Lin S.D. Water and Wastewater Calculations Manual

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mixed, and supernatant is not withdrawn. In a two-stage system, sludge

is stabilized in the first stage, whereas the second stage provides set-

tling and thickening. The digester is heated to 34 to 36⬚C.

An anaerobic digester is designed to provide warm, oxygen-free, and

well-mixed conditions to digest organic matter and to reduce patho-

genic organisms. The hardware include covers, heaters, and mixers.

Operation of the process demands control of food supply, temperature

(31 to 36⬚C, 88 to 97⬚F), pH (6.8 to 7.2), alkalinity (2000 to 3500 mg/L

as CaCO

), and detention time (60 days at 20⬚C to 15 days at 35⬚C).

Gas production. Gas production is one of the important parameters

for measuring the performance of the digester. Typically, gas production

ranges from 810 to 1120 L of digester gas per kg volatile solids (13 to

18 ft

gas/lb VS) destroyed. Gas produced from a properly operated

digester contains approximately 65% to 69% methane and 31% to 35%

carbon dioxide. If more than 35% of gas is carbon dioxide, there is prob-

ably something wrong with the digestion system (US EPA, 1991). The

quantity of methane gas produced can be computed by the following

equation (McCarty, 1964)

V ⫽ 350[Q(S

⫺ S)/(1000) – 1.42 P

] (SI units) (6.214)

V ⫽ 5.62[Q(S

⫺ S)8.34 ⫺ 1.42 P

] (British system) (6.215)

where V ⫽ volume of methane produced at standard

conditions (0⬚C, 32⬚F and 1 atm), L/d or ft

350, 5.62 ⫽ theoretical conversion factor for the amount of

methane produced per kg (lb) of ultimate BOD

oxidized (see Example 1), 350 L/kg or 5.62 ft

/lb

1000 ⫽ 1000 g/kg

Q ⫽ flow rate, m

/d or Mgal/d

⫽ influent ultimate BOD, mg/L

S ⫽ effluent ultimate BOD, mg/L

8.34 ⫽ conversion factor, lb/(Mgal ⭈ mg/L)

⫽ net mass of cell tissue produced, kg/d or lb/d

For a complete-mix high-rate two-stage anaerobic digester without recy-

cle, the mass of biological solids synthesized daily, P

, can be estimated

by the equation below

(SI units) (6.216)

(British system) (6.217)

Y[QsS

2 Sd8.34]

1 1 k

Y[QsS

2 Sd]

1 1 k

Wastewater Engineering 815

where Y ⫽ yield coefficient, kg/kg or lb/lb

⫽ endogenous coefficient, per day

⫽ mean cell residence time, day

Other terms are as defined previously.

Example 1: Determine the amount of methane generated per kg of ulti-

mate BOD stabilized. Use glucose, C

, as BOD.

solution:

Step 1. Use the balanced equation for converting glucose to methane and

carbon dioxide under anaerobic conditions

M.W. 180 48

1 x

x ⫽ 48/180 ⫽ 0.267

Step 2. Determine oxygen requirement for methane

Using a balanced equation of oxidation of methane to carbon dioxide and water

3CH

⫹ 6O

S 3CO

⫹ 6H

M.W. 48 192

Ultimate BOD per kg of glucose ⫽ (192/180) kg

⫽ 1.07 kg

Step 3. Calculate the rate of the amount of methane generated per kg of

BOD

converted

Note: 0.25 (or simply 48/192 ⫽ 0.25) kg of methane is produced by each kg of

BOD

stabilized.

Step 4. Calculate the volume equivalent of 0.25 kg of methane at the stan-

dard conditions (0⬚C and 1 atm)

Note: 350 L of methane is produced per kg of ultimate BOD stabilized, or 5.60

of methane is produced per lb of ultimate BOD stabilized (by a conversion

factor, L/kg ⫻ 0.016 ⫽ ft

/lb).

5 12.36 ft

5 350 L

Volume 5 s0.25 kg 3 1000 g/kgda

1 mole

16 g

b a

22.4 L

mole

or simply 48/192 5 0.25/1.0

kg CH

kg BOD

0.267

1.07

0.25

1.0

3CH

1 3CO

816 Chapter 6

Example 2: Determine the size of anaerobic digester required to treat pri-

mary sludge under the conditions given below, using a complete-mix reactor.

