Heat Pump Balance Point: Method

Heat Pump Balance Point

An air-source heat pump loses capacity and efficiency as the outdoor air gets colder while the building needs more heat. This tool puts the two against each other: it finds the temperatures at which the heat pump stops carrying the load and at which it stops being the cheaper source, sizes the backup, and adds the season up bin by bin. Every equipment value is an entered input: the capacity and power points come from the maker's expanded performance data, the bin hours from the analyst's weather file. The defaults are illustrative and are labelled so.

Units

The Entered Data Units select says what the typed temperatures and the pasted tables are in: °F with Btu/h, or °C with kW. Input power is always kW. The unit-system radio converts the design heat loss and UA fields through the site's unit dictionary and sets the units of the results; the entered data are converted to them before anything is calculated, so the same data give the same answer under either radio. The Appendix M section stays in °F, Btu/h and W, the units of the regulation.

Building load line

The heating load is taken as linear in outdoor temperature and zero at the building balance temperature T_bal, the outdoor temperature at which internal and solar gains make up the envelope loss. With a design heat loss Q_d at the design outdoor temperature T_d the line is BL(T) = Q_d (T_bal − T) / (T_bal − T_d); with an overall conductance it is BL(T) = UA (T_bal − T), and the page also shows the envelope loss UA (T_in − T_d) before the gains. The default balance temperature of 65 °F (18.3 °C) is the degree-day base and the zero-load temperature of Equation 4.2-2 of the DOE test procedure; a well-insulated building with steady gains balances lower, and the lower value should be entered when it is known. The design heat loss comes from a load calculation (Room Heating and Cooling Load, Envelope UA Compliance on this site; ACCA Manual J in practice).

Capacity and power curves

Each pasted point is an outdoor temperature, a heating capacity and either the input power in kW or the COP (then power = capacity / COP). Capacity and power are each fitted by least squares with a straight line or a parabola, y = c₀ + c₁ T + c₂ T²; the page prints the coefficients, R², the rms error and the fitted value beside every point. The fit is used at any temperature, so a result outside the pasted range is an extrapolation and is flagged. A two-stage or variable-speed unit takes a second set of points for its minimum (low-stage) capacity.

Frost on the outdoor coil and the defrost cycles that clear it lower the average capacity in a band of outdoor temperatures around freezing, roughly −7 to 4 °C (20 to 40 °F) in humid weather. Makers' tables are often steady-state values without it, so the tool takes a defrost derate table: outdoor temperature, capacity factor and, optionally, power factor, linear between rows and held outside them. The default rows (1.00 at 17 °F, 0.90 capacity and 0.985 power at 35 °F, 1.00 at 45 °F) follow the shape of the test procedure, which measures a frost-accumulation point at 35 °F, uses it between 17 and 45 °F, and where that test is not run sets the 35 °F capacity to 0.90 and the power to 0.985 of the values interpolated between the 17 and 47 °F tests. Clear the table when the pasted points already include defrost.

Thermal balance point

The thermal balance point is the outdoor temperature at which capacity equals load. The tool scans capacity − load from 10 °F (5.6 K) below the coldest of the points, the bins and the design temperature up to the balance temperature, and refines each sign change by bisection; it reports the warmest crossing at which the heat pump goes from deficit to surplus. With a straight-line capacity c₀ + c₁ T and load slope s the root is T = (s T_bal − c₀) / (c₁ + s). Below the balance point the heat pump runs continuously and the backup supplies the shortfall, load − capacity; at the design temperature that shortfall sizes the supplemental heat, reported in kW and as a count of resistance stages. For a unit with a minimum curve the same solve gives the crossover above which the load is below the minimum capacity and the unit must cycle.

Economic balance point

Against a fuel-fired backup the question is which source delivers a unit of heat for less. A kWh of heat costs p_el / COP from the heat pump and p_fuel / (η H) from the furnace, with η the furnace efficiency and H the heat content of a sale unit of fuel in kWh. They are equal at the break-even COP_be = η p_el H / p_fuel, and the economic balance point is the temperature at which the heat pump's COP at full capacity (defrost included) equals it. Above it the heat pump is cheaper. Against electric resistance the break-even COP is 1 and there is no economic balance point.

Lock-out settings

A dual-fuel system runs one source at a time, because the indoor coil sits in the furnace's air stream. Its switchover (compressor lock-out) is recommended at the warmer of the two balance points, rounded up to a whole degree: below the thermal balance point the heat pump cannot carry the load alone, and below the economic balance point the furnace is cheaper. The page says which of the two governs. An all-electric system keeps the compressor running down to the unit's own cut-out and locks the strip heat out above the thermal balance point. The settings do not allow for recovery from a night setback or for defrost tempering.

