Question

The tip speed ratio of a turbine is the ratio R = T/W, where T is the speed of the tip of a blade and W is the speed of the wind. (Engineers have found empirically that a turbine with n blades extracts maximum power from the wind when R = 2π/n.) Calculate dR/dt (t in minutes) if W = 75 km/h and W decreases at a rate of 4 km/h per minute, and the tip speed has constant value T = 150 km/h.

Answer 1

Answer:

$\displaystyle\frac{dR}{dt} \approx 0.107$      

Step-by-step explanation:

We are given the following information in the question:

$R = \displaystyle\frac{T}{W}$

where, R is the tip speed, T is the speed tip of blade and W is the speed of wind.

We have to find $\frac{dR}{dt}$, whem W = 75 km per hour, $\frac{dW}{dt} = -4\text{ km per hour}$, T = 150 km per hour

Formula:

$\displaystyle\frac{dR}{dt} = \displaystyle\frac{dR}{dW}\displaystyle\frac{dW}{dt} + \displaystyle\frac{dR}{dT}\displaystyle\frac{dT}{dt}\\\\= \displaystyle\frac{dR}{dW}\displaystyle\frac{dW}{dt}\\\\\text{ as T is constant with respect to t}\\\\= -\displaystyle\frac{T}{W^2}\displaystyle\frac{dw}{dT}$

Putting all the value, we have:

$\displaystyle\frac{dR}{dt} = -\displaystyle\frac{150\times -4}{75\times 75} = 0.106666666667 \approx 0.107$