Introduzione - Termodinamica

INTRODUCTION & THERMODYNAMICS

Thermodynamics describes the equilibrium and the kinetic describes the evolution to reach the equilibrium

The terms that characterize the reactor are: NC (3+ mixture) NR NP NP>3 SPECIE, i, AGGREGATION STATES -> L, L, P, C, S phases region of single component (water and ice) 2 phases and 2 off. state

How we calculate the amount of mass, molecules, atoms....? It is introduced the Avogadro Number: 6 10²³ = 1 mole -> N.A. can be applied for molecules or atoms etc...

Atoms of C -> 1 mole of C atoms = 6 10²³ atoms of C -> 12 g (atomic weight) -> Nc = A Wc / mc moles = molecules / NAv

Molecules of CO2 -> MWCO2 = (1 . WC) + (2 . WO)    1 12 g/mole 16 g/mole number of C in the CO2 molecule

Molar fractions are intensive variables > From Molar to mass: Yi = _nc/mass    Mi = x i MWi / sum xj MWj Viceversa: xi = yc / sum yj/MWj

How we calculate the volume? v = V(T,P,amount) -> volumetric equation of state

It is possible to use the following expression: Nm³ { T, P } 0 C 1 atm 25 C 1 atm

Intensive variables > specific volume > Vi/mi = v_i massic specific volume Vi/ Ni = v~i molar specific volume

Density = amount / volume = 1 / V (=) kg/m³

For ideal gases: P Vext = Ni Rg T -> p [Vext / Ni] = Rg T

For real gases there are more complicated function that connect the volume with p and T

In the course it is used the ideal gas expression because we work at high temperature, at which the reactions are activated For an ideal mixture Vmix = sum xi V~i + V~excess -> correction to obtain the real value for real mixture

To calculate p~mix, instead, we use the following formula: p~mix = sum p~i / xi

And p~i^ig = p / RgT

For ideal gas mixture p~^ig = p / RgT = p~i^ig and p~i^ig = p~i^ig mixi

p~All (0 C, 1 atm) = p~All - RWmix normal condition = p / RgT

Always = sum xi MWi

weighted average of MW

How we calculate the thermal properties? It must be identified a thermal equation of state: equation that gives us the enthalpy as a function of T and P

Typical approach of thermodynamics consists of trying to quantify the increase or decrease of h that follows a changing in T or P - then contributions are summed

d hⁱ = (partial hⁱ/partial T)_P dT + (partial hⁱ/partial P)_T dP

(sensitivity of hⁱ to T) (sensitivity of hⁱ to P)

the enthalpy generally doesn't change if the pressure increases or decreases, except when there is a extreme variation

so, it is obtained: d hⁱ = (partial hⁱ/partial T)_P dT

ideal heat capacity of species i at constant P

C_P,i = C_P,i(T,P) -> C_P,i(T)

∫ d hⁱ = ∫ C_P,i dT

Reference temp.

hi(T) = hi(T0) + ∫T0T CP,i(T) dT

when |T - T⁰| is small then C_P,i is constant

thermal equation of state

amount of i is constant

a) if i is not reacting (it may changes the phase) then hⁱ(T⁰) is always cancelled in Δhⁱ

It is used a relative scale of h - it is neglected the base point of the scale (h(T⁰))

b) if i is reacting (amount changes) it is necessary the use of an absolute scale of h -> we need to fix a reference to the scale:

hⁱ(T) = Δh^0,f_i(T_ref) + ∫_Tref^T C_P,i(T) dT

ENTHALPY OF species i formation at standard condition at reference T

formation reaction from elements

Base point

Pure species that goes from phase α to phase β -> h^i,β(T) = Δh^i,0,f(T_ref) + ∫_Tref^T C_P,i^β dT

where i is in α

Δh of formation from elements at standard conditions: p, phase and composition given at std, all is depending on T, more and any T

the standard condition doesn't fix T so Δhⁱ remain a function of T

h^i,β(T) = Δh^i,0,f(T_ref) + ∫_Tref^T_βα C_P,i^α dT + Δh^βα_i + ∫_Tαβ^T C_P,i^α dT

Δh of transition between β and α

contribution in β phase

contribution of transition

contribution in α phase

h^g_H2O(400K, 1bar) = Δh^0,f_H2O,l(27C) + ∫_373K^400K C_P,H2O^g dT + Δh^ev_H2O(100C) + ∫_333K^400K C_P,H2O^g dT

