Harnessing Water’s Anomalous Expansion for Green Energy

A Novel Heat Engine Apparently Defying Carnot’s Theorem
Surpassing Carnot's Efficiency

Introduction

Water’s anomalous expansion upon freezing is a fascinating phenomenon that has sparked interest in various fields. This document presents a novel heat engine concept that leverages the anomalous expansion of water upon freezing. This innovative approach has the potential to convert low-quality dissipated heat energy into useful work, challenging the fundamental limits imposed by Carnot’s theorem. It implies, quality of energy can be spontaneously improved.

Abstract

At standard conditions (0°C and 1 atm), water expands by approximately 9% when transitioning from liquid to solid state. Moreover, when water is spatially constrained, upon freezing, it exhibits a remarkable pressure increase up to 220 MPa before it becomes another form of ice. Notably, increased pressure further lowers the melting point, thereby amplifying the expansion. In other words, the expansion becomes even more significant due to the reduction in melting point caused by the increased pressure. For instance, at a pressure of 200 MPa, water freezes at about 253 Kelvin and undergoes a remarkable 16.8% expansion.

Image Source: Density Anomalies of Water

Contrary to typical liquids, water exhibits anomalous behaviour: its melting point decreases with increased pressure, as shown by the backward-sloping liquid-solid line in its phase diagram. This unique property enables a novel heat engine concept that leverages expanding ice to generate substantial work output. Notably, the engine can operate efficiently with extremely small temperature differences between its hot and cold reservoirs, as the phase change occurs at a constant temperature. By harnessing water’s anomalous expansion, this discovery offers a new way to convert low-quality heat energy into work, challenging traditional thermodynamic limits.

Arrangement and Working of the Proposed Heat Engine

Consider 1 kg of water at STP (approximately 1000 cc). When subjected to 200 MPa, its volume reduces to approximately 922 cc. Cooling this water to 253 K under 200 MPa pressure defines the initial condition.

The cycle begins with the freezing process when the water container contacts the sink at 252 K (T₂), 1 Kelvin below the freezing point. The latent heat fusion (q₂) of 1 kg of water is 334 kJ, released as heat to the sink (q₂ = 334 kJ/kg).

As the fluid freezes, it expands by approximately 16.8%, resulting in a volume increase to 1077 cc. The expanded ice delivers mechanical work:
$w_{1} = 200 MPa \times (1077 cc - 922 cc) = 200 MPa \times 155 cc = 31,000 Nm = 31 kJ$ .

The total change in internal energy (ΔU) during the freezing process is
$ΔU = w_{1} + q_{2} = 31 kJ + 334 kJ = 365 kJ$ .
The cycle completes when the frozen water absorbs energy $(q_{1} = ΔU = 365 kJ)$ from the hot body (source) at $254 K (T_{1})$ for melting.

Efficiency Comparison

The Carnot cycle efficiency $(E_{1})$ is calculated as
$E_{1} = 1 - (\frac{252}{254}) \approx 0.8 %$ .
Given the source temperature $(T_{1})$ of $254 K$ and sink temperature $(T_{2})$ of $252 K$ . In contrast, the proposed heat engine’s efficiency $(E_{2})$ is
$E_{2} = \frac{w_{1}}{q_{1}} = \frac{31 kJ}{365 kJ} \approx 8.5 %$ .
Notably, $E_{2}$ surpasses $E_{1}$ , appearing to defy Carnot’s theorem, which states that no heat engine can exceed the efficiency of a Carnot engine operating within the same temperature limits. Here, the source and sink temperatures are arbitrarily chosen to be 1 Kelvin apart from the freezing or melting point of the fluid to facilitate heat transfer and to prove the possibility of exceeding Carnot’s efficiency.

