Homework 7

Problem 1

Compute and plot the adiabatic variation of T, P, and $\rho$ with height $z$ in the atmosphere. Assume 1 atm, 15 $^o$ C air at sea level. Plot up to 10,000 m. The pressure-statics relation is

$$\frac{dP}{dz} = -\rho g.$$

The ideal gas law gives

$$\rho = \frac{MP}{RT}.$$

Also for adiabatic compression/expansion we have

$$T = \frac{T_0}{P_0^\alpha}P^\alpha,$$

where $\alpha = (\gamma-1)/\gamma$, and $\gamma=c_p/c_v=1.4$ for air. Inserting these expressions into the pressure-statics relation gives an ODE in pressure that can be integrated to give P(z). From this, $T(z)$, and $\rho(z)$ can be computed.

On each of the three graphs, also include plots of the US Standard Atmosphere for comparison.

Problem 2

If the wind is 1 m/s at 1 m above the ground, with some corresponding turbine power at that height, by what factors will the turbine power increase if located at 10, 20, 40, and 80 m above the ground?Assume constant air properties (temperature, pressure).

Problem 3

Problem 2 assumed constant air properties with height, but Problem 1 shows that air properties change with height. How much does the density change with the max height of 80 m considered in Problem 2? Does this matter to the answers in problem 2? What about turbines at the height of Denver?

Problem 4

Which US state produces the most wind power?
Which country produces the most wind power?
What is a typical wind power capacity factor?
What is a typical size in MW for a large power producing turbine?

Problem 5

Wind blows through a HAWT with a turbine diameter of 25 m. The combined efficiency of the gearbox and generator is 88%. Assume a turbine blade speed/wind speed ratio of 6. Compute the electrical power generated if the wind speed is 10 m/s.