The mathematical mold of the chemical processes represents an of import job for both the design phase and for the operation of the chemical and petrochemical workss. Among the chemical processes there is besides the shell-and-tube package heat money changer. Worldwide, there are avaible systems of chemical procedures simulation plans, including the heat money changers [ 1, 2 ] . These simulation plans treat globally the operation of the heat money changer, concentrating on the dimensioning of the heat transportation country in disfavour of the analysis of the operation of some already designed money changer. In this state of affairs, the writer has investigated the possibility of the simulation of the operation of the designed heat money changer, both for look intoing it and particularly for the undermentioned simulation of the control systems that have within the procedure construction a shell-and-tube package heat money changer [ 3, 4, 5 ] .

## The construction of the shell-and-tube package heat money changers

In figure 1-a there is presented a shell-and-tube package heat money changer, holding fluxes in counter flow. This heat money changer is characterized by four input and two end product variables, figure 1-b [ 5 ] . The input variables are the undermentioned: Th1, Qhot – the recess temperature and the hot fluid flow rate, Tcl, Qcold – the recess temperature and the cold fluid flow rate. The end product variables are represented by Th2 – the outlet temperature of the hot fluid and Tc2 – the outlet temperature of the cold fluid.

Fig. 1. The shell-and-tube package heat money changer: a ) cross subdivision country ; b ) the procedure block diagram.

## The mathematical mold of the heat money changer

The mathematical mold of the heat money changer presented in figure 1 has every bit chief mark the numerical concretion of the values associated to the end product variables when the input variables and exchanger geometry are know. Within the research activity, the writer has identified the following mold phases [ 3, 5 ] :

the mathematical mold of the heat transportation inside the tubings ;

the mathematical mold of the transportation in the shell ;

the planetary mathematical mold of the money changer heat transportation.

In order to put the planetary mathematical theoretical account of the heat money changer there is necessary the designation of the flows inside and outside the tubings. The mathematical theoretical account is developed harmonizing to the hypothesis that hot flow circulates outside the tubings, as place indexes out receive the value hot and the cold flow circulates inside the tubings, as the place indexes in receive the value cold.

The mathematical theoretical account of the heat money changer is defined by the non-linear equations system

. ( 1 )

From the mathematical point of position, the system ( 1 ) represents a system of two non-linear equations

. ( 2 )

The variables of the system ( 2 ) are the mercantile establishment hot fluid temperature and the mercantile establishment money changer cold fluid. The concrete looks of the maps f1 and f2 are:

; ( 3 )

. ( 4 )

The system of non-linear equations ( 1 ) can be solved utilizing the Newton-Raphson algorithm, where the Jacobean system has the undermentioned looks:

; ( 5 )

; ( 6 )

; ( 7 )

. ( 8 )

## The version of the mathematical theoretical account

The version of the mathematical theoretical account means the concrete specification of the hot fluid belongingss, of the cold fluid belongingss, every bit good as of the geometrical features of the heat money changer. Within the achieved survey, there has been chosen a heat money changer presented in [ 6 ] . Harmonizing to the quoted beginning, the heat exchange takes topographic point between the hot fluid ( the kerosine ) , that circulates in the money changer shell, and the cold fluid ( the petroleum oil ) , that circulates in the tubing. The belongingss of the cold fluid, the petroleum oil, and of the hot fluid, the kerosine, are presented in tabular arraies 1 and table 2.

Table 1. The belongingss of the cold fluid ( circulation inside the tubings )

## Variable

## Significance

## Measure units

## Value

Flow rate inside the tubings

50000

Fluid denseness inside the tubings

820

Fluid specific heat inside the tubings

2239

Fluid heat conduction inside the tubings

0.127

Fluid kinematic viscousness inside the tubings

Tc1

Inlet temperature of ( cold ) fluid in the tubings

& A ; deg ; C

103

Table 2. The belongingss of the hot fluid ( circulation outside the tubings )

## Variable

## Significance

## Measure units

## Value

Fluid flow rate outside tubings

163000

Fluid denseness outside tubings

660

Fluid specific heat outside tubings

2602

Fluid heat conduction outside tubings

0.1364

Fluid kinetic viscousness outside tubings

Th1

Inlet temperature fluid ( cold ) in shell

& A ; deg ; C

180

The geometrical features and the values of some parametric quantities of the heat money changer are presented in table 3 and 4.

