Chapter 7, Problem 7.26P

The conversion of the kinetic energy of wind to electricity may be an attractive alternative to the use of fossil fuels. Typically, wind causes the rotor of a turbine to turn, and a generator converts the rotational kinetic energy of the rotor into electricity. Power generated by a wind turbine $\left({\stackrel{\dot{}}{W}}_s\right)$can be estimated from the density of the air $\left(\rho \right)$, wind speed (u), and turbine rotor diameter (D) using the following formula:

   ${\stackrel{\dot{}}{W}}_s=\frac{1}{2}\stackrel{\dot{}}{m}{u}^2\times \eta$where $\stackrel{\dot{}}{m}=\rho Au=\rho \frac{\pi D^2}{4}u$

The conversion ef?ciency $\left(\eta \right)$is a function of many variables, including the electrical and mechanical properties of the turbine, the material of construction, and the blade design.

(a) Develop an equation for the density of air (kg/m³) as a function of the temperature (K) and relative humidity of the air. Use the Antoine equation for the vapor pressure of water, and assume atmospheric pressure equals 1.0 atm.

(b) A wind turbine with a 30.0-ft diameter and 35.0% conversion efficiency generates electricity on a day when the temperature is 75°F, the relative humidity is 78.0%, and the average wind velocity is 9.50 miles/h. Calculate the generated power in kW.

(c) Seasonal variations can cause signi?cant changes in the power obtained from a wind turbine. Your task is to calculate and analyze these variations over a year for three cities in the United States using historical averages recorded by the National Oceanic and Atmospheric Administration (NOAA). The table below and on the next page gives monthly average relative humidities, mean temperatures, and wind speeds at three different cities, one each from the south, northeast, and western regions of the country. Reproduce the table on a spreadsheet, assume a wind turbine diameter of 30.0 feet and conversion efficiency of 35.0%, and estimate the power generated (kW) for each city and month. (The calculated value for one month is given so you can check your calculations.)

Diameter (ft)	30.0
Ef?ciency	35.0%
City (Pop)		JAN	FEB	MAR	APR	MAY	JUN	JUL	AUG	SEP	OCT	NOV	DEC	Yr. Mean
Huntsville AL (179653)	hr(%)	80	79	78	81	85	87	89	89	88	86	82	81
T(°F)	39.8	44.3	52.3	60.4	68.6	76	79.5	78.6	72.4	61.3	51.2	43.1
u(mph)	9	9.4	9.8	9.2	7.9	6.9	6	5.8	6.7	7.3	8.1	9
T(K)
$\rho \left(kg/m^3\right)$
u(m/s)
${\stackrel{\dot{}}{W}}_s\left(\text{kW}\right)$	1.004
Bridgeport, CT (137912)	hr(%)	69	69	69	68	74	77	77	78	80	78	76	73
T(°F)	29.9	31.9	39.5	48.9	59	68	74	73.1	65.7	54.7	45.1	35.1
u(mph)	12.5	12.9	13	12.4	11.1	9.9	9.4	9.5	10.5	11.3	12	12.1
T(K)
$\rho \left(kg/m^3\right)$
u(m/s)
${\stackrel{\dot{}}{W}}_s\left(\text{kW}\right)$
Sacramento, CA (1394154)	hr(%)	90	88	85	82	82	78	77	78	77	79	87	88
T(°F)	51.2	54.5	58.9	65.5	71.5	75.4	74.8	71.7	64.4	53.3	45.8	51.2
u(mph)	7.3	8.4	8.6	9	9.6	8.9	8.4	7.4	6.4	6	6.4	7.3
T(K)
$\rho \left(kg/m^3\right)$
u(m/s)
${\stackrel{\dot{}}{W}}_s\left(\text{kW}\right)$

(d) Plot the power variation over the course of a year for all three cities. How do the cities compare as locations for wind turbines‘?

(e) The averageelectricity consumption in the United States is approximately 12,000 kWh per capita per year. On a wind—turbine farm, a single turbine occupies a space of 1000 m². Estimate the number of turbines that would be required to meet the electricity needs of each of the three cities listed in the table if the turbines were operated continuously. Then estimate how many acres and hectares each farm would occupy.

(f) The numbers of turbines actually put in place to meet the power requirements of the three cities would all be greater than the numbers calculated in Part (e). List three reasons for the calculated quantities to be underestimates.