study 3 to determine how neighboring channels modify the energy transfer from the electromagnetic source. 5.1 Excluding Radiative Losses and Convective Cooling An initial study was performed to model wall heating in a heat exchanger due to application of electromagnetic waves, omitting radiative losses and convective cooling, in a single channel. This study was evaluated in two parts; study 5.1a examined constant properties and study 5.1b examined temperature-dependent properties. In this set-up, one end of the channel was held at a constant temperature of 300 K and all the other walls of the channel were insulated. Because the radiative and convective losses were omitted, the channel continued to absorb the energy from the electromagnetic wave exposure without losing any energy so the temperature of the channel continued to increase for the entire evaluation period. It was desired for the channel to reach 2000 K at the end of the transient analysis, so rather than let the analysis continue to run for all time and exceed rational temperatures, each model was considered complete once the average temperature of the channel reached 2000 K. This initial study was used to spot check the electromagnetic and thermal prediction against simple analytical calculations to determine the relative importance of temperature-dependent properties and to establish the impact of the convective cooling and radiative losses, as well as to analyze the thermal response to dielectric heating. Study 1a: Constant Properties Using the properties listed in Table 1 and Table 2 of Chapter 3, the single channel geometry was modeled. Because the material properties for this analysis were constant, any results calculated

49 36 in HFSS remained constant throughout the study. Based on the S-parameters provided by HFSS, the reflectivity, transmissivity, and absorptivity of the system were calculated using the equations in Chapter 2 and are recorded in Table 7. These values are reported as percentages where the sum of all the radiative properties is 100%. Table 7. Radiation properties in percentages Material Reflectivity Transmissivity Absorptivity Silicon Carbide Alumina PTFE Teflon Borosilicate Glass Silicon From this table, it is clear silicon carbide has the highest absorptivity, followed by alumina, silicon, borosilicate glass, and lastly Teflon. This means silicon carbide will absorb the most incoming electromagnetic radiation followed by alumina, silicon, borosilicate glass, and lastly Teflon. By assessing these results against the loss factor calculations provided in Section 4.2, it is evident that the results produced in HFSS appear consistent with prior validation calculations. As a way to quantify the energy absorbed by each material, the total power loss of the system was calculated. In actuality, the total power loss of the entire system, calculated by the two Floquet ports in HFSS, is the total power absorbed by the channel. These power loss values are tabulated in Table 8 and have units of Watts. Table 8. Power calculations Material Total Loss [W] Silicon Carbide Alumina PTFE Teflon Borosilicate Glass Silicon

50 37 The electromagnetic power loss/thermal gain directly corresponds to the absorptivity of the material. The material with the highest absorptivity should have the largest power input and it does. HFSS also has the capability to calculate the magnitudes of the electric and magnetic fields within the volume of the channel. Using HFSS s fields calculator, the magnitudes for both the electric and magnetic fields were determined and tabulated in Table 9. Table 9. Electric and magnetic field calculations Material Electric Field Magnitude [V/m] Magnetic Field Magnitude [A/m] Silicon Carbide Alumina PTFE Teflon Borosilicate Glass Silicon It appears that the average magnitude of both the electric and magnetic field calculations vary between materials. This is expected because the materials have different dielectric properties which will cause the material to react differently to the incoming electromagnetic waves. From these results, it appears that alumina has the largest magnitudes followed by Teflon and silicon which have very similar values and lastly borosilicate glass and silicon which also have similar values. Materials will behave differently when exposed to electromagnetic radiation, as already seen from the results presented so far. This is because dielectric heating is highly positiondependent based on material properties. This means that the total power calculated previously will have a power distribution that will vary within the volume based on the characteristics of the material. As a way to see how the power is distributed within the materials, contours of the volumetric loss density, which is the main contributor in the calculation of power loss, are shown in Fig. 18 and have units of W/m 3.

