Torricelli’s equation

The most interesting applications of Bernoulli s equation include the effects of friction. Before we can solve these, we must learn how to evaluate the term, which we do in Chap. 6. However, in many flow problems the friction heating terms are small compared with the other terms and can be neglected. We can solve these by means of Bernoulli s equation without the friction heating term. A good example of this type of problem is the tank-draining problem, which leads to Torricelli s equation. 

This is Torricelli s equation, which says that the fluid velocity is exactly the same as the velocity the fluid would attain by falling freely from rest a distance h. Substituting the numerical values, we find... 

This is the classic tank-draining solution. It is correct only fpr situations in which the assumptions made in finding Torricelli s equation apply in Examples 5,3 and 5.4 we examine some situations in which they may not apply. 

This situation, in which the friction forces are dominant, is quite different from the situation shown in Fig. 5.5, from which we found Torricelli s equation, and is not covered by the frictionless assumption of Torricelli s equation. 

Ignoring the change in atmospheric pressure in Torricelli s equation for air and water causes an error of less than 0.1 percent (much less than the error introduced by some of the other assumptions). We are justified in leaving out this term if the ratio of the density of the surrounding fluid to that of the flowing fluid, pjp., is much less than 1. This is true in most hydraulics problems but not in two-liquid problems (Probs. 5.13 and 5.14). 

Here we assume that the dVIdi) is small we check that assumption later. Then the instantaneous flow rate is assumed to be given by Bernoulli s equation, which here takes the form of Torricelli s equation ... 

The maximum velocity in this tank is at the outlet, and all other velocities are proportional to it therefore, the maximum value of dVIdt) must occur at the outlet. Differentiating Torricelli s equation with respect to time yields... 

In Example 5.2 we computed the exit velocity by Torricelli s equation, which does not take into account the fact that at the bottom of the jet the velocity will be higher than at the top, as discussed in Sec. 5.11. How large an error are we likely to have made If the jet is passing through a perfectly rounded entrance with an outlet diameter of 0.5 ft, and if the centerline of the jet is 30 ft below the fluid surface, how much difference should there be between the velocities at the top and bottom of the jet ... 

If there had been no porous medium in the lowOr part of the apparatus in Fig. 12.3, then the exit velocity would have been given by Torricelli s equation, equal to about 9 ft/s. Here the calculated velocity is as large. Fluid friction effects in porous media are large j... 

One unit used to measure pressure is defined by using Torricelli s barometer. The standard atmosphere (atm) is defined as the pressure that supports a 760-mm column of mercury. This definition can be represented by the following equation. 

The mass flow term in can be taken as the product of the density of the fluid and its volume flow rate. The mass flow out can be specified by Torricelli s Law multiplied by the fluid density and the mass in the control volume is the product of the fluid density, the tank s cross-sectional area, and the level at any time t. This gives us the following equation ... 

In this expression Aoo is the nominal aperture size to deliver at the design flow rate based on the constant set input flow rate. The second term in the parenthetical expression is the product of a proportionality constant K and the difference between the set point level and the actual level as a function of time. We substitute this for Ao in Torricelli s Law and also in the equation describing a system with sinusoidally fluctuating input flow ... 

Recall that pi is just the density of the pure liquid solvent, and that h[t] = can be replaced into Torricelli s Law. These equations have become complicated enough that we shall define the density upfront and let NDSolve handle the work of solving the simultaneous equations (see the In statement that follows) ... 

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