Chemical Engineering Thermodynamics

1

This section covers the Gibbs energy, entropy and enthalpy changes of reaction, the general criteria of equilibrium, phase and chemical reaction equilibria, homogeneous and heterogeneous equilibria, and the effect of pressure, temperature and catalysts on reactions.

2

Thermodynamic Functions of Reaction • Enthalpy of reaction ΔHrxn measures the heat effect at constant pressure and is negative for an exothermic reaction; it is computed from standard heats of formation as in 1.4. • Entropy of reaction ΔSrxn = Σ νiS°products − Σ νiS°reactants.

3

Unlike enthalpy, absolute entropies are known, because the third law of thermodynamics fixes the entropy of a perfect crystalline substance at absolute zero as zero.

4

Entropy increases when the number of moles of gas increases, a solid or liquid becomes a gas, or a substance is heated or expanded. • Gibbs free energy of reaction ΔG = ΔH − TΔS, which combines the two tendencies of nature — towards minimum energy and towards maximum entropy — into a single criterion. ΔG negative means the reaction is spontaneous (feasible) in the forward direction; positive means it is non-spontaneous as written; zero means the system is at equilibrium.

5

The four possible sign combinations are worth memorising: ΔH negative with ΔS positive is spontaneous at all temperatures; ΔH positive with ΔS negative is never spontaneous; ΔH negative with ΔS negative is spontaneous only at low temperature; and ΔH positive with ΔS positive only at high temperature — which is why many endothermic decompositions require heating. • The essential caution: ΔG tells us whether a reaction is thermodynamically possible, not whether it will actually be observed — the rate may be immeasurably slow.

6

The conversion of diamond to graphite has a negative ΔG yet does not proceed.

7

Thermodynamics gives the destination; kinetics gives the speed of travel.

8

Criteria of Equilibrium • The general criterion: a system is at equilibrium when its total Gibbs free energy is at a minimum at the given temperature and pressure, that is (dG)T,P = 0.

9

Equivalent statements for other constraints are (dA)T,V = 0 for Helmholtz free energy and (dS)U,V = maximum for an isolated system. • Phase equilibrium: for a species distributed between phases, equilibrium requires that its chemical potential (partial molar Gibbs energy) be equal in every phase — μi α = μi β = …, which in practice is expressed as equality of fugacity, fi V = fi L.

10

This is the rigorous statement underlying Raoult's law and all the vapour-liquid equilibrium of 1.2. • Chemical reaction equilibrium: the corresponding condition is Σ νiμi = 0, which leads to the fundamental relation ΔG° = −RT ln K, where K = Π (ai)νi is the equilibrium constant written in terms of activities.

11

For gases the activity is the fugacity divided by the standard-state pressure, which for an ideal gas reduces to the partial pressure in bar; for pure solids and liquids the activity is unity, which is why they do not appear in the equilibrium expression. • Consequences: a large negative ΔG° gives a large K and a reaction that goes essentially to completion; a large positive ΔG° gives a very small K.

12

K depends only on temperature (through van 't Hoff), not on pressure or on the presence of inerts, although the equilibrium composition certainly does.

13

Homogeneous and Heterogeneous Equilibria • Homogeneous equilibrium exists in a single phase.

14

For a gas-phase reaction, Kp = Kc(RT)Δn = KyPΔn, where Δn is the change in the number of moles of gas.

15

The relation to composition shows immediately that pressure affects the equilibrium conversion only when Δn is not zero. • Heterogeneous equilibrium involves more than one phase, as in the decomposition of calcium carbonate, CaCO₃(s) ⇌ CaO(s) + CO₂(g).

16

Since the activities of the pure solids are unity, Kp = pCO₂ alone — so at a given temperature the equilibrium pressure of carbon dioxide is fixed, regardless of how much solid is present.

17

This is the principle of lime burning and is a standard examination example. • Multiple reactions at equilibrium require one equilibrium relation and one extent of reaction for each independent reaction, solved simultaneously.

18

Effect of Temperature, Pressure and Catalysts • Temperature acts on the equilibrium constant through the van 't Hoff equation, d(ln K)/dT = ΔH°/RT², integrated as ln(K₂/K₁) = −(ΔH°/R)(1/T₂ − 1/T₁).

19

Hence raising the temperature increases K for an endothermic reaction and decreases it for an exothermic one.

20

Temperature also acts on the rate, through Arrhenius, and always in the direction of increase — which creates the central industrial dilemma: for an exothermic reversible reaction, a high temperature gives a fast approach to a poor equilibrium and a low temperature a slow approach to a good one, so an optimum intermediate temperature (or a falling temperature profile) is used.

21

Ammonia synthesis at about 450 °C and sulphur dioxide oxidation with interstage cooling are the standard illustrations. • Pressure has essentially no effect on K itself for ideal gases, but a large effect on the equilibrium composition when Δn ≠ 0: increasing the pressure shifts the equilibrium towards the side with fewer moles of gas, which is why ammonia synthesis (4 moles → 2) is operated at 150-300 bar.

22

For reactions in the liquid or solid phase, pressure has little effect.

