New: Osmosis Desalination and Carnot

Loss 
Type 
Reduce by 
The price 
P_{s}∙V_{permeate} 
Thermodynamic transformation of mechanical energy into heat. 
Decrease the osmotic pressure P_{s} by reducing the water recovery ratio. 
Higher consumption of seawater. 
ΔP ∙V_{permeate} 
Dissipate heat of water flow through the membrane 
Lower water throughput. 
Lower utilization of the desalination plant 
P∙V_{concentrate} 
Dissipate heat of water flow through the pressure control valve. 
Application of energy recovery or pressure exchange devices. 
More equipment 
Note:
W = P∙V
Where W is the pump work, P is the pump pressure, and V is the volume of pumped
water.
W = P∙V ∙ 100
for W in Joules (Watt seconds), P in bars, and V in Liters.
Or,
W = P∙V / 36
for W in kWatt hours, P in bars, and V in cubic meters.
For example, the energy required to pump a volume of V = 1 cubic meter of salt water with osmotic pressure of
P_{s} = 27 bar, through a semi permeable membrane , is:
W = 27 ∙ 1 / 36 = 0.75 kWatt hour.
6. A spiral membrane module
Figure6 shows the water flow within a
spiral membrane module
Figure6: Spiral membrane module.
The membrane is shaped into a spirally wound flat sleeve (green) contained in a highpressure cylinder. The sleeve is closed at the spiral sides and outer end, and the inner end is connected to a central pipe. Pressurized seawater (red arrows) flows in the direction of the cylinder axis along the external surface of the membrane sleeve. Some water penetrates through the membrane and leaves the salt behind, thus it turns into permeate water within the sleeve. Permeate water (blue arrows) flows within the spiral sleeve towards the central pipe that leads it out of the module.
Water that penetrates through the membrane leaves behind it a locally highly concentrated salt at the external surface of the membrane. This concentrated salt immediately stops any further water flow through the membrane unless it is removed fast enough by a lateral seawater flow along the surface.
The membrane sleeve is supported from its inside with a porous spacer that prevents sleeve collapse by the osmotic pressure. Another porous spacer surrounds the sleeve and stabilizes the space of seawater flow.
The module manufacturer supplies the testing conditions of the membrane module. For example, the following data is given for "FILMTEC 8" Seawater RO Elements" (SW30HR380) by "DOW" [13].
Module size: Length 1016 mm, diameter 201 mm, diameter of central pipe 29 mm.
Operating pressure: 55.2 bar (800 psi), (max 70 bar (1015 psi).
Max Feed
Flow: 14 m^{3} / hour.
Product water flow rate: 23 m^{3} /
day at 25^{o} C.
Single element recovery (Permeate Flow to Feed
Flow): 0.08 (max 0.15 at lower feed flow).
Salt (NaCl) concentration: 32000
ppm (32 gram / liter).
These numbers give:
Feed water flow of 200 (max 233) liter / minute.
Permeate water flow of 16 liter / minute.
Osmotic pressure of 27 bar
(390 psi), calculated by van't Hoff formula (equation 1).
Flow rate factor
(equation 2):
K_{f} = F_{rate} / (P  P_{s}) = 16 / (55.2  27) = 0.57 (liter / minute) / bar (9)
7. Cyclic flow operation
Sections 2  6 describe the reverse
osmosis technology of seawater desalination. The rest of these pages are
theoretical considerations and calculations by the author.
Semi permeable membranes favor operation with continuous water flow and permanent operating pressure. Flow disturbances and unstable pressure stress the membranes and increase their wear. However, the continuous flow mode requires application of energy recovery devices for efficient operation.
An operation mode of cyclic flow may achieve, in principle, energy efficiency comparable to continuous flow and there is no need of energy recovery devices. Therefore, this possibility may not be ignored, even for a price of modifying the semi permeable membrane or the membrane module.
The system described in figure7 includes a low pressure circulating pump and a twostate valve.
Figure7: Seawater desalination in a cyclic water flow.
At one state of the valve the saltwater compartment of the module is closed. The highpressure pump pumps seawater into the membrane module and all the water penetrates through the membrane and turns into permeate water since there is no other water exit. The lowpressure pump circulates the water in the module at a flow rate required by the module manufacturer for proper operation. Since there is no exit for the salt it will accumulate within the module and steadily increase the osmotic pressure. At a pre determined osmotic pressure the valve revolves and relieves the pressure within the module.
