1

Steady state simulation of the Linde-Hampson Liquefaction

Process by sequential modular approach

Figure 1: Linde-Hampson liquefaction process (Open-cycle, without a compressor).

1.1. Problem definition

The Linde-Hampson liquefaction process shown in Fig. 1 was invented by Linde in Germany and

Hampson in England more than 100 years ago. The process was first used to liquefy air. The process is

used even today for cooling small cooling infra-red detectors on missiles. High pressure nitrogen is

supplied in a gas bottle (state point 2), and the unliquefied nitrogen is exhausted into atmosphere at

state point 5. The liquid product is taken out at state point f. The main aim of this exercise is to simulate

a Linde-Hampson air liquefaction process to determine (a) the flow rate of the product and (b) the

composition of the liquid product for a given feed (Pressure = 200 bar, temperature = 300 K, air

composition: nitrogen – 79 mol%, oxygen – 21 mol%).

1.2. Stream variables

It is important to understand all the governing equations before the process is simulated. Consider any

stream S. Every stream is defined by (a) flow rate (b) pressure (c) temperature (d) enthalpy (e) entropy

and (f) concentration of ‘n’ components in the mixture. While enthalpy and entropy can be determined

from pressure, temperature and concentration, it is convenient to define enthalpy and entropy as

stream variables since all unit operations involve enthalpy and entropy change.

S. No. Variable name Number of variables

1 Flow rate 1

2 Pressure 1

3 Temperature 1

4 Enthalpy 1

5 Entropy 1

6 Concentration n

Total Variables n+5

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1.3. Governing equations of different equipment of the process shown in

Fig. 1

Let us consider the sequential modular approach for the simulation of the process. In this approach the

input streams to the equipment are fully specified along with the design/operating parameters. The

output stream conditions are determined by solving appropriate governing equations.

The governing equations of different equipment are listed below. In this example, the use of a

regenerative heat exchanger results in a recycle. Therefore the stream g is torn as shown in Fig. 2. The

pressure, temperature, flow rate and concentration of the torn stream (g’) are assumed initially.

Figure 2: Linde-Hampson liquefaction process showing tear stream ‘g’.

1.4. Heat Exchanger

The following are fully specified:

1. Two input stream conditions (stream 2, g’)

2. One of the following operating/design parameters:

Effectiveness of heat exchanger

Temperature approach between streams at warm or cold end or even minimum

temperature approach between streams

Temperature drop or increase of any stream

Heat exchanger effectiveness

1.1.1. Heat exchanger effectiveness

The heat exchanger effectiveness is defined as follows:

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(1)

Case 1 : (T2 – T3) ≥ (T5 – Tg) ε = 1

32

32

hh

hh (2a)

Case 2 : (T2 – T3) < (T5 – Tg) ε = '

1

2

'5

g

g

hh

hh (2b)

Where h3‘ = h(P3, Tg’ , g’i) and h2

‘ = h(P5, T2 , 5i)

The heat exchanger effectiveness can be expressed in terms of temperature only when the specific

heat of the fluid is independent of temperature. Since specific heat varies with temperature for most

fluids, Eqs. (2a) and (2b) should be used for calculating the effectiveness of heat exchangers. Equations

(2a) and (2b), however, cannot be used when internal pinch points occur. Pinch points are common in

heat exchangers in which fluids undergo phase change. What to do when pinch points occur will be

taken up in the following classes. The main assumption while using effectiveness as the design

parameter is that there are no internal pinch-points in heat exchangers. In the present example, we

consider a heat exchanger with specified heat exchanger effectiveness.

1.1.2. Overall mass balance

The mass flow rate of the streams does not change along the length of the heat exchanger during steady

state operation. The mass flow rate of any stream entering and leaving may not be the same during

transient operation such as cool down because of increase/decrease in the liquid holdup in the heat

exchanger. In this exercise only the steady state case is considered.

(3)

(4)

1.1.3. Component mass balance:

The component balance equations during steady state can be written as follows:

ξ3i = ξ2i (i = 1 to n) (5)

ξ5i = ξg’i ( i = 1 to n) (6)

In the above equations, i represents the mole fraction of the ith component of the mixture with n

components. As with the overall mass balance, the component balance can be different at inlet and

outlet during a transient operation such as cool down. For example, during condensation of a mixture,

the concentration of the high boiling components in the liquid holdup in the heat exchanger increases,

therefore the high boiling components at the outlet can be less than that at inlet for a process

undergoing condensation.

