Plug flow is a simplified and idealized picture of the motion of a fluid, whereby all the fluid elements move with a uniform velocity along parallel streamlines.

This perfectly ordered flow is the only transport mechanism accounted for in the plug flow reactor model.

Because of the uniformity of conditions in a cross section the steady-state continuity equation is a very simple ordinary differential equation.

z z + zz = 0

z

z = L

z

MASS BALANCE

VAFVVAF

VrA

0FVrFVVAAVA

dt

dNA

VrFF AVAVVA

Rate of flow of A into a volume element

Rate of flow of A out of the volume element

Rate of generation of A by chemical reaction within the volume element

Rate of accumulation of A within the volume element

+ – =

For steady-state process: 0dt

dNA

Arz

FFA

zAzzA

0zlim

AVAVVA r

V

FF

AA r

dV

dF

VrFF AVAVVA

By definition of the conversion

X1FF 0AA

dXFdF 0AA

So that the continuity for A becomes:

dVrdXF A0A

A

0A

r

F

dX

dV

(1.1)

Design equation

To design an isothermal tubular/plug-flow reactor, the following information is needed:

1. Design equation

2. Rate law

(for first order reaction)

3. Stoichiometry (liquid phase)

A

0A

r

F

dX

dV

AA Ckr

X1CC 0AA

(1.1)

(1.2)

(1.3)

Combining eqs. (1.2) and (1.3) yields:

X1Ckr 0AA

Introducing eq. (1.4) into eq. (1.1) yields:

X

X

0A

X

X 0A0A0

0

X1lnCk

1

X1kC

dX

F

V

X1

X1ln

kC

F

X1

X1ln

kC

FV 0

0A

0A

00A

0A

(1.4)

X1

X1ln

k

v

X1

X1ln

k

vV 00

0

0

To design an isothermal tubular/plug-flow reactor, the following information is needed:

1. Design equation

2. Rate law

(for first order reaction)

3. Stoichiometry (liquid phase)

AA Ckr

X1CC 0AA

(1.1)

(1.2)

(1.3)

A

0A

r

F

dX

dV

T

1

T

1

R

Eexpkk

1

1

(1.5)

(1.6)

Combining eqs. (1.5) and (1.6) yields:

(1.7)

Combining eqs. (1.1), (1.3), and (1.4) yields:

00A

0A

v

X1k

F

X1kC

dV

dX

Recalling the Arrhenius equation:

T

1

T

1

T

EexpX1

v

k

dV

dX

10

1

In a closed system, the change in total energy of the system, dE, is equal to the heat flow to the system, Q, minus the work done by the system on the surrounding, W.

WQdE (1.8)Thus the energy balance for a closed system is:

Q

SW

0min 0mout

Q

SW

iniF

Open system:

iniH

outiF

outiH

For an open system in which some of the energy exchange is brought about by the flow of mass across the system boundaries, the energy balance for the case of only one species entering and leaving becomes:

Rate of accumu-lation of energy

within the system

+–=

Rate of flow of heat to the system from the

surrounding

Rate work done by the system on

the surrounding

Rate of energy added to the system by mass flow

into the system

Rate energy leaving

system by mass flow out of the system

–

outoutinin

sys EFEFWQdt

dE (1.9)

The unsteady-state energy balance for an open system that has n species, each entering and leaving the system at its respective molar flow rate Fi (mole of i per time) and with its respective energy Ei (joules per mole of i), is:

out

n

1iii

in

n

1iii

sys FEFEWQdt

dE

(1.10)

It is customary to separate the work term, , into:

flow work: work that is necessary to get the mass into and out of the system

other work / shaft work, .

For example, when shear stresses are absent:

W

SW

Sout

n

1iii

in

n

1iii WPVFPVFW

[rate of flow work]

(1.11)

where P is the pressure and Vi is the specific volume.

