Alcohol flows smoothly through a horizontal pipe that tapers in cross-sectional area from A1 = 41.9 cm2 to A2= A1/3. The pressure difference Δp between the wide and the narrow sections of the pipe is 11.1 kPa. What is the volume flow rate ΔV/Δt of the alcohol? The density of the alcohol is ρ = 838 kg/m3. Hint: Treat as an ideal fluid. You will need both the conservation of mass properties and the conservation of energy properties.
To solve this problem, we can apply the principles of conservation of mass and conservation of energy for an ideal fluid.
Step 1: Apply the conservation of mass
According to the conservation of mass, the mass flow rate (ṁ) of a fluid is constant throughout a horizontal pipe. The mass flow rate can be calculated using the following equation:
ṁ = ρ * A * V
where
ṁ is the mass flow rate,
ρ is the density of the fluid,
A is the cross-sectional area of the pipe, and
V is the velocity of the fluid.
Step 2: Relate the velocity and the cross-sectional area
Since the pipe tapers in cross-sectional area, the velocity of the fluid changes as it flows through the pipe. We can relate the velocities at the wide and narrow sections of the pipe using the conservation of energy.
According to Bernoulli's equation, the sum of the pressure energy, kinetic energy, and potential energy per unit volume is constant for ideal fluids. The equation is given as:
P + 0.5 * ρ * V^2 + ρ * g * h = constant,
where
P is the pressure of the fluid,
ρ is the density of the fluid,
V is the velocity of the fluid,
g is the acceleration due to gravity, and
h is the height above a reference point.
Step 3: Apply Bernoulli's equation
Since the pipe is horizontal, there is no change in height. Therefore, we can simplify Bernoulli's equation to focus on the pressure and velocity terms:
P1 + 0.5 * ρ * V1^2 = P2 + 0.5 * ρ * V2^2
where
P1 and P2 are the pressures at the wide and narrow sections, respectively,
V1 and V2 are the velocities at the wide and narrow sections, respectively.
Also, from the given information, we have:
A2 = A1/3
Δp = P1 - P2
ρ = 838 kg/m3
Using the equations and known values, we can now solve for the velocity V2 at the narrow section of the pipe.
Step 4: Calculate the volume flow rate
The volume flow rate (ΔV/Δt) is given by the equation:
ΔV/Δt = A2 * V2
where
A2 is the cross-sectional area of the narrow section of the pipe, and
V2 is the velocity at the narrow section.
By substituting the values into the equation, we can calculate the volume flow rate.