Editing Ocean thermal energy conversion (section)

== Thermodynamics ==

A rigorous treatment of OTEC reveals that a 20&nbsp;°C temperature difference will provide as much energy as a hydroelectric plant with 34&nbsp;m head for the same volume of water flow.{{Citation needed|date=January 2022}}
The low temperature difference means that water volumes must be very large to extract useful amounts of heat.  A 100MW power plant would be expected to pump on the order of 12 million gallons (44,400 tonnes) per minute.<ref>{{citation  |title=Challenge And Promise Of OTEC  |first1=Duke  |last1=Hartman  |journal=Ocean News  |date=October 2011  |access-date=11 June 2012  |url=http://virtual.ocean-news.com/display_article.php?id=864488  |archive-date=5 March 2016  |archive-url=https://web.archive.org/web/20160305082205/http://virtual.ocean-news.com/display_article.php?id=864488  |url-status=dead  }}</ref>  For comparison, pumps must move a mass of water greater than the weight of the ''[[German battleship Bismarck|battleship Bismarck]]'', which weighed 41,700 tonnes, every minute.  This makes pumping a substantial [[Losses in electrical systems#Parasitic loss|parasitic drain]] on energy production in OTEC systems, with one Lockheed design consuming 19.55&nbsp;MW in pumping costs for every 49.8&nbsp;MW net electricity generated.<!--Da Rosa, P.147 -->  For OTEC schemes using heat exchangers, to handle this volume of water the exchangers need to be enormous compared to those used in conventional thermal power generation plants,<ref name=" Da Rosa">{{cite book|last= Da Rosa|first= Aldo Vieira|title=Fundamentals of renewable energy processes|chapter-url=https://books.google.com/books?id=vDuec62ube8C&q=otec&pg=PA146|year=2009|publisher=Academic Press |isbn=978-0-12-374639-9|pages=139 to 152 |chapter=Chapter 4:Ocean Thermal Energy Converters}}</ref> making them one of the most critical components due to their impact on overall efficiency. A 100&nbsp;MW OTEC power plant would require 200 exchangers each larger than a 20-foot shipping container making them the single most expensive component.<ref>{{citation|contribution=Comparison Of Aluminum Alloys And Manufacturing Processes Based On Corrosion Performance For Use In OTEC Heat Exchangers|last1=Eldred|first1=M. |last2=Landherr|first2= A. |last3=Chen |first3=I.C. |doi=10.4043/20702-MS | title= Offshore Technology Conference 2010 (OTC 2010) |publisher= Curran Associates, Inc.  | isbn=9781617384264 | date= July 2010 }}</ref>

=== Variation of ocean temperature with depth ===
[[File:ThermoclineSeasonDepth.png|thumb|200px|right|Graph of different thermoclines (depth vs. temperature) based on seasons and latitude]]
The total [[insolation]] received by the oceans (covering 70% of the earth's surface, with [[clearness index]] of 0.5 and average energy retention of 15%) is: {{math|5.45&times;10<sup>18</sup> MJ/yr &times; 0.7 &times; 0.5 &times; 0.15 {{=}} 2.87&times;10<sup>17</sup> MJ/yr}}

We can use [[Beer–Lambert law|Beer–Lambert–Bouguer's law]] to quantify the solar energy absorption by water,

:<math>-\frac{dI(y)}{dy}=\mu I</math>

where, ''y'' is the depth of water, ''I'' is intensity and ''μ'' is the absorption coefficient.
Solving the above [[differential equation]],

:<math> I(y)=I_{0}\exp(-\mu y) \,</math>

The absorption coefficient ''μ''  may range from 0.05 m<sup>&minus;1</sup> for very clear fresh water to 0.5 m<sup>−1</sup> for very salty water.

Since the intensity [[exponential decay|falls exponentially]] with depth ''y'', heat absorption is concentrated at the top layers. Typically in the tropics, surface temperature values are in excess of {{convert|25|C}}, while at {{convert|1|km|mi|sp=us}}, the temperature is about {{convert|5|-|10|C|F}}. The warmer (and hence lighter) waters at the surface means there are no [[convection current|thermal convection currents]]. Due to the small temperature gradients, heat transfer by [[heat conduction|conduction]] is too low to equalize the temperatures. The ocean is thus both a practically infinite heat source and a practically infinite heat sink.{{Clarify|date=January 2013}}

This temperature difference varies with latitude and season, with the maximum in [[tropical]], [[subtropical]] and [[Equatorial climate|equatorial]] waters. Hence the tropics are generally the best OTEC locations.

=== Open/Claude cycle ===

In this scheme, warm surface water at around {{convert|27|C}} enters an evaporator at pressure slightly below the [[saturation pressure]]s causing it to vaporize.

:<math> H_{1}=H_{f} \,</math>

Where ''H{{sub|f}}'' is [[enthalpy]] of liquid water at the inlet temperature, ''T''{{sub|1}}.

[[File:otec oc t-s dia.jpg|frameless|upright=1.25]]

This temporarily [[superheating|superheated]] water undergoes volume boiling as opposed to pool boiling in conventional boilers where the heating surface is in contact. Thus the water partially flashes to steam with two-phase equilibrium prevailing. Suppose that the pressure inside the evaporator is maintained at the saturation pressure, ''T''{{sub|2}}.

:<math>H_{2}=H_{1}=H_{f}+x_{2}H_{fg} \,</math>

Here, ''x''{{sub|2}} is the fraction of water by mass that vaporizes. The warm water mass flow rate per unit [[turbine]] mass flow rate is 1/''x''{{sub|2}}.

