1. CHAPTER 2 MASS BALANCE and APPLICATION

2. Transport Phenomena
(1)   Material Balance
(2)   Momentum Transfer
(3)   Heat Transfer
Mass Transfer
Reactor Design
Particle Motion
Transport in Porous Media

3. Material balance
The materials balance (or mass balance) is a quantitative description of all materials that enter, leave, and accumulate in a system with defined boundaries.
A materials balance is based on the law of conservation of mass (i.e., mass is neither created nor destroyed).
The basic materials balance expression is developed on a chosen control volume and has terms for material entering, leaving, being generated, and being accumulated or stored within the volume.

4. Material balance
The basic materials balance expression is developed on a chosen control volume and has terms for material entering, leaving, being generated, and being accumulated or stored within the volume.

5. Material balance
The corresponding simplified word statement for a materials balance is:
Accumulation = Inflow – Outflow + Generation

7. Using rA to indicate the mass rate of generation of A within the volume and assuming that dispersion and diffusion are negligible, the mass balance equation can be written in quantitative symbolic form:

8. V = volume, m3
CA = mass concentration of A, g/m3
Q = volumetric flow rate in a single direction, m3/s
rA = mass rate of generation, g/m3×s

9. Remembering that
Qx = vxΔyΔz
Qy = vyΔxΔz
Qz = vzΔxΔy
where
vx = the velocity in the x direction
vy = the velocity in the y direction
vz = the velocity in the z direction

10. Taking the limit as Δx, Δy, Δz approach zero, then the mass balance equation becomes
The mass rate of generation can be positive or negative. Most of the materials of interest disappear, and therefore rA will be negative in most cases.

11. Steady and Transient States
In applying materials balances, two operational states must be considered: steady state and transient (unsteady) state.
The primary requirement for steady state is that there be no accumulation within the system (i.e., ∂CA∂t =0 ). In another words, all rates and concentrations do not vary with time at steady state.
In transient state, the rate of accumulation is changing with time (∂CA∂t ≠ 0). For example, the filling of a reservoir is a good example of system in the transient state.

12. Materials Balance Applications
Mass balances are of fundamental importance in the field of environmental engineering or environmental management.
􀁺 To evaluate water and wastewater treatment process performance
􀁺 To study the response of the aquatic environment to selected inputs, such as the discharge of wastewater.

13. One Dimensional Conditions
Continuous-flow stirred tank reactors (CFSTRs) have no concentration gradients within the system. Material entering is uniformly dispersed instantaneously throughout the reactor. The result is that the concentration of any material leaving the reactor is exactly the same as the concentration at any point in the reactor. A material balance for material A for the reactor would result in：

14. CFSTR Material Balance for A where
CA = concentration of material A in reactor, g/m3
V = reactor volume, m3
Q = volumetric flow rate, m3/s
CAi = input concentration of material A, g/m3
rA = rate of reaction of material A, g/m3s

15. 【Example 1】 A conservative (non-reactive) material is injected into the input flow of a CSFTR on a continuous basis, beginning at time t = 0 and resulting in a constant input tracer concentration of CTi. Determine CT the reactor output concentration, as a function of time, and plot the tracer-output response curve (i.e., reactor output versus time).

16. 【Solution 1】
1. Write the materials balance equation for the CFSTR.
where
CT = concentration of tracer in reactor, g/m3
V = reactor volume, m3
Q = volumetric flow rate, m3/s
CTi = input concentration of tracer, g/m3
rT = rate of reaction of tracer, g/m3s

17. 2. Because the tracer is conservative (i. e
2. Because the tracer is conservative (i.e., non-reactive), the generation term is zero since rT = 0. The material balance derived in step 1 can be rearranged can integrated as shown below. The limits of integration are from C = 0 to C = CT and from t = 0 to t = t.

18. 3. Plot the tracer response curve
3. Plot the tracer response curve. Because numerical data are not given, assume CTi = 1, and plot CT/CTi versus the term t(Q/V). The tracer response curve for the reactor output is plotted as follow:

19. Hydraulic detention time In example 1, the term V/Q appeared, which has units of time, and will be referred to as the mean hydraulic detention time θH. Note that the value of CT/CTi is 0.95 after three hydraulic detention times (t = 3θH). Therefore, in practice, the conditions existing after three hydraulic detention times are often considered to be a satisfactory approximation of the final, or steady state (dCT/dt = 0), conditions. Thus, if a CFSTR is subjected to a step change in input characteristics, the new steady-state value is assumed to have been achieved when three detention times have passed.

20. 【Example 2】 A reaction A → B, known to be first order (rA = -kCA), is to be carried out in a CFSTR. Water is run through the reactor at a flow rate Q m3/s, and at t = 0 the reaction A is added to the input stream on a continuous basis. Determine the output concentration of A as a function of time.

21. 【Solution 2】 1. Write the materials balance equation for the system 2
【Solution 2】 1.Write the materials balance equation for the system 2. Assume the rate of generation is rA = -kCA 3. Rearrange the materials balance given in step 1 and integrate the resulting expression to find CA. The limits of integration are from C = 1 to C = CA and from t = 0 to t = t.

22. 4. As t approaches infinity, the steady-state solution is approached:

23. Application of the CFSTR Model

24. Application of the CFSTR model in water quality management is made in two ways. Some treatment facilities are designed as stirred tanks (contents of the tank are mixed completely), and the model applies directly. Another use is made in describing natural systems. In the latter case, a physical system is divided conceptually into a series or group of stirred tanks. For example a reach of a river as shown in the above figure might be considered to behave as a series of CFSTRs. The river can be divided into several segments based on measured velocities and depths. Each segment can be viewed as a CFSTR. If a wastewater effluent discharge a contaminant A on the upstream and the river flow rate is Q, then we can use CFSTR model to predict the output concentration of A. Assuming the pollutant disappearing follows the first order reaction. Then, we can apply the CFSTRs model to predict the output concentration of A for each segment.

25. 【Example 3】 Using the conception of above, and dividing the target river into 5 segments, for the given data below, determine the steady-state pollutant concentration in each segment. A = pollutant V1 = 8.64×105 m3 CAi = 30 g/m3 V2 = 25.92×105 m3 rA = -kCA V3 = 17.28×105 m3 k = 0.2 d-1 V4 = 8.64×105 m3 Q = 5 m3/s V5 = 25.92×105 m3

26. 【Solution 3】
Assuming steady-state conditions (dCAi/dt = 0), write materials balance equations for each segment.

27. where Q = volumetric flow rate, g/m CAi = influent concentration of pollutant A, g/m3 CA1, CA2, CA3, CA4, CA5 = concentration of pollutant A leaving each reactor, g/m3 V1, V2, V3, V4, V5 = reactor volume, m3

28. 2. Rearranging the equations developed in step 1 and solve for CAn.

29. 3. Solve for the concentrations
3. Solve for the concentrations. Substituting numerical values and solving for the individual pollutant concentration yields.