Understanding Fluid Flow Calculations in Pipes: Practical Guide for Operators

When calculation errors cost downtime: a practical hook for operators

Imagine you’re halfway through a planned pipe-freezing job to isolate a branch for maintenance. The line cools, the trapped liquid can’t expand the way you expected, pressure climbs fast, and the freeze plug forms unevenly. A weld sees a short pressure spike, cracks, and suddenly you’re dealing with an emergency release, followed by hours of response and days of repair and cleanup. This isn’t a rare “edge case.” Post‑job reviews often trace unplanned shutdowns back to basic flow or pressure math that was skipped, rushed, or based on the wrong assumptions.

One fact drives most successful plans: you need a reliable way to predict how fast the fluid is moving, how much pressure you’ll lose through the line, and where the freeze plug is likely to form. That’s the practical value of fluid flow calculations, they help you protect isolation points and avoid surprises that turn a maintenance window into a crisis. Start by calculating trapped volumes (see How to calculate the amount of fluid in a length of pipe) early in planning. Then run worst‑case pressure‑rise scenarios before any valve is closed or liquid nitrogen is applied. In real operations, that upfront work can save hours of unplanned maintenance and reduce exposure to hazardous releases.

A common shortcut is “DIY pipe freezing,” and it’s where risk multiplies. If the freeze plug is placed in the wrong spot, or the cryogenic charge is misjudged, you can end up with slow leaks, hidden icing, or overpressure, problems that are expensive and dangerous. For complex systems, we recommend professional services. CryoTech Pipe Freezing is the U.S. Leader in liquid nitrogen and cryogenic pipe freezing, with certifications and nationwide experience; our team handles freeze‑plug design, safety protocols, and emergency mitigation so your maintenance window stays a window. Most practical operators do best with a mix of basic math, vendor guidance, and disciplined safety checks.

Here’s the caveat: no calculation guarantees a perfect outcome if the field conditions don’t match your inputs. Unknown pipe roughness, mixed fluids, partially closed valves, or unverified temperatures can all push real behavior away from the model. That’s why the best teams treat the numbers as a decision tool, and then verify with instrumentation in the field.

The good news is that the math behind these choices is well documented. Engineers typically consult a pipe‑flow reference before field work (see a tutorial on pipe‑flow equations from LBL) to validate assumptions and choose conservative safety margins.

How cryogenic temperatures change fluid properties operators must track

Why do cryogenic jobs feel “different” from normal piping work? Because cold temperatures can change how a fluid behaves in ways you can’t safely ignore. The key properties to track are density, viscosity, vapor pressure, thermal conductivity, and specific heat. Each one matters for a different reason: density affects inertia and static head, viscosity drives friction losses and flow regime, vapor pressure controls boiling and cavitation risk, and thermal conductivity plus specific heat influence how quickly the pipe cools and how the freeze front moves. In cryogenic work, these values can shift enough to break a room‑temperature assumption.

Cryogenic fluids can change quickly with temperature. Liquid nitrogen, for example, behaves very differently than water or typical oils. Its density and viscosity aren’t “close enough” to guess. Those changes affect the Reynolds number and, in turn, whether friction behaves more like smooth laminar flow or rough turbulent flow. Since Reynolds number equals ρ·V·D/μ, lower viscosity tends to increase Reynolds, while lower density tends to decrease it. Which effect wins depends on the fluid and the temperature, so quick lookups in temperature‑dependent tables are essential before you size pumps or estimate pressure losses at cryogenic conditions.

Two‑phase behavior (liquid + vapor) is where jobs get dangerous. If part of the liquid boils during cooldown, you can get pressure spikes, moving vapor pockets that hit like water hammer, or localized cavitation when pressure drops below vapor pressure. At low temperature and under pressure, boiling behavior can shift and narrow your safety margin. If you use liquid nitrogen for a freeze plug, plan for vapor generation and provide controlled venting and pressure relief. Isolation work that ignores two‑phase risk is a common DIY failure mode, and it can lead to hidden damage and extended downtime.

Practical guidance that holds up in the field: use published property tables and interpolate between tabulated points for small temperature steps; for larger ranges, spline interpolation can be more stable. NIST and cryogenics manufacturers publish downloadable data sets. For hydraulic checks and head‑loss estimates, engineers often keep a reference like the fundamentals of hydraulics (hydraulics basics from WVU) nearby, then overlay cryogenic property data for the actual temperature. The table below summarizes directional changes you’ll commonly encounter with cryogens.

