Convective Heat Transfer in Parallel Plate Heat Sinks. A thesis presented to. the faculty of. In partial fulfillment

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1 Convective Heat Transfer in Parallel Plate Heat Sinks A thesis presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Master of Science Gregory M. Holzaepfel March Gregory M. Holzaepfel. All Rights Reserved.

2 2 This thesis titled Convective Heat Transfer in Parallel Plate Heat Sinks by GREGORY M. HOLZAEPFEL has been approved for the Department of Mechanical Engineering and the Russ College of Engineering and Technology by Khairul Alam Moss Professor of Mechanical Engineering Dennis Irwin Dean, Russ College of Engineering and Technology

3 3 ABSTRACT HOLZAEPFEL, GREGORY M., M.S., March 2011, Mechanical Engineering Convective Heat Transfer in Parallel Plate Heat Sinks (67 pp.) Director of Thesis: Khairul Alam This research study explores the transition between developing and fullydeveloped flow to evaluate the heat transfer for single-phase convection in a parallel plate heat sink. A range of modified Reynolds number values between 14.0 and 44.8 was selected based upon pre-existing work to be representative of this transition region. The experimental results were found to be in reasonable agreement with the pre-existing model for high aspect ratio channels. Several numerical solutions for rectangular duct flow are also compared with the observed results. The experimental results for the Nusselt number were found to be within a maximum observed error of 16.5% of the published model for the transition between developing and fully-developed flow. This error exceeds the maximum standard deviation for the experimental results of Reasons for differences between the current experimental results and the selected models are discussed. Approved: Khairul Alam Moss Professor of Mechanical Engineering

4 4 ACKNOWLEDGMENTS First and foremost, I would like to acknowledge Dr. Khairul Alam, whose support and guidance have allowed me to finish this work. Additionally, I would like to thank Dr. Carole Wolmedorf and Dr. Israel Urieli for their aid and feedback as thesis committee members. Finally, I would like to thank my wife, Erin, for her gentle encouragement to finish this work.

5 5 TABLE OF CONTENTS Page Abstract... 3 Acknowledgments... 4 List of Tables... 7 List of Figures... 8 Nomenclature... 9 Chapter 1: Heat Transfer Mechanisms Concepts of Heat Transfer Conduction Heat Transfer Convection Heat Transfer Radiation Heat Transfer The Fundamental Laws Extended Surfaces Chapter 2: Related Convection Problems Flow Over a Flat Plate Flow in a Duct or Pipe Flow Between Infinite Parallel Plates Related Work by Others Fin Effectiveness Effects Chapter 3: A System Description Proposed Model of Limiting Conditions The System Major Assumptions Chapter 4: Experimental Results Raw Data Correlation to Proposed Models Raw Data Chapter 5: Conclusions and Recommendations Results of this Study Future Work... 65

6 References

7 7 LIST OF TABLES Page Table 1: Comparison of Heat Transfer Coefficients by Medium and Mechanisms Table 2: Numerical Solution of the Mean Nusselt Number at Different Aspect Ratios Table 3: Power and Flow Rate Combinations Sampled Table 4: Experimental Sample Result Summary of Air Stream Temperature Rise Table 5: Correlation of Experimental Results to Theoretical Predictions... 63

8 8 LIST OF FIGURES Page Figure 1: Conduction Heat Transfer Mechanism Figure 2: Convection Heat Transfer Mechanism Figure 3: Radiation Heat Transfer Mechanism Figure 4: 1-Dimensional Representation of Conduction Heat Transfer Figure 5: Boundary Layer Development Figure 6: Boundary Layer Transition Between Parallel Plates Figure 7: Heat Flow in a Typical Fin Figure 8: Boundary Layer Development Over a Stationary Plate Figure 9: Depiction of Transition between Developed and Developing Flow Figure 10: Heat Sink Under Study Figure 12: Heat Sink Assembly Geometry Figure 13: Thermocouple Arrangements (Shown in Red) Figure 14: Composite Nusselt Correlation to Experimental Observations... 61

9 9 NOMENCLATURE - Area for Conduction Mechanism [m 2 ] - Area for Convection Mechanism [m 2 ] - Area of Exposed Fins [m 2 ] - Total Exposed Surface Area [m 2 ] - Width of Rectangular Duct [m] - Specific Heat Capacity at Constant Pressure [J / kg K] - Hydraulic Diameter [m] - Height of Rectangular Duct [m] - Convection Heat Transfer Coefficient [W / m 2 -K] - Thermal Conductivity [W / m-k] - Thermal Conductivity of the Fin [W / m-k] - Thermal Conductivity of the Fluid [W / m-k] - Length of Rectangular Duct [m] - Entrance Length [m] - Mass Flow Rate [kg / s] - Number of Fins - Rate of Heat Transfer [W] - Actual Rate of Heat Transfer Including Fin Effects [W] - Rate of Conduction Heat Transfer [W] - Rate of Convection Heat Transfer [W] - Ideal Rate of Heat Transfer Excluding Fin Effects [W] - Power Supplied by (1) Heating Pad [W] - Fin Thickness [m] - Inlet Fluid Temperature [ C] - Film Temperature [ C] - Surface Temperature in Contact with Fluid [ C] - Mean Stream Velocity [m / s]

10 10 - Free Stream Velocity [m / s] - Distance Downstream from Incident Edge (Inlet) [m] - Distance for Energy Conduction From Source to Sink [m] - Simplification Parameter, Equivalent to, - Dimensionless Numbers: - Eckert Number: - Grashof Number: - Graetz Number: Characteristic Parameter Describing Flow in a Duct - Nusselt Number: - Nusselt Number for Characteristic Channel Width b - Nusselt Number for Characteristic Hydraulic Diameter - Mean Nusselt Number for Flat Plate Flow - Mean Nusselt Number for Channel Flow ( ) - Actual Mean Nusselt Number Including Fin Effects ( ) - Composite Mean Nusselt Number ( ) - Mean Developing Flow Nusselt Number ( ) - Experimental Nusselt Number ( ) - Mean Fully-Developed Flow Nusselt Number ( ) - Ideal Mean Nusselt Number Excluding Fin Effects - Localized Nusselt Number - Prandtl Number: - Reynolds Number: - Reynolds Number for Characteristic Channel Width b - Modified Reynolds Number for Characteristic Channel Width b - Hydraulic Diameter Reynolds Number - Modified Reynolds Number for Characteristic Hydraulic Diameter

11 11 - Reynolds Number for Full Plate Length L - Localized Reynolds Number Greek Symbols: - Fin Effectiveness Parameter - Fin Effectiveness Parameter for Straight Fin of Uniform Thickness - Total Surface Temperature Effectiveness - Density [kg / m3] - Kinematic Viscosity [m 2 / s] - Kinematic Viscosity at Film Temperature [m 2 / s]

