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In the case of macro-mixing, we are setting the minor loss coefficient to 1.3.  This value is assumed to be adequate to allow for macro-mixing.  The total head loss through the rapid mixer is set to 20 - 50 cm accounts for a total head loss of 40 cm throughout the entire plant, with most of the head loss in the rapid mix system belonging to the micro-mixer. This will be needed for the dose controller that will be integrated into the system.  The dose controller uses a flow measurement device which relates head loss to flowrate (See Lectures on Rapid Mix).  From the equation for a submerged orifice, it can be seen with the design assumption of K = 1.3, each pipe size will allow for one macro-mixing orifice size. 

 A A pipe size/macro-mixing combination can be used for a range of flow-rates.  An increase in flow-rates will result in a higher head loss for the macro-mixing orifice.   This can actually result in the macro-mixing orifice having a higher head loss than the micro-mixing orifice.  Due to this, the user can select a constraint value for head loss (usually 2 to 10 cm, currently set at 5 cm) that will cap how high the macro-mixing head loss can be.  This is done by selecting a larger pipe size and larger macro-mixing orifice.  In figure 2, the results for head loss through the macro-mixer are shown below.  At high flow-rates, macro-mixing head loss becomes higher than micro-mixing head loss. 

Figure 2:  Rapid mix sizing algorithm for lower flow rates with 20 cm through micro-mixer and no constraints 

 The The results show that above that with each pipe size and macro-mixing size, we have high head losses through the macro-mixing orifice which will increase to levels greater than 20 cm and hence give most of the head loss through the micro-mixing orifice. 
The algorithm can be summarized in these following steps:
1.  Determine the inner pipe size given the flow-rates, maximum pressure drop (20 to 50 cm). total minor loss coefficients, and the available pipe sizes. The user also sets a constraint or limit for the macro-mixing orifice head loss. 

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4.  Use the assigned total head loss value (20-50 cmcalculated to provide 40 cm of loss in the entire system) and maximum flow rate to determine the minor loss coefficient needed for the micro-mixing orifice.  Using the equation for minor loss coefficients for submerged orifices to determine the orifice diameter for the micro-mixing orifice. The number of orifices that can fit the area of the micro-mixing interface is returned based on the micro-mixing diameter.

 Running this algorithm for flow-rates under 50 L/s gave the following results as shown below on Figure 3.  
 

 Figure 3:  Rapid mix sizing algorithm for lower flow rates with 50 cm maximum head loss and 2 cm constraint on macro-mixer

 We also did runs where the constraint was raised to 10 cm head loss through the macomacro-mixing orifice. 

 Figure 4:  Rapid mix sizing algorithm for lower flow rates with 50 cm maximum head loss and 10 cm constraint on macro-mixer

 With this lower constraint, we find much smaller pipe sizes used.  The micro-mixing orifices are also larger.  We can also change the results to see what happens when we want a 20 cm head loss through the rapid mixer with our 2 cm constraint through the macro-mixer. 

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