Overview

Torricelli’s Law is a mathematical relationship between the flow rate of fluid from a draining tank and the height of fluid in the tank. The formal development of this relationship gives a lot of insight into the use of the resultant formula, including the inherent assumptions that place limitations on its application in certain situations.

We can develop the desired relationship by considering a tank containing a fluid that can be drained by removing a plug at the bottom of the tank -- a rough sketch of such a system is given below.

Assume that the exhaust pressure is a given constant, P_a, and that the inlet flow rate is zero for the current system. The fluid volume at any time is V(t) and the output of interest is the exit mass flow rate,. A set of mass and energy balances are performed on this system, as follows:

Mass Balance

As with any balance equation, the rate of change of the quantity of interest is equal to the difference in its production and loss rates within the desired volume. In this case, if we choose the mass of fluid in the tank as the quantity of interest, we have

The mass is simply the fluid density times the volume, , with units of (kg/m³)(m³) = kg. The mass flow rate at the tank exit is given in terms of the density, exit area, and exit velocity of the fluid, or

with consistent units, (kg/m³)(m²)(m/s) = kg/s.

For no inflow and constant density, eqn. (1) becomes

or

where V is the volume of fluid in the tank, A_e is the exit flow area, and v_e is the average exit fluid velocity.

Energy Balance (on control volume at tank exit)

Now we would like to relate the exit velocity, v_e, to the height of the fluid in the tank. This can be done by performing an energy balance on a small control volume at the tank exit,

or

where point 1 is at the inlet to the control volume (CV) and point 2 is at the exit as noted in the above sketch. If we assume no mass or energy accumulation in this small control volume at the bottom of the tank, then

The no mass accumulation assumption in the CV says that the mass flow rates at the inlet and exit of the CV are equal,

and, similarly, with no energy accumulation, the change in total energy from the inlet to the exit of the CV is zero.

The total energy is usually written as the sum of four major components -- internal energy (U), kinetic energy (KE), potential energy (PE), and, for problems involving flow, a term that accounts for the work associated with moving the fluid into and out of the CV [i.e. the flow work (FW) term]. Here we are implicitly neglecting electrical or magnetic energy effects. Thus, the change in energy from the inlet to the exit of the CV can be written as

Each of these terms is usually written as an energy flow rate with units of energy per unit time. For example, the internal energy per unit mass (u_i) times the mass flow rate gives the internal energy flow rate at position i (where the i subscript denotes either the inlet or exit of the CV), or

Using similar arguments, eqn. (9), written as a rate equation, becomes

where each of the four components have consistent units, as follows:

In the above expressions we have used the fact that work is the result of a force moving through some distance, or

Thus a joule (J) is simply a force times distance (N-m), and the rate of change of energy has units of J/s or N-m/s in the metric system of units.

Now to simplify things considerably, we make the following arguments:

1. Since there is no heat addition into the CV, the change in internal energy in the CV is zero.

2. Since, for this part of the problem, there is no change in elevation, there is also no change in potential energy in the CV.

3. The fluid velocity at the bottom of the tank (at point 1) is small compared to the velocity at the small exit opening (at point 2). Note that, since there is no mass accumulation in the control volume, then

If the density is essentially constant, , and

For a small discharge hole relative to the size of the tank, A₂ is usually much smaller than the cross-sectional area of the tank, A₁, and v₂ >> v₁ (usually). Thus, one usually lets .

With these arguments (i.e. and ), the energy balance in eqn. (11) becomes

This says that the change in pressure from the bottom of the tank to just outside the exit region is the driving force that increases the fluid velocity (i.e. the change in flow work is converted to an increase in kinetic energy of the fluid).

But the pressure at point 1 (at the bottom of the tank) is simply the sum of the pressure at the top of the tank plus the pressure due to the weight of a column of fluid of height y = h, or

If point 2 is open to the atmosphere, we have . Therefore,

and

where y is the height of fluid in the tank. Equation (18) is the desired relationship we were after and, with this expression for the exit velocity, the mass balance in eqn. (4) becomes

Finally, we note that actual flow from a nozzle varies somewhat from the above development. In particular, depending on the type of opening, the actual flow area tends to be smaller than the physical area, A_e. One often defines an orifice coefficient as the ratio of the observed area to the physical area of the exit nozzle, or

and the observed area, cA_e, is actually used in eqn. (19) instead of the physical area, A_e, giving

Note that, in practice, A_observed is difficult to obtain in realistic problems so that c is often determined experimentally.

Finally, letting , eqn. (21) can be simplified to give