12 12 CHAPTER 1: HEAT TRANSFER MECHANISMS The goal of this study is to design an experiment to examine the heat transfer phenomena that arises when a flow enters a heated channel. Many studies have been carried out on this topic, and results have been applied to thermal management systems in electronic cooling. The best example is the finned heat sink system used to cool electronic components which invariably produce waste heat. It is interesting to note that such heat sinks are quite complex from an analytical point of view, and theoretical results may not provide good predictions of heat transfer. The experimental work will focus on determination of the average heat transfer coefficient from a finned heat sink and comparison of the results with several analytical models. It will be shown that the analytical models differ greatly in their predictions, and the experimental results are best modeled by a semi-empirical approach that was developed specifically for heat sinks. This chapter will discuss the basic heat transfer concepts, with focus on conduction heat transfer. The following chapters will discuss convection in the channels of a finned heat sink, and the effect of fins on the heat transfer. Subsequent chapters will be devoted to description of the model, experimental system and results. 1.1 Concepts of Heat Transfer Traditionally, the science of thermodynamics is a study of energy and the various means of its transportation between systems. This energy being considered is described as taking many forms (i.e. kinetic, potential, and internal energy), and its transportation is of fundamental concern. If, on one hand, the energy exchange takes place without the transfer of mass, and not by means of a temperature difference, this exchange is described

13 13 as work. However, if this exchange is driven by a temperature difference, the exchange is described as a flow of heat or heat transfer (Chapman, 1984). This study will assume that the system under consideration will neither perform work nor have work performed on it by the surrounding environment. Thus, it will be solely concerned with mechanisms of heat transfer occurring within the system and between the system and its environment. The heat transfer process is categorized into three separate traditional mechanisms: conduction, convection, and radiation. Although these mechanisms can be quantitatively described separately, it is very rare for an exchange to occur that does not involve at least two, if not all three of these heat transfer mechanisms. Because of this simple fact, it is often practical to assume one, or two, of the exchange mechanisms are negligible in comparison to a dominant transfer means. Each heat transfer mechanism and the preceding statement will be discussed in the subsequent sections. 1.2 Conduction Heat Transfer Chapman (1984) describes conduction heat transfer as a mechanism of internal energy flow within a body, or between two bodies in direct contact, by the exchange of kinetic energy of motion of the between molecules by direct communication or by the drift of free electrons in the case of metals. Internal energy is a scientific description of the kinetic energy of individual atoms or molecules within a body. This kinetic energy is observed in the vibrational movement of individual atoms or molecules. As energy is supplied to a body, and the internal energy increases, the movement becomes more excited. By means of collisions between atoms and molecules, or via free electron drift

14 14 in metals, the momentum of individual atoms, or the internal energy, is diffused throughout the body from atoms or molecules of higher energy to atoms or molecules of a lower internal energy. This process is the conduction mechanism. When a skillet is placed on a stove top, and energy is supplied to its base from a heating element, this energy is transferred through the skillet and is observed when the cooking surface, away from the heating element, becomes hot. This process is an example of the conduction mechanism as the internal energy from the bottom of the skillet is diffused through the solid to the cooking surface which is initially at a lower internal energy. The conduction heat transfer mechanism is characterized by being the most scientifically understood of the three mechanisms mentioned. In many cases, concise mathematical models exist to describe the flow of energy within a body. These models tend to have parallels in disciplines outside of thermodynamics. For instance, the behavior of the conduction mechanism is sometimes compared to the flow of energy in an electrical circuit, or the pressure variation observed in a piping system. Figure 1.1 illustrates the conduction process. Regions of higher internal energy are represented in red hues, while areas of lower internal energy can be identified by a blue hue.

15 15 Figure 1: Conduction Heat Transfer Mechanism 1.3 Convection Heat Transfer Convection heat transfer is a mechanism of energy transfer that occurs primarily within a fluid system. Conceptually, it is both a combination of the physical movement of fluid portions described through the use of fluid dynamics and the internal energy transfer between portions of the fluid through the conduction mechanism. For this reason, it is fundamental to fully understand the conduction mechanism to be able to apply it to convective systems. Generically, convection can be separated into two distinct groupings based upon the driving force of the fluid motion, forced convection and free convection. Forced convection describes the convection mechanism observed in a fluid when the motion of the fluid is driven by an external source, such as a pump or fan, which imparts kinetic energy to the system causing fluid motion. In contrast, when the fluid motion is caused by the buoyancy forces existing within the fluid due to a temperature induced density differential, the convection process is labeled free convection or natural convection. Figure 2 attempts to illustrate this process. As before,

16 16 areas of higher internal energy are denoted in red, while those of lower energy are represented by a yellowish-blue hue. Physical fluid motion is shown by the overlaying directional arrows. Figure 2: Convection Heat Transfer Mechanism 1.4 Radiation Heat Transfer Radiation heat transfer is a mechanism of energy movement which does not require a medium in which to occur. All bodies that have internal energy, in other words all bodies above absolute zero, emit electromagnetic radiation. Radiation heat transfer is the net transfer of energy between two bodies through the means of exchanging electromagnetic waves. Each physical body radiates energy which is incident on the second body in question, however, of most interest is the net transfer by means of this mechanism. The spectrum of waves emitted is dependent upon the level of internal energy, oftentimes evaluated as temperature, of the body emitting the waves. In contrast, the amount absorbed or reflected is specific to the body receiving the electro-magnetic

17 17 waves. The most readily accessible example of this process, solar radiation, is diagrammed in Figure 3 (Bellona, 2009). Figure 3: Radiation Heat Transfer Mechanism (Bellona, 2009) 1.5 The Fundamental Laws In systems similar to the one of primary concern to this study, conduction and convection are the dominant mechanisms of heat transfer. The relatively small temperature differential between the heat source and atmospheric environment results in a negligible net amount of energy transfer via the radiation mechanism. In addition, the radiated energy passed between walls of the heat sink results in no net transfer. This allows the radiation effects present to be disregarded, as the rate of energy transfer between the system and the environment by the radiation mechanism is minimal in comparison to the two other dominant energy transfer processes. Because of this, the following discussion will take into account only the effects of the remaining two heat transfer mechanisms, conduction and convection.