Sizing ratios and the cold-climate check

The page reports the share of the design load carried at the design temperature, the capacity at 47 °F (8.3 °C) over the design load, and the capacity at 5 °F (−15 °C) as a percentage of that at 47 °F with the COP at 5 °F. The last two are compared with entered targets; the defaults, 70 percent and a COP of 1.75, are the low-ambient criteria of the ENERGY STAR Version 6.1 specification for its cold-climate mark, which are measured by the Appendix M1 tests. The values here come from the fitted curve and are not certified results.

Seasonal energy by the bin method

The heating season is entered as temperature bins: a bin temperature and the hours per year the outdoor temperature spends in that bin (5 °F or 3 °C bins are customary; the values come from a TMY file or NOAA bin data for the location). In each bin:

load L = BL(T); heat pump output = the smaller of L and the capacity; backup heat = L − output.

When the load is below the capacity of a single-stage unit it cycles. With the load factor X = L / capacity, the part-load factor is PLF = 1 − C_D (1 − X) and the average input is X · power / PLF. C_D is the cyclic degradation coefficient; 0.25 is the default the test procedure assigns when the cyclic test is not run. A two-stage or variable-speed unit cycles this way only below its minimum curve; between the minimum and maximum curves it delivers the load with an input interpolated on capacity between the two power curves, which is exact for a two-stage unit alternating between its stages and an approximation for a variable-speed compressor. Energy is the rate times the bin hours. The compressor does not run at or below an entered cut-out temperature, or below the switchover of a dual-fuel system, where the furnace carries the whole load.

The seasonal COP of an all-electric system is the seasonal load over all the electricity used, compressor and resistance; times 3.412 Btu/Wh it is an HSPF equivalent for the entered climate. A dual-fuel system has no single COP and is judged on cost and CO₂. The defrost penalty is the difference between the season as entered and the same season with every defrost factor set to 1. Input power should include the indoor blower, as rated values do; crankcase heat and standby power are not added.

The default bins are illustrative: the Region IV fractional bin hours of Table 17 of the test procedure multiplied by 5,643 hours, the season length at which that region's 2,250 heating load hours equal Σ n_j (65 − T_j) / (65 − 5).

Cost, CO₂ and the alternatives

Each alternative carries the same seasonal load: a fuel appliance burns load / (efficiency × heat content) sale units, an electric one draws load / COP. Cost is quantity times price, CO₂ quantity times the entered factor. The default factors are from the EPA GHG Emission Factors Hub (natural gas 53.06 kg per million Btu, No. 2 oil 10.21 kg per gallon, propane 5.72 kg per gallon) and the eGRID2023 U.S. average of 771.5 lb/MWh; the prices are illustrative and must be replaced with the local tariff.

The payback compares the heat pump with the first alternative of the list. The extra first cost is the heat pump's installed cost less the incentive and less the installed cost avoided, which may include the air conditioner the heat pump also replaces. The saving of year t is the difference of the two annual costs with electricity and fuel escalated to that year, taken at the end of the year and discounted; the page gives the simple payback at today's prices, the present value of the savings, the net present value, the savings-to-investment ratio, the internal rate of return, the discounted payback and the cash-flow table. Cooling savings, maintenance and replacement are left out.

Cooling check

A heat pump chosen for its heating capacity can be too large in cooling. The page compares the total capacity with the design total load, the sensible capacity (sensible heat ratio × total) with the sensible load and the latent capacity with the latent load. A seasonal estimate uses cooling bins with a load line from zero at the zero-load temperature to the design load at the design temperature, and divides the seasonal cooling by the SEER. It is a simple estimate: the rated SEER belongs to the test procedure's own bins and load line.

HSPF by the test-procedure method

The rated heating seasonal performance factor is itself a bin calculation, defined in 10 CFR 430 Subpart B Appendix M. The tool reproduces the edition of 1 January 2016: section 4.2.1 for a single-speed unit from its three steady tests (H1 at 47 °F, the H2 frost-accumulation test at 35 °F, H3 at 17 °F), section 4.2.3 for two-capacity units and section 4.2.4 for variable-speed units. The HSPF2 of Appendix M1, in force for ratings since 2023, is a second choice.

