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Chemical engineering Thermodynamics formula sheet...

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ENGR266

Formula Sheet

Thermodynamics for Chemical Engineering: Formula Sheet Useful constants Gas constant, R = 8.314 J mol-1 K-1 Temperature conversion, T℃ = T K - 273.15 Gravitational constant, g = 9.81 ms-2

Fundamental equations Newton’s Second Law

Pressure

Work

dW = −P dV t

F P= A

F = ma

F = Force m = mass a = acceleration

dW = work done P = pressure Vt = total volume

P = pressure F = force A = area

Kinetic energy

Potential energy

1 EK = mu 2 2

EP = mzg Ep = potential energy m = mass z = displacement in z g = acceleration due to gravity

EK = kinetic energy m = mass u = velocity Laws of thermodynamics First law of thermodynamics

dU = dQ + dW

Second law of thermodynamics

ΔStotal ≥ 0

dU = change in internal energy dQ = heat exchange with surroundings dW = work done on/by system

ΔStotal = total entropy change of the system

Third law of thermodynamics

S=∫

Tf

( C P )s

0

T

dT +

T (C P ) g Tv ( C P ) ΔH f ΔH v l dT + dT +∫ +∫ Tf Tv Tv Tf T T

S = entropy Tf = temperature of fusion Tv = temperature of vapourisation T = temperature CP = constant pressure heat capacity Miscellaneous The phase rule

F = 2 −π + N

F = degrees of freedom of the system 𝜋 = number of phases N = number of chemical species

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Formula Sheet

Energies and heat capacities Enthalpy

Helmholtz energy

Gibbs energy

A = U − TS

H = U + PV

H = enthalpy U = internal energy P = pressure V = volume

G = H − TS

A = Helmholtz energy U = internal energy T = temperature S = entropy

Internal energy change for a constant volume process

G = Gibbs energy H = enthalpy T = temperature S = entropy

Enthalpy change for a constant pressure process

dU = CV dT

dH = CP dT

dU = change in internal energy CV = constant volume heat capacity dT = temperature change

dH = change in enthalpy CP = constant pressure heat capacity dT = temperature change

Latent heat for a phase transformation

Temperature dependence of the heat capacity

sat

ΔH = T ΔV

dP dT

CPig = A + BT + CT 2 + DT −2 R

ΔH = latent heat T = temperature ΔV = volume change accompanying the phase change Psat = saturation pressure

CPig = constant pressure ideal gas heat capacity R = gas constant A, B, C, D = substance specific coefficients T = temperature

Heat capacity for a mixture

CigPmixture = yACPigA + yBCigPB + yC CPigC CigPmixture = constant pressure heat capacity for an ideal gas mixture yA,B,C = mole fractions of A, B and C CigPA ,B ,C = constant pressure ideal gas heat capacity for A, B and C

Pressure-volume-temperature relations Ideal gas law

Compressibility

PV = RT

Virial equations of state

PV Z= RT

P = pressure V = volume R = gas constant T = temperature

Z = Compressibility factor P = pressure V = volume R = gas constant T = temperature

Polytropic relationships

TV δ −1 = const P = pressure V = volume T = temperature δ = polytropic index

TP(1−

δ)/δ

= const

PV δ = const

Z = 1+ B' P + C ' P 2 + D ' P 3 +! B C D Z = 1+ + 2 + 3 +! V V V Z = compressibility V = volume P = pressure B’, C’, D’ = virial coefficients B, C, D = virial coefficients

ENGR266

Formula Sheet

Process calculations for ideal gases Isothermal process

Isobaric process

V P Q = −W = RT ln 2 = −RT ln 2 P1 V1

Q = ΔH = ∫ CP dT

Q = heat into/out of the system W = work done on/by the system R = gas constant T = temperature V = volume

Q = heat into/out of the system ΔH = enthalpy change CP = constant pressure specific heat dT = change in temperature

Isochoric process

Adiabatic process

Q = ΔU = ∫ CV dT

W=

Q = heat into/out of the system ΔU = internal energy change CV = constant volume specific heat dT = change in temperature