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1.1.4. Overall energy balance:

The overall energy balance over the heat exchanger is given by the expression:

(7)

Governing equations (2) and (3) can be used to determine the temperature of the outlet streams,

provided p3 , p5 are known. Normally the pressure drop for each of the hot and cold streams ( ph, pc) is

normally specified.

p3 = p2 – ph (8)

p5 = pg’ – pc (9)

1.1.5. Enthalpy and temperature of the two streams 3, 5

The temperature of the heat exchanger output streams (streams 3, 5) can be determined from the

pressure, enthalpy and concentration of the fluid, as shown in Eq. (9) using the equation of state and the

variation of ideal gas specific heat of the component fluids with temperature, using an enthalpy-

pressure flash routine. For the sake of simplicity the enthalpy-flash routine is considered as a function ft.

Similarly, the entropy of a fluid can be determined using the equation of state and the variation of ideal

gas specific heat with temperature. This procedure is considered as function fs. Similarly, the

determination of enthalpy from given pressure, temperature and concentration is termed as function fh.

The temperature, enthalpy and entropy of the two outlet streams 3, 5 can be expressed using Eqs. (2) to

(5) as follows:

Case-1 : (T2 – T3) ≥ (T5 – Tg)

h3 = h2 - ε(h2 - h3’) (10)

t3 = ft(p3 , h3 , ξ3i) (11)

(12)

t5 = ft(p5 , h5 , ξ5i) (13)

Case 2 : (T2 – T3) < (T5 – Tg)

h5 = hg’ + ε(h2’-hg’) (14)

t5= f(p5 , h5 , ξ5i) (15)

(16)

t3 = ft (p3 , h3 , ξ3i) (17)

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1.1.6. Entropy of the two outlet streams of the heat exchanger (streams 3, 5)

s3 = fs(p3 , t3 , ξ3i) (18)

s5 = fs(p5 , t5 , ξ5i) (19)

In the case of sequential modular or simultaneous modular approach, the governing equations of each

the equipment are solved separately. In the case of equation oriented approach the governing

equations of all the equipment are solved together. The present example uses the sequential modular

approach. The governing equations of the output streams of the heat exchanger (streams 3, 5) are

summarized below:

Table 1: Equations for determining the variables of the outlet stream of the heat exchanger (streams 3

and 5)

Variable Stream 3 Stream 5 No. of equations per

stream

Flowrate ( Eq. (3) Eq. (4) 1

Pressure (p) Eq. (8) Eq. (9) 1

Temperature (t) Eq. (11) (case-1)

Eq. (17) (case-2)

Eq. (13) (case-1)

Eq. (15) (case-2)

1

Enthalpy (h) Eq. (10) (case-1)

Eq. (16) (case-2)

Eq. (12) (case-1)

Eq. (14) (case-2)

1

Entropy (s) Eq. (18) Eq. (19) 1

Concentration( i i = 1,n) Eqs. (5) Eqs. (6) n

The number of governing equations is the same as the number of variables of the output streams (n+5

per stream, as shown in Sec. 1.1). The governing equations can therefore be solved once the

design/operating specifications namely effectiveness of heat exchanger and pressure drop for each

stream are specified. The governing equations of a two stream heat exchanger are not much different

from those shown in Table 1 when the temperature approach between the streams or the temperature

change of any stream are specified instead of the effectiveness.

1.5. Expansion valve

The expansion process in the valve can be treated as a constant enthalpy process (the kinetic energy

change is normally very small). The governing equations of the valve can therefore be written as follows:

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Governing equations:

Constant enthalpy process h4 = h3 (20)

Overall mass balance (21)

Component balance (22)

Temperature t4 = ft(p4 , h4 , ξ4i) (23)

Entropy s4 = fs(p4 , t4 , ξ4i) (24)

The temperature and entropy of the outlet stream of the valve (Eqs. (23) and (24)) can be determined

only when the valve outlet pressure p4 is specified. The valve outlet pressure is normally the

design/operating parameter of a valve. Alternately, one can specify the pressure drop across the valve.

1.6. Phase separator

Consider the phase separator shown in Fig. 2. The working fluid (stream 4) enters the phase separator in

a two-phase (liquid-vapour) condition. The job of the phase separator is to separate the inlet stream

into separate vapour and liquid streams. The governing equations of the phase separator can be

obtained by mass and energy balance as given below. Since the vapour and liquid phases will be in

equilibrium, the fugacity (gibbs free energy) of the different components of the mixture must be the

same in the vapour and liquid phases. The temperature of the coexisting phases will be the same.