During planning, document your source tables and interpolation method, add safety factors for vapor generation, and plan instrumentation so field readings can confirm (or challenge) your model. A simple checklist many teams use: verify temperature‑dependent ρ and μ. Compute Reynolds. Select a friction factor. Run a worst‑case vapor fraction check. Confirm pressure‑relief sizing. And record the planned freeze‑plug location. If any step is outside your comfort zone, bring in certified cryogenic pipe‑freezing professionals, this is one of the fastest ways to reduce maintenance risk and protect operational integrity.

Key Takeaways

• Don’t rely on room-temperature values. Interpolate cryogenic density and viscosity from property tables for calculations.
• Compute Reynolds and Darcy-Weisbach friction losses for fluid flow calculations, and don’t forget Moody, Colebrook, or explicit formulas for f.
• Follow the 7-step workflow. It’s how you’ll define goals, gather properties, compute losses, and validate measurements.
• Don’t skip calibration. Calibrate meters regularly, do on-site zero checks, and compare to weigh-scale balances.
• Before freezing, don’t skip venting, isolation, wall-thickness checks, and add temporary vents for trapped volumes.

Core equations operators use: continuity, Bernoulli, Reynolds and Darcy–Weisbach (with numeric examples)

If you only remember one relationship, make it this: flow rate equals area times velocity. The continuity equation, Q = A · V, is the starting point for most flow estimates. Compute area from the pipe’s inner diameter, then get velocity from your target volumetric flow. Convert to mass flow with ṁ = ρ·Q when you need pump sizing or mass‑based thermal load. For common pipe sizes, confirm actual ID (not nominal), see Pipe Dimensions: The Essential Guide for quick lookups and table conversions.

Bernoulli’s equation gives the ideal energy balance: p/ρ + V^2/2 + gz = constant. Use it to understand the “big picture” of where energy goes, especially when elevation changes matter. Real systems, however, always lose energy to friction and fittings, so you add head losses to make the result match plant reality. A practical approach is to use Bernoulli to map the layout, then append pressure losses using Darcy, Weisbach.

Reynolds number tells you what kind of flow you’ve: Re = ρVD/μ. Laminar flow is Re < ~2,000, transitional 2,000, 4,000, and turbulent above ~4,000. This matters because the friction factor can change a lot between regimes. In cold‑service lines, viscosity and density can shift with temperature, so don’t assume yesterday’s Reynolds number still applies, use property tables at the actual operating temperature.

Darcy, Weisbach is the workhorse for pressure drop: ΔP = f·(L/D)·(ρ V^2/2). Finding the friction factor f is the recurring task. Common options are summarized here:

When Re is <2,000, use f = 64/Re. For turbulent flow, use Colebrook or an explicit approximation (Swamee, Jain or Haaland). Many engineers run Swamee, Jain for a fast estimate and only iterate further if pump head, pressure ratings, or material limits are tight.

Worked numeric example (LN2): 100 m of 2″ NPS pipe (ID ≈ 0.0525 m), carrying liquid nitrogen at 77 K. Pick V = 2.0 m/s (Q = A·V = 0.002164·2 = 0.00433 m3/s). Using ρ = 808 kg/m3 and μ = 1.6·10−4 Pa·s, Re = ρVD/μ ≈ 5.3·10^5 (turbulent). Assume ε = 4.5·10−5 m (commercial steel). Swamee, Jain gives f ≈ 0.020. Then ρV^2/2 = 808·2^2/2 = 1,616 Pa. L/D = 100/0.0525 = 1,905. So ΔP = 0.020·1,905·1,616 ≈ 60.6 kPa. Add minor losses (valves, tees), e.g. K_total = 2 → extra ≈ 3.2 kPa, total ≈ 63.8 kPa. This is a single-phase LN2 example. Two-phase or flashing changes everything and must be avoided during pipeline isolation and freeze plug operations.

For further depth and classical formulations see Rennels and Hudson’s pipe-flow reference (catalog entry)Rennels and Hudson’s pipe-flow reference (catalog.princeton.edu).

