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Mechanical Energy Balance with Pump and Turbine Work

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8topics build on this

1,192prerequisites beneath it

Bernoulli Equation: Assumptions and Real Fluid Limitations First Law for Open Systems and Control Volumes +2 more→→Centrifugal Pump Performance Curves and System Selection Darcy-Weisbach Equation: Major Head Loss Calculation +2 more

Core Idea

The steady-flow mechanical energy equation (p₁/ρg + v₁²/2g + z₁ + H_pump = p₂/ρg + v₂²/2g + z₂ + H_turbine + H_loss) extends Bernoulli to include work interactions and irreversibilities. Pump head and turbine head represent useful work transfer; head loss represents energy dissipated as heat by viscous friction. This equation is the foundation for all piping system design.

Explainer

You already know that Bernoulli's equation is an energy balance along a streamline for an ideal, inviscid fluid: pressure energy, kinetic energy, and potential energy trade off while their sum stays constant. But Bernoulli breaks down when the fluid passes through a machine (pump or turbine) or when friction is significant. The mechanical energy equation is the corrected version: it adds terms for work added by pumps, work extracted by turbines, and energy destroyed by friction — all expressed in the same units of length called head.

Head is the most important concept here. By dividing each energy term by ρg, you convert joules per kilogram into meters — a "height equivalent" of energy. Pressure head (P/ρg) is the height a column of fluid would reach if all pressure energy were converted to elevation. Velocity head (V²/2g) is the equivalent height for kinetic energy. Elevation head z is the actual height. Pump head H_pump is the mechanical energy added to the fluid per unit weight of fluid — it increases the total head at the pump discharge. Turbine head H_turbine is the energy extracted. Head loss H_loss is energy permanently destroyed by viscous friction and converted to heat; it always appears on the right side of the equation because you always lose it, regardless of which way you write the balance.

The equation p₁/ρg + V₁²/2g + z₁ + H_pump = p₂/ρg + V₂²/2g + z₂ + H_turbine + H_loss reads as: total head at inlet, plus any head added by a pump, equals total head at outlet, plus any head extracted by a turbine, plus all head losses in between. This is an accounting statement: every joule of energy that enters a control volume must go somewhere. To use it in a piping system problem, pick two points (usually where conditions are known, like tank surfaces), write the equation, and solve for the unknown — typically pump head, flow rate, or pressure at some point.

The power required by or delivered by a machine follows directly from the head: P = ρgQH, where Q is volumetric flow rate. This connects the hydraulic head concept back to the first-law open-system analysis you learned earlier — power is the rate of energy transfer. Real pumps and turbines have efficiencies less than 1, so the shaft power input to a pump is P_shaft = ρgQH_pump/η_pump, and the shaft power output from a turbine is P_shaft = η_turbine · ρgQH_turbine. Correctly applying this equation is what allows engineers to size pumps for water distribution systems, calculate hydroelectric power output, or determine whether a pipe network can deliver the required flow rate.

Practice Questions 5 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Indefinite Integrals → Basic Integration Rules → Riemann Sums → Definite Integral Definition → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Work and Circulation → Line Integrals of Scalar and Vector Functions → Fundamental Theorem for Line Integrals → Conservative Vector Fields → Conservative Vector Fields and Potential Functions → Curl and Divergence of Vector Fields → Curl and Divergence → Divergence Theorem → Electric Flux and Divergence Theorem → Gauss's Law: Integral Form and Meaning → Solving Problems with Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → AC Circuits: Fundamentals → Impedance and Reactance → AC Power and Resonance → Electromagnetic Waves → Postulates of Special Relativity → Time Dilation → Length Contraction → Lorentz Transformation → Relativistic Velocity Addition → Relativistic Momentum and Energy → Mass-Energy Equivalence and E=mc² → Photons as Particles with Energy and Momentum → Planck-Einstein Relation: Energy and Frequency → Photoelectric Effect → The Photon: Light as Quanta → Compton Scattering → Wave-Particle Duality → de Broglie Wavelength → The Schrödinger Equation → State Vectors and Wavefunctions → Quantum Superposition → Quantum Entanglement → Bell Theorem and Bell Inequalities → Postulates of Quantum Mechanics → Observables and Quantum Operators → Commutators and Commutation Relations → Quantum Angular Momentum → Quantum Mechanical Treatment of Hydrogen → Solving the Schrödinger Equation for Hydrogen Atom → Quantum Numbers → Electron Configuration → Periodic Trends → Covalent Bonding → Electronegativity and Bond Polarity → Ionic Bonding → Lewis Structures → VSEPR Theory and Molecular Geometry → Molecular Geometry and Electron Pair Geometry → Molecular Polarity and Dipole Moments → Intermolecular Forces → States of Matter and Phase Changes: Melting, Boiling, and Sublimation → Gas Laws and the Ideal Gas Equation → Gas Stoichiometry and Volume-Volume Calculations → Thermochemistry and Enthalpy → Heat Capacity and Calorimetry → Entropy and Molecular Disorder → Spontaneity and ΔG → Entropy and Gibbs Free Energy → Chemical Equilibrium → Statistical Mechanics: Ensembles and the Boltzmann Distribution → Partition Function: Definition and Properties → The Canonical Partition Function and Thermodynamic Derivation → Maxwell-Boltzmann Distribution and Classical Limit → Transport Properties of Gases → Diffusion Coefficients and Kinetic Molecular Theory → Viscosity and Transport Properties → The Reynolds Number and Flow Regimes → Dimensional Analysis and Dynamic Similarity → Boundary Layer Theory → Flow Separation: Adverse Pressure Gradient Mechanics → Adverse Pressure Gradients and Flow Separation → Form Drag and Pressure Drag: Decomposition of Total Drag → Absolute, Gauge, and Atmospheric Pressure → Pitot Tube and Velocity Measurement → Flow Measurement: Venturi, Orifice, and Pitot Tube → Flow Visualization Techniques → Streamlines, Pathlines, and Flow Visualization → Control Volume and Mass Balance → Energy Equation for Steady Flow → Mechanical Energy and Head Forms → Mechanical Energy Balance with Pump and Turbine Work

Longest path: 195 steps · 1192 total prerequisite topics

Prerequisites (4)

First Law for Open Systems and Control Volumeshard Bernoulli Equation: Assumptions and Real Fluid Limitationshard Mechanical Energy and Head Formssoft Control Volume Analysis: Mass Balancesoft

Leads To (4)

Centrifugal Pump Performance Curves and System Selectionhard Darcy-Weisbach Equation: Major Head Loss Calculationhard Minor Loss Coefficients: Elbows, Valves, and Fittingshard Pipe Network Analysis: Hardy-Cross Iteration Methodhard