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Drexel University
Mechanical Engineering Department
Senior Design Final Report Titled:
Binary Refrigerant Refrigerator
Project 25
MEM-493
Samuel Beccaria Abhinav Duggal Mathew Smith
Xiaoyi Zhu
May 13, 2015 Advisory: Dr. Bakhtier Farouk
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Abstract The objectives are to design, retro-fit, and test a domestic refrigerator using a binary mixture of refrigerants. This concept is motivated by energy savings potential, derived from non-azeotropic refrigerant mixtures. While a single fluid remains at constant temperature during constant pressure evaporation/condensation processes, a non-azeotropic mixture undergoes evaporation/condensation processes at constant pressure over a temperature range, called temperature glide. The first phase of this project was to select two refrigerants and optimize their mixture fractions for the refrigeration cycle. Certain restrictions apply on the availability/production of some refrigerants due to their thermophysical and environmental properties, like high Global Warming Potential (GWP) and high Ozone Depletion Potential (ODP). Therefore, refrigerants R-125 and R-134A were selected on basis of low GWP and low ODP. The optimal mixture fraction was then calculated by using the method of Lagrange Multipliers and the Exhaustive Search method. Performance of the refrigerator, a Magic Chef Model MCBR270S mini-fridge, was simulated using the optimized binary mixture in the final phase of this project. Project stakeholders include consumers, appliance manufacturers, and government agencies.
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Table of Contents 1. Introduction ................................................................................................................................... 4
1.1. Problem Statement ................................................................................................................. 4
1.2. Background ............................................................................................................................ 4
1.3. Motivation .............................................................................................................................. 4
1.4. Refrigerant Selection .............................................................................................................. 4
2. Methods .......................................................................................................................................... 5
2.1. Thermodynamic System......................................................................................................... 5
2.2. Optimization Method ............................................................................................................. 7
2.2.1 Fall and Winter terms: Lagrange Multiplier Method ................................................... 7
2.2.2 Spring term: Exhaustive Search Method ....................................................................... 8
2.3. Experimental Studies ............................................................................................................. 9
2.4. Experimental Methods ......................................................................................................... 10
2.4.1. Generalized Experimental Procedure .......................................................................... 10
3. Conclusions and Discussions: ...................................................................................................... 10
3.1. Project Management ............................................................................................................ 10
3.1.1. Team Organization ....................................................................................................... 10
3.1.2. Project Budget .............................................................................................................. 11
3.2. Results of Optimization Procedure ...................................................................................... 11
3.2.1. Fall and Winter Terms: Lagrange Multiplier Results ................................................ 11
3.2.2. Spring Term: Exhaustive Search Results .................................................................... 12
3.2.3. Effect of LSHX on Reverse-Rankine System ............................................................... 14
3.3. Discussion of Experimental Results ..................................................................................... 16
3.4. Conclusions and Future Work ............................................................................................. 17
4. Works Cited and Referenced ...................................................................................................... 18
5. Appendix A: Reversed Rankine Cycle Analysis ......................................................................... 22
6. Appendix B: Optimization Procedure ......................................................................................... 26
7. Appendix C: Lagrange Multiplier Code ..................................................................................... 27
8. Appendix D: Refrigerant Justification ........................................................................................ 28
9. Appendix E: Daewoo Compressor HFL5Y-1 Datasheet ............................................................. 30
10. Appendix F: Exhaustive Solution ................................................................................................ 31
11. Appendix G: Exhaustive Solution Equations .............................................................................. 32
12. Appendix H: Water Calorimeter Test......................................................................................... 35
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1. Introduction 1.1. Problem Statement The objectives of the project are to design, retro-fit, and test a domestic vapor compression refrigerator that uses a binary refrigerant mixture instead of the more traditional single refrigerant refrigerators.
1.2. Background Refrigerators that are currently used in households operate on a single working fluid called a refrigerant. This refrigerant has to meet a certain safety and thermophysical criteria. The most ideal refrigerant to use should have low toxicity, low flammability, a high heat of vaporization, a small specific heat and specific volume and low ozone depletion and global warming potentials. Unfortunately, single refrigerants either offer poor thermophysical properties, causing little harm to the environment or offer excellent thermophysical properties, resulting in a high potential risk to the environment. Using a mix of both is one of the most effective ways to work around this problem [25], [36], [37], [39].
A mixture of two refrigerants is referred to as a binary mixture of refrigerants, which can experience one or two states simultaneously- liquid, vapor and liquid-vapor, depending upon pressure and temperature of the system and the saturation pressure of refrigerants in the mixture. In most cases this allows for a temperature glide effect (a range of saturation temperature for a given saturation pressure) that allows the freezer temperature to be reached at a lower pressure [36]. Additionally, by mixing a refrigerant with poor thermophysical properties that does little harm to the environment to the refrigerant with excellent thermophysical properties that does a lot of harm to the environment results in a binary mixture that can have a significantly reduced potential risk to the environment alongside observing gains in the Coefficient Of Performance COP [37].
1.3. Motivation Currently, domestic refrigerators in the United States make use of a refrigerant called HFC-134a (R-134A, 1,1,1,2-Tetrafluoroethane) as their working fluid, which does not deplete the ozone layer and has an acceptable set of thermophysical properties. However, R134a has high GWP of 1300. The Kyoto Protocol of the United Nations Framework Convention on Climate Change (UNFCCC) calls for reductions in emissions of six categories of greenhouse gases, including hydrofluorocarbons (HFCs) used as refrigerants. From the environmental, economic, social and ethical aspects, it is urgent to find a better alternative to HFC refrigerant [29]. This has been further explained in Appendix D. 1.4. Refrigerant Selection From a wide range of refrigerant pairs, R-125/R-152A and R-125/R-134A were chosen based on their thermophysical and environmental properties. The latter refrigerants were finally chosen because of their availability; as with the first choice of refrigerants, R-152A was not readily
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accessible in the United States. Also, there were certain restrictions placed on refrigerant use by the EPA, which states that- refrigerants more flammable than R-134A cannot be used as a replacement [30]. According to ASHRAE flammability classification, R-152A belongs to A2 class, which means this refrigerant has a lower flammability limit (LFL) of more than 0.10 kg/m3 at 21°C and 101 kPa and a heat of combustion of less than 19,000 kJ/kg [31]. Thus R-152A can’t be used, although the binary mixture of R-152A and R-125 may have higher COP. The mixture of R-125 and R-134A is an acceptable choice which has good thermophysical properties and is most importantly safe to use.
The refrigerator used for this project, which will be operating with the binary refrigerant selection is a Magic Chef 2.6 Model MCBR270S mini-fridge. This refrigerator employs the vapor-compression cycle and makes use of the standard compressor, evaporator, condenser, liquid suction line heat exchanger (LSHX) and throttling valve components.
2. Methods 2.1. Thermodynamic System In the fall and winter term the focus of the project was on an idealized Reverse-Rankine cycle using a binary refrigerant mixture. At the end of the winter term it was decided that a modified Reverse-Rankine cycle was to be used (a Reverse-Rankine cycle with a LSHX.) It was found from research that common household refrigerators utilize this modified Reverse-Rankine cycle which is shown in Figure 1.
FIGURE 1. DIAGRAM OF REFRIGERATION CYCLE STATE POINTS WITH LIQUID-SUCTION HEAT EXCHANGER
The working fluid enters the compressor as a saturated vapor and leaves as a superheated vapor. Due to the fact that superheated vapor is entering the condenser, state 3 is broken up into states 3a and 3b, where 3a is when the working fluid is a saturated vapor and state 3b is when it is a
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saturated liquid. At state 3b the working fluid leaves the condenser as a saturated liquid and becomes a subcooled liquid after leaving the LSHX. The working fluid is throttled and leaves as a saturated liquid-vapor mixture, where it enters the evaporator and partially evaporates. The remaining evaporation occurs over the LSHX and the cycle repeats as the working fluid leaves the LSHX as a saturated vapor. It is assumed no heat is lost to the environment during any other process other than inside the condenser, that there is a constant evaporator load, the pressures over states 5 to 1 and states 2 to 4 are constant, and that there is isentropic compression. The qualities of the working fluid remain on or under the liquid-vapor dome on its saturation line from states 5 to 1 and states 3a to 3b [18], [48]. Figure 2 shows the relative states in Figure 1 on the T-s diagram for the system where each state, states 1 through 6, are labeled at their respective quality.
FIGURE 2. T-S DIAGRAM FOR SYSTEM IN FIGURE 1, WITH A BINARY MIXTURE AS THE WORKING FLUID
For a binary refrigerant refrigerator that is utilizing a non-azeotropic mixture the phase change from a saturated vapor to a saturated liquid, and vice versa, experiences a temperature glide over the phase change [36]. The temperature glide characteristic for a non-azeotropic working fluid actually reduces the total energy needed to perform the cycle, hence why implementing this design is crucial to an increase in the coefficient of performance (COP) of the system [37], [24], [1]. For a binary mixture working fluid the T-s diagram is better understood by the accompanying concentration diagram (T-x diagram) shown in Figure 3.
