Sludge Disposal Effluent Discharge Standard

Chapter 6

SLUDGE DISPOSAL, EFFLUENT DISCHARGE STANDARD

Sludge and moisture content

Moisture content of sludge is the percentage of water present in the wet sludge. In this chapter moisture content is expressed on a wet basis, i.e., as the percentage of the total (wet) mass that is water.

For simplicity in many engineering problems the mass (or volume) of solids is taken as a unit quantity. Let the mass of dry solids be 1 (unit mass). Let the initial moisture content be p₁ (%) and the final moisture content after dewatering be p₂ (%). Let the mass of water initially be x and finally be y.

The percent moisture on wet basis is

p₁ = (mass of water / total mass) × 100 = (x / (1 + x)) × 100

Therefore, multiplying both sides by (1 + x):

100 x = p₁ + p₁ x

Rearranging gives

x (100 - p₁) = p₁

Hence

x = p₁ / (100 - p₁)

Similarly, for the final condition

y = p₂ / (100 - p₂)

These relations let you convert between moisture content (wet basis) and the mass of water per unit mass of dry solids. If the mass of solids is not unity but a known value M_s, multiply x and y by M_s to get actual masses of water.

Example application: If initial moisture p₁ = 98% and final moisture p₂ = 80%, the corresponding water masses per unit dry solids are

x = 98 / (100 - 98) = 98 / 2 = 49

y = 80 / (100 - 80) = 80 / 20 = 4

So water removed per unit dry solids = x - y = 45 units of mass.

Dilution and mixing (mass-balance) in a river

When sewage of concentration C_s (mass per unit volume) and discharge Q_s (volume per unit time) mixes with river water of concentration C_R and discharge Q_R, the combined concentration C immediately downstream (assuming complete instantaneous mixing across the cross section) is found by mass balance:

C = (C_s Q_s + C_R Q_R) / (Q_s + Q_R)

Here the numerator represents the total pollutant mass per unit time entering the mixing zone and the denominator is the total flow. This simple dilution formula is widely used to estimate initial pollutant concentrations after discharge.

Zones of pollution in a stream

• Zone of initial dilution - immediate mixing near the point of discharge; concentration falls rapidly due to dilution.
• Zone of decomposition (zone of active decomposition) - biological oxidation of organic matter causes biochemical oxygen demand (BOD) decay and oxygen consumption.
• Zone of recovery - reaeration from the atmosphere exceeds deoxygenation and dissolved oxygen (DO) begins to recover.
• Zone of clean water - DO reaches near saturation and river returns to normal quality.

Oxygen deficit and dissolved oxygen (DO)

Oxygen deficit at any time is the difference between saturation DO at the water temperature and the actual DO:

D = DO_sat - DO_actual

A plot of DO (or of deficit) with downstream distance (or time) typically shows a fall to a minimum (maximum deficit) followed by recovery, as illustrated conceptually below.

Streeter-Phelps model (deoxygenation and reaeration)

The Streeter-Phelps model describes the change in biochemical oxygen demand (BOD) and oxygen deficit with time downstream after a pollutant discharge. Two processes are modelled:

• Deoxygenation - oxidation of organic matter which consumes dissolved oxygen; characterised by deoxygenation rate constant k_d (time-1).
• Reaeration - transfer of oxygen from atmosphere to water; characterised by reaeration rate constant k_r (time-1).

BOD decay (deoxygenation)

Let L(t) be the biodegradable BOD remaining at time t and L₀ the initial biodegradable BOD at t = 0. First-order decay is assumed:

dL/dt = -k_d L

Solving (separation of variables and integration):

L(t) = L₀ e^{-k_d t}

Oxygen deficit differential equation

The rate of change of oxygen deficit D(t) equals oxygen demand generated by deoxygenation minus oxygen supplied by reaeration:

dD/dt = k_d L(t) - k_r D(t)

Solution for D(t)

To solve dD/dt + k_r D = k_d L₀ e^{-k_d t}, use an integrating factor μ(t) = e^{k_r t}.

Multiply both sides by μ(t):

e^{k_r t} dD/dt + k_r e^{k_r t} D = k_d L₀ e^{(k_r - k_d) t}

Left-hand side is derivative:

d/dt [e^{k_r t} D] = k_d L₀ e^{(k_r - k_d) t}

Integrate both sides from 0 to t:

e^{k_r t} D(t) - D(0) = k_d L₀ ∫₀^t e^{(k_r - k_d) τ} dτ

Performing the integral (consider k_r ≠ k_d):

e^{k_r t} D(t) - D(0) = k_d L₀ [e^{(k_r - k_d) t} - 1] / (k_r - k_d)

Therefore

D(t) = D(0) e^{-k_r t} + (k_d L₀ / (k_r - k_d)) [e^{-k_d t} - e^{-k_r t}]

In this expression:

• D(0) is the initial oxygen deficit at t = 0.
• L₀ is the initial biodegradable BOD immediately after mixing.
• k_d is the deoxygenation rate constant and k_r is the reaeration rate constant.

Critical time and maximum oxygen deficit

The oxygen deficit reaches a maximum at time t_c where dD/dt = 0. Using the differential equation dD/dt = k_d L(t) - k_r D(t), set the derivative to zero to get the condition for the critical time.

For the common case where the initial oxygen deficit is zero (D(0) = 0), the critical time simplifies to

t_c = (1 / (k_r - k_d)) ln (k_r / k_d)

Substituting t = t_c into the D(t) expression yields the maximum deficit D_max. For D(0) ≠ 0 the formula for t_c and D_max becomes more complex but follows from the same procedure of setting dD/dt = 0 and solving for t.

Where the symbols appearing in the expression are as defined above.

Practical notes and applications

• The Streeter-Phelps model is widely used for preliminary design and impact assessment of wastewater discharges to rivers; it predicts the downstream DO profile and helps determine safe dilution and treatment requirements.
• Rate constants k_d and k_r depend on temperature, turbulence, and characteristics of the organic load; they are typically reported for standard conditions and adjusted by correction factors.
• When applying the model, ensure L₀ is the biodegradable fraction of BOD immediately after mixing (non-biodegradable portion does not contribute to deoxygenation rate).
• Regulatory effluent discharge standards use combinations of limits on BOD, COD, suspended solids and sometimes concentration limits downstream; mass-balance dilution calculations and Streeter-Phelps predictions support compliance assessments.