Abstract
The efficiency of boiling water is set by the ratio of power to heat loss rather than by the appliance, and weaker heat wastes more (Paper 883), but that paper left out heat lost in steam and the lid. This paper adds to the model of Paper 883, which heats 1 kg of water from 20 to 100 degrees, the heat lost by evaporation from the water surface (saturated vapor density from the Antoine equation; area and mass-transfer coefficient set), treats a lid as stopping evaporation, and compares the least power that can boil, the time and the energy used. No new theorem or law is claimed.
First, heat lost from the water surface rises steeply toward boiling: the heat carried off by evaporation was $3.113036$ W at 20 degrees, $43.616210$ W at 60, $101.751082$ W at 80 and $\mathbf{209.656699}$ W at 100 (Section 2).
Second, without a lid, the evaporation heat at 100 degrees is added to the least power that can boil: for wall loss coefficients of 1, 3 and 6 W/K, the least power was 80, 240 and 480 W with a lid and $289.656699$, $449.656699$ and $689.656699$ W without (Section 2).
Third, the better the insulation, the more a lid matters: the ratio of least power without a lid to with a lid was $\mathbf{3.620709}$, $1.873570$ and $1.436785$ for coefficients 1, 3 and 6 W/K. At 3 W/K and 300 W, the lidded pot boiled in $2245.702$ s, and the open pot stopped at $82.5639$ degrees (Section 3).
Fourth, even when both boil, weaker heat makes the energy difference larger: at 3 W/K, the energy ratio without to with a lid was $\mathbf{1.510276}$ at 500 W and $1.023265$ at 3000 W. At 1 W/K and 300 W, $0.129147$ kg of water evaporated before boiling (Section 3).
Fifth, the separator is whether heat loss rises in proportion to the temperature difference or steeply toward the boiling point: heat lost through the wall is proportional to the temperature difference, reaching the coefficient times 80 at 100 degrees. Heat lost from the surface is proportional to the saturated vapor density, rising from $43.616210$ W at 60 degrees to $209.656699$ W at 100. That addition does not depend on the insulation of the vessel (Section 4).
Keywords: evaporation, lid, heat transfer, saturated vapor pressure, Antoine equation, boiling water.
Scope (what is not claimed)
No new theorem or law is claimed: that evaporation of water carries heat, the mass-transfer formula taking the evaporation rate as proportional to the difference between saturated vapor at the surface and vapor in the air, the Antoine equation for saturated vapor pressure, and that a lid stops steam and boils water faster, are all known in heat transfer and everyday experience. These are set values: a surface area of $0.02$ m², a mass-transfer coefficient of $0.008$ m/s, a latent heat of $2.26\times10^{6}$ J/kg (constant) and an ambient humidity of $0.5$ are set here. The coefficients and powers of Paper 883 are used as they are. They correspond to no pot or kettle. Boiling itself is not treated: the flow of vapor pushing air away near boiling, bubbles, heat lost through the lid and the heat capacity of the lid are not included. A lid is taken only to stop evaporation. Nothing is recommended, which appliance or lid should be used is not stated. Relation to earlier papers: Paper 883, in its scope note, did not include combustion and evaporation, leaving out the fraction of heat passing from a gas flame to a pot, the lid, heat lost in steam and the heat capacity of the vessel; this paper sits in the seat of the lid and heat lost in steam (no earlier occupant). Paper 883 showed that the efficiency of boiling water is set by the ratio of power to heat loss and that weaker heat wastes more. Here the steam heat, which rises steeply with temperature, is added to that heat loss, and how the least power that can boil changes with and without a lid is counted. Paper 392 showed that the boiling point is a property of substance and pressure together, and gave the vapor pressure at 100 degrees as $760.0864$ mmHg from the Antoine equation. Here the same equation gives the saturated vapor density at the water surface. What is added: computing the heat lost by evaporation at each temperature, comparing the least power that can boil with and without a lid, showing that this ratio is larger for better insulated vessels, setting out energy and evaporated water, and placing the separator on “proportional to the temperature difference, or steep toward boiling”.
