Rapid Mix Chamber Design Program

Abstract

The AguaClara plant design needed to be extended to accommodate high flow rates such as those at Gracias (3000 L/min). One of the new design issues was that we could no longer rely on having the alum adequately mixed with the raw water in the pipe connecting the entrance tank to the flocculator. The design used for smaller flow rates does not generate enough large scale mixing to evenly distribute alum throughout the flow or enough energy dissipation to ensure that alum is mixed at the molecular level. Consequently, a new entrance tank was designed to accommodate a rapid mix chamber that could appropriately mix alum with raw water before flocculation. The entrance tank and rapid mix chamber were designed to be used in conjunction with the Nonlinear Chemical Dose Controller between the maximum and minimum expected plant flow rates. The rapid mix chamber uses a combination of orifices and hydraulic drops to achieve desired even mixing.

Rapid Mix Chamber

The rapid mix chamber is assumed to be made of concrete and attached to one side of the entrance tank. The water within the entrance tank enters the rapid mix chamber's orifice collector channel through the flow meter orifice. Once water passes through this orifice it travels to the end of the channel and then experiences a hydraulic drop. By the time the water reaches the bottom of the waterfall mixer, enough large scale mixing has occurred. Finally, as the water passes through the fine scale turbulence generating orifice of the rapid mix chamber, it experiences a contraction followed by an expansion. This expansion generates a high enough energy dissipation rate for turbulence to blend the solution to a small enough scale that molecular diffusion can finish the blending thereafter.

Design Assumptions and Specifications

Design Assumptions Design Program

Entrance Tank Dimensions

Entrance Tank Design Program

Flow Meter Orifice

The flow meter orifice design is constrained by the maximum and minimum water height difference and the coefficient of discharge due to the vena contracta through the orifice. Taking these constraints into account and using the orifice equation the orifice area is calculated.

Given the flow meter orifice area requirement, the orifice height can be fixed and then subsequently the length can be calculated. The lowest measurable flow rate occurs when the orifice is barely submerged:

Where:

• A _EtOrifice is the actual orifice area
• Pi _{VenaContractaOrifice} is the coefficient of discharge due to the vena contracta
• g is the gravitational force
• HW _EtMin is the lowest water height allowed also known as the lowest attainable head loss

In Gracias, the plant had to operate at 20% of the maximum flow rate which corresponds to a specific head loss ratio as shown in the graph below:

This head loss ratio:

Where:

• Pi _EtFlow is the flow rate ratio (Q _min / Q _max)
• heightratio is the head loss or water height ratio (W _EtOrifice / HL _EtMax)

in turn gives us a specific width:

Knowing the width and area, the length is calculated:

Where:

• A _EtOrifice is the actual area of the orifice
• W _EtOrifice is the width of the orifice

Orifice Collector Channel

The Orifice Collector Channel is modeled as an open channel flow conduit. The water enters through the Flow Meter Orifice on the side of the channel and flows towards the hydraulic drop into the flocculator. The length of the channel is dependent on the length of the entrance tank orifice and the channel must be longer than this orifice. This approximation is conservative because it assumes that more water enters the channel at first than it actually does. The height of water along the channel is iteratively solved for by using the direct step method described below.

The flow at the top of the hydraulic drop is critical- potential energy and kinetic energy are in balance and the height of water in the tank can be solved for using the equation:

Where:

• h _crit is the critical depth
• q is the flow per unit width
• g is the acceleration dues due to gravity

The width of the channel is based on the baffle spacing in the first section of the flocculator.