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FIGURE 3. T-X DIAGRAM OF WORKING FLUID THROUGHOUT THE ENTIRE SYSTEM
The T-x diagram shows the non-azeotropic nature of the working fluid at two pressure states, at 720 kPa and at 1510 kPa, the pressure ranges for the refrigerator. The line in the graph represents an arbitrarily chosen mixture fraction. The working fluid, as it works through the system, rides along the line, starting at state 1 at a low pressure and going to a higher pressure. As it works through the system, the mass fractions will always remain the same, but the vapor and liquid fractions will change from state to state [36]. With these principles of how the system works the deterministic solution can be set up by creating a system of equations from state to state that represent the physics of the binary working fluid and its function over specific components. These equations are detailed in Appendix A.
2.2. Optimization Method 2.2.1 Fall and Winter terms: Lagrange Multiplier Method The goals of this project were to design, retro-fit and test an optimized domestic vapor-compression refrigerator that makes use of a binary refrigerant as its working fluid. The fall and winter term were an exercise in developing the method to find the optimal mass fractions of each refrigerant, R125 and R134A that minimize the work of compression of the Reverse-Rankine Cycle. In order to accomplish this task the method of Lagrange multipliers, an optimization algorithm (shown in Appendix B), was created in MATLAB (shown in Appendix C) and used to find the mass fractions that guarantee a minimized work of compression with regards to the objective function in Equation 1.
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𝑊𝑐 = �̇� �𝑥1 ∗ 189.3 ∗ �1−15.45− log(𝑃𝑡2)15.45− log(𝑃𝑡1)�+ (1− 𝑥1) ∗ 245.1
∗ �1 −15.91− log(𝑃𝑡2)15.91 − log(𝑃𝑡1)��
(1)
In Equation 1, mdot is the mass flow rate of the binary mixture, x1 represents the mass fraction of the refrigerant R125 in the binary working fluid, and Pt1 and Pt2 are the total pressures of the evaporator side and condenser side respectively. Pt1 and Pt2 are expressed as functions of x1, T1 and T3, and the derivation of the objective function and Pt1 and Pt2 are shown in Appendix A. The objective function is subject to the constraints shown in Equations 2 and 3 that were derived from the Reverse-Rankine cycle. 𝜙1 = 𝑄𝑒 − �̇� ∗ (ℎ1 − ℎ3) = 0 (2) 𝜙2 = 𝑄𝑐 − 𝑄𝑒 −𝑊𝑐 = 0 (3) The constraint in Equation 2 relates the cooling capacity (heat injected into the system by the evaporator), to the enthalpies from state 4 to state 1. Since the enthalpy at state 4 is equivalent to the enthalpy at state 3, due to the nature of the throttling valve, Equation 2 applies. Equation 3 describes the constraint that entails the entire energy balance of the system, where the heat rejected by the condenser is equivalent to the sum of the heat and work injected into the system. The condenser, evaporator and heat exchanger equations that relate the temperatures of different states to one another are shown in Appendix A. The log-mean temperature difference is mainly used to describe the effect of the energy exchange between fluids in a heat exchanger, however the full extent of this method is not shown, only the resulting equations from it.
Once the foundations were set for the Lagrange Multiplier method the procedure for obtaining the optimization result was created. Since constructing a third constraint to accompany the objective function in Equation 1, which is a function of four variables (and thus could have 3 constraint equations), would be tedious the optimization was done iteratively. As a result of this a value for the mass fraction of R125 in the mixture was chosen, between 0 and 1, and the optimization was run to produce the optimal result.
2.2.2 Spring term: Exhaustive Search Method The Lagrange Multiplier method was explored as an option to find the minimum work of compression for a modified Reverse-Rankine cycle shown in Figure 1 using R125 and R134A in mixture. However due to the complexity of the equations the Exhaustive Search method was explored as an option for optimization. In order to accomplish the task of the Exhaustive Search method a deterministic system was created, using a set of 8 equations and 8 unknowns, the formulation of which is shown in Appendices F and G. With a selected mass fraction of R125, the 8 variables are solved for using the deterministic system and the work of compression is computed. Then another mass fraction of R125 is selected and the process is repeated. This is carried over the range of 0 to 1 mass fraction of R125 and was used to find the mass fraction of R125 and R134A that produced a minimum work of compression.
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The deterministic system is a system of equations, with a number of variables the same as the number of equations, and solves the system of equations for the variables present. For the entire system there are 8 equations and 9 unknowns, however in this case the mass fraction of R125 was assumed, and the system contained 8 residual equations (equations whose value sums to be 0) and 8 unknown variables. These variables were the mass flow rate (�̇�) and the temperature at states 1 (T1), 2 (T2), 3a (T3a), 3b (T3b), 4 (T4), 5 (T5), and 6 (T6). The residual equations were setup using a variety of system balances from the energy balances and pressure balances derived from the modified Reverse-Rankine cycle. That is, the energy balances over the evaporator, condenser, the liquid to suction line heat exchanger and the overall system energy balance were used in conjunction with the assumption that the pressure of the high-side states were equivalent to one another and the pressure of the low-side states were equivalent to one another. This system of equations is then used to find the solution of each variable when the mass fraction of R125 is given. Then using the solved values for T1, T2, T3, etc the work of compression and the COP can be determined. This is done repetitiously for mass fractions of R125 ranging from 0 to 1 to obtain the temperature values needed to calculate the COP of the system when using that mixture. In order to solve the non-linear system of equations the Newton Raphson method was the method of choice in order to achieve results. The Newton-Raphson method was used because it is a numerical method capable of handling the bulk of the system of residual equations that resulted from attempting to create an optimization of the system. In order to use the Newton Raphson method an initial guess at what the temperature values and mass flow rate are must be provided, and have the added stipulation that those values must be very close to the actual value, otherwise the Newton Raphson method will fail.
2.3. Experimental Studies The experimental studies that pave the way for the testing procedures are drawn from several sources. Each one of these sources account for certain aspects, such as measuring temperature, energy exchange, pressures and work of compression for a binary refrigerant refrigerator. Two sources are PhD works performed at the University of Illinois, Urbana-Champaign’s Air-Conditioning and Refrigeration Center, where Dr. Stoecker performed his work. It is important to note that the United States has set specific procedures and conditions for refrigerator/freezer performance testing. While a calorimeter test is considered the most accurate for testing these appliances, other methods from a literature study will be presented [36]. One of these prior works by Launay [21], which was performed under supervision of Dr. Stoecker, describes experimental testing procedures for NARM based refrigerant systems in detail. They suggest doing performance testing with a fixed evaporator load for the fridge and freezer. Power consumption of the fridge, measured during compressor on-cycles, is found from a watt transducer or power meter. The mass flow rate of refrigerant can be measured using a mass flow meter placed in the refrigerant circuit [12], [13], [15]. The work by Mohanraj [27] performed on NARM refrigeration systems details testing of the system performance. Because the primary quantities of interest with respect to refrigerator system performance are evaporating temperature, condensing temperature, power consumption, and volumetric cooling capacity, experimental data can be determined from these quantities.
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The evaporating temperature can be measured by placing thermocouples along the tubing of the evaporator. For the condenser, the temperature of the refrigerant can be measured before and after entering the tubes. The air stream temperatures can be measured using temperature sensors at inlet and exit points with air blown over the evaporator. The mass flow rate of the air can be determined by the size of the fan used to blow the air, usually given in cubic foot per minute (CFM).
2.4. Experimental Methods The experimental procedure for designing, building, and testing the domestic refrigerator/freezer with the optimized binary mixture consists of three main stages: Data Acquisition and Reduction of Baseline System, Charging and Adjusting System with Binary Refrigerant Mixture, and Data Acquisition/Performance Testing of Optimized Binary Refrigerant System. Data acquisition and reduction for the baseline system was discussed in the experimental methods section.
2.4.1. Generalized Experimental Procedure In order to confirm the theoretical results from simulated cycle data, experimental validations were required to characterize the baseline refrigerator. The baseline refrigerator charged with R-134a was tested in the lab using a water calorimetry method along with compressor power data to estimate system several performance parameters. The criteria considered here was the power consumption of the compressor, the on-off cycling characteristics of the compressor, and the system COP.
The power input to the compressor was measured using a plug-in wattmeter. This provided compressor power consumption during compressor on-cycles. The cycle time of the compressor was the duration of time the compressor was turned on and provided insight into the total amount of energy supplied to the system. Temperature cycle information was obtained by observing when the cabinet temperature reached the upper limit for the thermostat, then cooled as the compressor was running, and finally shut off at the lower limit of the thermostat. By plotting the cabinet air temperature data over time, the compressor on-time, off-time, the total cycle time was determined.
