1. Chapter 3 Material & Energy Balance
Eskedar T. (AAiT-CED)

2. Materials and Energy Balances
Tools for quantitative understanding of the behavior of environmental systems.
For accounting of the flow of energy and materials into and out of the environmental systems.

3. Materials and Energy Balances
Material Balance
Energy Balance
production, transport, and fate
modeling
Pollutant
Energy

4. Unifying Theories Conservation of Matter
The law of conservation of matter states that (without nuclear reaction) matter can neither be created nor destroyed.
We ought to be able to account for the “matter” at any point in time.
The mathematical representation of this accounting system is called a materials balance or mass balance.
Conservation of Matter

5. Unifying Theories Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed.
Meaning that we should be able to account for the “energy” at any point in time.
The mathematical representation of this accounting system we use to trace energy is called an energy balance.
Conservation of Energy

6. Material Balances input output
The simplest form of a materials balance or mass balance
Accumulation = input – output
Accumulation
input
output
Environmental System
(Natural or Device)

7. The control volume Control Volume Food to Consumer people goods Solid
Accumulation
Food to
people
Consumer
goods
Solid
Waster

8. Examples of Control Volume

9. Time as a factor Mass rate of accumulation =
Mass rate of input – Mass rate of output

10. Example 3.1
Selam is filling her bathtub but she forgot to put the plug in. if the volume of water for a bath is m3 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? Assuming Selam shuts off the water when the tub is full and does not flood the house, how much water will be wasted? Assume the density of water is 1,000 kg/m3

11. Solution Vaccumulation We must convert volumes to masses.
Qin = 1.32 L/min
Qout = 0.32 L/min
We must convert volumes to masses.
Mass = (volume)(density)
Volume = (flow rate)(time) = (Q)(t)

12. Solution From mass balance we have Accumulation = mass in –mass out
(Vacc)() = (Qin)()(t) - (Qin)()(t)
Vacc = (Qin)(t) – (Qin)(t)
Vacc = 1.32t – 0.32t
350L = (1.00 L/min)(t)
t= 350 min or hr
The amount of wasted water is
Waste water = (0.32)(350) = 112 L

13. Efficiency
Mass flow rate =
Mass balance 
Efficiency of a system
OR 

14. Example
The air pollution control equipment on a municipal waste incinerator includes a fabric filter particle collector (known as a baghouse). The baghouse contains 424 cloth bags arranged in parallel, that is 1/424 of the flow goes through each bag. The gas flow rate into and out of the baghouse is 47 m3/s, and the concentration of particles entering the baghouse is 15 g/m3. In normal operation the baghouse particulate discharge meets the regulatory limit of 24 mg/m3. Calculate the fraction of particulate matter removed and the efficiency of particulate removal when all 424 bags are in place and the emissions comply with the regulatory requirements. Estimate the mass emission rate when one of the bags is missing and recalculate the efficiency of the baghouse. Assume the efficiency for each individual bag is the same as the overall efficiency for the baghouse.

15. Solution Cout = 24 mg/m3 Qout = 47 m3/s Cin = 15 g/m3 Qin = 47 m3/s
Baghouse
Cin = 15 g/m3
Qin = 47 m3/s
Accumulation =
particle
removal
Hopper

16. Cont’d 𝑑𝑀 𝑑𝑡 = (15,000mg/m3)(47m3/s)-(24mg/m3)(47m3/s)=703,872 mg/s
The fraction of particulates removed is
703,872𝑚𝑔/𝑠 (15,000𝑚𝑔/𝑚3)(47𝑚𝑔/𝑠) = 703,872𝑚𝑔/𝑠 705,000𝑚𝑔/𝑠 =0.9984
The efficiency of the baghouse is
η= 15,000 𝑚𝑔/𝑚3−24 𝑚𝑔/𝑚3 15,000 𝑚𝑔/𝑚3 (100%)
=99.84%

17. Cont’d Cemission= ? Cin = 15 g/m3 Qemission= 47 m3/s
QBypass =(1/424)( 47 m3/s)
Cemission= ?
Qemission= 47 m3/s
Baghouse
Bypass
Cout= ?
Qout= (423/424)47 m3/s
Cin = 15 g/m3
Qin =( )( 47 m3/s)
𝑑𝑀 𝑑𝑡 = ?
Control Volume

18. Cont’d
A control volume around the baghouse alone reduces the number of unknowns to two: Because we know the efficiency and the influent mass flow rate, we can solve the mass balance equation for the mass flow rate out of the filter. Solving for CoutQout CoutQout =(1- η)CinQin= ( )(15,000mg/m3)(47m3/s)( ) =1,125mg/s

