Energy Budget for Desalination

I’ve done a few postings (here, here, and here) on an idea for desalination (creating fresh water from saltwater) using a concrete sphere which is submerged to a great depth (~2,225 feet) to use the pressure at that depth to force saltwater through a reverse osmosis membrane.  The sphere -filled with fresh water- is then taken to the surface and the fresh water is harvested.

Today I’m looking at the “energy budget” necessary for this approach, and I compare it with existing reverse osmosis technology.  Due to global warming and rising sea levels, the availability of fresh water along coastal areas is a diminishing resource.  Humans have a growing need for potable water, which may be satisfied in part by desalination (desal) technology.  Because current desal technology requires large amounts of energy, any energy savings could be hugely beneficial.  Currently this energy comes primarily from burning fossil fuels to generate electricity: it is not a sustainable approach and will further contribute to anthropogenic global warming (AGW).

I begin by looking at the energy requirements for current desal technology.  In this source, the author reports that desal plants use around 5kWh of energy per cubic meter of fresh water produced.  Commenter’s feedback on this article report that greater efficiency can be expected: from between 3.4 – 4.5 kWh/m³; to possibly as low as 2.2 kWh/m³.

Main author says: “The most efficient (reverse osmosis based) desalination plants consume about 5 kWh of energy per cubic meter of fresh water produced. The fundamental thermodynamic limit for desalinating seawater is 0.86 kWh m⁻³.”

Commenter says: “One minor point to note is the energy numbers – your source of 5 kWh/m3 is from the IAEA in 1992. Today, this would be considered a very conservative number. Using energy recovery technology such as the PX (Pressure Exchanger) from ERI (www.energyrecovery.com), the SWRO process can consume under 2.2 kWh/m3. A 2008 study by the National Academy of Sciences (www.nap.edu/catalog.php?record_id=12184) puts the number at between 3.4 – 4.5 kWh/m3.”

If we look at my approach of sinking an empty concrete sphere, the main work needed for this is sinking the buoyant sphere to great depth.  I use a simplified approach, looking at 1 cubic meter of air, sunk to 2,225 feet below sea level, where a pressure of just over 1,000 psi is obtained; enough to force sea water through a semi-permeable membrane, removing the salt component.

On doing some more research, I've come to realize that apparently there are reverse osmosis (RO) filters that will perform this work at a depth of only 850 ft. (260 m) as developed by DXV Water Technologies.  This approach is described here (see pp. 2-3).  While this method uses the static head pressure at depth to help power the RO filtration, it still relies on a pump to bring the water to the surface.  This article states that "Many readers have grown numb to reports of new desalting techniques claiming energy reductions of 50 percent or more."  Furthermore, there is a history of people claiming the ability of "free" desal technologies which amount to perpetual motion machines (an impossibility: violating the 2nd Law of thermodynamics).  Low flow rates through an RO membrane achieve higher efficiency.  As the flow rate increases, efficiency drops exponentially.  Thus a large number of large volume spheres, operating at a low flow rate would achieve higher efficiency while still achieving higher volume. 

I simplified my analysis by considering the density of water as 1.0 g/cm³ (seawater has a density of around 1.025 g/cm³⁾.  I also consider the act of bringing the water to the surface as negligible.  This is because the water to be harvested is considered as neutrally buoyant.  Because it is neutrally buoyant, there is no gravitational acceleration, and no work is required to move the object (this is a simplification).  In fact, extra work will be required to overcome turbulence and drag (viscous effects): 

The total amount of work required to lift a nearly-neutrally buoyant object through a viscous medium has two terms, the gravitational term W = \Delta\rho V g \Delta h and a drag force term, the force required to overcome viscous effects. This has to be written like W =\int f dl , because the viscous drag will depend on the path taken- straight up, zig-zag, whatever. The drag force f can be written simply as \mathbf{F}_d= {1 \over 2} \rho \mathbf{v}^2 C_d A.   One simplification is to tow the body at constant speed, then you are left with a simple multiplication rather than an integration.

I am ignoring these viscous effects because it varies greatly depending on the size of the sphere.  I do not expect this effect to “break the budget” for energy used in this process.

If we crunch the numbers, here’s what I got:

Work = Force x Distance = Buoyant force x distance submerged

Distance =2,225 ft. = 678.18 meters

Work = Mgh = 1,000,000g x 9.8 m/s² x 678.18 m = 6646164000g m²/s² =6646164 Joules

Joule = kg x m²/s²   kWh = 3,600,000 J

(6646164J) x (1kWh/3,600,000J) = 1.846156 kWh (energy required to submerge 1m³ of air 2,225 ft.)

So there’s my number: 1.846 kWh to get 1 m³ of water at 2,225 ft. below sea level.  At  850 ft. below sea level, this works out to just over 0.7 kWh/m³, which is right around the thermodynamic limit.  You couldn't do much better.  How does this compare with existing technology?   The deeper we go, the faster water is produced; but much less efficiently.

If we compare 2,225 ft. below sea level with:

·         5 kWh/m³ it requires 36.92% of that energy.

·         4.5 kWh/m³ it requires 41.02% of that energy.

·         3.4 kWh/m³ it requires 54.29% of that energy.

·         2.2 kWh/m³ it requires 83.90% of that energy.

A few important notes on these preliminary numbers:  If more efficient reverse osmosis membrane filters are used, then the sphere does not need to be sunk to such a great depth, and energy savings translate directly to the sunken sphere approach.  This keeps my approach very competitive.

If it is possible to use weights (e.g., landfill, etc.) to sink the sphere, then NO ENERGY is required to obtain fresh water!  This may not be a realistic or sustainable approach.  It would involve dropping massive amounts of material on the ocean floor to sink the hollow spheres.  Still, it is a possibility.

I have neglected the weight of the concrete sphere itself.  This is because the ratio of the weight of the sphere to the volume of water varies extremely, depending on the size of the sphere.  This weight will reduce the amount of work done to sink a hollow sphere, but will add to the work needed to bring the freshwater to the surface.

I have also disregarded the two stages (minimal) necessary for producing potable water.  As described above, this process will merely produce brackish water.  There are methods using this same principle to achieve potable water; I'm just not sharing everything on this forum.  The energy analysis outlined still works for creating potable water.

This short mathematical exercise indicates that potentially significant energy savings can be obtained by using the approach I’ve described for desal.  If anyone out there wants to double-check my math and let me know if I’ve made any critical mistakes, it would be greatly appreciated!