The evaporator load was calculated using a water calorimeter placed inside the refrigerator; this consisted of a small and thin aluminum container filled with 500 milliliters of water at an initial temperature of 46.3 ℃. This was placed inside the refrigerator with a thermocouple probe submersed in the water to measure the temperature. The cabinet air temperature, ambient air temperature, and compressor power were measured as well; these measurements were recorded every 30 seconds until the water temperature reached 5 ℃. It is important to note that the compressor was running the entire time the water calorimeter data was recorded. With the evaporator load and the compressor power the COP of the refrigerator can be calculated.
3. Conclusions and Discussions: 3.1. Project Management 3.1.1. Team Organization Weekly meetings were held with the team’s advisor Dr. Farouk on Thursday at 11am in the Randell building. The group delivered weekly progress report to Dr. Farouk during this meeting
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and updated Dr. Farouk with any questions, comments, and concerns that may have developed in the past week. Dr. Farouk assigned weekly tasks to the group or to individual member in the group. Weekly team meetings amongst the individuals were held to work together on the tasks, which were assigned by Dr. Farouk. The group has four members. One member was held accountable for scheduling meetings, keeping meeting minutes, and keeping track of project deadlines and requirements. The group primarily uses Google drive to store all the work, data, research papers and the meeting minutes.
3.1.2. Project Budget The budget includes the cost of the refrigerator, refrigerants, and sensory instruments described in Table 1.
TABLE 1.EXPECTED BUDGETED LIST OF ITEMS
Item Vendor Qty Unit Cost
Total Cost
Magic Chef Mini Refrigerator 2.6 cu ft HomeDepot 1 $139.00 $139.00 R125 USA Refrigerants 25 $9.00 $282.00
Taylor 1443 Digital Thermometer Amazon 1 $10.00 $10.00 Blue LED Temperature Sensor Amazon 1 $16.00 $16.00
Kill-A-Watt Electric Usage Monitor Amazon 1 $19.00 $19.00 Aluminum pan Fresh Grocer 5 $1.50 $7.50
Enviro Safe Can-Tap Gauge Sears 1 $28.00 $28.00 Line Tap Valve Amazon 1 $6.00 $6.00
Grand Total: $507.50 The total budget for this project rests at $507.50. The actual items bought and budgeted for are listed in Table 2. Since the refrigerator was gifted to the group from Drexel’s IRT it is not included.
TABLE 2. ACTUAL BUDGETED ITEMS
Item Vendor Qty Unit Cost
Total Cost
R125 USA Refrigerants
25 $9.00 $282.00
Kill-A-Watt Electric Usage Monitor Amazon 1 $19.00 $19.00 Aluminum pan Fresh Grocer 5 $1.50 $7.50
Grand Total: $308.50 The cost of recharging the refrigerator was also not included in this cost since it was not budgeted for until the end of the spring term.
3.2. Results of Optimization Procedure 3.2.1. Fall and Winter Terms: Lagrange Multiplier Results The results of the optimization procedure for the Reverse Rankine cycle utilizing a binary mixture are shown in Table 3.
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TABLE 3. RESULTS OF OPTIMIZATION PROCEDURE FOR MIXTURE OF R125
Concentration of R125
0.2 0.4 0.6 0.8 0.9 1
T1 (°C) -79.0 -65 -51.9 -37.2 -29.7 -22.3 T3 (°C) 64.9 60 56.5 52.9 51.0 49.1 �̇� (kg/s) 0.00110 0.00105 0.000997 0.000953 0.000934 0.000918 Wc (kW) 0.111 0.0909 0.0723 0.0549 0.0466 0.0385
Table 3 shows that from 0 % to 100% of R125 in the mixture, the optimization favors R125 over R134A, which would mean that to optimize the simple vapor-compression cycle with regards to refrigerant mixture, only R125 should exist in the mixture. This was found to not be the case for the modified Reverse Rankine cycle analyzed with the Exhaustive Search method.
3.2.2. Spring Term: Exhaustive Search Results The optimization procedure using the deterministic system and the Newton Raphson solver were able to produce results for R125/R134A that had a minimum work of compression at 0.15/0.85 mass fractions of R125/R134A. With the Exhaustive Search method used for a system with the system parameters described in Table 4 the minimum work of compression for R125/R134A was found to exist at 0.15/0.85 mass fractions respectively, the graph of which is shown in Figure 5.
TABLE 4. DOMESTIC REFRIGERATOR SYSTEM PARAMETER VALUES
System Parameter Value Qe 50 Watts UA (Condenser) 0.012 W/K UA (Evaporator) 0.012 W/K UA (LSHX) 0.00075 W/K �̇� of air (Condenser) 0.0045 kg/s
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FIGURE 4. WORK OF COMPRESSION AT VARIOUS CONCENTRATIONS OF R125 (MINIMUM LOCATED AT 0.15.)
The minimum work of compression is located at 0.15/0.85 mass fractions of R125/R134A, although the location is obscured by the “shallowness” of the minimum point. The COP, in correspondence with the work of compression, has a maximum value of 2.38 located at 0.15 of R125, the graph of which is shown in Figure 6.
21.01 20.0021.0022.0023.0024.0025.0026.0027.0028.0029.0030.0031.0032.0033.0034.0035.0036.0037.0038.0039.0040.00
0 0.2 0.4 0.6 0.8 1
Wor
k of
Com
pres
sion
(W)
Mass Fraction of R125
Work of Compression vs. Concentration of R125
Work ofCompression vsConcentration of R125
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FIGURE 5. COP AT VARIOUS VALUES OF R125 IN MIXTURE (WITH A MAXIMUM AT 2.38).
The maximum COP and minimum work of compression corresponding to the mixture fractions of 0.15/0.85 for R125/R134A offer a 0.5% decrease in the work of compression and a 0.7% increase in COP with regards to pure R134A. The location of the minimum work of compression and maximum COP differ between refrigerant mixtures and can also change given specific system parameters. All system parameters have an effect on the location of the minimum work of compression, but the LSHX is the component of interest since it has been stated that the LSHX is what allows the smallest possible work of compression to occur for R125 and R134A.
3.2.3. Effect of LSHX on Reverse-Rankine System The effect of the LSHX on the system is that it drives the temperature of the working fluid down before it is throttled and allows for a dry saturated vapor quality of the working fluid to be present at state 1. This in turn seems to drive the work of compression to decrease further, which in turn drives the COP of the system to increase. A graph of this concept, with the system operating at different overall heat transfer coefficients, is shown in Figure 7.
2.380
1.300
1.500
1.700
1.900
2.100
2.300
2.500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
COP
(W/W
)
Mass Fraction of R125
COP vs. Concentration of R125
COP vsConcentrationof R125MaximumCOP
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FIGURE 6. COP OF THE SYSTEM VS. THE CONCENTRATION OF R125 AT DIFFERENT HEAT TRANSFER COEFFICIENTS OF THE LSHX
What was encountered in the winter term has been encountered for the deterministic solution of the optimization problem, that when the LSHX is absent, represented by a 0 overall heat transfer coefficient, there is no peak in the system, save for the maximum located at 0. However as the heat transfer coefficient is increased, the maximum COP location is shifted to the right and up, occurring at 0.15/0.85 and then 0.2/0.8. This means that the system itself could be designed to accept a specific mixture of R125/R134A that allows it to operate at the maximum COP, corresponding to a minimum work of compression.
This was done for two other types of refrigerants simply to see if this effect is constant across refrigerant mixtures. The refrigerants selected were R12 and R114. In Figure 8 are the results of modifying the overall heat transfer coefficient of the LSHX when the working fluid is R12 and R114 with different system parameters.
1.300
1.500
1.700
1.900
2.100
2.300
2.500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
COP
Mass Fraction of R125
COP vs. Mass Fraction of R125
Uax = 0.0025
Uax = 0.00075
Uax = 0.0
No LSHX
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FIGURE 7. COP VS. MASS FRACTION OF R114 FOR A MODIFIED REVERSE RANKINE CYCLE AND A CHANGING HEAT TRANSFER COEFFICIENT FOR THE LSHX
What is seen in Figure 7 is that the refrigerants R12 and R114 experience a maxima at around 50/50 mass fraction of R12/R114, despite the presence of a LSHX in the simulation model.
3.3. Discussion of Experimental Results Analysis of the experimental results obtained from the cabinet air temperature profiles showed a consistent compressor cycle behavior. The cabinet air temperature with the thermostat setting fixed at the lowest temperature produced an upper temperature value of -1.89 ℃ and a lower temperature value of -4.7 ℃. Figure # shows the plot of cabinet air temperature over time. It can be seen from this plot that the total compressor cycle time is approximately 16 minutes. The compressor on-time and off-time are approximately 7.5 and 8.5 minutes, respectively. A refrigerant mixture with better performance would be expected to show a shorter compressor on-time for a fixed evaporator load and thermostat setting.
