Turn a slurry pipeline spreadsheet into a calculator: velocity and head loss

Your slurry spreadsheet takes a pipe size, a flow, and the solids, and tells you whether the line will deposit and what head the pump needs. The project team wants that on a phone or tablet, for the client meeting or the site, without the workbook and its hidden columns. The page itself is one HTML file. The care goes into the correlations: they differ by a third or more on the same inputs, so the calculator has to say which ones it uses and where they hold.

This guide covers the mixture properties, the deposition velocity by Durand's equation and the other correlations, the carrier's friction loss, two named solids head models, and the pump's head and efficiency ratios as inputs. It ends with a test table and the prompt for your AI. The equations are as given in Sape Miedema's open textbook Slurry Transport (TU Delft, on LibreTexts), read on November 6, 2026, with the original papers named. Every number was computed with our own script and checked against a published worked value where one exists. It covers settling slurries in water-like carriers in horizontal pipe, not non-Newtonian slurries or pastes.

What the calculator computes

Losses are kept as a hydraulic gradient i, in metres of carrier liquid per metre of pipe: i = Δp / (ρ_l·g·L) (Miedema 6.4.2). The pump's head is in metres of slurry, so the page converts at the end.

Mixture properties: C_v, C_w, and density

With solids density ρ_s, carrier density ρ_l, and volume concentration C_v (Miedema 2.4.2 to 2.4.4):

ρ_m  = C_v·ρ_s + (1 − C_v)·ρ_l          mixture density, kg/m³
C_v  = (ρ_m − ρ_l) / (ρ_s − ρ_l)          from a measured mixture density
C_w  = C_v·ρ_s / ρ_m                      solids fraction by mass
C_v  = (C_w/ρ_s) / (C_w/ρ_s + (1 − C_w)/ρ_l)   from a mass fraction
R_sd = (ρ_s − ρ_l) / ρ_l = S − 1          relative submerged density; S = ρ_s/ρ_l

For sand (ρ_s = 2,650 kg/m³) in water (1,000 kg/m³), C_v = 0.15 gives C_w = 0.3186 and ρ_m = 1,247.5 kg/m³; C_w = 0.30 gives C_v = 0.1392 and ρ_m = 1,229.7 kg/m³. Miedema distinguishes the spatial concentration C_vs, the solids volume in a length of pipe, from the delivered concentration C_vt = Q_s/Q_m. Above the deposition velocity, in the heterogeneous regime, "there will not be much difference between the spatial and the transport concentration" (6.4.2). The head models below use the delivered concentration C_vt; Miedema writes the Wasp and Oroskar and Turian deposition correlations with C_vs. Above V_D the two are close, and the page takes one C_v for all of them. A mixture density measured in the pipe gives the spatial one (the density formula above uses C_vs), so label which one the sheet's input is.

Deposition velocity by Durand's equation

Durand and Condolios (1952) and Durand (1953) give the limit deposit velocity, the line speed below which a bed forms, as (Miedema 6.27.3):

V_D = F_L · √(2 · g · D · (S − 1))

with V_D in m/s, g = 9.81 m/s², D the pipe's inside diameter in m, and S the solids' density over the carrier's. All the solids' behaviour sits in F_L, which Durand read from a chart of particle size and concentration. The calculator takes F_L as an input from the chart your sheet uses.

Two traps make the same F_L give different velocities:

• The 1952 and 1953 charts differ by √R_sd. Miedema (6.27.3) puts the peak of Durand's 1953 chart, the one most textbooks copy, at "about 1.55 at a particle size of d=0.5 mm," and its value for very large particles at "about 1.34." Durand and Condolios' 1952 chart is lower by the square root of R_sd, "a factor of about 1.28." Note which chart your F_L came from.
• Some texts drop the 2. F_L is defined with √(2·g·D·R_sd) in the equation above, and with √(g·D·R_sd) in others, a factor of 1.414. Miedema's own text shows both: section 6.4.2.2 recommends "an asymptotic value of about 2.1," "leading to a Limit Deposit Velocity of about 3.3 m/s in a 0.1524 m (6 inch) pipe," which is 2.1·√(g·D·R_sd); with the 2, the same F_L gives 4.66 m/s.

