A Carnot engine between the Earth's poles and equator

[russ_watters]

I am not sure what kind of heat exchangers would be required to absorb/dissipate heat. But if the pressure difference is too high, we need thick heat exchangers, decreasing the rate of heat exchange.

@NTL2009 is correct that there is no pressure issue here. If the working fluid is water, the pressure is the same inside and outside the pipe/hx, so the pipe/hx is not under any stress.

 

[NTL2009]

@russ_watters confirmed my expectation that the pressures would be equal in/out, so this isn't a design restraint.

This is a rough estimate for quantity of fluid transported want to generate 1 GW of electricity.

Assuming a heat capacity of 4KJ per Kg per degree. We have heat capacity of 200KJ over 50 degrees.
If we generate electricity at 10 pc efficiency, we need to transport 10 GW of heat.
Which amounts to transport 50000 Kg of working fluid per second.
At a density of 1000Kg/m^3 we have to transport 50m^3/s of working fluid.
With a flow velocity of 10 m/s, we get a cross section area of 5m^2.I am not sure what kind of heat exchangers would be required to absorb/dissipate heat. But if the pressure difference is too high, we need thick heat exchangers, decreasing the rate of heat exchange.
But it gives a rough idea with regard to the scale/nature of the problem.

I didn't double check your heat capacity/flow numbers, but if correct, what kind of energy would be needed to pump that volume through ~ 2 km of pipe (down and back up)? That alone might make this an energy sink rather than source? But at least something on the order of a 5m^2 area pipe is not anything crazy (< 3 meter or 10 foot diameter). Supporting over 1 km of it in the ocean could be though!

Though the heat exchanger also might not be all that much of a challenge. The fluid would be losing heat to the ocean progressively on the trip down to that ~4~5C water. You would need to insulate the pipe on the way up, so as not to lose it back again. A 1 km pipe will have a lot of surface area, maybe just a relatively short distance of parallel paths (to increase surface area) horizontally at that 1 km depth would do the job? edit/add - OK, so at 10 m/s velocity, the 1 km uninsulated pipe would 'see' the working fluid for ~ 100 seconds. But those calculations would take me some time/research for even a rough estimate, and impossible on my first cup of coffee. I suspect this is a fairly easy calulation for someone with expertise in this area.

 

[Prathyush]

But those calculations would take me some time/research for even a rough estimate, and impossible on my first cup of coffee. I suspect this is a fairly easy calulation for someone with expertise in this area.

I suspect you can look at the density difference between hot and cold water and get an estimate.

I am not entirely clear about how density of water changes with temperature and pressure.(data is available though and it needs careful analysis)

For the purpose of estimation, I will assume density difference is 4 kg/m^3 based on https://www.open.edu/openlearn/science-maths-technology/the-oceans/content-section-3.2.
(thought it's probably a very rough estimate)

we have to transport 50m^3/s of working fluid

Assuming that estimate is not unreasonable, for a density difference of 4 kg/m^3, effectively 200 kg needs to be lifted by 1 KM every second. Which gives an estimate of 2 Mega watts of power.

( I will recheck my calculations once again though tomorrow. It feels suspiciously small. Perhaps I should worry about friction losses.)

 

[NTL2009]

I suspect you can look at the density difference between hot and cold water and get an estimate.
...
Assuming that estimate is not unreasonable, for a density difference of 4 kg/m^3, effectively 200 kg needs to be lifted by 1 KM every second. Which gives an estimate of 2 Mega watts of power.

( I will recheck my calculations once again though tomorrow. It feels suspiciously small. Perhaps I should worry about friction losses.)

Ahhh, I had not considered that the water being lifted would be cooler and more dense (by a few %, but could be a significant weight to lift at these scales), I'll try checking your numbers later.

What I was thinking of, but neglected to mention, was those friction losses - that's what I can't calculate w/o some research.

 

[gmax137]

With a flow velocity of 10 m/s, we get a cross section area of 5m^2.

Is there a basis for the 10 m/sec? In design for process piping, a value of 7 ft/sec or 2 m/sec is a typical maximum.

But, I don't know if that thumbrule applies to large concrete pipe. And it would be large: using 2 m/sec would make your area 25 m^2 (diameter 5.6 m or 18.5 ft).

 

[jrmichler]

The best way to calculate pressure drop through long pipes is by using the Moody chart (search the term). This Moody chart is from the Wikipedia article:

You can check your calculations by comparing to the following page from Cameron Hydraulic Data, 16th Edition:

Note that head loss is typically expressed as feet of head per 100 feet of pipe. One foot of head is the pressure from a column of liquid one foot high. Multiply feet of head by density in lbs/ft^3 to get pressure in lbs per ft^2.

It's a very important calculation to get right in OTEC (search the term) conceptual calculations.

 

[Prathyush]

It's a very important calculation to get right in OTEC (search the term) conceptual calculations.