18 18 It has been demonstrated historically that the rate of energy transfer in a solid, through the conduction mechanism solely, is directly proportional to the temperature difference and the cross-sectional area of material between the source and destination. In addition, the transfer process is indirectly proportional to the distance it is being transferred. In one dimension, the resulting transfer equation is Equation 1. [1] ( ) Equation 1 is known as Fourier s law. As can be seen in the presented equation, the entire process is also proportional to a material constant, k, the thermal conductivity of the material. A graphical representation of conduction in one dimension appears as Figure 4. Figure 4: 1-Dimensional Representation of Conduction Heat Transfer As described in Section 1.2, the convection mechanism requires fluid motion to disperse discreet portions of fluid, and thus transfer energy by both mass transfer and conduction mechanisms. With this in mind, it is easy to see that the behavior of the fluid

19 19 greatly influences the convection heat transfer process. To better understand convection, let s first examine several simple scenarios of fluid flow. First, consider a viscous fluid of uniform velocity approaching a flat plate. We will make the assumption that the fluid velocity is such that it will not produce a turbulent, random flow, but will instead produce a laminar flow condition as this will approximate the conditions being observed for this study. The fluid particles immediately adjacent to the plate adhere to the plate and become stagnant. In turn, these particles restrict the motion of the next layer of fluids passing adjacent to them through viscous forces. This restrictive force on the layer immediately adjacent reduces the velocity of this layer s motion. This immediately adjacent layer, which is moving at a reduced velocity relative to the free stream, also imposes a restrictive, viscous force on the next immediately adjacent layer because of the differential in velocities. This effect continues to pass from one layer to the next throughout the fluid. The duration of the restrictive impulse increases as the fluid passes along the flat plate in steady-state, causing the immediately adjacent layers to further slow relative to the free stream velocity. Because there is a greater gradient in velocity distribution, fluid layers further from the plate begin to demonstrate a reaction to the stationary plate. This is referred to as the development of the boundary layer, or border between the free stream fluid and that fluid which demonstrates a response to the flat plate. Note that the speed at which the fluid reacts to the stationary flat plate is dependent on both the approaching velocity and the interaction of individual fluid molecules demonstrated by fluid viscosity. An example of the velocity distribution within a developing boundary layer appears graphically in Figure 5.

20 20 Figure 5: Boundary Layer Development (Chapman, 1984) In brief, the fluid outside of the limits of the boundary layer does not display any reaction to the stationary plate and continues to travel at the free stream velocity. Fluid within the boundary layer demonstrates a restricted velocity due to the viscous effects imposed on the fluid by the plate. The thermal boundary layer develops in a similar manner along a flat plate which is at a temperature different than that of the free stream fluid. The layer of fluid immediately adjacent to the plate is assumed to be stagnant, thus the energy transfer from the plate to the fluid is present initially as conduction to the adjacent fluid layer. Recall from the previous discussion that the rate of conduction is proportionate to the magnitude of the temperature differential and the thermal conductivity of the medium. As the fluid passes further along the plate, the duration of energy transfer is longer, and subsequently more layers of fluid demonstrate a greater change in internal energy in the form of a temperature increase in the case of a heated plate or a temperature decrease for the case of a cooled stationary plate. The process of a developing thermal boundary layer is completely analogous to the process of a developing velocity boundary layer. Despite the

21 21 fact that these developing layers are analogous, it is important to note that these two boundary layers do not necessarily develop simultaneously. As Chapman (1984) mentions, the thermal boundary layer is generally not coincident with the velocity boundary layer, although it is certainly dependent on it (p. 12). Each process is dependent upon separate medium characteristics and system properties, and therefore, the rate at which these layers develop is partially independent. As with conduction, the convection mechanism is proportional both to the temperature differential present and the surface available for convection. In simplest form, the process complexities can be combined into a single variable, h, the convection coefficient. The result appears as Equation 2. [2] ( ) In Equation 2, note that the primary temperature differential in question is that between the solid surface and the fluid film temperature. The film temperature is defined as the mean temperature between the solid surface and the free stream. Next, consider a fluid approaching a pair of parallel plates with the same assumptions as before. The free stream fluid interacts with each plate separately in the manner described above. If the plates are positioned sufficiently close, the developing boundary layers described above associated with each plate will eventually interact. The region prior to this interaction is labeled the entrance, or developing region. After this point, there is a region where the developing boundary layers continue to interact while the fluid stabilizes into a truly laminar flow. This region is labeled the transition region. After a sufficient length, the fluid stabilizes into a truly laminar flow and is said to be fully-developed. As in the case

22 of a single stationary plate, the thermal layer will develop in an analogous manner. The distinction between developing and fully-developed regions appears below as Figure Figure 6: Boundary Layer Transition Between Parallel Plates (Chapman, 1984) This process can also be extended to pipe flow or flow within a channel. The distinguishing factor between these two arrangements and the previous systems described is how the developing boundary layers interact. In some scenarios, due to the geometry of the bounding conditions, a change in coordinate systems can result in simpler mathematical relationships; however, the physical system being described is unchanged. It is easy to see that all of these interactions between fluid motion and heat transfer quickly combine to create a truly complex system. The general equation for convective heat transfer presented as Equation 2 is known as Newton s Law of Cooling. Many of the system complexities are combined into a single parameter, h, the convection

23 23 heat transfer coefficient mentioned before. The general form of this law resembles that of conduction heat transfer. For convenience, Equation 2 has been reproduced. [2] ( ) Unlike the conduction equation s thermal conductivity parameter, the convection heat transfer coefficient is not solely dependent on the fluid s properties. In addition to specific fluid properties, the heat transfer coefficient takes into account the dynamics of the fluid and the system geometry. Table 1 has been reproduced from Heat Transfer by Chapman (1984) to provide a general quantitative comparison of the convection process within different mediums. Table 1 Comparison of Heat Transfer Coefficients by Medium and Mechanisms (Chapman, 1984) Typical Values of the Convective Heat Transfer Coefficient Situation h (W / m2- C) Free Convection in Air 5 25 Forced Convection in Air Free Convection in Water Forced Convection in Water ,000 By making the assumption that the fluid adjacent to the solid surface has no relative velocity, all energy transferred to the fluid at the convection surface can be assumed to be transferred via the conduction mechanism. This condition is represented as Equation 3. [3] ( )

24 24 By making the realization that the area available for convection is equivalent to the area available for conduction, Equation 3 can be solved for the convection heat transfer coefficient as shown in Equation 4. [4] ( ) For analysis purposes, it is often convenient to establish a dimensionless parameter. Chapman (1984) proceeds to take the above definition for the heat transfer coefficient and defines the dimensionless heat transfer coefficient, the Nusselt number, in Equation 5 for a characteristic dimension L. [5] ( ) Earlier discussion within the Chapman (1984) texts offers insight into the nature of the partial differential seen in the above equation. Chapman (1984) defines the parameter phi, φ, as a functional relationship shown in Equation 6. [6] { } * + This functional dependency is grouped into two dimensionless space variables, and, as well as four common parameters in heat transfer problems, the Reynolds, Grashof, Eckert, and Prandtl numbers. For a defined surface, the Equation 5 expressing the Nusselt number definition could be integrated over the body s surface to evaluate a mean Nusselt number. As Chapman (1984) points out, this mean Nusselt number is solely functionally dependent on the Reynolds number, the Grashof number, the Eckert number, and the Prandtl number. The Reynolds number represents the ratio of inertial force to viscous forces

