Horizontal and Inclined Pipe Fluid Element Analysis

Horizontal

In a horizontal pipe, the fluid element analysis assumes no gravitational influence in the flow direction

The force balance simplifies to

$$\Delta p\pi r^2=\tau \cdot 2\pi r\ell$$

where $\Delta p$ is the pressure difference across the pipe length $\ell$, $\tau$ is the shear stress, $r$ is the radius of the fluid element, and $\ell$ is the length of the fluid element

The velocity profile for fully developed laminar flow remains parabolic, as derived when we did laminar flow fluid element analysis:

$$u=\frac{\Delta p{D}^2}{16\mu \ell }\left[1-{\left(\frac{r}{R}\right)}^2\right]$$

Inclined

For a pipe inclined at an angle $\theta$, the gravitational component along the pipe’s axis introduces an additional force term.

In this case, the force balance equation becomes

$$\Delta p\pi r^2-\gamma \pi r^2\ell \sin (\theta )=\tau \cdot 2\pi r\ell$$

We now can find the effective pressure drop considering gravity as:

$$\Delta p’=\Delta p-\gamma \ell \sin (\theta )$$

so that we can simplify the expression to $\Delta p^{\prime }\pi r=\tau \cdot 2\pi rl$. We can also write:

$$u=\frac{\Delta p’D^2}{16\mu \ell }\left[1-{\left(\frac{r}{R}\right)}^2\right]$$

Velocity Volumetric Mass Flow Rate

The average velocity ($\overline{V}$) in both cases is calculated by integrating the velocity profile over the pipe’s cross-sectional area:

$$\overline{V}=\frac{\Delta p’D^2}{32\mu \ell }$$

Using the regular $p$ for the inclined case:

$$\overline{V}=\frac{(\Delta p-\gamma \ell \sin (\theta ))D^2}{32\mu \ell }$$

The volumetric flow rate is the product of average velocity and the cross-sectional area of the pipe:

$$\begin{array}{rl}Q& =\pi R^2\overline{V}\\ & =\frac{\pi D^4}{128\mu \ell }\Delta p’\\ & \\ & =\frac{\pi D^4}{128\mu \ell }(\Delta p-\gamma \ell \sin (\theta ))\end{array}$$