Thank you. A recent paper https://www.mdpi.com/2077-1312/9/4/356 does a review of the technology and possible implementations.

One sentence caught my attention "In a closed-loop OTEC system, the highest cost of implementation lies in the heat exchangers. These devices are responsible for transferring heat from one place to another,
and are separated by a contact wall."

The best way to calculate pressure drop through long pipes is by using the Moody chart

Thank you again, I will make basic calculations estimating pumping energy cost for various configurations.(in a bit)

Is there a basis for the 10 m/sec?

It was an arbitrary choice based on vague impressions.

 

[Prathyush]

At a density of 1000Kg/m^3 we have to transport 50m^3/s of working fluid.

This estimate is not reasonable for a pure OTEC system (where the only heat source is surface warm water. if heat can be harvested from another source and $\Delta T$ is increased, efficiencies can be increased)

Extracting heat from small thermal gradients is a technical challenge. The following paper reviews the topic.
(A review of research on the closed thermodynamic cycles of ocean thermal energy conversion).

I don't understand the details, but it seems typical efficiencies start at around 2.4pc for the simpler cycles and reaches upto 5.1 pc for more complicated multi component cycles. These are theoretical results, only recently obtained.

Assuming a heat capacity of 4KJ per Kg per degree. We have heat capacity of 200KJ over 50 degrees.
If we generate electricity at 10 pc efficiency, we need to transport 10 GW of heat.
Which amounts to transport 50000 Kg of working fluid per second.
At a density of 1000Kg/m^3 we have to transport 50m^3/s of working fluid.
With a flow velocity of 10 m/s, we get a cross section area of 5m^2.

To revise this estimate, We assume efficiency of 2.5 pc and $\Delta T = 20^{\circ}$, volume flow rate increases by 10 times to 500 m^3/s. And if we double the efficiency to 5 pc, it will be 250 m^3/s.

If we take the previously suggested 2 m/s flow rate, we get we get a pipe area between 125-250 m^2(6-9m radius). These calculations are for 1 GW of power generated before pumping costs.

 

[Prathyush]

I ran the numbers for a Carnot engine running between poles and equator and estimate diameter of the tunnel after accounting for pumping costs.

Calculation is done for an engine generating 1 Terawatt power, running at a temperature difference of 90 degrees, operating between 5 and 95 degrees. We assume 20 pc of the energy generated is used for pumping.

I will write down all the formulas so that it can be checked.

For the calculation I will assume a friction factor of .01. (The actual friction factor is slightly lower, which we will calculate at the end. We are overestimating friction by ~30pc, to leave some headroom)

$$ f = .01$$

The head loss per unit length is given by the formula

$$\frac{\Delta p}{L} = \frac{f \rho v^2}{2 d} $$

We assume the engine works at efficiency at peak power.( $T_h = 95$ and $T_c = 5$)

$$ \eta = 1- \sqrt{\frac{T_c}{T_h}} = .131$$

The potential to do work per Kg of water is ( c = specific heat of water )

$$ w = c \eta \Delta T = 49300 J/Kg$$

A fraction of the energy $p_f = .2$ is used for pumping. We assume pumping efficiency is 75 pc. In one cycle work can be extracted twice from the same mass, one at the pole when cooling, and another at the equator when heating, we will assume it is identical. ($h_{head} = \frac{p_f w}{g}$, L = 10000 Km} )

$$ \frac{\Delta p}{L} = .75 \frac{p g h_{head} }{L} = .739 pascal/m $$

mass flow rate M to generate 1 Terawatt of power is $M = \frac{10^{12}}{w (1-p_f)} = 2.5 \times 10^7 kg/s$ and Volume flow rate $Q = \frac{M}{\rho} = 25\times 10^3 m^3/s$

The formula $$\frac{\Delta p}{L} = \frac{f \rho v^2}{2 d} = \frac{ 8 f \rho Q^2 }{\pi^2 d^5}$$ can be inverted to get the diameter

$$d = \sqrt[5]{\frac{ 8 f Q^2 \rho }{ \pi^2 (\Delta P/L) } } = 93 m$$

velocity of flow is $v= \frac{Q}{\pi d^2} = .92 m/s$

We use Colebrook-White equation iteratively to estimate the friction factor for these flow parameters, with surface roughness $e = .25 mm$ and reynolds number $Re = 3.9 \times 10^8$. We get $f = 0.0067$ In principle the calculation can iterated over, and if that is done we get $d = 86m$. But some head room may be useful.

I will repeat the same for other cases of interest for different kinds of cases and make general comments on a separate post. But is the general method OK for the problem as stated?

 

[jrmichler]

Looking at some of your assumptions, but not your equations...

