{
  "version": 1,
  "topic": "DH09",
  "title": "When a laminar annular model stops being appropriate",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Concentric Laminar Annular Pressure Loss Calculator",
    "premise": "Fixed controls: Outer bore inside diameter = 0.2 m; Inner pipe outside diameter = 0.1 m; Newtonian dynamic viscosity = 0.1 Pa.s; Liquid density = 1200 kg/m3; Fully developed interval length = 1000 m.",
    "formula": "Ca = Ro^4 - Ri^4 - (Ro^2 - Ri^2)^2 / ln(Ro/Ri); Gp = 8*mu*Q/(pi*Ca); friction loss = Gp*L.; u(r) = Gp/(4*mu) * (Ro^2-r^2-(Ro^2-Ri^2)*ln(Ro/r)/ln(Ro/Ri)); DH = D-d; ReH = rho*mean_velocity*DH/mu; fD = 2*DH*Gp/(rho*mean_velocity^2).",
    "variables": "Actual annular volume flow rate is varied within the supported input range. Single incompressible Newtonian liquid, stationary concentric circular walls and fully developed laminar axial flow only. No eccentricity, pipe rotation, roughness-driven turbulence, washouts, yield stress, shear thinning, cuttings, gas or wall slip is modeled. Do not substitute a Herschel-Bulkley apparent viscosity and call this a non-Newtonian annular solution.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Actual annular volume flow rate (m3/s) = 0.001",
        "explanation": "Fixed controls: Outer bore inside diameter = 0.2 m; Inner pipe outside diameter = 0.1 m; Newtonian dynamic viscosity = 0.1 Pa.s; Liquid density = 1200 kg/m3; Fully developed interval length = 1000 m."
      },
      {
        "label": "Select the relation",
        "calculation": "Ca = Ro^4 - Ri^4 - (Ro^2 - Ri^2)^2 / ln(Ro/Ri); Gp = 8*mu*Q/(pi*Ca); friction loss = Gp*L.; u(r) = Gp/(4*mu) * (Ro^2-r^2-(Ro^2-Ri^2)*ln(Ro/r)/ln(Ro/Ri)); DH = D-d; ReH = rho*mean_velocity*DH/mu; fD = 2*DH*Gp/(rho*mean_velocity^2).",
        "explanation": "Actual annular volume flow rate is varied within the supported input range. Single incompressible Newtonian liquid, stationary concentric circular walls and fully developed laminar axial flow only. No eccentricity, pipe rotation, roughness-driven turbulence, washouts, yield stress, shear thinning, cuttings, gas or wall slip is modeled. Do not substitute a Herschel-Bulkley apparent viscosity and call this a non-Newtonian annular solution."
      },
      {
        "label": "Substitute the values",
        "calculation": "Q = 0.001 m3/s; with the fixed controls above, the workflow evaluates Annular friction pressure loss = 20212.7119 Pa; Hydraulic-diameter Reynolds number = 50.9295818 dimensionless",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Annular friction pressure loss (Pa) = 20212.7119; Hydraulic-diameter Reynolds number (dimensionless) = 50.9295818",
        "explanation": "The pressure output excludes hydrostatic elevation change, entrance/exit losses, fittings and changing geometry. Add appropriate separate components only with consistent signs and fluid properties. It is not total standpipe pressure, an ECD prediction or evidence of adequate hole cleaning. Positive inputs and the supported radius-ratio/Reynolds envelopes are enforced. Density affects the Reynolds and Darcy diagnostics, but not the exact viscous gradient at fixed rate and viscosity. Numerical agreement with the analytical solution does not verify a field fluid's Newtonian behavior or installation geometry."