Checks from our script, with sand (R_sd = 1.65):

• pipingtoolset.com's example. A 12.0 in. bore, F_L = 1.34, solids SG 2.65 in water, 5,000 gpm: V_D = 13.807 ft/s and V = 14.184 ft/s, a ratio of 1.027, as its page shows. At 4,000 gpm the ratio drops to 0.82.
• Durand and Condolios' own range. F_L = 1.34 in a 0.1524 m pipe gives 2.98 m/s, inside the 2.9 to 3.2 m/s that Miedema (6.4.5.6) says Durand and Condolios reported for that pipe.

Durand and Condolios' tests covered "a d50 between 0.18 mm and 22.5 mm in pipes with a diameter Dp from 40 to 700 mm and volumetric concentrations Cvt from 2% to 22%" (Miedema 6.4.2). Have the page flag inputs outside that range.

The other deposition correlations, side by side

No single correlation fits every slurry, so engineers often compute more than one. KASA Redberg's guide to slurry correlations says: "Wherever practical, an important slurry pipeline should be assessed using more than one suitable method," and "A deposit-velocity method should be compared with another deposit-velocity method." These are the ones the AI answers and the references name:

On our worked example below (a 200 mm pipe, sand with d50 = 0.5 mm, C_v = 0.15), Durand with F_L = 1.34 gives 3.41 m/s, Oroskar and Turian 2.73 m/s, and Wasp 2.57 m/s: a spread of a third. PNNL's report on Hanford slurries, citing Turian et al. (1987), says of the theoretical form of Oroskar and Turian that "Approximately 49% of the critical-velocity data have a deviation greater than 20% of the predicted value," and of the empirical forms, the kind this page computes, that "38% of the data have a deviation above 20%."

The margin above V_D is a design choice to enter, not a constant. Three published examples: KASA writes "If in doubt, a typical recommended margin is the greater of 10% or 0.3 m/s above the calculated deposit velocity"; pipingtoolset.com says V_D "is the floor, not the target" and "Common practice sits somewhere in the 1.1–1.5 ratio band"; the Hanford design guide that PNNL describes applies "a minimum design margin of 30%."

The carrier's friction loss

The carrier's gradient is Darcy-Weisbach's, as in Miedema 8.2:

Re  = V·D / ν
f   = 0.25 / [log10(ε/(3.7·D) + 5.74/Re^0.9)]²     Swamee and Jain (1976)
i_l = f·V² / (2·g·D)                              m of liquid per m of pipe
ν   = 497×10⁻⁶ / (42.5 + T)^1.5                   water, m²/s, T in °C (Miedema 2.4.5)

Miedema writes that "Over the whole range of Reynolds numbers above 2320 the Swamee Jain (1976) equation gives a good approximation for the Darcy Weisbach friction factor." His text prints 5.75; the form usually quoted has 5.74, and the difference is under 0.1% in f. A review of Colebrook approximations by Genić and others (FME Transactions, 2011, Table 1) lists Swamee and Jain (1976) for Re from 5,000 to 10⁸ and relative roughness from 0.000001 to 0.05; other secondary sources give 10⁻² as the upper roughness. A slurry line, at Re around 10⁵ to 10⁶ and ε/D near 10⁻⁴, sits well inside either. The ν(T) fit has no stated range; against Miedema's own table of water viscosity from 0 to 100 °C (Table 2.4-1) it is within 1.1%. If your sheet iterates Colebrook-White instead, have the page do the same, with a stated tolerance and an iteration cap. ν at 20 °C comes out 1.006×10⁻⁶ m²/s.

The solids' extra head: two named models

The solids add to the carrier's gradient. The two models below are both in Miedema's chapter 6 and give different answers on the same inputs, so the page should name the one it uses, and better, show both.

Durand-Condolios-Gibert (Miedema 6.4.2, equation 6.4-10):

i_m = i_l · (1 + Φ · C_vt)
Φ   = K · Ψ^(−3/2)
Ψ   = (V² / (g·D·R_sd)) · √C_x
C_x = g·d / v_t²                     Durand's virtual drag coefficient
v_t = (10·ν/d)·(√(1 + R_sd·g·d³/(100·ν²)) − 1)    Ruby and Zanke (1977), m and m/s

• K depends on the form. With R_sd inside Ψ, as here, K is about 83 (Durand and Condolios) or 85 (Gibert, 1960; Condolios and Chapus, 1963). Without R_sd, the same data give 176 and 180. Miedema: "The value of 180 is found in most publications after 1960." 180 in the form with R_sd overstates the solids term by 1.65^1.5 = 2.12, and on our worked example raises i_m by 65%.
• C_x is not the drag coefficient. Miedema (6.4.5.1) warns that Durand's particle Froude number "is mixed up in many text books" with the drag coefficient C_D, and the two differ by a factor 4·R_sd·ψ/3, ψ being the shape factor: about 1.1 for sand grains with a shape factor of 0.5, and 2.2 for spheres.
• Range. "310 test points cover a broad range of pipe diameters (Dp=40 to 580 mm), particle diameters (d50=0.2 to 25 mm) and concentrations (Cvt=2% to 22.5%)" (6.4.2).