Also check loading rate and estimate methane and total gas production.

solution:

Step 1. Calculate the sludge volume produced per day

Q ⫽ 11.24 m

⫽ 397 ft

Step 2. Calculate the total BOD

loading rate

BOD

loading ⫽ 0.14 kg/m

⫻ 3785 m

⫽ 530 kg/d

⫽ 1.170 lb/d

Step 3. Calculate the volume of the digester

From Step 1

Q ⫽ 11.24 m

V ⫽ Qu

⫽ (11.24 m

/d) (15 days)

⫽ 169 m

⫽ 5968 ft

Use the same volume for both the first and second stages.

Step 4. Check BOD

volumetric loading rate

⫽ 3.14 kg/(m

⭈ d)

BOD

530 kg/d

169 m

Sludge volume per day 5

s3785 m

/dds0.15 kg/m

1.01s1000 kg/m

ds0.05 kg/kgd

Average design flow 3785 m

/d (1 Mgal/d)

Dry solid removed 0.15 kg/m

(1250 lb/Mgal)

Ultimate BOD

removed 0.14 kg/m

(1170 lb/Mgal)

Sludge solids content 5% (95% moisture)

Specific gravity of solids 1.01

15 days

Temperature 35⬚C

Y 0.5 kg cells/kg BOD

0.04 per day

Efficiency of waste utilization 0.66

Wastewater Engineering 817

Step 5. Estimate the mass of volatile solids produced per day, using Eq. (6.216)

⫽ 530 kg/d (from Step 2)

QS ⫽ 530 kg/d (1 – 0.66) ⫽ 180 kg/d

⫽ 10.9 kg/d

Step 6. Calculate the percentage of stabilization

⫽ 63.1%

Step 7. Calculate the volume of methane produced per day, using Eq. (6.214)

V ⫽ 350 L/kg [(530 – 180) – 1.42 (10.9)] kg/d

⫽ 117,000 L/d

⫽ 117 m

⫽ 4130 ft

Step 8. Estimate total gas (CH

⫹ CO

) production

Normally 65% to 69% of gas is methane. Use 67%.

Total gas volume ⫽ (117 m

/d)/0.67

⫽ 175 m

⫽ 6170 ft

Example 3: Estimate the gas produced in a anaerobic digester treating the

secondary sludge from 1.0 MGD (3785 m

/d) of wastewater containing 240

mg/L of total suspended solids (TSS).

solution:

Step 1. Estimate TSS removed, assuming the over-all secondary treatment

process has TSS removal of 90%

TSS removed ⫽ 240 mgL ⫻ 0.9 ⫽ 216 mg/L

Step 2. Estimate volatile suspended solids (VSS) in raw sewage removed,

assuming VSS/TSS ⫽ 0.75

VSS removed ⫽ 216 mg/L ⫻ 0.75 ⫽ 162 mg/L

% stabilization 5

[s530 2 180d 2 1.42s10.9d] kg/d 3 100%

530 kg/d

0.05s530 2 180d kg/d

1 1 s0.04 day

d15 days

YQsS

2 Sd

1 1 k

818 Chapter 6

Step 3. Estimate volatile suspended solids (VSS) in the sludge reduced 65%

VSS reduced ⫽ 162 mg/L ⫻ 0.65 ⫽ 105.3 mg/L

Step 4. Calculate volatile solids (VS) reduced in 1 Mgal/d of sewage

VS reduced ⫽ (l Mgal/d) ⫻ 8.34 lb/(Mgal ⭈ mg/L) ⫻ 105.3 mg/L ⫽ 878 lb/d

Step 5. Estimate daily gas production for 1 MGD of sewage, assuming 15.0 ft

/lb

of VS

Gas produced ⫽ 15.0 ft

/lb ⫻ 878 lb/d

⫽ 13,170 ft

Example 4: The desired digester temperature is 91⬚F (32.8⬚C). The average

raw sludge temperature is 45⬚F (7.2⬚C). The volume of raw sludge is 1,800

gal/d (6,810 L/d) with a specific weight of 1.20. Determine the quantity of heat

needed for the digest, if SRT ⫽ 50 days and radiation heat loss of the inner

surface is 7 Btu/h/ft

solution:

Step 1. Calculate the heat required to raise the temperature of the sludge

Heat ⫽ 1800 gal/d ⫻ 8.34 lb/gal ⫻ (91 ⫺ 45) ⬚F

⫽ 690,600 Btu/d

Step 2. Determine the capacity of the digester with SRT ⫽ 50 days

Digester ⫽ 1800 gal/d ⫻ 0.1337 ft

/gal ⫻ 50 days

⫽ 12,030 ft

Use one digester 15 ft deep, thus the diameter (D)is

15 ⫻ (3.14/4)D

⫽ 12,030

D ⫽ 32.0 ft

Step 3. Calculate radiation heat loss, assuming 7 Btu/h/ft

from the inner

surface

Area of the tank floor or roof ⫽ 3.14 ⫻ 16 ft ⫻ 16 ft ⫽ 804 ft

Area of the wall ⫽ 32 ft ⫻ 3.14 ⫻15 ft ⫽ 1507 ft

Total area ⫽ 804 ft

⫹ 804 ft

⫹ 1507 ft

⫽ 3115 ft

Radiation loss ⫽ 7 Btu/h/ft

⫻ 24 h/d ⫻ 3115 ft

⫽ 523,300 Btu/d

Step 4. Calculate the total heat required

Sum of heat to raise sludge and radiation loss ⫽ 690,600 Btu/d

⫹ 523,300 Btu/d

⫽ 1,213,900 Btu/d

Wastewater Engineering 819

Adding extra heat (25%) for other losses

Total heat required ⫽ 1,213,900 Btu/d ⫻ 1.25 ⫽ 1,517,400 Btu/d

Egg-shaped digesters. The egg-shaped digester looks similar to an

upright egg and is different from the conventional American digester and

the conventional German digester. The American digester typically is a

relatively shallow cylindrical vessel with moderate floor and roof slopes.

The typical conventional German digester is a deep cylindrical tank

with steeply sloped top and bottom cones. The egg-shaped digester was

first installed in Germany in the 1950s.

The advantages of the egg-shaped digester over conventional digesters

include the elimination of abrupt change in the vessel geometry, reduced

inactive volume of the digester, more cost effectiveness and less energy

costs. The egg shape of the vessel provides unique advantages. Its mech-

anisms are as follows. Heavy solids cannot settle out over a large unman-

ageable bottom area; they must concentrate in the steep-sided bottom

cone. Light solids do not accumulate across a large diameter surface

area; they concentrate at the top of a steep-sided converging cone. The

settleable materials can be pumped to the top of the vessel to sink light

material into the digesting main mass. Likewise, light materials can be

pumped to the bottom zone to stir the heavier materials. Small amounts

of energy are required to manage the top and bottom zones. Moving

materials through the main digesting mass to assure homogeneous

digester conditions can be done more effectively (CBI Walker, 1998).

The egg-shaped digester is currently constructed of steel. It has

become popular in the United States and other countries. More detailed

description and design information may be found elsewhere (CBI

Walker, 1998; Stukenberg et al., 1990).

Aerobic digestion. Aerobic digestion is used to stabilize primary sludge,

secondary sludge, or a combination of these by long-term aeration. The

process converts organic sludge solids to carbon dioxide, ammonia, and

water by aerobic bacteria with reduction of volatile solids, pathogens,

and offensive odor.

In a conventional aerobic digester, concentration of influent VSS must

be no more than 3 % for retention times of 15 to 20 days. High-purity

oxygen may be used for oxygen supply.

Sludge is introduced to the aerobic digester on a batch (mostly), semi-

batch, or continuous basis. Aerobic digesters are typically a single-stage

open tank like the activated-sludge aeration tank. In the batch basis,

the digester is filled with raw sludge and aerated for 2 to 3 weeks, then

stopped. The supernatant is decanted and the settled solids are removed

(US EPA, 1991). For the semibatch basis, raw sludge is added every

couple of days; the supernatant is decanted periodically, and the settled

solids are held in the digester for a long time before being removed.