51 38 Silicon Carbide Alumina Teflon Borosilicate Glass Silicon Figure 18. Volumetric loss density contours of the materials under analysis. These contours show that the power distribution is highly position-dependent and varies greatly with the material. Some materials have a more uniform power distribution, Teflon and silicon, while others have areas of high concentration, silicon carbide, alumina, and borosilicate glass. Magnitude-wise, silicon carbide has the largest maximum value of 9e9 W/m 3 and Teflon has the smallest maximum value of 5e7 W/m 3. The other materials have a magnitude on the order of W/m 3. This position-dependent power distribution will determine the position-dependent heat generation for each channel. Take note of the areas with a high concentrations of volumetric loss density, these will contribute to hot spots in the thermal analysis. The thermal response of the system is completely dependent on the properties of the material. The dielectric properties determine the power absorption or the position-dependent generation within the material and the thermal properties determine how quickly the material reacts to the change in temperature caused by the generation. These two aspects coupled provide the total thermal response of the system.

52 39 The average and exit temperatures for each material as a function of time are shown in Fig. 19. Because there is a large time difference between some of the materials, the same plot has been split in two separate plots in order to see the time response of all the materials properly. The solid lines are the volume average temperatures and the dashed lines are the temperatures at the channel exits. Figure 19. Volume average (-) and exit temperatures (- -) of the channels as a function of time. From this figure, it is seen that silicon carbide responds the fastest to dielectric heating followed by silicon, alumina, borosilicate glass, and finally Teflon. It is also evident that the trends are linear in nature where the exit temperature of the channel reaches 2000 K slightly faster than the volume average temperature. This is due to the fact that heat moves from hot to cold so the exit of the channel is the hottest area and the heat moves towards the cooler inlet which produces the smaller average temperature. Energy absorption plays a big part in how quickly the material reaches 2000

53 40 K. If there is more input energy, the channel will require less time to heat up. Due to the dielectric properties of the materials, the amount of the incoming radiation that is absorbed and results in heating varies. As shown in Table 8 of this section, silicon carbide, alumina, and silicon have larger energy inputs which partly explains their quick thermal response. The thermal properties also play a big part in how quickly a material will respond. As stated in the validation section, the thermal diffusivity will also determine which materials will respond to the change in temperature quicker. For example, alumina has a larger power input but has a smaller thermal diffusivity than silicon and silicon is able to reach 2000 K about a second faster than alumina. Comparing these results in Fig. 19 against the power loss calculations in Table 8 and thermal diffusivity calculations in Chapter 4, the results produced in Fluent match well. To clearly understand the temperature distribution within the walls of the channel, temperature contours were taken at the exit of the channels at the completion of the transient analysis. The temperature contours for all five materials are shown in Fig. 20. Silicon Carbide Alumina Teflon Borosilicate Glass Silicon Figure 20. Temperature contours of the exits of the channels at the end of the transient run.

54 41 The thermal material properties contributed to the variations of temperature contours within the different materials under analysis. Materials with large thermal conductivities such as silicon carbide and silicon, will appear to have a more uniform distribution with small temperature ranges, < 9 K. Material with small thermal conductivities such as Teflon and borosilicate glass have large variations in temperature distributions and have a temperature range of +30 K. Alumina has fallen somewhere in between these two extremes. As predicted by the volumetric loss density contours, Fig. 18, the hottest spots in the channels correspond to areas that had a large concentration of power during the energy absorption process. For example, as seen in Fig. 18, silicon carbide and borosilicate glass had a large concentration of power in the side walls of the channel. This large power concentration has resulted in the hot spots seen, in those same regions, in Fig 20. It is also seen in Fig. 20 that Alumina has areas of hot spots resultant from its power distribution. This confirms that dielectric heating is position-dependent heat generation and varies with material. Study 1b: Temperature-Dependent Properties The simplified geometry model that excluded radiation and convective cooling of Study 1a was modified to incorporate temperature-dependent properties. Silicon carbide was the only material considered for this study because it was the only material that had adequate temperaturedependent properties at the high temperatures that were considered for this analysis. It also appears to be the only material suitable for this study due to the high absorption seen in the previous study. The temperature-dependent properties that were used in this study can be found in Chapter 3, Figs Because these properties varied with temperature, the results under analysis also varied with temperature and became a function of time. Based on the S-parameters provided by HFSS at each time step, the radiative properties of the temperature-dependent model are shown in Fig. 21. The reflectivity, the transmissivity, and the absorptivity for the model appear as percentages.