• Stirrer in a CSTR• Turbine in a PFR

Combining eqs. (1.10) and (1.11) yields:

out

n

1iiii

in

n

1iiiiS

sys PVEFPVEFWQdt

dE

(1.12)

The energy Ei is the sum of the internal energy (Ui), the kinetic energy , the potential energy (gzi), and any other energies, such as electric energy or light:

2u2

i

othergz2

uUE i

2

iii (1.13)

In almost all chemical reactor situations, the Kinetic, potential, and other energy terms are negligible in comparison with the enthalpy, heat transfer:

ii UE (1.14)

Recall the definition of enthalpy:

iii PVUH (1.15)

Combining eqs. (1.16), (1.15), and (1.13) yields:

out

n

1iii

in

n

1iiiS

sys HFHFWQdt

dE

(1.16)

We shall let the subscript “0” represent the inlet conditions. The un-subscripted variables represent the conditions at the outlet of the chosen system volume.

n

1iii

n

1i0i0iS

sys HFHFWQdt

dE (1.17)

The steady-state energy balance is obtained by setting (dEsys/dt) equal to zero in eq. (1.17) in order to yield:

0HFHFWQn

1iii

n

1i0i0iS

(1.18)

To carry out the manipulations to write eq. (1.18) in terms of the heat of reaction we shall use the generalized reaction:

(1.19)DdCcBbA

The inlet and outlet terms in Equation (1.19) are expanded, respectively, to:

0I0I0D0D0C0C0B0B0A0A0i0i FHFHFHFHFHFH

IIDDCCBBAAii FHFHFHFHFHFH

In:

Out:

(1.20)

(1.21)

We first express the molar flow rates in terms of conversion

X1FF 0AA

Xb

F

FFXFbFF

0A

0B0A0A0BB

XbFF B0AB (1.23)

(1.22)

Xc

F

FFXFcFF

0A

0C0A0A0CC

XcFF C0AC (1.24)

XdFF D0AD (1.25)

I0A

0A

0I0A0II F

F

FFFF

(1.26)

Substituting eqs. (1.23) – (1.27) into eq. (1.22) gives:

bXFHX1FHFH B0AB0AAii

I0AID0ADC0AC FHXdFHXcFH (1.26)

Subtracting eqs. (1.26) from eq. (1.20) gives:

BB0BA0A0A

n

1iii

n

1i0i0i HHHHFFHFH

II0IDD0DCC0C0A HHHHHHF

XFHHbHcHd 0AABCD (1.27)

The term in parentheses that is multiplied by FA0X is called the heat of reaction at temperature T and is designated HRx.

THTHbTHcTHdH ABCDRx (1.28)

All of the enthalpies (e.g., HA, HB) are evaluated at the temperature at the outlet of the system volume, and consequently, [HRx(T)] is the heat of reaction at the specific temperature ip: The heat of reaction is always given per mole of the species that is the basis of calculation [i.e., species A (joules per mole of A reacted)].

Substituting eq. (1. 28) into (1. 27) and reverting to summation notation for the species, eq. (1. 28) becomes

XFHHHFFHFH 0ARx

n

1ii0ii0A

n

1iii

n

1i0i0i

(1.29)

Substituting eq. (1.29) into (1.18) yields:

0XFHHHFWQ 0ARx

n

1ii0ii0AS

(1.30)

The enthalpy changes on mixing so that the partial molalenthalpies are equal to the molal enthalpies of the pure components.