W=

RT1 ⎡ ⎛ P2 ⎞ ⎢ δ −1 ⎢⎣ ⎜⎝ P1 ⎟⎠

(γ −1 )/γ

⎤ −1⎥ ⎥⎦

W = work done on/by the system R = gas constant T = temperature P = pressure 𝛾 = CP/CV

Work done during a general polytropic process ( δ −1)/δ

RT1 ⎡ ⎛ P2 ⎞ ⎢ γ −1 ⎢ ⎜⎝ P1 ⎟⎠ ⎣

Heat exchanged during a general polytropic process

⎤ −1⎥ ⎥⎦

Q=

W = work done on/by the system R = gas constant T = temperature δ = polytropic index P = pressure

( δ −γ ) RT1 ⎡⎢⎛ P2 ⎞ (δ −1 )(γ −1) ⎢⎣⎜⎝ P1 ⎟⎠

( δ −1) /δ

⎤ −1⎥ ⎥⎦

Q = heat into/out of the system R = gas constant T = temperature δ = polytropic index P = pressure 𝛾 = CP/CV

Cubic equations of state Reduced properties

T Tr = Tc

Acentric factor

Tr = reduced temperature T = temperature Tc = critical temperature

P Pr = Pc

ω = −1.0 − log (Prsat ) T =0.7 r

Pr = reduced pressure P = pressure Pc = critical pressure

ω = acentric factor Pr sat= reduced saturation pressure

General cubic equation of state

P=

a (T ) RT − V − b ( V + εb)(V + σ b)

P = pressure T = temperature V = volume

†α ‡α

a (T ) = Ψ

α ( Tr ) R 2Tc2 Pc

b=Ω

RTc Pc

Tc = critical temperature Pc = critical pressure

EOS

α(Tr)

𝜎

𝜀

Ω

𝚿

vdW

1

0

0

1/8

27/64

RK

Tr-1/2

1

0

0.08664

0.42748

SRK

αSRF(Tr,ω)†

1

0

0.08664

0.42748

PR

αPR(Tr,ω)‡

1+21/2

1-21/2

0.07780

0.45724

= [1+(0.480+1.574ω-0.176ω2)(1-Tr1/2)]2 1/2 2 2 PW(Tr,ω) = [1+(0.37464+1.54226ω-0.26992ω )(1-Tr )] SRK(Tr,ω)

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Formula Sheet

Generalised correlations Compressibility

Z = Z 0 + ωZ 1

Z = Compressibility factor Z0,Z1 = Lee and Kessler coefficients ω = acentric factor

Second virial coefficient

Bˆ = B 0 + ω B1 ˆB = reduced second viral coefficient 0.422 B 0 = 0.083 − 1.6 Tr 0.172 B1 = 0.139 − 4.2 Tr Tr = reduced temperature ω = acentric factor

Heat engines and refrigerators Thermal efficiency of a heat engine

η=

Q W = 1− C QH QH

η = thermal efficiency W = work done on/by the system QH = heat exchanged with the hot reservoir QC = heat exchanged with the cold reservoir Coefficient of performance for a Carnot refrigerator

ω=

TC TH − TC

Efficiency of a Carnot engine

η = 1−

TC TH

η = thermal efficiency for a Carnot engine TH = temperature of the hot reservoir TC = temperature of the cold reservoir

Coefficient of performance for the vapour compression cycle

ω=

H 2 − H1 H3 − H2

ω = coefficient of performance TC = temperature of the cold reservoir TH = temperature of the hot reservoir

ω = coefficient of performance H1 = Enthalpy before entering evaporator H2 = Enthalpy before entering compressor H3 = Enthalpy after leaving compressor

Rate of circulation of refrigerant

Entropy change in a heat engine

! = m

Q!C

H2 − H 1 ! = mass flow rate of refrigerant m ! C = rate of heat extraction from the cold Q reservoir H1 = Enthalpy before entering evaporator H2 = Enthalpy before entering compressor Work done by a heat engine

⎛ T ⎞ W = −TC ΔStotal + QH ⎜1− C ⎟ ⎝ TH ⎠ W = work done on/by the system ΔStotal = total entropy change QH = heat exchanged with hot reservoir TH = temperature of the hot reservoir TC = temperature of the cold reservoir

⎛ T − TC ⎞ ΔStotal = Q ⎜ H ⎝ TH TC ⎟⎠ ΔStotal = total entropy change Q = heat exchanged TH = temperature of the hot reservoir TC = temperature of the cold reservoir