Per NOAA, the average deep (below 200m) ocean temperature at the North pole is about 4 deg C, and the temperature near the equator is 30 deg C. There will be a temperature difference across the heat exchanger walls. The ocean temperature will vary as it travels through the heat exchanger, while the temperature of the condensing/boiling fluid will be roughly constant. The temperature for the Carnot efficiency is the hot and cold temperatures of the working fluid, not the incoming temperatures of the ocean water. The ocean water temperature leaving the condenser will be colder than the working fluid temperature, and the ocean water temperature leaving the evaporator will be warmer than the vapor temperature.

The average temperature difference across the heat exchanger is the average of the entering and leaving temperature differences. Increasing water flow reduces water temperature rise/fall, while increasing pumping power. Decreasing average temperature difference across the heat exchanger requires a combination of increased water flow and increased heat exchanger area. Pumping power must be calculated for the ocean water lines plus the heat exchanger pressure drop.

OTEC system design is an iterative process. Without fully thinking it through, here's how I would approach this problem:

1) Assume initial values for cold side ocean temperature and depth, warm side ocean temperature and depth, distance between the two, and cold and warm side working fluid temperatures. Assume a design for the heat exchanger, but not the heat exchange area.

2) Using the numbers from above, calculate the required heat exchanger size for different water flow rates. You will need to calculate the heat transfer coefficients.

3) Calculate water pressure loss in the pipelines and heat exchanger on both the hot and cold sides. This is a good place to assume so pipe diameters, and do the calculation for each. There will also be pressure losses in the working fluid.

4) You now have flow rates and pressure drops on the water sides. Calculate pumping power.

5) You now have hot and cold side temperatures of the working fluid at the engine after allowing for the pressure drops between the heat exchanger and engine. Calculate the Carnot efficiency, then search for actual efficiencies as a percentage of Carnot efficiency for this temperature range.

6) Show the results in a table or tables - temperatures, flow rates, pipe diameters, heat exchanger areas, pumping power, engine power at output shaft, and net output power.

7) Enjoy - it looks like a fun project.

 

[NTL2009]

I was assuming (maybe incorrectly) that the idea was to use solar collectors (maybe concentrated) to raise the temperature of the working fluid for the (Sterling? Turbine?) engine, and use the ocean at ~ 1 km depth (or at the poles) to cool the fluid back down to maintain a large temperature delta. Working with just the relatively small delta of surface water temperature to near freezing deep water temperatures is just not much to work with.

 

[Prathyush]

I was assuming (maybe incorrectly) that the idea was to use solar collectors (maybe concentrated) to raise the temperature of the working fluid for the (Sterling? Turbine?) engine, and use the ocean at ~ 1 km depth (or at the poles) to cool the fluid back down to maintain a large temperature delta.

Yes. That as the basic idea. I am hoping it would become economical at a large scale implementation. Once a tunnel/pipe is build it is a thermodynamic resource you can use for a very long time.

Per NOAA, the average deep (below 200m) ocean temperature at the North pole is about 4 deg C, and the temperature near the equator is 30 deg C.

One way to implement is to dig a tunnel 1-2 KM deep from deep ocean to a hot spot of solar radiation. You can let high pressure at deep sea to do most of the work in transporting water. It's an altitude sensitive thing.

The cost of tunneling is significantly lower(~100-1000KM) and can have benefits of network effects. You can then use a lower condensation temperature. You still need to deal with brine water, unless you dig another tunnel to discard the water ( or the extract salts and freshwater as a resource)

Assume a design for the heat exchanger, but not the heat exchange area.

I need to learn heat exchanger physics. I was hoping that practical efficiency numbers could abstract out the details.(but it is something I will look into slowly)

 

[Keith_McClary]

 

[Prathyush]

I did quick search on papers discussing extracting energy from the gulf stream.

They typical estimates seem to range around 10 GW .

https://www.researchgate.net/publication/256495742_Theoretical_Assessment_of_Ocean_Current_Energy_Potential_for_the_Gulf_Stream_System
"Assuming a 30% conversion efficiency from energy removal from the flow to electrical power, turbines yield
a peak energy potential for electricity production of about 13 GW from the Gulf Stream system"

EVALUATING THE POTENTIAL FOR ENERGY EXTRACTION FROM TURBINES IN THE GULF STREAM SYSTEM
"providing an estimate by assuming a 30% power conversion efficiency for energy removal from the flow to electrical power [19] yields an average potential for electricity production of about 1.5 GW from the Florida Current and 5.6 GW from the entire US portion of the Gulf Stream system or 13 and 49 TWh/yr, respectively. "

In comparison peak power recorded in US is about 700 GW. It seems (not sure) the gulf stream can only contribute a modest part of the energy mix.

 

[jbriggs444]

I was assuming (maybe incorrectly) that the idea was to use solar collectors (maybe concentrated) to raise the temperature of the working fluid for the (Sterling? Turbine?) engine, and use the ocean at ~ 1 km depth (or at the poles) to cool the fluid back down to maintain a large temperature delta. Working with just the relatively small delta of surface water temperature to near freezing deep water temperatures is just not much to work with.