25 25 within a fluid. Comparatively, the Grashof number is a non-dimensional ratio of the buoyancy force to the viscous forces in a fluid. Chapman (1984) defines the Eckert number as a measure of the thermal equivalent of the kinetic energy of flow to the temperature differential as caused by viscous dissipation (friction). The Prandtl number, combines fluid physical properties to measure the relative magnitude of the diffusion of momentum, through viscosity, and the diffusion of heat, through conduction, in a fluid. Free convection is said to exist when fluid motion is imparted by buoyancy forces existing in the fluid due to internal temperature differentials creating different localized densities. Under the system described for free convection, it is apparent that the free stream velocity is negligible or zero, thus the Reynolds number and the Eckert number, being strongly dependent on the stream velocity become negligible parameters as well. It follows that in free convection, the mean Nusselt number is a function of the Grashof number and the Prandtl number and the surface geometry. In contrast, forced convection describes a system of convection where the fluid motion is externally imposed. It is important to note that free convection can exist without forced convection; however, forced convection cannot exist uniquely. Although it would be incorrect to say that buoyancy forces are not present in forced convection, they may be negligible in comparison to externally imposed forces. In these cases, the influence of the Grashof number is insignificant. If we also neglect viscous dissipation for the case of air, as in this study, the Eckert number also is insignificant. In this manner, the Nusselt number for forced convection can be explored as simply a function of the Reynolds number and the Prandtl number. As with the mechanisms of energy transfer, in many cases, one driving force of fluid motion will dominate the other within

26 26 any given system. As shown in Table 1, the convection medium can also play an important role on the rate of heat transfer. For instance, convection in water, regardless of the driving force causing fluid motion, is typically one to two orders of magnitude greater than that in air. This, in itself, justifies the significant shift within the electronics industry towards liquid cooling for highly demanding applications. Eventually, though, waste heat produced by electronics must be expelled to a gaseous atmosphere, even in cases where the heat flux leaving critical components has been enhanced by liquid cooling to maintain lower temperatures. 1.6 Extended Surfaces A common method of increasing the rate of heat transfer from a solid surface to a fluid is by increasing the surface area of the solid in contact with the fluid. Recall the convection heat transfer equation presented in Section 1.4 presented as Equation 7 which follows. [7] ( ) The rate of convection heat transfer, q conv, varies proportionally to the amount of surface area available for the convection mechanism. In practical situations, the geometry of the heat producing component is relatively restricted, so the additional surface area is generated through the use of extended surfaces, or fins. Consider the flow of heat transfer in a typical fin of uniform cross-section shown in Figure 7.

27 27 Figure 7: Heat Flow in a Typical Fin The extended surface of the fins effectively increases the surface area in contact with the fluid, however, the additional material also introduces more conduction resistance to the flow of heat from the heat source to the ambient fluid. The energy being transferred from the heat producing component to the fluid must pass through the extended surface via the conduction mechanism prior to reaching the ambient fluid. Equation 1 dictates that for the conduction mechanism to occur, there must exist a temperature differential in the direction of the conductive heat transfer. This condition ensures that a temperature gradient exists in the axial direction of the extended surface, and that the portion of the solid immediately adjacent to the imposed heat source is at the highest temperature. From the convection heat transfer equation, the temperature differential drives the flow of energy. Thus the points further away from the heat source on the extended surface, at a lower net temperature due to the imposed conduction resistances, are less efficient at convective heat transfer. Fin geometry is crucial to ensuring that rate of heat transfer is improved by the gains associated with increasing the

28 area available for convection offset by the losses associated with additional heat transfer resistances and a reduced temperature differential with the ambient fluid. 28

29 29 CHAPTER 2: RELATED CONVECTION PROBLEMS Before attempting to tackle the complex system, it is oftentimes instructional to begin the approach by considering similar, simpler configurations and building upon the solutions to produce a sufficiently adequate system model. Many of the following configurations have been thoroughly explored through both integral analysis and experimental results. It will be noted as the system becomes more complex, thus presenting more difficulty in producing appropriate boundary conditions, empirical models based upon experimental data dominate the solutions. 2.1 Flow Over a Flat Plate One of the most recognizable problems in convective heat transfer is uniform flow of a fluid past a heated flat plate. This problem was discussed qualitatively in Section 1.4. The system configuration is shown again as Figure 8 for the laminar flow development of a free stream interacting with a flat plate. Figure 8: Boundary Layer Development Over a Stationary Plate (Chapman, 1984)

30 30 Note that although the velocity and thermal boundary layers develop simultaneously, these layers are not identical in thickness or developing length. If we make the assumption that the heated plate is isothermal across its entire length, and sufficiently short that the local Reynolds number does not transition into turbulent flow, Chapman, among others, presents the typical localized solution shown to be Equation 8. [8] The subscript x denotes the directional position being evaluated, with the incident edge of the flat plate assumed to be the origin of the coordinate system. Let s examine the implications of Equation 8. The Prandtl number, being a fluid specific property dependent only on temperature, is independent of the fluid motion, free stream velocity, and geometrical parameters. The localized Reynolds number, however, is defined by Equation 9. [9] It can be seen that, at the incident edge of the plate, the coordinate system s origin, the localized Reynolds value, and thus the Nusselt number, is zero and increases proportionally to the square root of distance along the plate s axis, x. However, it should be noted that the boundary layer theory, which is the basis for the above equations, is based on assumptions that are not valid at the edge of the plate. Although mathematically the above relation can approach infinity, the assumptions made to achieve this solution limit it to being applicable for laminar flow only. Laminar flow is generally assumed for.

31 31 The same approach is used to achieve the classical average solution over the length of the plate as Equation 10 which follows. [10] The characteristic Reynolds value present in Equation 10 is defined by Equation 11. [11] 2.2 Flow in a Duct or Pipe Just a small leap from the scenario described in Section 2.1, but of equal significance, is flow which occurs within a duct or pipe. Where the length of the plate in question from Section 2.1 was the geometric parameter of interest, for pipe flow a similarly significant parameter is the hydraulic diameter of the duct. In the case of pipe flow, boundary layers simultaneously form from all of the exposed surfaces consistent with the assumption of a stagnant fluid layer at the duct surface. In the entrance portions of the pipe, these layers do not significantly interact with one another. In contrast, as the fluid flow passes beyond a critical length, the entire flow is encompassed by the boundary layer development. This stabilized flow is described as fully-developed. We define this region where the characteristic fluid moves from developing to fullydeveloped as the transition region for the purposes of this study. This transition region is not as precisely understood as the two limiting extremes, as the fluid stream has not achieved a steady-state, however, the developing boundary layers have begun to have a significant impact on the behavior within the duct.

