h1. Calculating Orifice Sizes and the Dual Scale for the Lever Arm
!orifice.PNG|align=centre,width=400500,height=400!
+(Please label this caption and describe the components of it in this wiki)+h5. Figure 1: Constant Head Tank and Dosing Orifice
h3. Abstract:
During the fall semester of 2009, the Non-linear Chemical Alum Doser Team developed the dual orifice scale system for the lever arm in order to deliver turbulent flow of alum dosing. We have currently researched and identified the nonlinear relationship between plant flow rate and alum dosage and the movement of the lever arm. We utilized this relationship to develop the lever arm design to include the dual scale and the orifices. Attached is the Mathcad program that contains the calculations for our dosing system. Please see our [calculations|^Lever Arm Calculations 2009 NCDC TM.xmcd]
We have designed a 80cm long lever arm with equal lengths and two orifices of 0.122310 inchcm and 0.0610155 inchcm, 3/8" pvc tubing, and other associated hydraulic components listed in our [hydraulic components list|^NCDC Component List.xlsx] +does this list need to be updated? -karen+.
h3. Theory:
Our first step in developing a dosage system that can deliver both turbulent and laminar flow of alum was the selection of the orifice to control the flow of alum. We increased the tubing size connecting the constant head tank to the orifice to 3/8 inch which is wide and smooth enough to make the headloss (headloss is not a word, instead write head loss)+ from the tubing negligible +(compared to the head loss through each orifice)+ -This action essentially- , making the orifice the flow control component for the dosage system. The orifice equation, shown below, demostrates the nonlinear relationship between flowrate and the change in head headlossloss. The flow rate is the result of the square root of the head headlossloss, +the area of the orifice+, and -times a- the orifice constant.
{latex}
\large
$$
Q = K_{vc} A_{or} \sqrt {2gh}
$$
{latex}
-WeHead thenloss learnedoccuring thatin the orificeplant controls most of the other- Head loss occuring in the plant after after the entrance tank +including+ -since the- flow through -the- rapid mix, the flocculation tank, and the launderlaunders are all controlled by -the- orifices. The only source of head loss not controlled by an orifice is the weir. Because the majority of the head loss is dominated by the orifice we can link the flow rate of the plant with the flow rate of alum required for the plant using the same square root relationship mentioned above. +(You should show this by summing up the associated head loss with all parts of the plant and show that the deviation by the launders gives minimal error)+
-We then link- The relationship between the lever and -the- plant flow rate +are related+ by the following equations -mentioned- below. +(Label your variables in the equations. Which equation is referring to what?)+
{latex}
\large
$$
Q_C = K_C h_C^{n_C }
$$
$$
Q_P = K_P h_P^{n_P }
$$
{latex}
Above two equations show us that alumh5. Table 1: Head Loss Through the Plant
||Process||Head Loss||
|Rapid Mix Tube|10 cm|
|Flocculator|13.5 cm|
|Launder|5 cm|
|Weir|5 cm|
|*Total*|*33.5 cm*|
The lever arm dosing flow rate and plant flow rate are both governed by the same nonlinear relationship as a result of the use of orifice, see below. +(Isn't this obvious if we are using an orifice in both?)+
{latex}
\large
$$
CLabel your variables in the equations. Which equation is referring to what?)+
{latex}
\large
$$
Q_C = K_C h_C^{n_C }
$$
$$
Q_P = K_P h_P^{{C_Cn_P }
$$
{latex}
where Q_C } \over {Q_P }}
$$
{latex} represents the dosing flow rate, K_C is the orifice coefficient
The mass balance equation -above shows us how we- calculates the flow of alum +needed+ based on plant flow rate and the concentration of the alum chemical stock tank. stock tank.
{latex}
\large
$$
C_P = {{C_C Q_C } \over {Q_P }}
$$
{latex}
{latex}
\large
$$
C_P = {{C_C K_C h_C^{n_C } } \over {K_P h_P^{n_P } }}
$$
{latex}
We substitute the chemicaldosing flow rate equation in and link the two heights with a lever and cancel out any duplicate variables. +(Please label what each of these variables are, is L for level, C for chemical stock tank, and P for plant?)+
{latex}
\large
$$
h_C = K_L h_P
$$
$$
C_P = {{C_C K_C K_L^{n_C } h_P^{n_C } } \over {K_P h_P^{n_P } }}
$$
$$
C_P = {{C_C K_C K_L^{n_C } h_P^{n_C } } \over {K_P h_P^{n_P } }}
$$
$$
C_P \propto {{K_L^{n_C } h_P^{n_C } } \over {h_P^{n_P } }}
$$
$$
C_P = {{C_C K_C K_L^{n_C } h_P^{n_C } } \over {K_P h_P^{n_P } }}
$$
$$
C_P = \sqrt {K_L^{} } {{C_C K_C } \over {K_P }}
$$
$$
\sqrt {K_L } \propto C_P
$$
{latex}
The height of the water, or the flow rate to the plant via the entrance tank, is non-linearly related to the change in height of the scale. Because the relationship is defined by the orifice equation, square root of the change in height times a constant factor +(what exactly is the constant factor a function of?)+ in height gives us the change in flowrate required. Linking the different sets of heights to the scale, we generate a scale that an operator can use to adjust the dosage. Because the orifice equation that controls this relationship is nonlinear, the scale is nonlinear. An example of nonlinear scale is shown below.