The molal enthalpy of species i at a particular temperature and pressure, Hi, is usually expressed in terms of an enthalpy of formation of species i at some reference temperature TR, Hi(TR), plus the change in enthalpy that results when the temperature is raised from the reference temperature to some temperature T, HQi

QiR

0

ii HTHH (1.31)

The reference temperature at which Hi is given is usually 25°C. For any substance i that is being heated from T1 to T2 in the absence of phase change

2

1

T

TPQi dTCH (1.32)

A large number of chemical reactions carried out in industry do not involve phase change. Consequently, we shall further refine our energy balance to apply to single-phase chemical reactions. Under these conditions theenthalpy of species i at temperature T is related to the enthalpy of formation at the reference temperature TR by

T

TpiR

0

ii

R

dTCTHH (1.33)

The heat capacity at temperature T is frequently expressed in a quadratic function of temperature, that is,

2

iiipi TTC (1.34)

To calculate the change in enthalpy (Hi – Hi) when the reacting fluid is heated without phase change from its entrance temperature Ti0 to a temperature T, we use eq. (1.33)

0i

RR

T

TpiR

0

i

T

TpiR

0

i0ii dTCTHdTCTHHH

T

Tpi0ii

0i

dTCHH(1.35)

0XFHdTCFWQ 0ARx

n

1i

T

Tpii0AS

0i

Introducing eq. (1.35) into eq. (1.30) yields:

(1.36)

The heat of reaction at temperature T is given in eq. (1.28):

THTHbTHcTHdH ABCDRx (1.28)

where the enthalpy of each species is given by eq. (1.33):

T

TpiR

0

ii

R

dTCTHH (1.33)

If we now substitute for the enthalpy of each species, we have

R

0

AR

0

BR

0

CR

0

DRx THTbHTcHTdHH

T

TpApBpCpD

R

dTCbCcCdC (1.37)

The first set of terms on the right-hand side of eq. (1.37) is the heat of reaction at the reference temperature TR,

R

0

AR

0

BR

0

CR

0

DR

0

Rx THTHbTHcTHdTH (1.38)

The second term in brackets on the right-hand side of eq. (1.37) is the overall change in the heat capacity per mole of A reacted, Cp,

pApBpCpDp CbCcCdCC (1.39)

Combining Equations (1.38), (1.39), and (1.37) gives us

T

TpR

0

RxRx

R

dTCTHTH (1.40)

The heat flow to the reactor, Q , is given in terms of the overall heat-transfer coefficient, U, the heat-exchange area, A, and the difference between the ambient temperature, Ta, and the reaction temperature, T.

When the heat flow vanes along the length of the reactor, such as the case in a tubular flow reactor, we must integrate the heat flux equation along the length of the reactor to obtain the total heat added to the reactor,

V

0a

A

0a dVTTUadATTUQ

where a is the heat-exchange area per unit volume of reactor.

(1.41)

The variation in heat added along the reactor length (i.e., volume) is found by differentiating with respect to V:

TTUadV

Qda

(1.42)

For a tubular reactor of diameter D, a = D/4

For a packed-bed reactor, we can write eq. (1.43) in terms of catalyst weight by simply dividing by the bulk catalyst density

TTUa

dV

Qd1a

BB

(1.43)

Recalling dW = B dV, then

TTUa

dW

Qda

B

Substituting eq. (1.40) into eq. (1.36), the steady-state energybalance becomes

0XFdTCTHdTCFWQ 0A

T

TpR

0

Rx

n

1i

T

Tpii0AS

R0i

(1.44)

For constant of mean heat capacity:

n

1i0ipii0A0ARpR

0

RxS TTCFXFTTCTHWQ

(1.45)

EXAMPLE 1.1

Calculate the heat of reaction for the synthesis of ammonia from hydrogen and nitrogen at 150°C in kcal/mol of N2

reacted.

SOLUTION

Reaction: N2 + 3H2 2NH3

R

0

NR

0

HR

0

NHR

0

Rx THTH3TH2TH223

= 2 (– 11.02) – 3 (0) – 0 = – 20.04 kcal/mol N2

K.Hmolcal992.6C 2p2H

K.Nmolcal984.6C 2p2N

K.NHmolcal92.8C 3p3NH

2N3H3NH pppp CC3C2C

= 2 (8.92) – 3 (6.992) – 6.982

= – 10.12 cal/mol N2 reacted . K

RpR

0

RxRx TTCTHTH

29842312.1004.22423HRx

= – 23.21 kcal/mol N2