23

Adding an inert at constant total pressure increases the total moles and therefore acts like a reduction in pressure, shifting the equilibrium towards more moles of gas. • Catalysts, to repeat the point from 1.1 because it is examined so often: a catalyst increases the rates of the forward and reverse reactions equally by providing a path of lower activation energy, and therefore shortens the time taken to reach equilibrium — but it cannot change the equilibrium constant, the equilibrium composition, or ΔG, ΔH and ΔS, and it cannot make a thermodynamically infeasible reaction occur.

24

What a catalyst can do, and what makes it industrially decisive, is improve selectivity, by accelerating one of several possible reactions more than the others. • Concentration: adding a reactant or removing a product continuously drives the reaction forward — the principle behind reactive distillation and membrane reactors, in which removing a product allows conversion far beyond the normal equilibrium limit.

1

This section covers introductory concepts and definitions, energy and the first law, the properties of pure simple compressible substances, the second law and entropy, the analysis of thermal systems, and the air compressor.

2

Concepts and Definitions • System and surroundings: the system is the region under study, and it is closed (fixed mass, energy may cross), open (control volume) (mass and energy cross) or isolated (neither crosses).

3

Properties are intensive (independent of mass — T, P, density, specific volume) or extensive (proportional to mass — V, U, H, S); dividing an extensive property by mass gives a specific property, which is intensive. • State, process and cycle: the state is the condition described by the properties; a process is a change of state; a cycle returns the system to its initial state, so that all property changes over a cycle are zero.

4

Processes are named for what is held constant: isothermal, isobaric, isochoric, isentropic and adiabatic (Q = 0).

5

A quasi-static (reversible) process passes through a continuous succession of equilibrium states and can be exactly reversed leaving no trace; all real processes are irreversible, because of friction, unrestrained expansion, heat transfer across a finite temperature difference and mixing. • The zeroth law: if two bodies are each in thermal equilibrium with a third, they are in thermal equilibrium with one another — which is what makes temperature measurable and the thermometer possible.

6

The First Law • The first law is the principle of conservation of energy: energy can be neither created nor destroyed, only converted from one form to another.

7

It establishes internal energy U as a property, since for a cycle ∮δQ = ∮δW. • Closed system: ΔU = Q − W, with the convention that heat added to the system is positive and work done by the system is positive.

8

For a quasi-static process the work is W = ∫P dV, which is the area under the curve on a P-V diagram. • Open system at steady flow (the steady-flow energy equation):

9

Q − Ws = Δ[H + u²/2 + gz] per unit mass.

10

Applications follow by dropping the negligible terms: for a turbine or compressor, adiabatic and with negligible kinetic energy, Ws = −ΔH; for a heat exchanger or boiler, Q = ΔH; for a nozzle, adiabatic with no work, Δ(u²/2) = −ΔH; and for a throttling valve, which does no work and is adiabatic, ΔH = 0 — the process is isenthalpic, a result used constantly in refrigeration and in the Joule-Thomson expansion of 2.6. • Applications to ideal gases: ΔU = mCvΔT and ΔH = mCpΔT for any process; for a reversible adiabatic (isentropic) process, PVγ = constant with γ = Cp/Cv, and TVγ−1 = constant and T/P(γ−1)/γ = constant.

11

For an isothermal reversible process, W = nRT ln(V₂/V₁) = nRT ln(P₁/P₂). • The first law says nothing about direction: it is satisfied equally by a hot body warming a cold one and by the reverse, which is precisely the gap the second law fills.

12

Properties of a Pure Compressible Substance • A pure substance has a uniform and invariable chemical composition; steam and water together still constitute one pure substance.

13

For such a substance in a single phase, the state postulate says that two independent intensive properties fix the state — which is why tables and charts are two-dimensional. • The phase change of water at constant pressure, traced on a T-v diagram, passes through compressed (subcooled) liquid → saturated liquid (f) → the two-phase wet region → saturated vapour (g) → superheated vapour, with the horizontal portion at the saturation temperature corresponding to the given pressure.

14

The saturated liquid and saturated vapour lines meet at the critical point. • Quality (dryness fraction) x = mass of vapour/total mass, defined only inside the two-phase region, and any property there is found from y = yf + x·yfg, where yfg = yg − yf.

15

Thus h = hf + x·hfg, and hfg is the latent heat. • Steam tables are organised as saturation tables (by temperature and by pressure) and superheat and compressed-liquid tables; reading them correctly, and recognising from the given data whether a state is wet, saturated or superheated, is an assumed skill.

16

Mollier (h-s) and T-s diagrams are the graphical equivalents used for turbine and cycle work.

17

The Second Law and Entropy • Statements of the second law: the Kelvin-Planck statement — no device operating in a cycle can convert all the heat it receives from a single reservoir entirely into work, so a heat engine must reject heat to a low-temperature sink; and the Clausius statement — heat cannot of itself pass from a colder to a hotter body, so a refrigerator requires a work input.

18

The two statements are equivalent, and together they establish the direction of natural processes and a limit on the conversion of heat into work. • The Carnot cycle — two reversible isothermal and two reversible adiabatic processes — is the most efficient cycle possible between two reservoirs, with ηCarnot = 1 − TC/TH, temperatures in kelvin.