2.700
2.800
2.900
3.000
3.100
3.200
3.300
3.400
3.500
0 0.2 0.4 0.6 0.8 1 1.2
COP
Mass Fraction of R114
COP vs. Mass Fraction of R114
Uax = 1.0
Uax = 0.5
Uax = 0.3
Uax = 0.0
No LSHX
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FIGURE 8. CABINET AIR TEMPERATURE OVER TIME DEMONSTRATING THE COMPRESSOR ON/OFF CYCLES
The water calorimeter test was conducted with a shallow bowl of heated water that was placed inside of the refrigerator and made to be cooled from its high temperature to a temperature of 5°C using a thermocouple to record the temperature of the water every 30 seconds. The temperature of the water and the subsequent heat load calculations are shown in Appendix H. The end result was that the Magic Chef 2.6 cubic feet refrigerator, subject to a heated bowl of water, had a COP of 0.51 W/W, comparing this with the COP for the refrigerator data shown in Appendix E this is reasonable. This COP value from the experiment does not match with the simulated COP value for R134A (0% R125, 100% R134A). This is because of the assumptions made for the modified Reverse-Rankine cycle do not accurately match the Magic Chef refrigerator in the lab, which has pressure drops and superheated vapor at state 1.
3.4. Conclusions and Future Work The optimization due to the Lagrange Multiplier method favored 100% R125 for a Reversed-Rankine cycle. However for the Exhaustive Search method applied to the modified Reverse Rankine cycle the optimum occurred at 15% R125 and 85% R134A. The optimum value has not yet been tested in the refrigerator but in the future the refrigerator should be charged with the optimum mixture and the model results validated. A major component of the refrigerator considered in this analysis and almost every domestic refrigerator on the market is a liquid to suction line heat exchanger. This is a simple heat exchanger that transfers heat from the hot liquid refrigerant on the high-pressure side to the cold refrigerant on the low-pressure side. These are used on domestic refrigerators for two main reasons; the first reason is to increase the refrigeration effect in the evaporator by sub-cooling the liquid refrigerant before throttling, which reduces the amount of flash gas entering the evaporator. The heat lost from the liquid side is absorbed by the vapor side, which causes the refrigerant to enter the superheated state on the suction side. The other reason an LSHX is used on most domestic refrigerators is to prevent liquid from entering the compressor, which would eventually cause a burnout.
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
0 20 40 60 80 100 120 140
Tem
pera
ture
(\de
gc)
Time (minutes)
Temperature Data logged Once per Minute (Overnight 4-6-2015)
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From a thermodynamic standpoint, the LSHX produces two effects on system performance, which can cause an increase or decrease in efficiency, depending on the refrigerants considered. The LSHX allows the system to produce the same refrigeration effect at a higher evaporation temperature. At the same time, superheating the refrigerant vapor before compression reduces the specific volume, which can cause the compressor to operate less efficiently.
The impact on choice of refrigerants for a non-azeotropic refrigeration system comes from the performance of different working fluids with respect to the LSHX. For some fluids, such as R-12 and R-134a, the COP increases monotonically with the effectiveness of the LSHX, this is the ratio of the actual heat transfer to the maximum possible heat transfer. For fluids like R-22 and R-32, the LSHX is detrimental to system performance because the superheating effect outweighs the increased refrigeration effect, resulting in an overall decrease in COP.
The COP of the Magic Chef refrigerator was found, from the water calorimeter test, to be 0.52 W/W. For the future, the refrigerator should be charged with the simulated optimal mixture fraction and charged into the refrigerator, so that the COP could be tested for the optimal mixture. As well, a non-exhaustive optimization method, for the modified Reverse-Rankine system, using either the Lagrange Multiplier method or any other optimization method, should be explored. As well, the refrigerator could be tested at multiple mixture fractions in order to develop an experimental result for incremental mixture fractions of R125 and R134A. As well, for simulation modeling purposes pressure drops and superheating at state 1 in the modified Reverse-Rankine cycle could be accommodated for in order to fully represent to Magic Chef refrigerator.
4. Works Cited and Referenced
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[2] R. American Society of Heating, and Air-Conditioning Engineers (ASHRAE), "Designation and Safety Classification of Refrigerants," in Addendum A to ASHRAE 34-2004 vol. Standard 34-2004, ed. Atlanta, Georgia: ANSI/ASHRAE, 2006.
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Energy, vol. 35, pp. 3793-3798, 2010. [7] P. A. Domanski, D. A. Didion, and J. P. Doyle, "Evaluation of suction-line/liquid-line heat
exchange in the refrigeration cycle," International Journal of Refrigeration, vol. 17, pp. 487-493, // 1994.
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[8] P. A. Domanski and M. O. McLinden, "A simplified cycle simulation model for the performance rating of refrigerants and refrigerant mixtures," International journal of refrigeration, vol. 15, pp. 81-88, 1992.
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[12] C. J. L. Hermes, C. Melo, F. T. Knabben, and J. M. Gonçalves, "Prediction of the energy consumption of household refrigerators and freezers via steady-state simulation," Applied Energy, vol. 86, pp. 1311-1319, 7// 2009.
[13] G. B. Jain, C.W., "Simulation Analysis of Thermal Systems and Components," University of Illinois, Urbana-Champaign, Air-Conditioning and Refrigeration Center ACRC - TR-235, October 2004 2004.
[14] S. J. James, J. Evans, and C. James, "A review of the performance of domestic refrigerators," Journal of Food Engineering, vol. 87, pp. 2-10, 2008.
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[16] D. S. Jung and R. Radermacher, "Performance simulation of a two-evaporator refrigerator—freezer charged with pure and mixed refrigerants," International Journal of Refrigeration, vol. 14, pp. 254-263, 9// 1991.
[17] M. I. H. Khan, H. M. Afroz, M. A. Rohoman, M. Faruk, and M. Salim, "Effect of Different Operating Variables on Energy Consumption of Household Refrigerator," International Journal of Energy Engineering, 2013.
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[19] H. Kruse, "The advantages non-azeotropic refrigerant mixtures for heat pump application," International Journal of Refrigeration, vol. 4, pp. 119-125, 5// 1981.
[20] O. Laguerre, "Heat transfer and air flow in a domestic refrigerator," Mathematical Modelling of Food Processing, Mohammed M. Farid (ed.), pp. p. 445-p. 474, 2010.
[21] P. F. Launay, "Improving the Efficiency of Refrigerators and Heat Pumps by Using a Nonzaeotropic Mixture of Refrigerants," ed University of Illinois at Urbaba-Champaign: Oak Ridge National Laboratory, 1981.
[22] E. W. Lemmon and R. T. Jacobsen, "Equations of state for mixtures of R-32, R-125, R-134a, R-143a, and R-152a," Journal of physical and chemical reference data, vol. 33, pp. 593-620, 2004.
[23] D. Liangui, W. Wenchuan, Z. Danxing, and F. Jufu, "Optimization of the compositions for CFC alternative mixture refrigerants," Chin. J. Chem. Eng, vol. 3, pp. 32-38, 1995.
[24] B. Mathiprakasam and T. Sutikno, "Theoretical analysis of the use of binary refrigerant mixtures in the vapor compression cycle," in AIChE symposium series, 1984, pp. 122-127.
[25] M. M. Mathison, J. E. Braun, and E. A. Groll, "Approaching the performance limit for economized cycles using simplified cycles," International Journal of Refrigeration, vol. 45, pp. 64-72, 9// 2014.
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[26] M. O. McLinden and R. Radermacher, "Methods for comparing the performance of pure and mixed refrigerants in the vapour compression cycle," International Journal of Refrigeration, vol. 10, pp. 318-325, 11// 1987.
[27] M. Mohanraj, S. Jayaraj, and C. Muraleedharan, "Comparative assessment of environment-friendly alternatives to R134a in domestic refrigerators," Energy Efficiency, vol. 1, pp. 189-198, 2008.
[28] M. Mohanraj, S. Jayaraj, C. Muraleedharan, and P. Chandrasekar, "Experimental investigation of R290/R600a mixture as an alternative to R134a in a domestic refrigerator," International Journal of Thermal Sciences, vol. 48, pp. 1036-1042, 2009.
[29] H. Nishiumi, H. Akita, and S. Akiyama, "High pressure vapor-liquid equilibria for the HFC125-HFC152a system," Korean Journal of Chemical Engineering, vol. 14, pp. 359-364, 1997.