55 42 Figure 21. Radiative properties as a function of time. As seen from this figure, the radiative properties vary significantly with time. Both the reflectivity and transmissivity of the model peak and begin to decrease with time. Actually, transmissivity is almost non-existent at the end of the analysis. Absorptivity bottoms out rather quickly and begin to increase as time increases. This is an important trait because it is desired to have a high absorptivity within the body of the heat exchanger and it appears that absorptivity rises as temperature increases. However, it seems that near the end of the run, where the temperature begins to approach its maximum, the radiative properties begin to level out. As a way to quantify how much energy is being absorbed by the material, the total power in the system as the result of the applied electromagnetic radiation was calculated at each time step. This is shown in Fig. 22 and has units of Watts. Figure 22. Power calculations as a function of time.

56 43 The amount of power calculated within the material directly corresponds to the absorptivity so the trend seen in this figure for power directly resembles the absorption trend shown in Fig. 21. As absorptivity increases, the total power input increases in the same manner. At the end of the analysis, the power input reaches a maximum of ~1100 W. The magnitudes of the electric and magnetic field within the volume of the channel were also calculated as a function of time. Using HFSS s fields calculator, the magnitudes for both the electric and magnetic fields were determined at each time step and are shown in Fig. 23. Figure 23. Electric and magnetic field calculations as a function of time. Even with temperature-dependent properties, the fields are the same magnitude as the constant property case. The electric and magnetic fields appear to be decreasing with time due to the change in dielectric properties where both the relative permittivity and the loss tangent are increasing with temperature. For the sake of comparison, the thermal response of the temperature-dependent system was evaluated against the thermal response of the constant property system from the previous section. The comparison of these two different cases allows the impact of including temperature-dependent properties on the final results to be determined.

57 The volume average temperature for both the constant and temperature-dependent property cases are shown in Figure Figure 24. Volume average temperature of the channel as a function of time. It was expected and seen that the constant temperature case was linear because the power input was constant for all time. However, it is not expected that the temperature-dependent case was linear because the power input varies with temperature and therefore time. Figure 24 shows the expected response for both cases. Early on, the temperature-dependent case responds much faster than the constant case and reports hotter temperatures for the same instant in time. At about 0.9 second there is an approximate maximum temperature difference of 120 degrees between the two cases. As time increases the temperature margin between the two cases begins to decrease and the temperaturedependent case is not responding as quickly to the change in temperature as it was in the beginning. Because of this, both cases seem to reach 2000 K at moderately the same time with the temperaturedependent case just slightly quicker (~0.08 seconds). The difference in material properties is responsible for the variations between the two cases. As temperature increases, the thermal diffusivity of the temperature-dependent silicon carbide decreases. This means as time increases,

58 45 the temperature-dependent case is not able to respond as quickly to the change in temperature as it had previously. This explains why the two cases seem to reach 2000 K at the same time even though the temperature-dependent case initially responses faster and starts out being significantly hotter. To clearly understand the temperature distribution within the walls of the channel for these two cases, temperature contours were taken at the exits of the channel. These contours are shown in Fig. 25 where the constant case is on the left and the temperature-dependent case is on the right. These contours were taken at the end of the transient analysis where the average temperature had reached 2000 K. Figure 25. Temperature contours for both the constant (left) and temperature-dependent (right) property case Because both cases are being exposed to the same electromagnetic radiation and are made of the same material, it is expected that the temperature profiles should be similar in nature. This is shown in Fig. 25. Distribution wise the two profiles are identical. The two contours have the same areas of hot spots and cool areas which were determined by the position-dependent heat generation seen in the volumetric loss density contours of Fig. 18. It s in the temperature scales where these two cases differ. The constant property case reports that there is a 9 degree difference between the hot spots and the cool areas where the temperature-dependent case reports that there is a 44 degree difference. The difference in thermal conductivity is responsible for the variations in the temperature gradients. As temperature increases, the thermal conductivity of the silicon carbide