Wilson-GIW (1992), heterogeneous, with the simplified v50 (Miedema 6.20.2, in the Wilson-GIW chapter 6.17):

i_m = i_l + 0.22 · R_sd · C_v · (v50 / V)^M
v50 ≈ 3.93 · (d50 / 0.001)^0.35 · (R_sd/1.65)^0.45 · (ν/ν_20°C)^(−0.25)    d50 in m, v50 in m/s
M   ≈ 1 / ln(d85 / d50),  between 0.25 and 1.7 (1.7 for narrowly graded sand)

• The 0.22 is half a sliding friction coefficient of 0.44, at the line speed v50 where half the solids ride as a bed.
• Range. Miedema: "The model gives good results if the physics on which it is based occur, medium sized particles." It was developed "for operational conditions, which is above the LDV," and is invalid for very large particles, which are "almost always" more than half stratified.
• Check. The simplified v50 gives 3.93 m/s for 1 mm sand in 20 °C water, as Miedema states.

The forms differ, too: Durand's solids term multiplies i_l, so it rises with the carrier's friction factor; Wilson's is added to i_l and doesn't (Miedema 6.20.2.5). On the worked example, Durand-Condolios-Gibert gives i_m = 0.1358 and Wilson-GIW 0.0946: 43% apart. Both are for horizontal pipe; inclined runs need the corrections in Miedema 6.4.2.1.

Pump head and power, with the ratios as inputs

The page turns the gradient into head in metres of slurry, adds the lift, and converts to the water head you read on the pump's water curve:

H_m   = i_m · L · ρ_l/ρ_m + Δz                 m of slurry (friction plus static lift)
H_w   = H_m / HR                               m of water, for the pump's water curve
P     = ρ_m · g · Q · H_m / (η_w · ER)         W

HR and ER are the head ratio and efficiency ratio. US patent 6,033,187 (Graeme R. Addie, assigned to GIW Industries, 2000) defines them: "The head ratio H_r and efficiency ratio η_r are defined as H_m /H_w and η_m /η_w, respectively." Take both from the pump maker for your slurry. The Hydraulic Institute's standard ANSI/HI 12.1–12.6 for rotodynamic centrifugal slurry pumps defines the head ratio (12.2.9.24) and gives a derating method (12.3.2.7); the standard isn't free to read, so this page doesn't reproduce it. Leave them as inputs on the page, not constants in the code. Add minor losses if your sheet has them.

The worked example

Inputs, in the units of the prompt's fields below: sand, ρ_s = 2,650 kg/m³, in water, ρ_l = 1,000 kg/m³, at T = 20 °C; d50 = 0.5 mm, d85 = 0.8 mm; D = 200 mm, ε = 0.045 mm; Q = 0.12 m³/s; C_v = 15%; F_L = 1.34 (an example chart value); K = 85; L = 1,000 m; Δz = 10 m; HR = 0.90, ER = 0.90, η_w = 75% (example inputs, not typical values). Our script gives:

• Mixture: C_w = 0.3186, ρ_m = 1,247.5 kg/m³, R_sd = 1.65.
• Flow: V = 3.820 m/s, Re = 759,500, f = 0.01528, i_l = 0.05680.
• Deposition: Durand 3.410 m/s (V/V_D = 1.12), Oroskar and Turian 2.728 m/s, Wasp 2.566 m/s.
• Durand-Condolios-Gibert: v_t = 0.0721 m/s, C_x = 0.944, Ψ = 4.380, Φ = 9.273, i_m = 0.1358. Over 1,000 m that's 1,332 kPa, or 108.9 m of slurry; H_m = 118.9 m, H_w = 132.1 m, P = 258.6 kW.
• Wilson-GIW: v50 = 3.083 m/s, M = 1.70 (capped), i_m = 0.0946; H_m = 85.9 m, H_w = 95.4 m, P = 186.8 kW.
• Water alone: 56.8 m over the same 1,000 m.

