{ "version": 1, "topic": "FA03", "title": "Hydrostatic frictional and acceleration pressure terms", "createdAt": "2026-10-24", "case": { "title": "Homogeneous Gas-Liquid Pressure Profile Calculator", "premise": "Fixed controls: Inlet absolute pressure = 5 MPa; Constant absolute temperature = 320 K; Pipe inside diameter = 0.1 m; Length along flow = 200 m; Gas mass flow rate = 0.05 kg/s; Liquid mass flow rate = 5 kg/s; Constant liquid density = 850 kg/m3; Gas molecular mass = 20 kg/kmol; Constant gas Z-factor = 0.9 dimensionless; User-supplied constant Darcy factor = 0.02 dimensionless.", "formula": "x = mg/(mg+ml); G = (mg+ml)/area; cg = Z*R*T/M; rho_g = p/cg; v = (1-x)/rho_l + x*cg/p; rho_m = 1/v; u = G*v. R = 8314.46261815324 J/(kmol.K); p in Pa.; ap = G^2*x*cg/p^2; F = fD*G^2/(2*D*rho_m); H = rho_m*g*sin(theta); -dp/ds = (F+H)/(1-ap); acceleration loss = G^2*x*cg*(1/pout-1/pin). g = 9.80665 m/s2.", "variables": "Uphill angle from horizontal is varied within the supported input range. This is explicitly a homogeneous no-slip isothermal screening model, not Beggs-Brill, Hagedorn-Brown, drift-flux, a mechanistic flow-regime solver or three-phase PVT. It does not model slugging, slip, actual liquid holdup, gas liberation/dissolution, phase change, heat transfer, compositional properties or flow assurance.", "steps": [ { "label": "Define the baseline", "calculation": "Uphill angle from horizontal (deg) = 10", "explanation": "Fixed controls: Inlet absolute pressure = 5 MPa; Constant absolute temperature = 320 K; Pipe inside diameter = 0.1 m; Length along flow = 200 m; Gas mass flow rate = 0.05 kg/s; Liquid mass flow rate = 5 kg/s; Constant liquid density = 850 kg/m3; Gas molecular mass = 20 kg/kmol; Constant gas Z-factor = 0.9 dimensionless; User-supplied constant Darcy factor = 0.02 dimensionless." }, { "label": "Select the relation", "calculation": "x = mg/(mg+ml); G = (mg+ml)/area; cg = Z*R*T/M; rho_g = p/cg; v = (1-x)/rho_l + x*cg/p; rho_m = 1/v; u = G*v. R = 8314.46261815324 J/(kmol.K); p in Pa.; ap = G^2*x*cg/p^2; F = fD*G^2/(2*D*rho_m); H = rho_m*g*sin(theta); -dp/ds = (F+H)/(1-ap); acceleration loss = G^2*x*cg*(1/pout-1/pin). g = 9.80665 m/s2.", "explanation": "Uphill angle from horizontal is varied within the supported input range. This is explicitly a homogeneous no-slip isothermal screening model, not Beggs-Brill, Hagedorn-Brown, drift-flux, a mechanistic flow-regime solver or three-phase PVT. It does not model slugging, slip, actual liquid holdup, gas liberation/dissolution, phase change, heat transfer, compositional properties or flow assurance." }, { "label": "Substitute the values", "calculation": "theta = 10 deg; with the fixed controls above, the workflow evaluates Total pressure drop = 0.253516638 MPa; Integrated friction loss = 0.01164336 MPa; Integrated hydrostatic loss = 0.241868043 MPa; Acceleration pressure loss = 0.00000523531475 MPa", "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation." }, { "label": "Read the result", "calculation": "Total pressure drop (MPa) = 0.253516638; Integrated friction loss (MPa) = 0.01164336; Integrated hydrostatic loss (MPa) = 0.241868043; Acceleration pressure loss (MPa) = 0.00000523531475", "explanation": "One constant-ID interval, 0 to 90 degrees uphill from horizontal, constant phase masses, liquid density, Z, temperature and Darcy factor. Downhill flow, fittings and changes of direction/area are not supported. The zero-friction and single-phase options exist for limiting checks, not as recommended operating assumptions. All evaluated pressures must remain at least 0.1 MPa absolute, and the acceleration parameter must remain below 0.2, well away from the model's singular limit at one. These are conservative software bounds, not physical safety limits or a choking prediction. A successful numerical solve is not validation of no-slip behavior in a real petroleum system." } ], "result": "Total pressure drop (MPa) = 0.253516638; Integrated friction loss (MPa) = 0.01164336; Integrated hydrostatic loss (MPa) = 0.241868043; Acceleration pressure loss (MPa) = 0.00000523531475", "interpretation": "One