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Formula Sheet

Entropy Entropy change for a finite reversible process

ΔS t = ∫

dQrev T

ΔSt = total entropy change dQrev = differential heat change during reversible process T = temperature

Entropy change for an ideal gas ig

T C dT P ΔS =∫ P − ln T0 R T R P0

ΔS = entropy change R = gas constant CigP = constant pressure ideal gas heat capacity T = temperature P = pressure P0 = initial pressure

Balance equations for open systems Mass balance for an open system

dmcv + Δ (m ! ) fs = 0 dt dmcv = change in the total mass in the control volume in time dt Δ (m! ) fs = sum of mass entering/leaving control volume via all streams Energy balance for an open system

d( mU) cv + Δ ⎡⎣ H + 21 u 2 + zg m! ⎦⎤ = Q! + W! fs dt

(

d ( mU )cv dt

)

= change in internal energy in the control volume in time

H = enthalpy of the flow stream u = velocity of the flow stream z = displacement in z of the flow stream g = gravitational constant ! = mass flow rate for the flow stream m Q! = rate of heat transfer into/out of system W! = rate of work done on/by the system Entropy balance for an open system

d (mS )cv Q! + Δ (S m ! ) fs − ∑ j = S! G ≥ 0 dt j Tσ , j d (mS )cv dt

= change in internal energy in the control volume in time

S = entropy of the flow stream ! = mass flow rate for the flow stream m ! = rate of heat transfer into/out to the surroundings Q T𝜎,j = temperature S! G= rate of entropy generation

ENGR266

Formula Sheet

Flow processes Duct flow of a compressible fluid

(

V 1− M 2

⎛ βu + T ⎜1+ ) dP dx C ⎝

2

P

⎞ dS u 2 dA ⎟⎠ dx − A dx = 0

⎛ βu 2 ⎞ + M2 ⎜ CP ⎟ dS ⎛ 1 ⎞ u 2 dA du +⎜ =0 u −T ⎜ 2⎟ 2 ⎟ dx ⎜ 1− M ⎟ dx ⎝ 1− M ⎠ A dx ⎜⎝ ⎟⎠

V = volume M = Mach number (u/c, where c is the speed of sound in the fluid) u = velocity P = pressure x = displacement in x β = volume expansivity CP = constant pressure heat capacity S = entropy A = cross sectional area Velocity change in a nozzle

2γ RT1 ⎡ ⎛ P2 ⎞ ⎢ 1− u −u = γ −1 ⎢ ⎜⎝ P1 ⎟⎠ ⎣ 2 2

2 1

Pressure drop to ensure flow is sonic (γ −1)/γ

⎤ ⎥ ⎥⎦

P2 ⎛ 2 ⎞ = P1 ⎜⎝ γ +1 ⎟⎠

u = velocity 𝛾 = CP/CV R = gas constant T = temperature P = pressure

P = pressure 𝛾 = CP/CV

Work done in a turbine

Turbine efficiency

W!s = m ! ( H 2 − H 1) W!s = rate of shaft work ! = mass flow rate m

η=

γ /( γ −1)

Ws Ws ( isentropic)

H = enthalpy

η = turbine efficiency Ws = shaft work Ws(isentropic) = isentropic shaft work

Isentropic work

Compressor efficiency

⎤ ⎡⎛ P ⎞ R/CP −1⎥ Ws (isentropic ) = CPT1 ⎢⎜ 2 ⎟ ⎥⎦ ⎢⎣⎝ P1 ⎠

Ws(isentropic) = isentropic shaft work CP = constant pressure heat capacity T = temperature P = pressure R = gas constant