59 46 decreases. This means at the completion of the analysis, the temperature-dependent case has a smaller thermal conductivity than the constant property case which explains the difference in temperature gradients. For this study, two separate cases that evaluated temperature-dependent and constant properties were compared. It was found that temperature-dependent properties are important when evaluating a material that undergoes a large temperature change because these materials properties provide a more accurate depiction of how the material will behave, but this increase in accuracy still comes at a cost. By making the properties temperature-dependent, it added complexity to the model which in the end increases the computational memory and time necessary to solve the already complex coupled problem. With that being said, temperature-dependent properties may provide a more accurate representation but, depending on the time of the simulation, the constant property case provides a reasonable approximation with less computational exertion. 5.2 Including Radiative Losses and Convective Cooling As an extension to the initial study, a second study was performed to model wall heating in a heat exchanger due to application of electromagnetic waves in a single channel which included radiative losses and convective cooling. This study was evaluated in two parts; Study 2a evaluated constant properties and Study 2b, temperature-dependent properties. By including radiative losses and convective cooling, the model was able to simulate energy lost to the surroundings as well as to fluid flow through the channel. Even though the study is transient in nature, the channel was able to reach steady-state values after about 30 seconds. This study more accurately models the heat transfer in the heat exchanger and predicts a more realistic thermal response to dielectric heating. Study 2a: Constant Properties The same single channel geometry that was used in the previous study was also used for this study; however, radiation and convective cooling boundary conditions were employed. This model was also transient and results were collected as a function of time. Because the material

60 47 properties for this analysis are the same as those used for the constant property model with no radiative losses and convective cooling and no electromagnetic conditions were modified, the results from the HFSS portion of the modeling remain unchanged and are tabulated in Tables 7-9 of Section 5.1. These results include the reflectivity, the transmissivity, the absorptivity, the electric field magnitude, the magnetic field magnitude, and the total power. Unlike the previous model, these models will reach steady-state solutions due to the fact that these models will reached a point where they are losing energy just as quickly as it is being absorbed by the electromagnetic radiation. The average temperature and the exit temperature of channels composed of different materials are shown in Fig. 26 as a function of time. The solid lines are the average temperatures and the dashed lines are the temperatures at the channel exit. Figure 26. Volume average (-) and exit temperatures (- -) of the channels as a function of time. This figures reveals how the temperature of the channel, initially at 300 K, increases rather quickly and steadies out after 30 seconds. It is also noted that the exits of the channels reach a slightly

61 48 higher temperature than the average temperature. In this study, silicon carbide was able to reach the highest average temperature of 1142 K, followed by alumina at 856 K, silicon at 665 K, borosilicate glass at 369 K, and finally Teflon at 314 K. The maximum temperatures these materials achieve directly corresponds to how much power the material was able to absorb from the electromagnetic radiation. Those materials that were able to absorb a significant amount of energy reached higher temperature compared to those who were only able to absorb a small amount of energy and reached a temperature slightly about the initial temperature. Ideally, it was desired that the channel reach 2000 K, however these materials have fallen short of reaching this target. To clearly understand the temperature distribution within the walls of the channel, temperature contours were taken at the exit of the channel. Contours for all five materials are shown in Fig. 27 where the temperature profiles have reached steady-state solutions. Silicon Carbide Alumina Teflon Borosilicate Glass Silicon Figure 27. Temperature contours of the exits of the channels at steady state.

62 49 These temperature contours vary greatly based on material. Some materials appear to model the heat transfer within the channel as uniform generation, silicon and Teflon, while others appear to model the heat transfer within the channel as an applied surface heat flux, silicon carbide. Borosilicate glass, on the other hand, has a completely position-dependent heat source based on the power distribution within the volume of the channel. As predicated by the volumetric loss density contours, Fig. 18, areas with a high concentration of power result in thermal hot spots. These position-dependent hot spots can be seen in both the borosilicate glass and the alumina models. While these models vary in temperature magnitudes, the temperature gradients across the channel for all materials are small, less than 30 K. Temperature also varies down the length of the channel from its constant temperature inlet to the maximum temperature at its exit. In order to show this, temperature contours were taken down the length of the channel. The side view of the channel walls are shown in Fig. 28.