The low pressure in the evaporator is maintained by a [[vacuum pump]] that also removes the dissolved non-condensable gases from the evaporator. The evaporator now contains a mixture of water and steam of very low [[vapor quality]] (steam content). The steam is separated from the water as saturated vapor. The remaining water is saturated and is discharged to the ocean in the open cycle. The steam is a low pressure/high [[specific volume]] working fluid. It expands in a special low pressure turbine.

:<math>H_{3}=H_{g} \,</math>

Here, ''H{{sub|g}}'' corresponds to ''T''{{sub|2}}. For an ideal [[isentropic process|isentropic]] ([[reversible adiabatic]]) turbine,

:<math>s_{5,s}=s_{3}=s_{f}+x_{5,s}s_{fg} \, </math>

The above equation corresponds to the temperature at the exhaust of the turbine, ''T''{{sub|5}}. ''x''{{sub|5,''s''}} is the mass fraction of vapor at state 5.

The enthalpy at ''T''{{sub|5}} is,

:<math> H_{5,s}=H_{f}+x_{5,s}H_{fg} \,</math>

This enthalpy is lower. The adiabatic reversible turbine work = ''H''{{sub|3}}-''H''{{sub|5,''s'' }}.

Actual turbine work {{Math|1=''W''{{sub|T}} = (''H''{{sub|3}}-''H''{{sub|5,''s''}}) x ''polytropic efficiency''}}

:<math>H_{5}=H_{3}-\ \mathrm{actual}\ \mathrm{work} </math>

The condenser temperature and pressure are lower. Since the turbine exhaust is to be discharged back into the ocean, a direct contact condenser is used to mix the exhaust with cold water, which results in a near-saturated water. That water is now discharged back to the ocean.

''H''{{sub|6}}=''H{{sub|f}}'', at ''T''{{sub|5}}. ''T''{{sub|7}} is the temperature of the exhaust mixed with cold sea water, as the vapor content now is negligible,

:<math>H_{7}\approx H_{f}\,\ at\ T_{7} \,</math>

The temperature differences between stages include that between warm surface water and working steam, that between exhaust steam and cooling water, and that between cooling water reaching the condenser and deep water. These represent external [[Reversible process (thermodynamics)|irreversibilities]] that reduce the overall temperature difference.

The cold water flow rate per unit turbine mass flow rate,

:<math>\dot{m_{c}=\frac{H_{5}-\ H_{6}}{H_{6}-\ H_{7}}} \,</math>

Turbine mass flow rate, <math>\dot{M_{T}}=\frac{\mathrm{turbine}\ \mathrm{work}\ \mathrm{required}}{W_{T}} </math>

Warm water mass flow rate, <math> \dot{M_{w}}=\dot{M_{T}\dot{m_{w}}} \,</math>

Cold water mass flow rate <math>\dot{\dot{M_{c}}=\dot{M_{T}m_{C}}} \,</math>

=== Closed Anderson cycle ===

As developed starting in the 1960s by J. Hilbert Anderson of Sea Solar Power, Inc., in this cycle, ''Q{{sub|H}}'' is the heat transferred in the evaporator from the warm sea water to the working fluid. The working fluid exits the evaporator as a gas near its [[dew point]].

The high-pressure, high-temperature gas then is expanded in the turbine to yield turbine work, ''W{{sub|T}}''. The working fluid is slightly superheated at the turbine exit and the turbine typically has an efficiency of 90% based on reversible, adiabatic expansion.

From the turbine exit, the working fluid enters the condenser where it rejects heat, ''-Q{{sub|C}}'', to the cold sea water. The condensate is then compressed to the highest pressure in the cycle, requiring condensate pump work, ''W{{sub|C}}''. Thus, the Anderson closed cycle is a Rankine-type cycle similar to the conventional power plant steam cycle except that in the Anderson cycle the working fluid is never superheated more than a few [[degrees Fahrenheit]]. Owing to viscosity effects, working fluid pressure drops in both the evaporator and the condenser. This pressure drop, which depends on the types of heat exchangers used, must be considered in final design calculations but is ignored here to simplify the analysis. Thus, the parasitic condensate pump work, ''W{{sub|C}}'', computed here will be lower than if the heat exchanger pressure drop was included. The major additional parasitic energy requirements in the OTEC plant are the cold water pump work, ''W{{sub|CT}}'', and the warm water pump work, ''W{{sub|HT}}''. Denoting all other parasitic energy requirements by ''W{{sub|A}}'', the net work from the OTEC plant, ''W{{sub|NP}}'' is

:<math> W_{NP}=W_{T}-W_{C}-W_{CT}-W_{HT}-W_{A} \,</math>

The thermodynamic cycle undergone by the working fluid can be analyzed without detailed consideration of the parasitic energy requirements. From the first law of thermodynamics, the energy balance for the working fluid as the system is

:<math> W_{N}=Q_{H}-Q_{C} \,</math>

where {{math|1=''W{{sub|N}}'' = ''W{{sub|T}}'' + ''W{{sub|C}}''}} is the net work for the thermodynamic cycle. For the idealized case in which there is no working fluid pressure drop in the heat exchangers,

:<math> Q_{H}=\int_{H}T_{H}ds \,</math>

and

:<math> Q_{C}=\int_{C}T_{C}ds \,</math>

so that the net thermodynamic cycle work becomes

:<math> W_{N}=\int_{H}T_{H}ds-\int_{C}T_{C}ds \,</math>

Subcooled liquid enters the evaporator. Due to the heat exchange with warm sea water, evaporation takes place and usually superheated vapor leaves the evaporator. This vapor drives the turbine and the 2-phase mixture enters the condenser. Usually, the subcooled liquid leaves the condenser and finally, this liquid is pumped to the evaporator completing a cycle.