19

The Carnot principles state that no engine can be more efficient than a reversible engine between the same reservoirs, and all reversible engines between the same reservoirs have the same efficiency, independent of the working fluid — which is what allows the thermodynamic temperature scale to be defined. • Entropy is defined by dS = δQrev/T and is a property (state function), as the Clausius inequality ∮δQ/T ≤ 0 establishes.

20

The entropy balance for any process is ΔSsystem + ΔSsurroundings = Sgen ≥ 0, with Sgen = 0 only for a reversible process.

21

Hence the principle of increase of entropy: the entropy of an isolated system can never decrease. • Entropy changes for an ideal gas: Δs = Cvln(T₂/T₁) + R ln(v₂/v₁) = Cpln(T₂/T₁) − R ln(P₂/P₁). • Isentropic efficiency compares a real device with a reversible adiabatic one: for a turbine, η = actual work/isentropic work = (h₁ − h₂)/(h₁ − h₂s); for a compressor or pump the ratio is inverted, η = isentropic work/actual work, since the real device requires more work. • Availability (exergy) is the maximum useful work obtainable as a system comes to equilibrium with its environment; irreversibility I = T₀Sgen is the work lost through irreversibility, and exergy analysis locates where in a plant the thermodynamic losses actually occur — often revealing that the greatest loss is at a point that energy analysis alone shows as efficient.

22

Analysis of Thermal Systems and the Air Compressor • Power cycles: the Rankine cycle (steam — pump, boiler, turbine, condenser) is the basis of all thermal power generation, improved by superheating, reheating, regeneration (feedwater heating) and raising the boiler pressure; the Brayton cycle for gas turbines; and the Otto and Diesel cycles for internal combustion engines, whose efficiencies depend on the compression ratio.

23

Refrigeration cycles are treated in 2.6. • The air compressor is the standard worked example of the section.

24

Reciprocating compressors are positive-displacement machines; rotary, centrifugal and axial machines are the dynamic alternatives for large volumes. • Work of compression: for a polytropic process PVn = constant, the work required per cycle is W = [n/(n − 1)]·P₁V₁[(P₂/P₁)(n−1)/n − 1].

25

The important consequence, and the one examined, is that the work is least for isothermal compression (n = 1), greatest for adiabatic compression (n = γ), and intermediate for the polytropic case — so cooling during compression saves work, which is why compressors are jacketed, fitted with fins or water-cooled. • Multistage compression with intercooling is the practical expression of this.

26

Its advantages are: reduced total work (approaching the isothermal ideal), lower discharge temperature so that lubricating oil does not degrade or ignite, reduced thermal stress, better volumetric efficiency, and lower leakage past the piston.

27

For minimum total work with perfect intercooling back to the inlet temperature, the pressure ratio should be the same in each stage, that is Pintermediate = √(P₁P₂) for two stages — the single most examined result in compressor theory. • Clearance volume and volumetric efficiency: the gas trapped in the clearance space re-expands on the suction stroke and so reduces the volume of fresh air induced.

28

Volumetric efficiency ηv = 1 + c − c(P₂/P₁)1/n, where c is the clearance ratio — showing that volumetric efficiency falls as the pressure ratio or the clearance increases, which is a further reason for staging.

1

This section covers the volumetric properties of pure fluids, PVT relations, the thermodynamic property relations and residual properties, the thermodynamics of solutions, flow processes, heat effects and the production of power from heat.

2

Volumetric Properties and PVT Relations • The PVT behaviour of a pure fluid, described in 1.2, is the starting point for all property calculation: once V = f(P,T) is known, every other thermodynamic property can be derived from it together with the ideal-gas heat capacity. • The P-V diagram shows the isotherms, the two-phase dome, and the critical isotherm which has a point of inflection at the critical point, so that (∂P/∂V)T = 0 and (∂²P/∂V²)T = 0 there — the two conditions used to evaluate the constants of a cubic equation of state. • The tools available, in order of sophistication: the ideal gas law; the generalised compressibility chart with reduced properties; cubic equations of state (van der Waals, Redlich-Kwong, Soave, Peng-Robinson), which are the practical choice because they represent both liquid and vapour and can therefore predict vapour-liquid equilibrium; and the virial equation, sound at moderate pressure.

3

For liquids, the coefficient of volume expansion β = (1/V)(∂V/∂T)P and the isothermal compressibility κ = −(1/V)(∂V/∂P)T describe the small changes of volume.

4

Property Relations and Residual Properties • The four fundamental property relations for a closed system of constant composition: dU = T dS − P dV, dH = T dS + V dP, dA = −S dT − P dV and dG = −S dT + V dP.

5

From these follow the Maxwell relations, of which the most used are (∂T/∂V)S = −(∂P/∂S)V, (∂S/∂V)T = (∂P/∂T)V and (∂S/∂P)T = −(∂V/∂T)P.

6

Their value is that they express entropy derivatives, which cannot be measured, in terms of PVT derivatives, which can — which is the whole basis of practical property calculation. • Residual properties are defined as the difference between the real value and the ideal-gas value at the same T and P:

7

The practical route to any property is therefore compute the ideal-gas value from heat capacity data, then add the residual obtained from an equation of state or a generalised chart.

