Question

To maintain the passenger compartment of an automobile traveling at \(13.4 \mathrm{~m} / \mathrm{s}\) at \(21^{\circ} \mathrm{C}\) when the surrounding air temperature is \(32^{\circ} \mathrm{C}\), the vehicle's air conditioner removes \(5.275 \mathrm{~kW}\), by heat transfer. Estimate the amount of engine horsepower required to drive the air conditioner. Referring to typical manufacturer's data, compare your estimate with the actual horsepower requirement. Discuss the relationship between the initial unit cost of an automobile air-conditioning system and its operating cost.

Short answer

The estimated engine horsepower required is 7.07 hp. Manufacturer data typically indicates a range of 5-10 hp, making the estimate reasonable. Higher initial costs generally correlate with lower operating costs.

Step-by-step solution

- Understand the problem

We need to estimate the engine horsepower required to drive the air conditioner, which removes 5.275 kW of heat. One horsepower is equivalent to 746 watts.

- Convert kW to horsepower

First, convert the power removed by the air conditioner from kilowatts (kW) to horsepower (hp) using the conversion factor: \ \[ 1 \text{ kW} = 1000 \text{ Watts (W)} \] \ \[ 1 \text{ hp} = 746 \text{ W} \] \ \[ 5.275 \text{ kW} = 5.275 \times 1000 \text{ W} = 5275 \text{ W} \] \ \[ \text{ Power required (hp)} = \frac{5275 \text{ W}}{746 \text{ W per hp}} \approx 7.07 \text{ hp} \]

- Estimate actual horsepower from manufacturer's data

Referring to manufacturer data, the actual horsepower requirement for a vehicle’s air conditioning system typically ranges between 5-10 hp. The estimated 7.07 hp fits within this range. Therefore, the estimate seems reasonable.

- Discuss initial unit costs vs. operating costs

The initial unit cost of an automobile air-conditioning system reflects its design, efficiency, and components. Systems with higher efficiency often have higher initial costs but lower operating costs because they require less horsepower to remove the same amount of heat, conserving fuel and reducing long-term expenses.

Key concepts

Power Conversion

Understanding power conversion helps you determine how much energy is needed for different tasks. In our exercise, we need to estimate the engine horsepower required to drive the air conditioner, which removes 5.275 kW of heat. Kilowatts (kW) are a unit of power that tells us how much energy is transferred per second. To convert kilowatts into horsepower (hp), we use the fact that 1 horsepower is equivalent to 746 watts.

• 1 kW = 1000 Watts (W)
• 1 hp = 746 W

For this exercise:
5.275 kW = 5.275 × 1000 W = 5275 W
Power required (hp) = 5275 W / 746 W per hp ≈ 7.07 hp.

Heat Transfer

Heat transfer is crucial in understanding how an air conditioning system works. When the air conditioner removes heat from the passenger compartment, it ensures a comfortable temperature inside your vehicle. In our case, the air conditioner removes 5.275 kW of heat to keep the interior at a pleasant 21°C, even though the outside is 32°C. This difference is achieved through three main methods of heat transfer: conduction, convection, and radiation.

• Conduction: Direct transfer of heat through physical contact.
• Convection: Transfer of heat through fluid (air) movement.
• Radiation: Transfer of heat through electromagnetic waves.

Understanding these methods helps explain how the air conditioner efficiently maintains a cooler environment within the car.

Engine Horsepower Estimation

Accurately estimating engine horsepower is essential for assessing an air conditioner's impact on a vehicle's performance. From our calculations, we found that the air conditioner requires approximately 7.07 hp. Referring to typical manufacturer's data, the horsepower required for vehicle air conditioning systems typically ranges between 5-10 hp. Our estimate of 7.07 hp fits well within this range, affirming the calculation's accuracy. Knowing the required horsepower helps you understand how much additional load the air conditioning system places on the engine and how it affects overall vehicle performance.

Energy Efficiency

Energy efficiency measures how well a system converts energy into useful work. For automobile air conditioning, higher efficiency means less fuel is used to remove the same amount of heat, reducing operating costs. Systems with greater efficiency often have higher initial costs due to advanced design and components but prove cost-effective over time. Here's what to consider for energy efficiency:

• Initial unit cost: Higher for efficient systems but reduces long-term costs.
• Fuel conservation: Efficient air conditioning uses less fuel, reducing expenses.
• Long-term savings: Lower operating costs balance out the high initial investment.

Energy efficiency is a critical factor influencing your choice of an air conditioning system, especially considering long-term operational costs and environmental impact.