Sludge Treatment and Disposal

Sludge digestion

Sludge digestion is the biological decomposition of the organic (volatile) matter of sludge in the absence of air (anaerobic) by bacteria, which converts it into stable inert matter, water and gases (methane and CO₂, about 65% and 30%). It reduces the volume and putrescibility of sludge and kills most pathogens. It is done in closed digestion tanks.

Stages (periods) of digestion

 Gas / pH
  CH4 ^                         ______  alkaline fermentation
      |                  ______/
  pH  |      _______----/  acid regression
  5   |_____/  acid fermentation
      +-----------------------------------> time
       (about 15 days)   (about 2 months)  (completion)

1. Acid fermentation (acid formation): acid-forming bacteria hydrolyse carbohydrates, fats and proteins into volatile fatty acids (acetic, propionic, butyric), alcohols, CO₂, H₂S and ammonia. The pH falls to about 5-6, the sludge is acid with foul odour, and little gas is produced.
2. Acid regression: the acids and nitrogen compounds are partly broken down; ammonia and amines are formed from proteins, the pH rises slowly and the sludge becomes less offensive. Sludge becomes grey and the volatile acids begin to decrease.
3. Alkaline fermentation (methane formation): methane bacteria (slow growing) break down the acids to CH₄ and CO₂. pH rises to 6.8-7.4, the sludge is stable and smells like tar, the gas production is highest, and the digested sludge is black and settles and dries easily.

Factors affecting digestion

1. Temperature: mesophilic range 30-38°C (best at 35°C, digestion in 20-30 days) and thermophilic 50-57°C. Digestion time doubles for every 10°C fall; sudden changes of temperature more than 1°C/day upset the process.
2. pH: must be kept at 6.8-7.4; below 6.2 methane bacteria are inhibited. Alkalinity of 1500-5000 mg/l (as CaCO₃) gives buffering.
3. Volatile acids / alkalinity ratio: should be below 0.1-0.2.
4. Solids content and loading: feed of 4-8% solids with volatile loading 1.6-3.2 kg VS/m³/day; overload causes acid conditions.
5. Seeding: mixing fresh sludge with 10-25% ripe digested sludge keeps the active methane bacteria.
6. Mixing: gas or mechanical mixing improves contact of sludge and bacteria.
7. Toxic substances: heavy metals, detergents, chlorine, disinfectants and very high ammonia stop digestion.
8. Retention time: 30-60 days for conventional-rate and 10-20 days for high-rate digestion.
9. Nutrients (C : N ratio): should be about 10-20 : 1.

Aerobic digestion

Sludge (usually waste activated sludge or mixed primary and secondary sludge) is aerated for 10-20 days (at 15-20°C) in an open tank, in the same way as in the activated sludge process. Without fresh food the aerobic bacteria oxidise their own cell mass (endogenous respiration):

$$C_5H_7NO_2+5O_2\rightarrow5CO_2+2H_2O+NH_3+\text{energy}$$

It reduces volatile solids by 35-50%.

• Advantages: simple operation, low capital cost, no odour problems, a supernatant of lower BOD, good for small plants.
• Disadvantages: high power for aeration, no methane gas is produced, and the sludge dewaters less easily.

Anaerobic digestion

Sludge is held in a closed, heated tank without air, with decomposition in three stages by two groups of bacteria. Acid-forming bacteria convert complex organics to volatile acids, and then methane-forming bacteria convert the acids to CH₄ and CO₂:

$$\text{organics}\rightarrow\text{volatile acids}\rightarrow CH_4+CO_2+H_2S$$

It is the common method for primary and mixed sludge in medium and large plants: 40-60% volatile solids are destroyed, and the biogas (65% CH₄, calorific value about 22 MJ/m³) is used for heating and power. The digested sludge is stable, inoffensive and can be dried on beds and used as manure.

