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h1. MATHCAD Design of Filter and Clear Well
[MATHCAD File. | ^Fluidization Velocity Experiment-Single Media.xmcd]
[PDF version of our MATHCAD File. | ^Fluidization Velocity Experiment-Single Media.xmcd]<!-- /* Font Definitions */ @font-face {font-family:SimSun; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-alt:宋体; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:1; mso-generic-font-family:roman; mso-font-format:other; mso-font-pitch:variable; mso-font-signature:0 0 0 0 0 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-1610611985 1073750139 0 0 159 0;} @font-face {font-family:"\@SimSun"; panose-1:2 1 6 0 3 1 1 1 1 1; mso-font-charset:134; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:3 135135232 16 0 262145 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin-top:0in; margin-right:0in; margin-bottom:10.0pt; margin-left:0in; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:SimSun; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:Arial; mso-bidi-theme-font:minor-bidi;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:SimSun; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:Arial; mso-bidi-theme-font:minor-bidi;} .MsoPapDefault {mso-style-type:export-only; margin-bottom:10.0pt; line-height:115%;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.0in 1.0in 1.0in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} -->*{+}Target Input:+*
We are determining the parameters of the plant that we have, and then the basics of the filter design, such as number of filters, backwash time, and the amount of bed expansion.
*{+}Condition Input:+*
These are where the parameters of the construction materials are entered (ex. Media and pipes). The design is for a sand filter resting on about half as much gravel.
*{+}Calculations:+*
Filter Calculations: First we determine the size of the filter and head loss in normal filtration. Then we use two approachs to find the area of the clear well . The first is based upon conventional knowledge that the back wash velocity needs to be 10 times the filter velocity. From there we know that the space between the top of the filter bed and the bottom of the clear well is linked directly to the amount of headloss between those two points ( headloss from the pipe, elbows, expansion and contraction, expansion of the bed. To do this, we used information from Okun (what book?\!?\!) and the "Minor Losses in Pipes" Manual.
The second approach is to use the empirical formula, called the Weber Equation. The velocity for filtration is based on the sand's d60 (only works with sand) and the specific weight of both water and sand. Then the velocity for back wash is based on the filtration velocity, the sand's porosity and the ratio of height of the expanded and unexpanded filter bed. Once the back wash velocity is determined, solving the clear well and filter dimensions are the same process.
*{+}Design Output:+* *{+}Mathcad Conclusions:+{*}1) Our design based on simple hydraulics will work. However, it is a very large filter and will not be sustainable economically. The material cost for construction will be too high.
2) The design based on the empirical Weber equation (instead of the conservative approach) is much smaller and less expensive. However, the validity of the empirical equations is not yet certain. Therefore testing needs to be done in bench and pilot scale models.
3) If the empirical equations are valid, then we can change parts of the design, by changing the sand parameters in the following equation:
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| We can lower BW velocity by lowering the d60 and specific weight of the media. | |
Bench Scale Modeling through MATHCAD
The basic concept of bench scale modeling is to test the effectiveness of an actual filtration design by shrinking the design parameters of the system (filter bed depth, filter bed surface area, and etc) to a smaller scale that is easier to test. For example, we would simulate a filter bed of 50 cm of sand with a 5 cm of sand with the porosity and specific gravity of the sand being constant. This would also enable us to test the validity of the empirical equations that are behind our design.
Our MATHCAD files are divided into three parts: input, calculation, and output. In the input section, the user is prompted to enter in the properties of the filter media, water, filter dimensions, and the degree of bed expansion targeted. The calculation section contains the empirical equations from Schulz, Okun, and Weber. The output section produces the fluidization velocity and the corresponding flow rate required for the targeted bed expansion.
Click here for our MATHCAD File.
Click here for the PDF version of our MATHCAD File.
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