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Heat Pump Cycles and Heating Applications

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Heat Pump Systems for Heating and CoolingThrottling and Isenthalpic Expansion Processes+1 moreAbsorption and Adsorption Refrigeration Cycles
heat-pump heating cop coefficient-performance

Core Idea

Heat pumps reverse refrigeration cycles to deliver heating; coefficient of performance COP_heating = Q_h/W_in = COP_cooling + 1. Modern air-source and ground-source heat pumps achieve seasonal COP of 2.5-4.0, making them 2-4 times more efficient than electric resistance heating. Performance degrades at low outdoor temperatures unless supplemented with auxiliary heating.

Explainer

A heat pump is mechanically identical to a refrigerator — the same vapor-compression cycle, the same compressor, the same refrigerant. The difference is which heat exchange you care about. A refrigerator removes heat from a cold space (the interior) and rejects it to a warm space (the kitchen); you value Q_cold, the heat removed from inside. A heat pump does the same thermodynamic cycle but you value Q_hot, the heat delivered to the warm space (the room being heated). The compressor work W_in drives the cycle, and by the first law, Q_hot = Q_cold + W_in. This single energy balance is the source of the key relationship: COP_heating = Q_hot / W_in = (Q_cold + W_in) / W_in = COP_cooling + 1.

That "+1" is the crucial insight. Since COP_cooling is always positive, COP_heating is always greater than 1. An electric resistance heater converts exactly 1 kWh of electrical work into 1 kWh of heat — a COP of 1 by definition. A heat pump delivers *more* heat energy than the electrical energy it consumes, because it is not converting electricity to heat; it is using electricity to *move* heat from outside to inside. A COP of 3 means 3 kWh of heat delivered per kWh of electricity consumed — three times as efficient as resistance heating. The "extra" energy comes from the outdoor air or ground, which cools down slightly as the heat pump extracts heat from it.

Air-source heat pumps extract heat from outdoor air. The Carnot limit for heating is COP_Carnot = T_hot / (T_hot − T_cold) in absolute temperatures. As outdoor temperature T_cold drops, the denominator grows and the Carnot limit falls — meaning real performance must also fall. At −10°C outdoors with 20°C indoors, T_hot = 293 K and T_cold = 263 K gives Carnot COP_heating = 293/30 ≈ 9.8, but real systems achieve only 2–2.5 at such temperatures due to compression irreversibilities, heat-exchanger temperature differences, and defrost cycles needed to prevent ice buildup on the outdoor coil. Below some balance-point temperature, the heat pump cannot meet the full heating load and auxiliary resistance heating supplements it.

Ground-source heat pumps (geothermal) avoid this degradation by extracting heat from the ground or groundwater, which stays near 10–15°C year-round in temperate climates. The more stable source temperature means the temperature difference across the cycle stays nearly constant, so COP remains in the 3–5 range regardless of outdoor air temperature. The trade-off is installation cost: buried ground loops require excavation or boreholes. Analysis of a heat pump system involves computing the seasonal COP over a range of operating conditions, integrating across the heating season's temperature distribution, and comparing total electrical consumption to the equivalent resistance-heating baseline. The crossover — where the capital cost of a heat pump is recovered through operating savings — depends on local electricity and fuel prices and climate.

