PH NEUTRALIZATION REACTOR: A CSTR EXAMPLE

4.4.1 Process Description

The control of pH is quite common in the chemical process and biotechnological industries. For instance, the pH of effluent streams from wastewater treatment plants should be maintained within stringent environmental limits. When a stream is acidic, some source of OH– ions (e.g., NaOH) is used to bring the pH up to the specification of seven. Here, an example is taken to know more about a pH process.

Figure 4.9 depicts a CSTR, where an acidic solution having a volumetric flow rate, FA, with a composition, x1,i, is neutralized with an alkaline solution having a volumetric flow, FB, with a composition of base, x2,i, and buffer agent, x3,i.

We must know that the resistance of a solution to changes in H+ ion concentration upon the addition of small amounts of acid or alkali is referred to as buffer action (Rakshit, 1993). Solutions that possess such properties are called buffer solutions or simply, buffers. Usually, the buffer solutions consist of mixtures of solutions of a weak acid or base and its salt.

FIGURE 4.9 The pH neutralization reactor.

4.4.2 Mathematical Model

Assumptions

The model (Galan et al., 2000) of a pH neutralization reactor is developed based on the following assumptions:

The tank is assumed to be perfectly mixed and isothermal.

The reactor volume is maintained constant using an overflow weir.

The chemical and electrical equilibrium conditions are prevailed due to the fact that the reaction rates in acid-base systems between dissolved compounds are extremely high.

The acid used is strong acid, and the base and buffer agents are both highly soluble salts.

The solution in the reactor is electrically neutral.

Reactions

The neutralization reaction takes place between a strong acid (say, HA) and a strong base

(say, BOH) in the presence of a buffer agent (say, BX). The system experiences the following chemical reactions:

...(4.26a) ...(4.26b) ...(4.26c) ...(4.26d) ...(4.26e)

First three equations (i.e., 4.26a–c) represent the acid, base and buffer dissociation, respectfully. Next reaction (4.26d) defines the buffer action that works as a weak base and final reaction (4.26e) represents the water dissociation. The following chemical equilibrium conditions are obtained by supposing the high reaction rates of the acid-base reactions (instantaneous):

...(4.27a)

...(4.27b)

...(4.27c)

...(4.27d)

...(4.27e)

In the above expressions of equilibrium constants, the concentration of any species, say s, is represented by [s].

For water dissociation, the equilibrium constant is given in terms of activity (denoted by a) as:

(4.28)

The activity of water, in the pure state or in dilute solution is constant and is taken as unity.

Therefore, for the dilute acid-base solution in the example reactor, we have:

...(4.29)

Expressing activity as the product of concentration and activity coefficient (), we obtain:

...(4.30)

In dilute solutions or pure water, the activity coefficients and are almost unity. As a result, Equation (4.30) yields Equation (4.27e).

At this point, it is important to mention that usually the acid-base systems are modelled by the use of invariant chemical species. The term invariant is used for process state variables, which are not affected by the chemical reactions occurring in the system (Fjeld et al., 1974). For the example system, the invariant chemical species are selected as material balances on the irreducible ions associated with the acid (A–), base (B+) and buffer agent (X–).

Charge balance: The pH equation

Supposing the electroneutrality condition, the charge balance establishes that the net charge in an ionic solution is zero. The charge balance yields:

...(4.31)

The sources of B+ are the base as well as the buffer agent. Accordingly, ...(4.32)

The invariant species for the example system are given below:

x1 = [HA] + [A–]...(4.33a)

x2 = [BOH] + [BX] + [B+]...(4.33b) x3 = [BX] + [HX] + [X–]...(4.33c)

Notice that the invariant species [(4.33a), (4.33b) and (4.33c)] are associated with the species A (acid), B (base) and X (buffer), respectively. Since HA is considered as a strong acid, it is completely dissociated. It implies,

[HA] = 0

...KA  

Similarly, the base and buffer agents are both highly soluble. Therefore, [BOH] = 0

...KB   and

[BX] = 0

...KC   Now Equations (4.33a–c) give:

x1 = [A–]...(4.34a) x2 = [B+]...(4.34b) x3 = [HX] + [X–]...(4.34c) We can rewrite Equation (4.32) as:

[B+]total = [B+]from BOH + [B+]from BX...(4.35)

= x2 + x3

The second right-hand term in the above equation indicates the same number of moles of BX added, i.e., [HX] + [X–]. Substituting [HX] from Equation (4.27d) into Equation (4.34c) and rearranging, we obtain:

...(4.36)

Substituting [B+]total from Equation (4.35), [A–] from Equation (4.34a), [OH–] from Equation (4.27e) and [X–] from Equation (4.36) into Equation (4.31), we obtain:

...(4.37)

This correlation is referred to as the pH equation. Once [H+] is known, the pH can be determined from the following expression:

...(4.38)

Material balance equations Total material balance

Since the reactor volume is assumed constant, the overall balance yields for the case of constant density:

FA + FB = Fo...(4.39) Component material balance

It is easy to write for acid:

...(4.40)

Substituting Equation (4.39) and rearranging, we get:

...(4.41)

In the same fashion, we obtain two component material balance equations, respectively, for base and buffer:

...(4.42)

...(4.43)

The dynamic model structure of the representative pH neutralization reactor system derived using conservation equations and equilibrium relations is summarized below:

Model of the pH neutralization reactor

FA + FB = Fo...(4.39) ...(4.41)

...(4.42) ...(4.43)

...(4.37) with

pH = –log10 [H+]...(4.38)

In the present study, the computer simulation of the reactor model is not covered. A set of model parameters is given in Table 4.4 (Galan et al., 2000) for interested readers for simulation. The pH system for which the data are given receives an acid stream (HCL solution) and an alkaline stream (NaOH and NaHCO3 solution).

Table 4.4 Model parameters for the pH system

Parameter Value x1,i

x2,i x3,i KD KE FA FB V

0.0012 mol/l HCL 0.002 mol/l NaOH 0.0025 mol/l NaHCO3 10–7

mol/l 10–14 mol2/l2 1 l/min (16.67 ml/s) 0.14 l/min

2.5 l

4.5 SUMMARY AND CONCLUSIONS

The detailed development of a mathematical model for a non-isothermal continuous stirred

tank reactor is presented in this chapter. The process model consists of balance ordinary differential equations supported by algebraic form of equations for the calculation of rate of reaction and heat transfer area. In order to predict the reactor dynamics, the process simulator is also developed by solving the model equations. Next, the CSTR model has been

simplified and used to analyze the steady state multiplicities. The control of the simplified CSTR process is also covered. Finally, the model structure for a pH neutralization reactor is formulated.

EXERCISES