The prompt for ChatGPT, Claude, or Gemini

Paste your sheet's formulas with each cell's name and unit (in Excel, Formulas > Show Formulas), the F_L chart values you use as a table, and the test table below. Then add:

Write one self-contained HTML file: a settling-slurry pipeline calculator my project team opens on a phone or tablet. Use the formulas and tables I pasted.

Rules
- Plain JavaScript functions only. No eval, no new Function, no formula-parsing library, no fetch or other network request. Write every table (pipe sizes, roughness, and my F_L chart values as a reference table) into the file.
- Use Math.pow or ** for powers, never ^. Use Math.log10 where a formula says log10 (Excel's LOG with no base is base 10; JavaScript's Math.log is natural).
- SI inputs, each field labelled with its unit: D (mm), Q (m³/s), rho_s and rho_l (kg/m³), a switch for C_v or C_w (%), d50 and d85 (mm), roughness eps (mm), water temperature T (°C), F_L (a field, prefilled from the chart table and editable; the test cases set it), K, pipe length L (m), static lift dz (m), head ratio HR and efficiency ratio ER (such as 0.90), and the pump's efficiency on water eta_w (%). Convert mm to m and % to a fraction once, at input. g = 9.81 m/s².
- Equations, in this order (d is d50 in m):
  nu = 497e-6 / (42.5 + T)^1.5; R_sd = (rho_s - rho_l)/rho_l
  C_v from C_w = (C_w/rho_s) / (C_w/rho_s + (1 - C_w)/rho_l); rho_m = C_v rho_s + (1 - C_v) rho_l; C_w = C_v rho_s / rho_m
  V = Q / (pi D²/4); Re = V D / nu; f = 0.25 / (log10(eps/(3.7 D) + 5.74/Re^0.9))²; i_l = f V² / (2 g D)
  Durand V_D = F_L sqrt(2 g D R_sd)
  Wasp V_D = 4 C_v^(1/5) (d/D)^(1/6) sqrt(2 g D R_sd)
  Oroskar-Turian V_D = 1.85 C_v^0.1536 (1 - C_v)^0.3564 (D/d)^0.378 (D sqrt(g d R_sd)/nu)^0.09 sqrt(g d R_sd)
  v_t = (10 nu/d) (sqrt(1 + R_sd g d³/(100 nu²)) - 1); C_x = g d / v_t²; Psi = V²/(g D R_sd) sqrt(C_x); Phi = K Psi^(-1.5)
  Durand-Condolios-Gibert i_m = i_l (1 + Phi C_v)
  v50 = 3.93 (d/0.001)^0.35 (R_sd/1.65)^0.45 (nu/nu20)^(-0.25), nu20 = nu at 20 °C; M = 1/ln(d85/d50), kept between 0.25 and 1.7
  Wilson-GIW i_m = i_l + 0.22 R_sd C_v (v50/V)^M
  For each of the two i_m: H_m = i_m L rho_l/rho_m + dz; H_w = H_m/HR; P = rho_m g Q H_m / (eta_w ER)
- Outputs, each with its unit: nu, R_sd, C_v, C_w, rho_m, V, Re, f, i_l; V_D by each of the three correlations and V / V_D for each; v_t, C_x, Psi, Phi and i_m by Durand-Condolios-Gibert; v50, M and i_m by Wilson-GIW; H_m, H_w and P for each of the two head models.
- Name the correlation on every result. Flag inputs outside Durand and Condolios' tested range (d50 0.18 to 22.5 mm, D 40 to 700 mm, C_v 2 to 22%), d50 below 0.1 mm for Oroskar and Turian, and any V below a deposition velocity.
- Phone layout: one column, inputs first, then a results card. Put the version number, the date, and the list of correlations used in the footer.
- A "Run the test cases" button at the bottom of the page: it runs my test cases and shows expected, computed, and the difference in percent for each.

The page computes in the phone's browser from what's in the file. On Convika, a script's fetch and XHR requests reach only the page's own address, plus, on Pro, the analytics tools Convika supports and the hosts you list for your own ad network, so the chart values have to be in the file. Scripts from a CDN, such as a charting library, do run; eval and new Function don't.

The test table

All cases use the worked example's inputs, in the prompt's units, unless the row says otherwise. Expected values come from a script that follows the prompt's equations step by step; compare to 4 significant figures, and add cases from jobs you have already checked.