There's steam

-285 kJ/mole

2 kJ/kgK

(25+273)K

-2260 kJ/kg

4,1 kJ/kg K

ΔH nuclear rxn > ΔH rxn > ΔH phase transition > ΔH sensible heat

Hmix = sum Xi hi ext

qmix = sum Xi h^i int = [energy / molemix]

h^i(l) = h^i(Tref) + integral cp,i dT

Smix = sum Xi s^i - Rg sum xi ln xi

For ideal mixture, for xi < 1 (mixture) Xi > 0 and xi < 0 so -Rg sum xi ln xi > 0

Therefore Smix > sum pure,i xi => in the mixture there is more disorder than the one we have with separated species

G = H - TS

qmix^id = sum Xi h^i - T [sum Xi s^i - Rg sum xi ln xi]

There is a non linear connection between Xi and qmix

The free Gibbs Energy rules the equilibrium at constant T and P

equilibrium of reactions -> so q tends to some species will move into other species by reaction and will move into other phases

ATTENTION ppm = parts per million (based on mass, moles, volume...)

molar fraction of i: xi = moles i / total moles

how many moles of i are in a mixture of many species

If moles is very small it is convenient using a scale factor: ppm mass

ppm mass = (moles i / total moles) 10^6

100 ppm CO2, molar = (100 moles CO2 / moles total) 10^6 = 100 xco2 10^6

xCO2 = ppm / 10^6

m and n are absolute amount while x, y and ppm are relative amount

Measure of composition = amount of each species in the mixture

To measure the composition it is used also the concentration (or density) = amount / volume

massive conc. = mi / V

molar conc. = Ni / V

p and T even without adding more species

Generally the reactions are faster at high pressure, that is because at high P the molecules are close to each other, so their interactions are more likely to occur (reactive interactions)

c^i = xi ctot

c^i = psi ctot

for gas ci ~ pi = xi P

remember that p^liq = p^liq => So, for an ideal gas: c^i^ig = xi ctot^ig = xi P / (Rg T) = pi / (Rg T)

ppm molar,i = ppm volume,i = how flows 100 ppm -> mg/m^3 density concentration

For ideal gas xCO2 = 400 / 10^6 = 4 * 10^-4 moles CO2 / moles tot

Henry: xCO2 p^ig / H = cCO2 xH2O = cCO2 = mg/m^3

g CO2 / moles CO2 * moles air / m^3 air = g CO2 / m^3 air = mg CO2 / m^3

Let us consider N > 1 Np = 1 Nr > 0

Reacting mixture

The amount of reactants and products is expressed by stoichiometry, using some multiplying factors called stoichiometric coefficients and obtained from atomic balances (ABC, ABO, ABK)

Ex. a CkHn + b O2 = d CO + c H2

Critical equation/expression to formulate the statement of atomic conservation (not molecules, the atoms will be rearranged in the other ways)

Setting a = 1 we obtain b = 1

ARBITRARY

d = 2c = 2

The values |a|, |b|... are immaterial, what is significant is the ratio a/b, a/d... The ratio is the same despite the fact that the abs values change

The stoichiometric coefficients are characterized by the sign, which is + for products and - reactantsWhen there are more reactions than 1 the st. coefficients are represented with a matrix

For any reaction it is possible to calculate the summation i v_ii N? y?i any n_i, ?i, ?i, ?i

Progress of reaction

2 CO + O2 -> 2 CO2

At t = 0 Nco = 100 moles (10²⁵ molecules)At t = 1 Nco = 54 moles

Delta Nco = Nco(t1) - Nco(t0) = -46 moles

CONSUMPTION OF CO

Delta Nco / v_co = -46 moles / -2 = 23 moles = -23 moles / -1 = Delta No2 / v_o2

Delta Ni / v_i = Delta Nj / v_j

Delta No2 = -46 / 2 moles o2 = -23 moles o2

CONSUMPTION OF O2

We define Delta N_i / v_i = Delta E - Degree of Reaction Progress, Property of the reaction, [moles]Extensive variable

Actual - initial

Property of the species

In our case: Delta E = 23 moles = e(t1) - e(t0)

= 0

So, E is the coordinate of the reaction; it tells you the reaction is at the beginning, if it is increasing the reaction is going from reactants to products

E ↑ lots of products and few reactants

E ↓ Lots of reactants and few products

REACT     E     PROD

Ni(t) - Ni(t = 0) = Delta Ni = vi E

Ni(t) = Ni(t = 0) + vi E

Setting equal to zero Ni(t) of limiting reactant, it is possible to find E max i = Ni / vi highest value

We can't go beyond the Emax because at that point the entire reactant is all consumedAt Emax it is possible to have only products if the initial amounts of species are stoichiometric

With NR > 1 there are as many E as there are reactions (bc of different initial extents)

Plot of N(t) vs E:

N(t)↑

Equations: Ni(E) = Ni⁰ - vi·Ei: limiting reactant

In open system the flow rates and the degree of reaction progress have the following unit:moles/s (Ni, Ei)

Attention. A x B = C(a x b)(c x d) = (a x d)only if b = c

When NR > 1Ni(t) = Ni(t) - ∑ vi,r Ei

Plot of qtot = ∑i Ni vs E

At the lowest value of qtot the system reaches the equilibrium

The Eq can be very close to Emax and generally when it happens the reaction is called IRREVERSIBLEwhile if Eq < Emax the reaction is considered REVERSIBLE, so at some point the products will move backwards to reform reactants

So, the Thermodynamics tells us where the reaction will end (Eq)

But how do we determine the progress at equilibrium?

