{
  "version": 1,
  "topic": "DH05",
  "title": "Annular pressure loss through several hole sections",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Two separately solved HB annular intervals",
    "premise": "First interval D/d = 0.2/0.1 m, L = 500 m; second D/d = 0.25/0.1 m with variable length. Q 0.001 m³/s, K 0.2 Pa·s^0.5, n 0.5, τy 10 Pa, ρ 1,200 kg/m³.",
    "formula": "Δptotal = G1 L1 + G2 L2",
    "variables": "Both gradients are separately solved by the supported HB annulus engine. Pressure units are Pa.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Second interval length (m) = 500",
        "explanation": "First interval D/d = 0.2/0.1 m, L = 500 m; second D/d = 0.25/0.1 m with variable length. Q 0.001 m³/s, K 0.2 Pa·s^0.5, n 0.5, τy 10 Pa, ρ 1,200 kg/m³."
      },
      {
        "label": "Select the relation",
        "calculation": "Δptotal = G1 L1 + G2 L2",
        "explanation": "Both gradients are separately solved by the supported HB annulus engine. Pressure units are Pa."
      },
      {
        "label": "Substitute the values",
        "calculation": "First loss at 500 m = 226381.448; second at 500 m = 145796.177; sum = 372177.625 Pa",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "First loss (Pa) = 226381.448; Second loss (Pa) = 145796.177; Total loss (Pa) = 372177.625",
        "explanation": "Serial sections share the same Q but not the same gradient. This is a laminar HB calculation with its stated regime assumptions."
      }
    ],
    "result": "First loss (Pa) = 226381.448; Second loss (Pa) = 145796.177; Total loss (Pa) = 372177.625",
    "interpretation": "Serial sections share the same Q but not the same gradient. This is a laminar HB calculation with its stated regime assumptions.",
    "control": "Second interval length (m)",
    "columns": [
      "Second interval length (m)",
      "First loss (Pa)",
      "Second loss (Pa)",
      "Total loss (Pa)"
    ],
    "rows": [
      [
        0,
        226381.44815682608,
        0,
        226381.44815682608
      ],
      [
        250,
        226381.44815682608,
        72898.08865377077,
        299279.5368105968
      ],
      [
        500,
        226381.44815682608,
        145796.17730754154,
        372177.6254643676
      ],
      [
        750,
        226381.44815682608,
        218694.2659613123,
        445075.7141181384
      ],
      [
        1000,
        226381.44815682608,
        291592.3546150831,
        517973.80277190916
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Second interval length (m) = 0. First loss (Pa): 226381.448; Second loss (Pa): 0; Total loss (Pa): 226381.448.",
      "Second interval length (m) = 250. First loss (Pa): 226381.448; Second loss (Pa): 72898.0887; Total loss (Pa): 299279.537.",
      "Second interval length (m) = 500. First loss (Pa): 226381.448; Second loss (Pa): 145796.177; Total loss (Pa): 372177.625.",
      "Second interval length (m) = 750. First loss (Pa): 226381.448; Second loss (Pa): 218694.266; Total loss (Pa): 445075.714.",
      "Second interval length (m) = 1000. First loss (Pa): 226381.448; Second loss (Pa): 291592.355; Total loss (Pa): 517973.803."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline First loss (Pa) without rounding intermediate values.",
        "answer": "First loss at 500 m = 226381.448; second at 500 m = 145796.177; sum = 372177.625 Pa. First loss (Pa) = 226381.448. Serial sections share the same Q but not the same gradient. This is a laminar HB calculation with its stated regime assumptions."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Second interval length (m) changes from 0 to 1000. Other controls remain as stated in the premise. First loss (Pa) changes from 226381.448 to 226381.448."
      },
      {
        "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. Both gradients are separately solved by the supported HB annulus engine. Pressure units are Pa. Serial sections share the same Q but not the same gradient. This is a laminar HB calculation with its stated regime assumptions."
