U-values, insulation and heat pumps

The U-value of a wall is $0.35\ \text{W m}^{-2}\text{K}^{-1}$.

1. a) State what a U-value measures.
2. b) Explain whether a low or a high U-value is better for a well-insulated house.
3. c) Calculate the rate of heat loss through $20\ \text{m}^2$ of this wall when the temperature difference between inside and outside is $15\ \text{K}$, using $P = UA\Delta T$.

A house in Waterford has the following elements, with U-values and areas: walls $90\ \text{m}^2$ at $0.30\ \text{W m}^{-2}\text{K}^{-1}$, roof $50\ \text{m}^2$ at $0.16\ \text{W m}^{-2}\text{K}^{-1}$, windows $18\ \text{m}^2$ at $1.4\ \text{W m}^{-2}\text{K}^{-1}$, and floor $50\ \text{m}^2$ at $0.25\ \text{W m}^{-2}\text{K}^{-1}$.

1. a) Calculate the heat loss rate through each element for a temperature difference of $18\ \text{K}$.
2. b) Calculate the total heat loss rate.
3. c) State which element should be improved first to reduce heat loss, and justify your answer.

A homeowner in Kilkenny adds insulation to the attic of her house.

1. a) Name two ways in which heat is lost through an uninsulated roof.
2. b) Explain why trapped air in fibreglass insulation reduces heat loss.
3. c) State one advantage of insulation, other than reducing a household's energy bills.

An attic conversion in Dundalk has a sloped roof of area $30\ \text{m}^2$. The roof consists of $12\ \text{mm}$ plasterboard ($k = 0.25\ \text{W m}^{-1}\text{K}^{-1}$), $150\ \text{mm}$ mineral wool ($k = 0.035\ \text{W m}^{-1}\text{K}^{-1}$) and $20\ \text{mm}$ of timber boarding ($k = 0.13\ \text{W m}^{-1}\text{K}^{-1}$). Ignore surface resistances. For a layered material, the U-value is $U=\dfrac{1}{\sum L/k}$.

1. a) Calculate the U-value of the roof.
2. b) Calculate the heat loss rate when the inside is at $20\,^\circ\text{C}$ and the outside at $-2\,^\circ\text{C}$.
3. c) Building regulations require a U-value of at most $0.16\ \text{W m}^{-2}\text{K}^{-1}$. Calculate the minimum thickness of mineral wool needed to meet this, keeping the other layers unchanged.
4. d) Explain why increasing the thickness gives diminishing returns in reducing heat loss.

A thermal imaging camera shows that the windows of a house in Letterkenny are much brighter (hotter) than the walls on a cold night.

1. a) Explain how the camera forms an image of the house, and why it can detect hot surfaces in the dark.
2. b) A window has U-value $2.8\ \text{W m}^{-2}\text{K}^{-1}$ and area $2.0\ \text{m}^2$, while a wall section of the same area has U-value $0.25\ \text{W m}^{-2}\text{K}^{-1}$. For a temperature difference of $20\ \text{K}$, calculate the heat loss through each and the ratio of losses.
3. c) Explain why the outer surface of the window is warmer than the outer surface of the wall.
4. d) A heat pump is proposed to heat the house. Explain, with reference to the first and second laws of thermodynamics, how it can deliver more heat energy than the electrical energy it uses without violating conservation of energy.

A heating system uses $5.0\ \text{kW}$ of power for $4.0$ hours on a cold winter day in Sligo.

1. a) Calculate the energy used in $\text{kWh}$.
2. b) Calculate the energy used in joules.
3. c) At a cost of €0.35 per $\text{kWh}$, calculate the cost of running the system for this time.

A house in Athlone has external walls of total area $120\ \text{m}^2$ with U-value $1.20\ \text{W m}^{-2}\text{K}^{-1}$. After retrofitting with external insulation the U-value falls to $0.20\ \text{W m}^{-2}\text{K}^{-1}$. The average temperature difference between inside and outside over the heating season is $10\ \text{K}$.

1. a) Calculate the rate of heat loss through the walls before and after retrofitting.
2. b) The heating season lasts $200$ days with the heating on continuously. Calculate the energy saved in $\text{kWh}$.
3. c) The heating system is oil-fired and delivers heat at a cost of €0.12 per $\text{kWh}$. Calculate the money saved over the season.

A school hall in Ennis has a floor area of $200\ \text{m}^2$. The building envelope (walls, roof, windows and floor) has a combined $UA$ value of $350\ \text{W K}^{-1}$. In addition, ventilation causes a heat loss of $1.2\times10^{2}\ \text{W K}^{-1}$. The hall is kept at $19\,^\circ\text{C}$ while the average outside temperature during a school term of $100$ days is $7\,^\circ\text{C}$. The hall is heated for $10$ hours on each of $100$ days.

1. a) Calculate the total heat loss coefficient and the heating power required.
2. b) Calculate the energy needed in $\text{kWh}$ over the term.
3. c) The school considers installing a heat pump with an average COP of $3.2$ to replace a gas boiler of efficiency $90\%$. Gas costs €0.09 per $\text{kWh}$ and electricity €0.34 per $\text{kWh}$. Calculate the cost of each option over the term.
4. c) Comment on the result and on one non-financial factor the school should consider.

A heat pump takes in $1.0\ \text{kW}$ of electrical power and delivers $3.5\ \text{kW}$ of heating power to a house in Tralee.

1. a) Calculate the coefficient of performance (COP) of the heat pump, defined as the heat delivered divided by the electrical energy input.
2. b) Calculate the power the heat pump extracts from the outside air.
3. c) Compare the running cost over $10$ hours with that of a resistive electric heater delivering $3.5\ \text{kW}$, if electricity costs €0.34 per $\text{kWh}$.