Effect of temperature and pH

• Temperature: the rate of digestion increases with temperature. Anaerobic digestion works at the mesophilic range 30-38°C (optimum about 35°C, digestion time about 20-30 days) and the thermophilic range 50-57°C (faster, 10-15 days, but less stable). Below 20°C digestion becomes very slow (60 days or more); the digestion time about doubles for each 10°C fall. Sudden changes of more than 1°C per day upset the methane bacteria, so digesters are heated and insulated.
• pH: acid formers tolerate pH 5-6 but methane formers need 6.8-7.4. If volatile acids accumulate, pH falls below 6.2 and digestion stops ("sour" digester, foul odour, little gas). The pH is corrected by reducing the feed rate and adding lime or bicarbonate to maintain alkalinity of 1500-5000 mg/l.

Thickening

Thickening (concentration) is the process of increasing the solids content of sludge by removing part of its water, so that the volume to be handled is reduced. For example, raising the solids from 2% to 5% reduces the volume to $2/5$, i.e. by 60%.

Purpose

• To reduce the volume of sludge, so that digesters, pumps, pipes, dewatering units and transport are smaller and cheaper.
• To increase the solids loading and so reduce the heat required for heating digesters.
• To reduce chemical demand in conditioning.
• To remove supernatant and give a more uniform feed.

Methods of sludge thickening

1. Gravity thickening (most common for primary and mixed sludge).
2. Dissolved-air flotation (DAF) for waste activated sludge: air bubbles carry the light solids to the surface.
3. Centrifugal thickening (for waste activated sludge).
4. Gravity belt or drum thickeners.
5. Simple settling in the digester itself (supernatant withdrawal) and co-settling in the primary tank.

Gravity sludge thickener

 sludge in -> [feed well]   motor
              |    |        |
  supernatant |    v        | picket fence rake
  <-- weir ~~~~~~~~~~~~~~~~~~~~~~~~~~ water level
      |      \      thickening zone     /|  3-4 m
      |       \_ ___  ___  __/
      |    slope 1 in 6 to 1 in 4 floor
      |            \_/  sludge hopper
      +-------------> thickened sludge to digester

• A circular tank (like a sedimentation tank), 3-4 m side depth, with a floor slope of 1 in 6 to 1 in 4 and a deep central hopper.
• A slowly rotating (about 1 rpm) rake with vertical pickets stirs the sludge gently, releases trapped water and gas, and moves the sludge to the hopper.
• Dilute sludge enters at the centre feed well. The solids settle and compress, and clear supernatant overflows through peripheral weirs (returned to the plant inlet).
• The thickened sludge (5-10% solids for primary sludge) is drawn off from the hopper by pump.
• Design uses solids loading of about 100-150 kg/m²/day for primary sludge and 20-40 kg/m²/day for activated sludge, and hydraulic loading 15-30 m³/m²/day.

Aims of sludge treatment

1. To reduce the volume (water content) so handling, transport and disposal become cheaper.
2. To stabilise the organic matter so that it does not putrefy and cause odour.
3. To destroy pathogens and make the sludge safe to handle and reuse.
4. To make the sludge easy to dewater and dry.
5. To recover resources: biogas (methane) for energy and the dried sludge as manure/soil conditioner.
6. To protect the environment and public health from nuisance, flies and water pollution.

Digester volume

For a conventional-rate digester the capacity with parabolic variation of sludge volume is

$$V=\left[V_1-\frac{2}{3}(V_1-V_2)\right]t_d+V_2\,t_s$$

where $V_1$ = raw sludge volume per day, $V_2$ = digested sludge volume per day, $t_d$ = digestion period, $t_s$ = storage period.

Assumption: 50% of the volatile solids are destroyed in the digester (usual 40-60% for conventional rate).

Raw sludge (500 kg dry solids/day, 70% volatile, moisture 94% i.e. 6% solids):

$$\begin{aligned} VS &= 350\ \text{kg},\quad FS=150\ \text{kg}\\ \text{Mass of wet sludge} &= \frac{500}{0.06}=8333\ \text{kg},\ \text{water}=7833\ \text{kg}=7.833\ \text{m}^3\\ \text{Volume of solids} &= \frac{350}{1.02\times1000}+\frac{150}{2.65\times1000}=0.3997\ \text{m}^3\\ V_1 &= 7.833+0.3997=8.233\ \text{m}^3/\text{day} \end{aligned}$$