HSPF = Σ (n_j/N) BL(T_j) / [Σ e_h(T_j)/N + Σ RH(T_j)/N] · F_def (4.2-1)

BL(T_j) = (65 − T_j) / (65 − T_OD) · C · DHR, C = 0.77 (4.2-2)

e_h/N = X E_h δ / PLF · n_j/N (4.2.1-1); RH/N = [BL − X Q_h δ] / 3.413 · n_j/N (4.2.1-2); X = BL / Q_h not above 1; PLF = 1 − C_D (1 − X)

Capacity Q_h and power E_h are linear between the H3 and H1 tests at and below 17 °F and at and above 45 °F, and between H3 and H2 from 17 to 45 °F (4.2.1-4, 4.2.1-5). The cut-out factor δ is 0 where the compressor is off or its COP is below 1, ½ between the cut-out and cut-in temperatures, and 1 otherwise. Table 17 gives six generalized climatic regions with their outdoor design temperatures, heating load hours and fractional bin hours in 5 °F bins from 62 to −23 °F. The design heating requirement DHR is, at its minimum, Q_h(47) (65 − T_OD) / 60 (Q_h(47) in Region V) and at its maximum twice that (2.2 Q_h(47) in Region V), each rounded to the nearest standardized value of Table 18. F_def is the demand-defrost credit of section 3.9.2, between 1 and 1.03. Published ratings are the Region IV value at the minimum DHR.

The compressor cut-out temperature T_off and cut-in temperature T_on are entered in the HSPF group; left blank, δ depends on the COP alone. The Procedure and Compressor Type selects choose among the cases below. The six H1, H2, H3 fields serve the single-speed unit; the other two take a test table, one test per line (name, capacity in Btu/h, power in W).

Two-capacity units (section 4.2.3)

Tests: H0₁ (low capacity, 62 °F), H1₁ (low, 47 °F), H1₂ (high, 47 °F), H2₂ (high, 35 °F), H3₂ (high, 17 °F), and H2₁, H3₁ (low at 35 and 17 °F) when low capacity runs below 40 °F. A missing H2₁ is approximated as section 3.6.3 allows: Q(35) = 0.90 {Q(17) + 0.6 [Q(47) − Q(17)]}, with 0.985 for the power. The low-capacity curves Q₁, E₁ follow the H1₁ to H0₁ line at and above 40 °F, the H3₁ to H2₁ line from 17 to 40 °F and the H3₁ to H1₁ line below 17 °F; the high-capacity curves Q₂, E₂ are those of the single-speed unit (4.2.2-3, 4.2.2-4). Each bin falls in one case:

4.2.3.1, Q₁ ≥ BL, cycling at low capacity: X = BL / Q₁, e_h/N = X E₁ δ′ / PLF · n_j/N (4.2.3-1), RH/N = BL (1 − δ′) / 3.413 · n_j/N (4.2.3-2)

4.2.3.2, Q₁ < BL < Q₂, alternating between the stages: X₁ = (Q₂ − BL) / (Q₂ − Q₁), e_h/N = [X₁ E₁ + (1 − X₁) E₂] δ′ · n_j/N, RH/N by 4.2.3-2

4.2.3.3, low capacity locked out and BL < Q₂, cycling at high capacity: X = BL / Q₂, e_h/N = X E₂ δ′ / PLF · n_j/N with C_D(k=2), RH/N by 4.2.3-2

4.2.3.4, BL ≥ Q₂, continuous at high capacity: e_h/N = E₂ δ″ · n_j/N, RH/N = [BL − Q₂ δ″] / 3.413 · n_j/N

δ′ is 0 at and below T_off, ½ up to T_on and 1 above (4.2.3-3); δ″ is the same and also 0 where Q₂ / (3.413 E₂) is below 1. The design heating requirement uses the H1₂ capacity; the northern two-capacity option (H1₁) is not offered. C_D(k=2) takes the low-capacity value, the procedure's default.

Variable-speed units (section 4.2.4)

Tests: H0₁ and H1₁ at minimum speed (62 and 47 °F), H2_V at the intermediate speed (35 °F), H1₂ and H3₂ at maximum speed (47 and 17 °F), and optionally H2₂ (35 °F), otherwise approximated by the 0.90 and 0.985 form of section 3.6.4. Minimum speed: the straight line through H1₁ and H0₁ at every temperature (4.2.4-1, 4.2.4-2). Maximum speed: the single-speed curves with k = 2. Intermediate speed:

Q_v(T) = Q_v(35) + M_Q (T − 35), E_v(T) = E_v(35) + M_E (T − 35) (4.2.4-3, 4.2.4-4)

M_Q = [Q₁(62) − Q₁(47)] / 15 · (1 − N_Q) + N_Q [Q₂(35) − Q₂(17)] / 18, N_Q = [Q_v(35) − Q₁(35)] / [Q₂(35) − Q₁(35)], and the same with E

