EXAMPLE 3.1A: DRAG FORCE ON A TRUCK & CAR (ELEMENTARY)

A car and truck drive on the highway. We wonder: what is the drag force on both vehicles? To compare both, we assume that velocity of the truck and the car are the same: this means 20 m/s (72 km/hr).

For this analysis, the drag coefficient C_D and surface area A perpendicular to the driving direction are two important terms. Generally, the value of C_D can be found in test reports. Typical values for modern cars are C_D= 0.33 and for trucks are around 0.85. It must be noted that these values differ significantly if the model (geometry) varies.

We may assume that the the front surface area of a car or truck can be calculated by the following formula: A = α·H·W, where α is a constant dependent on the type of car/truck, H the height and W the width of a car/truck. For the car, you may use α = 0.8 and for the truck α = 1. Assume that the truck is 2×2 meters and the car is 1.8×1.6 meters.

Which factors affect the drag force F, and what is the drag force on the car and the truck we described above?

EXAMPLE 3.1B: DRAG ON A CAR (MEDIUM)

Modern cars have low Cdrag coefficients compared to the first ones. This is achieved by streamlining the shape of the car. Furthermore, compared to the good old T-Ford, the frontal area of modern cars is also smaller, at least for standard cars (for SUV, for example, this is not true).

Both the lower drag coefficeint and the smaller frontal area mean that cars, in princple, can have a much better mileage (M ) than older types, i.e. modern cars use much less gasoline per 100km than older types. Of course, improvements in engines has also helped significantly.

In this exercise, we will concentrate on the drag force and compare the energy use of a T-Ford and a modern car based on the drag. For the T-Ford, we have C_D=1 and A_⊥=28ft²; for a modern car these numbers are 0.3 and 7 ft², resp.

1. Show that the mileage of a car, if we ignore any other effects other than drag, is proportional to the drag force: M∝F_D
2. If a T-Ford and a modern car both drive at the same velocity, compute the ratio of their mileage.
3. Environmentalists make a pley for lower maximum velocities at high ways. They argue that driving at 90 km/h saves a lot more energy than driving at 120 km/h. Compute the ratio of the mileages for these two cases. Do you agree with the environmentalists?

EXAMPLE 3.1C: DIMENSION ANALYSIS (ADVANCED)

If you have processes where many variables are involved, it can be useful to apply dimension analysis. It reduces the variables in related dimensionless groups which are related to each other. In the field of transport phenomena, many dimensionless quantities are used. A few examples are the: Reynolds number (Re), Nusselt number (Nu) and Fourier (Fo) number.

Generally, problems with many variables are solved using matrix algebra. If there are not too many physical quantities involved, however, you can do it on paper. Remember that you use SI-units and work precisely. Otherwise, mistakes are made quickly.

As a result of condensation, droplets are formed on a horizontal plate. This can be your bath room, or on steel pipes. The phenomena is that these droplets grow to a certain critical volume (Vcrit) and then fall down. Use dimension analysis and derive a dimensionless group relevant for this process.