Figure 6.33 shows a storage tank for sugar syrup that has a...

Figure 6.33 shows a storage tank for sugar syrup that has a viscosity of 15.2 centiposes at $25^{\circ} \mathrm{C}$, and a density of $1008 \mathrm{~kg} / \mathrm{m}^3$. Friction loss includes an entrance loss to the drain pipe that equals the kinetic energy gain of the fluid, and resistance to flow through the short section of the drain pipe. (a) Formulate an energy balance equation for the system. (b) Formulate the continuity equation that represents the increment change in fluid level in the tank as a function of the fluid velocity through the drain pipe. (c) Solve simultaneously equations formulated in (a) and (b) to calculate the time to drain the tank to a residual level $1 \mathrm{~m}$ from the bottom of the tank. (d) Calculate the amount of residual fluid in the tank including the film adhering to the side of the tank. (FIGURE CAN'T COPY)

Figure 6.33 shows a storage tank for sugar syrup that has a viscosity of 15.2 centiposes at $25^{\circ} \mathrm{C}$, and a density of $1008 \mathrm{~kg} / \mathrm{m}^3$. Friction loss includes an entrance loss to the drain pipe that equals the kinetic energy gain of the fluid, and resistance to flow through the short section of the drain pipe.
(a) Formulate an energy balance equation for the system.
(b) Formulate the continuity equation that represents the increment change in fluid level in the tank as a function of the fluid velocity through the drain pipe.
(c) Solve simultaneously equations formulated in (a) and (b) to calculate the time to drain the tank to a residual level $1 \mathrm{~m}$ from the bottom of the tank.
(d) Calculate the amount of residual fluid in the tank including the film adhering to the side of the tank. (FIGURE CAN'T COPY)

Key Concepts

Energy Conservation in Fluid Mechanics

This concept involves applying the energy balance principle to fluid flow, where the conversion between potential energy, kinetic energy, and losses (such as friction or entrance losses) is considered. It is fundamental in understanding how energy is distributed and transformed as fluid flows from one level to another, which is essential in many applications including draining tanks or pipelines.

Continuity Equation

The continuity equation represents the conservation of mass in fluid systems. In scenarios involving unsteady flow or changing free surface conditions, like in a draining tank, it is used to relate the rate of change of fluid volume (or level) to the velocity of fluid through an outlet. This concept ensures that the mass of fluid entering and leaving a system is accounted for over time.

Fluid Friction Losses

Friction losses in conduits arise due to the viscous effects of the fluid interacting with the walls of the pipe and other features like abrupt contractions or entrances. These losses are critical in accurately modeling energy dissipation in a system and must be quantified to correctly assess the available energy for fluid motion. This concept includes both distributed friction loss along the pipe and localized losses such as entrance losses.

Transient Flow Analysis in Draining Tanks

This concept involves setting up and solving differential equations that describe the time-dependent behavior of fluid draining from a tank. It typically combines the energy balance and continuity equations to develop an expression for the rate of change of the fluid level. Integration of these equations then provides the time required for the fluid level to drop to a desired value.

Residual Fluid Assessment

Residual fluid assessment relates to determining the volume of fluid that remains in a system after drainage, including fluids adhering to surfaces due to viscosity and wetting effects. This calculation is important in practical applications where complete emptying of a tank is either not possible or not desired, and it influences system designs and process considerations.

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