η=

Ws ( isentropic ) Ws

η = turbine efficiency Ws = shaft work Ws(isentropic) = isentropic shaft work

ENGR266

Formula Sheet

Properties of fluids Fundamental property relations

dU = T dS − P dV

dH = T dS + V dP

dA = −P dV − S dT

dG = V dP − S dT

H = enthalpy V = volume A = Helmholtz energy G = Gibbs energy

U = internal energy T = temperature P = pressure S = entropy Maxwells equations

⎛ ∂T ⎞ = − ⎛ ∂P ⎞ ⎜⎝ ∂V ⎟⎠ ⎜⎝ ∂S ⎟⎠ S V

⎛ ∂T ⎞ = ⎛ ∂V ⎞ ⎜⎝ ∂P ⎟⎠ ⎜⎝ ∂S ⎟⎠ S P

⎛ ∂P ⎞ = ⎛ ∂S ⎞ ⎜⎝ ∂T ⎟⎠ ⎜⎝ ∂V ⎟⎠ V T

⎛ ∂V ⎞ = ⎛ ∂S ⎞ ⎜⎝ ∂T ⎟⎠ ⎜⎝ ∂P ⎟⎠ P T

V = volume T = temperature P = pressure S = entropy Gibbs energy as a generating function

H ⎛ G⎞ V d⎜ dP − = dT ⎝ RT ⎟⎠ RT RT 2 G = Gibbs energy T = temperature R = gas constant V = volume P = pressure H = enthalpy Residual enthalpy P ∂Z dP H = −T ∫ ⎛⎜ ⎞⎟ 0 ⎝ ⎠ ∂T RT P P R

HR = residual enthalpy R = gas constant T = temperature Z = compressibility P = pressure Residual Gibbs energy P dP GR = ∫ ( Z −1) 0 RT P

GR = residual Gibbs energy R = gas constant T = temperature Z = compressibility P = pressure

Residual properties

M R = M − M ig

MR = residual molar property M = molar property Mig = molar property of ideal gas

Residual entropy P ∂Z P dP dP SR = −T ∫ ⎛⎜ ⎞⎟ − ∫ ( Z −1) 0 0 ⎝ ⎠ ∂T R P P P

SR = residual entropy R = gas constant T = temperature Z = compressibility P = pressure

ENGR266

Formula Sheet

Vapour/liquid equilibrium Raoult’s law

Henry’s law

yi P = xi Pi sat

yi P = xi H i

yi = vapour phase mole fraction P = pressure xi = liquid phase mole fraction Pi sat= saturation pressure of pure species i

yi = vapour phase mole fraction P = pressure xi = liquid phase mole fraction Hi = Henry’s constant for species i

Modified Raoult’s law

yi P = xiγ i Pi sat

yi = vapour phase mole fraction P = pressure xi = liquid phase mole fraction 𝛾i = activity coefficient for species i Pi sat= saturation pressure of pure species i Solution thermodynamics Gibbs energy for a multicomponent system

Chemical potential

⎡ ∂ (nG )⎤ µi = ⎢ ⎥ ⎣ ∂ni ⎦ P,T ,n j

d ( nG ) = (nV )dP − (nS ) dT + ∑ µi dni i

n = number of moles G = Gibbs energy V = volume P = pressure S = entropy T = temperature μi = chemical potential of i

μi = chemical potential of i n = number of moles G = Gibbs energy P = pressure T = temperature nj = number of moles of j

Partial property

Summability relations

⎡ ∂ (nM )⎤ Mi = ⎢ ⎥ ⎣ ∂ni ⎦ P,T ,n j Mi = partial molar property of i M = molar property n = number of moles P = pressure T = temperature nj = number of moles of j

M = ∑ xi M i

nM = ∑ ni M i

i

i

M = molar property xi = mole fraction of i Mi = partial molar property of i n = number of moles ni = number of moles of i

Gibbs Duhem equation

⎛ ∂M ⎞ dP + ⎛ ∂M ⎞ ⎜⎝ ∂P ⎟⎠ ⎜⎝ ∂T ⎟⎠ dT − ∑ xi dM i = 0 i T ,x P,x T = temperature xi = mole fraction of i

Mi = partial molar property of i M = molar property P = pressure Ideal gas mixture model

H ig = ∑ yi H iig i ig

S ig = ∑ yi Sigi − R ∑ yi ln yi i

H = enthalpy of an ideal gas mixture yi = mole fraction of i H iig= enthalpy of species i T = temperature

i

G ig = ∑ yi Giig − RT ∑ yi ln yi i

i

S ig= entropy of an ideal gas mixture Siig= entropy of species i G ig= Gibbs energy of an ideal gas mixture Giig= Gibbs of energy of species i

