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Pipe flow rate (Q = A * V) and continuity nozzle velocity formulas
Calculate volumetric pipe flow in Liters/sec from inner diameter and velocity using PI()*(D/2)^2*V*1000, and scale nozzle velocity via V1*(D1/D2)^2.
Read the guide →Vacuum Metrology & Semiconductor Load-Lock Safety, Marcello Pirani (1906) Gauge PhysicsPirani Vacuum Gauge Heat Loss & Gas Calibration Formula
Calculate Nitrogen-Equivalent Indicated Pirani Pressure (mbar), Gas-Conductive Filament Heat Loss (mW), Total Wheatstone Bridge Power (mW), and Gas Signal Share (%).
Read the guide →Engineering & ScientificPitot-Static Tube Velocity: Equivalent Airspeed (EAS) vs True Airspeed (TAS)
Convert Pitot-static differential pressure dP = P_total - P_static into sea-level calibrated Equivalent Airspeed and altitude-density-corrected True Airspeed V_TAS = SQRT(2 * dP / rho).
Read the guide →Healthcare & Clinical CalculationsPitt Bacteremia Score (PBS) for Bloodstream Infection Acute Mortality
Calculate the 0–14 Pitt Bacteremia Score on the day of positive blood culture draw across Temperature, Hypotension, Mechanical Ventilation, Cardiac Arrest, and Mental Status (PBS >= 4 Critical Illness).
Read the guide →Healthcare & ClinicalPittsburgh Knee Rule Acute Knee Fracture X-Ray Decision Formula
Evaluate David Seaberg and Dan Tandberg’s validated Pittsburgh Knee Rule—ordering knee radiographs after blunt trauma or fall ONLY when Age < 12, Age > 50, or inability to walk 4 full weight-bearing steps in the ED.
Read the guide →Engineering & SciencePitzer / Lee-Kesler Second-Virial Compressibility Factor Z = Z0 + ω·Z1
Calculate real-gas compressibility factor Z = 1 + (B^0 + ω * B^1) * (P_r / T_r) using Abbott’s Pitzer-virial correlations B^0 = 0.083 - 0.422 / T_r^1.6 and B^1 = 0.139 - 0.172 / T_r^4.2.
Read the guide →Engineering & SciencePitzer Acentric Factor (ω) from Reduced Vapor Pressure at Tr = 0.7
Calculate Kenneth Pitzer’s acentric factor ω = -1.000 - LOG10(P_sat(0.7 * T_c) / P_c) and invert it to estimate saturation vapor pressure at T_r = 0.7 for cubic equations of state.
Read the guide →EngineeringPlanck Photon Energy (1239.84 / lambda_nm) & Bragg X-Ray Diffraction Law
Convert optical/LED wavelength in nanometers directly to photon energy in electron-volts (eV) and solve Bragg's Law d = n*lambda / (2*sin(theta)) for crystal d-spacing.
Read the guide →EngineeringPlane-stress von Mises and Tresca equivalent stress and yield ratios
Combine multiaxial normal stresses sigma_x, sigma_y and shear stress tau_xy into FEA Von Mises distortion-energy stress SQRT(sig_x^2 - sig_x*sig_y + sig_y^2 + 3*tau_xy^2) and Yield Factor of Safety.
Read the guide →Engineering & SciencePlanetary Atmospheric Scale Height (H) & Barometric Altitude Pressure
Calculate isothermal planetary atmospheric scale height H = (R * T) / (M * g) (km) and exponential barometric pressure P(z) = P_0 * EXP(-z / H) (kPa) across Earth, Mars, Venus, and Titan.
Read the guide →Engineering & Fluid MechanicsPlank’s Freezing Time Equation: Blast-Freezer & Biopharma Cryopreservation Slab Formula
Calculate phase-change freezing time for infinite slabs, cylinders, and spheres using Rudolf Plank’s moving-boundary equation t_f = (rho_f * L_f / (T_f - T_inf)) * (P * a / h + R * a² / k_f) alongside the freezing Biot number.
Read the guide →Engineering & SciencePlasma Debye Screening Length (λ_D) & Debye Sphere Parameter (N_D)
Calculate electrostatic charge-shielding length λ_D = 7433.94 * SQRT(T_e_eV / n_e_m3) and weakly coupled plasma parameter N_D = (4/3) * π * n_e * λ_D^3 in RF etching tools and tokamaks.