Digested sludge (VS destroyed $=0.5\times350=175$ kg; remaining VS $=175$ kg, FS $=150$ kg; total solids $=325$ kg at 10% solids):

$$\begin{aligned} \text{Mass of wet sludge} &= \frac{325}{0.10}=3250\ \text{kg},\ \text{water}=2.925\ \text{m}^3\\ \text{Volume of solids} &= \frac{175}{1020}+\frac{150}{2650}=0.2282\ \text{m}^3\\ V_2 &= 2.925+0.2282=3.153\ \text{m}^3/\text{day} \end{aligned}$$

Volume of digester ($t_d=30$ days, $t_s=12$ days):

$$V=\left[8.233-\frac{2}{3}(8.233-3.153)\right]\times30+3.153\times12=183.2\ \text{m}^3$$

Answer: digester volume $\approx 183.2$ m³ (say 183 m³).

For a conventional-rate digester the capacity with parabolic variation of sludge volume is

$$V=\left[V_1-\frac{2}{3}(V_1-V_2)\right]t_d+V_2\,t_s$$

where $V_1$ = raw sludge volume per day, $V_2$ = digested sludge volume per day, $t_d=30$ days (digestion), $t_s=12$ days (storage during monsoon).

Assumption: 50% of the volatile solids are destroyed in digestion (usual 40-60%).

Raw sludge (500 kg dry solids/day, 70% volatile, moisture 92% i.e. 8% solids):

$$\begin{aligned} VS &= 350\ \text{kg},\quad FS=150\ \text{kg}\\ \text{Mass of wet sludge} &= \frac{500}{0.08}=6250\ \text{kg},\ \text{water}=5.750\ \text{m}^3\\ \text{Volume of solids} &= \frac{350}{1020}+\frac{150}{2650}=0.3997\ \text{m}^3\\ V_1 &= 5.750+0.3997=6.150\ \text{m}^3/\text{day} \end{aligned}$$

Digested sludge (VS left $=175$ kg, FS $=150$ kg, solids $=325$ kg, 90% moisture so 10% solids):

$$\begin{aligned} \text{Mass of wet sludge} &= \frac{325}{0.10}=3250\ \text{kg},\ \text{water}=2.925\ \text{m}^3\\ \text{Volume of solids} &= \frac{175}{1020}+\frac{150}{2650}=0.2282\ \text{m}^3\\ V_2 &= 2.925+0.2282=3.153\ \text{m}^3/\text{day} \end{aligned}$$

Volume of digester:

$$V=\left[6.150-\frac{2}{3}(6.150-3.153)\right]\times30+3.153\times12=162.4\ \text{m}^3$$

Answer: digester volume $\approx 162.4$ m³ (say 162 m³).

Sludge is the semi-solid material (95-99% water) that settles out in the primary and secondary sedimentation tanks, or is removed from screens and filters, in a sewage treatment plant. It is putrescible and contains pathogens, so it needs treatment before disposal.

Need of sludge treatment

1. Control of odour and nuisance: fresh sludge from a primary tank putrefies within hours, giving hydrogen sulphide and other foul gases and attracting flies. Example: untreated sludge dumped near a settlement gives a smell nuisance; digestion makes it inoffensive.
2. Pathogen destruction: raw sludge from domestic sewage carries bacteria, helminth eggs and viruses. Example: using raw sludge as manure on vegetables can transmit worm infection; digestion and drying kill most of them.
3. Volume reduction: the sludge from a plant is about 0.25-0.5% of the sewage volume but contains 95% water. Thickening and dewatering reduce the volume from, say, 100 m³ at 5% solids to 25 m³ at 20%, which lowers the cost of transport.
4. Stabilisation of organic matter: anaerobic digestion converts about half of the volatile solids to gas, leaving inert humus-like matter.
5. Energy and resource recovery: digester gas (about 65% methane) is used for heating and power; dried sludge is used as soil conditioner.
6. Environmental protection: untreated sludge discharged into a river creates sludge banks and oxygen depletion.

Necessity of sludge treatment

Sludge is the semi-solid residue (about 95-99% water) removed from screens, sedimentation tanks, filters and activated sludge units. Raw sludge is a highly putrescible mixture of organic matter, pathogens and heavy metals, so it must be treated before disposal.