      }
    ],
    "result": "Annular friction pressure loss (Pa) = 20212.7119; Hydraulic-diameter Reynolds number (dimensionless) = 50.9295818",
    "interpretation": "The pressure output excludes hydrostatic elevation change, entrance/exit losses, fittings and changing geometry. Add appropriate separate components only with consistent signs and fluid properties. It is not total standpipe pressure, an ECD prediction or evidence of adequate hole cleaning. Positive inputs and the supported radius-ratio/Reynolds envelopes are enforced. Density affects the Reynolds and Darcy diagnostics, but not the exact viscous gradient at fixed rate and viscosity. Numerical agreement with the analytical solution does not verify a field fluid's Newtonian behavior or installation geometry.",
    "control": "Actual annular volume flow rate (m3/s)",
    "columns": [
      "Actual annular volume flow rate (m3/s)",
      "Annular friction pressure loss (Pa)",
      "Hydraulic-diameter Reynolds number (dimensionless)"
    ],
    "rows": [
      [
        0.0002,
        4042.542372156868,
        10.1859163578813
      ],
      [
        0.0005,
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        25.464790894703253
      ],
      [
        0.001,
        20212.71186078434,
        50.929581789406505
      ],
      [
        0.002,
        40425.42372156868,
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      ],
      [
        0.005,
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      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Actual annular volume flow rate (m3/s) = 0.0002. Annular friction pressure loss (Pa): 4042.54237; Hydraulic-diameter Reynolds number (dimensionless): 10.1859164.",
      "Actual annular volume flow rate (m3/s) = 0.0005. Annular friction pressure loss (Pa): 10106.3559; Hydraulic-diameter Reynolds number (dimensionless): 25.4647909.",
      "Actual annular volume flow rate (m3/s) = 0.001. Annular friction pressure loss (Pa): 20212.7119; Hydraulic-diameter Reynolds number (dimensionless): 50.9295818.",
      "Actual annular volume flow rate (m3/s) = 0.002. Annular friction pressure loss (Pa): 40425.4237; Hydraulic-diameter Reynolds number (dimensionless): 101.859164.",
      "Actual annular volume flow rate (m3/s) = 0.005. Annular friction pressure loss (Pa): 101063.559; Hydraulic-diameter Reynolds number (dimensionless): 254.647909."
    ],
    "provenance": "Original recorded synthetic cases solved with the site's laminar-annular-pressure-loss engine. The full inputs, outputs and diagnostic rows are in the downloadable JSON. These are discrete solved states; the lesson does not interpolate a new solve.",
    "questions": [
      {
        "question": "Reproduce the baseline Annular friction pressure loss (Pa) without rounding intermediate values.",
        "answer": "Q = 0.001 m3/s; with the fixed controls above, the workflow evaluates Annular friction pressure loss = 20212.7119 Pa; Hydraulic-diameter Reynolds number = 50.9295818 dimensionless. Annular friction pressure loss (Pa) = 20212.7119. The pressure output excludes hydrostatic elevation change, entrance/exit losses, fittings and changing geometry. Add appropriate separate components only with consistent signs and fluid properties. It is not total standpipe pressure, an ECD prediction or evidence of adequate hole cleaning. Positive inputs and the supported radius-ratio/Reynolds envelopes are enforced. Density affects the Reynolds and Darcy diagnostics, but not the exact viscous gradient at fixed rate and viscosity. Numerical agreement with the analytical solution does not verify a field fluid's Newtonian behavior or installation geometry."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Actual annular volume flow rate (m3/s) changes from 0.0002 to 0.005. Other controls remain as stated in the premise. Annular friction pressure loss (Pa) changes from 4042.54237 to 101063.559."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. Actual annular volume flow rate is varied within the supported input range. Single incompressible Newtonian liquid, stationary concentric circular walls and fully developed laminar axial flow only. No eccentricity, pipe rotation, roughness-driven turbulence, washouts, yield stress, shear thinning, cuttings, gas or wall slip is modeled. Do not substitute a Herschel-Bulkley apparent viscosity and call this a non-Newtonian annular solution. The pressure output excludes hydrostatic elevation change, entrance/exit losses, fittings and changing geometry. Add appropriate separate components only with consistent signs and fluid properties. It is not total standpipe pressure, an ECD prediction or evidence of adequate hole cleaning. Positive inputs and the supported radius-ratio/Reynolds envelopes are enforced. Density affects the Reynolds and Darcy diagnostics, but not the exact viscous gradient at fixed rate and viscosity. Numerical agreement with the analytical solution does not verify a field fluid's Newtonian behavior or installation geometry."
      }
    ]
  },
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        "result": {
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              "unit": "Pa"
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          "d": 0.1,
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          "mu": 0.1,
          "rho": 1200,
          "L": 1000
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        "result": {
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            {
              "label": "Annular friction pressure loss",
              "value": 20212.71186078434,
              "unit": "Pa"
            },
            {
              "label": "Friction pressure gradient",
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