Ideal Fluid vs the Anomalous Real

Carnot’s efficiency limit relies on ideal gas laws, but real fluids deviate from this ideal behavior. Unlike typical fluids, which require energy to expand under constant pressure, ice uniquely expands while releasing heat. Water’s anomalous behavior during solidification when volumetrically constrained by excess pressure enables cyclic phase changes, (solidifying, expanding, melting etc. intermittently) generating work under seemingly constant pressure and temperature conditions.

One of the sources of inspiration: "Freezing water expands. What if you do not let it?"

Also, the deduced Clausius-Clapeyron relation:
$\frac{d P}{d T} = \frac{Δ s}{Δ v}$
explains the anomalous behaviour of water as $\frac{d P}{d T}$ is negative of the curve separating solid-liquid states.

Source: Clausius–Clapeyron relation - Wikipedia

Perpetual Motion Machine of the Second Kind (PMM2) by the Augmented Heat Pump

When combined with a heat pump operating between $251 K$ and $255 K$ , the Coefficient of Performance (CoP) is about 63.5 from $(\frac{255}{255 - 251})$ . To pump energy from the sink $252 K$ back to the source $254 K$ , the required work $w_{2}$ is approximately $5.3 kJ$ , calculated as the latent heat of fusion $334 kJ {kg}^{-1}$ divided by the CoP. The combined system achieves a net output of
$w_{1} - w_{2} = 31 - 5.3 = 25.7 kJ {kg}^{-1}$ of water.
This is supplied by the atmosphere to the source as heat.

Maximising the Work Extraction

When the room temperature is sufficiently high compared to the Source temperature, we can make use of this difference to drive an additional Stirling Heat Engine (added to the PMM2 pack), to produce additional work.

Practical and the Evidence

Utilize freezing water to generate energy

Notes on Efficiency and Power Density

A Thought Experiment

Let us begin with the assumption that the volumetric change in the phase transition occurs rapidly at the solid, meaning, the transition is like the path $A \to D \to C \to D \to A$ as shown above. Consider 1 kg of ice $(1090 cc)$ that has fully formed at standard atmospheric pressure $(1 atm)$ and a temperature of approximately $273 K$ . At this stage, the ice has expanded by about 9%. This state is marked as point ‘ $a$ ’ along the phase change path ‘ $A a$ ’ on the volume-internal energy (V-U) diagram shown below.

Now, let us analyse the freezing process under pressure. When water freezes under a pressure of $200 MPa$ at $253 K$ , it follows the phase change path ‘ $B b$ .’ Applying a pressure of $200 MPa$ to the ice compresses it, performing mechanical work and increasing its internal energy. This results in the melting of the ice, shifting the melting point to $253 K$ . Consequently, the fluid state must align with a point on the high-pressure path ‘ $B b$ ’ denoted as ‘ $c$ .’ This transition signifies that the system moves from the low-pressure phase change path ‘ $A a$ ’ to the high-pressure path ‘ $B b$ .’

To complete the cycle, the system is brought into contact with a heat source at $254 K$ . The source provides approximately $30 kJ$ of energy, equivalent to the sum of the heat rejected to the sink and the mechanical work output. This input energy allows the fluid to return to its original state, thus completing the cycle. Consequently, the system operates in a cyclic manner between states ‘ $b$ ’ and ‘ $c$ ’ producing $15 kJ$ of work output for every $30 kJ$ of heat input.

Compared to the earlier setup where source supplies $365 kJ$ for a net of $25.7 kJ$ ; the thought experiment results in a net of $14.76 kJ$ for $30 kJ$ supplied. Therefore, the Power Density Improvement is
$\frac{14.76 / 30}{25.7 / 365} = \frac{5387}{771} =$ about 7 times .

Since this is related to PMM2 (Perpetual Motion Machine of the Second Kind), efficiency is not the primary concern, but rather the power density and equipment’s life. Further, to minimize the number of moving parts and to enhance the life of the equipment we can use solid-state electronics to convert pressure to ‘charge’ by piezoelectric effect and Peltier’s phenomenon for heat pump.