Exercises: Refrigerators and heat pumps

Explain in your own words how a refrigerator can 'remove cold' if what it actually does is move heat: where it takes it from, where it expels it, and which machine provides the necessary work.

The figure shows the refrigeration cycle with its four components, but one is covered. State which component is missing, what happens to the refrigerant inside it (change of state and pressure), and where it is physically located in your refrigerator at home.

Put the four steps of the refrigeration cycle in order as the refrigerant goes through them: (a) it evaporates at low pressure, absorbing heat; (b) it expands and its temperature drops sharply; (c) it is compressed, increasing pressure and temperature; (d) it condenses, releasing heat to the outside. Explain which component performs each step.

Touch the back grid of your refrigerator: it is lukewarm or hot. Explain where this heat comes from and why the kitchen heats up slightly when the refrigerator is running.

Explain what a refrigerant or refrigerating fluid is and what two properties it must have for the cycle (low boiling point at low pressure and the ability to absorb heat when evaporating). Why is water not simply used?

The same air conditioning unit can cool in summer and heat in winter. Explain what changes between both modes (which heat exchanger acts as the evaporator and which as the condenser) and what the valve that reverses the cycle is called.

Define the coefficient of performance (COP) of a refrigerator and explain what a COP of $3$ means in terms of heat extracted per unit of electricity consumed. Why does a higher COP indicate a more efficient appliance?

Explain why it is not advisable to leave the freezer door open 'while deciding what to take out': what happens to the heat that enters, to the frost, and to the work the compressor will have to do afterwards.

A refrigerator extracts $600\ \text{kJ}$ of heat from its interior every hour, consuming $200\ \text{kJ}$ of electrical energy. a) Calculate its efficiency or coefficient of performance, $\mathrm{COP} = \frac{Q_1}{W}$. b) Calculate the total heat released to the kitchen in that hour. c) Explain why the COP can be greater than $1$ without violating the conservation of energy.

A refrigerator that extracts $600\ \text{kJ}$ of heat from its interior every hour consuming $200\ \text{kJ}$ of electrical energy is used in winter as a heat pump to heat a dwelling. Calculate its COP as a heat pump, $\mathrm{COP}_{bc} = \frac{Q_2}{W}$, and explain why a heat pump heats with more thermal energy than an electric heater of the same power output.

Explain why the freezer performs worse if the kitchen is very hot or if the condenser coils are covered in dust, reasoning with the cycle: why it is harder for the compressor when the temperature of the hot reservoir rises and what happens to the COP.

An air conditioner consumes $1{,}2\ \text{kWh}$ of electricity in an afternoon and extracts $3\,600\ \text{kcal}$ from the air in the house. a) Convert the heat to $\text{kWh}$ ($1\ \text{kWh} = 860\ \text{kcal}$) and calculate the COP. b) Calculate the heat released to the outside. c) Compare with a fan: where does the electrical energy go in each appliance?

Compare an air-source heat pump (takes heat from outdoor air) with a ground-source heat pump (takes heat from the ground via probes): what temperature cold reservoir each has in January, how that affects the COP, and why the ground-source heat pump costs much more to install.

The evaporator of a refrigerator becomes covered in frost over time. Explain where this water comes from, why frost hinders heat exchange (insulating effect), and how frost-free ("no frost") systems solve this (periodic defrost heater and fan).

Two identical refrigerators operate, one next to the oven and the other in a cool pantry. Reason which one will consume more energy and why, relating it to the temperature difference the cycle must overcome and the cabinet insulation.

Four cans of soft drink at $25\ ^\circ\text{C}$ are placed in a refrigerator that can extract $100\ \text{W}$ of heat from its interior. Estimate the minimum time to cool them to $5\ ^\circ\text{C}$ (total mass of liquid $1{,}4\ \text{kg}$, approximate specific heat capacity of water $4\,180\ \text{J}/(\text{kg}\cdot\text{K})$) and comment on two reasons why the actual time will be longer.