At this state of the valve the two pumps drive the concentrated salt water out of the module and replace it with fresh seawater. The valve then revolves again and the operation is repeated.
Pressure release of concentrated salt water by valve revolution does not waste energy, similarly to the case of the "rotating door" (section 4), since water is incompressible and does not accumulate energy. However, there are other energylosses that will be considered later.
In cyclic operation the highpressure pump pumps a volume of seawater equal to the volume of delivered permeate water. In this respect it is equivalent to continuous operation with an energy recovery device. Only here there is no such a device. Efficient continuous operation without energy recovery is achieved with deep sea deslination by reverse osmosis [14  15].
8. Desalination energy, salinity and cycle time in cyclic flow operation
Since the pressure increases with the salt concentration of
saltwater within the module, the work of pumping water through the membrane
will be:
W = ∫p∙dV = ∫(P_{s} + ΔP) ∙dV (10)
where P_{s} is the increasing osmotic pressure. The over pressure ΔP is determined by the flow rate of the highpressure pump ΔP = F_{rate} / K_{f}.
The salt concentration c_{s} within the module is given by:
c_{s} = c_{sea}∙ (V + V_{0}) / V_{0} (11)
where c_{sea} is the salt concentration of seawater, V_{0} is the saltwater volume within the module, and V is the delivered permeate water. Since the osmotic pressure is proportional to the salt concentration it is given by a similar equation:
P_{s} = P_{sea}∙(V + V_{0}) / V_{0} (12)
The work of desalinating a volume V of permeate water will be (by inserting equation 12 into equation 10 and integration):
W = P_{sea}∙(0.5∙V^{2} / V_{0} + V) + ΔP∙V (13)
Or:
W = (P_{sea}∙(0.5∙V / V_{0} + 1) + ΔP)∙V (14)
Or:
W = (P_{sea}∙(1  α / 2) / (1  α) + ΔP)∙V (15)
Where α = V' / (V' + V_{0}) is the recovery ratio. V' is the volume of permeate water delivered in one cycle.
In cyclic operation there is no need to connect modules in series. This is an advantage that leads to higher permeate water throughput.
Salinity, the salt concentration in permeatewater for 1% salt penetration through a semi permeable membrane, is:
Salinity = 0.01∙ [∫c_{s}∙dV] / V (16)
where c_{s} is the salt concentration of salt water within the module
and V is the volume of permeate water.
c_{s} =(c_{sea} /
P_{sea})∙P_{s} by using equations (10)  (12), therefore:
Salinity = 0.01∙ (c_{sea} / P_{sea})∙ [∫P_{s}∙dV] / V = 0.01∙ c_{sea} ∙ (1  α / 2) / (1  α) (17)
The cycle time in cyclic operation depends on the seawater volume within the module. Using the module dimensions in section 6 its internal volume is estimated to be 32 liter. Assuming that half of this volume is solid material, membrane and spacers, and the rest is divided to equal volumes of saltwater and permeatewater, the saltwater volume will be V_{0} = 8 liter. This is a coarse estimate.
The permeatewater recovery ratio is α = V' / (V' +
V_{0}) , where V' is the permeatewater delivered per cycle.
V' = F_{rate} ∙ t, where F_{rate} is the permeatewater flow rate and
t is the cycle time. Therefore, the cycle time in seconds is:
t = 60 ∙ (V_{0} / F_{rate}) ∙ α / (1  α) (18)
Calculated values of the desalination energy, salinity, cycle time and water throughput are given in the next section.
The cycle time may be increased by connecting an auxiliary tank in series to the salt water side of the membrane module. It is also possible to alternately connect two tanks so that in one tank pressurized water circulates with increasing salt content while the other tank is flushed with seawater and vice versa. In this case the membrane module may be loaded under permanent pressure. Such a system, however, requires the operation of more valves.
9. Comparison of continuous flow to cyclic flow
The comparison is done for the testing parameter values mentioned in section 6. Higher values may
be applied to practical operation, though they should not exceed the operating
limits.
a. Continuous flow system equipped with 6 modules connected in series, and
with a 100% efficient energy recovery device.