32 32 It is customary (Chapman, 1984) to define the Reynolds number in terms of a characteristic duct diameter. In a similar manner to the definition presented for the flat plate solution, the duct or pipe flow Reynolds number is defined by Equation 12. [12] For duct flow, there is no free stream present, as the entire flow will ultimately fall within the boundary layer development. As such, the Reynolds number is defined by the mean stream velocity of the flow within the duct. Additionally, the characteristic dimension of this flow is the hydraulic diameter of the duct, D Hyd, defined as follows in Equation 13. [13] As with the flat plate solution, a critical Reynolds value exists that corresponds to a transition from laminar to turbulent flow. From experimental results presented in texts, this transitional Reynolds value is assumed to be 2,300 (Chapman, 1984). For values less than this threshold, the flow is customarily assumed to be laminar in nature; whereas, above this threshold the flow tends to be turbulent. Take note, however, that unlike the case of the flat plate solution, the eventual turbulent flow expected by the Reynolds number definition shown as Equation 9 is not a given beyond a finite length of pipe. The Reynolds value defined by Equation 12 does not vary over the axial length of the duct in question. Rather, the diameter of the pipe and fluid velocity establish whether the transition criterion has been reached. As mentioned in earlier portions of this discussion, at the beginning of the duct or pipe, the boundary layers do not significantly interact. Various theories exist trying to

33 33 define the length of this initial portion; however, due in part to the fact that the boundary layers approach each other on an asymptote, it is difficult to define the mathematical point at which this occurs. Bejan (1995) proposes three such solutions for laminar pipe flow. (Blasius Approximation) (Sparrow Integral Solution) (Schlichting Series Solution) In contrast, Chapman (1984) defines the entrance length as follows. (Langhaar Solution) No such approximations are available for turbulent flow starting length, since many other factors such as pipe entrance, the pipe roughness, etc can have significant effects. (Chapman, 1984) An approximate evaluation of the average Nusselt number for channel flow accurate within ±3% of a model that was developed by Sparrow (1955) is shown in Equation 14. [14] [ ( ) ] The relationship presented above takes into account an approximation of the entrance effects of the duct flow. The Graetz number, Gz, is defined as follows in Equation 15. [15]. /

34 34 In Equation 12, the variable can be approximated by Pr -1/3 for Prandtl numbers approximately equal to 1 or greater; therefore this approximation is generally used for systems with air flow. 2.3 Flow Between Infinite Parallel Plates Consider a special case of the scenario described in Section 2.2 above. Specifically, consider a rectangular shaped duct shaped in a manner where one aspect dimension, for instance the height, H, is much greater than the other aspect dimensions, in this case, the width, b. If the height is allowed to approach infinity, the duct may be considered a pair of infinitely wide parallel plates. The hydraulic diameter can be defined by Equation 16. [16] ( ) The average Nusselt number is still evaluated as in Section 2.2 for alternative duct geometries. In the particular case used for this study, one aspect ratio is not significantly greater than another, so the hydraulic diameter is not found from the simplification shown in Equation 16. However, in many common heat sink applications the above simplification may be employed with limited loss of accuracy. 2.4 Related Work by Others In a paper prepared for the Semi-Therm Symposium of the Institute of Electrical and Electronics Engineers, Inc, otherwise known as IEEE, Teertstra, Yovanovich, and Culham presented a study entitled Analytic Forced Convection Modeling of Plate Fin Heat Sinks. This study s primary focus was the basis for the further work discussed in this document. Teertstra et al (1999) define a solution for forced convection of an air

35 35 medium through an enclosed array of fins found in a parallel plate heat sink. The goal of their work was to produce an analytic solution to this problem capable of predicting the mean heat transfer coefficient for this heat sink array throughout the extremes of flow development. Teertstra et al begin (1999) their discussion by bounding the problem at its two extremes. Bejan and Sciubba (1992, as cited in Teertstra, 1999) define the following enthalpy balance for fully-developed flow shown in Equation 17. [17] ( ) By replacing the mass flow rate with an equivalent expression, Equation 17 can be rewritten as follows in Equation 18. [18] ( ) ( ) Bejan and Sciubba (1992) suggested that the enthalpy balance is based upon the assumption that the Nusselt number for fully-developed flow occurs when the average temperature of the air exiting the channel equals the wall temperature. This assumption is consistent with the problem definition used by Teertstra et al (1999) to simplify the behavior to an isothermal wall condition before applying the fin effectiveness effect. This assumption results in the definition of the fully-developed condition occurring when the net heat transfer from the heat sink to the fluid approaches zero. Using the typical non-dimensional relationships for the Reynolds number and Nusselt number associated with this particular system arrangement, Teertstra (1999) rearranges the above mass flow rate enthalpy balance to the following relationship shown as Equation 19. ( ) ( )

36 36 [19] ( ) [20] ( ) [ ] [ ] By substituting for the definition of the Nusselt, Reynolds, and Prandtl numbers, Teertstra (1999) simplifies the relationship in Equation 20 to define the solution for the fully-developed flow as Equation 21. [21] At this point in their analysis, Teertstra (1999) introduces a new, non-dimensional value, the channel Reynolds number, which is comparable to the Elenbaas Rayleigh number used in natural convection. The Elenbaas Rayleigh number from natural convection is a buoyancy driven flow parameter for natural convection augmented by a factor of b/l for a plate fin heat sink geometry (Teertstra, 1999). The definition introduced for the modified Reynolds number is as follows in Equation 22. [22] ( ) This new number allows further simplification of Equation 18 to the final, fullydeveloped flow limit for the Nusselt number presented by Teertstra et al (1999) shown below as Equation 23. [23] In addition to the fully-developed limit shown in Equation 23, Teertstra et al (1999) explore the alternate extreme for developing flow; however, much of this discussion is based upon the work of Sparrow (1955). Sparrow (1955) presents a solution for the problem defined by Teertstra, including an approximation for the effect of the entrance region of a rectangular duct, as follows.

37 37 [24] [ ( ) ] For the range of Prandtl numbers applicable to this study (i.e < Pr < 2), Sparrow (1955) suggests that the value of the function φ shown in Equation 24 above to be equal to Pr -1/3. In addition, the Graetz number is defined as follows. [25]. / Using these relationships, the value of the Nusselt number for developing flow can be redefined using more consistent terms to that used for fully-developed flow. This simplification is as follows: [ ( ) ]. / [ * + ]. / [26] ( ) 0 ( ) 1 By defining a modified Reynolds number parameter as Equation 27, the preceding expression shown as Equation 26 can be further simplified. [27] ( ) ( ) Applying this new definition, Equation 26 becomes Equation 28. [28] * +

