h1. Calculating Orifice Sizes and the Dual Scale for the Lever Arm.
h3. Abstract:
During the fall semester of 2009, the Non-linear Chemical Alum Doser Team worked on developing the dual orifice scale system for the lever arm in order to handle turbulent flow 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 lever arm design parameters to include the dual scale and the orifices. Attached is the Mathcad program that contains the calculations for our dosing system. Please see our [Mathcad filecalculations|^Lever Arm Calculations 2009 NCDC TM.xmcd].
h3.We have Theory:designed a
Our80cm firstlong steplever inarm developingwith aequal dosagelengths systemand thattwo canorifices deliverof both0.122 turbulentinch and laminar flow of alum was the selection of the orifice to control the flow of alum. We increased the tubing sizing connecting the constant head tank to the orifice to a large 3/8 inch in order to make the headloss from the tubing negligible. This action essentially make 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 headloss. The flowrate is the result of the square root of the headloss times a constant factor, which in this case is 2 times the force of gravity and the area of the orifice, which we can manipulate.
{latex}
\large
$$
Q = K_{vc} A_{or} \sqrt {2gh}
$$
{latex}
We then recognized that the orifice control most of the headloss occuring in the plant after the entrance tank since the flow through the macro and micro mixer, the flocculation tank, and the launder are all controlled by the orifice. The only source of headloss not controlled by an orifice is the weir but because the majority of the headloss 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 orifice relationship mentioned above.
We then link the relationship between the lever and the plant flow rate.
{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 alum and plant flow rate are governed by the same nonlinear relationship as a result of the use of orifice0.061 inch, 3/8" pvc tubing, and other associated hydraulic components listed in our [hydraulic components list|^NCDC Component List.xls].
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 sizing connecting the constant head tank to the orifice to a large 3/8 inch in order to make the headloss from the tubing negligible. This action essentially make 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 headloss. The flow rate is the result of the square root of the headloss times a constant factor, which in this case is 2 times the force of gravity and the area of the orifice, which we can manipulate.
{latex}
\large
$$
Q = K_{vc} A_{or} \sqrt {2gh}
$$
{latex}
We then recognized that the orifice control most of the headloss occuring in the plant after the entrance tank since the flow through the macro and micro mixer, the flocculation tank, and the launder are all controlled by the orifice. The only source of headloss not controlled by an orifice is the weir but because the majority of the headloss 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 orifice relationship mentioned above.
We then link the relationship between the lever and the plant flow rate.
{latex}
\large
$$
Q_C_P = {{CK_C Qh_C^{n_C } \over {
$$
$$
Q_P }}
$$
= K_P h_P^{n_P }
$$
{latex}
TheAbove masstwo balance equation above showsequations show us how we calculate the flow ofthat alum based onand plant flow rate andare thegoverned concentrationby of the alum chemical stock tanksame nonlinear relationship as a result of the use of orifice.
{latex}
\large
$$
C_P = {{C_C K_C h_C^{nQ_C } } \over {K_P h_P^{nQ_P } }}
$$
{latex}
WeThe substitutemass thebalance chemicalequation flowrateabove equationshows inus how andwe linkcalculate the two heights with a lever and cancel out necessary variables flow of alum based on plant flow rate and the concentration of the alum chemical stock tank.
{latex}
\large
$$
hC_CP = {{C_C K_LC h_P
$$
$$
C_P = {{C_C K_C K_L^{n_C } h_P^{C^{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 } }}{latex}
We substitute the chemical flow rate equation in and link the two heights with a lever and cancel out necessary variables.
{latex}
\large
$$
h_C = K_L h_P
$$
$$
C_P = {{C_C K_C K_L^{n_C } h_P^{n_C } } \over {K_P h_P^{n_P } }}
$$
$$
C_P = \sqrt {{K_L^{}C_C K_C K_L^{n_C } h_P^{{Cn_C K_C} } \over {K_P h_P^{n_P } }}
$$
$$
C_P \sqrtpropto {{K_L^{n_LC } \propto C_P
$$
{latex}
The height of the water, or the flow rate, is nonlinearly 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 gives us the change in flowrate required. Linking the different sets of heights to the scale, we generate a scale that 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, is nonlinearly 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 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 as shown below is nonlinearis nonlinear. An example of nonlinear scale is shown below.