Step-by-step workflow operators should use for reliable flow calculations

A repeatable process beats “hero math” every time. Here is a 7‑step checklist many operators use to keep results consistent and defensible:

Operational checklist (7 steps):

1. Define objective, target flow, allowable pressure drop, or pump head. Be explicit: is the goal to maintain throughput during pipeline isolation or to create a freeze plug for maintenance? That choice changes acceptable ΔP and allowable temperatures.
2. Gather fluid/property data: density, viscosity, vapor pressure at operating temperature. Thermal properties if cryogenic. For liquid nitrogen, this is non‑negotiable because temperature changes can drive viscosity shifts and phase change.
3. Select pipe geometry: nominal size, schedule, and ID (use Pipe Dimensions: The Essential Guide during this step) and list fittings.
4. Assume flow regime and compute V and Reynolds. Pick an initial friction‑factor method.
5. Compute frictional losses (Darcy, Weisbach) and minor losses (K‑factors), then translate results into pump/compressor requirements.
6. Validate against instrumentation and safety protocols, including pressure‑relief ratings and structural limits for thermal contraction.
7. Document assumptions and iterate if pump speed, temperature, or valve positions change.

Fittings can quietly dominate pressure loss in short runs. Use manufacturer K‑factors when you can, or convert to equivalent length Le = K·(D/f). For quick scoping, some teams assume 0.5, 2 m equivalent length per common elbow depending on radius, then refine later. When a line has many fittings or tight bends, model them explicitly instead of lumping everything into one generic K. If you’re planning a freeze plug or deliberate pipe freezing, include thermal stresses from cooling in your material and joint checks.

Iteration isn’t a sign you “did it wrong”, it’s normal. Iterate when pump curves intersect system curves, or when speed‑controlled compressors are involved. Escalate to CFD when you’ve swirling flow, complex manifolding, or any meaningful two‑phase behavior. For cryogenic pipe freezing designs, engineering support is strongly recommended because DIY hazards can include leaks, brittle fractures, and uncontrolled boiling. CryoTech Pipe Freezing provides professional liquid nitrogen freeze‑plug services across the USA and helps reduce downtime, preserve system integrity, and enforce safety protocols.

Finally, run fast error checks before you trust the output. Sanity‑check velocities (typical liquid velocities might be in the 0.5, 3 m/s range for many LN2 lines), verify units at every step, and compare calculated pressure drops to component pressure ratings. If your pressure drop per 100 m comes out as “tens of bars” for a line that normally runs gently, you likely have a unit conversion error. Log assumptions so maintenance teams can repeat the work during future outages and reduce unexpected downtime.

Selecting and calibrating flow measurement for low-temperature and frozen-pipe operations

Meter choice matters more in cold service than many people expect. Coriolis meters measure mass flow directly and stay accurate even when density changes, but they need heated electronics and solid supports to handle contraction and vibration. Ultrasonic transit‑time meters are popular because they can be non‑invasive and have no moving parts, but they need good sensor contact and can struggle with multiphase flow or surface ice. In plain terms: a clamp‑on ultrasonic meter is only as good as its contact with the pipe. Thermal‑mass meters are simple for gases but can be misleading in two‑phase conditions. Differential‑pressure and Pitot methods are still common for rugged service, but they require temperature compensation and careful piping geometry to keep errors under control. For geometry-sensitive effects, see a computational study by ScienceDirect (sciencedirect.com) on how flow profiles change with pipe and sensor geometry.

Reliable installation beats convenient installation. Give meters the straight pipe runs the vendor specifies, and keep inlet and outlet runs clear of elbows, valves, and supports that create uneven flow. Add trace heating or insulated enclosures for transducers and wiring, and route sensor cables away from cold-air pockets where condensation can freeze. Put sensor warm‑up steps and temperature‑compensation tables into the commissioning checklist, and add strain relief so thermal contraction doesn’t shear a fitting or damage wiring.

Calibration and verification are operational controls, not paperwork. Set intervals based on risk and stability: high‑risk cryogenic services every 6 months. Stable liquid lines every 12 months is a common practice. Perform on‑site zero checks before every shift on differential meters, and do daily checks on thermal devices during freeze operations. When possible, validate with a mass‑balance or weigh‑scale test during commissioning to tie meter readings back to actual mass moved, then record results in the maintenance log for traceability.