8

Departure functions and the generalised charts for enthalpy and entropy departure are tabulations of exactly this. • The Joule-Thomson coefficient μJT = (∂T/∂P)H describes the temperature change on throttling.

9

It is positive below the inversion temperature, so the gas cools on expansion — the basis of the Linde liquefaction process of 2.6 — and negative above it, so the gas warms.

10

For an ideal gas μJT is zero, and hydrogen and helium have inversion temperatures below ambient, so they must be pre-cooled before throttling will liquefy them — a favourite examination point.

11

Thermodynamics of Solutions • For a mixture, every extensive property is apportioned among the species by its partial molar property, M̄i = (∂(nM)/∂ni)T,P,nj , so that M = Σ xiM̄i.

12

The partial molar Gibbs energy is the chemical potential, μi = Ḡi, the central quantity of phase and reaction equilibrium. • The Gibbs-Duhem equation, Σ xidM̄i = 0 at constant T and P, constrains the partial molar properties and is used to test the thermodynamic consistency of experimental VLE data — data that fail the test are rejected. • Fugacity and fugacity coefficient: fugacity f is a 'corrected pressure' defined so that dG = RT d(ln f) at constant T, with the fugacity coefficient φ = f/P → 1 as P → 0.

13

For a species in solution, activity ai = f̂i/fi° and activity coefficient γi = ai/xi. • Ideal solution: obeys Raoult's law and is characterised by ΔVmix = 0 and ΔHmix = 0, with ΔSmix = −R Σ xiln xi and ΔGmix = RT Σ xiln xi — mixing is spontaneous purely because of the entropy of mixing.

14

Excess properties, ME = M − Mideal solution, measure the departure, and excess Gibbs energy models — Margules, van Laar, Wilson, NRTL and UNIQUAC, with UNIFAC for prediction from group contributions — are what process simulators use to obtain activity coefficients.

15

Thermodynamics of Flow Processes and Heat Effects • Flow processes are analysed with the steady-flow energy equation of 2.2 together with the entropy balance.

16

Standard applications: nozzles and diffusers (enthalpy to kinetic energy and back, with choking at sonic velocity); turbines and compressors (isentropic efficiency); throttling (isenthalpic); ejectors; and pipe flow, where the mechanical energy balance and friction loss appear. • Heat effects collected together: sensible heat from a temperature change, requiring the mean heat capacity or the integral of Cp; latent heat of phase change, correlated by Trouton's rule (ΔHvap/Tb ≈ 88 J/mol·K) and Watson's relation for its variation with temperature; heat of mixing and solution (2.5); and heat of reaction (1.4 and 2.5). • Production of power from heat: the second law fixes the ceiling at the Carnot efficiency, and the practical cycles approach it as closely as materials allow.

17

The Rankine cycle with steam is the workhorse — its efficiency is raised by superheating (which also protects the turbine blades from erosion by wet steam), reheating, regenerative feedwater heating and higher boiler pressure, and lowered by a higher condenser pressure; the Brayton cycle serves gas turbines and, in combined cycle with a Rankine bottoming cycle, reaches the highest efficiencies of any thermal plant; the Otto and Diesel cycles govern engines; and cogeneration recovers the rejected heat for process use, raising the overall utilisation of fuel far above the electrical efficiency alone.

1

This section covers the kinetics of homogeneous reactions, single and multiple reactions in ideal reactors, non-ideal reactors, non-isothermal reactors, batch and continuous flow reactors, the kinetics of heterogeneous catalytic reactions and diffusional effects in catalysis.

2

Theories of Reaction Rate • Collision theory holds that reaction requires molecules to collide with an energy at least equal to the activation energy and with the correct orientation, giving k = PZe−E/RT, where Z is the collision frequency and P the steric factor.

3

It works well for simple gas-phase bimolecular reactions but poorly for complex molecules. • Transition state (activated complex) theory assumes the reactants form an activated complex in quasi-equilibrium with them at the top of the energy barrier, which then decomposes to products, giving the Eyring equation, k = (kBT/h)·e−ΔG‡/RT, with ΔG‡ the free energy of activation.

4

It accounts for both the energy and the entropy of activation, and so explains the steric factor. • The temperature dependence in both is essentially Arrhenius; the theories differ mainly in how the pre-exponential factor is interpreted.

5

Ideal Reactors and Their Design Equations Reactor Description Design equation and character Batch Charged, reacted for a fixed time, discharged; unsteady, well mixed, composition uniform but varying with time t = NA0∫0 X dX/(−rAV).

6

High conversion per unit volume, flexible, suited to small-scale and multi-product operation (pharmaceuticals, speciality chemicals); but labour-intensive with down-time for charging and cleaning, and variable product quality between batches CSTR (mixed flow, backmix) Continuous, perfectly mixed, so the contents and the exit stream are at the same composition — the reaction therefore occurs throughout at the lowest (exit) concentration V = FA0X/(−rA)exit, with space time τ = V/v₀.

7

Good temperature control and easy heat removal, simple construction, suited to liquid-phase and slow reactions; but requires the largest volume for a given conversion with positive-order kinetics PFR (plug flow, tubular) Continuous, no axial mixing, complete radial mixing, so composition varies continuously along the length — equivalent to a batch reactor travelling down the tube V = FA0∫0 X dX/(−rA).

