Question

A water-cooled engine produces \(1000 .\) W of power. Water enters the engine block at \(15.0^{\circ} \mathrm{C}\) and exits at \(30.0^{\circ} \mathrm{C}\). The rate of water flow is \(100 . \mathrm{L} / \mathrm{h}\). What is the engine's efficiency?

Short answer

Answer: The efficiency of the water-cooled engine is approximately 57.98%.

Step-by-step solution

Calculate the heat transferred by the water

We need to find the energy transferred by the water when it goes through the engine block. To do this, we need to calculate the specific heat capacity of water and the mass flow rate of the water. First, let's convert the flow rate of water from L/h to kg/s. We know that: 1 L = 0.001 m³ and the Density of water = 1000 kg/m³ Flow rate (in mass) = (100 L/h) * (0.001 m³/L) * (1000 kg/m³) * (1 h/3600 s) = 0.0278 kg/s Now, we can calculate the heat transferred by the water using the specific heat capacity of water and the mass flow rate: q = (mass flow rate) * (specific heat capacity) * (ΔT) The specific heat capacity of water (c) is 4190 J/(kg·°C), and the temperature difference (ΔT) is (T2 - T1) = (30.0 - 15.0) °C.

Calculate the heat transferred by the water

Using the provided data and the calculation from step 1, the heat transferred by the water (q) is: q = (0.0278 kg/s) * (4190 J/(kg·°C)) * (30.0 - 15.0) °C q = 1724.41 W This is the amount of heat transferred by the water per second. Now, we'll find the efficiency of the engine.

Calculate the efficiency of the engine

Efficiency (η) is the ratio of the useful work output (W) to the heat input (Q), so we can write: η = (W / Q) * 100% The power output (W) is given: 1000 W. The heat input (Q) is calculated in step 2 as approximately 1724.41 W. Now, finding the efficiency (η): η = (1000 W / 1724.41 W) * 100% η ≈ 57.98% The efficiency of the water-cooled engine is approximately 57.98%.

Key concepts

Specific Heat Capacity

Understanding the specific heat capacity is crucial when calculating engine efficiency, as it's a measure of the amount of heat energy required to raise the temperature of one kilogram of a substance by one degree Celsius. For water, which is the coolant in our exercise, the specific heat capacity is relatively high at 4190 J/(kg°C). This means water can absorb a lot of heat before it gets hot, making it an excellent medium for cooling engines. When calculating the heat transferred by water in our problem, we multiply the mass flow rate of water by its specific heat capacity and the temperature change. This allows us to determine how much energy is absorbed by the water as it flows through the engine, which is a key step in calculating engine efficiency.

Heat Transfer

Heat transfer is a foundational concept in thermodynamics, involving the movement of thermal energy from one place to another. It can occur in three ways: conduction, convection, and radiation. In the context of our exercise, we're concerned with convective heat transfer, where the engine heat is transferred to the water coolant that circulates through the engine block. This process is quantified using the formula for heat transfer \( q = (\text{mass flow rate}) \times (\text{specific heat capacity}) \times (\Delta T) \). By calculating the amount of heat transferred to the water, we can infer the amount of energy not used for work, which is essential for determining the engine's efficiency.

Mass Flow Rate

The mass flow rate is a term describing the mass of a substance passing through a given surface per unit time. It's expressed typically in kilograms per second (kg/s) and is a vital variable in the calculation of heat transfer in thermodynamic systems, such as engines. In our exercise, converting the volumetric flow rate of water (given in liters per hour) to mass flow rate (in kg/s) requires understanding water's density. Knowing that 1 liter of water has a mass of approximately 1 kilogram, we use the density of water and the conversion from liters to cubic meters to determine how much water, by mass, is flowing through the engine every second. This value is essential to measure how much energy the water absorbs as it heats up while passing through the engine.

Thermodynamics

Thermodynamics is the branch of physics concerned with heat, work, temperature, and energy. The principles of thermodynamics guide the operation of heat engines and the study of energy conversion, which is at the heart of our exercise. When considering engine efficiency, we are dealing with the first law of thermodynamics, which states that energy cannot be created or destroyed, only transformed. In this scenario, the total energy input to the engine is partly converted to work (useful energy) and partly absorbed as heat by the water coolant. By comparing the work output to the total energy input, we can calculate the efficiency of the engine. An engine's efficiency, in simple terms, indicates how well it converts heat into work, with the remaining energy typically lost as waste heat.