Understanding Hydraulic Grade Line (HGL) and Energy Grade Line (EGL)

In fluid mechanics, particularly when analyzing fluid flow in pipes or open channels, the concepts of Hydraulic Grade Line (HGL) and Energy Grade Line (EGL) are very important. They represent the energy levels of the fluid at different points along the flow path.

What is Hydraulic Grade Line (HGL)?

The Hydraulic Grade Line (HGL) represents the sum of the pressure head and the elevation head of the fluid. It indicates the height to which liquid would rise in a piezometer tube (an open tube used to measure pressure) inserted into the flow.

Mathematically, HGL is given by:

\( HGL = \frac{P}{\rho g} + z \)

• \( P \) is the pressure of the fluid
• \( \rho \) is the density of the fluid
• \( g \) is the acceleration due to gravity
• \( \frac{P}{\rho g} \) is the pressure head
• \( z \) is the elevation head (height above a reference datum)

The HGL essentially shows the potential energy per unit weight of the fluid (related to pressure and elevation).

What is Energy Grade Line (EGL)?

The Energy Grade Line (EGL), also known as the Total Energy Line (TEL), represents the total energy per unit weight of the fluid. According to Bernoulli's equation (assuming steady, incompressible, inviscid flow along a streamline), the total energy is the sum of pressure head, elevation head, and velocity head.

Mathematically, EGL is given by:

\( EGL = \frac{P}{\rho g} + z + \frac{V^2}{2g} \)

• \( V \) is the velocity of the fluid
• \( \frac{V^2}{2g} \) is the velocity head

The EGL includes all forms of mechanical energy: potential energy due to pressure and elevation (represented by HGL) and kinetic energy due to velocity.

Relationship between HGL and EGL

Comparing the formulas for HGL and EGL:

\( HGL = \frac{P}{\rho g} + z \)

\( EGL = \frac{P}{\rho g} + z + \frac{V^2}{2g} \)

We can see that:

\( EGL = HGL + \frac{V^2}{2g} \)

The velocity head \( \frac{V^2}{2g} \) represents the difference between the EGL and the HGL. Since the velocity head \( \frac{V^2}{2g} \) is always non-negative (assuming real flow where velocity \(V\) is real), the EGL is always greater than or equal to the HGL. The only case where EGL equals HGL is when the velocity is zero, such as in a large reservoir or at a stagnation point.

Therefore, the Hydraulic Grade Line (HGL) can never be greater than the Energy Grade Line (EGL) in standard fluid flow scenarios where energy losses are considered as a drop in EGL.

Analyzing the Statements

Let's evaluate each given statement:

1. HGL shows the height corresponding to the elevation and pressure head of the Bernoulli equation.

   This statement is true. HGL is defined as the sum of elevation head (\(z\)) and pressure head (\(\frac{P}{\rho g}\)).

2. HGL can be greater than Energy Grade Line (EGL) in some cases

   This statement is false. As explained above, \( EGL = HGL + \frac{V^2}{2g} \). Since the velocity head (\(\frac{V^2}{2g}\)) is always non-negative, EGL is always greater than or equal to HGL. HGL cannot be greater than EGL.

3. HGL is never greater than Energy Grade Line (EGL).

   This statement is true. This is the direct consequence of the relationship \( EGL = HGL + \frac{V^2}{2g} \).

4. HGL is the height to which liquid would rise in a piezometer tube attached to the flow

   This statement is true. A piezometer measures the pressure head. The liquid level in a piezometer rises to the height equivalent to the pressure head above the point of attachment, plus the elevation of the attachment point. This combined height is the HGL.

The question asks for the statement that is NOT true about Hydraulic Grade Lines (HGL). Based on our analysis, the statement "HGL can be greater than Energy Grade Line (EGL) in some cases" is the false statement.