The page should flag three of them: case 3 is outside Durand and Condolios' tested concentrations, V in case 4 is below Durand's V_D, and V in case 8 is below Durand's and Oroskar and Turian's. Case 4 is the one to watch: Wasp and Oroskar and Turian say the line runs, and Durand with this F_L says it deposits. Wilson-GIW was developed for line speeds above the deposition velocity, so in cases 4 and 8 its head is a number to flag, not to use.

Where a sheet and a page drift apart

• LOG and Math.log. Microsoft's LOG help: "If base is omitted, it is assumed to be 10." JavaScript's Math.log is natural, as MDN puts it: "returns the natural logarithm (base e) of a number." A Swamee-Jain line copied with Math.log gives a friction factor 5.3 times too small (ln 10 squared).
• ^ is not a power in JavaScript. It's bitwise XOR, which "converts both operands to 32-bit integers" (MDN), so C_v ^ 0.2 returns 0 for any concentration below 1. The prompt rules it out.
• Units of d. Ruby and Zanke's settling velocity, Oroskar and Turian, and v50 take d in metres, and Wasp needs d and D in the same unit. A sheet that stores mm and divides by 1,000 in one cell but not another gives a v_t off by orders of magnitude.
• Percent cells. Excel stores a cell showing 15% as 0.15. A page field labelled % must divide by 100 once.
• Chart lookups. If your F_L comes from a chart in the sheet, the page needs the same table and the same interpolation (linear in what: d, log d, or C_v?), or its F_L differs from the sheet's.
• Goal Seek for pipe size or flow. Microsoft lists Goal Seek for desktop Excel only. In a given pipe the three deposition velocities don't depend on Q, so the lowest flow that keeps your margin needs no solver: Q = margin × V_D × π·D²/4. Finding the pipe size for a flow does need one: have the page solve it by bisection with a stated tolerance and step limit.

Publishing it for a project team

• Who can open it. A slurry sheet often holds a client's design data. When you publish, choose Only people with the password: Convika's server checks the password before it sends the page, and each device remembers it for 7 days. Private link ("Anyone with the link can open it, with no password or account. It stays out of search results and out of the gallery.") keeps it out of search but not away from anyone the link is forwarded to. Both are Pro, at $12 a month. Public is free, with the Convika bar, and "may show up in search results and in Convika's gallery." Anyone who opens the page can read its source, including every table in it.
• One link for every revision. When an F_L table or a pump changes, upload the corrected file as a new version of the same page and click Publish v2 at the same link (with your version number). The team's link stays the same. Keep the version number, date, and correlation list in the footer, so a screenshot in a meeting says what produced it. Updating a page at the same link covers versions and rollback.
• More than one calculator. On Free, 1 page can be live at a time; Pro keeps up to 100.
• What Convika doesn't do. It doesn't limit a page to your company's email addresses or named accounts, and it doesn't make the page work offline. Everyone with the password opens the same page.

You stay responsible for checking the page's results against your own design standard and the pump maker's data.

Put your slurry sheet on the project team's phones in 5 steps

1. Collect the formulas, the tables, and the test cases

Show the sheet's formulas, list each input and output with its unit, and note which correlation each result comes from. Export the F_L chart values as a table. Run the 8 cases above through the sheet.

2. Ask the AI for one HTML file

Give ChatGPT, Claude, or Gemini the formulas, the tables, the test cases, and the prompt above, and ask for the whole file in one reply.

3. Run the test table

Open the file in the free HTML viewer, which runs its scripts in your browser at phone width without uploading it. Press the page's "Run the test cases" button, look at the intermediate values for any case that's off (C_v, V, f, Ψ), and fix the formula that differs.

4. Upload it and publish behind a password

Drop the file into the box below and click Get my preview link. That preview runs no scripts, so the calculator shows no results there; don't test it there. Click Keep this page — free, no card and sign up: the page moves to your dashboard, and its scripts run when you open it from there. Click Publish page, choose Only people with the password, type a password, and click Next. The first time, that opens an order for Pro: click Subscribe, free for 1 day and add your card, and the page goes live behind the password once the trial starts.

5. Send the link and the password separately

Send the link in one message and the password in another. For each revision, click Upload a new version under Update from an HTML file, run the test cases in the new version's preview, and click Publish v2 at the same link.

Frequently asked questions

How do I calculate the critical deposition velocity of a slurry?