8

Highest conversion per unit volume for positive-order kinetics, good for gas-phase and fast reactions and for high-capacity continuous production; but temperature control is harder and hot spots may form Packed bed / fluidised bed Heterogeneous catalytic reactors: the fixed bed approximates plug flow, the fluidised bed approximates a CSTR Fixed bed gives high conversion but poor temperature control and difficult catalyst replacement; fluidised bed gives excellent heat transfer and continuous catalyst regeneration at the cost of attrition and backmixing • The central comparison, which is examined every year: for reactions of positive order, a PFR always requires a smaller volume than a CSTR for the same conversion, because the PFR operates at concentrations that are high at the inlet and fall gradually, whereas the entire CSTR operates at the low exit concentration.

9

The exception is autocatalytic reactions, and reactions of negative order, for which the CSTR can be smaller.

10

A series of CSTRs approaches PFR behaviour as the number of tanks increases, and N tanks in series is a standard way of obtaining near-plug-flow performance with the mechanical convenience of stirred vessels. • Multiple reactions: here selectivity, not conversion, governs the choice.

11

For parallel reactions, if the desired reaction has the higher order, keep the concentration high — use a batch or PFR and avoid dilution; if it has the lower order, keep the concentration low — use a CSTR, dilute the feed or add reactant slowly.

12

For series reactions A → B → C with B the desired product, there is an optimum time or space time beyond which B is destroyed, so a PFR or batch reactor with careful timing is preferred and a CSTR gives a lower maximum yield.

13

Non-Ideal and Non-Isothermal Reactors • Real reactors deviate from the ideal patterns because of channelling and bypassing, stagnant (dead) zones, short-circuiting, recycling and axial dispersion.

14

The deviation is characterised experimentally by the residence time distribution (RTD), measured by injecting a tracer as a pulse (giving the E curve) or as a step (giving the F curve).

15

From the E curve are computed the mean residence time t̄ = ∫tE dt and the variance σ², which measures the spread — zero for ideal plug flow and equal to the square of the mean for an ideal CSTR. • Models for non-ideal flow: the dispersion model, characterised by the vessel dispersion number D/uL — zero for plug flow and infinite for a CSTR; the tanks-in-series model, characterised by N = t̄²/σ², with N = 1 a single CSTR and N → ∞ plug flow; and compartment models with dead zones and bypass. • Non-isothermal operation: because the rate depends exponentially on temperature and the reaction releases or absorbs heat, the energy balance must be solved simultaneously with the mole balance.

16

Adiabatic operation gives a straight line relation between conversion and temperature, X = (T − T₀)Cp/(−ΔHrxn).

17

The dangerous phenomenon is multiple steady states in a CSTR, where the heat generation curve (S-shaped, from the Arrhenius term) intersects the heat removal line (straight) at up to three points, of which the middle one is unstable; and thermal runaway, in which the heat generated by an exothermic reaction exceeds the heat that can be removed, so the temperature rises, further accelerating the reaction — the mechanism of several major industrial disasters, controlled by adequate cooling area, dilution, semi-batch addition of the limiting reactant, temperature alarms, emergency quench and relief systems.

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Heterogeneous Catalysis and Diffusional Effects • The seven steps of a heterogeneous catalytic reaction, which must be known in order:

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(1) external (film) diffusion of reactant from the bulk fluid to the catalyst surface;

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(2) internal (pore) diffusion into the pores;

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(3) adsorption onto the active site;

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(4) surface reaction;

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(5) desorption of product;

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(6) internal diffusion out of the pores;

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(7) external diffusion back to the bulk fluid.

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Steps 3, 4 and 5 are the intrinsic chemical steps;

27

1, 2, 6 and 7 are physical transport steps, and the slowest step controls the overall rate. • Adsorption: physisorption is weak, by van der Waals forces, multilayer, reversible and low in heat of adsorption; chemisorption involves an actual chemical bond, is monolayer, often irreversible, has a much larger heat of adsorption, and is the step responsible for catalysis.

28

The Langmuir isotherm, θ = KP/(1 + KP), describes monolayer adsorption on uniform sites, and Langmuir-Hinshelwood-Hougen-Watson (LHHW) kinetics builds rate expressions from adsorption, surface reaction and desorption steps, giving the characteristic form rate = (kinetic term × driving force)/(adsorption term). • Diffusional limitation is the practically important topic.

29

When pore diffusion is slow relative to reaction, the reactant is consumed near the pore mouth and the interior of the pellet is starved.

30

This is quantified by the Thiele modulus φ, the ratio of the reaction rate to the diffusion rate, and by the effectiveness factor η, the ratio of the actual rate to the rate that would occur if the whole interior were at the surface concentration.

31

For small φ (fast diffusion or slow reaction) η approaches 1 and there is no limitation; for large φ, η ≈ 1/φ and the reaction is strongly pore-diffusion limited. • The observable consequences of diffusion control are examinable: the apparent activation energy falls to about half the true value in the strong pore-diffusion regime, and to a very low value (a few kJ/mol) under external film control; the apparent reaction order shifts towards one; the rate becomes sensitive to pellet size (smaller pellets react faster) and, for film control, to flow velocity.

