From the information given;
- the height of the water stream = 50 m
- the efficiency of conversion from potential energy to electrical energy is 91%
- loss of power transmission = 8%
To determine the mass flow rate, let's start by understanding some concepts and parameters.
The power is known to be the energy per unit of time. Mathematically, it can be written as:
[tex]\mathbf{Power = \dfrac{Energy}{Time}}[/tex]
[tex]\mathbf{P =\dfrac{E_p}{time}}[/tex]
[tex]\mathbf{P =\dfrac{m\times g\times z}{time}}[/tex]
where;
- [tex]\mathbf{E_p}[/tex] is the potential energy of the stream
- m = mass flow rate
- g = acceleration under gravity
- z = height
Thus;
[tex]\mathbf{E_p}[/tex] = m × 9.81 m/s² × 50 m
[tex]\mathbf{E_p}[/tex] = m × 490.5 (m²/s²)
Recall that:
- The power P = 200 W, and;
- the conversion of the P.E = 91% = 0.91
∴
[tex]\mathbf{E_p}[/tex] = 0.91 × 490.5m (m²/s²)
[tex]\mathbf{E_p}[/tex] = 446.355m (m²/s²)
Since the resulting power transmission is said to be 8%
Then;
the loss in the power transmission (P) = 100% - 8% × 446.355m (m²/s²)
the loss in the power transmission (P) = 92% × 446.355m (m²/s²)
the loss in the power transmission (P) = 0.92 × 446.355m (m²/s²)
the loss in the power transmission (P) = 410.65m (m²/s²)
Finally;
P = 410.65m (m²/s²)
[tex]\mathbf{P = 410.65 \times m (\dfrac{m^2}{s^2})}[/tex]
replacing the values, we have:
[tex]\mathbf{200 = 410.65 \times m (\dfrac{m^2}{s^2})}[/tex]
[tex]\mathbf{m = \dfrac{200 watt}{410.65\times (\dfrac{m^2}{s^2})}}[/tex]
[tex]\mathbf{m = \dfrac{200 \dfrac{J}{s}}{410.65\times (\dfrac{m^2}{s^2})}}[/tex]
since 1 J/s = 1 kgm²/s²)
Then:
[tex]\mathbf{m = \dfrac{200 \dfrac{\dfrac{kg\times m^2}{s^2}}{s}}{410.65\times (\dfrac{m^2}{s^2})}}[/tex]
[tex]\mathbf{m = \dfrac{200 \ {kg}}{410.65 \ s}}[/tex]
mass flow rate of the water (m) = 0.487 kg/s
Therefore, we can conclude that the mass flow rate of the water required to power a 200 W bulb light is 0.487 kg/s
Learn more about the hydroelectric plant here:
https://brainly.com/question/2635539?referrer=searchResults