63 50 Silicon Carbide Alumina Teflon Borosilicate Glass Silicon Figure 28. Full length temperature contours at steady state (side view).

64 51 Once again, the temperature profiles down the length of the channel varies based on material. Silicon carbide, alumina, and silicon appear to reach a uniform temperature a short distance away for the constant temperature inlet. It is also seen that Teflon and borosilicate glass do not reach uniform temperature but rather have a non-uniform temperature variation down the length of the channel. These results can once again be tied back to the power distribution seen in the volumetric loss density contours in Fig. 18. For example, the borosilicate glass has a large power distribution in the side walls of the channel which has made it hottest at the center of the channel. Similarly, Teflon has a different power distribution that has caused the side walls to be coolest at the center of the channel. These characteristics can be seen down the length of the channel in Fig. 28. Study 2b: Temperature-Dependent Properties For this analysis, it was required to modify the pervious study, where radiation and convection were included, to also implement temperature-dependent properties. As in the previous temperature-dependent study, silicon carbide was the only material considered and the properties can be found in Fig Because these properties varied with temperature, all the results under analysis are also a function of temperature and therefore time. Based on the S-parameters provided by HFSS at each time step, the radiative properties of the temperature-dependent model including radiation and convective cooling were determined and are shown in Fig. 29. The reflectivity, the transmissivity, and the absorptivity as reported as percentages.

65 52 Figure 29. Radiative properties as a function of time. From this figure, it is clear that the radiative and convective cooling conditions have caused a significant impact on the radiative properties and have produced significantly different results than the analysis that excluded their effects, Fig. 21. It can be seen from Fig. 29 that reflectivity continues to increase before maxing out at 53.6%. Transmissivity peaks out before beginning to decrease and steadying out at 26.1%. These values are quite large for a system that needs to minimize reflection and transmission in order to increase absorption. In the figure, absorptivity valleys out before slightly increasing to reach a steady value just over 20%. Ideally, a large absorptivity is desired but an absorptivity of 20% should produce adequate heating. It should be noted that each radiative property reaches steady values after about 10 seconds. As a way to quantify how much energy corresponded to a reported absorptivity, the total power of the system was calculated at each time step. This is shown in Fig. 30 and has units of Watts.

66 53 Figure 30. Power calculations as a function of time. As to be expected, the trend for power seen in Fig. 30 directly corresponds to the absorptivity trend seen in Fig. 29. Based on this relationship, an absorptivity of 20.3% corresponds to a steady-state power input of 352 Watts. This is the power input the model retains at its steady-state solution. The magnitudes of the electric and magnetic field within the volume of the channel were also calculated as a function of time. Using HFSS s fields calculator, Figure 31 shows the average magnitudes for both the electric and magnetic fields at each time step. Figure 31. Electric and magnetic field calculations as a function of time.

67 54 The electric field reaches a peak in magnitude and begins to decrease before reaching a steady-state solution after about 10 seconds. The magnetic field quickly increases before leveling off and reaching its steady-state value after about 5 seconds. Comparatively, the steady-state value of the electric field (2.6e4 V/m) is slightly lower than of the value that was calculated in the constant property case (4e4 V/m). Conversely, the value for the magnetic field (250 A/m) is slightly higher than the value that was calculated for the constant property case (199 A/m). The differences in these magnitudes are due to the fact that the dielectric properties are increasing with temperature which have altered the electromagnetic fields within the volume of channel. As in the previous temperature-dependent case, the thermal response of the temperaturedependent system was evaluated against the thermal response of the constant property system to see how the different material set-ups would vary the final results. For the thermal analysis performed in Fluent, the average and exit temperatures for both the cases are shown in Figure 32. The solid lines are the average temperatures and the dashed lines are the exit temperatures. Figure 32. Volume average (-) and exit temperatures (- -) of the channel as a function of time.