32

Remedies: smaller or egg-shell catalyst pellets, larger pore size, higher velocity past the pellet. • Catalyst deactivation occurs by poisoning (sulphur, lead, chlorine occupying active sites), fouling or coking (carbon deposition, reversible by burning off), sintering (thermal loss of surface area, irreversible) and loss of active phase by volatilisation or attrition.

1

This section covers heat capacity calculations, the heats of dissolution and mixing, the laws of thermochemistry, and the effect of pressure and temperature on the heat of reaction.

2

Heat Capacity Calculations • Heat capacity is the heat required to raise the temperature of a substance by one degree:

3

Cp at constant pressure and Cv at constant volume, with Cp = (∂H/∂T)P and Cv = (∂U/∂T)V.

4

It may be expressed per mole (molar heat capacity), per unit mass (specific heat) or per unit volume. • Relations: for an ideal gas, Cp − Cv = R, and γ = Cp/Cv is 1.67 for a monatomic, 1.40 for a diatomic and about 1.3 for a triatomic gas.

5

For liquids and solids the two are nearly equal because the volume change on heating is small. • Temperature dependence: heat capacity is not constant, and is correlated as a polynomial, Cp = a + bT + cT² + dT³ or Cp/R = A + BT + CT² + DT−2.

6

The enthalpy change is then ΔH = ∫T₁ T₂ Cp dT.

7

For hand calculation a mean heat capacity Cpm over the interval is used, so that ΔH = Cpm(T₂ − T₁) — but it must be the mean over that particular interval, since a mean quoted for a different range is not interchangeable. • Mixtures: the heat capacity of a gas mixture is the mole-fraction-weighted sum, Cp,mix = Σ yiCpi.

8

Kopp's rule estimates the heat capacity of a solid compound as the sum of atomic contributions when data are unavailable.

9

Useful values: water 4.18 kJ/kg·K (75.3 J/mol·K), steam about 2.0, air about 1.0 kJ/kg·K. • Solids at low temperature are described by the Dulong-Petit rule (molar heat capacity of a solid element ≈ 25 J/mol·K at ordinary temperature) and by the Debye theory, which shows Cv falling to zero as T³ near absolute zero.

10

Laws of Thermochemistry • Lavoisier-Laplace law: the heat change accompanying a reaction in one direction is exactly equal in magnitude and opposite in sign to that of the reverse reaction.

11

Hence the heat of formation of a compound is the negative of its heat of decomposition. • Hess's law of constant heat summation: the total enthalpy change of a reaction is the same whether it takes place in one step or in several, because enthalpy is a state function.

12

Its practical importance is that heats of reaction that cannot be measured directly can be obtained by algebraic combination of reactions that can — the formation of carbon monoxide from carbon, which always yields some carbon dioxide, being the classic example, obtained by subtracting the combustion of CO from that of carbon. • Kirchhoff's law: the effect of temperature on the heat of reaction, d(ΔH)/dT = ΔCp = Σ νiCp,products − Σ νiCp,reactants, integrated as ΔHT₂ = ΔHT₁ + ∫ΔCp dT.

13

If ΔCp is positive the heat of reaction becomes more positive (less exothermic) as the temperature rises; if ΔCp is small, the heat of reaction is nearly independent of temperature. • Standard states and conventions, restated for completeness: the standard state is 25 °C and 1 bar, with each substance in its most stable form; the heat of formation of an element in its standard state is zero; and the standard heat of reaction is products minus reactants using formation enthalpies, or reactants minus products using combustion enthalpies.

14

Heats of Dissolution and Mixing • Heat of solution is the enthalpy change when one mole of solute is dissolved in a specified quantity of solvent at constant temperature and pressure.

15

It depends on the amount of solvent, so it must always be quoted with the dilution, and the limiting value at very great dilution is the integral heat of solution at infinite dilution.

16

The differential heat of solution is the enthalpy change when one mole of solute is added to so large a quantity of solution that the concentration does not change appreciably. • Heat of dilution is the enthalpy change on adding more solvent to an existing solution, and equals the difference between the integral heats of solution at the two concentrations. • Physical origin: dissolution involves breaking the solute lattice (endothermic), breaking solvent-solvent interactions (endothermic) and forming solute-solvent interactions, that is solvation or hydration (exothermic).

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The sign of the overall heat of solution is the balance of these: dissolving sulphuric acid, sodium hydroxide or calcium chloride in water is strongly exothermic, whereas dissolving ammonium nitrate or potassium nitrate is endothermic and produces a cooling effect — the basis of the instant cold pack. • The practical safety rule, which is examinable and important: always add concentrated sulphuric acid to water, never water to acid, because the heat of dilution is very large and adding water to acid produces localised boiling that can eject acid violently. • Enthalpy-concentration diagrams (for sulphuric acid-water, sodium hydroxide-water, ammonia-water) tabulate all of this and are used directly in the energy balances of absorbers, evaporators and mixing vessels. • Heat of mixing is the corresponding quantity for two liquids; it is zero for an ideal solution, and excess enthalpy HE = ΔHmix measures the departure from ideality (2.3).

