Nonlinear Theory

The nonlinear chemical doser is part of the evolution of alum dosing techniques in an effort to increase the maximum flow rate capacity of AguaClara plants as well robustness of design and implementation. Additionally, a frequent problem in AguaClara plants was the occurrence of foam in the flocculator after the addition of alum to the entrance tank. It was postulated that the previous free fall method of dosing alum was contributing to the occurrence of foam. Any new dosing mechanism would need to have alum dosed below the level of water in the entrance tank. Previous linear chemical dosers have relied upon the major head losses caused by the friction of the dosing tube to control the flow of alum. The important point in distinguishing the need to move to nonlinear flow is recognizing the relationship between head loss and the flow rate of alum in the two dosing methods. In previous linear dosing designs, the flow of alum is proportional to major head losses in the dosing tube. If the flow rate of alum were to increase into the turbulent range then the head loss is proportional to the flow rate squared, as shown in the figure below. Since the relationship between flow rate and head loss is not linear for turbulent, significant dosing errors would result in the linear dosing scheme. In order to allow there to be turbulent alum flow in the dosing tube, an orifice controlled nonlinear doser is now being used. In the nonlinear orifice controlled doser, the majority of the head losses is due to the minor losses caused by the orifice. In the nonlinear system the flow rate of alum is proportional to the square root of h in both laminar and turbulent ranges. This homogeneity in relationships allows there to be reliable dosing even in turbulent ranges.

where:

= proportional to

The connection between the flow control and the flow measurement aspect of the dosing mechanism is important to understand the evolution of dosing mechanisms in AguaClara plants.
Since the nonlinear chemical doser has the same relationship between flow and head at the turbulent ranges; AguaClara plants can be scaled up to much higher flow rates without being limited by the turbulence in the dosing tube. This is a huge advantage of the nonlinear system because it expands the AguaClara plants capabilities to serve much larger communities. The size of AguaClara plants is no longer limited to the flow limitations in the dosing tube.

As mentioned, the nonlinear doser uses the minor losses caused by the orifice instead of a dosing tube (major losses) to control the relationship between changing plant flow rates and chemical dose. The flow rate through the Chemical Dose Controller (CDC) is related to the available head by the equation:

where

• Unknown macro: {latex}

  $$Q_

  Unknown macro: {Cdc}

  $$

  is the chemical flow rate
• Unknown macro: {latex}

  $$ K_

  Unknown macro: {orifice}

  $$

  is the orifice coefficient
• h is the available head

The desired chemical dose to the plant can be determined by a mass balance:

where

• C _c is the chemical stock concentration
• C _p is the chemical dose

The influent raw water leaves the entrance tank through the Rapid Mix Tube which is the entry point for the dosing of alum. In the Rapid mix tube, an orifice is located in the tube to generate macro and micro-scale mixing. The entry point for the dosing of alum with the raw water has been redesigned to be submerged in the entrance tank in order to possibly reduce the occurrence of foam in the flocculator. After conversations with Dan Smith, the AguaClara engineer in Honduras, there doesn't appear to be foam forming in the Agalteca plant, where this dosing system has been implemented.
The flow of water through an AguaClara plant can be modeled as minor losses due to flow expansions. The relationship between plant flow rate and head loss through the plant is governed by the minor loss equation shown below.

where:

= plant minor loss coefficient

The equation above is important since it defines the relationship between plant flow and head loss through the plant, which is important as well for our dosing apparatus. The CDC uses a lever arm with a float and counterweight to relate the dosing to the changes in the entrance tank water level, which is a function of the influent flow rate. An increase in head loss links the chemical flow rate to the plant flow rate and the chemical dose (mg/L) will be constant as plant flow varies.As can be seen above, the equation which describes the flow of water through plant and the orifice equation which relates flow of alum in the dosing system both have the same relationship between flow rate and head loss. Since the flow rate for both the plant and alum flow are each related by the square root of the head loss, the two flows can be linked through the float in the entrance tank. Any rise
For a detailed step by step description of the steps involved with measuring the plant flow rate please see flow measurement section of the Chemical Dose Controller Manual.
The dosing tube must be designed to minimize major losses so that minor losses dominate head loss. The amount of major losses due to the friction in the dosing tube will vary based on the flow of alum going through the dosing tube, with higher head loss occurring at higher alum flow rates. As a result, it is not desirable to set the size of the orifice based on the varying major losses through the dosing tube, so the head loss caused by the orifice needs far outweigh major losses so the friction in the tube can be ignored. As a result, the dosing tube needs to have a large enough diameter so that the effects of major losses are not significant when compared to the minor losses causes by the orifice. The graph below illustrates the relative contributions of head loss caused by a large 2 mm orifice cap as opposed to the major losses through the dosing tube.

INSERT MAJOR losses graph

As can be seen in the graph above, the relative contribution of major losses to the total head loss in the dosing system, using a 2 mm diameter orifice (the large orifice), is minor. The percent contribution of the major losses never is around 1.5% of the total head losses, so it can effectively be ignored in the sizing of the dosing orifice.