DETERMINATION OF EQUILIBRIUM CONFIGURATION IN A REACTIVE MIX

Thermodynamics describes the evolution to the equilibrium in the second principle, which sets that the reaction will stop when total extensive entropy

Entropy / K {S^totU,V -> maximum

Isolated system: closed to mass and energy

If we maximize the entropy function of T, p and N at U and V constant respect to composition, we will obtain the composition at equilibrium

max S^tot(T,p,N) |_U,V -> N^eq

that expression can be reformulated in terms of free energy of Gibbs

min G^tot(N) |_T,p

G^tot = sum_alpha G^alpha = G^l + G^g = sum_c^Nc,l N_c^l gbar_c^l + sum_j^Nc,g N_j^g gbar_j^g

mu_i^l

G_i^alpha = G_i^alpha(T,p,N^alpha)

amount of a species, not just of i

The function that connects mu_i^alpha to N^alpha is not linear

The combination of N_i that provides the minimum G^tot is the composition at EQ.

The atomic concentration must be respected at the equilibrium

Ex.

Co   CO2   O2   C

N(0) = [ 1   0   3   2 ] = moles of molecules

N_{C atoms}(0) = [ 1   0   0   2 ] = 3 moles of C-atoms

N_{O atoms}(0) = [ 1   0   6   0 ] = 7 moles of O-atoms

[1 0 3 2] [ 1 2 2 0 ] O2

Any combination must have the exact number of atoms of the species that is represented at initial time

N_k(0) = sum_c N_i(0) a_ik = N_k(t) = sum_j N_j(t) a_jk

Number of atoms k in species c

So the criteria for equilibrium is searching in the space of compositions of the species the one with which it is possible to achieve the minimum of the free Gibbs energy and to respect the atoms conservation constraint

{

min G^tot(N) |_T,p -> G is a function of Nc variables

input: that reduce requires the species

free at equilibrium, T,p,N(0)

N . A = N(0) atoms

}

Amount of atoms expressed in terms of moles

=> CEA

The vector of the composition contain reactants, expected and unexpected products. There aren't intermediates bc they are not stable molecules, which are at equilibrium.

The minimum of G^tot could be determined also using the equilibrium constant. Instead of the composition, it is convenient to write vectors of progress degree. So, the minimum of G^tot, expressed in terms of degree of progress, is found exploring the E space.

min G^tot(N) -> min G^tot(E)

The input are T, p, N(0) -> species and reactions -> we can calculate the equilibrium constant and then the degree of progress at equilibrium for each reaction

not atoms

the disadvantage (we might forget some reactions)

The advantage to use that method consist of obtaining a qualitative prediction of the eq. position

K^eq ↔ a ≈ P

product

react

If the reaction is endothermic ΔH_R > 0 T↑ E^eq -> Emax

exothermic ΔH_R < 0 T↑ E^eq -> 0

If ∑_i=1^nc ν_i < 0 (decrease of moles) P↑ E^eq -> Emax

> 0 (expansion of moles) P↑ E^eq -> 0 at the equilibrium we will have more reactants than products

The effects of inert species are the opposite of ions:

∑ν_i < 0 ↑ inerts E^eq -> 0

∑ν_i > 0 ↑ inerts E^eq -> Emax

Difference between variation of Gibbs free energy of the reaction and variation of Gibbs free energy standard of the reaction

ΔG_R⁰ tells us the position of equilibrium -> if the reaction at given temperature is likely to be close to maximum or not enough to be seen

ΔG_R⁰ = ∑_i=1^nc ν_iΔG_i⁰(T)

ΔG of formation at standard conditions

If ΔG_R⁰ < 0 the products are more stable than the reactants, so the reaction is more likely to shift to the products (ΔG large)

If 0 < ΔG_R⁰ < 40 kJ/mole the reaction can be run (E stable)

If ΔG_R⁰ > 40 kJ/mole NO

ΔG_R = ∑ν_iG_i and it's equal to zero when the system is at equilibrium

Gtot

React.

equilibrium

Prod