Table 2 summarizes the
operation parameters.
Feed Flow 
Permeate Flow (liter / min) 

Pressure 
Feed (L/min) 
1st Module 
Water Recovery 
Energy 
Salt 

bar 
Pump 
Recovery 
% 
V_{1} 
α (%) 
V 
kWh/m^{3} 
mg/L 
55.2 
78 
155 
7 
16 
33.5 
78 
1.53 
377 
55.2 
74 
136 
8 
16 
37 
74 
1.53 
385 
55.2 
50 
50 
16 
16 
50 
50 
1.53 
414 
45.4 
50 
150 
5.2 
10.5 
25 
50 
1.26 
360 
The table is calculated by the equations:
P_{s}(1) = P_{sea} (19)
Permeate(i) = K_{f} ∙ (P_{pump}  P_{s}(i)) (20)
Supply(i) = Σ(j = 1 to i) Permeate(j) (21)
P_{s}(i + 1) = P_{sea} ∙ Feed / (Feed  Supply(i)) (22)
W = P_{pump}∙V (23)
Salinity = 0.01 ∙ c_{sea} ∙ (Σ Permeate(i) ∙ Ps(i) / Psea) / Supply(6) (24)
P_{s}(i) is the osmotic pressure in the i'th module.
Permeate(i)
is the permeate water flow of the i'th module.
Supply(i) is the sum of
permeate water flows of the first i modules.
P_{sea} = 27 bar is the
osmotic pressure of seawater at 300 K (27^{o} C).
c_{sea} =
32 gram / liter is the salinity of seawater.
P_{pump} is the pump
pressure in bars, given in the table.
Kf = 0.57 (liter / minute) / bar is
the flow rate factor.
α is the permeatewater
recovery ratio.
V is the volume of delivered permeatewater.
V_{1} is the volume of permeatewater delivered by the first module.
Feed, given in the table, is the water feed flow through a module. Feed is
the same for all modules since they are connected in series.
The work of desalination per 1 m^{3} of permeatewater is:
W/V = P_{pump}∙100 (Joule / liter = Watt second / liter) = P_{pump}∙100 / 3600 (kW hour / m^{3}) (25)
Salinity, the amount of salt in permeatewater is calculated for %1 salt penetration through semi permeable membrane.
Notes:
I. The calculation is somewhat
inaccurate since it assumes uniform salt concentration within each module while
the concentration does change within each one.
II. The desalination energy calculated in the table assumes 100% energy recovery. In practical systems, with lower energy recovery, the desalination energy will be higher than the table values, and the difference will increase as the water recovery ratio decreases.
III. Comparison of lines 1  3 demonstrates the effect of increasing the water recovery ratio by reducing the overall feed rate of seawater. Higher ratio saves preosmosis seawater but reduces the throughput of permeatewater.
IV. Comparison of lines 2  4 demonstrates the effect of pump pressure on the system performance. Higher pressure saves preosmosis seawater and increases the throughput of permeatewater, but also increases the energy of desalination.
b. Cyclic flow system equipped with 6 modules connected in parallel.
In a cyclic system there is no need to connect modules in series. The
modules are connected in parallel and the flow in each module is 1 / 6 of the
overall flow.
Table 3 summarizes the operation parameters for permeate water supply similar to table 2.
Feed Flow 
Permeate Flow (liter / min) 

Pressure (bar) 
Feed (L/min) 
1 module 
Recovery 
Energy 
Salt 
cycle 

ΔP 
P_{start} 
P_{end} 
Pump 
Flush 
% 
V_{1} 
α( ) 
V 
kWh/m^{3} 
mg/L 
sec 
22.8 
49.8 
63.4 
78 
155 
5.6 
12.8 
33.5 
78 
1.57 
401 
20.5 
21.6 
48.6 
64.5 
74 
136 
6.2 
12.3 
37 
74 
1.57 
414 
25.1 
14.6 
41.6 
68.6 
50 
50 
8.3 
8.3 
50 
50 
1.53 
480 
63.4 
14.6 
41.6 
50.6 
50 
150 
4.2 
8.3 
25 
50 
1.28 
373 
21.1 
28.2 
55.2 
70 
96 
173 
8.3 
16 
35.7 
96 
1.74 
409 
16.6 
The table is calculated for the pumping period only. The period required to flush the concentrated salt water in the module and replace it with fresh seawater is about 10% of the pumping period. Therefore, the overall cycle is about 10% longer than the table values, and the flow rates per overall cycle are about 10% lower than the table values.