[30] W. Paper, "Revisiting Flammable Refrigerants," EPA, USA2011. [31] N. Refrigerants, "Material Safety Datasheet R-152a," ed: National Refrigerants, 2008. [32] N. Refrigerants, "Material Safety Datasheet R-134a," ed, 2008. [33] N. Refrigerants, "Material Safety Datasheet R-125", ed, 2008. [34] C. K. Rice and J. Sand, "Initial Parametric Results Using Cyclez-An LMTD-Specified, Lorenz-
Meutzner Cycle Refrigerator-Freezer Model," 1990. [35] M. K. Smith, M. C. Heun, R. R. Crawford, and T. A. Newell, "Thermodynamic performance limit
considerations for dual-evaporator, non-azeotropic refrigerant mixture-based domestic refrigerator-freezer systems," International Journal of Refrigeration, vol. 13, pp. 237-242, 7// 1990.
[36] W. Stoecker, "Improving the energy effectiveness of domestic refrigerators by the application of refrigerant mixtures," Urbana, vol. 101, p. 61801, 1978.
[37] W. Stoecker, "Internal performance of a refrigerant mixture in a two-evaporator refrigerator," ASHRAE transactions, vol. 91, pp. 241-249, 1985.
[38] W. F. Stoecker, Design of Thermal Systems, 3 ed. New York, NY: McGraw Hill, 1989. [39] W. F. Stoecker and D. J. Walukas, "Conserving energy in domestic refrigerators through the use
of refrigerant mixtures," International Journal of Refrigeration, vol. 4, pp. 201-208, 7// 1981. [40] B. Tashtoush, M. Tahat, and M. A. Shudeifat, "Experimental study of new refrigerant mixtures to
replace R12 in domestic refrigerators," Applied Thermal Engineering, vol. 22, pp. 495-506, 4// 2002.
[41] Q. Wang, D. Li, J. Wang, T. Sun, X. Han, and G. Chen, "Numerical investigations on the performance of a single-stage auto-cascade refrigerator operating with two vapor–liquid separators and environmentally benign binary refrigerants," Applied Energy, vol. 112, pp. 949-955, 2013.
[42] D. G. Westra, "COP improvement of refrigerator/freezers, air-conditioners, and heat pumps using nonazeotropic refrigerant mixtures," 1993.
[43] S. Wongwises and N. Chimres, "Experimental study of hydrocarbon mixtures to replace HFC-134a in a domestic refrigerator," Energy conversion and management, vol. 46, pp. 85-100, 2005.
[44] C. Wu, "Non-azeotropic mixture energy conversion," Energy Conversion and Management, vol. 25, pp. 199-206, // 1985.
[45] C. Wu, "Choice of working fluids for non-azeotropic mixed-refrigerants air conditioning systems," Energy Conversion and Management, vol. 27, pp. 385-387, // 1987.
[46] W. Wu, L. Zhao, and T. Ho, "Experimental investigation on pinch points and maximum temperature differences in a horizontal tube-in-tube evaporator using zeotropic refrigerants," Energy Conversion and Management, vol. 56, pp. 22-31, 4// 2012.
[47] M. Yilmaz, "Performance analysis of a vapor compression heat pump using zeotropic refrigerant mixtures," Energy Conversion and Management, vol. 44, pp. 267-282, 1// 2003.
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[48] W. J. Yoon, K. Seo, H. J. Chung, E.-J. Lee, and Y. Kim, "Performance optimization of a Lorenz–Meutzner cycle charged with hydrocarbon mixtures for a domestic refrigerator-freezer," International Journal of Refrigeration, vol. 35, pp. 36-46, 1// 2012.
22
5. Appendix A: Reversed Rankine Cycle Analysis Figure 9 refers to the system diagram for the Reversed Rankine Cycle T-s diagram.
FIGURE 9. REVERSE RANKINE CYCLE T-S DIAGRAM
The Equations 1, 2 and 3 are derived from the properties of the thermodynamic system. Starting with the objective function in Equation 1, the objective function stems from the model of isentropic compression for a single working fluid in Equation 5.
𝑊𝑐 = �̇� ∗ 𝑐1 ∗ �1−𝐴1 − log(𝑃𝑡2)𝐴1 − log(𝑃𝑡1)�
(5)
Where mdot is the mass flow rate of the working fluid, A1 is the saturated pressure-temperature constant derived from the method of least squares, Pt2 and Pt1 are the total pressure on the evaporator and condenser sides respectively and the c constant is derived for a specific fluid undergoing isentropic compression. These constants are derived specifically for R125 and R134A. The objective function for a single refrigerant can be modified, using the constants derived for each refrigerant in mixture, shown in Equation 6. 𝑊𝑐 = 𝑥1 ∗ 𝑊𝑐 ,125 + (1 − 𝑥1)𝑊𝑐,134𝐴 (6)
23
Where Wc,125 and Wc,134A are the work of compression associated with each working fluid and x1 is the mass fraction of R125 in solution. Substituting Equation 5 into Equation 6 for the respective refrigerant mixtures gives Equation 7.
𝑊𝑐 = 𝑥1 ∗ 𝑐1 ∗ �1 −𝐴1 − log(𝑃𝑡2)𝐴1 − log(𝑃𝑡1)� + (1 − 𝑥1) ∗ 𝑐2 ∗ �1−
𝐴2 − log(𝑃𝑡2)𝐴2 − log(𝑃𝑡1)�
(7)
Equation 7 is then modified by substituting the constant values for R125 and R134A returns the equation given in Equation 1 which is the objective function that is being minimized. In order to calculate the pressures, Pt1 and Pt2, the mole fraction of each component must be calculated from the mass fraction. The relationship between mole fraction and mass fraction is shown in Equation 8.
𝑦𝑥,𝑛 =𝑥1
�𝑥1 + (1 − 𝑥1)𝑀1𝑀2� (8)
The variable yx,n represents the mole fraction at a specific quality (vapor or liquid) and at a specific state, M1 and M2 are the molecular weights of R125 and R134A respectively. Since the quality of the mixture is a liquid at state 3, this equation applies to state 3. However since the quality of the mixture at state 1 is a saturated liquid, Equation 9 must be used to calculate the mole fraction of liquid at state 1.
𝑦𝑙,1 =𝑦𝑣 ,1 ∗ 𝑃134𝐴,1
�𝑃125,1 − 𝑦𝑣,1 ∗ �𝑃134𝐴,1 − 𝑃125,1�� (9)
And since yv,1 is equivalent to the mole fraction at that state, equation 7 applies in the calculation of yv,1 using the mass fraction of R125 in mixture, x1. PR,1 represents the saturated vapor pressure of each refrigerant, R125 and R134A at states 1, and thus with these relationships Pt1 and Pt2 are expressed as functions of y1 and T1 and y1 and T3 respectively. From Dalton’s law the relationship between Pt1 and Pt2 to the mass fraction and the temperatures are shown in Equations 10 and 11 respectively. 𝑃𝑡1 = 𝑦𝑙,1𝑃125,1 + �1 − 𝑦𝑙,1� ∗ 𝑃134𝐴,1 (10) 𝑃𝑡2 = 𝑦𝑙,3𝑃125,3 + �1 − 𝑦𝑙,3� ∗ 𝑃134𝐴,3 (11)
Two more equations for the saturated pressure of each refrigerant R125 and R134A are given in Equation 12 and 13 respectively, the culmination of a least squares fit of saturated pressure-temperature data from NIST [need reference].
𝑃125,𝑛 = exp �15.45−2446.3𝑇𝑛
� (12)
𝑃134𝐴,𝑛 = exp �15.91−2810.9𝑇𝑛
� (13)
Where PR,n is the saturated pressure for R125 and R134A at states 1 or 3 (where R is R125A or R134A and n is 1 or 3). Thus by substituting Equations 8, 9, 12 and 13 into Equations 10 and 11,
24
Pt1 and Pt2 can be expressed as functions of x1, T1 and x1, T3 shown in Equations 14 and 15 respectively.
