FYS200, Assignment 2
Diana Quintero Castro
19 October 2021
Consider the adiabatic wind tu
ine, shown in the figure below, within the
following assumptions:
Poo Pi pressures
0 : velocities*É 0*0*8 A : Areas -
gild
Poo
Ad Aw
No
Aoo
wakeful
Upstream lad
• The flow is steady and the fluid is incompressible.
• The flow is uniform through the disc and in the wake.
• The stream is horizontal.
• The actuator disc models the tu
ine blades and the disc extracts energy
from the flow.
• The actuator disc creates a pressure discontinuity across the disc (P+,
P−).
• The disc does not impart any swirl to the flow.
1. Write a short introduction explaining how wind tu
ines produce electric-
ity (include due references).
2. Write the continuity equation for the three relevant points highlighted in
the sketches (far stream: ∞, at the actuator disc: d and in the wake: w.)
3. Write the Bernoulli equation for the top and the bottom stream lines
(green lines in the sketch).
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4. Calculate the change in the momentum rate (∆ṗ = Ṁ∆v ) at the actuato
disc, consider the velocity at the disc as an average velocity with the form
vd =
1
2
(vinf + vw)
Rewrite the momentum rate as a function of the axial induction factor,
defined as:
a =
vinf − vd
vinf
5. The force causing this change in momentum rate comes from the pressure
difference across the actuator disc (P+, P−), as
Force = change in momentum rate = change in pressure× actuator area
Use the equations found above to rewrite the force equation.
6. Use the expression above to find a relation between the speed in the wake
and in the far stream.
7. Calculate the power output as
Power = Force× velocity at the disc
and plot the power as a function of a (you can use a plotting program fo
that eg. mathematica, matlab).
8. Calculate the power coefficient as
Cpow =
Powe
Power of wind in a free stream
where
Power of wind in a free stream =
1
2
ρv3infAd
Find for which value of a the power coefficient has a maximum. This
maximum is called the Betz limit.
9. Calculate the enthalpy change in this process for amax.
10. Write a short discussion of your results and conclusions, for example,
discuss in which settings the power output can be increased, e.g. down by
the sea, up in a mountain, etc.
In order to pass this assignment, you will need to answer co
ectly at least
50% of the questions above. Deadline: 9th November 2021.
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