20.2 Filter Loading Rate, Backwash Rise Rate & Washwater Volume

Filter Loading Rate and Forward Filtration Hydraulics

Granular media filtration removes non-settleable particulate matter, chemical floc carryover, and pathogenic protozoan cysts (Giardia lamblia and Cryptosporidium) by depth filtration. The rate at which water passes downward through the media bed is termed the Filter Loading Rate or Filtration Rate, expressed in gallons per minute per square foot ($gpm/sq\ ft$).

                    [ Influent Flow Rate (gpm) ]
                                 |
                                 v
           +---------------------------------------------+
           | ~ ~ ~ ~ ~ ~ ~ Water Head ~ ~ ~ ~ ~ ~ ~ ~ ~  |
           |---------------------------------------------|
           | Anthracite Coal Layer (0.8 - 1.2 mm)        |  Filter Loading Rate
           |---------------------------------------------|  = Flow (gpm) / Area (sq ft)
           | Silica Sand Layer     (0.45 - 0.55 mm)      |  
           |---------------------------------------------|  Target: 3.0 to 5.0 gpm/sq ft
           | Garnet / Support Gravel Layer               |
           |=============================================|
           | Underdrain Lateral & Nozzle System          |
           +---------------------------------------------+
                                 |
                                 v
                    [ Filter Effluent (Turbidity < 0.10 NTU) ]

The Filtration Rate Formula

To compute the filtration rate, convert plant flow into gallons per minute ($gpm$) and divide by the total surface area of the active filter media bed:

$\text{Filtration Rate }(gpm/sq\ ft) = \frac{\text{Operating Flow Rate }(gpm)}{\text{Filter Surface Area }(sq\ ft)}$

$\text{Operating Flow Rate }(gpm) = \frac{\text{Flow Rate }(MGD) \times 1,000,000}{1,440\ min/day} = \text{Flow }(MGD) \times 694.4$

$\text{Filter Surface Area }(sq\ ft) = \text{Length }(ft) \times \text{Width }(ft)$

Regulatory Loading Standards by Media Type

1. Rapid Sand Filters: Single-media silica sand beds ($24\text{ to }30\text{ inches}$ depth). Historically rated under the Ten States Standards at $2.0\text{ gpm/sq ft}$. Particles deposit almost entirely within the top $1\text{ to }2\text{ inches}$ of sand, leading to rapid head loss accumulation.
2. High-Rate Dual-Media Filters: Anthracite coal ($18\text{ to }24\text{ in}$) layered atop silica sand ($8\text{ to }12\text{ in}$). Coarser, less-dense coal allows floc to penetrate deeply into the bed before reaching the fine polishing sand. Standard rating: $3.0\text{ to }5.0\text{ gpm/sq ft}$ (up to $6.0\text{ gpm/sq ft}$ under state pilot-study approvals).
3. Multi-Media (Mixed Media) Filters: Anthracite coal ($18\text{ in}$), silica sand ($9\text{ in}$), and dense garnet sand ($3\text{ in}$). Provides true three-dimensional depth filtration. Standard rating: $4.0\text{ to }6.0\text{ gpm/sq ft}$.
4. Deep-Bed Monomedia Filters: Coarse anthracite ($48\text{ to }72\text{ in}$, $1.5\text{ to }2.0\text{ mm}$ effective size) operated at rates up to $6.0\text{ to }8.0\text{ gpm/sq ft}$.

Table 20.2.1: Granular Media Filter Types and Operating Rates

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Backwash Hydraulics and Rise Rate Calculations

As solids accumulate within media pores, operational head loss increases across the bed. When head loss reaches the terminal setpoint (typically $6\text{ to }9\text{ feet}$), effluent turbidity spikes above $0.10\text{ NTU}$, or filter run time expires (typically $36\text{ to }72\text{ hours}$), the filter must be backwashed.

Fluidization and Bed Expansion

During backwashing, clean treated water is pumped upward through the underdrain system at high velocity, reversing flow through the media. This upward flow lifts and fluidizes the media particles, allowing them to abrade against each other (inter-particle scouring) to release trapped solids. Proper backwashing requires $20%\text{ to }50%$ bed expansion.

