2. Mass Balance Calculations

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 Konzzumer purchase 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

dM dt

d ( In ) d ( Out ) = − dt dt

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

C

mix

=

C

A

Q Q

A A

+ C + Q

B

Q

B

B

• Non-conservative substances – there is reaction. Eg the reaction of an acid and a base, the degradation of organic matter

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. byproducts 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

dM d ( In ) d ( Out ) = − ±r dt dt dt • The reaction rate (r) is often a complex function of temperature, pressure, the reacting components, and/or the products of reaction

• For the decay of non-conservative substances, a convenient model that can be used is the firstorder 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 dC dt

=

− kC

Where, k = reaction rate constant, s-1 or d-1 C = concentration of substance

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.