A shop owner claims that, during a heatwave, he leaves the door of a $400\ \text{W}$ electric power chest freezer open to "cool the back room" ($V = 35\ \text{m}^3$, $\rho_{\text{air}} = 1,2\ \text{kg/m}^3$, $c_e = 1005\ \text{J/(kg}\cdot\text{K)}$). Assuming the room is closed and thermally insulated from the outside: a) Refute his claim by applying the first law of thermodynamics to the system formed by the entire back room; calculate the rate of temperature change over time in the room ($\Delta T/\Delta t$ in $^\circ\text{C/h}$). b) Explain what will happen to the refrigeration $\mathrm{COP}$ of the unit as the room temperature increases. c) Predict what will happen to the compressor motor if this continuous operation without thermostat cut-off is maintained.

A commercial facility must replace an $\text{R-404A}$ central refrigeration unit ($\mathrm{GWP} = 3922$) that has a charge of $50\ \text{kg}$, annual leaks of $8\,\%$, and an electricity consumption of $40\,000\ \text{kWh/year}$. Two options are being studied: 1) Retrofit to $\text{R-449A}$ ($\mathrm{GWP} = 1397$), which maintains the leakage rate but increases electricity consumption by $3\,\%$ due to lower thermodynamic performance. 2) New installation of $\text{R-290}$ (propane, $\mathrm{GWP} = 3$), which reduces leaks to $1\,\%$ and electricity consumption by $6\,\%$. With an electricity emission factor of $0,25\ \text{kg CO}_2/\text{kWh}$: a) Calculate the total annual environmental impact in equivalent $\text{CO}_2$ emissions (direct from leaks and indirect from consumption) of each option and compare them with the original situation. b) Justify why the concept of Total Equivalent Warming Impact (TEWI) shows that a gas with a lower $\mathrm{GWP}$ is not always the best solution if it impairs efficiency. c) Explain the safety requirements imposed by the F-Gas regulation and the safety regulations for refrigeration systems when choosing propane.

A single-family home has a heating demand of $8\ \text{kW}$. Two alternatives are evaluated: a natural gas boiler (seasonal efficiency of $90\,\%$, fuel cost $0,07\ \text{€/kWh}$) and an air-source heat pump (electricity cost $0,18\ \text{€/kWh}$). a) Determine the minimum $\mathrm{COP}$ that the heat pump must achieve so that the hourly economic cost is lower than that of the boiler. b) If the electricity emission factor is $0,25\ \text{kg CO}_2/\text{kWh}$ and that of natural gas is $0,20\ \text{kg CO}_2/\text{kWh}$, calculate the $\mathrm{COP}$ above which the heat pump reduces $\text{CO}_2$ emissions. c) Knowing that at $-7\ ^\circ\text{C}$ outdoors the actual $\mathrm{COP}$ drops to $2,1$, evaluate which of the two systems is more advantageous under these extreme conditions and what technical solution is usually added to air-source heat pumps in very cold climates.

The commercial datasheet for a domestic freezer claims that «thanks to its inverter control, the unit achieves a cooling $\mathrm{COP}$ of $6,5$ while maintaining the chamber at $-18\ ^\circ\text{C}$ with an ambient kitchen temperature of $25\ ^\circ\text{C}$». a) Prove that this claim is thermodynamically impossible by calculating the theoretical maximum Carnot $\mathrm{COP}$ between both thermal reservoirs. b) In a real vapor-compression cycle, to transfer heat with a reasonable heat exchanger size, the refrigerant must evaporate at $5\ ^\circ\text{C}$ below the chamber and condense at $10\ ^\circ\text{C}$ above ambient temperature. Calculate the new Carnot $\mathrm{COP}$ with the actual refrigerant temperatures. c) If the actual measured $\mathrm{COP}$ on a test bench is $1,9$, calculate the exergetic efficiency relative to the Carnot cycle in part b and justify two internal irreversibilities that cause this discrepancy.