ENGR266

Formula Sheet

Solution thermodynamics cont….. Gibbs energy of a pure species

Residual Gibbs energy of a pure species

Gi = Γ i (T ) + RT ln fi

R i =

GRi = RT ln φ i

G excess Gibbs energy of pure species i R = gas constant T = temperature ϕi = fugacity coefficient of pure species i

Gi = Gibbs energy of pure species i 𝛤i(T) = integration constant at T R = gas constant T = temperature fi = fugacity of pure species i Fugacity coefficient of a pure species

φi =

fi P

P

lnφ i = ∫ ( Z −1) 0

dP P

ϕi = fugacity coefficient of pure species i fi = fugacity of pure species i P = pressure

ϕi = fugacity coefficient of pure species i Z = compressibility P = pressure

Fugacity of a pure liquid

Chemical potential of a species in mixture

⎛ V P − Pi fi = φisat Pi sat exp ⎜ RT ⎝ l i

(

sat

⎞ ⎟ ⎠

)

µi = Γ i ( T ) + RT ln ˆfi μi = chemical potential of species i in solution 𝛤i(T) = integration constant at T R = gas constant T = temperature fˆi = fugacity of species i in solution

φ isat = fugacity coefficient of pure species i at saturation Pi sat = saturation pressure of i Vi l = volume of the liquid i P = pressure R = gas constant T = temperature Partial residual Gibbs energy

Fugacity coefficient in solution

ˆf φˆi = i yi P

GiR = RT ln φˆ i GRi = partial residual Gibbs energy of

φˆi = fugacity coefficient of species i in solution fˆi = fugacity coefficient of species i in solution yi = mole fraction of i P = pressure

species i in solution R = gas constant T = temperature φˆi = fugacity coefficient of species i in solution Ideal solution model

H id = ∑ yi H i

S id = ∑ xi Si − R∑ xi ln xi

i id

H = enthalpy of an ideal solution xi = mole fraction of i H i = enthalpy of species i T = temperature Lewis/Randall rule

fˆi id = xi fi fˆiid = ideal solution fugacity of i xi = liquid phase mole fraction fi = fugacity of pure species i

i

i

G id = ∑ xi Gi + RT ∑ xi ln xi i

i

S id= entropy of an ideal solution Si = entropy of species i G id= Gibbs energy of an ideal solution Gi = Gibbs of energy of species i Excess properties

M E = M − M id ME = molar excess property M = molar property Mid = ideal solution molar property

ENGR266

Formula Sheet

Solution thermodynamics cont….. Activity coefficient

Excess Gibbs energy

GE = ∑ xi lnγ RT i

ˆf γi= i xi fi 𝛾i = activity coefficient for species i fˆi = fugacity coefficient of species i in solution xi = liquid phase mole fraction fi = fugacity of pure species i

i

GE = excess Gibbs energy R = gas constant T = temperature xi = liquid phase mole fraction 𝛾i = activity coefficient for species i

Margules equations

lnγ 1 = x22 ⎡⎣A12 + 2 (A21 − A12 ) x1 ⎤⎦ lnγ 2 = x12 ⎡⎣A21 + 2 ( A12 − A21 ) x2 ⎦⎤ 𝛾1 = activity coefficient for species 1 𝛾2 = activity coefficient for species 2 xi = liquid phase mole fraction of i A12, A21 = Margules constants

Chemical-reaction equilibria Mole fraction in a single reaction

n + υi ε yi = i0 n0 + υε

ni0 + ∑υ i, j ε j

yi = mole fraction of species i ni0 = initial number of moles of species i 𝝊i = stoichiometric number of species i 𝜀 = reaction co-ordinate n0 = initial total number of moles 𝝊 = total stoichiometric number

Equilibrium constant

⎛ −ΔG ! ⎞ K = exp ⎜ ⎝ RT ⎟⎠

Mole fraction in a single reaction

yi =

j

n0 + ∑ υ j ε j j

yi = mole fraction of species i ni0 = initial number of moles of species i 𝝊i,j = stoichiometric number of species i in reaction j 𝜀j = reaction co-ordinate for reaction j n0 = initial total number of moles 𝝊j = total stoichiometric number for reaction j Gas phase reactions

−ΔG ! ln K = RT

...