Mass balances and reactors

Matter is neither created nor destroyed. That simple law of conservation of mass is the most powerful tool an environmental engineer has. Draw a boundary around anything, whether a lake, a treatment tank, a city's air or a landfill, and the mass of a substance inside can change only because the substance flows in, flows out, is produced or is consumed by reactions inside. Writing that statement as an equation, a mass balance, lets engineers predict concentrations in rivers below outfalls, size treatment tanks, estimate how fast a polluted lake will clean itself and check whether laboratory data make sense. This lesson develops the general mass balance, applies it to mixing problems and to lakes, and introduces the two ideal reactors, the completely mixed flow reactor and the plug flow reactor, that are used to model everything from treatment basins to rivers.

The general mass balance

Choose a control volume, a region of space with a clearly defined boundary. For any substance, the rate at which its mass inside the control volume changes equals the rate of mass flowing in, minus the rate flowing out, plus the rate of production by reactions inside, minus the rate of consumption by reactions inside:

The flow terms are flow times concentration, as in lesson 1. The reaction terms are written per unit volume and multiplied by the volume. A substance that does not react, such as chloride or a dye tracer, is called conservative; for it, the reaction terms are zero. A substance that decays, such as biodegradable organic matter, bacteria or a radioactive isotope, is non-conservative.

Most problems simplify in one of two ways. At steady state, nothing changes with time, so dM/dt is zero and the inflows balance the outflows and reactions. Many design calculations assume steady state using average flows and loads. For time-varying problems, such as a spill or a change in a discharge, the full equation is solved as a differential equation.

Steady-state mixing

The simplest mass balance is the mixing of two streams with no reaction, such as an effluent entering a river. If the river and the effluent mix completely at the junction, the flow downstream is the sum of the two flows and the downstream concentration is the flow-weighted average:

Complete mixing at the outfall is an idealization. In reality, an effluent plume hugs one bank and may travel kilometres before it spreads across a wide river. Permits often define a mixing zone, a limited area near the outfall where water quality objectives may be exceeded, and require the objectives to be met at its edge. Diffusers, pipes with many ports across the riverbed, speed up mixing.

The completely mixed flow reactor

An ideal completely mixed flow reactor (CMFR), also called a continuously stirred tank reactor, is perfectly mixed: the concentration is the same everywhere inside, and the outflow has the same concentration as the contents. Well-stirred tanks and many lakes and ponds behave approximately this way. The key parameter is the hydraulic residence time, the average time water spends inside:

For a substance that decays by a first-order reaction, where the rate of loss is proportional to the concentration (rate = kC, lesson 4), the steady-state mass balance on a CMFR is 0 = QC_in − QC − kCV. Solving for the outlet concentration gives:

When things change: the transient CMFR

Suppose the creek suddenly begins carrying herbicide into a lake that had none. The mass balance is then a first-order differential equation, V dC/dt = QC_in − QC − kCV, and its solution starting from zero is:

Here C∞ is the steady-state value found above. The concentration approaches steady state exponentially, reaching about 95 per cent of it after three time constants, where one time constant is 1/(1/τ + k). For Willow Lake, 1/τ + k = 0.0216 + 0.10 = 0.122 per day, so the lake would reach 95 per cent of its steady value in about 3/0.122 = 25 days. When the source stops, the concentration falls with the same exponential rate. A lake with a long residence time and slow decay takes years to recover after a pollutant is removed, a hard lesson learned on many lakes polluted by phosphorus or mercury.

# Willow Lake: herbicide concentration after the creek starts carrying it
import math
V = 2.0e6               # lake volume, m3
Q = 0.50 * 86400        # inflow, m3/d
Cin = 8.0               # herbicide in creek, mg/L
k = 0.10                # first-order decay, 1/d
tau = V / Q
Css = Cin / (1 + k * tau)
rate = 1 / tau + k      # 1/d
print(f"tau = {tau:.1f} d, steady state = {Css:.2f} mg/L")
for t in (0, 5, 10, 20, 30, 50):
    C = Css * (1 - math.exp(-rate * t))
    print(f"day {t:3d}: C = {C:.2f} mg/L ({100 * C / Css:.0f}% of steady state)")

tau = 46.3 d, steady state = 1.42 mg/L
day   0: C = 0.00 mg/L (0% of steady state)
day   5: C = 0.65 mg/L (46% of steady state)
day  10: C = 1.00 mg/L (70% of steady state)
day  20: C = 1.30 mg/L (91% of steady state)
day  30: C = 1.38 mg/L (97% of steady state)
day  50: C = 1.42 mg/L (100% of steady state)

The plug flow reactor

In an ideal plug flow reactor (PFR), fluid moves through a long channel or pipe as a series of plugs that do not mix with the plugs ahead or behind. Each plug behaves like a small batch reactor travelling through the system, so a parcel of water spends exactly the residence time τ inside. Long, narrow basins, pipelines and some rivers approximate plug flow. For a first-order reaction, the outlet concentration is:

For the same volume and flow, a plug flow reactor removes more of a substance that decays by first-order reaction than a CMFR does. The reason is that a CMFR immediately dilutes the incoming water down to the low outlet concentration, so the reaction runs at its slowest rate everywhere, while in a plug flow reactor the reaction runs fast near the inlet where concentrations are high.

Real reactors lie between the two ideals. Dead zones, short-circuiting and back-mixing make a tank behave differently from its design. Engineers measure the actual behaviour with a tracer test: a pulse of a conservative dye or salt is added at the inlet and its concentration is recorded at the outlet. The shape of the resulting curve shows how close the tank is to plug flow, and the time at which 10 per cent of the tracer has passed, called T10, is used to give credit for disinfection (lesson 7).

Mass balances as a check on data

Mass balances are also an auditing tool. If a treatment plant reports that 150 kg/d of phosphorus enters, 15 kg/d leaves in the effluent and only 60 kg/d leaves in the sludge, then 75 kg/d is unaccounted for, which suggests a sampling problem, a flow meter error or a mistake in the sludge data. Joy checks every set of plant data this way before Nadia uses it for design. Mass balances on water itself, comparing metered inflows with billed consumption, are how utilities find leaks in distribution systems.

Mass balances also apply to whole regions. Air quality agencies build emission inventories by adding up the mass of each pollutant released by every source in an airshed. Watershed managers compare the phosphorus delivered by farms, towns and the atmosphere with the mass leaving at the river mouth. In each case the same equation, in minus out plus production minus consumption, organizes the problem.

Choosing the control volume well is half the work. A good boundary cuts through places where flows and concentrations are known or can be measured, such as pipes, channels and stacks, and encloses the process of interest. Nadia draws a box around the whole treatment plant to check the plant's phosphorus data, a smaller box around one clarifier to size it, and a box around a reach of river to study an outfall. The same physical system can be analysed with many control volumes, and comparing them is often how errors are found.