𝑃𝑡1 =𝑥1 ∗ 𝑃134𝐴,1
�𝑃125,1 − 𝑥1 ∗ �𝑃134𝐴,1 − 𝑃125,1��∗ exp �15.45−
2446.3𝑇1
�
+ �1 −𝑥1 ∗ 𝑃134𝐴,1
�𝑃125,1 − 𝑥1 ∗ �𝑃134𝐴,1 − 𝑃125,1��� ∗ exp �15.91−
2810.9𝑇1
�
(14)
𝑃𝑡2 =𝑥1
�𝑥1 + (1− 𝑥1)𝑀1𝑀2�∗ exp �15.45−
2446.3𝑇3
�+ �1 −𝑥1
�𝑥1 + (1− 𝑥1)𝑀1𝑀2��
∗ exp �15.91−2810.9𝑇3
�
(15)
Having the relationships between Pt1 and Pt2 and x1, T1 and T3 allows the creation of the objective function as a function of the mass flow rate, mass fraction of R125 in mixture, the temperature at state 1 and the temperature at state 3. The constraint equations are then defined over the condenser and over the compressor. Starting with the easiest of the constraints, Equation 2 stems from the energy balance over the evaporator. Since the energy put into the system by the heating fluid in the evaporator is constant, at 0.05 kW, the enthalpies at each state are left to be calculated in terms of x1, T1 and T3. The enthalpies at states 4 and 1 must be derived from the thermodynamic system. The enthalpy at state 4 is known from the throttling valve, where it is assumed that the expansion is isenthalpic. Thus Equation 16 relates the enthalpy at state 3 and the enthalpy at state 4. ℎ3 = ℎ4 (16)
Equations relating the enthalpy of state to a quadratic function of Tn, where n is the state, are shown for vapor and liquid in Equations 17 through 20 for R125 and R134A respectively. ℎ125,𝑙,𝑛 = 200 + 1.268 ∗ 𝑇𝑛 + 0.00219 ∗ 𝑇𝑛2 (17) ℎ125,𝑣,𝑛 = 333.2 + 0.449 ∗ 𝑇𝑛 − 0.00232 ∗ 𝑇𝑛2 (18) ℎ134𝐴,𝑙,𝑛 = 200 + 1.349 ∗ 𝑇𝑛 + 0.00142 ∗ 𝑇𝑛2 (19) ℎ134𝐴,𝑣,𝑛 = 398.6 + 0.574 ∗ 𝑇𝑛 − 0.000117 ∗ 𝑇𝑛2 (20)
Where hR,x,n represents the enthalpy for refrigerant R125 or R134A, the state n, and the quality at that state, vapor, liquid or liquid-vapor. Liquid-vapor mixtures yield an equation that is a combination of hR,l,n and hR,v,n dependent upon mixture fraction. Since the quality of state 4 is a subcooled liquid the enthalpy at that state can be approximated by Equation 21. ℎ3 = 𝑥1 ∗ ℎ125,𝑙,3 + (1− 𝑥1) ∗ ℎ134𝐴.𝑙,3 (21)
25
For the sensible heat equation, it is necessary to express T2 and T3’ as functions of T3 and the temperature of the cooling fluid entering the condenser. The analysis of the condenser requires that the condenser be analyzed in sections, denoted by Figure 5.
FIGURE 10. MODEL OF CONDENSER ANALYSIS
The sensible heat Equations for the sensible heat of vaporization and the initial sensible heat them become Equations 22 and 23. 𝑄𝑠,1 = �̇�𝑐𝑝 ∗ (𝑇2 − 𝑇3′) (22) 𝑄𝑠,2 = �̇�𝑐𝑝 ∗ (𝑇3′ − 𝑇3) (23)
The only additional information needed are the equations for T2 and T3’. These equations are given in Equations 24, 25 and 26.
𝑇3′ =𝑇3 − 25 ∗ �1− exp�𝑈𝐴 ∗ � 1
𝑊1− 1𝑊2
�� �
𝑊1𝑊2
− 1
(24)
𝑇𝑚 = 25− (25− 𝑇3′) ∗1 − exp �𝑈𝐴 ∗ � 1
𝑊2− 1𝑊1��
𝑊2𝑊1
− exp �𝑈𝐴 ∗ � 1𝑊2
− 1𝑊1��
(25)
26
𝑇2 =𝑇3′ − 𝑇𝑚 ∗ �1− exp �𝑈𝐴 ∗ � 1
𝑊1− 1𝑊2
���
𝑊1𝑊2
− 1
(26)
Where W1 is the product of the mass flow rate and the specific heat of the working fluid, W2 is the product of the mass flow rate and the specific heat of the cooling fluid, and the UA is the heat transfer coefficient of the condenser. Substituting all of these equations into the relevant preceding equations yields the equation that describes the energy out of the system, given in Equation 27. 𝑄𝑐 = �̇�𝑐𝑝 ∗ (𝑇2 − 𝑇3′) + �̇�𝑐𝑝 ∗ (𝑇3′ − 𝑇3) + �̇��ℎ3,𝑣 − ℎ3,𝑙� (27) A third constraint equation could be created to accommodate the fourth variable, x1, in the objective function could be created, however due to the nature of the process, where x1 is selected prior to the optimization, a third constraint equation is unnecessary.
6. Appendix B: Optimization Procedure The method of Lagrange multipliers is a constrained optimization algorithm that takes a function to be minimized (or maximized), the objective function, and, along with a set of constraints, finds the values within the system that minimize the objective function. The method of Lagrange multipliers states that the optimum occurs when the Equations 28, 29 and 30 are satisfied.
𝛻𝐹(𝑥1,𝑥2, … , 𝑥𝑛)− �𝜆𝑖∇𝜙𝑖
𝑚
𝑖=1
= 0 (28)
𝜙1(𝑥1, 𝑥2, … ,𝑥𝑛) = 0 (29) .................................. 𝜙𝑚(𝑥1,𝑥2, … , 𝑥𝑛) = 0 (30)
In this set of equations, F is the objective function, λi is the set of Lagrange multipliers and φi is the set of constraint equations. The set of Lagrange multipliers is always the same size as the set of constraint equations, but the number of constraint equations is always less than the number of variables, n, in the objective function. If the set of equations resulting from Equations 38, 39 and 40 are linear then linear algebra can be used to solve that respective set. The set of resulting equations is non-linear so the Newton-Raphson method must be implemented on the set of Lagrange multiplier equations. Regardless of the process, the solution to the system of equations given in Equations 38, 39 and 40 produce the values x1, x2,…,xn and λ1, λ2,…,λm that result in the minimum (or maximum) output of the objective function. The resulting set of Lagrange multipliers shed light on how the minimum (or maximum) changes with a perturbation in the control parameters (the constants in the system).
27
7. Appendix C: Lagrange Multiplier Code function [f,err] = LagrangeComplete(objf,conf,mx,ml,xt,step)
F = objf - ml*conf;
hF = hessian(F,[mx(1:numel(mx)-1) ml]);
A = eval(subs(hF,[mx(1:numel(mx)-1) ml],xt));
jF = jacobian(F,[mx(1:numel(mx)-1) ml]);
B = eval(subs(jF,[mx(1:numel(mx)-1) ml],xt));
Sol = [xt eval(subs(objf,mx(1:numel(mx)-1),xt(1:numel(mx)-1)))];
xc = xt' - A\B';
err = step;
while err>1E-6;
xt = xc';
A = eval(subs(hF,[mx(1:numel(mx)-1) ml],xt));
B = eval(subs(jF,[mx(1:numel(mx)-1) ml],xt));
SolL = [xt eval(subs(objf,mx(1:numel(mx)-1),xt(1:numel(mx)-1)))];
Sol = [Sol;SolL];
xc = xt' - A\B';
err = [err;max(abs((A\B')./xt'))];
end
si = size(Sol);
f = [xc;double(vpa(subs(objf,mx(1:numel(mx)-1),xc(1:numel(mx)-1)')))];
tab(f',mx,ml)
check = eval(subs(conf,[mx(1:numel(mx)-1) ml],f(1:numel(f)-1)'));
display(check)
hold on
for i = 1:numel(xt)+1
plot(1:si(1),Sol(:,i),'Color',rand(1,3))
end
m = [mx(1:numel(mx)-1) ml mx(numel(mx))];
legend([mx(1:numel(mx)-1) ml mx(numel(mx))],'Location','SouthEastOutside');
28
for i = 1:numel(m)
text(si(1),Sol(si(1),i),[char(m(i)) ' converges to ' num2str(Sol(si(1),i))],...
'FontSize',5);
end
xlabel('No of Iterations')
ylabel('Variable')
title('Convergence Chart')
hold off
Nam = cellstr(num2str([1:si(1)]'))' ;
array2table(Sol,...
'VariableName',[mx(1:numel(mx)-1) ml mx(numel(mx))],...
'RowNames',Nam)
Published with MATLAB® R2014b
The purpose of the Lagrange Multiplier code is to take any set of objective function, constraint equations, system variables and Lagrange multipliers, as well as an initial guess and error, and solve for the minimum value of the given objective function subject to the given constraints. The output of the code is an iteration table showing how much iteration the function took to converge, a convergence chart, the optimized values for the set of x and λ and the final error output of the code.
8. Appendix D: Refrigerant Justification The selections of the refrigerants to use in the binary mixture were made such that the resultant refrigerant mixture was actually usable as a replacement to R134A according to ASHRAE and EPA standards [28]. That is, any refrigerant with flammability class A2 or higher cannot be used as a replacement to a refrigerant with an A1 flammability class. The properties for the four best, and available, refrigerants that were up for selection are in Table 3.
