Question

two reservoirs are connected by three clean cast-iron pipes in series with lengths 440 m, 460 m, and 1,050 m, and diameters 220 mm, 420 mm, and 450 mm, respectively. when the flow rate through the system is 250 l/s, determine the difference in water level between the reservoirs. ignore minor losses

Answer 1

To determine the difference in water level between the two reservoirs, we can apply Bernoulli's equation for steady, incompressible flow. Bernoulli's equation relates the pressure, velocity, and elevation of a fluid at two points along a streamline.

Step-by-step explanation:

It's given by:

P1 + 0.5 * ρ * v1^2 + ρ * g * h1 = P2 + 0.5 * ρ * v2^2 + ρ * g * h2,

where:

P1 and P2 are the pressures at points 1 and 2 (in Pa or N/m²),

ρ is the density of the fluid (in kg/m³),

v1 and v2 are the velocities at points 1 and 2 (in m/s),

g is the acceleration due to gravity (in m/s²),

h1 and h2 are the elevations at points 1 and 2 (in meters).

Given the flow rate (Q) as 250 liters per second (l/s), we can convert it to cubic meters per second (m³/s):

Q = 250 * 10^-3 m³/s.

Since the pipes are in series, the flow rate through each pipe remains constant. Therefore, we can calculate the velocity of water in each pipe using the formula:

Q = A * v,

where A is the cross-sectional area of the pipe and v is the velocity.

The cross-sectional area (A) of a pipe with diameter (D) is given by:

A = π * (D/2)^2.

Now, let's solve for the velocities in each pipe using the given flow rate and pipe diameters:

For the first pipe (D = 220 mm):

A1 = π * (0.22 / 2)^2,

v1 = Q / A1.

For the second pipe (D = 420 mm):

A2 = π * (0.42 / 2)^2,

v2 = Q / A2.

For the third pipe (D = 450 mm):

A3 = π * (0.45 / 2)^2,

v3 = Q / A3.

With the velocities known, you can now apply Bernoulli's equation between the two reservoirs (assuming the pressure at both reservoirs is atmospheric pressure) to find the difference in water level (h2 - h1). Since the elevation change is negligible, the equation simplifies to:

0.5 * ρ * (v2^2 - v1^2) = ρ * g * (h2 - h1).

Solving for h2 - h1:

h2 - h1 = (v2^2 - v1^2) / (2 * g).

Substitute the calculated values of v1 and v2 into the equation, and you'll get the difference in water level between the two reservoirs. Note that the density of water (ρ) is approximately 1000 kg/m³, and the acceleration due to gravity (g) is approximately 9.81 m/s².