Table 3. The heat features associated to the heat money changer

## Variable

## Significance

## Measure units

## Value

Tube heat conduction ( tubings of C steel )

40

Specific heat opposition of the sedimentation inside tubings

0.0011

Specific heat opposition of the sedimentation outside tubings

0.0004

Table 4. The geometrical features of the heat money changer

## Variable

## Significance

## Measure units

## Value

Liter

Tube length

m

6

Number of base on ballss of tubings

## –

2

Number of tubings

## –

900

Number of tubings in window

## –

112

The interior diameter of tubings

millimeter

20

The exterior diameter of tubings

millimeter

25

Shell diameter

m

1.1

Window diameter

m

1.06

Cavil diameter

m

1.095

ten

Distance between quibbles

m

0.4

s

Side of equilateral trigon of the tubings

millimeter

32

Holes diameter

m

0.026

The angle at the centre of the chord of quibble

## & A ; deg ;

106

Number of braces of scaling longitudinal quibbles

## –

2

Number of the tubes rows placed between the Windowss

## –

24

H

Cavil tallness

m

0.88

## The simulation of the heat money changer utilizing Unisim plan

The plan Unisim Shell Tube Exchanger Modeler R380 is used to patterning and to imitating the shell-and-tube package heat money changers. The most of import constructive categorization of the heat money changer with shell and tubing has proposed by Tubular Exchanger Manufacturers Association ( abbreviation TEMA ) [ 7 ] . This categorization uses the undermentioned standards [ 8 ] :

the front terminal caput building ;

the circulation type of the watercourse between the tubings and shell ;

the type of the front terminal caput.

The writer has studied the chief installations of the Unisim Shell Tube Exchanger Modeler R380 plan and has identified the undermentioned concretion phases:

Choose the Simulation map of Start Up subdivision.

Choose the geometrical specifications of the heat money changer in the Exchanger General subdivision. The heat money changer has the undermentioned features: the front terminal caput type is demountable ( TEMA A ) , the shell type has two tubing base on balls into shell ( TEMA F ) , the rear terminal caput type with demountable nomadic caput ( TEMA S ) , the shell orientation is horizontal and the side for hot watercourse is the shell side. An image of this phase is presented in figure 2.

Choose the subdivision Tube Details for specification of the geometrically tubes features.

Choose the subdivision Transverse Baffles for specification of the geometrically features of the baffles. The flow subdivision between the baffle and the shell is calculated utilizing the dealingss presented in table 5.

The features of the cold and the hot watercourse are specificities into Physical Proprieties subdivision.

Fig. 2. The geometrical specifications of the heat money changer.

Table 5. The expressions used for the flow subdivision between the baffle and the shell

Variable

Formula

Circle country

Area of the circle section with angle

Triangle country

Flow subdivision country

Flow subdivision per centum

## Numeric consequences

The writer has simulate the heat transportation trough shell and tubing heat money changer utilizing two ways: first manner is dedicated to work out the mathematical theoretical account ( 1 ) and 2nd manner contains the heat money changer simulation utilizing the Unisim Shell Tube Exchanger Modeler R380.

For solve the mathematical theoretical account ( 1 ) , the writer has lucubrate a specially plan, which use the Newton – Raphson algorithm for work outing the non-linear equations systems [ 9 ] . There has implemented two versions of simulation plans: one version use the analytically Jacobean matrix and the 2nd version use the numerically Jacobean matrix rating [ 4 ] . In table 6 there are presented relatively the consequences obtained for the resolution of the mathematical theoretical account of the heat money changer by agencies of the two algorithms.

Table 6. The Newton – Raphson comparative consequences

## Iteration

## Equation index

## Newton-Raphson algorithm based on analytical derived functions

## Newton-Raphson based on numerical derived functions

0

1

1.3701357466

7.0651772244E+05

1.3701357466

7.0651772244E+05

2

1.1701357466

2.5760903814E+05

1.1701357466

2.5760903814E+05

1

1

1.3786567675

0.0000000000E+00

1.3880450337

-2.7160644531E-03

2

1.1896271727

1.8637047361E+05

1.1860703994

2.3748612976E+03

2

1

1.3837998035

0.0000000000E+00

2

1.1876787178

8.5532275970E+04

The 2nd manner to imitate the heat money changer simulation has used the Unisim Shell Tube Exchanger Modeler R380. The consequences obtained with this simulation plan are presented in figure 3.

Fig. 3. The numerical consequences obtained by Unisim Shell Tube Exchanger Modeler R380

In table 7 are presented the comparative end product temperatures of the heat money changer. There are three value beginnings:

the original illustration, presented in [ 6 ] ;

the consequences obtained by work outing the mathematical theoretical account ( 1 ) ;

the consequences obtained by use the Unisim Shell Tube Exchanger Modeler R380 simulation plan.

Thesiss consequences validate the mathematical theoretical account proposed by the writer and the detailed simulation plan. In future, the mathematical theoretical account will be used to imitate the control systems what contain the heat money changer.

Table 7. The comparative consequences of the heat money changer simulation

## Simulation consequence

## Outlet hot temperature [ & A ; deg ; C ]

## Outlet cold temperature [ & A ; deg ; C ]

Original illustration [ 6 ]

140.0

118.0

Simulation on the mathematical theoretical account ( 1 )

138.4

118.7

UNISIM simulation

131.3

121.5