Pool Heat Pump Cost per Day: With and Without a Cover

The water is a well-mixed, constant-inventory control volume with specific liquid enthalpy $h(T,p)$:

Both cases start at the same temperature and use identical hourly weather and heating windows. A midpoint integration uses two-minute intervals by default; operating-window overlaps are integrated exactly within each interval. The half-sine solar profile is integrated analytically over each interval, retaining short and fractional-hour daylight windows. All fluxes use these interval means in the same midpoint energy balance as stored enthalpy.

For an inactive outdoor pool, Smith, Löf and Jones (1994), equations (4)/(9), give $\dot m/A=(30.6+32.1v)(p_w-p_a)/h_{fg}$ with pressure in mmHg, latent heat in kJ/kg and mass flux in kg/(m²·h). The code converts these coefficients to SI before use. Wind is measured near the pool surface. Negative vapour flux represents condensation. ASHRAE HVAC Applications (2023), Chapter 6 provides the underlying vapour-pressure-deficit load framework; its indoor activity factors are not substituted for the outdoor correlation.

Convection follows a Lewis-number-one heat/mass-transfer analogy: $h_c=c_e\,p\,c_{p,air}/(\beta\,h_{fg})$, where $c_e$ is the evaporation heat coefficient and $\beta$ is the water/dry-air molar-mass ratio. Long-wave exchange is $\epsilon\sigma(T_s^4-T_{sky}^4)$, using the entered effective sky depression.

On the fitted fraction, Newton iteration solves $(T_w-T_c)/R_c+\alpha_cG=h_c(T_c-T_a)+\epsilon_c\sigma(T_c^4-T_{sky}^4)$. Solar gain includes transmitted sunlight absorbed by the water plus cover absorption; outer-surface convection and radiation then close this balance. Residual evaporation uses the entered suppression. Ground transfer is $UA_g(T_w-T_g)$. Automatic top-up and additional replacement carry $\dot m_{make}[h(T_w)-h(T_{make})]$.

Heat output at 26°C water is interpolated between the entered 15°C and 26°C air ratings. Each COP rating is divided by its Carnot heating limit to obtain second-law efficiency. This efficiency is interpolated with air temperature and bounded to 0–1, then multiplied by the operating Carnot limit to reconstruct COP. The Carnot limit is $T_{cond}/\max(10\,K,T_{cond}-T_{evap})$, using absolute temperatures, entered heat-exchanger approaches and the model's 10 K minimum refrigerant lift. COP is bounded below by one; all supported operating Carnot limits exceed one. NIST, Didion and Brown (2000), PDF p.3, provides the reversible-cycle limit and second-law framework. At fixed air temperature, capacity changes with COP while full-load electrical input stays fixed. Low-air lockout removes capacity. Thermostat duty is limited by available capacity and window time; full-power mode uses all available time. Cost includes heat pump electricity and incremental circulation electricity.

Fresh water, negligible salinity, no swimmers or fountains, and uniform water temperature. Volume is fixed using initial density; evaporation is immediately replaced, and condensate overflows at pool temperature. Ground temperature and overall ground coefficient remain fixed.

Air, humidity and wind consist of 25 generated hourly states with shape-preserving interpolation and a sinusoidal daily cycle. Sunlight follows the daylight-limited half-sine directly for snapshots and plots, with exact interval means for integration; hourly weather rows show samples of that profile. Wind and humidity extrema occur at 03:00/15:00. Solar input already accounts for shade and clouds; effective sky temperature is an independent input.

The cover has negligible heat capacity, uniform resistance and a dry outer surface. Its resistance excludes external air and radiation films. Default resistance, emissivities, absorption and 95% evaporation suppression are illustrative adjustable assumptions, not certified cover properties. DOE guidance supports using covers but no fixed cost-saving percentage is imposed.

Water enthalpy, saturation pressure, latent heat, density and molar mass use CoolProp at this calculation's states; humid-air properties use the hourly air temperature, humidity and pressure. The convection analogy, cover balance, controller and bounded second-law interpolation are engineering models implemented here.

Use for inactive outdoor pools with predicted liquid-water temperatures of 2–45°C. Smith et al. measured water near 29°C, air 14.4–27.8°C, humidity 27–65% and winds up to 3.2 m/s; conditions beyond these are extrapolations. Cover properties and local surface wind often dominate uncertainty. Wet covers, rain, splashing and detailed sky/cloud radiation are not resolved.

Default heat pump anchors are the MUPIR-17-H9 brochure's maximum-capacity and minimum-COP endpoints. These endpoints approximate full load; the brochure's separate input-power extrema are not exactly Q/COP. Replace ratings for your equipment. Humidity dependence beyond the rating conditions, defrost, inverter part-load benefits, standby and cycling losses are not modeled. Second-law efficiency is interpolated between ratings and clipped to 0–1 when extrapolated; this maintains the thermodynamic bound but does not replace measured performance data. Below 10°C or above 35°C air, a condition requests better manufacturer data.

Water properties are interpolated on a 0.1 K grid; the enthalpy inversion uses that same table. Midpoint integration does not resolve sub-step cycling. Increase integration steps per hour to check time convergence; refinement checks compare 30 versus 60 steps per hour. The illustration's arrow lengths are schematic, and its play control reads the already solved day.

The saving is for the entered first day. Compare final temperatures and stored heat before interpreting it as a like-for-like maintenance saving. Fixed-runtime comparisons can deliver warmer covered water with little electricity saving. If uncovered electricity is zero, the relative-saving helper returns zero; no percentage saving is claimed.