Practice Questions 5 questions

Prerequisite Chain

Understanding ZeroThe Number ZeroCounting to FiveCounting to 10Counting to 20Counting a Set of Objects Up to 20Cardinality: The Last Number CountedMatching Numerals to QuantitiesSubitizing Small QuantitiesAddition Within 10Number Bonds to 10Addition Within 20Doubles and Near DoublesDoubles Facts Within 10Near Doubles Facts Within 20Mental Math Strategies for AdditionMental Math: Adding and Subtracting TensAddition Within 100Repeated Addition as MultiplicationMultiplication as Equal GroupsMultiplication: ArraysBasic Multiplication Facts (0s, 1s, 2s, 5s, 10s)Multiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Multiplication and Division Fact FamiliesRelationship Between Multiplication and DivisionDivision Facts as Inverse of MultiplicationRemainders and Quotients in DivisionDivision Word ProblemsMulti-Step Word ProblemsSolving Multi-Step Word ProblemsMultiplication Word ProblemsDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueIntegers and the Number LineComparing and Ordering IntegersAbsolute ValueAdding IntegersSubtracting IntegersMultiplying IntegersDividing IntegersUnit RatesProportionsPercent ConceptConverting Between Fractions, Decimals, and PercentsOperations with Rational NumbersTwo-Step EquationsSolving Multi-Step EquationsEquations with Variables on Both SidesAngle Pairs: Complementary, Supplementary, and VerticalParallel Lines and TransversalsCorresponding AnglesAlternate Interior AnglesTriangle Angle Sum TheoremExterior Angle TheoremTriangle Inequality TheoremSimilar Triangles: AA SimilaritySimilar Triangles: SSS and SAS SimilarityProportions in Similar TrianglesRight Triangle Trigonometry IntroductionSine, Cosine, and Tangent RatiosTrigonometric Ratios ReviewRadian MeasureConverting Between Degrees and RadiansThe Unit CircleGraphing Sine and CosineGraphing Tangent and Reciprocal Trigonometric FunctionsDerivatives of Trigonometric FunctionsAntiderivativesIndefinite IntegralsBasic Integration RulesRiemann SumsDefinite Integral DefinitionDouble Integrals: Definition and SetupIterated Integrals and Fubini's TheoremDouble Integrals over Rectangular RegionsDouble Integrals over General RegionsApplications of Double Integrals: Area, Mass, and MomentsTriple Integrals in Cartesian CoordinatesTriple Integrals in Cylindrical and Spherical CoordinatesChange of Variables and the Jacobian DeterminantApplications of Triple Integrals: Volume and MassVector Fields and Their RepresentationsLine Integrals of Vector FieldsWork and CirculationLine Integrals of Scalar and Vector FunctionsFundamental Theorem for Line IntegralsConservative Vector FieldsConservative Vector Fields and Potential FunctionsCurl and Divergence of Vector FieldsCurl and DivergenceDivergence TheoremElectric Flux and Divergence TheoremGauss's Law: Integral Form and MeaningSolving Problems with Gauss's LawConductors in Electrostatic EquilibriumCapacitance and CapacitorsDielectricsDielectric Constant and Relative PermittivityElectric Field Inside Dielectric MaterialsDielectric Materials and PolarizationDielectric Susceptibility and PermittivityEnergy Density in Electric FieldsElectric Current and Current DensityElectrical Resistance and ResistivityOhm's Law and Circuit ElementsElectromotive Force (EMF) and BatteriesKirchhoff's Circuit Laws: Voltage and CurrentDC Circuit Network Analysis MethodsTransient Response in RC CircuitsRC CircuitsLC and RLC CircuitsAC Circuits: FundamentalsImpedance and ReactanceAC Power and ResonanceElectromagnetic WavesPostulates of Special RelativityTime DilationLength ContractionLorentz TransformationRelativistic Velocity AdditionRelativistic Momentum and EnergyMass-Energy Equivalence and E=mc²Photons as Particles with Energy and MomentumPlanck-Einstein Relation: Energy and FrequencyPhotoelectric EffectThe Photon: Light as QuantaCompton ScatteringWave-Particle Dualityde Broglie WavelengthThe Schrödinger EquationState Vectors and WavefunctionsQuantum SuperpositionQuantum EntanglementBell Theorem and Bell InequalitiesPostulates of Quantum MechanicsObservables and Quantum OperatorsCommutators and Commutation RelationsQuantum Angular MomentumQuantum Mechanical Treatment of HydrogenSolving the Schrödinger Equation for Hydrogen AtomQuantum NumbersElectron ConfigurationPeriodic TrendsCovalent BondingElectronegativity and Bond PolarityIonic BondingLewis StructuresVSEPR Theory and Molecular GeometryMolecular Geometry and Electron Pair GeometryMolecular Polarity and Dipole MomentsIntermolecular ForcesStates of Matter and Phase Changes: Melting, Boiling, and SublimationGas Laws and the Ideal Gas EquationGas Stoichiometry and Volume-Volume CalculationsThermochemistry and EnthalpyHeat Capacity and CalorimetryEntropy and Molecular DisorderSpontaneity and ΔGEntropy and Gibbs Free EnergyChemical EquilibriumStatistical Mechanics: Ensembles and the Boltzmann DistributionPartition Function: Definition and PropertiesThe Canonical Partition Function and Thermodynamic DerivationFree Energy and Thermodynamic Relations from Partition FunctionsPhase Transitions and Equilibrium Phase DiagramsLandau Theory of Phase TransitionsSpontaneous Symmetry BreakingOrder Parameters and Phase TransitionsMean Field Theory and Self-ConsistencyVan der Waals Equation from Statistical MechanicsCritical Point and Supercritical Fluid BehaviorReal Gas Thermodynamics and Equations of StateJoule-Thomson Coefficient and Inversion CurveHeat Pump Systems for Heating and CoolingHeat Pump Cycles and Heating Applications

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