$\text{Backwash Loading Rate }(gpm/sq\ ft) = \frac{\text{Backwash Pumping Rate }(gpm)}{\text{Filter Surface Area }(sq\ ft)}$

Typical backwash loading rates range from $15\text{ to }25\text{ gpm/sq ft}$—approximately five to ten times greater than the forward filtration rate.

Derivation of the Rise Rate Conversion Factor (1.605)

In the field, backwash intensity is often monitored as the rise rate—the vertical velocity of the water level rising in the filter box, measured in inches per minute ($in/min$) or feet per minute ($ft/min$).

1. One cubic foot ($cu\ ft$) contains $7.48\ gallons$.
2. Dividing $1.0\ gpm/sq\ ft$ by $7.48\ gal/cu\ ft$ converts flow to vertical velocity in feet per minute:
   $\text{Rise Velocity }(ft/min) = \frac{1.0\ gpm/sq\ ft}{7.48\ gal/cu\ ft} = 0.13369\ ft/min$
3. Multiplying feet per minute by $12\ inches/ft$ yields inches per minute:
   $\text{Rise Velocity }(in/min) = 0.13369\ ft/min \times 12\ in/ft = \mathbf{1.60428\ in/min} \approx \mathbf{1.605\ in/min}$

$\mathbf{\text{Rise Rate }(in/min) = \text{Backwash Loading Rate }(gpm/sq\ ft) \times 1.605}$

$\mathbf{\text{Backwash Rate }(gpm/sq\ ft) = \frac{\text{Rise Rate }(in/min)}{1.605}}$

Direct Field Measurement of Rise Rate

Operators calibrate backwash flow meters by directly measuring water rise during a backwash sequence:

1. Close the filter influent and effluent valves.
2. Open the backwash supply valve to establish steady backwash flow, keeping the waste drain valve closed.
3. Using a staff gauge or hook gauge mounted on the filter wall, record the time ($T$ in seconds or minutes) required for the water surface to rise a measured vertical distance ($D$ in inches):

$\text{Rise Rate }(in/min) = \frac{\text{Rise Distance }(inches)}{\text{Elapsed Time }(seconds)} \times 60\ sec/min = \frac{\text{Rise Distance }(inches)}{\text{Elapsed Time }(minutes)}$

                                  Water Level at Time T2
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - - - - - - - - - - - - - - - - -  ^
                                                                        | Rise Distance
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - - - - - - - - - - - - - - - - -  v (Inches)
                                  Water Level at Time T1
  +------------------------------+
  |   Fluidized Media (Expanded) |
  |   ========================== |   Rise Rate (in/min) = Rise (in) / Time (min)
  |   Fixed Media Bed (Resting)  |
  +------------------------------+

Temperature Effects on Backwash Hydraulics

Water viscosity increases as water temperature drops. Cold water ($4^\circ\text{C}$) is significantly more viscous and dense than warm summer water ($25^\circ\text{C}$), exerting greater hydraulic drag on media grains. Consequently:

• In winter, lower backwash rise rates ($15\text{ to }18\text{ gpm/sq ft}$) achieve the required $30%$ bed expansion. Over-washing with high rates in winter sweeps expensive anthracite media over the washwater troughs into the drain.
• In summer, warmer water is less viscous and provides less lifting force. Operators must increase backwash pumping rates ($20\text{ to }25\text{ gpm/sq ft}$) to achieve the same bed fluidization and prevent mudball formation.

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Filter Run Performance and Washwater Efficiency

Monitoring filter efficiency involves tracking water volume filtered per run, total backwash water consumed, and the net clean water recovery ratio.

1. Total Backwash Water Volume

$\text{Backwash Water Volume }(gal) = \text{Backwash Pump Rate }(gpm) \times \text{Backwash Pumping Duration }(min)$

If auxiliary surface wash or filter-to-waste (ripening) is utilized, those volumetric totals are added to the backwash volume to obtain total non-potable process washwater consumption.