constant-ID interval, 0 to 90 degrees uphill from horizontal, constant phase masses, liquid density, Z, temperature and Darcy factor. Downhill flow, fittings and changes of direction/area are not supported. The zero-friction and single-phase options exist for limiting checks, not as recommended operating assumptions. All evaluated pressures must remain at least 0.1 MPa absolute, and the acceleration parameter must remain below 0.2, well away from the model's singular limit at one. These are conservative software bounds, not physical safety limits or a choking prediction. A successful numerical solve is not validation of no-slip behavior in a real petroleum system.", "control": "Uphill angle from horizontal (deg)", "columns": [ "Uphill angle from horizontal (deg)", "Total pressure drop (MPa)", "Integrated friction loss (MPa)", "Integrated hydrostatic loss (MPa)", "Acceleration pressure loss (MPa)" ], "rows": [ [ 0, 0.011594347714448894, 0.01159411989392738, 0, 2.2782052619916184e-7 ], [ 5, 0.13327666443847175, 0.011618461603777708, 0.12165551856745614, 0.0000026842672296291874 ], [ 10, 0.253516638306035, 0.011643359998206105, 0.24186804299307366, 0.000005235314753032115 ], [ 20, 0.486034783902082, 0.01169408908560419, 0.4743301408080721, 0.000010554008407054777 ], [ 30, 0.7022250217206203, 0.011744626916823701, 0.6904643792883867, 0.000016015515410781293 ] ], "baseline": 2, "rowNotes": [ "Uphill angle from horizontal (deg) = 0. Total pressure drop (MPa): 0.0115943477; Integrated friction loss (MPa): 0.0115941199; Integrated hydrostatic loss (MPa): 0; Acceleration pressure loss (MPa): 2.27820526e-7.", "Uphill angle from horizontal (deg) = 5. Total pressure drop (MPa): 0.133276664; Integrated friction loss (MPa): 0.0116184616; Integrated hydrostatic loss (MPa): 0.121655519; Acceleration pressure loss (MPa): 0.00000268426723.", "Uphill angle from horizontal (deg) = 10. Total pressure drop (MPa): 0.253516638; Integrated friction loss (MPa): 0.01164336; Integrated hydrostatic loss (MPa): 0.241868043; Acceleration pressure loss (MPa): 0.00000523531475.", "Uphill angle from horizontal (deg) = 20. Total pressure drop (MPa): 0.486034784; Integrated friction loss (MPa): 0.0116940891; Integrated hydrostatic loss (MPa): 0.474330141; Acceleration pressure loss (MPa): 0.0000105540084.", "Uphill angle from horizontal (deg) = 30. Total pressure drop (MPa): 0.702225022; Integrated friction loss (MPa): 0.0117446269; Integrated hydrostatic loss (MPa): 0.690464379; Acceleration pressure loss (MPa): 0.0000160155154." ], "provenance": "Original recorded synthetic cases solved with the site's homogeneous-multiphase-profile 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 Total pressure drop (MPa) without rounding intermediate values.", "answer": "theta = 10 deg; with the fixed controls above, the workflow evaluates Total pressure drop = 0.253516638 MPa; Integrated friction loss = 0.01164336 MPa; Integrated hydrostatic loss = 0.241868043 MPa; Acceleration pressure loss = 0.00000523531475 MPa. Total pressure drop (MPa) = 0.253516638. One constant-ID interval, 0 to 90 degrees uphill from horizontal, constant phase masses, liquid density, Z, temperature and Darcy factor. Downhill flow, fittings and changes of direction/area are not supported. The zero-friction and single-phase options exist for limiting checks, not as recommended operating assumptions. All evaluated pressures must remain at least 0.1 MPa absolute, and the acceleration parameter must remain below 0.2, well away from the model's singular limit at one. These are conservative software bounds, not physical safety limits or a choking prediction. A successful numerical solve is not validation of no-slip behavior in a real petroleum system." }, { "question": "Compare the first and last recorded scenarios. Which input changes?", "answer": "Uphill angle from horizontal (deg) changes from 0 to 30. Other controls remain as stated in the premise. Total pressure drop (MPa) changes from 0.0115943477 to 0.702225022." }, { "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. Uphill