TABLE 5. LIST OF REFRIGERANT PROPERTIES AT 140 KPA [5],[22],[24],[25]
Refrigerant Hfg (kJ/kg)
Cp(L,V) (kJ/kg)
Density(L,V) (kg/m^3)
Tsat@Pe=140 (C)
GWP/ODP Flammability Class
R-32 374.6057 1.5978;0.91044 1194.6;4.0502 -45.093 675;0 A2 R-125 160.3147 1.1294;0.71763 1488.7;9.2095 -41.267 3500;0 A1 R-134A 212.08 1.2957;0.82037 1354.5;7.1353 -18.760 1100;0 A1 R-152A 323.061 1.6454;1.0104 995.22;4.5770 -16.464 124;0 A2
Of the best performing refrigerants in this table, R125 and R134A are the only two refrigerants that meet the criteria set by ASHRAE standard 15 [28]. Thusly these were the two refrigerants that have been selected regardless of harm to the environment. For the scope of this project, which must abide by the standards that it purports to follow, this refrigerant mixture is acceptable
29
to be used based off of that standard. However a conundrum arises from ASHRAE standard 34 and the Kyoto Protocol which require that the GWP from hydrofluorocarbons (which includes refrigerants), be reduced. With the mixture of R125/R134A it appears that the GWP actually increases for these mixtures, which are both HFCs. The contradiction could only be annulled by creating an entirely new refrigerant with a low GWP, however this is beyond the scope of this project. The conundrum is only rectified by the fact that complying with ASHRAE standard 15 guarantees immediate safety (i.e. the refrigerator won’t explode), and of the four refrigerants in Table 3 only R125 and R134A are easily available.
30
9. Appendix E: Daewoo Compressor HFL5Y-1 Datasheet Table 6. SPECIFICATION OF DAEWOO COMPRESSOR
Model HFL5Y-1 Date of Approval 2002-07-22 Voltage 110-115V Frequency 60Hz
APPLICATION
Evaporating temp. range -35¡É~-10¡É Refrigerant control Capillary tube
-31¢µ~14¢µ Compressor cooling Static cooling
Refrigerant
R134a Voltage range 94~132 V
PERFORMANCE kcal/h 43 (50W)
Test condition (ASHRAE L.B.P)
Cooling capacity Btu/h 171 Evaporating temp. ¡É -23.3 (-10 ¢µ)
Power input W 65 Condensing temp. ¡É 54.4 (130 ¢µ)
Current A 0.9 Liquid subcooled to ¡É 32.2 (90 ¢µ)
Efficiency : COP W/W 0.77 Gas superheated to ¡É 32.2 (90 ¢µ)
Efficiency : EER Btu/Wh 2.63 Ambient temp. ¡É 32.2 (90 ¢µ)
Test power source : 115V 60Hz Test equipment : DAEWOO Calorimeter
COMPRESSOR DATA MOTOR DATA
Design Recipro. ball joint Motor type RSIR, 1¥Õ 2-pole
Displacement cc/rev 2.3 Voltage range V 94~132
Bore size (¥Õ) mm 18.0 High potential test V/sec 1800
Speed r.p.m. 3520 Rated current A 0.9
Oil charge cc 230 Starting current (LRA) A 4.3
Weight (incl. oil) kg 6.3 Running capacitor VAC/§Þ NONE
Residual moisture mg ¡é 70 Starting capacitor VAC/§Þ NONE
Impurities mg ¡é 25 LRA : IEC Standard method (335-2-34)
OVERLOAD PROTECTOR (OLP) STARTER Type 4TM
174SHBYY-52
Type : PTC
Opening temp. ¡É 135 (275 ¢µ) RSIR
Closing temp. ¡É 69 (156 ¢µ) Resistance at 25¡É §Ù 6.8
Trip current at 70¡É A 1.4 Recovery time sec ¡é 80 Trip current at 25¡É A
5.0 [5~15sec]
31
10. Appendix F: Exhaustive Solution Due to the size of the equations that are produced from the reduction of equations the optimization problem was turned into a deterministic problem by assuming that a variable in the equation set is known. In this case the mass fraction of R125 is assumed to be known, thus leaving a total of 8 equations and 8 unknowns. The unknown variables are listed below:
1. �̇� (Mass flow rate of the binary refrigerant) 2. T1 (Temperature at state 1) 3. T2 (Temperature at state 2) 4. T3a (Temperature at state 3a) 5. T3b (Temperature at state 3b) 6. T4 (Temperature at state 4) 7. T5 (Temperature at state 5) 8. T6 (Temperature at state 6)
The equations themselves are too large to put into this Appendix. They will be attached in the second appendix.
32
11. Appendix G: Exhaustive Solution Equations The 8 equations are shown in Equations 31 through 38.
𝑥1.4135−0.4135∗𝑥∗exp�14.861− 2498.3
𝑇3𝑎+273�
exp�15.407− 2993.2𝑇3𝑎+273
�− 𝑥1.4135−0.4135∗𝑥∗�exp�15.407− 2993.2
𝑇3𝑎+273�−exp�14.861− 2498.3
𝑇3𝑎+273��∗ exp �15.407− 2993.2
𝑇3𝑎+273� +
�1−𝑥
1.4135−0.4135∗𝑥∗exp�14.861− 2498.3𝑇3𝑎+273
�
exp�15.407− 2993.2𝑇3𝑎+273
�− 𝑥1.4135−0.4135∗𝑥∗�exp�15.407− 2993.2
𝑇3𝑎+273�−exp�14.861− 2498.3
𝑇3𝑎+273��� ∗ exp �14.861− 2498.3
𝑇3𝑎+273� = 𝑥
1.4135−0.4135∗𝑥∗ exp �15.407−
2993.2𝑇3𝑏+273
�+ �1− 𝑥1.4135−0.4135∗𝑥
� ∗ exp �14.861− 2498.3𝑇3𝑏+273
�
(31)
𝑋𝑙,5(𝑇1,𝑇5) ∗ (200 + 0.954 ∗ 𝑇5 + 0.00116 ∗ 𝑇52) + �1 − 𝑋𝑙,5(𝑇1,𝑇5)� ∗ (200 + 0.925 ∗ 𝑇5 + 0.00081 ∗ 𝑇52) + 𝑋𝑣,5(𝑇1,𝑇5) ∗ ( 337.4 +
0.623 ∗ 𝑇5 − 0.000086 ∗ 𝑇52) ∗ �1 − 𝑋𝑣,5(𝑇1,𝑇5)� ∗ (351.5 + 0.428 ∗ 𝑇5 − 0.00071 ∗ 𝑇52) = 0.1 ∗ (200 + 0.954 ∗ 𝑇5 + 0.00116 ∗ 𝑇52) +(1 − 0.1) ∗ (200 + 0.925 ∗ 𝑇5 + 0.00081 ∗ 𝑇52)
(32)
20 = 5 ∗ 10−𝑇6−𝑇5log−15−𝑇6−25−𝑇5
(33)
5 ∗ 𝑓𝑆𝐻 ∗𝑇2−𝑇𝑐𝑜−𝑇3𝑎+𝑇𝑚log� 𝑇2−𝑇𝑐𝑜𝑇3𝑎−𝑇𝑚
�+ 5 ∗ (1 − 𝑓𝑆𝐻) ∗ 𝑇3𝑎−𝑇𝑚−𝑇3𝑏+25
ln𝑇3𝑎−𝑇𝑚𝑇3𝑏−25=
20 + �̇� ∗ �0.1 ∗ 158 ∗ �1− 15.407−ln�𝑃3𝑎(𝑇3𝑎)�
15.407−ln�𝑃𝑡,6(𝑇1,𝑇6)�� + 0.9 ∗ 188 ∗ �1− 14.861−ln�𝑃3𝑏(𝑇3𝑏)�
14.861−ln�𝑃𝑡,5(𝑇1,𝑇5)���
(34)
20 = �̇� ∗ �𝐹𝑟𝑙6(𝑇1,𝑇6) ∗ �𝑋𝑙,6(𝑇1,𝑇6) ∗ (200 + 0.954 ∗ 𝑇6 + 0.00116 ∗ 𝑇62) + �1− 𝑋𝑙,6(𝑇1,𝑇6)� ∗ (200 + 0.925 ∗ 𝑇6 + 0.00081 ∗ 𝑇62)� +
�1 − 𝐹𝑟𝑙6(𝑇1,𝑇6)� ∗ �𝑋𝑣,6(𝑇1,𝑇6) ∗ ( 337.4 + 0.623 ∗ 𝑇6 − 0.000086 ∗ 𝑇62) ∗ �1− 𝑋𝑣,6(𝑇1,𝑇6)� ∗ (351.5 + 0.428 ∗ 𝑇6 − 0.00071 ∗ 𝑇62)� −
�𝐹𝑟𝑙5(𝑇1,𝑇5) ∗ �𝑋𝑙,5(𝑇1,𝑇5) ∗ (200 + 0.954 ∗ 𝑇5 + 0.00116 ∗ 𝑇52) + �1− 𝑋𝑙,5(𝑇1,𝑇5)� ∗ (200 + 0.925 ∗ 𝑇5 + 0.00081 ∗ 𝑇52)�+
(35)
33