2. Unit Filter Run Volume (UFRV)

Unit Filter Run Volume (UFRV) is the premier operational metric for evaluating filter performance, defined as the total volume of treated water produced per square foot of media surface area during an entire filter run:

$\text{UFRV }(gal/sq\ ft) = \frac{\text{Total Gallons Filtered During Operating Run}}{\text{Filter Surface Area }(sq\ ft)}$

$\text{UFRV }(gal/sq\ ft) = \text{Filtration Rate }(gpm/sq\ ft) \times \text{Filter Run Time }(minutes)$

• Operational Benchmarks:
  • $< 3,000\ gal/sq\ ft$: Poor filter performance (short filter runs caused by chemical over-dosing, high coagulant carryover, or severe algae blinding).
  • $5,000\text{ to }8,000\ gal/sq\ ft$: Acceptable standard operational performance.
  • $> 10,000\ gal/sq\ ft$: Superior, highly optimized filtration performance.

3. Percent Backwash Water Used

To ensure a treatment plant is operating economically, the percentage of treated finished water consumed for backwashing must be strictly controlled:

$\%\text{ Backwash Water} = \frac{\text{Total Backwash Water Consumed }(gal)}{\text{Total Finished Water Produced in Filter Run }(gal)} \times 100$

• Benchmark: Industry best-practice standards require backwash consumption to remain below $2%\text{ to }5%$. Operating with backwash consumption above $5%$ wastes finished water, increases pumping costs, and overloads washwater reclaim facilities.

Table 20.2.2: Master Formulas for Filter Operations and Efficiency

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Step-by-Step Worked Multi-Step Calculations

Example 1: Filtration Loading Rate across Multi-Cell Filter Gallery

Problem Statement: A conventional water treatment facility operates four identical dual-media filters, each measuring $20\text{ feet}$ long by $15\text{ feet}$ wide. The facility treats a steady total plant flow rate of $5.76\text{ MGD}$ with all four filters operating in parallel. Calculate:

1. The total surface area of one individual filter bed.
2. The filtration rate across the filters in $gpm/sq\ ft$ with all four units in service.
3. The resulting filtration rate if one filter is taken offline for backwashing while total plant flow remains at $5.76\text{ MGD}$.
4. Whether the three-filter operating rate satisfies standard regulatory guidelines ($3.0\text{ to }5.0\text{ gpm/sq ft}$).

Solution Procedure:

• Step 1: Calculate Surface Area of One Filter
  $\text{Area}_1 = 20\ ft \times 15\ ft = \mathbf{300\ sq\ ft}$ Combined area of 4 filters:
  $\text{Area}_{total} = 4 \times 300\ sq\ ft = 1,200\ sq\ ft$

• Step 2: Convert Plant Flow from $MGD$ to $gpm$
  $\text{Flow Rate } (gpm) = \frac{5,760,000\ gpd}{1,440\ min/day} = \mathbf{4,000\ gpm}$

• Step 3: Calculate Filtration Rate (All 4 Filters in Service)
  Flow per filter:
  $\text{Flow per Filter} = \frac{4,000\ gpm}{4\ filters} = 1,000\ gpm/filter$ Filtration loading rate:
  $\text{Loading Rate} = \frac{1,000\ gpm}{300\ sq\ ft} = \mathbf{3.33\ gpm/sq\ ft}$ (Or: $4,000\ gpm / 1,200\ sq\ ft = 3.33\ gpm/sq\ ft$).

• Step 4: Calculate Filtration Rate with 1 Filter Offline
  With one filter isolated, the remaining 3 filters share the entire $4,000\ gpm$: $\text{Flow per Active Filter} = \frac{4,000\ gpm}{3\ filters} = 1,333.33\ gpm/filter$ $\text{Loading Rate}_{3\text{-filters}} = \frac{1,333.33\ gpm}{300\ sq\ ft} = \mathbf{4.44\ gpm/sq\ ft}$ (Or: $4,000\ gpm / 900\ sq\ ft = 4.44\ gpm/sq\ ft$).

• Step 5: Operational Evaluation
  The loading rate increases from $3.33\text{ gpm/sq ft}$ to $4.44\text{ gpm/sq ft}$. Because $4.44\text{ gpm/sq ft}$ is below the $5.0\text{ gpm/sq ft}$ state regulatory limit for dual-media filters, the plant can safely backwash one filter at full production without triggering hydraulic overload.