Read the guide →Statistical & Clinical AnalysisPLASMIC Score Formula: Severe ADAMTS13 Deficiency & Emergent TTP Triage (0 to 7)
Identify immune Thrombotic Thrombocytopenic Purpura (ADAMTS13 <= 10%) within minutes of admission Labs using Pavan Bendapudi’s 7-point PLASMIC score for life-saving therapeutic plasma exchange (TPE) & caplacizumab.
Read the guide →Machine Design, Friction Clutches & Automotive Disc BrakesPlate Clutch & Disc Brake Uniform-Wear vs Pressure Torque Formula
Calculate Uniform-Wear Effective Friction Radius (mm), Conservative Uniform-Wear Torque Capacity (N·m), Brand-New Uniform-Pressure Torque (N·m), and Max Inner-Edge Lining Pressure (kPa).
Read the guide →Thermal Process Engineering, Gasketed Plate Heat Exchangers & Anti-Fouling DesignPlate Heat Exchanger (PHE) Velocity & Wall Shear Formula
Calculate Parallel Channels Per Fluid, Enlargement-Adjusted Hydraulic Diameter D_h (mm), Inter-Plate Channel Velocity (m/s), and Self-Cleaning Wall Shear Stress tau_w (Pa).
Read the guide →StatisticalPlus-or-Minus (+/-) Tolerance Check & Count Out-of-Spec Values
Classify measurements using an explicit six-decimal comparison policy and count rejected parts from the same classifications.
Read the guide →Financial analysisPMT function: calculate a fixed monthly loan payment
Convert an annual interest rate and term into matching monthly units inside PMT, interpret the negative cash-outflow sign, and test zero-interest financing.
Read the guide →Pneumatic Conveying, Powder Transport, Saltation Plugging & Blower Line SizingPneumatic Conveying Gas Velocity & Saltation Margin Formula
Calculate Required Conveying Air Mass Rate (kg/h), Line Volumetric Gas Flow at Pick-Up Pressure (m³/min), Superficial Pick-Up Gas Velocity U_gas (m/s), and Saltation Safety Margin U_gas / U_salt.
Read the guide →Health & FitnessPneumonia Severity Index (PSI / PORT): Score and Risk Classes I–V
Practice PSI scoring from verified clinical point subtotals, apply the separate Class I screen before score bands, and validate missing inputs. This worksheet does not prescribe admission or ICU care.
Read the guide →Engineering & SciencePockels Electro-Optic Modulator Half-Wave Voltage (Vpi) & Transmission
Calculate transverse lithium-niobate (LiNbO3) Pockels modulator half-wave voltage V_pi = (lambda_0 * d) / (2 * n_e^3 * r_33 * L) and crossed-polarizer optical transmission I / I_0 = SIN((PI()/2) * (V / V_pi))^2.
Read the guide →Biostatistics & Clinical TrialsPocock Win Ratio, Win Odds, and Net Treatment Benefit Formula
Analyze hierarchical composite clinical endpoints (death before hospitalization before biomarker) using Pocock’s Win Ratio, Dong’s tie-adjusted Win Odds, and Buyse’s Net Benefit.
Read the guide →Statistical AnalysisPoint-Biserial Correlation (r_pb) for Binary Groups & Exam Item Discrimination
Compute Point-Biserial Correlation r_pb = ((M_1 - M_0) / SD_pop) * SQRT(p * q) from subgroup means and verify algebraic identity with CORREL on 0/1 binary indicators.
Read the guide →StatisticalPoisson probability formula: P(X = k) arrivals with EXP and FACT
Calculate the exact probability of k random arrivals, defects, or outages per interval given average rate lambda using (lambda^k * EXP(-lambda)) / FACT(k).
Read the guide →Charts & Data VisualizationPolar Plot Coordinate Formula: Convert Radius & Angle (r, theta) to XY Scatter
Plot true continuous-angle Polar Charts (antenna radiation lobes, wind roses, acoustic microphone patterns) on an XY Scatter grid by converting Polar (r, theta_deg) into Cartesian X = r * COS(RADIANS(theta)) and Y = r * SIN(RADIANS(theta)).