Governing equations:

Overall mass balance (25)

Component mass balance (i = 1,n) (26)

Enthalpy balance (27)

Overall mole fraction of stream f

(28)

Overall mole fraction of stream g

(29)

Equal fugacity of coexisting phases (i = 1,n) (30)

Enthalpy of stream f hf = fh(pf, tf, fi) (31)

Enthalpy of stream f hg = fh(pg, tg, gi) (32)

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Entropy of stream f sf = fs(pf, tf, fi) (33)

Entropy of stream f sg = fs(pg, tg, gi) (34)

Temperature of coexisting phases tg = tf (35)

Equations (25) to (35) fully describe a phase-separator shown in Fig. 2. The number of equations is 2n+9,

while the total variables of the two output streams f and g are 2n+10 (see section 1.1). Therefore one

more equation is required. The difference between the number of variables and number of equations is

also known in literature as the degrees of freedom. In the above case the degree of freedom is one. The

number of variables to be specified as operating/design parameters is the same as the number of

degrees of freedom.

Different types of equipment can be used to separate gas and liquid phases. The most common of them

is the cyclone separator shown in Fig. 3. Normally the pressure at the exit of the cyclone separator is

lower than that at inlet due to expansion and contraction losses. In other words, p4 ≥ pf or pg in the

phase separator shown in Fig. 2.. In an ideal phase separator, the pressure at the exit of the phase

separator is the same as that at the inlet, or the pressure drop is zero. Normally the pressure at the exit

of the phase separator is specified or the pressure drop is assumed to be zero.

Figure 3: Cyclone separator

Different methods can be used to solve the governing equations of the phase separator (Eqs. (25) to (35)

along with the specified pressure drop). These routines are normally known as isothermal-isobaric flash

routines. Two most commonly used approaches are: (a) the Rachford-Rice approach, also known as the

inside-out approach or (b) the gibbs energy approach. The difference between the two approaches will

be taken up at a later stage. It suffices at this stage to know that the inside-out approach is generally

faster for most flow sheeting problems involving two or three phases.

1.7. Convergence block for converging streams g” and g’

In the sequential modular approach the stream g is torn and stream g’ assumed as shown in Fig. 2,

because of the recycle. After solving the phase separator block and determining all the variables of

stream g (flowrate, pressure, temperature, enthalpy, entropy, and concentration), a convergence block

is used to modify the parameters of stream g’ such that all parameters of the calculated stream

Feed

Vapour

Liquid

8

(stream g”) are identical to that of the assumed stream (stream g’). The governing equations of the

convergence block are given below:

Flowrate (36)

Pressure pg’ = pg” (37)

Temperature tg’ = tg” (38)

Enthalpy hg’ = hg” (39)

Entropy sg’ = sg” (40)

Concentration g’i = g”i (41)

Either Newton-Raphson method or Wegstein method can be used to update stream variables of stream

g’. Please refer to any numerical methods book (say, numerical recipes in C or Fortran,

http://www.nr.com) to understand the Wegstein method.

2. Solution methodology Figure 4 shows the method adopted for solving the flowsheet using the sequential modular approach. In

this approach the equipment modules are solved one at a time sequentially till convergence of the tear

streams is achieved. When there is no convergence, the variables of the torn stream (g’) are reassumed

to achieve convergence.

The governing equations described above (Eqs. (2) to (35)) for the sequential modular (SM) approach

are the same as that for the equation oriented (EO) approach, except that in the case of the heat

exchanger the variables of g are used instead of stream g’ in Eqs. (2) to (19) since no stream is torn in

EO approach. In the case of EO approach, the governing equations (2) to (35) are solved simultaneously.

Since these are non-linear equations, one needs an initial estimate to solve the routines.

Consider the Newton – Raphson method to solve a non-linear equation in one variable, f(x) = 0 for x. In

the Newton approach, the variable x is updated in the i+1th iteration using the ith iteration values using

the following expression:

The x value is updated till | Xi+1 - XI| <= ε, a specified error. Newton-Raphson method can also be used

with a set of non-linear equations. In the case of the EO approach, we need a starting value for all the

stream variables (n,P,T,h,s, i). It is therefore customary to start EO with a sequential modular step, and

then solve them with EO after obtaining an initial estimate.

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Figure 4: Solution methodology adopted to solve the flowsheet shown in Fig. 2 using sequential modular

approach