A single-family home in a continental zone requires $10\ \text{kW}$ of heating in winter ($T_{\text{ext}} = -5\ ^\circ\text{C}$, $T_{\text{int}} = 20\ ^\circ\text{C}$) and $6\ \text{kW}$ of cooling in summer ($T_{\text{ext}} = 36\ ^\circ\text{C}$, $T_{\text{int}} = 24\ ^\circ\text{C}$). Installing a reversible air-to-air heat pump is being considered. a) Explain the role of the four-way reversing valve and the behavior of the outdoor and indoor units in each operating mode. b) The equipment catalog indicates that its thermal power is $10\ \text{kW}$ at $+7\ ^\circ\text{C}$ outdoor temperature, but it drops to $6\ \text{kW}$ at $-5\ ^\circ\text{C}$. Explain the thermodynamic reasons for this loss of power and define the concept of "balance point". c) At $-5\ ^\circ\text{C}$, describe the physical problem of frost formation on the outdoor coil and how the machine acts to eliminate it without interrupting service for an extended period.

A technician checks a storage cold room ($T_{\text{setpoint}} = +4\ ^\circ\text{C}$) whose compressor does not stop, but the indoor temperature does not drop below $+12\ ^\circ\text{C}$. The technician takes the following measurements: suction pressure is very low ($1,0\ \text{bar}$, when normal is $2,6\ \text{bar}$); discharge pressure is low ($8,5\ \text{bar}$, normal: $15\ \text{bar}$); superheat at the evaporator outlet is abnormally high ($30\ ^\circ\text{C}$, normal: $5\ ^\circ\text{C}$); subcooling at the condenser outlet is zero ($0\ ^\circ\text{C}$, normal: $5\ ^\circ\text{C}$); and motor current is $40\,\%$ below nominal. a) Deduce with justification the cause of the failure among: 1) refrigerant leak, 2) dirty condenser, 3) thermostat stuck in the closed position. b) Rule out the other two hypotheses using thermodynamic arguments based on the pressure and temperature readings. c) Justify why the compressor runs the risk of burning out due to overheating despite consuming less current than nominal.

A logistics center must design its refrigeration installation: frozen food storage rooms ($-25\ ^\circ\text{C}$) and an order preparation area ($+4\ ^\circ\text{C}$). Three natural refrigerants are evaluated against HFCs: $\text{NH}_3$ (R-717), $\text{CO}_2$ (R-744), and $\text{R-290}$ (propane). a) Justify the choice of each fluid according to its safety classification (toxicity/flammability according to standards): explain why $\text{NH}_3$ is reserved for isolated machinery rooms and what charge limitations apply to $\text{R-290}$. b) $\text{CO}_2$ has a critical temperature of $31,1\ ^\circ\text{C}$. Explain how a transcritical $\text{CO}_2$ cycle operates in summer if the ambient temperature reaches $38\ ^\circ\text{C}$, indicating what element replaces the condenser and how this affects the design pressures. c) Design and justify an $\text{NH}_3/\text{CO}_2$ cascade scheme, indicating which of the two fluids must act in the high-temperature stage and which in the low-temperature stage.

An NGO tests evaporative double-clay-pot refrigerators (Zeer system) to store supplies in rural areas without electricity. a) Justify thermodynamically why the water evaporation process cools the interior: identify the latent heat transfer and explain why the ambient air wet-bulb temperature ($T_{\text{wb}}$) represents the theoretical lower limit of cooling. b) Two prototypes are tested with identical outdoor dry-bulb temperature ($T = 38\ ^\circ\text{C}$): Prototype A in an arid climate ($T_{\text{wb}} = 18\ ^\circ\text{C}$, $\mathrm{RH} = 15\,\%$) and Prototype B in a tropical monsoon climate ($T_{\text{wb}} = 32\ ^\circ\text{C}$, $\mathrm{RH} = 75\,\%$). If both exhibit an evaporative efficiency of $70\,\%$, defined as $\eta = \frac{T_{\text{ext}} - T_{\text{int}}}{T_{\text{ext}} - T_{\text{wb}}}$, calculate the indoor temperature obtained in each case. c) Using the data obtained, critically evaluate the feasibility of using this device to preserve thermosensitive medications (range $+2\ ^\circ\text{C}$ to $+8\ ^\circ\text{C}$) versus vegetables (temperature $< 20\ ^\circ\text{C}$) in both climates.