The Feed and Water Recovery columns are identical to table 2 (except the last line), so that the two processes are compared for the same permeatewater recoveryratio and throughput.
The table is calculated by the equations:
ΔP = (V / 6) / K_{f }(26)
P_{start} =P_{sea} + ΔP (27)
P_{end} = P_{s} +ΔP = P _{sea} / (1  α) + ΔP (28)
W / V = (P_{sea}∙ (1  α / 2) / (1  α) + ΔP) / 36 (29)
ΔP is the over pressure that drives water flow
through the membrane.
V is the delivered volume of permeatewater.
V_{1} is the volume of permeatewater delivered by one module.
Kf = 0.57 (liter / minute) / bar is the flow rate factor.
P_{start} is the pressure at the start of the pumping cycle.
P_{sea} = 27 bar is the osmotic pressure of seawater.
P_{end} is the pressure at the end of the pumping
cycle.
α = V' / (V' + V_{0}) is the permeatewater recovery
ratio.
V' is the volume of permeate water delivered in one cycle.
V_{0} = 8 liter is the volume of salt water within a module.
W /
V is the desalination energy per 1 m^{3} of permeatewater (equation 15,
section 8).
The permeate water salinity is calculated by equation 17,
section 8.
c_{sea} = 32 gram / liter is the salt concentration of
seawater.
c. Conclusion
Comparison of the two tables indicates that the
energy of desalination in the two processes, operated at similar permeatewater
recovery ratios and throughputs, is practically the same. However, the two
processes have further energy losses not considered in the tables.
In the continuous flow process there is a full permeatewater flow only at the first module and the flow drops at each successive module. Therefore the capacity of permeatewater flow is not fully utilized. Compared to that, in the equivalent cyclic process the modules are connected in parallel and the permeatewater flow per module is lower than the permitted limit value. Alternatively (line 5 of table 3), the cyclic process can operate at the highest permitted permeatewater flow and achieve higher permeatewater throughput per module, though, at a cost of a higher desalination energy.
10. Difficulties with cyclic flow operation
Apart from variable pressure operation that might wear or even damage the
membrane, other factors should be considered as well. Any part of the system
that accumulates energy will waste it in the cyclic process.
Consider a possible expansion of the highpressure cylinder that contains the
membrane unit by the pressurized water in it. If the 201 mm diameter cylinder
expands by one millimeter its inner volume will increase by V = 0.4 liter. The
energy accumulated in the cylinder is equal to p∙ V / 2 and it is lost when the
pressure is relieved. Inserting
p = P_{sea} = 27 bar, and V = 0.4 ∙
10^{3} m^{3}, the energy will be E = (27 / 36) ∙ 0.4 ∙
10^{3} / 2 = 0.15 ∙ 10^{3 }kW hour per cycle. If a cycle
delivers about 8 liters of permeatewater, the energy loss will be 0.15 ∙
10^{3 }∙ 1000 / 8 = 0.02 kW hour per one m^{3} of permeate
water.
Similar loss might come from pressure squeezing of the permeatewater spacer within the membrane sleeve, and the loss can be calculated in a similar way. A more rigid spacer material, and possibly, mechanically pre squeezing the membrane unit within the cylinder, may reduce the loss.
When a number of modules are connected in parallel to one pump it is important to have similar water flow in each of them to within tight tolerance. Otherwise, in some modules the replacement of concentrated salt water with seawater will not be complete, while in other modules there will be excessive flow and loss of seawater.
The concentrated salt water within the membrane module is replaced by fresh seawater when the pressure is relieved. During this time permeate water will start to flow back through the membrane towards the saltwater. According to specs, the flow rate of saltwater, in parallel to the membrane, is at least ten times higher than the flow rate of permeatewater through the membrane. Therefore, the time of seawater replacement will be about ten times shorter than the time of permeatewater pumping, and the permeate water loss will be less than 10%. The back flow of permeatewater is not completely negative since it automatically flushes the membrane during each cycle.
11. Utilization of the energy accumulated within concentrated salt water
Figure8 presents a scheme for utilizing energy from concentrated salt water.