angle from horizontal is varied within the supported input range. This is explicitly a homogeneous no-slip isothermal screening model, not Beggs-Brill, Hagedorn-Brown, drift-flux, a mechanistic flow-regime solver or three-phase PVT. It does not model slugging, slip, actual liquid holdup, gas liberation/dissolution, phase change, heat transfer, compositional properties or flow assurance. One constant-ID interval, 0 to 90 degrees uphill from horizontal, constant phase masses, liquid density, Z, temperature and Darcy factor. Downhill flow, fittings and changes of direction/area are not supported. The zero-friction and single-phase options exist for limiting checks, not as recommended operating assumptions. All evaluated pressures must remain at least 0.1 MPa absolute, and the acceleration parameter must remain below 0.2, well away from the model's singular limit at one. These are conservative software bounds, not physical safety limits or a choking prediction. A successful numerical solve is not validation of no-slip behavior in a real petroleum system." } ] }, "recorded": { "kind": "workflow", "workflow": "homogeneous-multiphase-profile", "parameter": "theta", "runs": [ { "inputs": { "p0": 5, "T": 320, "D": 0.1, "L": 200, "m_g": 0.05, "m_l": 5, "rho_l": 850, "M": 20, "Z": 0.9, "f_D": 0.02, "theta": 0 }, "result": { "outputs": [ { "label": "Outlet absolute pressure", "value": 4.988405652285551, "unit": "MPa" }, { "label": "Total pressure drop", "value": 0.011594347714448894, "unit": "MPa" }, { "label": "Integrated friction loss", "value": 0.01159411989392738, "unit": "MPa" }, { "label": "Integrated hydrostatic loss", "value": 0, "unit": "MPa" }, { "label": "Acceleration pressure loss", "value": 2.2782052619916184e-7, "unit": "MPa" }, { "label": "Outlet gas void fraction (no slip)", "value": 0.16944288691792717, "unit": "dimensionless" }, { "label": "Outlet mixture velocity", "value": 0.9017615119810518, "unit": "m/s" }, { "label": "Relative momentum closure error", "value": 9.370735553990573e-16, "unit": "dimensionless" } ], "columns": [ "Distance along flow (m)", "Absolute pressure (MPa)", "Mixture density (kg/m3)", "Mixture velocity (m/s)", "Gas void fraction", "Friction loss (MPa)", "Hydrostatic loss (MPa)", "Acceleration loss (MPa)", "Acceleration parameter" ], "rows": [ [ 0, 5, 713.3135538534294, 0.9014071955001388, 0.16911641950677989, 0, 0, 0, 0.000019603710852698172 ], [ 4, 4.99976815767378, 713.3079600692647, 0.9014142643646104, 0.16912293527167774, 0.0002318377810392555, 0, 4.545180679688871e-9, 0.00001960552896712261 ], [ 8, 4.999536313528939, 713.3023658101616, 0.901421333940145, 0.16912945158979426, 0.0004636773802422211, 0, 9.090818562908616e-9, 0.000019607347348750552 ], [ 12, 4.9993044675652945, 713.2967710760558, 0.9014284042268547, 0.16913596846120463, 0.0006955187977917926, 0, 1.3636913721813102e-8, 0.000019609165997636306 ], [ 16, 4.999072619782664, 713.291175866883, 0.9014354752248519, 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constant Darcy friction factor, temperature and Z. It does not predict actual liquid holdup, flow regime, phase transfer or heat loss." ] } }, { "inputs": { "p0": 5, "T": 320, "D": 0.1, "L": 200, "m_g": 0.05, "m_l": 5, "rho_l": 850, "M": 20, "Z": 0.9, "f_D": 0.02, "theta": 5 }, "result": { "outputs": [ { "label": "Outlet absolute pressure", "value": 4.866723335561528, "unit": "MPa" }, { "label": "Total pressure drop", "value": 0.13327666443847175, "unit": "MPa" }, { "label": "Integrated friction loss", "value": 0.011618461603777708, "unit": "MPa" }, { "label": "Integrated hydrostatic loss", "value": 0.12165551856745614, "unit": "MPa" }, { "label": "Acceleration pressure loss", "value": 0.0000026842672296291874, "unit": "MPa" }, { "label": "Outlet gas void fraction (no slip)", "value": 0.17294675401058915, "unit": "dimensionless" }, { "label": "Outlet mixture velocity", "value": 0.905581885702552, "unit": "m/s" }, { "label": "Relative momentum closure error", "value": 1.6544169598739847e-15, "unit": "dimensionless" } ], "columns": [ 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