1. To prevent nuisance: raw sludge decomposes quickly and gives foul odours, flies and insects.
2. To protect public health: it carries pathogens (bacteria, viruses, helminth eggs) that could spread disease.
3. To reduce volume: untreated sludge has a very large volume; thickening and digestion cut the volume to 1/3-1/10 and make transport and disposal cheaper.
4. To stabilise the organic matter: digestion changes the unstable organic matter into inert stable matter.
5. To prevent pollution of land, groundwater and streams from leachate.
6. To recover resources: biogas for energy and dried sludge as manure.

Sludge drying bed (dewatering)

Digested sludge (about 90-95% water) is dewatered by spreading it in thin layers on sand drying beds, where water is lost by drainage and evaporation.

 Section                          Plan
  sludge  ->  _____ 20-30 cm layer     inlet pipes
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~    +----+----+----+
  sand 15-30 cm (0.3-1.2 mm)      |bed |bed |bed |  6-8 m x 30 m
  gravel 15-30 cm (graded)        +----+----+----+
  open-jointed underdrain pipes -> filtrate back to plant inlet

Construction:

• Beds are rectangular (6-8 m wide, 20-30 m long), with brick or concrete walls 0.5-0.9 m high.
• Layers from the top: 10-15 cm sand (effective size 0.3-1.2 mm), 15-30 cm graded gravel, and an underdrain of open-jointed or perforated pipes at 1 in 100 slope. The filtrate goes back to the treatment plant.
• Area allowance is about 0.1-0.25 m² per person (covered beds in rainy climate).

Working:

1. Digested sludge is applied 20-30 cm deep.
2. Water drains through the sand in 1-3 days (about 50-75% of the water), and the rest evaporates by sun and wind.
3. After 10-15 days (in dry weather) the cake has 30-40% solids and cracks; it is removed by hand with forks and the bed is reused. Dried cake is used as manure or landfill.
4. The sand is renewed after about 10-12 cycles.

Necessity of sludge treatment

Sludge is the semi-solid residue (about 95-99% water) removed from screens, sedimentation tanks, filters and activated sludge units. Raw sludge is a highly putrescible mixture of organic matter, pathogens and heavy metals, so it must be treated before disposal.

• Prevent odour, flies and insect nuisance from putrescible sludge.
• Destroy pathogens and protect public health.
• Reduce the volume and make handling and disposal easier.
• Stabilise the organic matter and recover biogas.

Sludge digester (anaerobic, floating-cover type)