      }
    ]
  },
  "recorded": {
    "first": {
      "inputs": {
        "D": 0.2,
        "d": 0.1,
        "Q": 0.001,
        "K": 0.2,
        "n": 0.5,
        "tau_y": 10,
        "rho": 1200,
        "L": 500
      },
      "result": {
        "outputs": [
          {
            "label": "Annular friction pressure loss",
            "value": 226381.44815682608,
            "unit": "Pa"
          },
          {
            "label": "Friction pressure gradient",
            "value": 452.76289631365216,
            "unit": "Pa/m"
          },
          {
            "label": "Ideal yield-onset gradient",
            "value": 400,
            "unit": "Pa/m"
          },
          {
            "label": "Inner plug boundary radius",
            "value": 0.052297760072153504,
            "unit": "m"
          },
          {
            "label": "Outer plug boundary radius",
            "value": 0.09647099105648967,
            "unit": "m"
          },
          {
            "label": "Plug / maximum velocity",
            "value": 0.043803963081069326,
            "unit": "m/s"
          },
          {
            "label": "Mean axial velocity",
            "value": 0.042441318157838755,
            "unit": "m/s"
          },
          {
            "label": "Inner-wall shear rate",
            "value": 58.04893612376097,
            "unit": "1/s"
          },
          {
            "label": "Outer-wall shear rate",
            "value": 37.00958592700127,
            "unit": "1/s"
          },
          {
            "label": "Wall-viscosity inertia screening number",
            "value": 25.654808286419534,
            "unit": "dimensionless"
          },
          {
            "label": "Relative flow-rate residual",
            "value": 8.265721440636753e-12,
            "unit": "dimensionless"
          },
          {
            "label": "Relative two-wall velocity residual",
            "value": 3.854829886273189e-12,
            "unit": "dimensionless"
          }
        ],
        "columns": [
          "Radius (m)",
          "Axial velocity (m/s)",
          "Shear-rate magnitude (1/s)",
          "Shear-stress magnitude (Pa)"
        ],
        "rows": [
          [
            0.05,
            0,
            58.04893612376097,
            11.523797048478057
          ],
          [
            0.051250000000000004,
            0.039749764736277386,
            11.68535877065011,
            10.683677080810819
          ],
          [
            0.052500000000000005,
            0.043803963081069326,
            0,
            9.87008773968983
          ],
          [
            0.053750000000000006,
            0.043803963081069326,
            0,
            9.081178051170001
          ],
          [
            0.055,
            0.043803963081069326,
            0,
            8.315265311664895
          ],
          [
            0.05625,
            0.043803963081069326,
            0,
            7.570816391240188
          ],
          [
            0.0575,
            0.043803963081069326,
            0,
            6.846431475608032
          ],
          [
            0.058750000000000004,
            0.043803963081069326,
            0,
            6.140829883611454
          ],
          [
            0.060000000000000005,
            0.043803963081069326,
            0,
            5.452837657523234
          ],
          [
            0.06125,
            0.043803963081069326,
            0,
            4.781376672900005
          ],
          [
            0.0625,
            0.043803963081069326,
            0,
            4.12545505525386
          ],
          [
            0.06375,
            0.043803963081069326,
            0,
            3.484158724174386
          ],
          [
            0.065,
            0.043803963081069326,
            0,
            2.8566439131288894
          ],
          [
            0.06625,
            0.043803963081069326,
            0,
            2.2421305360777155
          ],
          [
            0.0675,
            0.043803963081069326,
            0,
            1.6398962911322834
          ],
          [
            0.06875,
            0.043803963081069326,
            0,
            1.0492714074504677
          ],
          [
            0.07,
            0.043803963081069326,
            0,
            0.46963395496457383
          ],
          [
            0.07125000000000001,
            0.043803963081069326,
            0,
            0.0995943521778165
          ],
          [
            0.07250000000000001,
            0.043803963081069326,
            0,
            0.6589519180461957
          ],
          [
            0.07375000000000001,
            0.043803963081069326,
            0,
            1.2089406447392417
          ],
          [
            0.07500000000000001,
            0.043803963081069326,
            0,
            1.7500289742157205
          ],
          [
            0.07625000000000001,
            0.043803963081069326,
            0,