68 55 From this figure, it is seen that both profiles follow that same trend but there is a slight variation between the two cases. The figure shows how the temperature for both cases, initially at 300 K, increases rather quickly and steadies out after 30 seconds; however, the temperature-dependent case responds slightly faster than the constant property case and reaches its steady-state value earlier. Magnitude-wise, the cases reach slightly different end temperatures. Based on the volume average temperature of the channels, the temperature-dependent case is hotter (1149 K) compared to the constant property case (1142 K). Based on exit temperature of the channels, the constant case is hotter (1182 K) compared to the temperature-dependent case (1181 K). This means the temperature-dependent case has a smaller temperature margin between its average temperature and the temperature at the exit. It is interesting to note that even though the properties vary greatly for the temperature-dependent case, both cases converge on very similar end solutions. To clearly understand the temperature distribution within the walls of the channel for these cases, temperature contours were taken at the exits of the channels. These contours are shown in Fig. 33 where the constant case is on the left and the temperature-dependent case is on the right. Figure 33. Temperature contours at the channel exit for both the constant (left) and temperature-dependent case (right). Both cases are being exposed to the same electromagnetic radiation. This radiation will produce the same position-dependent heat generation but will vary in magnitude. Because the exposure is

69 56 the same, it is expected that the temperature distribution should be similar but the power inputs are different so the magnitudes should differ for one another. This is seen in Fig. 33. The profiles within the material are practically identical but the temperature scales are slightly different. In fact, the temperature-dependent case has a larger temperature gradient (17 K) compared to the constant property case (6 K). As stated previously, the difference in thermal conductivities are responsible for the variations in the temperature gradients seen in these contours. As temperature increases, the thermal conductivity of the temperature-dependent silicon carbide decreases. This means at the steady state solutions, the temperature-dependent case has a smaller thermal conductivity compared to the constant property case explaining the larger temperature gradient. Temperature should also vary down the length of the channel from its constant temperature inlet to the maximum temperature at its exit. In order to show the temperature variation between the two cases, temperature contours were taken down the length of the channel, side view, and is shown in Fig. 34. Constant Temperature-Dependent Figure 34. Full length temperature contours (side view) of the constant (top) and temperature-dependent case (bottom).

70 57 Besides the slight difference in maximum temperatures (3 K), the full length contours show the same temperature characteristics. The contours for both cases appear to have almost vertically uniform temperature across the channel and reach a uniform maximum temperature a short distance away for the constant temperature inlet. This uniform maximum temperature continues down the length of the channel. This is actually ideal in heat exchanger applications. If the channel reaches uniform maximum temperatures for most of the channel length, the heat exchanger will be more efficient at transferring heat to the working fluid. For this study, radiative and convective losses were included in the heat transfer analysis of the heat exchanger. Even though the study was transient in nature, the channels, composed of different materials, were able to reach steady-state values after about 30 seconds and were able to reach various levels of temperature increases. The results from this study suggest that it is essential to include both radiative losses and convective cooling in order to more accurately model the heat transfer in the heat exchanger. With the addition of these losses, some complexity was added to the problem but it had little impact on computational time. By incorporating radiation and convection boundary conditions to the heat exchanger, the model received minimal additional computational resources but gained accuracy because it allows the model to predict a more realistic thermal response due to dielectric heating. 5.3 Channel Array By expanding the simplified coupled model of a single channel to include an array of channels, this study is able to examine how neighboring channels modify the energy transfer from the electromagnetic source. Taking the channel array set-up that was described in Chapter 3 and assigning silicon carbide constant properties, this model analyzed the thermal response of a heat exchanger when multiple channels were exposed to electromagnetic radiation. This model also included radiative and convective losses to more accurately model the heat exchanger. The channel array was exposed to the same electromagnetic radiation as the single channel geometry with a frequency of 94 GHz and a power input of Watts. For the sake of comparison, the channel