38 38 Teertstra s study, focusing on large aspect ratio heat sinks, utilizes the assumption that the hydraulic diameter is equal to 2b. By applying this assumption, Teertstra et al show that the developing flow Nusselt number limit may be defined as follows in Equation 29, using the modified Reynolds value. [29] * + By recognizing the mentioned assumption for the hydraulic diameter, the results for the characteristic Nusselt value indicated may be extended to the mean hydraulic diameter Nusselt number as follows in Equation 30. [30] * + The assumptions used to achieve the above results can be verified using the standard solution for isothermal flat plate heat transfer. As the Reynolds number is allowed to increase, the interaction of the bounding walls of the heat sink channels becomes negligible. The derived solution based upon the Sparrow (1955) model approaches Equation 31 as the modified Reynolds number increases, i.e.: [31] The above equation is consistent with that presented by Teertstra in Equation 30 when it is recognized that the assumptions made regarding heat sink geometry by Teertstra et al result in the Reynolds value used to be equal half that of the hydraulic diameter Reynolds value present in Equations 28 and 31. This limit approaches a similar form to that of the classic isothermal flat plate solution shown as Equation 32, but it must be recognized that

39 the basis Reynolds values in Equation 32 is fundamentally different than that in Equation 31, so direct comparison is not appropriate. 39 [32] NuL 1/ Re L Pr 1/ 3 In this manner, the two extremes of the system under observation have been defined. This establishes the boundaries for further work in the transition region of flow rates. The study presented by Teertstra et al (1999), due to its similarities, creates a baseline for approaching this complex problem. Kakaҫ et al. (1987) presented the numerical models of Wibulswas (1966) for rectangular duct flow at various aspect ratios in the thermal entrance region of rectangular ducts. For the isothermal wall condition, the numerical results for the mean Nusselt number have been reproduced in Table 2.

40 40 Table 2 Numerical Solution of the Mean Nusselt Number at Different Aspect Ratios X * -1 * Mean Nusselt Number Re D_hyd 1:1 (b:h) 1:2 1:3 1:4 1:5 1: Note: Modified Reynolds Value Shown for Pr = As the fin aspect ratio is increased and approaches the parallel flat plate approximation, the mean Nusselt number observed increases. The tabled parameter X * -1 is defined as follows in Equation 33. [33] As cited by Jouhara and Axcell (2009), Hwang and Fan (1963) also performed a numerical approximation of the mean Nusselt number for simultaneously developing velocity and thermal boundary layers. Their solution, reportedly accurate within 3%, is presented as follows in Equation 34. [34]

41 Hwang and Fan (1963) reference an additional asymptotical approximation presented by Stephan (1959) which follows in Equation [35] ( ) Equations 34 and 35 are very similar for the case of an air flow. Only Equation 34 will be plotted with respect to the experimental results for comparison. 2.5 Fin Effectiveness Effects The discussion presented in Section 2.4 is based upon the assumption that the walls of the heat sink assembly channels are of uniform temperature. This assumption is not valid for real materials, and the temperature gradient which drives the conduction mechanism within the fin material must be accounted for in the proposed solution. Thus, the relationships presented in Equations 23 and 28. Equations 31, 34, and 35 are mathematical models for experimentally observed results, and thus represent real results as opposed to the ideal heat transfer coefficients in Equations 23 and 28. When considering the mean physical heat transfer coefficient of a heat sink assembly, conventional methods follow the process of establishing a fin heat transfer effectiveness parameter,. Kays (1993) defines the fin heat transfer effectiveness in terms of the heat transfer rate by Equation 36 below, which is redefined by Teertstra (1999) in terms of the observed and ideal Nusselt values. [36] ( ) ( )

42 42 Consequently, by rearranging the terms in Equation 36, the actual Nusselt number can be found from the ideal Nusselt number derived in the prior discussion from the following relationship shown in Equation 37. [37] ( ) ( ) Chapman (1984) presents the solution for the total surface temperature effectiveness of an array of fins, η, defined as follows in Equation 38. [38] ( ) In this relationship, Chapman (1984) defines A fins as the exposed surface area of the fins only, A total as the total exposed surface area, and as the effectiveness parameter for the particular shape of fin under study. For the parallel plate heat sink geometry under consideration in this study, assuming the fin tips are insulated, the preceding relationship can be expressed as follows in Equation 39. [39] ( ) ( ) ( ) ( ) ( ) In the case of high aspect parallel plate heat sinks, where fin height is much greater than fin spacing, that is 2H >> b, the above relationship simplifies to simply the fin effectiveness parameter for a single fin,. This approximation will be used in this study. For a straight fin of uniform thickness found in the parallel plate heat sink under study, Chapman (1984) defines the fin effectiveness parameter of this particular geometry as Equation 40. [40] ( ) where: ( ) ( )

43 43 By applying the definition of the Nusselt number and combining with the definition above, the parameter m can be alternatively expressed as Equation 41. [41] ( ) ( ) ( ) ( ) ( ) For the specific case of rectangular fin profile, parallel plate heat sinks, having a high aspect ratio, where H >> b, t fin, the total effectiveness of the array of fins can be expressed as approximately equivalent to the effectiveness of a single fin as follows in Equation 42. [42] ( ) ( ( ) ( ) ( ) ( ) ) ( ) ( ) The actual Nusselt number can thus be expressed in terms of the ideal Nusselt number from the following relationship shown as Equation 43. [43] ( ) ( ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) The relationship shown in Equation 43 accounts for the fin geometry s effectiveness when evaluating the theoretical Nusselt number for a given parallel plate heat sink system. In this manner, it is possible to analyze the system based upon a uniform wall temperature assumption, applying the fin effectiveness definition shown above to calculate the mean heat transfer coefficient.

44 44 CHAPTER 3: A SYSTEM DESCRIPTION This study is particularly concerned with the transition region between developing and fully developed flow specific to a thermal heat sink typical of those used to cool large electronic circuit boards. The rectangular array of flat plate protrusions is a common, cost-effective way to increase the area available for thermal convection of the waste heat of the electronic circuits. At the entrance region of this arrangement, where the velocity and thermal layers are still developing, there is little interaction between the individual velocity and thermal boundary layers associated with each parallel plate. Because of this, the solution to the energy transfer equation most clearly resembles that of the single flat plate system. As the fluid progresses, under the proper system circumstances, the fluid boundary layers will interact and the energy transfer equation will more closely resemble that of fluid flow within a pipe. By understanding the two extremes, the transition region between can be better approximated. 3.1 Proposed Model of Limiting Conditions First, let s consider the developing region. In this entrance portion of the heat sink, the interactions between the two parallel plates forming a flow channel are negligible. For a single channel, the heat transfer of the flow should approach that of the flat plate heat transfer solution. The Teertstra model follows as Equation 44. [44] * + The solution shown as Equation 41 is the limiting extreme of the developing flow region and is accurate for the entrance region in which neither the thermal boundary layer nor the flow velocity boundary layer of the opposing walls has interacted, or more