820 Chapter 6

The design standard for aerobic digesters varies with sludge character-

istics and method of ultimate sludge disposal. Sizing of the digester is com-

monly based on volatile solids loading of the digester or the volatile solids

loading rate. It is generally determined by pilot and/or full-scale study. In

general, volatile suspended solids loading rates for aerobic digesters vary

from 1.1 to 3.2 kg VSS/(m

⭈ d) (0.07 to 0.20 lb VSS/(ft

⭈ d)), depending on

the temperature and type of sludge. Solids retention time is 15 to 20 days

for activated sludge only, and that for primary plus activated sludge is 20

to 25 days (US EPA, 1991). An aeration period ranges from 200 to 300

degree-days which is computed by multiplying the digester’s temperature

in ⬚C by the sludge age. According to Rule 503 (Federal Register, 1993), if

the sludge is for land application, the residence time requirements range

from 60 days at 15⬚C (900⬚ ⭈ d) to 40 days at 20⬚C (800⬚ ⭈ d). Desirable aer-

obic digestion temperatures are approximately 18 to 27⬚C (65 to 80⬚F). A

properly operated aerobic digester is capable of achieving a volatile solids

reduction of 40% to 50% and a pathogens reduction of 90%.

GLUMRB (1996) recommended volume requirements based on pop-

ulation equivalent (PE). Table 6.25 presents the recommended digestion

tank capacity based on a solids content of 2% with supernatant sepa-

ration performed in a separate tank. If supernatant separation is per-

formed in the digestion tank, a minimum of 25% additional volume is

required. If high solids content (>2%) is applied, the digestion tank

volume may be reduced proportionally. A digestion temperature of 15⬚C

(59⬚F) and a solids retention time of 27 days (405⬚ ⭈ d) is recommended.

Cover and heating are required for cold temperature climate areas.

The air requirements of aerobic digestion are based on DO (1 to 2 mg/L)

and mixing to keep solids in suspension. If DO in the digestion tank

falls below 1.0 mg/L, the aerobic digestion process will be negatively

Wastewater Engineering 821

TABLE 6.25 Recommended Volume Required for Aerobic Digester

Volume per population equivalent

Type of sludge m

Waste activated sludge—no 0.13

∗

4.5

primary settling

Primary plus activated sludge 0.11

∗

4.0

Waste activated sludge 0.06

∗

2.0

exclusive of primary sludge

Extended aeration activated 0.09 3.0

sludge

Primary plus fixed film 0.09 3.0

reactor sludge

∗

These volumes also apply to waste activated sludge from single-stage nitrification

facilities with less than 24 h detention time based on design average flow.

SOURCE: Greater Lakes Upper Mississippi River Board, 1996

impacted. The diffused air requirements for waste activated sludge only

is 20 to 35 ft

/min per 1000 ft

(20 to 35 L/(min ⭈ m

)) (US EPA, 1991).

The oxygen requirement is 2.3 kg oxygen per kg VSS destroyed (Metcalf

and Eddy, Inc. 1991).

The aerobic tank volume can be computed by the following equation

(Water Pollution Control Federation, 1985b)

(6.218)

where V ⫽ volume of aerobic digester, ft

⫽ influent average flow rate to digester, ft

⫽ influent suspended solids concentration, mg/L

Y ⫽ fraction of the influent BOD

consisting of raw

primary sludge, in decimals

⫽ influent BOD

, mg/L

X ⫽ digester suspended solids concentration, mg/L

⫽ reaction-rate constant, day

–1

⫽ volatile fraction of digester suspended solids,

in decimals

u ⫽ solids retention time, day

The term YS

can be eliminated if no primary sludge is included in the

influent of the aerobic digester. Equation (6.218) is not to be used for a

system where significant nitrification will occur.

Example 1: The pH of an aerobic digester is found to have declined to 6.3.