Freeze work has predictable failure modes. Watch for ice bridging on clamp‑on probes, frozen pressure taps, thermocouple lead fractures, and transmitter drift from repeated thermal cycling. Reduce risk with redundant sensing, duplicate pressure taps, and a contingency plan that includes bypass metering and alarmed rate‑of‑change limits. Strong safety programs also include automated shutdowns for rapid pressure spikes to protect system integrity and reduce unplanned downtime.

Infographic of core fluid flow calculations showing continuity, Bernoulli, Reynolds number and Darcy–Weisbach with a numeric example for operator use.

Quick checklist (meters & calibration)

• Choose meter type: Coriolis for mass accuracy, ultrasonic for non-intrusive installs.
• Install: vendor straight runs, trace heat, secure sensors.
• Cal: zero checks daily/shift, full calibration 6, 12 months.
• Monitor: redundancy, pressure alarms, maintenance logs.

Common operational pitfalls, DIY kit downsides, and integrity checks before freezing

What usually goes wrong first: people assume the pipe will “just get cold” and stay put. In reality, metal shrinks as it cools. Over a long run, that shrinkage can pull on joints and supports. Plan for restraint (so the pipe can’t jump or twist) and leave room for small movements so fittings aren’t forced out of alignment.

Here’s another common surprise: trapped air or gas. As a freeze plug forms, any pocket of vapor can squeeze down and then rebound, creating sudden pressure changes. To prevent that, set clear vent points and confirm you’ve removed trapped volumes before you start. Valve order matters too. If valves are opened and closed in the wrong sequence, you can accidentally create pockets, pull flow backward, or isolate pressure where you didn’t mean to. Write down the exact open/close sequence and do a quick “walk-through” with the crew before the first drop of coolant is used.

DIY pipe-freezing kits can look like a bargain, but the tradeoff is control. Many kits struggle to make a consistent, reliable plug. They also tend to lack engineered pressure monitoring for spikes, and the instructions are often thin. The result is a higher chance of a leak or rupture, longer downtime, and costly repairs. Pointing out DIY risks isn’t alarmism, it’s basic risk management. A failed plug can flood a system, damage equipment downstream, and turn a planned maintenance window into a multi-day outage.

Before any freeze, run integrity checks like you’d before a pressure job. Hydrostatic or pneumatic tests to 1.3, 1.5× MAWP, held for 30 minutes, help confirm the line can handle stress without leaking. If you can, do leak checks at operating temperature (materials can seal differently when hot vs. Cold). Confirm isolation and valve closure with physical verification, not just a drawing or a handle position. Then set explicit venting and relief paths for any trapped vapor.

Wall condition is the quiet deal-breaker. Verify minimum wall thickness and corrosion allowance before attempting a freeze plug. A brittle or heavily corroded pipe may not survive a pressure spike, even if it “looks fine” from the outside.

When fluid flow calculations or overall system behavior are beyond what your on-site team can confidently handle, bring in professionals. Our team at CryoTech Pipe Freezing Services focuses on liquid nitrogen and cryogenic pipe freezing in the USA, with certifications, association memberships, and field experience on major industrial systems. For a practical prep checklist consult Cryogenic Freezing Pipeline Prep as part of the planning process. DIY may save money up front, but it can cost far more in repairs, safety incidents, and unplanned downtime. For critical systems, use engineered freezing with monitored execution.

One honest caveat: even with good planning and monitoring, field conditions can differ from drawings, unknown dead-legs, partially blocked vents, or unreported modifications can change results. Build time into the plan for verification checks before committing to the freeze.

Quick reference: formula cheat-sheet, common conversions and a troubleshooting checklist

A useful fact: most “pipe math” is just three ideas, how much moves, how fast it moves, and what it costs in pressure. The formulas below are standard, and they’re the same ones used in many industry references.

Core formulas (variables defined inline):

• Continuity: Q = A·V, where Q is volumetric flow (m3/s), A = πD2/4 (m2), V is velocity (m/s).
• Bernoulli with losses: P1/γ + V1²/(2g) + z1 = P2/γ + V2²/(2g) + z2 + hf, where γ = ρg, g = 9.81 m/s².
• Darcy, Weisbach (major losses): hf = f·(L/D)·(V²/(2g)), or in flow terms hf = f·(L/D)·(8Q²/(π²D4g)).
• Reynolds: Re = ρVD/μ (use ν = μ/ρ for kinematic viscosity if easier).
• K-factor (local): ΔP = K·½ρV² for valves, tees, entrances.

