MASS CONSERVATION AND MOLE BALANCE EQUATIONS

G. MASS CONSERVATION AND MOLE BALANCE EQUATIONS

In Chapter 2 we derived the conservation equations for mass and energy. In multicomponent reacting systems, species are both consumed and generated due to chemical reactions. For instance, if a mole of CO and 2 moles of O 2 are admitted into a combustor, and if we will find that 1 kmole of CO 2 is produced and 1.5 kmoles of O 2 remain, then we can assume that complete combustion has occured. Even though the mass leaving (1 × 44 + 1.5 ×32

= 92 kg) is same as the mass entering (1 ×28+2×32 = 92 kg) the combustor, the total moles exiting the reactor (1+1.5 = 2.5) are different from those entering it (1+2 = 3), implying that

moles (or species) have been generated. The reactor produced 1 kmole of CO 2 (or 44 kg) but consumed 1 kmole of CO (= 28 kg) and 1/2 kmole of O 2 (= 16 kg). In chemically reacting systems, the mass production rate of species k is m ˙ k,gen and mass conservation the species is

written in the for

σ σ meter σ σ

G meter

412 kg

Figure 7: Illustration of entropy generation and decrease in G of a fixed mass with reaction. The value of G keeps decreas- ing as a reaction proceeds.

dm k /dt = ( m ˙ k,i + m ˙ k,gen - m ˙ k,e ), (27) where dm k /dt denotes the mass accumulation rate within the reactor volume V and m ˙ k,gen the

generation rate of species k due to chemical reaction. Summing over all species, dm cv /dt = Σdm k /dt = ( Σ m ˙ k,i + Σ m ˙ k,gen - Σ m ˙ k,e )= m ˙ i - m ˙ , e (28)

where m ˙ i = Σ m ˙ k,i , m ˙ e = Σ m ˙ k,e , Σ m ˙ k,gen = 0, since mass is conserved. Writing Eq. (27) in vectorial form,

(29) where ∫ m ˙ k,gen ´´´ denotes the mass generated per unit volume of reactor and ρ k the density of

(d/dt) ∫ρ k dV = ∫ρ k · V k d A + ∫ m ˙ k,gen ´´´dV

species k. Using the Gauss divergence theorem, we can write the species conservation equation in differential form as

(30) Summing over all species

∂ρ k / ∂t + ∇·ρ k V k = m ˙ k,gen ´´´.

(31) Since m k =N k M k , Eq. (27) has the following form in terms of mole balance dN k /dt = N ˙ k,i + N ˙ k, gen - N ˙ k, e

∂ρ /∂t + ∇·ρ V = 0.

(32) where N ˙ k, gen denotes the number of moles of species k produced by the chemical reaction. As

mentioned before Σ N ˙ k,gen = N ˙ gen ≠ 0. Summing over all species ΣdN k /dt = dN/dt = Σ N ˙ k,i + Σ N ˙ k, gen - Σ N ˙ k, e = N ˙ i + N ˙ gen - N ˙ e (33) Similarly proceeding as in Chapters 2 to 4 for the energy conservation, entropy and availabil-

ity balance equations, we write the mole balance equation in integral form as

(34) where n k = moles of species k per unit volume. Further using the Gauss divergence theorem,

(d/dt) ∫n k dV = ∫n k · V d A + ∫ N ˙ k gen ′′′ , dV,

∂n k / ∂t + ∇.n k V = N ˙ ′′′ k gen . ,

1. Steady State System

Under steady state conditions dN k /dt = 0, dN/dt = 0,dm k /dt = 0, dm/dt = 0 and from

Eq. (28), m ˙ i = m ˙ e . Likewise, from Eq.(33), N ˙ i + N ˙ gen = N ˙ e .

n. Example 13

A combustor is fired with 2 kmole of CO and 3 k mole of O 2 . When the combustor is just started, very little CO burns. As it warms up, more and more CO are burnt. As- sume that mass does not accumulate within the reactor. At the point when the com- bustor achieves 40% efficiency, write the mass conservation, mole balance and en- ergy conservation equations. If the combustor reaches a steady state with combustion efficiency of 90%, write the mass conservation and mole balance equations.

Solution The reaction equation is written as follows:

(A) Since mass does not accumulate within the reactor, the atom balance for C and O at-

2 CO + 3 O 2 →N CO,e CO + N CO2,e CO 2 +N O2,e O 2 .

oms yields 2=N CO,e +N CO2,e , and

(B) 2+6=N CO,e +2N CO2,e +2N O2,e , i.e.,

(C)

0.4 = (2 - N CO,e )/2. (D) With the three equations Eqs. (B) through (D), we can solve for the three unknowns

N CO,e ,N CO2,e and N O2,e , i.e., N CO,e = 1.6, N CO2,e = 0.4, and N O2,e = 2.8. The mass conservation implies

dm CO /dt = 2 ×28+ m ˙ CO,gen - 1.6 ×28 = 11.2 + m ˙ CO,gen . Similarly, dN CO /dt = 2 + N ˙ CO, gen - 1.6 = 0.4 + N ˙ CO gen

Normally N ˙ CO, gen is a negative quantity since CO is consumed. The term dN CO /dt represents the accumulation (destruction, since negative) rate of CO in the reactor.

Similarly for CO 2 , dN CO2 /dt = 0 + N ˙ CO2 gen - 0.8, where N ˙ CO2, gen > 0, since CO 2 is a product that is generated. In the initial periods

when the combustor is being fired, the CO 2 concentration gradually increases due to the term dN CO2 /dt. Similarly for 90% efficiency,

N CO,e = 0.2 N CO2,e = 1.8, and N O2,e = 2.1 dm CO /dt = 2 ×28+ m ˙ CO,gen - 0.2 ×28×28 = 50.4 + m ˙ CO,gen .

The combustor is operating steadily so that dm cO /dt = 0, m ˙ CO,gen = - 50.4 kg/s Similarly , dN CO /dt = 2 - 0.2 + N ˙ CO, gen = 1.8+ N ˙ CO, gen . Since dN CO /dt = 0,

N ˙ CO, gen = - 1.8 k mole s -1 , and N ˙ CO, gen = - 1.8 k mole s -1 , and

CO 2,gen = 1.8 k mole s , N ˙

O 2,gen = -0.9 kmole s .