+(Your scale is a good visual representation. I would include a table as well of these values and perhaps a sample calculation for what you described above.)+
!scale.PNG|align=centre!
h3. Method:
We utilized [Mathcad|^Lever Arm Calculations 2009 NCDC TM.xmcd] to calculate our design parameters. The rest of this section illustrates our method in designing the orifice diameter and the dual scale for the plant.
The maximum movement of the water height in the entrance tank is determined by the sum of the the headloss from the flocculator, macro and micro mixer, weir, and launder orifices as shown below. +(Please manually create tables in the wiki and put the label Table 1. (with a brief description above the table. For figures put this below. Please apply this to all future work on the wiki) )+ is nonlinear, the scale is nonlinear. An example of a nonlinear scale is shown below.
+(Your scale is a good visual representation. I would include a table as well of these values and perhaps a sample calculation for what you described above.)+
!scale.PNG|align=centre!
h3. Method:
We utilized [Mathcad|^Lever Arm Calculations 2009 NCDC TM.xmcd] to calculate our design parameters. The rest of this section illustrates our method in designing the orifice diameter and the dual scale for the plant.
The maximum movement of the water height in the entrance tank is determined by the sum of the the headloss from the flocculator, macro and micro mixer, weir, and launder orifices as shown in Table 1.
!HL Chart.PNG|align=centre,width=300,height=300!
This vertical distance +of 3633.5 cm+ establishes the upper and lower limit of flow to thatwhich our lever arm must be designed to respond to.
In our calculations, we first define the applicable -scale- of -the- alum dose range -that we want the plant to operate around- +of plant operation+. -We have- Two different orifices -to- handle two different -levels- +scales+ of alum +dosage+: high +(indicate range)+20-100 mg/L) and low +(indicate range)+5-25 mg/L). They are offset to each other by a factor of 4four. Consequently, as dictated by the area portion of the orifice equation relationship, the diameter of the larger orifice is twice the diameter of the smaller orifice.
+(Please label and describe your mathcad screenshot)+
!input.PNG|align=centre!
DiameterThe input +(how did you specifyoperator of this or does the user specify this?)+ is correct but this Mathcad program is designed so that we can findwill change the diameter of the rightprogram sizeuntil diameterthe throughdual ascale seriesutilizes ofmaximum trialspace andon errorthe untillever wearm utilze the entire length of the scale which corresponds to the maximum and minimum height of the plant flow. The operator of this program will change the diameter(40 cm). This Mathcad program is designed to find the right size diameter through a series of trial and error iterations until we utilze the entire length of the programscale untilwhich thecorresponds dualto scalethe takesmaximum upand asminimum muchheight of the scale (40 cm) as possible.plant flow.
After we have input the tentative diameter for the larger orifice and the dual series of doses we conduct the calculations mentioned in the theory section to produce the nonlinear dual scale. We convert the dosage to the alum flow rate required by utilizing the mass balance equation.
!angle flow.PNG|align=centre!
The above snap shot also shows how the maximum headloss dicatates the maximum angle that our lever arm will be operating around.
!h scale.PNG|align=centre!
We then calculate the necessary headlosshead loss in the dosage system to produced the required flow rates.
Utilizing the sine function we convert the headlosshead loss to points along the lever scale arm. +(Show the equation if possible)+
We can also calculate the dual scale through a quicker method by utilizing the orifice equation relationship governing this whole process by linking the dosage directly to the scale. The resulting scale is shown below with the rest of the results for comparison. +(Did you compare your two methods here? I didn't see this.)+
!k.PNG|align=centre!
We know that the governing equation for this system dictates that the flow is equal to square root of distance times a constant K. We solve for the K by divingdividing the maximum dosage by the square root of the maximum length +(can you refer to an equation here?)+. We then utilize the K to solve for the dual scale.
The snapshot below shows us our final results. We manipulated the larger orifice size until we fully utilize the lever arm scale as shown below.
!output.PNG|align=centre!
Shown below is the scale that is generated from our calculations.
+(Please connect the calculations you performed in MathCAD to some sort of diagram of the system and refer to it. Also, make sure that your mathcad screenshots and graphs are separate and that you have a caption for both figures and that you refer to your caption in the text.)+
!Scale 2.PNG|align=centre!
h3. Results and Discussions
Currently, our orifices are 0.122310 inchcm for alum dosage of 20 to 100 mg/L offset by 10 mg/L and 0.0610155 inchcm for alum dosage of 5 to 25 mg/L offset by 2.5 mg/L. Our lever arm is 80 cm in length with equal lengths on each side. Tubing is made up of PVC and of 3/8 inch diameter which is wide and smooth enough to produce negligible head headlossloss on the alum flow. The scale developed by linking the dosage to the scale is very similiar to our dual scale developed via our longer mathcadMathcad programs, validating that the nonlinear relationship dicateddictated by the orifice equation governs plant flow rate, the change in head headlossloss, and the dual scale. We currently have purchased or acquired all hydraulic components needed for our lever arm.
For ease of operation, whenever this lever arm is used in the field, this lever arm can be delivered to the Aguaclara Plant with the dual scales already engraved on the arm. The operator simply has to calibrate the maximum dosage to the maximum flow rate and the lever arm will be ready for operation.
Our near goal is to build the [lever arm prototype|Prototype Design] and set up our hydraulic components and engrave our dual scale. We also plan on conducting experiments to test our lever arm and find a solution to the [clogging problem|Clogging Experiment] currently besetting actual alum dosers in Honduras.
h2. Bibliography
* CEE4540 Flow Control Measurement Notes at https://confluence.cornell.edu/display/cee4540/Syllabus
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