45 45 specifically, neither the thermal nor the flow velocity boundary layer are theoretically greater than half of the channel width. Next, let s consider the limiting assumption of fully-developed flow in the heat sink channel. Teertstra et al (1999) define the fully-developed limiting case of flow through a two-dimensional channel and the use of an enthalpy balance using a uniform wall temperature assumption. The resulting solution is shown as Equation 45. [45] ( ) In Equation 45, the heat transfer coefficient is dependent on the Reynolds number, system geometry, and the Prandtl number. Further in their analysis, Teertstra (1999) recommends combining the system geometry with the base Reynolds number to produce a single, dimensionless value analogous to the channel, or Elenbass Rayleigh number for natural convection. The Elenbass Rayleigh number is defined as the traditional Rayleigh number, augmented by a channel geometry factor of b/l to account for channel entrance effects due to natural convection within a parallel plate heat sink. This limiting case is accurate for the fully-developed region in which both the thermal and flow velocity boundary layers are fully-defined. By equating the relationships shown as Equations 44 and 45, the transition point, and the associated Reynolds value, between the two asymptotes defined for the fullydeveloped and developing Nusselt numbers can be evaluated as follows. [ ] [ ]

46 46 If a Prandtl value of is assumed, as is used later in this work, the resulting real root of the above equality is as follows in Equation 46. [46] Maintaining the same assumptions used up to this point in the analysis, the Reynolds value shown in Equation 46 can be rewritten in terms of the hydraulic diameter Reynolds number as follows in Equations 47 and 48. [47] ( ) For the specific heat sink being studied, the transition point in terms of the hydraulic diameter Reynolds number is as follows in Equation 48. [48] ( ) The Sparrow integral solution (as cited in Bejan, 1995) for the laminar entrance length is reproduced from Section 2.2 as Equation 49. [49] The Teertstra (1999) solution is valid for flows which display a sufficiently short entrance length that the flow reaches a fully-developed flow in the flow length of the heat sink. The end result is the establishment of two fully independent limiting cases valid for the extremes of flow within the channel. The arbitrary intermediate region, the transition region, where both the thermal boundary layer is fully-developed and the velocity boundary layer is still developing, or vice-versa, is less well defined. It is assumed that this region exhibits a smooth transition from the two limiting cases, but the

47 47 actual nature of that transition has not been established. As many flows, particularly in electronic cooling applications, can exhibit this behavior, the intent of this experiment is to establish an accurate numerical model for this region. The transition region between the two limiting conditions will be established similarly to the solution presented by Teertstra et al (1999) using the Churchill and Usagi composite solution technique. That is, the transition region will be defined by the form shown in Equation 50. [50] ( ) *( ( ) ) ( ( ) ) + Figure 8 shows the interaction of the two limiting conditions for the actual heat sink being studied. This figure is a visual representation of the interaction between Equations 28 and 30, which were defined in Section 2.4 as the limiting extremes of developed and developing flow. The Transition Function is merely an application of Equation 44, to effectively create a smooth transition. The Teertstra Solution is given for reference purposes, and follows the same theory as discussed above.

48 48 Figure 9: Depiction of Transition between Developed and Developing Flow 3.2 The System The system under study is a rectangular fin profile, flat plate heat sink. The physical heat sink can be seen in Figure 10.

49 49 Figure 10: Heat Sink Under Study This particular heat sink configuration is common throughout the electronics cooling industry due to its relative simplicity and cost-effective manufacturing profile. Aluminum construction is chosen to be representative of industry standard due to its relatively high thermal conductivity to cost ratio. This choice of material also lends itself to being formed by the extrusion process, a valuable characteristic for manufacturability. The air stream, typically produced by means of a fan to improve thermal characteristics in an operating scenario, has been simulated by means of a compressed air source, which expels to the ambient room. The production of waste heat by an electronic circuit board is replicated using a flat sheet heater of the same physical dimensions as the base of the heat sink. A general schematic of the experimental set-up follows in Figure 11.

50 50 Figure 11: System Schematic Air flow enters the system from a regulated air supply controlled by a restrictive ball valve. It then passes through a rotameter calibrated for air flow with a valve upstream of the assembly. Finally, the flow is diffused and exits the system after passing through the heat sinks under study. A schematic of the particular geometry of the heat sink under study appears in Figure 12.

51 51 Figure 12: Heat Sink Assembly Geometry. Two individual heat sinks are arranged back to back with a separating insulation layer. Each heat sink is attached to a dedicated heat source, specifically a flexible heating pad capable of generating 320 W of power (5 W / in 2 ). Two thermocouples located upstream of the assembly record the approaching free stream air temperature. An additional two thermocouples suspended in the center of the tunnel and located approximately 8 inches downstream from the heat sink assembly are used to record the exiting air temperature. Three thermocouples are attached directly to the base plate of a heat sink component in the thermal grease layer between the heat sink base and the heating pad. This arrangement allows the monitoring of the base plate temperature and the isothermal assumption. Finally, another thermocouple is located in the center of the opposing heat sink. This allows monitoring of the cross-insulation layer temperature

52 52 gradient to ensure that all heat produced by a heating pad is directed into the attached heat sink. The arrangement of all thermocouples is diagramed in Figure 13 for clarity. Figure 13: Thermocouple Arrangements (Shown in Red) Measurements intended to be used for calculation purposes are recorded when thermal steady-state conditions are achieved. Thermal steady-state is assumed to be achieved based upon the criteria assumed in the Teertstra study (1999). That is, thermal steady state is assumed when thermocouple measurements do not vary by more than one percent of their recorded values over a 15 minute interval (Teertstra, 1999). The

53 53 approach velocity is indirectly monitored via a rotameter instrument (volumetric flow meter). By dividing the volumetric flow rate of the approaching air stream by the crosssectional area available for flow, less the obstructions imposed by the fins and heat sink base, the mean air stream velocity is evaluated. The required air flow is driven by the release of compressed air. This air is first passed through a regulator mechanism, followed by a ball-valve used to induce a mechanical restriction to manually control the volumetric flow rate of air. Temperature measurements taken upstream of the assembly and on the base of the heat sink provide the required information to experimentally calculate the average convection heat transfer coefficient. These results were then compared to the predicted values of Sparrow (1955) and Teertstra (1999). In addition, knowledge of the air temperatures upstream and downstream of the heat sink components provides enough information to conduct an energy balance on the system. This will ensure that alternative means of heat dissipation, which allow waste heat to escape the system using mechanisms other than heat transfer to the air stream, do not significantly influence the results. Approaching air flow rates of 5, 10, and 15 cubic feet per minute were specifically chosen to provide information about the transition region between fullydeveloped and developing flow. Based upon the research results of Teertstra (1999), the channel velocities (0.307 m/s, m/s, and m/s respectively) imparted by these volumetric flow rates are linked to Reynolds values in the transition region of flow rates. Each of these air streams is sampled three independent times at a low and high power