Given this initial height in the channel, the height of water in the rest of the channel can be theoretically determined. The energy equation for an open channel flow can be rearranged to give the relationship:

Where:

• S _o is the slope of the channel which is zero in this case
• S _f is the friction slope
• y is the depth along the channel
• x is the distance from the hydraulic drop to the end of the channel

The friction slope (Figure 3) was found using the equation:

Where:

• f is the friction factor
• V is velocity
• g is acceleration due to gravity
• R _h is the hydraulic radius defined as the cross sectional area over the wetted perimeter

Finally, to solve for the heights, the critical depth is used as the initial V ₁ and a change in depth of 0.0001 cm is used to find V ₂. The modified energy equation is solved to find the corresponding distance upstream in the channel, given the calculated friction slope that corresponds to a change in water depth of 0.0001 cm. This process was repeated until the upstream end of the channel was reached and the height of water at the upstream end of the channel was found.

Waterfall Mixer

The waterfall mixer is the first mixing step after alum is added in the orifice collector channel. This hydraulic drop provides a region of energy dissipation for large scale mixing. The minor loss coefficient can be directly related to the mixing length and in this case it is assumed that a minor loss coefficient of 1.3 is adequate for large scale eddies to evenly mix the alum. The velocity can be found with the critical depth at the orifice collector channel exit and channel width. The equation for minor loss is then used to determine the height of the free fall necessary to obtain the desired minor loss:

Where:

• V is 1.07 m/s; the critical velocity of the drop
• K is 1.3; the minor loss coefficient
• g is acceleration due to gravity

Note: These values will change slightly with the square channel geometry.

First Baffle Section

The hydraulic drop deposits water into the first baffle section. The water level in this section is determined by the water level in the flocculator and the head loss through the flocculator entrance orifice(described below). The height of the channel includes the water level and the lower hydraulic drop height.

Flocculator Entrance Orifice

The orifice that leads into the flocculator must provide enough energy dissipation so that turbulence will mix the alum with the water to a sufficiently small scale so that molecular diffusion can finish the mixing process in a few seconds. Therefore, energy dissipation rate from minor head loss has to be greater than diffusion requirements. The following equation is used to determine the minimum energy dissipation required to meet the diffusion requirements.

Where:

• pi _Dmix is 2.4; the ratio between the diffusion length scale and the Kolmogrov length scale
• nu _water is the kinematic viscosity of water
• D _m is 0.673 cm ²/s; the diffusion coefficient for aluminum hydroxide
• tau _diffusion is 10s; time allowed for diffusion to finish the mixing process

The minimum energy dissipation is calculated to be approximately 0.733 W/kg. This number is then rounded to 0.8 W/kg (conservative estimate of energy dissipation) in the flocculator entrance orifice equation.

In order to design the dimensions of the flocculator entrance orifice, the velocity of the flow in the vena contracta is calculated. Once the flow through the vena contracta has been determined, the area of the flow through the vena contracta is calculated with the following equation.

Where:

• Q is the flow rate
• K _exp is 1.3; the global mixing coefficient
• V is the velocity in the vena contracta
• &Theta& is the resonance time

By using the coefficient of discharge, the area of the orifice was calculated. A rectangular orifice is determined to be practical for this design. An orifice height of 13 cm is assumed, but will eventually need to be less than the width of the baffle spacing. Based on the calculated orifice area, the length of the rectangular flocculator entrance orifice will be determined.

After final calculations, the rectangular orifice was determined to have a height of 13 cm and a width of 51.59 cm.

Alum Stock Tank

As with previous designs, the Gracias design calls for two alum stock tanks. This way, while one of the tanks is operating, the other tank may be filled with the appropriate amount of chemicals. Therefore, the tanks are designed to be big enough for the operator to have enough time to mix the chemical before the other tank has drained. It is assumed that a supply for 30 hours is adequate. The following formula is then used to determine what the maximum flow of alum out of the stock tank needs to be:

Where:

• C _FcmDoseMax is 90 mg/L; the maximum concentration needed for flocculation
• C _FcmAlum is 120 gm/L; the concentration of alum in the stock tank

  Perhaps a note that higher concentrations (up to 500 g/L) could potentially be used.

From this equation, the maximum flow of alum is determined to be 2.25 L/min; and the volume required to sustain this flow rate for 30 hours is 4050 L.