Example 2: Backwash Loading Rate, Rise Rate, and Washwater Volume

Problem Statement: A high-rate filter bed measures $24\text{ feet}$ long by $18\text{ feet}$ wide. The backwash pumping system supplies treated water to the bed at a constant rate of $8,640\text{ gpm}$ for a total backwash duration of $12\text{ minutes}$. Calculate:

1. The backwash loading rate in $gpm/sq\ ft$.
2. The vertical backwash rise rate in inches per minute ($in/min$) and feet per minute ($ft/min$).
3. The total volume of backwash water consumed in gallons.

Solution Procedure:

• Step 1: Calculate Filter Bed Surface Area
  $\text{Area} = 24\ ft \times 18\ ft = \mathbf{432\ sq\ ft}$

• Step 2: Calculate Backwash Loading Rate
  $\text{Backwash Loading Rate } (gpm/sq\ ft) = \frac{8,640\ gpm}{432\ sq\ ft} = \mathbf{20.0\ gpm/sq\ ft}$

• Step 3: Calculate Backwash Rise Rate in Inches per Minute and Feet per Minute
  Using the 1.605 conversion factor:
  $\text{Rise Rate } (in/min) = 20.0\ gpm/sq\ ft \times 1.605 = \mathbf{32.1\ in/min}$ (Using exact factor $1.60428$: $20.0 \times 1.60428 = 32.09\ in/min$).
  Converting to feet per minute:
  $\text{Rise Rate } (ft/min) = \frac{32.1\ in/min}{12\ in/ft} = \mathbf{2.675\ ft/min}$ (Or: $20.0\ gpm/sq\ ft / 7.48\ gal/cu\ ft = 2.674\ ft/min$).

• Step 4: Calculate Total Backwash Water Volume Consumed
  $\text{Volume } (gal) = 8,640\ gpm \times 12\ min = \mathbf{103,680\ gallons}$

Example 3: Evaluating UFRV and Percent Backwash Water Efficiency

Problem Statement: A dual-media filter box has a surface area of $350\text{ sq ft}$. The filter operates at a steady filtration rate of $1,400\text{ gpm}$ for a continuous $48\text{-hour}$ filter run before reaching a terminal head loss setpoint of $8.0\text{ feet}$. At the conclusion of the run, the backwash and filter-to-waste cycle consumes a combined total of $94,500\text{ gallons}$ of water. Calculate:

1. The total volume of drinking water filtered during the operating cycle in gallons.
2. The Unit Filter Run Volume (UFRV) in gallons per square foot ($gal/sq\ ft$).
3. The percentage of produced water consumed by backwash operations.
4. Evaluate overall operational efficiency against industry standards.

Solution Procedure:

• Step 1: Calculate Total Water Filtered During the Run
  Convert run duration from hours to minutes:
  $\text{Run Duration} = 48\ hours \times 60\ min/hour = 2,880\ minutes$ $\text{Total Water Filtered } (gal) = 1,400\ gpm \times 2,880\ min = \mathbf{4,032,000\ gallons}$

• Step 2: Calculate Unit Filter Run Volume (UFRV)
  $\text{UFRV } (gal/sq\ ft) = \frac{4,032,000\ gallons}{350\ sq\ ft} = \mathbf{11,520\ gal/sq\ ft}$ (Verification: Filter rate in gpm/sq ft = $1,400 / 350 = 4.0\ gpm/sq\ ft$. UFRV = $4.0\ gpm/sq\ ft \times 2,880\ min = 11,520\ gal/sq\ ft$).

• Step 3: Calculate Percent Backwash Water Consumed
  $\%\text{ Backwash Water} = \frac{94,500\ gal}{4,032,000\ gal} \times 100 = \mathbf{2.34\%}$

• Step 4: Operational Performance Assessment
  The UFRV of $11,520\text{ gal/sq ft}$ exceeds the $10,000\text{ gal/sq ft}$ benchmark for excellence, and backwash consumption ($2.34%$) remains well below the $5.0%$ upper threshold, confirming optimal chemical dosing, effective solids retention, and high net water recovery.