Question

1. Compare the power output of a wind turbine when the temperature is 45 °C and when the temperature is 0 °C (using the same wind speed distribution). At the higher temperature, the power output will be:

Answer 1

Given data:

Temperature, $T_{1}=45^{\circ}C$ = 318 K

Temperature, $T_{2}=0^{\circ}C$ = 273 K

Solution:

The formula for the power output of windturbine is

$P = \frac{1}{2}\rho AV^{2}$

Where,

$\rho$ = density of the air

A = rotor swept area

V = wind speed.

Keeping everything constant except air density,

Air density varies with temperature

The relation between air density, pressure enad temperature is,

p=pRT

where,

P = air pressure

$\rho$ is air density

R = universal gas constant of dry air

T = Air temperature in K

$\rho=\frac{p}{RT}$

The formula of power output of wind turbine becomes

$P = \frac{1}{2} AV^{2}\left ( \frac{p}{RT} \right )$

The output power at $T_{1}=273 \text{ K}$ is

$P_{1} = \frac{1}{2} AV^{2}\left ( \frac{p}{R*273} \right )$

The output power at $T_{2}=318 \text{ K}$ is

$P_{2} = \frac{1}{2} AV^{2}\left ( \frac{p}{R*318} \right )$

Therefore the ration of P₁ and P₂ is

$\\\frac{P_{1}}{P_{2}} =\frac{ \frac{1}{2} AV^{2}\left ( \frac{p}{R*273} \right )}{\frac{1}{2} AV^{2}\left ( \frac{p}{R*318} \right )}\\ \\ \\ \frac{P_{1}}{P_{2}} =\frac{318}{273}\\ \\ P_{2} =0.86P_{1}\\ \\$

Therefore, at higher temperature $45^{\circ}C$ , the power output will be 86% of power at $0^{\circ}C$