A process fluid with a constant density, p kg/m³, enters a vertical cylindrical tank through an inlet pipe at its top. The tank is Dm in diameter and 2.4 m tall. The fluid leaves the tank at the flat bottom through an outlet pipe with an internal diameter dm. When operated at a steady state with 1.2 m liquid level inside the tank, the outlet velocity of the fluid through the bottom pipe, v(t) m/s, can be estimated from Bernoulli equation as:v(t) = 4h(t)where h(t) m is the liquid level inside the tank.(a) Obtain the transfer function between the liquid level inside the tank and the inlet volumetric flow rate when the tank is half full of the liquid.(b) Determine how long it will take for the liquid level inside the tank to increase by 0.1 m when the inlet volumetric flow rate is suddenly increased by 0.01 m³/s from the steady state value. Data: p = 1000 kg/m³, D = 1 m and d = 0.1 m.
Question
A process fluid with a constant density, p kg/m³, enters a vertical cylindrical tank through an inlet pipe at its top. The tank is Dm in diameter and 2.4 m tall. The fluid leaves the tank at the flat bottom through an outlet pipe with an internal diameter dm. When operated at a steady state with 1.2 m liquid level inside the tank, the outlet velocity of the fluid through the bottom pipe, v(t) m/s, can be estimated from Bernoulli equation as:v(t) = 4h(t)where h(t) m is the liquid level inside the tank.(a) Obtain the transfer function between the liquid level inside the tank and the inlet volumetric flow rate when the tank is half full of the liquid.(b) Determine how long it will take for the liquid level inside the tank to increase by 0.1 m when the inlet volumetric flow rate is suddenly increased by 0.01 m³/s from the steady state value. Data: p = 1000 kg/m³, D = 1 m and d = 0.1 m.
Solution
(a) To obtain the transfer function between the liquid level inside the tank and the inlet volumetric flow rate when the tank is half full, we can start by considering the equation of continuity. The equation of continuity states that the mass flow rate into the tank must be equal to the mass flow rate out of the tank.
The mass flow rate into the tank can be expressed as the product of the inlet volumetric flow rate and the density of the fluid, which is given as p kg/m³. Therefore, the mass flow rate into the tank is given by:
m_in = Q_in * p
where Q_in is the inlet volumetric flow rate.
The mass flow rate out of the tank can be expressed as the product of the outlet velocity and the cross-sectional area of the outlet pipe. The cross-sectional area of the outlet pipe can be calculated using the formula for the area of a circle:
A_out = π * (dm/2)^2
where dm is the internal diameter of the outlet pipe.
The mass flow rate out of the tank is then given by:
m_out = v(t) * A_out
where v(t) is the outlet velocity of the fluid through the bottom pipe.
Since the tank is at a steady state, the mass flow rate into the tank must be equal to the mass flow rate out of the tank. Therefore, we can equate the two expressions for the mass flow rates:
Q_in * p = v(t) * A_out
Substituting the expression for A_out, we have:
Q_in * p = v(t) * π * (dm/2)^2
Simplifying the equation, we can solve for v(t):
v(t) = (Q_in * p) / (π * (dm/2)^2)
Now, we can substitute the given values for p, D, and d to obtain the transfer function. Since the tank is half full, the liquid level inside the tank is h(t) = 1.2 m. Therefore, the transfer function is:
v(t) = (Q_in * 1000) / (π * (0.1/2)^2)
(b) To determine how long it will take for the liquid level inside the tank to increase by 0.1 m when the inlet volumetric flow rate is suddenly increased by 0.01 m³/s, we can use the transfer function obtained in part (a).
First, we need to find the initial outlet velocity, v(t0), at the steady state. We can substitute the given values for p, D, and d, and the steady state liquid level h(t0) = 1.2 m into the transfer function to find v(t0).
Next, we can calculate the new outlet velocity, v(t1), when the inlet volumetric flow rate is increased by 0.01 m³/s. We can use the transfer function and substitute the new inlet volumetric flow rate, Q_in + 0.01, and the liquid level h(t0) + 0.1 into the transfer function to find v(t1).
Finally, we can calculate the time it takes for the liquid level to increase by 0.1 m by using the equation:
Δt = Δh / Δv
where Δh = 0.1 m and Δv = v(t1) - v(t0).
By substituting the values into the equation, we can find the time it takes for the liquid level to increase by 0.1 m.
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