Appunti di combustion

Chemical Equilibrium

A new approach to thermodynamics is necessary and useful to put a base on torebuild the theory of combustion. To do that the number of postulates that we'llbe used will be lower than usual giving essentially a basic set.

For first, must be noted that thermodynamics is not a rigid science in the sensethat there is no dynamics and work on time, but only the study ofequilibrium states of macroscopic system. This means that I would be better tocall the thermodynamics of processes should be linked as quasi-staticbecause they occur in finite time but almost infinite. Because of that, processes arereal processes that move systems in time and so time derivatives means physicallya rate of change:

equilibrium processes that do not work in time but simply drive one systemfrom an equilibrium state to another. There is a change, but time is nottaken into account.

By what we said thermodynamics doesn't say anything about how a systemchanges in time but only the direction and eventually, some informationon equilibrium processes.

thermodynamics

• equilibrium (thermostatics);
• quasi-static
• infinite time
• reversible/irreversible equilibrium processes

• non-equilibrium:
• non quasi-static
• real processes of near equilibrium or far from equilibrium
• finite time

By the way, the basic set of postulates is for thermostatics.1st postulate says that for a system there exist particular states, equilibrium states, that from a macroscopically point of view are completely characterized bythe extensive variables, V, V, and n_k. Those states are reached because the system interacts with the surroundings only by the exchange of mechanicalwork and by thermal and chemical interactions. A particular type of equilibriumstate are the metastable states which are related to the presence of some constraintthat doesn't allow the system to reach the equilibrium state (ex O₂ and the heatrooms, when the temperature is fixed to 20° then moves O₂ and H₂ can reach themor otherwise the walls, those walls can be

Premixed combustion can be of two types:

• deflagration, if it is subsonic
• detonation, if flame velocity is supersonic

The shape of a flame is such that the normal velocity of the flow is exactly equal to the flame velocity.

• irreversibile or reversibile, which lead or not exange of matter and so a change of chemical composition;

• adiabatic or diathermal, allowing or not the exange of heat.

Because of this, wells an equivalence between the conservation of energy ΔU= W+Q and the 1° postulate can be seen.

• 2° postulate is related to the fact that exist even internal constraints and not only external ones. We know how a system involves due to those constraints, the 1° postulate says that there exists a function the entropy, S, that depends on the external parameters of the system and this function is such that if the internal constraints are removed then the extensive parameters will assume values that maximize the entropy over the constraint equilibrium states (metastable states). This postulate therefore tell to know how a system change by the fundamental equation.

S=S(U,V,n_k);

• 3° postulate states that an entropy of a composite system is extensive and additive over all the subsystems. Moreover it is continuous and differentiable and it is monotonically increasing function of U. Because of that I seem possible to say that U=U(S,V,n_k) so the 2° postulate can be expressed in terms of minimization of its internal energy. At the end because of the additive property S(U,V,V,n_k) = ΣS(U,v,n_k).

Because of the 3° postulate and the fundamental equation it's possible to write: dU= T dS − P dV + Σ μ_k dN_k, where each contribution has a physical meaning:

• the first represents an exange of heat from/to the surroundings T dS = δQ
• the second an exange of work ∫U∫V∫S = -P
• the third a variation of chemical composition, so its chemical energy, ∫U∫N,V,S=N, where μ_k: chemical potential

• 4° postulate states that at T=0 K the entropy is equal to zero which corresponds to P=0 Pa and 0 K = 273.15°C.

Consider an isolated system so that dU=0 and internal energy can't change. By removing the internal constraint, I will arrange in such a way to reach max S, where S=S₁+S₂. So perturbating the system will necessarily be dS≥0, so that dS=dS₁+dS₂+

dS_i ¹dU_i ²dU_i

because U=∑n_iU_i and dU=∑dU_i + dE=0 then dU=-dU₂ and so:

dS=∫_1/T1dU₁+∫_1/T2dU₂

= (∫_{1/T1 - 1/T2})dU₁

= (∫_2/T1)dU if dU>0

to ∫₂ warm to cold) in a spontaneous way.

If the walls are instead adiabatic, rigid and permeable (dn_k=0), then it will be:

dS ^jdU_j + ∫^jdU_j

= T¹dU₁ + ∑₁μ_k∫_kdn_k + ∑_iμ_kdn_k = 0, if the overall n_k can't change then dn_k = 0

= (1/T₁ - ∑μ_k∫_idn_k)

By removing T₁=T₂ then the equilibrium state (dS=0) it must be such that

dn_k=0 and so there must be uniform chemical potential.

By perturbing the system μ_i>μ_{i' then dS=∫1/T1 (μi-μi')dni>0 and is only possible if dnk<0, so the system will spontaneously respond by changing the matter from high chemical potential to low one state.}

we saw that there exist ̅_a and ̅_b limits of so let's see how to calculate these

limits for a generic reaction aA + bB ⇄ γC + δD

n_a-n_a₀ ^n_a-n_a₀ n_c-n_c₀ n_d-n_d₀

----- = ------ = ------------ = -----------

a b γ

so to calculate we must think that the reaction becomes aA + bB ⇄ C + δD

and is related to the unchanging of the first reactant i which will be the

limiting product

n_eⁱ ͢^n_i-n_i₀ ͢ as = ------ = _L m the n_eⁱ

case that C is the limiting product

⇒ m general _E = n_E ͢ where n_E is the

----------------

V_i V

limiting product

in an similar ﻳit is possible to calculate ̅_R by the reaction

of the type aA + bB ⇄ γC + δD then m general one

easily obtains:

_R = n_R

---------------

V_i ͨ Vⁱ

For example the reaction H₂, O₂ ⇄ H₂O is such that

[3,2] ²n_B

So B is the limiting reactant

n_eⁱ

Its now usellul to go back to the Gibb's energy minimizations problem for

which we have the relation G : G(T, p, n) U - T.S - pV Atom which:

dG = SdT + Vdp + Σₜμ_i dn_i

----- ----- ------

δG_T δG/p δG/δn_i

can we saurt for a reservon at P.T = constant minG is such that in general

dG: Σⁱμ_i dn_i e represents the eq^ilibrium state

dG: [ Vⁱ ̤]ⁱ[Q - dS] intensive

δn_i

If want to finj min{G$\}, we need to express G differentially, remembering G:G(T, p, n)

so that G(T', p ͨ p_k) G(T', p, n)

thus Euler's Theorem (G[ T, p n) = Σⁱ (∂G/∂n_i), Σⁱ = μ_i [ µⁱ] Σⁱ μ_i n_i extracos

d( )];

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