Figure8: Utilizing energy from concentrated salt water.
A lowpressure pump flushes one compartment of a membrane module with seawater, while a mediumpressure pump pumps concentrated salt water via the other compartment. The pressurized water drives a turbine that supplies mechanical energy. The pressure difference that drives water through the membrane is:
ΔP = P_{pump} + P_{sea}  P_{s} (30)
where P_{pump} is the pump pressure, P_{sea }is the osmotic pressure of seawater and P_{s} is the osmotic pressure of the concentrated salt water. If the pressure difference is negative, ΔP < 0, or, P_{pump} < P_{s}  P_{sea}, water will flow from the seawater side of the membrane towards the concentrated water side. The volume of water that drives the turbine is then equal to the sum of a volume V delivered by the pump, and a volume V_{1} that flows through the membrane. The work consumed by the pump is P_{pump}∙ V and the work that drives the turbine is P_{pump}∙ (V + V_{1}). Therefore there is a net energy profit of P_{pump}∙ (V + V_{1})  P_{pump}∙ V = P_{pump}∙ V_{1} that comes from dilution of the concentrated salt water.
The size of a membrane module for utilizing concentrated salt water is similar to that of a desalination module, and, as seen in figure8, it has four different water outlets instead of three. Therefore, addition of salt utilizing ability to a desalination plant practically means using two different types of membrane modules and doubling their number. In addition to that the energy utilizing process consumes more seawater.
Apart from investing in more membrane modules of a type that doesn't exist yet, the consumption of extra seawater makes energy utilization of concentrated salt water a nonbeneficial process. The same amount of extra seawater may alternatively be added to a standard desalination system and save more energy by the reduction of the water recovery ratio. The use of more seawater in a desalination system reduces the osmotic pressure within it, and the reduced pressure saves energy consumption in systems equipped with an energy recovery device, as discussed in section 4.
In summary of this section, there is no benefit in utilizing the (free) energy accumulated within the concentrated salt water. The same amount of seawater, required to dilute the concentrated salt, will achieve higher energy saving by adding it into a standard desalination system, without the need to invest in extra equipment.
12. Summary and conclusions
A cyclic operated system that does not
apply energy recovery devices is suggested for seawater desalination by reverse
osmosis. The desalination energy, product water salinity and system throughput
are comparable to that of continuous water flow systems that do apply energy
recovery devices.
Appendix: Energy recovery efficiency below 100%
Consider a system operating with a water recovery ratio α and with an energy recovery
device of efficiency Ef.
V = α ∙ V_{sea} is a volume of permeate water
and V_{sea} is the overall volume of seawater used to produce it. Out of the volume V_{sea}, a
volume V is delivered by the pump, and the rest of the seawater volume
V_{sea}  V = V∙ (1 / α  1) is delivered by the energy recovery device.
The work done by the pump is P ∙ V where P is the pump pressure. For the volume V ∙ (1 / α  1) delivered by the energy recovery device there is a need to add an energy (1  Ef) ∙ P ∙ V ∙ (1 / α  1) to compensate for the incomplete efficiency. Adding together the work of the pump and the energy added to the recovery device yields:
W = P ∙ V ∙ [1 + (1  Ef) ∙ (1 / α  1)] (31)
For example, the work for efficiency Ef = 0.95 and water recovery ratio α = 0.1 is
W = P ∙ V ∙ [1 + 0.05 ∙ 9] = P ∙ V ∙ 1.45, compared to P ∙ V, for the efficiency Ef = 1. Therefore, for a recovery ratio of 0.1, a system
with 95% efficient energy recovery device consumes 45% more energy than a system without any recovery loss.
The minimal desalination energy for recovery without loss is given by equation 8, P ∙ V = P_{sea} ∙ V / (1  α). Therefore the minimal desalination energy for a system including the energy recovery loss will be:
W_{min} = P_{sea} ∙ V ∙ [1 + (1  Ef) ∙ (1 / α  1)] / (1  α) (32)
See:
Osmosis Reverse Osmosis and Osmotic Pressure what they are
Desalination machine
Energy of Seawater Desalination
A Pipe of Fresh Water instead of "Canal of the Seas"
references
On the net: May, revised September 2002, Appendix added January, references added March 2003.
By the author:
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