Read the guide →OperationsPollaczek-Khinchine M/G/1 Queue Waiting Time & Service Variance
Calculate exact M/G/1 expected queue wait W_q = (lambda * (sigma_s^2 + (1/mu)^2)) / (2 * (1 - rho)) and queue length L_q = lambda * W_q across deterministic (M/D/1), exponential (M/M/1), and high-variance servers.
Read the guide →Polymer Processing & Battery Electrode Coating, Gaskell Two-Roll Calendering NipPolymer Two-Roll Calendering Gaskell Nip Force Formula
Calculate Post-Nip Exit Sheet Gauge 1.226*H0 (mm), Peak Hydrodynamic Nip Pressure (MPa), Gaskell Roll Separating Force (kN), and Two-Roll Viscous Drive Power (kW).
Read the guide →Turbomachinery, Pipeline Gas Compression & Process EngineeringPolytropic Centrifugal Gas Compressor Temp & Power Formula
Calculate Centrifugal Gas Compressor Polytropic Temperature Exponent, Discharge Temperature T2 (K), Polytropic Specific Enthalpy Rise (kJ/kg), and Gas Compression Power (kW).
Read the guide →Engineering & SciencePoole-Frenkel Trap Barrier Lowering vs. Schottky Emission Formula
Calculate bulk dielectric Coulomb trap barrier lowering Delta_phi_PF = SQRT((e * E) / (PI() * eps_dyn * eps_0)) (eV) and compare directly against electrode-limited Schottky image-force lowering (0.5 * Delta_phi_PF).
Read the guide →Charts & Visual Data ModelingPopulation Pyramid Chart Formula: Negated Male Share, Sex Ratio & Age Dependency
Prepare an age-sex Population Pyramid bar chart: negate the left-side Male cohort share (-Male / Total_Pop * 100), compute Cohort Sex Ratio (Males per 100 Females), and Total Age-Dependency Ratio.
Read the guide →Statistical AnalysisPopulation Variance & Standard Deviation (VAR.P & STDEV.P vs. Bessel N-1 Correction)
Choose accurately between N-denominator population dispersion (VAR.P, STDEV.P) for complete censuses and N-1 sample estimators (VAR.S, STDEV.S), and verify the exact N/(N-1) Bessel scaling ratio.
Read the guide →StatisticalPopulation weighted standard deviation with frequency or probability weights
Calculate weighted variance and weighted standard deviation around the weighted mean in one LET + SUMPRODUCT formula.
Read the guide →Cement Pyroprocessing, ASTM C150 Clinker Mineralogy & Sulfate-Resistance ChemistryPortland Cement Bogue Clinker Phases & LSF Formula
Calculate ASTM C150 Bogue Potential Alite C_3S (%), Belite C_2S (%), Tricalcium Aluminate C_3A (%), and Raw-Mix Lime Saturation Factor LSF (%) from XRF Oxide Percentages.
Read the guide →Statistical & Clinical AnalysisPortsmouth P-POSSUM Surgical Mortality Formula: Physiology & Operative Logit
Calculate predicted 30-day surgical inpatient mortality (%) from 12-factor Physiology Score (12–88) and 6-factor Operative Severity Score (6–48) using Prytherch’s Portsmouth P-POSSUM logistic equation.
Read the guide →Lookup & ReferencePosition & Exact Cell Address of Max or Min Value (XMATCH & ADDRESS)
Find both the relative list position (1..N) and exact worksheet cell address (such as "B4") of the maximum or minimum value in a column using XMATCH(MAX(range), range, 0) and ADDRESS.
Read the guide →Venture Capital & Seed FinancingPost-Money SAFE vs Pre-Money SAFE Dilution Formula
Compare stacked YC Post-Money SAFE ownership locks (alpha_i = I_i / Cap_i) against Legacy Pre-Money SAFEs that mutually dilute each other prior to a priced Series A round.
Read the guide →MathPOWER function: calculate exponents, reciprocals, and Nth roots
Raise numbers to positive, zero, negative, and fractional (1/N) powers with POWER—and guard even roots of negative numbers.