54 Flow Rate (cfm) 54 supply. For more information, please refer to Table 2. Air supply limitations prevent the verification of the fully-developed flow asymptote experimentally, as a stable air flow could not be achieved and recorded at a low enough flow rate. Table 3 Power and Flow Rate Combinations Sampled Power (Watts) Major Assumptions Significant assumptions have been made in the experimental model presented. Firstly, it is assumed that the flow behavior of the air stream within the heat sink channels remains laminar. The relatively thin channels of the heat sink structure in conjunction with the Reynolds values of the air streams being evaluated support the validity of this assumption. The channel width, 0.40 inches (10.2 mm) assists in the production of an organized flow. The Reynolds values chosen for the approaching air stream velocity are significantly below the generally accepted threshold for turbulent flow. As can been observed in Figure 10, the fin profile of the heat sink being studied is not truly uniform. To ease in manufacturing, the heat sink fins possess a slight draft angle. This draft is assumed to not significantly influence the experimental results. A mean rectangular profile is assumed based upon a random sampling of fin base

55 55 thicknesses and fin tip thickness caliper measurements. However, this does represent a potential for variation between theoretical and experimental results. Secondly, it is assumed that the buoyancy forces associated with natural convection are negligible in comparison to the impact of forced convection on the air stream under consideration. This assumption is discussed in greater detail further in the experimental results section. In addition, the air flow is assumed to be relatively well mixed, or possessing a uniform temperature profile. This carries with it the assumption that the air stream under consideration has not developed a significant temperature gradient after exiting the heat sink assembly. Finally, it is assumed that heat generated by the power supply is dissipated solely to the air stream. In other words, the mounting arrangement is such that it provides significantly greater thermal resistance to the flow of thermal energy.

56 56 CHAPTER 4: EXPERIMENTAL RESULTS As discussed in the Section 2.2 above, system observations were recorded for 18 separate sampling instances covering a range of volumetric flow rates between 5 and 15 cubic feet per minute ( cubic meters per minute) passing through the heat sink selected for this study. Each of the volumetric flow rates was sampled at both a low and high power supply to examine the effect of net energy being transferred. In addition, the power supplies chosen included a common sampling magnitude of 75 W. Each combination of volumetric flow rate and power supply was sampled three distinct times. The results of the experiment are included in the following sections. 4.1 Raw Data Table 3 details the observed temperature rise in the air flow passing through the heat sink assembly. Inlet and outlet temperatures reported represent the mean thermocouple measurement recordings over the final 15 minutes observed in each thermocouple. Inlet measurements were taken approximately 8 inches upstream of the assembly, while outlet thermocouples were located approximately the same distance downstream of the heat sink assembly. In addition to the observed temperature readings, the experimental voltage, and corresponding power, applied across the heating pads used to simulate waste heat produced by circuitry is included.

57 57 Table 4 Experimental Sample Result Summary of Air Stream Temperature Rise By examining the data shown in Table 4, a significant discrepancy can be seen between the observed and predicted temperature rises. There are several potential sources for this difference, including: thermal system losses due to conduction effects within the test apparatus, thermal losses due to radiation effects, or inaccurate measurement of either the inlet or outlet temperatures. Energy exiting the system through the conduction mechanism is minimized by the natural insulation properties of the wooden enclosure and through suspending the heat sink assembly within polystyrene thermal insulation. The additional thermal resistance ensures that the amount of heat transferred via the convection mechanism is significantly greater than that passing through the walls of the enclosure.

58 58 The second possible explanation presented for the discrepancy between observed and predicted is thermal system losses due to radiation effects of the heat sink assembly. In a similar study performed by Teertstra et al (1999), the authors investigate the radiation effects of a parallel plate heat sink assembly. In all cases explored, the radiation component of the total heat transfer was found to be less than 1% of the total heat transferred (Teertstra, 1999). This was identified as typical in cases of forced confection. As such, this is unlikely to be the source of the significant differences observed. The final suggested source of this variation was inaccurate measurements of the inlet and/or outlet air stream temperatures. A closer examination of the individual thermocouple readings over the final fifteen minutes of observation for each sample reveals significant variation in the readings of the outlet air stream temperatures only. It is apparent that the thermocouples located downstream of the heat sink assembly do not accurately observe the mean temperature of the exiting air stream. There is an apparent correlation between the temperature differences observed in the downstream thermocouples and the observed error associated with the predicted temperature rises. Regardless of the potential explanations for this observed phenomenon in the differences associated with predicting the temperature rise of the exiting air stream, it does not fully rule out the validity of any other observations. The goal of this study was to evaluate the average heat transfer coefficient of the system in the transition region between developing and fully-developed flows. This was accomplished by monitoring the temperature of the actual heat sink assembly. The thermocouple readings associated

59 59 with this measurement did not display the same fluctuations as discussed above with regards to the outlet air stream. Recall the model presented in Section 2.4 to evaluate the composite Nusselt number in the transition region between developing and fully-developed flow. For convenience, this model is reproduced as Equation 51. [51] ( ) *( ( ) ) ( ( ) ) + { ( ) * + ( ) The Prandtl number is relatively inflexible in the temperature region under question, and may be evaluated as a constant value assumed to be equal to at the mean film temperature of the air flow. Thus, the relationship above becomes solely dependent on the modified Reynolds number. The experimentally observed Nusselt number, Nu exp, is evaluated from the following definition shown as Equation 52. [52] ( ) ( ) ( ) The thermal conductivity of the air is evaluated at the mean film temperature. The experimental temperature readings are the mean observed temperatures for the given parameter. For instance, the surface temperature was evaluated as the mean of the four thermocouple readings recorded for the base temperature. As previously discussed, the outlet thermocouple readings questionably evaluate the mean exiting air stream temperature. As such, the mean film temperature was found from the predicted rise in

60 60 temperature for the system, assuming all heat is transferred to the air flow and losses from conduction and radiation mechanisms are minimal. The power shown in Table 4 is the total heat energy produced by a single heating pad as a result of the applied voltage. 4.2 Correlation to Proposed Models Raw Data Recall the Teertstra (1999) model discussed in the initial stages of the development of the current system model. Based upon the presented limiting conditions of the Teertstra (1999) work, the predicted transition region for the developed to developing Nusselt numbers occurs at a modified hydraulic diameter Reynolds number of approximately 18. The Teertstra (1999) model, the experimental data, the Stephan s (1959, as cited by Hwang and Fan, 1963) model, and the Wibulswas numerical results are shown graphically as Figure 14.

61 61 Figure 14: Composite Nusselt Correlation to Experimental Observations It is apparent in Figure 14 that the Teertstra (1999) model shows the behavior of the experimental results more accurately than either the Wibulswas (1966, as cited in Kakaҫ, 1987) numerical results or the Hwang-Fan model (Hwang & Fan, 1963). However, the correlation is less accurate in the fully-developed regime. The fully developed flow model used in the Teertstra (1999) study is based upon the assumption that the exiting fluid stream reaches a mean temperature equivalent to the channel walls. A high aspect ratio heat sink will more closely match this assumption; but the heat sink used in this study does not quite meet the criteria for a high aspect ratio. As such, it is expected that this model represents a maximum limit to the experimental Nusselt number. The actual experimental results for the heat sink falls below this curve in the fully-

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