# Why Output is proportional to Velocity squared

2020年4月27日

It is said that Force [N] is proportional to velocity squared and
Output [W] is proportional to velocity squared.

This article proves that Output [W] is proportional to velocity squared
from theoretical formula of hydroelectric power output and Bernoulli’s principle.

### Situation

To prove the matter, the situation of the following water storage tank is used.
The position (1) is the surface of water and the position (2) is the outlet of tank.

The variables in Figure are defined at the last chapter.

### Theoretical formula of hydroelectric power output at outlet

Firstly, let’s consider theoretical formula of hydroelectric power output at outlet position (2).
The formula is defined as follows.
$$P_{out}=9.8Q(z_1-z_2)ρ$$
$$P_{out}=9.8Q(h)ρ$$
The density of water is approximately $${1ton/m}^3$$ so that the formula is expanded into $$P_{out}=9.8Qh$$.
Flow Q can be expressed by the outlet port area A and fluid velocity v as follows.
$$Q=Av_2$$
Therefore, the formula expansion is carried out as below.
$$P_{out}=9.8Av_2h{…Formula 1}$$

### Applying Bernoulli’s principle.

The below formula can be obtained from applying Bernoulli’s principle with Situation.
$$\frac{1}{2}v_1^2+\frac{P_1}{ρ}+gz_1=\frac{1}{2}v_2^2+\frac{P_2}{ρ}+gz_2$$
The second term means the formula regarding pressure and can be omitted
because the both points are under atmosphere.

Velocity at position (1) is almost zero and height at position (2) is a point of reference.
The following can be obtained.

$$g(z_1-z_2)=\frac{1}{2}v_2^2$$
$$gh=\frac{1}{2}v_2^2$$
$$h=\frac{v_2^2}{2g}{…Formula 2}$$

### Output is proportional to velocity squared

Finally, Formula 2 is substituted for Formula 1 and Formula 3 which indicate output is proportional to velocity cubed is obtained.

$$P_{out}=9.8Av_2h{…Formula 1}$$
$$P_{out}=9.8\frac{Av_2^3}{2g}{…Formula 3}$$

### Variable definition

Pout[kW]:Theoretical power at outlet port Position (2)
z1[m]：Height between water surface position (1) and tank bottom
z2[m]：Height between outlet port position (2) and tank bottom
v1[m/sec]：Fluid velocity at water surface position (1)
v2[m/sec]：Fluid velocity at outlet port Position (2)
P1[Pa or N/m2]：Fluid pressure at water surface position (1)
P2[Pa or N/m2]：Fluid pressure at outlet port Position (2)
ρ[ton/m3]：Water density
Q[m3/sec]：Flow at outlet port Position (2)
h[m]：Height between water surface position (1) and outlet port Position (2)
A[m2]:Area of outlet port Position (2)
g[m2/sec]：Acceleration of Gravity