Relation between conservation of energy and conservation of mass of a fluid flow

Question

Let us consider the following hypotheses for fluid flow:

1. Incompressible flow
2. Steady flow

That being stated, let us consider a horizontal pipe with constant diameter, thus constant cross-sectional area. According to conservation of energy, we have

\begin{equation} \frac{P_1}{\rho g} + \frac{V_1^2}{2g} = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + h_L \tag{1} \label{1} \end{equation}

where the term $h_L$ relates all losses between point 1 and 2. Applying Eq. \ref{1} to calculate $V_2$, it is possible to define the following equation

\begin{equation} \frac{V_2^2}{2g} = \frac{P_1 - P_2}{\rho g} + \frac{V_1^2}{2 g} - h_L \tag{2} \label{2} \end{equation}

It is also possible to calculate $V_2$ by applying conservation of mass

\begin{equation} V_2 A_2 = V_1 A_1 \tag{3} \label{3} \end{equation}

As the pipe has the same cross-sectional area, Eq. \ref{3} can be written as \begin{equation} V_2 = V_1 \tag{4} \label{4} \end{equation}

Comparing Eq. \ref{2} and \ref{4}, the unique way to Eq. \ref{2} be true is if \begin{equation} \frac{P_1 - P_2}{\rho g} - h_L = 0 \tag{5} \label{5} \end{equation}

Is it Eq. \ref{5} valid, exclusively, due to hypothesis 2? As the flow is steady, it shouldn't be accelerating. Is it correct?

Answer 1

Is it Eq. 5 valid, exclusively, due to hypothesis 2? As the flow is steady, it shouldn't be accelerating. Is it correct?

Not exclusively, no. You also need $(3)$ and $(4)$.

Regarding:

$$\frac{P_1 - P_2}{\rho g} - h_L = 0\tag{5}$$

It can be re-written as:

$$P_1-P_2=\rho g h_L$$

This can then be seen as a balance of forces because $h_L$ is really just a mathematical device$^{\dagger}$, where $\rho gh_L$ is a pressure equal to $P_1-P_2$. With the forces on the fluid balancing, there can be no acceleration (Newton's 2nd Law)

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$^{\dagger}$ I call $h_L$ a 'mathematical device' because the sum total of the pressure losses (between points $1$ and $2$), i.e. $P_L$, can be written as:

$$P_L=\rho g h_L$$

In pipe system engineering circles $h_L$ is often referred to as the head loss.