A tank containing 30 m^3 of kerosene is to be emptied by a...

A tank containing $30 \mathrm{m}^{3}$ of kerosene is to be emptied by a gravity feed using a drain hose of diameter $15 \mathrm{mm}$ roughness $0.2 \mathrm{mm},$ and length $1 \mathrm{m}$. The top of the tank is open to the atmosphere and the hose exits to an open chamber. If the kerosene level is initially $10 \mathrm{m}$ above the drain exit, estimate (by assuming steady flow) the initial drainage rate. Estimate the flow rate when the kerosene level is down to $5 \mathrm{m},$ and then down to $1 \mathrm{m}$. Based on these three estimates, make a rough estimate of the time it took to drain to the 1 -m level.

Key Concepts

Drainage Time Estimation

Estimating drainage time involves combining the instantaneous flow rates—which depend on the varying head as the fluid level drops—with the total volume to be drained. By integrating the flow rate over time or using average flow values for intermediate levels, one can obtain a rough estimate of the total time required for the liquid level to fall to a certain height, acknowledging that conditions change during the process.

Steady Flow Assumption

This concept presumes that the flow variables (velocity, pressure, etc.) do not change with time at any given point in the system for the purpose of analysis. In the context of tank drainage, it allows us to use static energy principles and engineering formulas to estimate the flow rate at a specific moment based solely on the instantaneous head, neglecting transient effects that occur as the level changes.

Bernoulli’s Principle

Bernoulli’s equation is central to fluid mechanics and embodies the energy conservation principle for flowing fluids. It relates the pressure, velocity, and elevation (head) between two points in a streamline flow and is used to determine the conversion of gravitational potential energy into kinetic energy, which drives the flow from the tank.

Gravity-Driven Flow

This concept refers to fluid motion induced by gravitational forces, where a height difference (or gravitational head) causes the fluid to accelerate. In drainage problems, the elevation difference between the free surface of the fluid and the outlet determines the available energy to overcome resistance and drive the flow through the hose.

Friction Losses in Pipe Flow

Friction losses are the loss of mechanical energy due to the viscous effects of the fluid interacting with the walls of the pipe. In any pipeline system, including a drain hose, these losses must be accounted for because they reduce the effective driving head available for flow, and they depend on factors such as pipe length, diameter, roughness, and the flow regime.

Darcy-Weisbach Equation

The Darcy-Weisbach equation provides a method to quantify the head loss due to friction in a pipe. By incorporating parameters like the friction factor (which is a function of the Reynolds number and relative roughness), pipe length, diameter, and velocity, this equation allows the accurate estimation of pressure drop and is crucial for determining the actual velocity and flow rate of the draining liquid.

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