Read the guide →EngineeringPower-Plant Net Heat Rate (BTU/kWh & kJ/kWh) & Stirling Cycle Efficiency
Convert a supplied net thermal efficiency to heat rate in Btu/kWh or kJ/kWh, and separately calculate the ideal perfectly regenerated Stirling temperature limit.
Read the guide →EngineeringPoynting Vector Electromagnetic Intensity & Solar Sail Radiation Pressure
Calculate electromagnetic wave irradiance I = E_0^2 / (2 * c * mu_0) from peak electric field E_0 and compute absorbed (I / c) vs perfectly reflected (2 * I / c) radiation thrust.
Read the guide →Engineering & SciencePoynting-Robertson Effect: Interplanetary Dust Spiral Decay Lifetime & Beta Ratio
Calculate the solar radiation-to-gravity ratio beta = 0.574 * Qpr / (rho * s) and Poynting-Robertson orbital spiral decay lifetime t_PR = 700 * s * rho * a^2 / Qpr (years) for zodiacal dust grains.
Read the guide →FinancialPPMT function: calculate principal paid down in a specific loan period
Measure how much loan principal each installment retires—and verify that summing all periods recovers the exact original principal.
Read the guide →HealthcarePractice mg/kg/day to mg per Dose and mL Conversions
A fictional spreadsheet exercise for a supplied daily regimen, daily cap, dose count and liquid concentration. Distinguish daily amounts from per-dose quantities.
Read the guide →LogicalPractice text-complexity checks with dummy password-like strings
Combine length, case and digit checks in a spreadsheet using synthetic examples; this is not a password-strength test.
Read the guide →Economics & Market MicrostructurePrais-Shorrocks Income Mobility Index M(P) Formula: Markov Transition Matrix Trace
Quantify intergenerational or credit-rating transition mobility from a K x K Markov matrix P using the Prais-Shorrocks Trace Index M(P) = (K - trace(P)) / (K - 1) and Mean Class Stay Duration 1 / (1 - p_ii).
Read the guide →Statistical & Clinical AnalysisPRAM Score Formula: Pediatric Respiratory Assessment Measure for Acute Asthma (0 to 12)
Grade acute pediatric asthma exacerbation severity across toddlers and teenagers (ages 1–17) using Ducharme & Chalut’s 5-sign PRAM score (0–12 points) without spirometry.
Read the guide →EngineeringPrandtl-Meyer Supersonic Expansion Function nu(M) & Maximum Turn Angle
Calculate the isentropic Prandtl-Meyer expansion angle nu(M) (in degrees) from Mach 1 to supersonic Mach M and the vacuum limiting turn angle nu_max = (SQRT((gamma+1)/(gamma-1)) - 1) * 90.
Read the guide →Healthcare & Clinical CalculatorsPRECISE-DAPT Score (0–100): Post-PCI Bleeding Risk & DAPT Duration
Calculate Francesco Costa et al.’s (Lancet 2017) 5-item PRECISE-DAPT score (Hemoglobin, Age, WBC, CrCl, Prior Bleeding) to guide short (3–6 month, score >= 25) vs. standard/extended DAPT after coronary stenting.
Read the guide →StatisticalPrecision, Recall, F1 Score & Matthews Correlation (MCC) from a Confusion Matrix
Evaluate ML classifiers, spam filters, and QA inspection models on imbalanced datasets where raw Accuracy misleads.
Read the guide →FinancialPrepare a Trial Balance from journal entries (SUMIFS + balance check)
Aggregate general-ledger Debit and Credit lines by account with SUMIFS and verify double-entry balance to the cent.
Read the guide →FinancialPresent and future value of an annuity: Ordinary (type 0) vs Annuity Due (type 1)
Value leases, pensions, and savings plans accurately by choosing end-of-period (type = 0) vs beginning-of-period (type = 1) cash flows in PV and FV.
Read the guide →Sheet-Metal Forming, Hydraulic Press-Brake Tooling & Tonnage ProtectionPress-Brake Air-Bending Tonnage & V-Die Opening Formula
Calculate V-Die Shoulder Opening V (mm), Floated Air-Bent Natural Inside Radius I_r (mm), Forming Force F (kN), and Required Hydraulic Press-Brake Capacity (metric tons-force).
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