You are in charge in designing a blimp that will move at 15 mph through the air To decide
on what engines to use, u=you need to predict the drag force on the full scale blimp.
To do so, you place a scale model that is 9 times smaller then the full blimp in a moving
water in the laboratory.
You measure the force of drag in that blimp.
First off we need to determine the water velocity needed to insure the flow characteristics
seen between the model blimp and the full scale blimp.
If you do the dimensionless analysis on this problem.
You would find that there are only 2 pi terms.
The Re number and the coefficient of drag.
So in this case if the Re for the model is equal to the Re of the prototype then the
coefficient of drag for the model would equal the coefficient of drag for the prototype.
So if the Re number are the same even though this full scale blimp is in air.
In this model blimp is it 9 times smaller and it is flowing through water.
If we get the Re numbers to be the same.
The flow characteristics would be around the same.
So lets expand this out.
Here there is a density, velocity, and length scale for the model, and a velocity.
So in this case it would be the density of water, the viscosity of water, and the velocity
of water that we are interested in.
Here the Re number for the prototype, the density of air, the viscosity of air, the
length of the full size blimp, and the velocity of which it is flying, which is 15 mph.
So lets rearrange this equation to figure out the velocity of the water we need to flow
the water to flow through.
Rearranging it we find that the velocity of the prototype times the ratio of viscosity,
the ratios of the densities, and the ratios of the length scales, and if I plug in numbers
the full size blimp will travel at 15 mph.
Here is the viscosity of water.
So the viscosity of the model case, and the viscosity of the air is this value.
Here is the density of air, and the density of water.
Here is the length scale.
The length of the prototypes divided by the length of the model.
I come up with a value of 6.75 mph.
This is the speed we need to have water moving over the model blimp to insure flow characteristics
that are similar between the model and the full size.
That will insure that the coefficient of drag for model is the same as the coefficient of
drag for the prototype.
So you ran the experiment, and we have measured a value of 700 lb, is the drag force on the
model.
In using that lets estimate the drag on the prototype, and the power of the engines needed
to move the blimp through the air.
To do that we will use the coefficient of drag and here is how the coefficient of drag
relates to the drag force.
So if we maintain the same Re number the coefficient of drag for the model is equal to the coefficient
drag for the prototype.
When I am going to do, and I will solve the coefficient drag for both the model and the
prototype and I will equate those 2.
This is the rearrange form of it.
Here on the left is the force of drag on the model, which we will measure in the laboratory,
and similarly for the prototype the important thing to recognize because we have maintained
similitude the Re numbers are the same.
That means the the coefficient of drag of both the model and the prototype are the same,
so they cancel out.
Lets calculate the force of drag on the prototype by rearranging.
So we just need the ratios of the densities, ratios of the two velocities, which we now
know, and the ratio of the length scale times the force of drag acting on the model.
So I will plug numbers into this.
So the actual drag on the prototype is 280 lb, and it may seem strange because we are
talking about something that is 9 times bigger, and the drag force we just found is you know
about half, less then half of that of the prototype.
We can also consider that the blimp is flowing through air instead of water.
Inviscid fluid the density is 1000 times less.
So in reality even though it is 9 times bigger then drag force is actually less, but on a
laboratory scale it would be easier to handle something that is 9 times smaller and it would
be easier to make more sophisticated measurements in the laboratory.
Finally the power need is the drag force on the prototype times its velocity.
So the drag force we estimate is 280 lb, and its speed is 15 mph.
If we used some conversion factors we have mph crossed out, we have ft*lb/s, which is
dimensions of power.
Here is the conversion we can calculate it in hp.
What you calculate the power needed by the blimps engines is 11 hp.