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Related quantities are the heat of crystallisation, heat of hydration of a salt, heat of wetting and heat of adsorption.

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Effect of Pressure and Temperature on Heat of Reaction • Temperature acts through Kirchhoff's law as above, and the correction is made by the hypothetical path of 1.4 — cool the reactants to 25 °C, react, then heat the products — which is exactly equivalent to integrating ΔCp. • Pressure has little effect on the heat of reaction for reactions involving only ideal gases, liquids and solids, because the enthalpy of an ideal gas is independent of pressure and that of condensed phases is nearly so.

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It becomes significant only at high pressures, where real-gas behaviour matters, and the correction is made through the enthalpy departure (residual enthalpy) of 2.3, evaluated from an equation of state or a generalised chart. • Phase of the reactants and products must always be stated, because the heat of reaction differs by the latent heat — the distinction between the gross and net calorific value of a fuel (1.4) being the standard instance, the two differing by the latent heat of the water formed. • Measurement: heats of reaction and combustion are measured in a calorimeter — the bomb calorimeter at constant volume, which therefore gives ΔU and requires the correction ΔH = ΔU + ΔngasRT, and the flow or flame calorimeter at constant pressure, which gives ΔH directly.

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This conversion between ΔU and ΔH is a standard examination calculation.

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This section covers the refrigeration cycle and its performance, the vapour compression cycle, eco-friendly refrigerants, absorption and adsorption refrigeration, and the processes used for gas liquefaction.

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Principles and Coefficient of Performance • Refrigeration is the continuous removal of heat from a body at low temperature and its rejection to the surroundings at a higher temperature.

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By the Clausius statement of the second law this cannot happen unaided, so a work or heat input is always required. • Coefficient of performance replaces efficiency, because the useful output exceeds the work input.

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For a refrigerator, COP = QC/W = desired cooling/work input; for a heat pump, COP = QH/W; and since QH = QC + W it follows that COPheat pump = COPrefrigerator + 1, a relation frequently asked. • Carnot (maximum) values:

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COPref,Carnot = TC/(TH − TC) and COPHP,Carnot = TH/(TH − TC), with temperatures in kelvin.

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The immediate consequence is that the COP falls sharply as the temperature difference increases, which is why deep refrigeration is expensive and why the condenser should be as cool and the evaporator as warm as the duty permits. • Units: the tonne (ton) of refrigeration is the rate of heat removal required to freeze one short ton of water at 0 °C in 24 hours, equal to 3.517 kW (211 kJ/min, 12,000 BTU/h).

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The Vapour Compression Cycle • This is the cycle of virtually every domestic refrigerator, air conditioner and cold store.

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Its four components and four processes must be known exactly: • (1) Compressor — the low-pressure saturated vapour from the evaporator is compressed to high pressure and temperature; ideally isentropic, and this is where the work is supplied. • (2) Condenser — the hot high-pressure vapour is desuperheated and condensed at constant pressure, rejecting heat QH to the surroundings, leaving as saturated (or slightly subcooled) liquid. • (3) Expansion (throttle) valve or capillary tube — the liquid is expanded to the low pressure.

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This is an isenthalpic throttling process, not an isentropic expansion: a valve is used rather than a turbine because the work recoverable from expanding a liquid is negligible and a turbine handling a two-phase mixture is impractical.

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The temperature falls sharply and part of the liquid flashes to vapour. • (4) Evaporator — the cold two-phase mixture absorbs the refrigeration load QC at constant pressure and temperature, evaporating to saturated vapour, and returns to the compressor. • Representation: the cycle is drawn on the pressure-enthalpy (P-h) diagram, on which three of the four processes are straight lines — the throttling is vertical, and the condensation and evaporation are horizontal, which is why this diagram rather than T-s is used in refrigeration practice.

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From it, COP = (h₁ − h₄)/(h₂ − h₁) — refrigerating effect divided by compressor work. • Practical modifications: subcooling the liquid leaving the condenser increases the refrigerating effect at no extra work and so raises the COP; superheating the vapour entering the compressor protects it from liquid slugging; multistage compression with flash intercooling is used for large temperature lifts; and cascade systems, in which two cycles with different refrigerants are thermally coupled, are used for very low temperatures.

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Refrigerants and Environmental Considerations • Desirable properties of a refrigerant: a high latent heat of vaporisation (so that little needs to circulate), an evaporator pressure above atmospheric (so that air and moisture cannot leak in), a moderate condenser pressure, a low specific volume of vapour (a smaller compressor), chemical stability, non-toxicity, non-flammability and non-corrosiveness, miscibility with the lubricating oil, easy leak detection, low cost, and zero ozone depletion potential and low global warming potential. • The environmental history, which is examinable:

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CFCs (R-11, R-12) were chemically ideal but release chlorine in the stratosphere and were the principal cause of ozone depletion; they were phased out under the Montreal Protocol (1987).

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HCFCs (R-22) were the transitional substitutes, with a lower but non-zero ozone depletion potential, and are now also being phased out.