19. Cont’d
Effluent
𝑑𝑀 𝑑𝑡 =CinQin from bypass +CinQin from baghouse-CemissionQemission
Because there is no accumulation in the junction
𝑑𝑀 𝑑𝑡 =0 and the mass balance equation
CoutQout=CinQin from bypass +CinQin from baghouse
= (15,000mg/m3)(47m3/s)(1/424)+1,125=2788 mg/s
The concentration in the effluent is
𝐶𝑜𝑢𝑡𝑄𝑜𝑢𝑡 𝑄𝑜𝑢𝑡 = 2,788 mg/s 47𝑚3/𝑠 = 59 mg/m3
The overall efficiency of the baghouse with missing bag is
η= 15,000 𝑚𝑔/𝑚3−59 𝑚𝑔/𝑚3 15,000 𝑚𝑔/𝑚3 (100%)= 99.61%
Bypass
From baghouse

20. Exercise
A storm sewer is carrying snow melt containing g/L of sodium chloride into a small stream. The stream has a naturally occurring sodium chloride concentration of 20 mg/L. If the storm sewer flow rate is 2.00 L/min and the stream flow rate is 2.0 m3/s, what is the concentration of salt in the stream after the discharge point? Assume that the sewer flow and the storm flow are completely mixed, that the salt is a conservative substance (it does not react) and that the system is at steady state.

21. Reaction Order and Reactors
kinetic reactions : reactions that are time dependent.
Reaction kinetics: the study of the effects of temperature, pressure, and concentration on the rate of a chemical reaction.

22. Rate of reaction
The rate of reaction, ri, the rate of formation or disappearance of a substance.
𝑑𝑀 𝑑𝑡 = 𝑑(𝑖𝑛) 𝑑𝑡 − 𝑑(𝑜𝑢𝑡) 𝑑𝑡 +r, r=-KCn
Homogenous reactions. single phase reactions
Heterogeneous reactions : multiphase reactions (between phases surface)

23. Rate of reaction ri = kf1(T,P);f2([A],[B], …)
Rate constant
Concentration of reactant
Assuming that the pressure and temperature are constant
aA + bB cC
Rate of reaction  rA = - k[A]α[b]β = k[C]γ

24. Order of reaction Rate of reaction  rA = - k[A]α[b]β = k[C]γ
the order with respect to reactant A is α, to B is β, and to product C is γ.
rA = -k zero-order reaction
rA = -k[A] first-order reaction
rA = -k[A2] second-order reaction
rA = -k[A][B] second-order reaction

25. Order of reaction

26. material flows into, through, and out of the reactor
Types of Reactors
batch reactors and flow reactors.
fill-and-draw
material flows into, through, and out of the reactor
Unsteady state

27. Flow reactors
IDEAL REACTORS
REAL REACTOR

28. Steady-state conservative system
Conserved system: where no chemical or biological reaction takes place and no radioactive decay occurs for the substance in the mass balance.
Steady-state:
Input rate = Output rate  Accumulation =0

29. Steady-state conservative system
Decay rate = 0
Accumulation rate = 0
Mixture
Wastes
Stream Qm Cm Qw Cw Qs Cs
Q = flow rate
C = concentration
CsQs + QwCw = QmCm

30. Including reactions For non-conservative substances
Accumulation rate = input rate – output rare ± transformation rate

31. Simple completely mixed systems
With first-order reactions
Total mass of substance = concentration x volume
when V is a constant, the mass rate of decay of the substance is
first-order reactions can be described by r = -kC=dC/dt, 

32. Example
A well-mixed sewage lagoon is receiving 430 m3/d of sewage out of a sewer pipe. The lagoon has a surface area of 10 ha and a depth of 1.0m. The pollutant concentration in the raw sewage discharging into the lagoon is 180 mg/L. The organic matter in the sewage degrades biologically in the lagoon according to first-order kinetics. The reaction rate constant is 0.70 d-1. Assuming no other water losses or gains and that the lagoon is completely mixed, find the steady-state concentration of the pollutant in the lagoon effluent.

