Domestic installations and utilities

A home's electric water heater is rated at $1,500\ \text{W}$ and operates for about $2\ \text{h}$ per day. (a) Calculate its monthly energy consumption ($30$ days), in $\text{kW} \times \text{h}$. (b) Calculate its monthly cost at $0{,}\$20$ per $\text{kW} \times \text{h}$. (c) Check that the breaker panel's $16\ \text{A}$ circuit is sufficient for the water heater, by calculating its current at $230\ \text{V}$.

A family evaluates whether to replace their traditional butane gas cooker with an electric induction cooktop. To prepare their monthly meals, they need to transfer a net useful thermal energy of $100\ \text{kW}\times\text{h}$ to the cookware.\n\nTechnical and economic data:\n- Gas cooker: supplied by $12.5\ \text{kg}$ butane cylinders ($LCV = 12.8\ \text{kW}\times\text{h}/\text{kg}$). Due to thermal losses from convection and radiation to the environment, the efficiency of the gas burner is only $42\,\%$. The price of a cylinder is $17{,}\$50$.\n- Induction cooktop: directly heats the ferromagnetic base of the vessel, with an overall energy efficiency of $84\,\%$. The cost of the electricity consumed is $0{,}\$19$ per $\text{kW}\times\text{h}$.\n\n(a) Calculate the gross primary energy contained in a full cylinder and determine how many cylinders per month (or fraction of a cylinder) the family needs with the gas cooker.\n(b) Calculate the monthly electricity consumption in billed $\text{kW}\times\text{h}$ needed to provide exactly the same $100\ \text{kW}\times\text{h}$ of useful heat with the induction cooktop.\n(c) Determine the monthly cost of fuel/electricity for both alternatives and justify why comparing only the price per gross $\text{kW}\times\text{h}$ of gas versus electricity leads to an erroneous conclusion.\n(d) Analyze what impact the installation of the induction cooktop (typical maximum power of $7\ \text{kW}$) could have on the electrical service capacity of the home and the fixed charge on the electric bill.

An electric water heater with an $80\ \text{L}$ capacity ($P = 1,800\ \text{W}$) supplies domestic hot water (DHW) to a home. Four people take showers consecutively in the morning. Each shower lasts $6$ minutes with a mixed water flow rate at the showerhead of $8\ \text{L/min}$ at a comfort temperature of $38\ ^\circ\text{C}$. Mains cold water enters at $14\ ^\circ\text{C}$, while the heater keeps the stored water at $62\ ^\circ\text{C}$ (recommended temperature for Legionella prevention). The standing thermal losses of the heater due to imperfect insulation are equivalent to $1.1\ \text{kW}\times\text{h}$ per day. ($c = 4,186\ \text{J}/(\text{kg}\times^\circ\text{C})$, water density $= 1\ \text{kg/L}$, $1\ \text{kW}\times\text{h} = 3.6\times 10^6\ \text{J}$).\n\n(a) Using a thermal mixing balance ($V_{\text{mezcla}} \times T_{\text{mezcla}} = V_{\text{termo}} \times T_{\text{termo}} + V_{\text{red}} \times T_{\text{red}}$), calculate how many liters of water at $62\ ^\circ\text{C}$ are drawn from the heater during each shower.\n(b) Deduce with reasoned arguments whether the heater will be able to supply hot water at the desired temperature to the third and fourth person if they shower consecutively without recovery time.\n(c) Calculate the total monthly electrical energy expenditure attributable to DHW (including the four daily showers and standing thermal losses) and its economic cost at $0{,}\$18/\text{kW}\times\text{h}$.\n(d) A family member suggests: "If we turn down the water heater thermostat to $40\ ^\circ\text{C}$, we will save $50\,\%$ energy without noticing any difference". Quantitatively refute this statement by evaluating the volume of hot water that each shower would require at $40\ ^\circ\text{C}$ compared to $62\ ^\circ\text{C}$, and warn about the associated microbiological risk.

A detached single-family home of $150\ \text{m}^2$ located in the mountains (continental climate zone with cold winters) needs to replace its comprehensive heating and DHW system. The estimated annual thermal demand for the comfort of the home is $15,000\ \text{kW}\times\text{h}$ of heat. Three engineering alternatives are compared:\n- Alternative A (Bulk piped propane): Condensing boiler with a seasonal average efficiency of $91\,\%$. Propane cost: $0{,}\$095/\text{kW}\times\text{h}$ of primary energy. Initial investment in boiler and tank: $\$3,200$.\n- Alternative B (Biomass - Pellets): Pellet boiler with a seasonal efficiency of $84\,\%$. Calorific value of pellets: $4.8\ \text{kW}\times\text{h}/\text{kg}$; pellet price: $0{,}\$34/\text{kg}$. Initial investment in boiler, hopper, and installation: $\$8,800$.\n- Alternative C (Air-source heat pump with underfloor heating): Air-to-water heat pump with a seasonal average coefficient of performance ($\text{SCOP}$) of $2.75$ (affected by low outdoor temperatures in winter). Cost of electric $\text{kW}\times\text{h}$: $0{,}\$17$. Initial investment (equipment and integration): $\$11,500$.\n\n(a) Calculate the annual economic expenditure on fuel or electricity for each of the three alternatives to fully cover the demand of $15,000\ \text{kW}\times\text{h}$ of useful heat.\n(b) Calculate the simple payback period of the investment (amortization) for the heat pump compared to the propane option based on the annual fuel savings.\n(c) Draft a technical assessment giving a reasoned choice for a family living in the house year-round, weighing not only economic cost and payback, but also operational constraints (space for a silo and ash maintenance in biomass vs reliability of COP during severe frosts in aerothermal systems).