Yes. That as the basic idea. I am hoping it would become economical at a large scale implementation. Once a tunnel/pipe is build it is a thermodynamic resource you can use for a very long time.

If you have a high temperature solar heat source then I do not see the point of going to huge expense to put your cooling towers one kilometer deep in an ocean that is (perhaps) tens or hundreds of miles away.

If you are gathering solar input from a large surface area then air cooling over that surface area is clearly sufficient -- because it was already sufficient before building the solar collector.

You only need low temperature cooling if you have a low temperature heat source.

 

[NTL2009]

If you have a high temperature solar heat source then I do not see the point of going to huge expense to put your cooling towers one kilometer deep in an ocean that is (perhaps) tens or hundreds of miles away.
...
You only need low temperature cooling if you have a low temperature heat source.

That makes sense - I think I got drawn into the OPs idea of using near freezing water on the sink side.

The ocean would make a very good sink, due to size and thermal characteristics of water. Whether that water is at surface temperatures, or near freezing wouldn't be that significant if the hot side is near or above boiling. As an example, 200F - 70F is a 130F delta, get that water cooled to 35F, and you've only increased the delta to 165F, unlikely to be worth the construction costs and ongoing pumping costs to push that water through the heat exchange, plus maintenance.

Let's see, convert to Kelvin for that efficiency value... 366C, 294K, 275K

1 − ((294 ∕ 366)^0.5) ≈ 0.10374184
1 − ((275 ∕ 366)^0.5) ≈ 0.13318623

About a 28% improvement in efficiency (relative improvement, ~ 10% to 13%, ~ 3 points, not 28 points!)
Significant, but I'm certain it would not warrant the expense and complexity. Even less of an improvement if you use an intermediate fluid with a higher boiling point, or pressurize water as the working fluid in a concentrated solar collector.

 

[jbriggs444]

if the hot side is near or above boiling

After a little Googling on concentrated solar power generation, it seems that the hot side temperature is often higher than this. There are a variety of designs.

[In a parabolic trough design]
Common fluids are synthetic oil, molten salt and pressurized steam.
[...]
[In a power-tower design]
Within the receiver the concentrated sunlight heats molten salt to over 1,000 °F (538 °C)

 

[Prathyush]

Significant, but I'm certain it would not warrant the expense and complexity.

I will do a detailed estimate sometime in the future. The economics roughly works like this.

Consider, Ouarzazate Solar Power Station.

Water consumption for the Ouarzazate Noor complex is estimated at 2.5 to 3 million $m^3$ per year for cleaning and cooling. It produces 1470 GWH of energy annually.

China built, Dahuofang water tunnel 85.3 Km of water tunnel at 750 million USD. It's cross section is 50 $m^2$ and diameter is 8m.

Assume we need 10x the size of tunnel and we can estimate the cost at 7.5 billion USD. At 10 cents a KWh. We need to produce 75 Billion KWh = 75,000 GWh of extra energy energy.

Ouarzazate Noor complex produces 1470 GWh of energy annually.

Estimating efficiency gain between cooling between 45 degrees and 5 degrees as (there is no water in Sahara, so should be efficiency should be compared to Air cooled thermal plants which I believe is less efficient.)

$$ \frac{ 1- \sqrt{\frac{278}{600}} }{1- \sqrt{\frac{318}{600}} } - 1 = .174 $$

If we can produce 17.4 pc of 1470 GWh ~ 255 GWh extra annually. You have a pay back period of 294 years.
If you should dump the brine water back in the ocean. You need 2 tunnels, so 588 years

If you can make a Plant 25X in size (assuming same sized tunnel), you get a pay back period of 23.5 years.

At .5 m/s a 50 m^3 cross section tunnel is capable of transporting 788 million $m^3$ of water a day.
Well above 3 million*25 = 75 million $m^3$ obtained from extrapolating.

That is a very very rough estimate, which I will sharpen slowly. I have not estimated the cost of pumping in this estimate.(I will work it out sometime. If you can collect energy when the water is pumped back into the ocean it can be made lower. Typical altitude of Sahara is under a KM)

One you have a tunnel network the incremental cost of more plants should should also be considered. For instance Libya built Great Man made river. It's a tragedy that the country has now been destroyed.

You can also extract fresh water from such a setup using a vapor pump. So economic value of water in Sahara can also be estimated. In principle concentrated brine can be used to extract certain minerals using osmotic methods, further optimizing economic value.

It requires more careful consideration, especially with a detailed project on where and how. The construction cost of Ouarzazate Solar Power Station is 2.5 billion. A plant 25X larger plant is 62.5 billion. So it may start making economical sense if the project size is in the order of 70 Billion.