# A Primer on RO Application: Wastewater Concentration Calculations

By Tanya Lubner, PhD

Reverse osmosis technology finds numerous applications at the point of use as a final-barrier treatment for a series of anionic contaminants that are either already regulated or are being considered for regulation. Among these contaminants are nitrite/nitrate (NO2-/NO3-), arsenate (H2AsO4-), chromate (CrO42-) and perchlorate (ClO4-). The advantage of employing a final-barrier technology is the ability to treat only the water used for consumption and to allow customers receiving water that already meets federal regulations to reduce contaminants identified as health hazards to an even lower concentration. Any water treatment application will eventually need to address the impact of the treatment process on the waste stream. Municipalities will often have disposal limits on a number of contaminants, especially those that are regulated health hazards. Commercial and industrial applications may require that the wastewater undergo additional treatment, such as denitrification, prior to disposal, and so the question may arise: “How much of contaminant X will be in a certain volume of wastewater?”

The answer to this question is based on the key concept that the passage of dissolved inorganic substances (salts) through the RO membrane is independent of the passage of water through the membrane. This means that the calculations for concentration of contaminant in the effluent (treated water or wastewater) and volume of wastewater are also independent of each other.

Consider the simplified diagram illustrating the RO process (Figure 1). Compartment A contains the raw water to be treated, compartment B is the treated water, the permeate. The two compartments are separated by the semi-permeable RO membrane. Regardless of whether or not water comes through the membrane from the raw-water side to the permeate side, equilibrium processes will allow some portion of salts to diffuse through the membrane at a regular rate. This rate is a function of the difference in salt concentration between the raw-water side and the permeate side and is illustrated in the following equation:

Js = Ks∆C
Where:
Js = rate of ionized substance (salt) passage
Ks = permeability coefficient for the salt for a particular membrane
C = concentration of the salt

Thus, the concentration of salt that will end up in the product water will depend on the membrane’s percent rejection rating. The overall concentration of salt in the product water is going to reflect how much water comes through the membrane to dilute the dissolved substances not rejected by the membrane.

For water to pass through the membrane, the treatment system has to overcome the natural phenomenon of osmosis that is trying to push the treated water in the permeate back through the membrane to dilute the higher concentration of salts in the raw-water side. The pressure with which the natural process is pushing the water back is the osmotic pressure, the value of which is a function of the specific composition of salts and their concentrations in the permeate and in high-percent recovery operations. It is also based on the average salt concentration of the permeate and concentrate. For initial calculations, such as those illustrated here, the osmotic pressure can be estimated as 1 psi for every 100 mg/L of total dissolved solids.

Any pressurized storage tanks on the system will also interfere with the passage of water by exerting a back pressure against the membrane on the permeate side. Therefore, the rate of water passage through the membrane from the raw-water side to the treated side will depend on the difference between the forward pressure applied to the membrane by the customer’s distribution system and the sum of the osmotic and back pressures, as illustrated in the following equation:

Jw = Kw [Pf – (π + Pb)]

Where:
Jw = rate of water passage
Kw = permeability coefficient of water for a particular membrane
Pf = forward pressure exerted by the distribution system
π = osmotic pressure
Pb = maximum back pressure exerted by the storage tank

Only a portion of the raw water coming into the RO system goes into the permeate. The remainder of the water acts to wash the surface of the membrane in a process called crossflow and is collected as the reject or the concentrate stream. The volume of this stream is the volume of the wastewater and it can be calculated from the system’s percent recovery. Percent recovery is the ratio of the treated water (permeate) volume to the raw-water (reject/concentrate) volume, as illustrated in the following equation:

% Recovery = (Volume permeate/volume raw water) * 100.

Now let’s look at how these concepts might be applied to answer a customer’s question on wastewater quantity and quality. The concentration of nitrate was reported on the customer’s water analysis as 78 mg/L as nitrogen (N). The RO membrane used has a 90-percent rejection rating. The system recovery is 50 percent. The customer wants to know how much wastewater will be needed to treat and what the concentration of nitrate will be in the wastewater on a daily basis. The calculations below are a good way to estimate the final concentration of contaminants in the wastewater, although pilot testing should be employed for any situation where regulations have to be met.

In this example, percent recovery for the RO process is 50. If the customer’s plant processes 1,000 gallons per day at a 50-percent recovery, the calculation for the volume of the permeate would be:

50 = (Volume permeate/1,000 gallons)*100; rearranging to solve for volume permeate: (50/100)*1,000 gallons = 500 gallons.

Whatever percent of contaminants is not able to pass through the membrane will end up in the reject stream. This volume of the reject is lower than the initial raw-water volume, which in effect, concentrates the rejected contaminants. The concentration factor formula, based on the percent recovery for an RO system, is useful for calculating the concentration of dissolved substances in the concentrate stream: concentration factor = 1/(1-percent recovery as a decimal).

Using the values from the example applications, where the percent recovery is 50, the concentration factor is calculated as: concentration factor = 1/(1-0.50) = 2. What the formula assumes, however, is that the membrane will reject 100 percent of the dissolved substances and everything that was in the raw water will end up in the reject after treatment. In this example, the membrane is rejecting 90 percent of the nitrate, so it is necessary to account for 90 percent of the original 78 mg/L going into the reject (78 mg/L * 0.90 = 70 mg/L). Multiplying the result by the concentration factor of 2: 70 mg/L * 2 = 140 mg/L.

The answers to the customer’s question are that he can expect to generate 500 gallons/day of RO wastewater, which will have a nitrate concentration of 140 mg/L. If necessary, this can be taken a step further to determine the amount of nitrate that’s collected each day by converting the concentration to pounds per gallon and multiplying by the 500 gallons of daily waste volume.

Conclusion
Considering the salt passage and the water passage as independent events allows for calculations of the results of each process that can then be recombined as needed to determine quantity and quality of the product water and the wastewater.

References

1. Reverse Osmosis for Point of Use Applications, Water Quality Association. 2001.
2. Slovak, R. Reverse Osmosis: A Practical Application Manual for Residential Point-of-Use Systems, Lisle: Water Quality Association. 2002.