            2.282654630928904
          ],
          [
            0.07750000000000001,
            0.043803963081069326,
            0,
            2.8072270990447614
          ],
          [
            0.07875000000000001,
            0.043803963081069326,
            0,
            3.324129863734591
          ],
          [
            0.08000000000000002,
            0.043803963081069326,
            0,
            3.83372244234649
          ],
          [
            0.08125,
            0.043803963081069326,
            0,
            4.3363422280840505
          ],
          [
            0.0825,
            0.043803963081069326,
            0,
            4.832306166077934
          ],
          [
            0.08375,
            0.043803963081069326,
            0,
            5.321912279361432
          ],
          [
            0.085,
            0.043803963081069326,
            0,
            5.805441060201182
          ],
          [
            0.08625000000000001,
            0.043803963081069326,
            0,
            6.283156740442564
          ],
          [
            0.0875,
            0.043803963081069326,
            0,
            6.755308452968359
          ],
          [
            0.08875,
            0.043803963081069326,
            0,
            7.222131295005996
          ],
          [
            0.09,
            0.043803963081069326,
            0,
            7.683847302825811
          ],
          [
            0.09125,
            0.043803963081069326,
            0,
            8.140666346327166
          ],
          [
            0.0925,
            0.043803963081069326,
            0,
            8.592786951090542
          ],
          [
            0.09375,
            0.043803963081069326,
            0,
            9.040397054665451
          ],
          [
            0.095,
            0.043803963081069326,
            0,
            9.483674703151445
          ],
          [
            0.09625,
            0.043803963081069326,
            0,
            9.922788693500037
          ],
          [
            0.0975,
            0.04270353255432863,
            3.202295332903184,
            10.357899166408819
          ],
          [
            0.09875,
            0.03192869232546696,
            15.569264807941883,
            10.789158154185634
          ],
          [
            0.1,
            0,
            37.00958592700127,
            11.216710087522928
          ]
        ],
        "chart": {
          "xLabel": "Radius from common axis (m)",
          "yLabel": "Axial velocity (m/s)",
          "series": [
            {
              "name": "Concentric HB profile",
              "x": [
                0.05,
                0.051250000000000004,
                0.052500000000000005,
                0.053750000000000006,
                0.055,
                0.05625,
                0.0575,
                0.058750000000000004,
                0.060000000000000005,
                0.06125,
                0.0625,
                0.06375,
                0.065,
                0.06625,
                0.0675,
                0.06875,
                0.07,
                0.07125000000000001,
                0.07250000000000001,
                0.07375000000000001,
                0.07500000000000001,
                0.07625000000000001,
                0.07750000000000001,
                0.07875000000000001,
                0.08000000000000002,
                0.08125,
                0.0825,
                0.08375,
                0.085,
                0.08625000000000001,
                0.0875,
                0.08875,
                0.09,
                0.09125,
                0.0925,
                0.09375,
                0.095,
                0.09625,
                0.0975,
                0.09875,
                0.1
              ],
              "y": [
                0,
                0.039749764736277386,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.043803963081069326,
                0.04270353255432863,
                0.03192869232546696,
                0
              ]
            }
          ]
        },
        "warnings": [
          "Laminar stationary concentric HB solution only. The inertia screen is not a transition prediction; confirm laminar conditions independently. No hydrostatic, eccentricity, rotation, turbulence or cuttings losses are included."
        ]
      }
    },
    "rows": [
      [
        0,
        226381.44815682608,
        0,
        226381.44815682608
      ],
      [
        250,
        226381.44815682608,
        72898.08865377077,
        299279.5368105968
      ],
      [
        500,
        226381.44815682608,
        145796.17730754154,
        372177.6254643676
      ],
      [
        750,
        226381.44815682608,
        218694.2659613123,
        445075.7141181384
      ],
      [
        1000,
        226381.44815682608,
        291592.3546150831,
        517973.80277190916
      ]
    ]
  }
}