!scale.PNG|align=right!
h3. Method:
We utilized the Mathcad program [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.
We then define the boundary !HL Chart.PNG|align=centre,width=300,height=300!
We then define the scake of the alum dosage that we want
the plant to operate around. We have two different orifices to handle two different levels of alum: high and smalllow. They are offset to each other by a factor of 4. Consequently, as dictated by the orifice equation relationship, the diameter of the larger orifice is twice the diameter of the smaller orifice.
!Mathcad1.PNG|align=centre,width=400,height=300!
After we have inputed 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 flowrate required by utilizing the mass balance equation.
!Mathcad2.PNG|align=left,width=400,height=300!
Diameter input is correct but this Mathcad program is designed so that we can find the right size diameter through a series of trial and error until we utilze the entire length of the scale which corresponds to the maximum and minimum height of the plant flow.
!Mathcad1.PNG|align=centre,width=600,height=600!
The above snap shot also shows how the maximum headloss dicatates the maximum angle that our lever arm will be oeprating around.
!Mathcad3After we have inputed 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.
!Mathcad2.PNG|align=leftcentre,width=400300,height=400!300!
WeThe then calculateabove snap shot also shows how the necessarymaximum headloss indicatates the maximum angle dosagethat systemour tolever producedarm thewill requiredbe flowoeprating ratesaround.
!Mathcad4Mathcad3.PNG|align=leftcentre,width=400300,height=400300!
UtilizingWe then calculate the sinenecessary functionheadloss wein convertthe thedosage headlosssystem to points alongproduced the leverrequired scaleflow armrates.
By manipulating
!Mathcad6Mathcad4.PNG|align=leftcentre,width=400300,height=400300!
The snapshot shows us our final results. We manipulate the larger orifice size until we fully utilizeUtilizing the sine function we convert the headloss to points along the lever arm scale as shown above.
We can also validate the existence of the orifice equation relationship governing this whole process by linking the dosage to scale directly and observing how close such a scale matches the scales produced from steps mentioned above. arm.
!Mathcad6.PNG|align=centre,width=1000,height=1000!
The snapshot shows us our final results. We manipulate the larger orifice size until we fully utilize the lever arm scale as shown above. Shown below is the scale that is generated from our calculations.
!Mathcad5.PNG|align=leftcenter,width=400300,height=400300!
!Mathcad7.PNG|align=left,width=400,height=400We can also validate the existence of the orifice equation relationship governing this whole process by linking the dosage directly to the scale and observing how close such a scale matches the scales produced from steps mentioned above.
!Mathcad5.PNG|align=centre,width=300,height=300!
We solve for the K by diving the maximum dosage by the square root of the maximum length. We then utilize the
K to solve for!Mathcad7.PNG|align=centre,width=300,height=300!
We solve for the K by diving the maximum dosage by the square root of the maximum length. We then utilize the
K to solve for the dual scale.
h3. Results and Discussions
Currently, our orifices are 0.122 inch for alum dosage of 20 to 100 mg/L offset by 10 mg/L and 0.061 inch 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 headloss on the alum flow. The scale developed by linking the dosage to the scale is very similiar to our dual scale developed via our mathcad programs, validating that the nonlinear relationship dicated by the orifice equation governs plant flow rate, the change in headloss, and the dual scale.
h3. Future Challenge
Besides actually building and testing the prototype for the design challenge in 2010. We plan on conducting experiments to find a solution to the clogging problem currently occurring with actual alum dosers in the fields of Honduras.
A future challenge for us will be to figure out a way to automate our array calculation method so that the design tool could be used to also calculate these parameters. This is possible in Mathcad if we utilize a while loop and slowly increasing the diameter of the larger orifice until we fully utilized the lever arm scale.
h2. Bibliography
* CEE4540 Flow Control Measurement Notes at https://confluence.cornell.edu/display/cee4540/Syllabus
h2. Deliverables
*[Final hydraulic component list| ^NCDC Component List.xlsx]
*Final orifice and dual scale calculations-2009 NCDC Lever Arm Calculations
*Hydraulic components for the lever arm prototype
*Dual scale engraved on the lever arm prototype |