Quick unit conversions and tips:

To avoid unit mistakes, convert everything to SI before using Darcy, Weisbach. Also confirm whether the pipe size you’ve is inside diameter (ID) or outside diameter (OD). Drawings and field markings don’t always match what people assume. For a quick explanation of common OD/ID mix-ups, see “When does a pipes od actually match its id” as part of your hour‑one checks. If you’re dealing with gas, density can change as pressure changes. The LBL tutorial gives practical guidance on when that matters (a tutorial by LBL).

Troubleshooting checklist (on-site, quick):

• Unrealistic flow numbers: check unit mismatch, wrong D, or blocked tap.
• Unexpected pressure rise during freeze: isolate, don’t continue freeze. Consider staged venting.
• Sensor disagreement: swap to a known-good transducer, verify impulse lines.
• Visual triggers: frost on fittings, condensate, or abnormal valve movement, stop and reassess.

Real-world case study: calculating flow and freeze risk during an LN2 shutdown

Picture a common maintenance window. A plant needs to work on a section of nitrogen line, but a full shutdown would be expensive. The plan is to isolate a short section using an LN2-based pipe freeze (creating a temporary “ice plug” inside the pipe) so the rest of the system can stay online.

Scenario: a plant will isolate a 50 m length of 4‑inch (ID 0.102 m) process line carrying nitrogen gas for maintenance. The plan is an LN2-based pipe freezing operation to create a freeze plug and avoid full pipeline isolation and excessive downtime. We need an LN2 flow calculation and a pressure rise calculation for the trapped volume to decide valve sequencing and relief setpoints.

Step 1, trapped volume and baseline: A = πD²/4 = 0.00817 m², Vtrap = A·L = 0.408 m³ (408 L). Assume initial conditions 6 bar abs and 20 °C (293 K). If trapped gas thermally equilibrates to -50 °C (223 K) during cooling, ideal‑gas gives P2 = P1·(T2/T1). That predicts a pressure drop while cooling. The reverse on warm‑up gives Prise = P1·(293/223) ≈ 1.315·P1, so a 31.5% rise on rewarming. If P1 = 6 bar abs, P after warming ≈ 7.9 bar abs. Gauge increase ~1.9 bar. If relief setpoints or flange ratings are below that, abort or vent.

Step 2, liquid nitrogen carryover and vaporization risk: vaporizing even small masses of LN2 can pressurize the trapped pocket rapidly. Rough check: ideal gas specific volume for N2 ~0.86 m³/kg at 1 atm and 20 °C, so 0.5 kg of vapor is on the order of the trap volume. That means any liquid ingress that vaporizes will spike pressure, hence the need for a trapped‑volume vent path or staged venting before warmup.

Step 3, venting flow and frictional loss check: if you plan to vent at Q = 0.1 m³/s, V = Q/A ≈ 12.2 m/s. Take f ≈ 0.02 (turbulent, commercial steel), L/D ≈ 490. Darcy head hf ≈ 75 m, giving ΔP = ρ·g·hf. At operating density ~7.2 kg/m³ (scaled from 1 atm to 6 bar), ΔP ≈ 5.3 kPa (0.053 bar). Friction losses here are small compared with the thermal pressure swing, but they matter for vent line sizing and relief valve selection.

Decision thresholds and actions:

• If predicted warm-up rise > relief or joint rating, don’t proceed with freeze until you provide staged vents or pressure relief.
• If any LN2 ingress is possible, assume worstcase vaporization and size relief accordingly.
• If sensor array shows divergent pressures >0.2 bar between locations, stop and verify.

Instrumentation and monitoring (minimum):

• Redundant pressure transducers on both sides of freeze, one independent mechanical gauge, and a high‑speed recorder.
• Multiple thermocouples across the freeze zone to detect a migrating cold front.
• Flow meter and LN2 supply rate limiter. The USACE piping manual has practical selection tables for transducer placement and materials (see the USACE engineer manual for guidance) (USACE manual). For advanced diagnostics, consider installing intelligent flow measurement or pressure‑based analytics, as suggested by recent research on measurement systems (a study on intelligent flow measurement from Nature).

