Question

An engine block with a surface area measured to be \(0.95 \mathrm{~m}^{2}\) generates a power output of \(50 \mathrm{~kW}\) with a net engine efficiency of \(35 \%\). The engine block operates inside a compartment at \(157^{\circ} \mathrm{C}\) and the average convection heat transfer coefficient is \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If convection is the only heat transfer mechanism occurring, determine the engine block surface temperature.

Short answer

Answer: The engine block surface temperature is approximately 3430.15 K.

Step-by-step solution

Calculate the rate of heat generation (Q_gen)

Use the given power output (P_out) and net engine efficiency (η) to determine the rate of heat generation. The formula is given by: Q_gen = P_out / η Where, P_out = 50 kW η = 35% First convert the power output (P_out) from kW to W: P_out = 50,000 W Now calculate the rate of heat generation (Q_gen): Q_gen = P_out / η = 50,000 W / 0.35 = 142,857.14 W

Apply the convection heat transfer equation

Given that convection is the only heat transfer mechanism, we can use the following equation to relate the heat transfer rate (Q_trans), surface area (A_s), convection heat transfer coefficient (h_c), and temperature difference (ΔT) between engine surface (T_s) and ambient compartment (T_c) temperatures: Q_trans = h_c * A_s * ΔT Where, h_c = 50 W/m²·K A_s = 0.95 m² ΔT = T_s - T_c T_c = 157°C We already know that the heat generated by the engine (Q_gen) is equal to the heat transferred by convection (Q_trans). Hence, we can write the equation as: Q_gen = h_c * A_s * (T_s - T_c)

Solve for engine surface temperature (T_s)

Now, let's solve the equation to determine T_s: 142,857.14 W = (50 W/m²·K) * (0.95 m²) * (T_s - 157°C) Divide both sides by the product of h_c and A_s: T_s - 157°C = 142,857.14 W / (50 W/m²·K * 0.95 m²) T_s - 157°C ≈ 3000 K Now, add T_c to both sides of the equation: T_s ≈ 3000 K + 157°C T_s ≈ 3000 K + 157 + 273.15 (Converting Celsius to Kelvin) Finally, calculate T_s: T_s ≈ 3430.15 K The engine block surface temperature is approximately 3430.15 K.

Key concepts

Heat Generation in Engines

Understanding how heat is generated in an engine is crucial for grasping the fundamentals of engine operation and what influences the temperature of engine components like the engine block. An engine converts the chemical energy stored in fuel into mechanical energy. During this conversion process, not all the energy from the fuel is converted into useful work. A significant portion is lost as heat due to inefficiencies in the engine process.

The efficiency of an engine, given in percentage, represents how well an engine converts the input energy into work. It's calculated by dividing the power output by the total energy input from the fuel. However, because the efficiency is never 100%, the rest, which is substantial, is dissipated as heat. This heat needs to be managed effectively since excessive heat can damage engine components, reduce performance, and increase emissions.

In the example given, by dividing the power output by the engine efficiency, we can calculate the rate of heat generation. This amount represents the heat energy per unit time that must be dealt with via various engine cooling methods to ensure the engine runs within an optimal temperature range.

Convection Heat Transfer

When it comes to cooling an engine, convection heat transfer is a key mechanism for moving heat away from the engine block to the surrounding environment. It occurs when a moving fluid, which can be a liquid or gas, removes heat from a solid object. The rate at which heat is transferred by convection is influenced by several factors, including the temperature difference between the object's surface and the fluid, the surface area of the object, and the convection heat transfer coefficient.

The convection heat transfer coefficient is a measure of the efficiency with which the fluid can absorb heat from the object's surface—it quantifies the ability of the fluid flow to remove heat and is often provided in units of Watts per square meter per Kelvin \( W/m^2\cdot K \)

Applying the Convection Heat Transfer Equation

To calculate how much heat is being carried away via convection, we use the convection heat transfer equation: \( Q_{trans} = h_c \times A_s \times \Delta T \), where \( Q_{trans} \) is the heat transfer rate, \( h_c \) is the convection heat transfer coefficient, \( A_s \) is the surface area, and \( \Delta T \) is the temperature difference.

By knowing these factors and realizing that the heat generated within the engine should equal the heat transferred by convection, we can establish an equality that allows us to solve for unknowns such as the surface temperature of the engine block.

Engine Efficiency

Engine efficiency represents the ratio of the useful work that an engine can produce to the total energy supplied from the fuel. It is a critical indicator not only for performance but also for determining how much energy is expended as heat. High efficiency implies that more of the fuel's energy is being converted into work, and less is wasted as heat. Conversely, an engine with lower efficiency will dissipate more heat, requiring effective cooling systems to handle the increased thermal output.

In thermal engines, efficiency is often limited by the engine design, material limitations, and the thermodynamic cycle it operates on. To enhance engine efficiency, various strategies can be implemented such as improving combustion, reducing frictional losses, and optimizing engine tuning. Increasing engine efficiency is beneficial as it results in better fuel economy, less heat generation, and consequently, a less stressed cooling system.

Efficiency in the Heat Transfer Context

As shown in the exercise provided, the engine's efficiency directly impacts the rate of heat generation. With a net engine efficiency of 35%, a significant amount of the energy from the fuel is not converted into work but instead becomes excess heat. This heat must be transferred away efficiently to prevent overheating which can be detrimental to both engine performance and longevity.