71 58 array will be evaluated against the single channel geometry. Because the channel array was limited in length compared to the single channel, the results were taken at the same channel position (43.33 mm). The length of the channel array may be shorter than desired but as seen from the previous study, there was little change in the temperature profile after about a third of the distance, so mm is actually sufficient for this channel array study. Based on the S-parameters provided by HFSS, the reflectivity, transmissivity, and absorptivity of both the single channel and the channel array were calculated using the equations in Chapter 2 and are recorded in Table 10. As a way to quantify the energy absorbed for each model, the total power loss of the system was calculated. These power loss values are tabulated in Table 11. Finally using HFSS s fields calculator, the magnitudes for both the electric and magnetic fields were determined and tabulated in Table 12. Table 10. Radiation properties in percentages Geometry Reflectivity Transmissivity Absorptivity Single Channel Channel Array Table 11. Power calculations Geometry Power [W] Single Channel Channel Array Table 12. Electric and magnetic field calculations Geometry Electric Field [V/m] Magnetic Field [A/m] Single Channel Channel Array

72 59 The reflectivity, the transmissivity, and the absorptivity for both models are identical. Considering that both models used the same dielectric material properties, this was expected. Given that both systems have the same power input of Watts and it was previously determined in Table 10 that both cases had the same absorptivity, it was entirely expected that the power loss for both models be identical as well. The magnitude of the electric and magnetic field for both the single channel and the channel array geometries were also essentially the same and this was to be expected as well. As a way to see the power distribution within the single channel compared to the channel array, contours of the volumetric loss density are shown in Fig. 35 and has units of W/m 3. Figure 35. Volume loss density of the single channel (top) and the channel array (bottom). The contours of the volumetric loss density for both cases are indistinguishable between one another. This means that the power is distributed in the same manner for the channel array as it was for the single channel which will give the models the same position-dependent generation for the thermal analysis in Fluent.

73 60 The average temperature and the exit temperature of both the single channel and the channel array geometry are shown in Fig. 36 as a function of time. The solid lines are the volume average temperatures and the dashed lines are the temperatures at the exits of the channels. Figure 36. Volume average (-) and exit temperatures (- -) of the single channel and channel array geometry. Volume-wise, the average temperatures of the single channel and the channel array differ from one another. Keeping in mind that the channel array is 1/3 the length of the single channel, it is expected that the channel array should be cooler than the single channel because the channel array is missing 2/3 of its length which would in fact have the larger temperatures for its volume average temperature calculation. However, if the exit temperatures of the two cases are examined, it appears that these models are at the same temperature. This is because these temperatures were taken down towards the end of the channel where the temperature has become a uniform maximum temperature. To clearly understand the temperature distribution within the walls of the channel for these two cases, temperature contours were taken at the exit of the channel array and at 1/3 length of the single channel. These contours are shown in Fig. 37 where the single channel is on top and the channel array is on bottom.

74 61 Figure 37. Temperature contours for both the single channel (top) and channel array (bottom) geometries. From the electromagnetic response analysis, it was determined that the both cases had identical position-dependent generations. When the thermal response was analyzed in Fluent, the identical positon-dependent generations terms produced identical thermal profiles. Magnitude-wise, the two cases only vary less than half a degree. Temperature also varies down the length of the channel from its constant temperature inlet to the maximum temperature at its exit. In order to show the temperature variation between the two cases, temperature contours were taken down the length of the channel and is shown in Fig. 38.

75 62 Single Channel Channel Array Figure 38. Full length temperature contours of the single channel (top) and channel array (bottom). Besides the slight difference in maximum temperatures (1 K), the full length contours show the same temperature characteristics up to the difference in lengths. Up to the point where the lengths match, contours for both cases appear to be the same with vertically uniform temperature across the channel and reaches an almost uniform maximum temperature a short distance away for the constant temperature inlet. By examining the results from the channel array, it can be concluded that including the neighboring channels in the simulation will not modify the model predictions. In hindsight, these results should have been expected with the use of the Floquet port excitation. This excitation allows periodic structures, like the channels of a heat exchanger, to be modeled as a unit cell. With appropriate boundary conditions, HFSS assumes the unit cell (the single channel) has an infinite amount of structures (the entire heat exchanger) of infinite length. By simply adding more channels to the analysis, we simply made the unit cell larger but did not modify the energy transfer that was previously determined in the single channel.