        gas dome / floating cover
      gas out <-  ____|____
   sludge in ->  |  gas    |<- supernatant draw-off
   ~~~~~~~~~~~~~~|~~~~~~~~~|~~~~~~~~~ liquid level
   (heated)      | scum    |
   [heating coil] supernatant
                 | digesting sludge |
                 \  digested sludge /
                  \_______________/ slope 1:1 hopper
                     digested sludge out

Construction: a closed circular tank (diameter 6-30 m, depth 6-12 m), with a conical hopper bottom for sludge withdrawal, a fixed or floating gas-collecting cover (to avoid air entering), inlet for raw sludge, draw-off pipes at different levels for supernatant, a pipe for digested sludge, a gas outlet, a heating coil or heat exchanger to keep 35°C, and a mixer or recirculation pump (in high-rate type).

Working (three stages):

1. Acid fermentation: acid-forming bacteria break down the complex organic matter into volatile acids; pH falls to 5-6.
2. Acid regression: the acids are partly broken down, and the pH rises slowly.
3. Alkaline (methane) fermentation: methane bacteria convert the acids to CH₄ and CO₂ at pH 6.8-7.4. The gas (65% CH₄) collects under the cover and is used for heating.

Raw sludge is fed regularly; after 30-60 days (conventional rate), the stabilised sludge is withdrawn from the bottom to drying beds, the supernatant is drawn off and returned to the plant inlet.

Sludge treatment is arranged in stages so that its volume is reduced, its organic matter is stabilised and it is made safe for disposal.

Brief description

1. Thickening: raw sludge of 1-3% solids is settled in a gravity thickener (picket-fence rake), so that supernatant is removed and a sludge of 5-8% is obtained.
2. Digestion: in a closed, heated tank (35°C) anaerobic bacteria convert volatile organics to methane and CO₂ in 20-60 days. The sludge turns black, inoffensive and settles easily.
3. Conditioning: coagulants or polymers flocculate the solids so the water is released.
4. Dewatering: on sand drying beds (natural drainage and evaporation, for small plants) or by mechanical means (filter press, centrifuge) for large plants.
5. Final disposal: dewatered sludge is used as manure, or put in sanitary landfill, or incinerated. Grinding and blending of sludge are also done to get a uniform feed for digesters and pumps.

After treatment (digestion and dewatering) sludge must be disposed of in a safe way. The common methods are:

1. Dumping: the sludge is disposed of in a large water body (sea disposal or deep water) or on a low-lying open land. In the sea, it is carried by barge to deep water beyond the coast. It is cheap, but it may pollute the water, so it is now banned in many countries. On land, it is spread or dumped in a designated site; only digested and dried sludge should be used.
2. Land filling: dewatered sludge (cake) is buried in trenches or mixed with municipal solid waste in a sanitary landfill and covered daily with 15-30 cm of soil. The decomposition goes on slowly. It needs a site with low groundwater and a leachate collection system. The land can be reclaimed afterwards.
3. Lagooning: digested sludge is run into a shallow earthen basin (lagoon) 1-2 m deep, in which it is stored for 6-24 months. The solids settle and decompose further, the supernatant is drained and returned to the plant, and the dried sludge is removed. It needs a large land area and is suited to small plants in sparsely populated places; odour may occur and the lagoon needs lining to protect groundwater.

Other methods are use as manure/soil conditioner on agricultural land (after digestion and drying) and incineration. The choice depends on the sludge quality, land available, climate and cost.

Volume-moisture relationship