1. Notation and the Question
Setting: as in Paper 883, heating $1$ kg of water (heat capacity $4186$ J/K) from $20$ to $100$ degrees takes $334880$ J, with power $P$ and wall heat loss $h(T-20)$. This paper adds the heat lost by evaporation from the water surface.
$$E(T)=L\,\beta\,A\,\bigl(\rho_s(T)-\phi\,\rho_s(20)\bigr),\qquad \rho_s(T)=\frac{p_s(T)\,M}{R\,(T+273.15)}$$The saturated vapor pressure $p_s$ follows the Antoine equation (mmHg, degrees) $\log_{10}p_s=8.07131-1730.63/(233.426+T)$, giving $760.0864$ mmHg at 100 degrees and $17.4733$ mmHg at 20 degrees, and saturated vapor densities of $0.588412$ and $0.017218$ kg/m³. $A=0.02$ m², $\beta=0.008$ m/s, $L=2.26\times10^{6}$ J/kg and $\phi=0.5$ are set values. The water temperature $T$ and mass $m$ follow
$$m\,c\,\frac{dT}{dt}=P-h(T-20)-E(T),\qquad \frac{dm}{dt}=-\frac{E(T)}{L}$$with $E=0$ under a lid. If the water falls below half, the run is stopped (not counted as boiling).
Question: how much do the least power that can boil and the energy to boil change with and without a lid?
2. Steam Heat and the Least Power That Can Boil
Computation 2.1: heat lost by evaporation is $3.113036$ W at 20 degrees, $43.616210$ W at 60, $101.751082$ W at 80 and $209.656699$ W at 100.
Computation 2.2: the least power that can boil is where power balances losses at 100 degrees: $80h$ with a lid and $80h+E(100)$ without.
| coefficient $h$ (W/K) | with lid (W) | without lid (W) | without / with |
|---|---|---|---|
| $1$ | $80.000000$ | $289.656699$ | $\mathbf{3.620709}$ |
| $3$ | $240.000000$ | $449.656699$ | $1.873570$ |
| $6$ | $480.000000$ | $689.656699$ | $1.436785$ |
At 3 W/K without a lid, the least power found without the formula, by narrowing the heating model, was $449.657$ W, within 1 W of the formula.
Observation 2.3 steam heat adds directly to the least power that can boil. The addition of $209.656699$ W is the evaporation heat at 100 degrees and does not depend on the wall coefficient: the smaller the wall loss of a vessel, the larger that addition was in proportion.
3. Time and Energy to Boil
Computation 3.1: time (s) and energy (kWh) at 3 W/K.
| power | with lid | without lid | evaporated water (kg) |
|---|---|---|---|
| $300$ W | $2245.702$ s, $0.187142$ | stops at $82.5639$ degrees | (none) |
| $500$ W | $912.445$ s, $0.126729$ | $1378.044$ s, $0.191395$ | $0.061703$ |
| $1000$ W | $382.931$ s, $0.106370$ | $419.659$ s, $0.116572$ | $0.013649$ |
| $2000$ W | $178.370$ s, $0.099095$ | $185.044$ s, $0.102802$ | $0.005566$ |
| $3000$ W | $116.345$ s, $0.096954$ | $119.052$ s, $0.099210$ | $0.003504$ |
All lidded times agreed with the closed form of Paper 883 within 0.05 s. The energy ratio without to with a lid was, at 1 W/K, $1.963091$ at 300 W, $1.208163$ at 500 W, $1.076679$ at 1000 W, $1.034103$ at 2000 W and $1.021942$ at 3000 W; at 3 W/K, $1.510276$ at 500 W, $1.095914$ at 1000 W, $1.037417$ at 2000 W and $1.023265$ at 3000 W; at 6 W/K, $1.154048$ at 1000 W, $1.043786$ at 2000 W and $1.025577$ at 3000 W. At 6 W/K, 500 W stopped at $83.7519$ degrees without a lid, and 300 W stopped at $70.0000$ degrees with a lid and $62.0232$ degrees without. At 1 W/K and 300 W without a lid, boiling took $2548.698$ s with water loss included (evaporating $0.129147$ kg) and $2663.933$ s with the water held at 1 kg.