4.2.4.1, Q₁ ≥ BL: cycling at minimum speed, as 4.2.3.1. 4.2.4.3, BL ≥ Q₂: continuous at maximum speed, as 4.2.3.4. In between (4.2.4.2) the speed matches the load:

e_h/N = BL / [3.413 COP_i(T_j)] · δ′ · n_j/N, RH/N by 4.2.3-2, COP_i(T) = A + B T + C T²

A, B and C are fixed once per unit: the parabola passes through the minimum-speed COP at T₃, the intermediate-speed COP at T_vh and the maximum-speed COP at T₄, the outdoor temperatures at which the minimum-speed, intermediate-speed and maximum-speed capacity each equal the building load. The regulation writes them as D = (T₃² − T₄²) / (T_vh² − T₄²), B = [COP₂(T₄) − COP₁(T₃) − D (COP₂(T₄) − COP_v(T_vh))] / [T₄ − T₃ − D (T₄ − T_vh)], C = [COP₂(T₄) − COP₁(T₃) − B (T₄ − T₃)] / (T₄² − T₃²), A = COP₂(T₄) − B T₄ − C T₄². The page prints the three temperatures, the three COPs and A, B, C. When the load line meets the maximum-speed capacity inside its step at 45 °F, T₄ is taken as 45 °F (the tool's choice; the regulation is silent). The multiple-split alternative (two straight COP lines) is not offered.

HSPF2 by Appendix M1

Appendix M1 (edition of 1 January 2023) keeps Equation 4.2-1 and the bin structure and changes the load line and the regional data:

BL(T_j) = (T_zl − T_j) / (T_zl − 5 °F) · C · Q_c(95 °F) (4.2-2)

Q_c(95 °F) is the cooling capacity from the A or A₂ test; for a heating-only unit the 47 °F heating capacity takes its place (H1, H1₂, or H1_N for a variable-speed unit). Table 20 gives each region its zero-load temperature T_zl (55 to 58 °F), slope factor C (C_VS for variable-speed units) and heating load hours; bins at and above T_zl carry no hours. There is no design heating requirement. Single-speed and two-capacity units follow sections 4.2.1 and 4.2.3 as above, with one addition: an optional test at 5 °F (H4, H4₂) replaces the capacity and power below 17 °F by the line through the 5 °F and 17 °F tests (4.2.1-6, 4.2.2-5).

Variable-speed units (section 4.2.4 of Appendix M1) differ more:

• Full speed uses 47 °F values Q_calc, E_calc that belong to the H3₂ compressor speed (section 3.6.4.1 b.): those of a 47 °F test run at that speed, or Q_calc = Q₂(17) (1 + 30 CSF), E_calc = E₂(17) (1 + 30 PSF), with CSF = 0.0204 per °F for split systems, 0.0262 per °F for single-package systems, and PSF = 0.00455 per °F. At and above 45 °F the H3₂ to 47 °F line is multiplied by the H1_N value over the calculated value; from 17 to 45 °F the H3₂ to H2₂ line applies; below 17 °F the H3₂ to 47 °F line, or with an H4₂ test the line through 5 and 17 °F and, below 5 °F, the H4₂ value continued at the slope of the H3₂ to 47 °F line.
• A minimum-speed-limiting unit takes its minimum-speed curve from the H1₁ to H0₁ line at and above 47 °F, the line from H2_V to H1₁ between 35 and 47 °F, and the intermediate-speed curve below 35 °F (4.2.4-5, 4.2.4-6).
• At an intermediate speed the COP is interpolated on the load at the bin temperature: COP_i = COP₁ + (COP_v − COP₁) / (Q_v − Q₁) · (BL − Q₁) while BL < Q_v, and COP_i = COP_v + (COP₂ − COP_v) / (Q₂ − Q_v) · (BL − Q_v) from there to Q₂.

Not calculated under either appendix: single-speed units with a variable-speed indoor blower (4.2.2), heat comfort controllers (4.2.5), triple-capacity northern units (4.2.6), multiple indoor blowers (4.2.7), and the variable-speed non-communicating coil-only units of Appendix M1. The HSPF of two-capacity and variable-speed units follows the 2016 edition of Appendix M; later editions of that appendix changed the variable-speed calculation (full-speed 47 °F values by its section 3.6.4 b.) and are not reproduced. Neither appendix prints a worked numerical example; the tool's arithmetic is checked by hand on single bins and against an independent implementation. The test tables that fill the form are illustrative units, not a product.

Limits

The load line ignores wind, solar and thermal-mass effects hour by hour; the bin method has no memory between hours. Duct losses, setback recovery and the indoor temperature droop below the balance point are not modelled. The economic balance point uses the full-capacity COP. Equipment selection limits belong to ACCA Manual S and the load to Manual J; rated values to AHRI 210/240 and the DOE test procedure.