With Durand's equation, V_D = F_L·√(2·g·D·(S − 1)), where D is the pipe's inside diameter, S the solids' density over the carrier's, and F_L a factor read from Durand's chart for your particle size and concentration. For F_L = 1.34, sand, and a 0.1524 m pipe, V_D is 2.98 m/s. Run a second correlation beside it, such as Oroskar and Turian or Wasp et al., because they can differ by a third.

What is the Durand equation for slurry flow?

There are two. Durand's deposition velocity is V_D = F_L·√(2gD(S − 1)). The Durand-Condolios head loss is i_m = i_l·(1 + Φ·C_v), with Φ = K·[(V²/(g·D·(S − 1)))·√C_x]^(−3/2) and K about 83 to 85 in that form, where C_x = g·d/v_t². Both come from Durand and Condolios' 1952 tests on sand and gravel.

What F_L value should I use?

The one from the chart or correlation your design standard names, for your particle size and concentration. Durand's 1953 chart peaks near 1.55 at about 0.5 mm and levels off near 1.34 for large particles, and Durand and Condolios' 1952 chart is lower by about 1.28. Check whether your source defines F_L with √(2gD(S − 1)) or √(gD(S − 1)): they differ by 1.414.

How do I calculate head loss in a slurry pipeline?

Compute the carrier's gradient with Darcy-Weisbach, i_l = f·V²/(2gD), then add the solids' effect with a named model: Durand-Condolios-Gibert, i_m = i_l·(1 + Φ·C_v), or Wilson-GIW, i_m = i_l + 0.22·(S − 1)·C_v·(v50/V)^M. Multiply by the length for metres of carrier liquid and convert to metres of slurry with ρ_l/ρ_m. On the same inputs the two models can be 40% apart.

How do I convert weight concentration to volume concentration?

C_v = (C_w/ρ_s) / (C_w/ρ_s + (1 − C_w)/ρ_l). For sand at 2,650 kg/m³ in water, C_w = 30% is C_v = 13.9%, and the mixture density is 1,229.7 kg/m³.

How do I size a slurry pipeline?

Pick the diameter that keeps the line velocity above the deposition velocity, with your margin, at the lowest flow you'll run, then check the head the pump needs at the highest. A smaller pipe raises the velocity but also the friction. A calculator page can step through standard pipe sizes and show both for each.

Can I build a slurry calculator from my Excel spreadsheet?

Yes. Give ChatGPT, Claude, or Gemini the sheet's formulas with their units, your F_L table, and test cases, and ask for one HTML file with plain JavaScript functions. Check it against the sheet case by case, then publish it as a link, behind a password if it holds a client's data.

Do slurry pump head and efficiency drop with solids?

Yes, and the head ratio HR = H_m/H_w and efficiency ratio ER = η_m/η_w express it. Take both from the pump maker for your slurry, or from the Hydraulic Institute's ANSI/HI 12.1–12.6 method, and enter them in the calculator. H_w = H_m/HR is the head to read on the pump's water curve.

Sources

• Sape A. Miedema, Slurry Transport: Fundamentals, a Historical Overview and the Delft Head Loss & Limit Deposit Velocity Framework (TU Delft Open Textbooks), on LibreTexts: 2.4 Other Important Parameters, 4.4 Terminal Settling Velocity Equations, 6.4 The Durand and Condolios (1952) School, 6.17 The Wilson-GIW (1979) Models, 6.27 The Limit Deposit Velocity, and 8.2 Default Equations Used In This Book. Read November 6, 2026.
• A. P. Poloski et al., Deposition Velocities of Newtonian and Non-Newtonian Slurries in Pipelines (PNNL-17639, March 2009): Tables 3.1 and 3.3, the design-guide steps in section 3, and section 9. Read November 6, 2026.
• S. Genić et al., A Review of Explicit Approximations of Colebrook's Equation (FME Transactions 39, 2011, pp. 67–71): Table 1. Read November 6, 2026.
• KASA Redberg, Slurry Pipeline Calculations: Which Correlation Should You Use? (dated August 3, 2026). Read November 6, 2026.
• pipingtoolset.com, Slurry Deposition Velocity Calculator, Durand Method. Read November 6, 2026.
• Graeme R. Addie, US 6,033,187, Method for controlling slurry pump performance to increase system operational stability (2000). Read November 6, 2026.
• Hydraulic Institute, ANSI/HI 12.1–12.6-2021 table of contents. Read November 6, 2026.
• Microsoft, LOG function and Use Goal Seek; MDN, Math.log() and Bitwise XOR (^). Read November 6, 2026.