SAE 30W oil at 20C flows through a straight pipe 25 m long, with diameter 4cm. The avg velocity is 2m/s. calculate the pressure drop and power required to drive the flow. By what factor would required power increase if the pipe diameter were doubled?
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
-
PURPOSE OF THE EXERCISE:
Demonstrate how to use Poiseuille’s law to estimate pressure drop and pumping power in a viscous (laminar) flow and to appreciate the counterintuitive scaling of required pumping power with pipe diameter at fixed average velocity. -
ENGINEERING CONTEXT:
This analysis arises in the design of piping systems for oil transport. Once you complete this calculation, you will be equipped to select pipe sizes in systems handling viscous fluids such as SAE 30W oil, ensuring that pumps are neither oversized (wasting energy) nor undersized (failing to overcome viscous dissipation losses). -
PHYSICAL CONTEXT:
The flow is dominated by viscous shear stresses, and inertia is negligible, so the fully developed laminar flow (Poiseuille) approximation applies. -
ASSUMPTIONS:
Assume steady, incompressible, fully developed, one‐dimensional laminar flow in a straight circular pipe (entrance effects are ignored). These assumptions are valid because the calculated Reynolds number is well below the typical transition threshold for turbulence. -
SOLUTION TECHNIQUE:
First, note the given parameters:
- Fluid: SAE 30W oil at 20 °C
- Pipe length, L=25\,\mathrm{m}
- Pipe diameter, d=0.04\,\mathrm{m}
- Average velocity, U=2\,\mathrm{m/s}
For a laminar flow in a circular pipe, Poiseuille’s law gives the pressure drop
\Delta p = \frac{32\,\eta\,L\,U}{d^2}\,.
A typical value for SAE 30W oil at 20 °C is a dynamic viscosity of about \eta=0.29\,\mathrm{Pa\,s} (and although the density is not needed for \Delta p, it is approximately \rho\approx870\,\mathrm{kg/m^3}).
Substitute the values:
\Delta p = \frac{32\times0.29\,\mathrm{Pa\,s}\times25\,\mathrm{m}\times2\,\mathrm{m/s}}{(0.04\,\mathrm{m})^2}\,.
Since (0.04)^2=0.0016\,\mathrm{m^2} and the numerator evaluates as
32\times0.29\times25\times2\approx464\,\mathrm{Pa\cdot m^2/s}\,,
we find
\Delta p \approx \frac{464}{0.0016}\approx2.9\times10^5\,\mathrm{Pa}\,.
Next, compute the volumetric flow rate
Q = U\,A = U\left(\frac{\pi d^2}{4}\right)=2\,\mathrm{m/s}\times\frac{\pi (0.04)^2}{4}\,.
Since \frac{\pi (0.04)^2}{4}\approx0.00126\,\mathrm{m^2},
Q\approx2\times0.00126 \approx0.00252\,\mathrm{m^3/s}\,.
The required pumping power is estimated by
P = \Delta p \; Q \,,
so
P \approx 2.9\times10^5\,\mathrm{Pa}\times0.00252\,\mathrm{m^3/s}\approx730\,\mathrm{W}\,.
A student is encouraged to sketch a control volume that spans the pipe length to visualize the pressure drop acting over the cross-sectional flow.
Now consider the effect of doubling the pipe diameter to d'=0.08\,\mathrm{m} while maintaining the same average velocity U. For Poiseuille flow:
- The pressure drop scales as
\Delta p\propto\frac{1}{d^2}\,,
so doubling d reduces \Delta p by a factor of 1/4.
- However, the cross-sectional area scales as d^2, so the flow rate
Q = \frac{\pi d^2}{4}U
increases by a factor of 4.
Thus, the pumping power, given by
P =\Delta p\,Q\,,
remains invariant because
\Delta p \propto \frac{1}{d^2}\quad \text{and}\quad Q \propto d^2\quad\Longrightarrow\quad P\propto \text{constant}\,.
- REFLECTION:
The calculation confirms that for fully developed laminar flow at a prescribed average velocity, the reduction in pressure drop due to a larger diameter is exactly offset by the increased volumetric flow rate, so the required pumping power remains essentially the same. This underscores the importance of understanding the scaling laws in viscous flows when designing piping systems.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 6.4: Bernoulli constant drop in pipes Part 2 (HIGHLY RELEVANT)
- Chapter 6.4: Bernoulli constant drop in pipes Part 1 (HIGHLY RELEVANT)
- Chapter 6.4: Bernoulli constant drop in pipes Part 3 (HIGHLY RELEVANT)
- Appendix C.2: Dimensional Analysis to Generalize Experiments (HIGHLY RELEVANT)
- Appendix C.5: Viscous Dissipation Factor (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
how do i know its laminar. also what are entrance effects
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
For determining whether the flow is laminar, you check the Reynolds number defined as
Re=\frac{\rho\,U\,d}{\eta}\,.
In our example with SAE 30W oil (using, for instance, \rho\approx870\,\mathrm{kg/m^3}, U=2\,\mathrm{m/s}, d=0.04\,\mathrm{m}, and \eta\approx0.29\,\mathrm{Pa\cdot s}), we find
Re\approx\frac{870\times2\times0.04}{0.29}\approx240\,.
Since Re\ll2300, the flow is well within the laminar regime.
Entrance effects refer to the region at the beginning of the pipe where the velocity profile is still “developing.” When fluid enters a pipe it often has a nearly uniform (plug) profile. As the flow proceeds downstream, viscous forces near the wall gradually build up the shear needed to form the fully developed parabolic profile predicted by Poiseuille’s law. In this entrance region the velocity gradients are not solely normal to the flow direction, so the convective terms in the Navier–Stokes equations are nonzero and the analytical solution is not as straightforward. For practical purposes, provided the pipe length exceeds the entrance length the fully developed assumption applies. In laminar flow this entrance length can be estimated as
L_e\sim0.05\,Re\,d\,,
which, in our case, is very short compared to the 25\,\mathrm{m} pipe length.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 11.3: Poiseuille Flow Part 3 (HIGHLY RELEVANT)
- Appendix C.2: Dimensional Analysis to Generalize Experiments (HIGHLY RELEVANT)
- Chapter 6.4: Bernoulli constant drop in pipes Part 3 (HIGHLY RELEVANT)
- Chapter 6.4: Bernoulli constant drop in pipes Part 1 (HIGHLY RELEVANT)
- Chapter 6.2: Modified Bernoulli equation Part 2 (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.