Question

A building for which the heat transfer rate through the walls and roof is \(400 \mathrm{~W}\) per degree temperature difference between the inside and outside is to be maintained at \(20^{\circ} \mathrm{C}\). For a day when the outside temperature is \(4^{\circ} \mathrm{C}\), determine the power required at steady state, \(\mathrm{kW}\), to heat the building using electrical resistance elements and compare with the minimum theoretical power that would be required by a heat pump. Repeat the comparison using typical manufacturer's data for the heat pump coefficient of performance.

Short answer

Power required using electrical resistance: 6.4 kW. Minimum theoretical power required by heat pump: 0.3491 kW. Power required by heat pump (manufacturer's COP): 2.133 kW.

Step-by-step solution

Determine the temperature difference

Find the difference between the inside temperature (20℃) and the outside temperature (4℃). Temperature difference, 𝛥T: 20℃ - 4℃ = 16℃.

Calculate the power required using electrical resistance elements

Given that the heat transfer rate per degree temperature difference is 400 W/℃,Power required, P = 400 W/℃ * 16℃P = 6400 WConvert to kW: 6400 W = 6.4 kW.

Determine the minimum theoretical power required by a heat pump

To find the minimum power required by a heat pump, use the equation: COP (Carnot) = T_hot / (T_hot - T_cold)where T_hot = 20+273.15 = 293.15 K, and T_cold = 4+273.15 = 277.15 K.COP (Carnot) = 293.15 / (293.15-277.15) = 293.15 / 16 = 18.32.Minimum power required by heat pump = 6400 W / 18.32 =349.1 WConvert to kW: 349.1 W = 0.3491 kW.

Calculate the power required by a typical heat pump manufacturer’s value of COP

Assume COP (manufacturer) = 3.0Power required by heat pump = Power required by electrical resistance / COP(manufacturer)= 6400 W / 3.0 = 2133.33 WConvert to kW: 2133.33 W = 2.133 kW.

Key concepts

heat transfer rate

Heat transfer rate is a measure of the amount of heat energy transferred per unit of time. In the context of buildings, it represents how quickly heat passes through the walls and roof. This rate is typically measured in watts per degree Celsius \(W/°C\). For example, if a building has a heat transfer rate of 400 W/°C, it means that for every degree difference between the inside and outside temperature, 400 watts of heat energy will either enter or leave the building. This concept is crucial for understanding energy efficiency and energy requirements for heating or cooling structures.

temperature difference

The temperature difference is the driving factor for heat transfer. It is calculated by taking the difference between the internal and external temperatures. In our exercise, the building is maintained at an inside temperature of 20°C, while the outside temperature is 4°C. So, the temperature difference \( \Delta T \) = 20°C - 4°C = 16°C. The larger the temperature difference, the higher the rate of heat transfer. This is important for determining the heating or cooling load required to maintain a comfortable indoor environment.

coefficient of performance (cop)

The Coefficient of Performance (COP) is a measure of a heating system's efficiency. It is the ratio of heat delivered to the amount of work input. For electrical resistance heating, the COP is typically 1. However, for heat pumps, COP varies and can be much higher. The theoretical maximum COP for a heat pump, also known as the Carnot COP, can be calculated using the formula: \(COP_{Carnot} = \frac{T_{hot}}{T_{hot} - T_{cold}}\) where \(T_{hot}\) and \(T_{cold}\) are the absolute temperatures of the hot and cold reservoirs in Kelvin. For instance, with \(T_{hot} = 293.15K\) and \(T_{cold} = 277.15K\), \(COP_{Carnot} = \frac{293.15}{(293.15 - 277.15)} \approx 18.32\). This means the heat pump theoretically requires significantly less power to achieve the same heating compared to electrical resistance. Actual heat pumps have a lower COP due to real-world inefficiencies, but they still perform better than resistance heaters. A typical manufacturer COP value might be around 3.0, showing the practical efficiency available in standard heat pumps.