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The mathematical modeling of the chemical processes represents an important problem for both the design stage and for the operation of the chemical and petrochemical plants. Among the chemical processes there is also the shell-and-tube bundle heat exchanger. Worldwide, there are avaible systems of chemical processes simulation programs, including the heat exchangers [1, 2]. These simulation programs treat globally the operation of the heat exchanger, focusing on the dimensioning of the heat transfer area in disfavor of the analysis of the operation of some already designed exchanger. In this situation, the author has investigated the possibility of the simulation of the operation of the designed heat exchanger, both for checking it and especially for the following simulation of the control systems that have within the process structure a shell-and-tube bundle heat exchanger [3, 4, 5].
The structure of the shell-and-tube bundle heat exchangers
In figure 1-a there is presented a shell-and-tube bundle heat exchanger, having fluxes in counter flow. This heat exchanger is characterized by four input and two output variables, figure 1-b . The input variables are the following: Th1, Qhot - the inlet temperature and the hot fluid flow rate, Tcl, Qcold - the inlet temperature and the cold fluid flow rate. The output 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 bundle heat exchanger: a) cross section area; b) the process block diagram.
The mathematical modeling of the heat exchanger
The mathematical modeling of the heat exchanger presented in figure 1 has as main target the numerical calculus of the values associated to the output variables when the input variables and exchanger geometry are know. Within the research activity, the author has identified the following modeling stages [3, 5]:
the mathematical modeling of the heat transfer inside the tubes;
the mathematical modeling of the transfer in the shell;
the global mathematical modeling of the exchanger heat transfer.
In order to set the global mathematical model of the heat exchanger there is necessary the identification of the flows inside and outside the tubes. The mathematical model is developed according to the hypothesis that hot flow circulates outside the tubes, as position indexes out receive the value hot and the cold flow circulates inside the tubes, as the position indexes in receive the value cold.
The mathematical model of the heat exchanger is defined by the non-linear equations system
From the mathematical point of view, the system (1) represents a system of two non-linear equations
The variables of the system (2) are the outlet hot fluid temperature and the outlet exchanger cold fluid . The concrete expressions of the functions f1 and f2 are:
The system of non-linear equations (1) can be solved using the Newton-Raphson algorithm, where the Jacobean system has the following expressions:
The adaptation of the mathematical model
The adaptation of the mathematical model means the concrete specification of the hot fluid properties, of the cold fluid properties, as well as of the geometrical characteristics of the heat exchanger. Within the achieved study, there has been chosen a heat exchanger presented in . According to the quoted source, the heat exchange takes place between the hot fluid (the kerosene), that circulates in the exchanger shell, and the cold fluid (the crude oil), that circulates in the tubes. The properties of the cold fluid, the crude oil, and of the hot fluid, the kerosene, are presented in tables 1 and table 2.
Table 1. The properties of the cold fluid (circulation inside the tubes)
Flow rate inside the tubes
Fluid density inside the tubes
Fluid specific heat inside the tubes
Fluid heat conductivity inside the tubes
Fluid kinematic viscosity inside the tubes
Inlet temperature of (cold) fluid in the tubes
Table 2. The properties of the hot fluid (circulation outside the tubes)
Fluid flow rate outside tubes
Fluid density outside tubes
Fluid specific heat outside tubes
Fluid heat conductivity outside tubes
Fluid kinetic viscosity outside tubes
Inlet temperature fluid (cold) in shell
The geometrical characteristics and the values of some parameters of the heat exchanger are presented in table 3 and 4.
Table 3. The heat characteristics associated to the heat exchanger
Tube heat conductivity (tubes of carbon steel)
Specific heat resistance of the deposit inside tubes
Specific heat resistance of the deposit outside tubes
Table 4. The geometrical characteristics of the heat exchanger
Number of passes of tubes
Number of tubes
Number of tubes in window
The interior diameter of tubes
The exterior diameter of tubes
Distance between cavils
Side of equilateral triangle of the tubes
The angle at the center of the chord of cavil
Number of pairs of scaling longitudinal cavils
Number of the tubes rows placed between the windows
The simulation of the heat exchanger using Unisim program
The program Unisim Shell Tube Exchanger Modeler R380 is used to modeling and to simulating the shell-and-tube bundle heat exchangers. The most important constructive classification of the heat exchanger with shell and tube has proposed by Tubular Exchanger Manufacturers Association (abbreviation TEMA) . This classification uses the following criterions :
the front end head construction;
the circulation type of the stream between the tubes and shell;
the type of the front end head.
The author has studied the main facilities of the Unisim Shell Tube Exchanger Modeler R380 program and has identified the following calculus stages:
Select the Simulation function of Start Up section.
Select the geometrical specifications of the heat exchanger in the Exchanger General section. The heat exchanger has the following characteristics: the front end head type is demountable (TEMA A), the shell type has two tube pass into shell (TEMA F), the rear end head type with demountable mobile head (TEMA S), the shell orientation is horizontal and the side for hot stream is the shell side. An image of this stage is presented in figure 2.
Select the section Tube Details for specification of the geometrically tubes characteristics.
Select the section Transverse Baffles for specification of the geometrically characteristics of the baffles. The flow section between the baffle and the shell is calculated using the relations presented in table 5.
The characteristics of the cold and the hot stream are specificities into Physical Proprieties section.
Fig. 2. The geometrical specifications of the heat exchanger.
Table 5. The formulas used for the flow section between the baffle and the shell
Area of the circle segment with angle
Flow section area
Flow section percent
The author has simulate the heat transfer trough shell and tube heat exchanger using two ways: first way is dedicated to solve the mathematical model (1) and second way contains the heat exchanger simulation using the Unisim Shell Tube Exchanger Modeler R380.
For solve the mathematical model (1), the author has elaborate a specially program, which use the Newton - Raphson algorithm for solving the non-linear equations systems . There has implemented two versions of simulation programs: one version use the analytically Jacobean matrix and the second version use the numerically Jacobean matrix evaluation . In table 6 there are presented comparatively the results obtained for the solving of the mathematical model of the heat exchanger by means of the two algorithms.
Table 6. The Newton - Raphson comparative results
Newton-Raphson algorithm based on analytical derivatives
Newton-Raphson based on numerical derivatives
The second way to simulate the heat exchanger simulation has used the Unisim Shell Tube Exchanger Modeler R380. The results obtained with this simulation program are presented in figure 3.
Fig. 3. The numerical results obtained by Unisim Shell Tube Exchanger Modeler R380
In table 7 are presented the comparative output temperatures of the heat exchanger. There are three value sources:
the original example, presented in ;
the results obtained by solve the mathematical model (1);
the results obtained by use the Unisim Shell Tube Exchanger Modeler R380 simulation program.
Theses results validate the mathematical model proposed by the author and the elaborated simulation program. In future, the mathematical model will be used to simulate the control systems what contain the heat exchanger.
Table 7. The comparative results of the heat exchanger simulation
Outlet hot temperature [°C]
Outlet cold temperature [°C]
Original example 
Simulation on the mathematical model (1)