�1 − 𝐹𝑟𝑙5(𝑇1,𝑇5)� ∗ �𝑋𝑣,5(𝑇1,𝑇5) ∗ ( 337.4 + 0.623 ∗ 𝑇5 − 0.000086 ∗ 𝑇52) ∗ �1− 𝑋𝑣,5(𝑇1,𝑇5)� ∗ (351.5 + 0.428 ∗ 𝑇5 − 0.00071 ∗ 𝑇52)���
�̇� ∗ �0.1 ∗ (337.4 + 0.623 ∗ 𝑇1 + 0.000086 ∗ 𝑇12) + 0.9 ∗ (351.5 + 0.428 ∗ 𝑇1 − 0.00071 ∗ 𝑇1
2) − �𝐹𝑟𝑙6(𝑇1,𝑇6) ∗ �𝑋𝑙,6(𝑇1,𝑇6) ∗ (200 + 0.954 ∗
𝑇6 + 0.00116 ∗ 𝑇62) + �1− 𝑋𝑙,6(𝑇1,𝑇6)� ∗ (200 + 0.925 ∗ 𝑇6 + 0.00081 ∗ 𝑇62)�+ �1 − 𝐹𝑟𝑙6(𝑇1,𝑇6)� ∗ �𝑋𝑣,6(𝑇1,𝑇6) ∗ ( 337.4 + 0.623 ∗
𝑇6 − 0.000086 ∗ 𝑇62) ∗ �1− 𝑋𝑣,6(𝑇1,𝑇6)� ∗ (351.5 + 0.428 ∗ 𝑇6 − 0.00071 ∗ 𝑇62)��� = 0.3 ∗ 𝑇3𝑏−𝑇1−𝑇4+𝑇6ln𝑇3𝑏−𝑇1𝑇4−𝑇6
(36)
�̇� ∗ �0.1 ∗ (337.4 + 0.623 ∗ 𝑇1 + 0.000086 ∗ 𝑇12) + 0.9 ∗ (351.5 + 0.428 ∗ 𝑇1 − 0.00071 ∗ 𝑇1
2) − �𝐹𝑟𝑙6(𝑇1,𝑇6) ∗ �𝑋𝑙,6(𝑇1,𝑇6) ∗ (200 + 0.954 ∗
𝑇6 + 0.00116 ∗ 𝑇62) + �1− 𝑋𝑙,6(𝑇1,𝑇6)� ∗ (200 + 0.925 ∗ 𝑇6 + 0.00081 ∗ 𝑇62)�+ �1 − 𝐹𝑟𝑙6(𝑇1,𝑇6)� ∗ �𝑋𝑣,6(𝑇1,𝑇6) ∗ ( 337.4 + 0.623 ∗
𝑇6 − 0.000086 ∗ 𝑇62) ∗ �1− 𝑋𝑣,6(𝑇1,𝑇6)� ∗ (351.5 + 0.428 ∗ 𝑇6 − 0.00071 ∗ 𝑇62)��� = 0.1 ∗ (200 + 0.954 ∗ 𝑇3𝑏 + 0.00116 ∗ 𝑇3𝑏2 ) + 0.9 ∗
(200 + 0.925 ∗ 𝑇3𝑏 + 0.00081 ∗ 𝑇3𝑏2 ) − �0.1 ∗ (200 + 0.954 ∗ 𝑇4 + 0.00116 ∗ 𝑇42) + 0.9 ∗ (200 + 0.925 ∗ 𝑇4 + 0.00081 ∗ 𝑇42)�
(37)
5 ∗ 𝑓𝑆𝐻 ∗𝑇2−𝑇𝑐𝑜−𝑇3𝑎+𝑇𝑚log� 𝑇2−𝑇𝑐𝑜𝑇3𝑎−𝑇𝑚
�+ 5 ∗ (1 − 𝑓𝑆𝐻) ∗ 𝑇3𝑎−𝑇𝑚−𝑇3𝑏+25
ln𝑇3𝑎−𝑇𝑚𝑇3𝑏−25= 3.352 ∗ (𝑇𝑐𝑜 − 25)
(38)
Where Tm and Tco are shown in equations 39 and 40.
𝑇𝑚 =
𝑇3𝑎−𝑇2∗�1−exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352��
�̇�∗0.993.352 +exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352���
1−1−exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352��
�̇�∗0.993.352 +exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352��
(39)
34
𝑇𝑐𝑜 =
𝑇3𝑎−𝑇2∗�1−exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352��
�̇�∗0.993.352 +exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352���
1−1−exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352��
�̇�∗0.993.352 +exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352��
+
⎝
⎜⎜⎛𝑇3𝑎 −
𝑇3𝑎−𝑇2∗�1−exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352��
�̇�∗0.993.352 +exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352���
1−1−exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352��
�̇�∗0.993.352 +exp�5∗𝑓𝑠ℎ∗� 1
�̇�∗0.99−1
3.352�� ⎠
⎟⎟⎞∗
1−exp�5∗𝑓𝑠ℎ∗� 13.352−
1�̇�∗0.99��
3.352�̇�∗0.99+exp�5∗𝑓𝑠ℎ∗�
13.352−
1�̇�∗0.99��
(40)
In order to turn this into an optimization problem, the concentration is set between 0 and 1 and the concentration value that produces the highest COP is the optimal value for the system.
35
12. Appendix H: Water Calorimeter Test The process of acquiring a crude approximation of the COP of the Magic Chef 2.6 cu ft refrigerator involved setting a heated bowl of water inside the refrigerator and taking the change in the temperature of the water over a period of time using a K-type thermocouple. By using a thin and shallow aluminum pan the heat load of the pan and heat transfer due to vertical components were neglected. The volume of the water, time, temperature of the water, and the power used by the refrigerator were recorded. Figure 11 shows the graph of the temperature of the water with respect to time.
FIGURE 111. TEMPERATURE OF HOT WATER WITH RESPECT TO TIME
The pan was filled with 500 mL of water, corresponding to 0.5 kg of water. The water exchanges heat with the air inside the refrigerator and in turn the air exchanges heat with the evaporator. This relationship is symbolized in Equation 41. 𝑄𝑒 = 𝑄𝑎𝑖𝑟 = 𝑄𝑤 (41)
Where Qe is the evaporator heat load, Qair is the air heat load and Qw is the water heat load. Since the heat load of interest is the water heat load Qw, and the temperature-time response is inherently nonlinear an exponential function is fitted to the data, given in Equation 42.
𝑇 = 48.11 ∗ exp �−8 ∗ 10−25∗𝑡� (42)
Evaluating the function at the 0 and 2460 second marks, multiplying by the mass and specific heat of the water, and then dividing by the total time gives the heat load of the water, and subsequently the evaporator load, the entire process is shown in Equation 43.
y = 48.11e-8E-04x R² = 0.9901
0
10
20
30
40
50
60
0 500 1000 1500 2000 2500 3000
Axis
Title
Axis Title
Thermocouple in Water Temperature vs. Time Data
ThermocoupleData
Expon.(ThermocoupleData)
36
𝑄𝑒 = 𝑄𝑤 = 0.5 ∗ 4.19 ∗ 48.11 ∗�exp(−8 ∗ 10−4 ∗ 𝑡0) − exp�−8 ∗ 10−4 ∗ 𝑡𝑓��
2460 (43)
With the evaporator load approximated, from this equation Qe is 0.0351 kW, the compressor power is needed in order to calculate the COP, which is the ratio of the evaporator load to the work of compression. The compressor power graph is shown in Figure 11.
FIGURE 11. COMPRESSOR POWER VS. TIME
The compressor power varies with time, and as the heat from the water is experienced inside the refrigerator the compressor power increases. However as the heat load dissipates and is cooled the compressor power decreases to match the effect. Thus the average compressor power of 68.04 W is used to calculate the COP. From the calculated total heat load and the average work of compression the COP was found to be 0.52 W/W. Attached at the end of this report is the Daewoo data sheet for the Magic Chef 2.6 cu ft refrigerator, shown in Table 1, where the COP for an ambient temperature of 32.2 °C is given as 0.77 W/W. (in the case of the group’s testing the ambient temperature is approx. 27-28 °C)
65
66
67
68
69
70
71
72
73
0 500 1000 1500 2000 2500 3000
Com
pres
sor P
ower
(W)
Time (s)
Compressor Power vs. Time
Compressor Power