Sludge consists of water and solids. If the amount of solids does not change, the volume of sludge is inversely proportional to the percentage of solids (i.e. $100-P$ where $P$ is the moisture content):

$$\frac{V_1}{V_2}=\frac{100-P_2}{100-P_1}=\frac{S_2}{S_1}$$

where $V_1,V_2$ are the volumes at moisture contents $P_1,P_2$ (%) and $S$ is the percentage solids. This assumes the specific gravity of the sludge stays the same (about 1.02, since solids are so dilute).

Example: sludge of 98% moisture dewatered to 90% moisture gives $V_2/V_1=\dfrac{100-98}{100-90}=0.2$, so the volume falls to one-fifth (80% reduction). A fall from 98% to 96% moisture halves the volume.

Because a small drop of moisture content gives a large drop in volume, thickening and dewatering are very effective in reducing handling and disposal cost.

Methods of sludge disposal

1. Dumping on land or in the sea.
2. Sanitary landfill (burial of dewatered sludge in trenches).
3. Lagooning (storage in shallow basins).
4. Use as manure or soil conditioner on land (after digestion and drying).
5. Incineration.

Most suitable method: land application of digested and dried sludge (with landfill as a back-up)

For a town in Nepal, I choose digested sludge, dried on sand drying beds, and used as manure/soil conditioner on agricultural or forest land, with surplus sent to a sanitary landfill. The justification is:

• Digested and dried sludge is stable, inoffensive and has few pathogens.
• It contains nitrogen, phosphorus and organic matter that improve soil, so it is a resource, not a waste.
• It is cheap: drying beds need little energy or skill, and the climate allows drying for much of the year.
• Landfilling or lagooning needs large land and gives odour risk, dumping in rivers is unacceptable, and incineration needs high capital and fuel cost.
• Heavy-metal content must be checked, and use must be avoided on crops eaten raw.

Volume-moisture relationship

Sludge consists of water and solids; when the solids are constant, the volume varies inversely with the solids percentage:

$$\frac{V_1}{V_2}=\frac{100-P_2}{100-P_1}$$

where $P$ is the moisture content (%). For example, thickening sludge from 98% to 95% moisture reduces the volume to $\dfrac{2}{5}=40\%$ of the original. The relation holds when the sludge specific gravity does not change much, and it explains why a small loss of water greatly reduces volume.

The specific gravity of sludge $S_s$ follows from the masses and specific gravities of its parts:

$$\frac{W_s}{S_s}=\frac{W_{sol}}{S_{sol}}+\frac{W_w}{S_w}$$

where $W$ is the weight fraction.

Specific gravity of the given sludge

Moisture $=95\%$, so solids $=5\%$ and water $=95\%$ by weight. Of the solids, 20% are fixed ($G=2.65$) and 80% volatile ($G=1.0$).

Specific gravity of the solids:

$$\frac{1}{G_{sol}}=\frac{0.2}{2.65}+\frac{0.8}{1.0}=0.8755\ \Rightarrow\ G_{sol}=1.142$$

Specific gravity of the sludge:

$$\frac{1}{S_s}=\frac{0.05}{1.142}+\frac{0.95}{1.0}=0.9938\ \Rightarrow\ S_s=1.0063$$

Answer: specific gravity of the sludge $\approx 1.0063$ (solids $\approx 1.142$).

Method

Sludge volume $=$ volume of water $+$ volume of volatile solids $+$ volume of fixed solids, with solids $=600$ kg dry. Water is 1000 kg/m³; volatile solids $G=1.0$ (1000 kg/m³); fixed solids $G=2.5$ (2500 kg/m³).

Primary sludge (6% solids, 65% volatile)

$$\begin{aligned} VM &= 0.65\times600=390\ \text{kg},\quad FM=210\ \text{kg}\\ \text{Mass of wet sludge} &= \frac{600}{0.06}=10000\ \text{kg}\ \Rightarrow\ \text{water}=9400\ \text{kg}=9.400\ \text{m}^3\\ \text{Volume of solids} &= \frac{390}{1000}+\frac{210}{2500}=0.4740\ \text{m}^3\\ V_{primary} &= 9.400+0.4740=9.874\ \text{m}^3 \end{aligned}$$

Digested sludge (12% solids, 65% of volatile matter removed)