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HFCs (R-134a, R-410A) have zero ozone depletion potential but high global warming potential, and are being phased down under the Kigali Amendment (2016). • Eco-friendly refrigerants now in use or under development:

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HFOs (R-1234yf, R-1234ze) with very low GWP; and the natural refrigerants — ammonia (R-717), which has an excellent latent heat and zero ODP and GWP but is toxic and attacks copper, and is the standard in large industrial plants; carbon dioxide (R-744), non-toxic and non-flammable but requiring transcritical operation at very high pressure; hydrocarbons (propane R-290, isobutane R-600a), excellent thermodynamically and now standard in domestic refrigerators, but flammable; and water (R-718) and air for special applications. • Regulatory status changes with each amendment and national schedule, so the current position of any refrigerant must be verified before it is specified.

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Absorption and Adsorption Refrigeration • Absorption refrigeration replaces the compressor — the only work-consuming component — with a thermally driven circuit, so that the system is driven by heat rather than by work.

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Its components are the absorber, solution (liquid) pump, generator, condenser, expansion valve and evaporator, together with a heat exchanger between the strong and weak solution streams. • Operation: refrigerant vapour leaving the evaporator is absorbed into a liquid absorbent in the absorber, releasing heat; the solution pump raises the pressure of the liquid, which requires far less work than compressing a vapour; in the generator, heat is supplied to drive the refrigerant vapour out of solution; the vapour passes to the condenser and expansion valve as usual, while the weak solution returns to the absorber through a throttling valve. • The two systems: ammonia-water, in which ammonia is the refrigerant and water the absorbent, capable of temperatures below 0 °C and requiring a rectifier to remove water from the ammonia vapour; and lithium bromide-water, in which water is the refrigerant and lithium bromide solution the absorbent, limited to above 0 °C and therefore used for air conditioning and chilled water, with the advantages of a non-toxic refrigerant and no rectifier but the problems of crystallisation of the salt and the need for a deep vacuum. • Advantages: it uses low-grade heat — waste heat, steam, solar or a gas flame — so it is attractive where electricity is expensive or unavailable, has almost no moving parts and is therefore quiet and reliable, and uses environmentally benign fluids.

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Disadvantages: a much lower COP (typically 0.5-0.8 for single-effect, against 3-5 for vapour compression), larger and more expensive equipment, and the need for a cooling water supply.

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The comparison of COP is a standard examination question, and the point to make is that the two COPs are not directly comparable, because one is per unit of work and the other per unit of low-grade heat. • Adsorption refrigeration uses a solid adsorbent — silica gel, zeolite or activated carbon — instead of a liquid absorbent, with pairs such as silica gel-water, zeolite-water and activated carbon-methanol.

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The bed alternately adsorbs refrigerant vapour (releasing heat) and is regenerated by heating (desorbing the vapour), so the operation is inherently intermittent and two beds are used alternately for continuous output.

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It can use very low-grade heat (60-90 °C, and hence solar energy), has no moving parts and no corrosion or crystallisation problems, but has a low COP and a low specific cooling power, so the equipment is bulky. • Other refrigeration methods worth naming: steam-jet (ejector) refrigeration, air (Bell-Coleman) refrigeration, used in aircraft, thermoelectric (Peltier) cooling for small loads, vortex tube and magnetic refrigeration.

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Liquefaction Processes • To liquefy a gas it must be brought below its critical temperature, since above that no pressure will condense it; the industrial problem is therefore one of cooling, and the three mechanisms available are heat exchange against a colder stream, Joule-Thomson throttling, and expansion in a turbine with the production of external work. • The Linde-Hampson process relies on Joule-Thomson throttling alone.

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The gas is compressed, cooled against the returning cold stream in a counter-current heat exchanger, and throttled through a valve, where it cools further because μJT is positive; part liquefies and is withdrawn, and the remaining cold vapour returns through the heat exchanger to pre-cool the incoming gas — the regenerative cooling that progressively lowers the temperature until liquefaction begins.

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It is simple and has no moving parts in the cold section, but has a low liquid yield and a high work requirement, and — the examinable point — it cannot be used for hydrogen, helium or neon at ambient temperature, because their inversion temperatures are below ambient, so throttling would warm them; they must first be pre-cooled with liquid nitrogen. • The Claude process improves on this by diverting part of the stream through an expansion engine or turbine, where it performs external work and therefore cools far more than by throttling alone, the remainder being throttled as before.

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It gives a much higher yield and lower energy consumption at the cost of a machine operating at low temperature; the Heylandt process is a variant in which the expander works from ambient temperature, used for air. • Cascade liquefaction uses a series of refrigeration cycles with progressively lower-boiling refrigerants — for example propane, ethylene and methane — each condensing the next, and is the classical basis of LNG production; modern LNG plants use mixed refrigerant and propane-precooled mixed refrigerant (C3MR) processes. • Applications: air separation into oxygen, nitrogen and argon by cryogenic distillation of liquid air (the Linde double column);

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LNG for transport and storage of natural gas, reducing its volume by a factor of about 600; liquid hydrogen and helium; storage and transport of industrial gases; and cryogenic preservation and superconductivity.

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Cryogenic safety requires attention to cold burns, embrittlement of materials, oxygen enrichment and the asphyxiation hazard of an evaporating inert gas in a confined space.