Lessons learned and practical adjustments:

• Valve sequencing: open a small vent upstream and downstream before you form the plug. Seal only after vents show nominal pressure.
• Temporary vents: fit a small blowdown line that can be throttled to limit the vent rate and reduce frictional overpressures.
• Redundant monitoring: two independent pressure channels, plus a local mechanical gauge to catch electronic failures. In one job we stopped a freeze when a redundant transducer read 0.6 bar higher than the other. The extra check saved a flange.
• Safety protocols: enforce lockout, thermal dynamics briefings, and emergency relief plans. LN2 introduces cryogenic hazards and rapid thermal contraction. Don’t DIY. A failed freeze plug or unexpected vaporization can damage joints and compromise system integrity. Use professional, certified CryoTech Pipe Freezing services to minimize downtime and protect system integrity, they specialize in liquid nitrogen cryogenic isolation and have the certifications and field experience to run these operations safely.

Final note: start with the simple ideal‑gas check (P·T ratio), then model possible LN2 carryover scenarios. If the numbers suggest more than a small pressure rise, or if you see any sign of liquid getting into the trapped space, stop and call a specialist.

Frequently Asked Questions

What is the first calculation operators should do when planning a pipe freeze?

Start with the trapped volume: how much fluid (gas or liquid) will be sealed in once valves are closed and the plug forms. From there, estimate how pressure changes as temperature changes, then compare the result to allowable limits (relief settings, flange ratings, and equipment limits). In many systems, fluid flow calculations support this step because you need a realistic picture of what can compress, expand, or vaporize during cooling and warm-up.

How does cryogenic temperature affect frictional pressure loss calculations?

Cold temperatures can change density and viscosity, which affects Reynolds number and the friction factor. Use temperature-specific properties and recalculate friction factor with Colebrook or an explicit correlation instead of relying on room-temperature assumptions. Keeping these inputs current makes your head-loss estimates more realistic, especially when the flow may shift between laminar and turbulent behavior.

Are DIY pipe-freezing kits acceptable for critical systems?

A practical rule: if a leak would cause major downtime, safety risk, or equipment damage, DIY kits aren’t the right tool. They can produce uneven plug quality, provide limited monitoring, and leave you exposed if something goes wrong. They’re best reserved for simple, low-risk situations where consequences are minor and the system is easy to verify.

When should operators call an engineering firm or specialist like CryoTech?

Bring in a specialist when predicted pressures or temperatures get close to material limits, when vaporization or two-phase behavior is likely, or when you don’t have reliable instrumentation. Our team can provide engineered freezing, real-time monitoring, and contingency planning to reduce risk. It’s also smart to engage experts for unfamiliar systems, regulatory requirements, or any condition that raises safety or asset-risk concerns.

References

1. “A Tutorial on Pipe Flow Equations” (www-eng.lbl.gov) https://www-eng.lbl.gov/~shuman/NEXT/MATERIALS&COMPONENTS/Pressure_vessels/FM/compressible_pipe_flow.pdf
2. “Pipe flow : a practical and thorough guide / Donald C .” (catalog.princeton.edu) https://catalog.princeton.edu/catalog/99125360389706421
3. “A computational study of the impact of fluid flow .” (sciencedirect.com) https://www.sciencedirect.com/science/article/abs/pii/S0263876224000959
4. “An intelligent flow measurement system based on pressure .” (nature.com) https://www.nature.com/articles/s41598-025-19401-z
5. “Fundamentals of Hydraulics: Flow” (actat.wvu.edu) https://actat.wvu.edu/files/d/267c3249-4d00-4131-9803-12a3bb199b60/fund-of-hyd-flow.pdf
6. “Probabilistic Descriptions of Fluid Flow: A Survey” (link.springer.com) https://link.springer.com/article/10.1007/s00021-023-00800-z
7. “em_1110-1-4008.pdf – USACE Publications – Army.mil” (publications.usace.army.mil) https://www.publications.usace.army.mil/portals/76/publications/engineermanuals/em_1110-1-4008.pdf
8. “Volumetric flow rate” (en.wikipedia.org) https://en.wikipedia.org/wiki/Volumetric_flow_rate
9. “Statistical Analysis of Hydrodynamic Forces in Power-Law .” (frontiersin.org) https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.830408/full
10. “A Study of the Data on the Flow of Fluids in Pipes” (discovery.researcher.life) https://discovery.researcher.life/article/a-study-of-the-data-on-the-flow-of-fluids-in-pipes/47c3453efca730fab17a95a785a84b3c