The Bernoulli and the energy equations

These are perhaps the most important equations for the transport of fluids, in practical engineering, because these are a mechanical balance of the forces acting on the fluid. In fact, the energy equation is a generalization of the Bernoulli equation.

The equation of Bernoulli

The Bernoulli equation is the oldest one, as you may suppose, and it was not always written in the way in most fluid mechanics courses is used:

$\dfrac{p_1}{\gamma} + z_1 +  \dfrac{v_1^2}{2g} = \dfrac{p_2}{\gamma} + z_2 + \dfrac{v_2^2}{2g}$        Eq. (1)

but in a more confusing fashion as is shown in the seminal book of Daniel Bernoulli, Hydrodynamica. Different authors use different presentations of Eq. (1) but this is the most widely used because it can be easily related to the conduit through which the fluid is being transported and on which the mechanical balance is made. Below is shown the mechanical balance represented by the equation of Bernoulli

where as you can see, all terms of Eq. (1) are schematically represented. The mechanical balance of the equation of Bernoulli establishes that the energy at point 1 should be the same at point 2, and it is implicit that there are no changes in temperatue or properties of the fluid.

Understanding the mechanical balance

For short the mechanical balance states that the total mechanical energy at point 1 should be the same at point 2, and that if there are losses or additions of energy between these two points these should be included.

This is why the Bernoulli equation has pressure and speed measured at two different points. However, if you look carefully Eq. (1) you will notice that all terms have units of length instead of energy and this is in agreemen with the sketch if the figure above. This is because the energy is considered as that required to make the fluid travel upwards a given distance (length) as shown in the sketch above.

Consideration of mechanical energy in terms of the length that the fluid may travel upwards gives origin to an important engineering concept  very useful from the professional point of view: the hydraulic grade line HGL. This is mathematically defined as follows:

$HGL_1 = \dfrac{p_1}{\gamma} + z_1$

$HGL_2 = \dfrac{p_2}{\gamma} + z_2$ Eq. (2)

for points 1 and 2 in the sketch above, respectively. The idea of $HGL$ is key to design pipe systems. For example, by knowing the value of $HGL$ you may learn in which direction the fluid is moving. For the case presented in the sketch above $HGL_1>HGL_2$ since the fluid moves downwards.

Trying to infer the flow direction from pressure will not always work. If you take that pressure should always be larger at 1 that at 2, errors can appear along with physical inconsistencies as well. For the present case, for example $p_1<p_2$ which could be contradictory if you not take into account the role played by gravity. On the other hand $HGL$ gives a neat appreciation of the flow direction since it includes the pressure and gravity, due to $z$ height, effects.

Going back to the sketch above. It is noticeable that the mechanical balance given by the equation of Bernoulli is incomplete since the mechanical energy at 1 should be same at 2. For exmple, if the inclination of the pipe were of just a few degrees, say $3^\circ$ for example, the pressure $p_2$ would be slightly larger than $p_1$ and if the pipe diameter remains constant, as is shown, the fluid velocities would be the same at 1 and 2, $v_1=v_2$. However the increment of $p_2$, at point 2, would no be larger enough to equal the total energy at 1. This means that mechanical energy was lost some how , which is represented by the question mark (?). If the inclination of the pipe were too much the situation would go worst since for that case we coulod have an excess of energy!

Some disadvantages of the equation of Bernoulli

Despite the equation of Bernoulli is still in use for rough estimations we should be aware of its disadvantages. These are, mainly:

• the effects or changes occurring between 1 and 2 are not considered. This means that you could have bends or equipment or accesories between these two points and none of these will be considered for the overall mechanical balance;
• there is loss of mechanical energy as indicated in the sketch above not being accounted. Perhaps the origin of the general energy equation has its origin at this point;
• changes in the properties of the fluid, due to temperature variation for eample, are not taken into account. In other words, viscosity, density, etc. are to be kept constant;
• its use is only advised for rough estimations in very simple and small pipe situations. Very large errors may arise if its is not used with caution.

The general energy equation

The general equation of mechanical energy comes to deal with some of the weaknesses of the Bernoulli equation and in particular those related with the mechanical energy loss or won due to accessories or equipment between points 1 and 2. The general energy equation is then written as:

$\dfrac{p_1}{\gamma} + z_1 + \dfrac{v_1^2}{2g} + h_A - h_R - h_L = \dfrac{p_2}{\gamma} + z_2 + \dfrac{v_2^2}{2g}$ Eq. (3)

where as you can see Eq. (3) is in fact a generalization of Eq. (1). The terms $h_A$, $h_R$ and $h_L$ were introduced to account for three different possibilities of sinks or sources of mechanical energy. These are:

• mechanical energy added to the fluid by equipment such as pumps, for liquids, or fans or compressors, for gases. In many applications could happen that the fluid losses mechanical energy or momentum and slows down before reaching point 2, and the solution is to introduce an equipment capable of providing more energy to the fluid so that its velocity increases again and arrive to point 2. Since this is added mechanical energy the equation has the $+h_A$ term;
• when the fluid increases its mechanical energy or momentum, due to pipe inclinations (when travelling down) and gravity, the excess of energy should be removed for safety reasons, mostly. Then it is usual to install equipments performing the inverse job of a pump or fan, such as turbines. Then, since the energy is to be removed from the system the equation has the $-h_R$ term;
• any pipe system will provoke mechanical energy losses due to friction between the fluid and the pipe wall or to small obstacles created at the joints between the pipes and the accessories, for example. These losses of energy can not be avoided and should be taken into account, specially in large pipe systems. Since this is a loss of energy the equation has the $-h_L$ term.

The sketch above shows a partial representation of general energy equation were only $h_L$ is presented since this is the most common in textbooks.

to be continued...

Any questtion? Write in the comments and I shall try to help.

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Ildebrando.