Basic calculations of reverse osmosis

1.Recovery rate

recovery rate =Permeate?flow/ Feed flow

2.Concentrate flow?

=?Feed?–?Permeate

3.Recovery Rate Vs. NO of Stages

4. SDI Vs. Membrane Flux

5. Number of element NOE

NOE=Permeate?Flow?(m3/h)/?(Average?Flux?(m3/h)Membrane?Surface?Area?(m2))

6. Number of Pressure vessels (NOP)

NOP =?NO?of?Total?elements/?No?of?elements?per?vessel

The number?of?stages?can?be?determined?depending?on?the?recovery?rate.

7.Number?of?stages?(For?6?element?vessels)? (Brackish?Water)

8. Salt rejection (SR)

SR=(Cf –Cp/Cf )100%

SR= 1– (Cp /Cf) *100%

9. Salt passage( SP)

SP= Cp /Cf*100%

Salt passage is the opposite of salt rejection.

SP=100%– SR

10.RECOVERY AND CONCENTRATION FACTOR

Recovery (R) (conversion). Recovery is by definition the part of feed water that is converted in product water and is expressed as percentage.

R=Qp/Qf *100%

Where:

Qp= permeate or Product water flow rate (m3/h)

Qf= Feed water flow rate (m3/h)

Balance in a RO system

Qf = Qp + Qc

Then multiplying by the concentration of salts in each stream, we have:

Qf .Cf = Qc Cc + Qp. Cp

=1– R

11.The concentration factor (CF) in a RO system is by definition:

CF =Cc/Cf

Cp = Cf · (1 – SR)

Cc =Cf [1– R *(1– SR)]/ (1– R )

result:

CF = [1– R *(1– SR)]/ (1– R )

Since the salt passage (SR) is usually low?≈ 0, the result:

CF =1/1– R

Cc =Cf/1– R

Assuming salt rejection 100 %, the concentration factor for various recoveries is presented in table .

Table Concentration factor versus recovery

Recovery% ?????????? Concentration factor

30?????????????????????????????????????????? 1.4

40?????????????????????????????? ??????????? 1.7

50 ????????????????????????????????????????? 2.0

75 ????????????????????????????????????????? 4.0

90 ????????????????????????????????????????? 10.0

12.PRESSURE DROP

Pressure drop in feed – concentrate channel can be calculated with the formula of Schock and Miquel.

Δ P= 0.5·f ·ρ ·v^2 ·L/Dh

f= 6.23 *Re^-0.3

Re = ρ v Dh/μ

P= 0.5·ρ ^0.7 ·v^1.7 ·μ ^0.3/Dh^0.7

Where:

ΔP = pressure drop across spacer f = friction coefficient

ρ = density water v = velocity Re Reynolds number

L = length membrane Dh = hydraulic diameter

13.Concentration polarization factor

The concentration polarization factor (β) can be calculated with the following formula:

β = Cm/Cb= e^(J/k)

flux= [J] &[k]=mass transfer coefficient

In practice, the formula is simplified to:

β =Kp · e ^(Qp/Qfavg )

Where: Kp is a proportionality constant depending on the module geometry.

This simplification is justified by the fact that:

i)??????????????? Qp is proportional to J

ii)?????????????? Qf avg is the average feed flow and is proportional to k and k is almost proportional to the cross flow velocity (v).

Using the arithmetic average of feed and concentrate flow as average feed flow, the concentration polarization factor can be expressed as a function of the permeate recovery rate of a membrane element Ri.

β =Kp *e^(2·Ri/2?Ri)

Where:

β = CPF = concentration polarization factor

Kp =constant depending on type (manufacturer) membrane (usually 0.99)

Qp =permeate flow of an element (m3/h)

Qc =concentrate flow in an element (m3/h)

Ri =recovery of a membrane element

The value of the concentration polarization factor of 1.2, which is the recommended

Hydranautics limit, corresponds to 18 % permeate recovery for a 40 inches long membrane element

In sea water RO systems, the concentration polarization will decrease with increasing recovery. Consequently, the CPF is the last element is lower than in the first element. The reason for that is that the flux is reducing dramatically with increasing recovery.

14. TEMPERATURE AND WATER QUALITY

The higher is the water temperature the higher the permeability will be. The change in permeability is about 3 % per °C. Kw is linked with the viscosity of water.

TCF = 1.03^(t -25)

When dealing with the membrane permeability, the correction will be as follows:

Kwt = K25C *1.03^(t -25)

Where:

TCF temperature correction factor

t = temperature in°C

Kwt= membrane permeability at temperature [t]

Kw25 =membrane permeability at 25 °C

As a result of the temperature effect of viscosity and therefore on membrane permeability, required pressure to achieve or keep a certain flux (capacity) will be lower at higher temperatures.

15. ENERGY CONSUMPTION

To pressurize water cost energy. The theoretical minimum energy can be calculated with the formula:

E = 0.0275 ·*P

Taking into account the efficiency of the pump the formula changes into

E= 0.0275 *P/Npump

In RO and NF the recovery is less than 100%. As a consequence, the energy consumption per m3 water produced will be higher according the formula:

E= 0.0275 P/ηpump R

Where:

E = energy consumption in kWh/m3

P = pressure in bar

η pump = efficiency pump + motor

R = recovery

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