How much sodium hydroxide must be added to raise the pH to 7.0? The

volume of the digester is 378 m

(0.1 Mgal). Results from jar tests show that

36 mg of caustic soda will raise the pH to 7.0 in a 2-L jar.

solution:

NaOH required per m

⫽ 36 mg/2 L ⫽ 18 mg/L

⫽ 18 g/m

NaOH to be added ⫽ 18 g/m

⫻ 378 m

⫽ 6804 g

⫽ 6.8 kg ⫽ 15 lb

Example 2: A 50 ft (15 m) diameter aerobic digester with 10 ft (3 m) side-

wall depth treats 13,000 gal/d (49.2 m

/d) of thickened secondary sludge. The

sludge has 3.0% of solids content and is 80% volatile matter. Determine the

hydraulic digestion time and volatile solids loading rate.

V 5

1 YS

XsK

1 1/u

822 Chapter 6

solution:

Step 1. Compute the effective volume of the digester

V ⫽ 3.14 (50 ft/2)

⫻ 10 ft ⫻ 7.48 gal/ft

⫽ 147,000 gal

⫽ 556 m

Step 2. Compute the hydraulic digestion time

Digestion time ⫽ V/Q ⫽ 147,000 gal/13,000 gal/d

⫽ 11.3 days

Step 3. Compute the volatile solids loading rate

Solids ⫽ 3.0% ⫽ 30,000 mg/L

VSS ⫽ 30,000 mg/L ⫻ 0.8 ⫽ 24,000 mg/L

Volume ⫽ 147,000 gal/(6.48 ft

/gal) ⫽ 19,650 ft

⫽ 0.132 lb/(d ⭈ ft

)

⫽ 2.11 kg/(d ⭈ m

)

Example 3: Design an aerobic digester to receive thickened waste acti-

vated sludge at 1500 kg/d (3300 lb/d) with 2.5% solids content and a specific

gravity of 1.02. Assume that the following conditions apply:

solution:

Step 1. Calculate the quantity of sludge to be treated per day, Q

⫽ 58.8 m

1500 kg/d

1000 kg/m

s1.02ds0.025d

Minimum winter temperature 15⬚C

Maximum summer temperature 25⬚C

Residence time (land application, 900⬚

⭈ d at 15⬚C

Rule 503) 700⬚

⭈ d at 25⬚C

VSS/TSS 0.80

SS concentration in digester 75% of influent SS

at 15⬚C 0.05 day

⫺1

Oxygen supply and mixing diffused air

0.013 Mgal/d 3 24,000 mg/L 3 8.34 lb/sMgal/ddsmg/Ld

19,650 ft

VSS loading 5

VSS applied, lb/d

Volume of digester, ft

Wastewater Engineering 823

Step 2. Calculate sludge age required for winter and summer conditions

During the winter:

Sludge age ⫽ 900⬚d/15⬚⫽60 days

During the summer:

Sludge age ⫽ 700⬚d/25⬚⫽28 days

The winter period is the controlling factor.

Under these conditions, assume the volatile solids reduction is 45% (normally

40% to 50%).

Note: A curve for determining volatile solids reduction in an aerobic digester

as a function of digester liquid temperature and sludge age is available

(WPCF, 1985b). It can be used to compare winter and summer conditions.

Step 3. Calculate volatile solids reduction

Total mass of VSS ⫽ 1500 kg/d ⫻ 0.8

⫽ 1200 kg/d

VSS reduction ⫽ 1200 kg/d ⫻ 0.45

⫽ 540 kg/d

Step 4. Calculate the oxygen required

Using 2.3 kg O

/kg VSS reduction

⫽ 540 kg/d ⫻ 2.3

⫽ 1242 kg/d

Convert to the volume of air required under standard conditions, taking

the mass of air as 1.2 kg/m

(0.75 lb/ft

) with 23.2% of oxygen.

⫽ 4460 m

Assume an oxygen transfer efficiency of 9%. Total air required is

Air ⫽ (4460 m

/d)/0.09

⫽ 49,600 m

⫽ 34.4 m

/min

⫽ 1210 ft

/min

Step 5. Calculate the volume of the digester under winter conditions

Using Eq. (6.218)

⫽ 2.5% ⫽ 25,000 mg/L

X ⫽ 25,000 mg/L ⫻ 0.75 ⫽ 18,750 mg/L

Air 5

1242 kg/d

1.2 kg/m

3 0.232

824 Chapter 6