$$\begin{aligned} VM_d &= 390\times(1-0.65)=136.5\ \text{kg},\quad FM=210\ \text{kg}\ (\text{unchanged})\\ \text{Dry solids} &= 136.5+210=346.5\ \text{kg}\\ \text{Mass of wet sludge} &= \frac{346.5}{0.12}=2887.5\ \text{kg}\ \Rightarrow\ \text{water}=2541.0\ \text{kg}=2.541\ \text{m}^3\\ \text{Volume of solids} &= \frac{136.5}{1000}+\frac{210}{2500}=0.2205\ \text{m}^3\\ V_{digested} &= 2.541+0.2205=2.761\ \text{m}^3 \end{aligned}$$

Percentage reduction

$$\frac{9.874-2.761}{9.874}\times100=72.0\%$$

Answer: volume before digestion $=9.874$ m³, after digestion $=2.761$ m³, reduction $=72.0\%$.

Data and assumptions

Flow $Q=6500$ m³/day; suspended solids (SS) $=250$ mg/l; removal in the primary tank $=55\%$; fresh sludge moisture $=95\%$; digested sludge moisture $=85\%$; specific gravity of sludge $=1.02$; digestion period $=60$ days (2 months). The digested sludge is taken to hold the same dry solids as the fresh sludge (solids loss in digestion is ignored, which is conservative); no extra storage period is added.

Step 1: Dry solids removed per day

$$W_s=Q\times SS\times\text{removal}=6500\times250\times0.55\times10^{-3}=893.8\ \text{kg/day}$$

Step 2: Volume of fresh sludge

$$V_1=\frac{W_s}{(1-0.95)\times S\times1000}=\frac{893.8}{0.05\times1.02\times1000}=17.52\ \text{m}^3/\text{day}$$

Step 3: Volume of digested sludge (volume-moisture relation)

$$V_2=V_1\times\frac{100-95}{100-85}=17.52\times\frac{5}{15}=5.84\ \text{m}^3/\text{day}$$

Step 4: Capacity of the digester (parabolic digestion)

$$V=\left[V_1-\frac{2}{3}(V_1-V_2)\right]t=\left[17.52-\frac{2}{3}(17.52-5.84)\right]\times60=584\ \text{m}^3$$

Step 5: Dimensions

Adopt a circular digester with side water depth 6 m (usual 6-12 m) plus 0.5 m free board and a hopper bottom with 1 : 1 slope:

$$A=\frac{584}{6}=97.4\ \text{m}^2,\qquad D=\sqrt{\frac{4A}{\pi}}=11.13\ \text{m}\ (\text{adopt 11.5 m})$$

Answer: digester capacity $\approx 584$ m³, circular, diameter 11.5 m, side water depth 6 m (plus free board, hopper and gas dome).

Flow from the sedimentation tank

The tank capacity is 812.5 m³ with detention time of 3 h, so

$$Q=\frac{812.5}{3}\times24=6500.0\ \text{m}^3/\text{day}$$

Data and assumptions

Flow $Q=6500$ m³/day; suspended solids (SS) $=250$ mg/l; removal in the primary tank $=55\%$; fresh sludge moisture $=95\%$; digested sludge moisture $=80\%$; specific gravity of sludge $=1.02$; digestion period $=60$ days (2 months). The digested sludge is taken to hold the same dry solids as the fresh sludge (solids loss in digestion is ignored, which is conservative); no extra storage period is added.

Step 1: Dry solids removed per day

$$W_s=Q\times SS\times\text{removal}=6500\times250\times0.55\times10^{-3}=893.8\ \text{kg/day}$$

Step 2: Volume of fresh sludge

$$V_1=\frac{W_s}{(1-0.95)\times S\times1000}=\frac{893.8}{0.05\times1.02\times1000}=17.52\ \text{m}^3/\text{day}$$

Step 3: Volume of digested sludge (volume-moisture relation)

$$V_2=V_1\times\frac{100-95}{100-80}=17.52\times\frac{5}{20}=4.38\ \text{m}^3/\text{day}$$

Step 4: Capacity of the digester (parabolic digestion)

$$V=\left[V_1-\frac{2}{3}(V_1-V_2)\right]t=\left[17.52-\frac{2}{3}(17.52-4.38)\right]\times60=526\ \text{m}^3$$

Step 5: Dimensions

Adopt a circular digester with side water depth 6 m (usual 6-12 m) plus 0.5 m free board and a hopper bottom with 1 : 1 slope:

$$A=\frac{526}{6}=87.6\ \text{m}^2,\qquad D=\sqrt{\frac{4A}{\pi}}=10.56\ \text{m}\ (\text{adopt 11.0 m})$$

Answer: digester capacity $\approx 526$ m³, circular, diameter 11.0 m, side water depth 6 m (plus free board, hopper and gas dome).

Sludge volume $V = \dfrac{W_s}{\rho_w \, G_{sl}\,(1-p)}$, where $W_s$ is the dry solids mass, $G_{sl}$ the specific gravity of wet sludge and $p$ the water fraction.

Assumptions (not given in the data): suspended solids are 70% organic and 30% inorganic; trickling filter removes 85% of the BOD reaching it; humus (SST sludge) produced is 0.4 kg dry solids per kg BOD removed; SST sludge also has 95% water.

