Introduction to Basic Mass Balance Calculations in Environmental Engineering

In your bank account:

Balance = Deposit – Withdrawal

Similarly, in an environmental system or subsystem:

Accumulation = Input – Output

Question 1

In an average week, Mr and Mrs Konzzumerpurchase and bring into their house approximately 50kg of consumer goods (food, magazines, newspapers, appliances, furniture and associated packaging). Of this amount, 50% is consumed as food. Half the food is used for biological maintenance and ultimately released as CO2; the remainder is discharged to the sewer system. Approximately 1kg accumulates in the house. The Konzzumers recycle approximately 25% of the solid waste that is generated. Estimate the amount of solid waste that they place at the curb each week.

• For many environmental problems, time is an important factor in establishing the degree of severity of the problem or in designing a solution

• In these instances, the equations are modified to the following term:

Rate of accumulation = Rate of input – Rate of output

dt

Outd

dt

Ind

dt

dM )()(−=

Question 2

Ms Crystal Clearwater is filling her bathtub, but

she forgot to put the plug in. If the volume of

water for a bath is 0.35 cubic meters and the tap

is flowing at 1.32 L/min and the drain is running at

0.32 L/min, how long will it take to fill the tub to

bath level? How much water will be wasted?

Assume the density of water is 1000 kg/m3.

Mixed systems

Completely mixed systems - those in which every

drop of fluid is homogenous with every other

drop of fluid; every drop contains the same

material or physical property (concentration,

temperature, etc) within its boundary

Steady state – when the rate of accumulation is

zero; Where, Rate in = Rate out

Conservative and Non-conservative substances

• Conservative substances – no reaction; only

mixing. Eg the mixing of sugar or salt in water

• Non-conservative substances – there is reaction.

Eg the reaction of an acid and a base, the

degradation of organic matter

BA

BBAAmix

QQ

QCQCC

+

+=

Question 3

A storm sewer is carrying snow melt containing

1.2 g/L NaCl into a small stream. The stream has

a naturally occurring NaCl concentration of 20

mg/L. If the storm sewer flow rate is 2000 L/min

and the stream flow rate is 2.0 m3/s, what is the

concentration of salt in the stream? Assume that

the sewer flow and stream flow are completely

mixed, and that salt is a conservative substance,

and that the system is at steady state.

• In most environmental systems – reactions or transformations occur within the system Eg. by-products are formed, compounds destroyed, etc. Therefore, it is important to study its reaction kinetics

Rate of accumulation = Rate of input – Rate of output ± Rate of transformation

• The reaction rate (r) is often a complex function of temperature, pressure, the reacting components, and/or the products of reaction

rdt

Outd

dt

Ind

dt

dM±−=

)()(

• For the decay of non-conservative substances,

a convenient model that can be used is the first-

order reaction. In this model, it is assumed that

the rate of loss of the substance is proportional

to the amount of substance present at any given

time, t

Where, k = reaction rate constant, s-1 or d-1

C = concentration of substance

kCdt

dC−=

Question 4

A well mixed sewage lagoon is receiving 430 m3/d

of sewage. The lagoon has a surface area of 105 m2

and a depth of 1 m. The pollution concentration in

the raw sewage is 180 mg/L. The organic matter in

the sewage degrades biologically (decays) in the

lagoon according to first-order kinetics. The reaction

rate constant (decay coefficient) is 0.70 d-1.

Assuming no other water losses or gains

(evaporation, seepage, rainfall) and that the lagoon

is completely mixed, find the steady state

concentration of the pollutant in the effluent.