Question

Consider an electric motor with a shaft power output of \(20 \mathrm{kW}\) and an efficiency of 88 percent. Determine the rate at which the motor dissipates heat to the room it is in when the motor operates at full load. In winter, this room is normally heated by a \(2-\mathrm{kW}\) resistance heater. Determine if it is necessary to turn the heater on when the motor runs at full load.

Short answer

Answer: The electric motor dissipates heat to the room at a rate of 2.73 kW when operating at full load. Since this is greater than the power of the resistance heater (2 kW), it is not necessary to turn the heater on when the motor is running at full load, as the motor is already providing enough heat for the room.

Step-by-step solution

Find the electric power input to the motor

To find the electric power input to the motor, we can use its shaft power output and efficiency. We know that efficiency is the ratio of output power to input power: \(Efficiency = \frac{Power_{output}}{Power_{input}}\) In this case, the efficiency is 88%, which is 0.88 in decimal form. We also know the shaft power output, which is 20 kW. Therefore, we can rearrange the equation to solve for the input power: \(Power_{input} = \frac{Power_{output}}{Efficiency} = \frac{20 \mathrm{kW}}{0.88} = 22.73 \mathrm{kW}\)

Calculate the heat dissipated by the motor

The motor dissipates heat to the room due to its inefficiencies. We can calculate the heat dissipated by multiplying the input power of the motor by (1 - efficiency): \(Heat_{dissipated} = Power_{input} \times (1 - Efficiency) = 22.73 \mathrm{kW} \times (1 - 0.88) = 2.73 \mathrm{kW}\)

Compare the heat dissipated by the motor to the resistance heater's power

Now we should compare the heat dissipated by the motor to the power of the resistance heater in the room: \(Heat_{dissipated} = 2.73 \mathrm{kW}\) \(Resistance\, heater\, power = 2\mathrm{kW}\) Since the heat dissipated by the motor is greater than the power of the resistance heater (2.73 kW > 2 kW), it is not necessary to turn the heater on when the motor is operating at full load. The motor is already providing enough heat for the room.

Key concepts

Electric Motor Efficiency

When describing electric motor efficiency, we refer to the ratio of mechanical power output to electrical power input. In other words, it tells us how well a motor converts electrical energy, supplied from a power source, into mechanical energy, which does useful work like turning a fan or driving a pump.

Motor efficiency is key in determining the overall cost-effectiveness of the motor over its lifecycle. A higher efficiency means less energy is wasted, typically in the form of heat, which can result in lower operating costs. For instance, the motor from our exercise has an efficiency of 88%, indicated by the calculation \( Efficiency = \frac{Power_{output}}{Power_{input}} \). This high efficiency is indicative of a well-designed motor that wastes minimal energy.

Heat Dissipation

Heat dissipation in electric motors occurs due to inefficiencies in energy conversion. These inefficiencies are inherent to the motor's operation and can be caused by various factors such as friction, electrical resistance, and stray losses. Heat must be managed effectively to avoid overheating and prolong the motor's lifespan.

In the case of our exercise, the heat dissipated by the motor when it's operating at full load is calculated to be 2.73 kW. This is significant when considering environmental temperature control. For instance, in a heated room, the waste heat from a motor can influence whether additional heating is required. Proper understanding and management of heat dissipation can lead to more energy-efficient and cost-effective solutions, like the scenario outlined where the motor's heat output negates the need for an extra heater.

Energy Conversion

Energy conversion within an electric motor is the process of changing electrical energy into mechanical energy. This process is never 100% efficient due to energy losses primarily from heat. The efficiency of energy conversion is a function of the motor design, materials, and operating conditions.

In practical applications, such as the one presented in the exercise, understanding energy conversion can help in making decisions on equipment usage and energy management. The balance of energy input, useful mechanical output, and wasted heat is essential for optimizing the operation of electrical devices and systems. In our example, energy conversion efficiency plays a central role in understanding whether an electric motor can solely handle the dual role of performing work while also warming the environment.