Specific gravity of sludge

$$\frac{1}{G_s} = \frac{0.7}{1.02} + \frac{0.3}{2.65} \Rightarrow G_s = 1.251$$ $$\frac{1}{G_{sl}} = \frac{0.05}{1.251} + \frac{0.95}{1} \Rightarrow G_{sl} = 1.0101$$

Sludge from PST

• Suspended solids in raw sewage: $20\times10^6 \times 280\times10^{-6} = 5,600$ kg/d
• Removed in PST (60%): $W_s = 3,360$ kg/d

$$V_{PST} = \frac{3,360}{1000 \times 1.0101 \times 0.05} = 66.5\ \text{m}^3\text{/d}$$

Sludge from SST (humus)

• BOD in raw sewage: $20\times10^6 \times 220\times10^{-6} = 4,400$ kg/d
• BOD after PST (30% removed): $3,080$ kg/d
• BOD removed in trickling filter (85%): $2,618$ kg/d
• Dry humus solids: $0.4 \times 2,618 = 1,047$ kg/d

$$V_{SST} = \frac{1,047}{1000 \times 1.0101 \times 0.05} = 20.7\ \text{m}^3\text{/d}$$

Answer: PST sludge $\approx 66.5\ \text{m}^3$/d; SST sludge $\approx 20.7\ \text{m}^3$/d (total about 87.3 m³/d).

The mass of dry solids stays the same while the water content falls, so $V = \dfrac{W_s}{\rho_w\,G_{sl}\,(1-p)}$.

Fresh sludge (95% moisture)

• Solids in raw sewage: $4\times10^6 \times 220\times10^{-6} = 880$ kg/d
• Removed (55%): $W_s = 484.0$ kg/d

$$V_1 = \frac{484.0}{1000 \times 1.02 \times 0.05} = 9.49\ \text{m}^3\text{/d}$$

Digested sludge (80% moisture)

$$V_2 = \frac{484.0}{1000 \times 1.02 \times 0.20} = 2.37\ \text{m}^3\text{/d}$$

(solids loss during digestion is neglected, and the sludge specific gravity is taken as 1.02 in both cases.)

Digester capacity (T = 80 days)

$$\begin{aligned} V &= \left[V_1 - \tfrac{2}{3}(V_1 - V_2)\right] T \\ &= \left[9.49 - \tfrac{2}{3}(9.49 - 2.37)\right] \times 80 \\ &= 4.75 \times 80 = 380\ \text{m}^3 \end{aligned}$$

Adopt effective depth 6 m (assumed):

$$\frac{\pi}{4} D^2 \times 6 = 380 \Rightarrow D = 8.98\ \text{m} \approx 9.0\ \text{m}$$

Provide a circular digester of 9.0 m diameter and 6 m effective depth, plus 0.5 m free board and a hopper bottom (slope 1 vertical to 1.5 horizontal) with a sludge withdrawal pipe.

Answer: fresh sludge 9.49 m³/d; digested sludge 2.37 m³/d; digester capacity 380 m³ (diameter about 9.0 m).

Volume of sludge $V = \dfrac{W_s}{\rho_w \, G_{sl}\,(1-p)}$

• Solids in raw sewage: $10\times10^6\ \text{L/d} \times 250\ \text{mg/L} = 2500$ kg/d
• Solids removed (60%): $W_s = 1500$ kg/d
• Solids fraction of sludge: $1 - 0.95 = 0.05$

$$V = \frac{1500}{1000 \times 1.02 \times 0.05} = 29.41\ \text{m}^3\text{/d}$$

Answer: volume of sludge $\approx 29.4\ \text{m}^3$ per day.

The specific gravity of sludge is not given, so take $G_{sl} = 1.02$ (usual value for sewage sludge).

• Solids in raw sewage: $6\times10^6 \times 300\times10^{-6} = 1800$ kg/d
• Solids removed (65%): $W_s = 1170$ kg/d

$$V = \frac{W_s}{\rho_w\,G_{sl}\,(1-p)} = \frac{1170}{1000 \times 1.02 \times 0.05} = 22.94\ \text{m}^3\text{/d}$$ $$V_{year} = 22.94 \times 365 = 8,374\ \text{m}^3$$

Answer: about 22.9 m³/day, i.e. about 8,374 m³ of sludge per year.

Flow

$$Q = \frac{\text{Volume}}{\text{detention time}} = \frac{500}{4} = 125\ \text{m}^3\text{/h} = 3000\ \text{m}^3\text{/d}$$

Sludge from the sedimentation tank

• Solids in sewage: $3000\times1000 \times 200\times10^{-6} = 600$ kg/d
• Removed (60%): $W_s = 360$ kg/d

$$V_1 = \frac{360}{1000 \times 1.02 \times (1-0.98)} = 17.65\ \text{m}^3\text{/d}$$

Digested sludge

$$V_2 = 0.40 \times 17.65 = 7.06\ \text{m}^3\text{/d}$$

Digester volume (T = 30 days)

$$\begin{aligned} V &= \left[V_1 - \tfrac{2}{3}(V_1 - V_2)\right] T \\ &= \left[17.65 - \tfrac{2}{3}(17.65 - 7.06)\right] \times 30 \\ &= 10.59 \times 30 = 317.6\ \text{m}^3 \end{aligned}$$

Diameter (effective depth 6 m)

$$\frac{\pi}{4}D^2 \times 6 = 317.6 \Rightarrow D = \sqrt{\frac{4 \times 317.6}{\pi \times 6}} = 8.21\ \text{m}$$

Answer: diameter of digester $\approx 8.2$ m (say 8.3 m), digester volume 318 m³.