Observation 3.2 even when both boil, weaker heat makes the lid matter more. At 3 W/K the energy ratio fell from $1.510276$ at 500 W to $1.023265$ at 3000 W: near the least power that can boil, the open pot takes longer, and steam heat keeps escaping all that time.
4. The Separator
Observation 4.1 the separator is whether heat loss rises in proportion to the temperature difference or steeply toward the boiling point. Heat lost through the wall is proportional to the temperature difference and reaches $80h$ at 100 degrees: heat lost from the surface is proportional to the saturated vapor density and rose from $43.616210$ W at 60 degrees to $209.656699$ W at 100. This addition does not depend on the wall coefficient, so for a well insulated vessel (1 W/K) the least power that can boil became $3.620709$ times as large.
Observation 4.2 what is not said. Values for real pots or kettles, the flow of vapor near boiling, bubbles, heat lost through the lid, the fraction of flame heat reaching the pot, and which appliance to use are not treated. What is shown is how far, in a model adding steam heat with set area and mass-transfer coefficient, a lid changes the least power that can boil and the energy used.
5. Roll Call and Closing
- two heat losses, wall and surface (Section 1): saturated vapor from the Antoine equation.
- evaporation heat at 100 degrees $\mathbf{209.656699}$ W (Computation 2.1): steep toward boiling.
- least-power ratios $\mathbf{3.620709}$, 1.873570, 1.436785 (Computation 2.2): larger for better insulation.
- at 3 W/K and 300 W the open pot stops at $82.5639$ degrees (Computation 3.1): the lidded pot boils.
- energy ratio from $\mathbf{1.510276}$ (500 W) to 1.023265 (3000 W) (Computation 3.1): weaker heat, larger difference.
- proportional to the temperature difference, or steep toward boiling (Observation 4.1): the separator; the scope note of Paper 883.
Closing: In a pot without a lid, steam adds to the least power that can boil, and the better the insulation the more a lid matters. Adding surface evaporation to the model of Paper 883, the heat lost reached $209.656699$ W at 100 degrees, and the least power that can boil without a lid became $3.620709$, $1.873570$ and $1.436785$ times that with a lid for coefficients 1, 3 and 6 W/K: at 3 W/K and 300 W, the lidded pot boiled and the open pot stopped at $82.5639$ degrees. Even when both boiled, the energy ratio was $1.510276$ at 500 W. The separator is whether heat loss rises in proportion to the temperature difference or steeply toward the boiling point. Placed among the earlier papers: to the efficiency that Paper 883 counted with wall loss alone, the steam and lid that Paper 883 did not include were added, with vapor from the same Antoine equation as Paper 392. To be honest: the area and mass-transfer coefficient are set values, and vapor flow near boiling is not included. With steam heat added, where the lid matters, and no further.
References
- Incropera, F. P., DeWitt, D. P., Bergman, T. L., & Lavine, A. S. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). Wiley.
- Yamagishi, Y. (2026). The Boiling Point Is Not a Property of the Substance, but of the Substance and the Pressure Together [Paper 392]. YOI: https://www.yamagishiyuuki.com/yoi/392; DOI: 10.5281/zenodo.22699237
- Yamagishi, Y. (2026). The Efficiency of Boiling Water Is Set by the Ratio of Power to Heat Loss, Not the Appliance, and Weaker Heat Wastes More [Paper 883]. YOI: https://www.yamagishiyuuki.com/yoi/883
Reference
Yamagishi, Y. (2026). In a Pot Without a Lid, Steam Adds to the Least Power That Can Boil, and the Better the Insulation the More a Lid Matters [Paper 965]. YOI: https://www.yamagishiyuuki.com/yoi/965
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 965. This paper is a preprint and has not been peer reviewed.