91Ó°ÊÓ

Develop an equation for the power required, in hp, for a vehicle traveling on a level road to overcome (a) the aerodynamic drag force of the surrounding air on the vehicle and (b) the rolling resistance force on the tires. The drag force, \(F_{d}\), imposed on a vehicle by the surrounding air is given by $$ F_{d}=C_{d} A \frac{1}{2} \rho V^{2} $$ where \(C_{d}\) is a constant for the vehicle called the drag coefficient, \(A\) is the projected frontal area of the vehicle, \(\rho\) is the air density, and \(V\) is the velocity of the vehicle. The rolling resistance of the tires, \(F_{r}\), opposing the motion of a vehicle is given by $$ F_{r}=f W $$ where \(f\) is a constant called the rolling resistance coefficient and \(W\) is the vehicle weight. What is the equation for the power required in \(\mathrm{kW}\) ?

Step by step solution

01

Determine power to overcome aerodynamic drag force

First recognize that Power \( (P) \) is equal to Force \( (F) \) times Velocity \( (V) \). Therefore the power to overcome the aerodynamic drag can be calculated by substituting the equation for drag force (\( F_{d}=C_{d} A \frac{1}{2} \rho V^{2} \)) into the equation for power. This results in the following equation: \( P1=C_{d} A \frac{1}{2} \rho V^{3} \). This is the power required to overcome aerodynamic drag in Watts.

02

Determine power to overcome rolling resistance

For the rolling resistance, use the same method. By substitifying the equation for rolling resistance (\( F_{r}=f W \)) into the power equation we get the equation for the power required to overcome rolling resistance: \( P2=f W V \). This is the power required in Watts.

03

Convert the power from Watts to kilowatts and horsepower

In order to convert the power to kilowatts (\( kW \)), divide the power calculated in Watts by 1000. For the power \(\ P1 \), we have \( P1 (kW)= \frac{C_{d} A \frac {1}{2} \rho V^{3}}{1000}= 0.001 C_{d} A \rho V^{3} \). And for the power \(\ P2 \), we have \( P2 (kW)= \frac{f W V}{1000}= 0.001 f W V \). Since the question also requires the answer in horsepower (\( hp \)), we need to convert kW into hp by dividing the kW power by 0.7457. Therefore, the powers \( P1 \) and \( P2 \) in hp are given by \( P1 (hp)= \frac{0.001 C_{d} A \rho V^{3}}{0.7457} \) and \( P2 (hp)= \frac{0.001 f W V}{0.7457} \) respectively.

04

Add both powers to get total power

Adding both powers to each other provides the needed total power for both overcoming the aerodynamic and the rolling resistance. The final equations for the total power are: \( P (kW)= P1 (kW) + P2 (kW)= 0.001 C_{d} A \rho V^{3} + 0.001 f W V \) or in hp: \( P (hp)= P1 (hp) + P2 (hp)= \frac{0.001 C_{d} A \rho V^{3}}{0.7457} + \frac{0.001 f W V}{0.7457} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Aerodynamic Drag Force

When a vehicle moves through the air, it encounters a force called aerodynamic drag. This force opposes the motion of the vehicle and increases with speed. The aerodynamic drag force, denoted as \( F_{d} \), depends on factors such as the shape of the vehicle, the density of the air, and how fast the vehicle is traveling.
The mathematical expression for the aerodynamic drag force is:

• \( F_{d}=C_{d} A \frac{1}{2} \rho V^{2} \)

Where:

• \( C_{d} \) is the drag coefficient, reflecting how smooth or streamlined the vehicle is.
• \( A \) is the frontal area facing the oncoming air.
• \( \rho \) is the air density, a measure of mass per unit volume.
• \( V \) is the velocity of the vehicle.

As the vehicle speeds up, the aerodynamic drag force rises, making it key to designing efficient vehicles that reduce this drag.

Rolling Resistance Force

Rolling resistance is another force that opposes the movement of a vehicle. It occurs due to the deformation of the tires and road as they come into contact. While this force is smaller than aerodynamic drag at high speeds, it becomes significant at lower speeds or when the vehicle is accelerating.
The rolling resistance force \( F_{r} \) can be calculated using:

• \( F_{r}=f W \)

Where:

• \( f \) is the rolling resistance coefficient, a constant that depends on the tire and road surface materials.
• \( W \) is the weight of the vehicle.

This formula highlights the importance of tire selection and maintenance, as minimizing rolling resistance can greatly enhance fuel efficiency and reduce the power required to drive the vehicle.

Drag Coefficient

The drag coefficient, \( C_{d} \), is a critical factor in aerodynamic calculations. It represents how aerodynamic a vehicle is. A lower \( C_{d} \) indicates a more aerodynamic, or streamlined shape, allowing the vehicle to move more easily through the air.
Different vehicles have different drag coefficients, often ranging from:

• 0.25 for highly aerodynamic sports cars,
• to about 0.35 for typical sedans,
• and even higher for larger vehicles like SUVs.

Improving the drag coefficient involves design changes such as smoothing surfaces, optimizing shapes, and reducing sharp angles, all of which contribute to reducing energy consumption and enhancing performance.

Rolling Resistance Coefficient

The rolling resistance coefficient, \( f \), is a measure of the frictional force between a vehicle's tires and the road surface. It plays a crucial role in how much power is needed to move a vehicle. Lowering the rolling resistance coefficient means that less power is necessary to keep a vehicle moving.

• \( f \) is typically affected by factors such as tire material, tread pattern, and road surface texture.

To achieve lower rolling resistance, manufacturers often develop special tires designed to minimize deformation and thus reduce friction. These advancements can lead to substantial energy savings, especially over long distances, making them an important consideration for fuel-efficient vehicle design.