33. Solution Accumulation=input rate – output rate – decay rate
Sewage
Lagoon
Cin = 180 mg/L
Qin = 430 m3/d
Ceff = ?
Qeff = 430 m3/d
Decay
Control volume
Accumulation=input rate – output rate – decay rate
Assuming steady-state condition, accumulation = 0
input rate = output rate + decay rate
CinQin = CeffQeff + (K)(Clagoon)(V)
Ceff=1.10mg/ L

34. Batch reactors
dM/dt = ?

35. Plug-flow reactors Mass balance for each plug element
No mass exchange occurs across the plug boundaries, d(in) and d(out) = 0

36. Plug-flow reactors The residence time for each plug: Residence time
L=length
The residence time for each plug:

37. Example
A wastewater treatment plant must disinfect its effluent before discharging the wastewater to a near-by stream. The wastewater contains 4.5 x 105 fecal coliform colony-forming units (CFU) per liter. The maximum permissible fecal coliform concentration that may be discharged is 2,000 fecal coliform CFU/L. It is proposed that a pipe carrying the wastewater be used for disinfection process. Determine the length of the pipe required if the linear velocity of the wastewater in the pipe is 0.75 m/s. Assume that the pipe behaves as a steady-state plug-flow system and that the reaction rate constant for destruction of the fecal coliforms is 0.23 min-1.

38. Solution
Using the steady-state solution on the mass-balance equation, we obtain
Solving for the length of pipe, we have
ln(4.44x10-3)=-0.23min-1 𝐿 45𝑚/𝑚𝑖𝑛
L=1,060m
Cin=4.5x105 CFU/L
U=0.75 m/s
Cout=2,000 CFU/L
U=0.75 m/s

39. Waste Concentration (mg/L)
Example
A contaminated soil is to be excavated and treated in a completely mixed aerated lagoon. To determine the time it will take to treat the contaminated soil, a laboratory completely mixed batch reactor is used to gather the following data. Assuming a first-order reaction, estimate the rate constant, k, and determine the time to achieve 99 % reduction in the original concentration.
Time (d)
Waste Concentration (mg/L) 1
280 16
132

40. Solution Using the 1st and 16th day, the time interval t= 16-1 = 15 d
Solving for k, we have k = d-1
To achieve 99 % reduction the concentration at
Time t must be 1 – 0.99 of the original concentration
t = 92 days

41. Step function response
Concentration, Cin
Time
Step increase
Time
Step decrease
Time
pulse/spike increase

42. Batch reactor Concentration, Cin Decay Formation Time 1/k Time 1/k
Time
1/k
Time
Decay
1/k
Time
Formation

43. Influent concentration Effluent concentration
CMFR Conservative C1
0 t= C1 C0
-0.37C C1 C0
Concentration, Cin
Concentration, Cin
Time
Influent concentration
Time
Effluent concentration C0
Concentration, Cin C0
Concentration, Cin
0.37C0

44. CMFR Conservative
For balanced flow (Qin = Qout) and no reaction, the mass balance becomes
Where M = CV. The solution is
Where  = V/Q

45. CMFR Conservative
Flushing of nonreactive contaminant from a CMFR by a contaminant-free fluid
Which means Cin = 0 and the mass balance becomes
Where M = CV. The initial concentration is C0=M/V
For time t  0 we obtain

46. Exercise
Before entering an underground utility vault to do repairs, a work crew analyzed the gas in the vault and found that it contained 29mg/m3 of H2S. Because the allowable exposure level is 14 mg/m3 the work crew began ventilating the vault with a blower. If the volume of the vault is 160 m3 and the flow rate of contaminant-free air is 10 m3/min, how long will it take to lower the H2S level to that will allow the work crew to enter? Assume the manhole behaves as CMFR and that H2S is nonreactive in the time period considered.

47. CMFR NonConservative
For balanced flow (Qin = Qout) and first-order reaction the mass balance becomes
Where M = CV. By dividing with Q and V we have

48. CMFR NonConservative OR For stead-state conditions dC/dt=0 Decay
Formation Co Co
Concentration, Cin
Concentration, Ceff
Time
Time

49. CMFR NonConservative A step decrease in influent concentration (Cin=O)
for non-steady-state conditions with first-order decay
Where M = CV. By dividing with Q and V we have

50. Effluent concentration Influent concentration
CMFR NonConservative C0
Concentration, Cin C0
Concentration, Cin
Time
Effluent concentration
Time
Influent concentration

51. Example
A chemical degrades in a flow-balanced, steady-state CMFR according to first-order reaction kinetics. The upstream concentration of the chemical is 10 mg/L and the downstream concentration is 2 mg/L. Water is being treated at a rate of 29 m3/min. The volume of the tank is 590 m3. What is the rate of decay? What is the rate constant?

52. Solution
For a first-order reaction, the rate of decay, r =-kC, thus we have to solve for kC from
For steady-state, dM/dt = 0 and for balanced flow, Qin = Qout
r=kC=0.4

53. Solution… For a first-order reaction in a CMFR
The mean hydraulic detention time is
Solving for the rate constant we get

54. Report format Mini Project 20 to 30 pages
Cover page: Title and Group member names
Table of content
References
Last Date of Submission: June , 2011