## Brazilian Journal of Chemical Engineering

*Print version* ISSN 0104-6632

### Braz. J. Chem. Eng. vol.17 n.1 São Paulo Mar. 2000

#### http://dx.doi.org/10.1590/S0104-66322000000100002

**Fluid dynamical considerations on heat exchanger networks**

^{ }

**A. J. M. Vieira , F. L. P. Pessoa And E. M. Queiroz ^{* } **Universidade Federal do Rio de Janeiro (UFRJ), Departamento de Engenharia Química,

Escola de Química, Centro de Tecnologia, Bloco E-209, Cidade Universitária, CEP 21949-900,

Phone: (55) (021) 590-3192, Fax: (55) (021)590-4991, Rio de Janeiro - RJ, Brasil

Email: mach@h2o.eq.ufr.br

*(Received: April 6, 1999; Accepted: August 1 ,1999)*

Abstract– The synthesis and analysis of heat exchanger networks are issues of great industrial interest due to the possibilities of decreasing plant costs, through the reduction of the utilities consumption and/or the number of equipments, in a grassroot design or retrofitting an existent network. The present paper explores a new design algorithm based on the Total Annual Cost (TAC) optimization for a thermal equipment, with mean tubeside and shellside flow velocities constraints, studying also the influence of pumping cost in the network’s final cost.

Keywords: heat exchanger network, pressure drop optimization, heat exchanger design.

INTRODUCTION

Traditionally, Heat Exchanger Network (HEN) synthesis involves only the definition of the combination streams, performing a simplified thermal design, with assumed countercurrent configuration and pre-specified heat transfer coefficients. It is not taken into account any kind of information relative to the streams pressure drops, which are considered as constraints to be satisfied during the design of the equipments later stage. Such procedure usually leads to networks with equipments of unviable construction.

Recently, some authors have begun to take into account the influence of the equipment design during the network synthesis.

Jegede and Polley (1992) present relationships among the specified tubeside and shellside pressure drops for a heat exchanger, the correspondent heat transfer coefficients and the exchanger area, therefore showing the existence of a trade-off between pumping and capital costs. It’s worth to note that the correlations proposed by Kern (1950) are used for shellside flow description, although nowadays they are considered to give poor shellside pressure drop predictions. Nevertheless, the methodology proposed, in theory, can be adapted to any correlation between the shellside pressure drop and heat transfer coefficient.

Ravagnani (1994) uses the Pinch Technology (Linnhoff and Hindmarsh, 1983) to synthesize a HEN that satisfies the minimum utility consumption, followed by loop breaking as a means to reduce the number of units. Each exchanger in the proposed HEN is then designed by checking geometrical configuration tables, top to bottom, for a combination that fulfills the thermal and fluiddynamical constraints (as for a pre-specified pressure drop), stopping on the very first one that accomplishes both. This sequential procedure neglects the possibility of other viable, optimum design. Moreover, all synthesis effort is wasted if only one unit cannot fulfill its design requirements.

Oliveira (1995) develops a program in which the network synthesis follows heuristical rules and the equipment design is accomplished simultaneously, step by step, based on the methodology proposed by Jegede and Polley (1992). The program selects the stream pair for exchanging heat and immediately the thermofluiddynamical relationships are used to design the equipment, from specified shellside and tubeside pressure drops. The method is adapted for designing heat exchangers with phase change in the hot stream, flowing in the shellside with negligible mean velocity (e.g., saturated vapor). In case of a combination with a strong temperature crossing, it is performed the division of the equipment in a series of shells, all with the same area. However, the proposed method indeed doesn’t test if the results were in accordance with some design constraints, such as the values of the mean flow velocities.

The procedures adopted by the preceding authors don’t take into account the pumping cost in the network’s cost optimization. Moreover, some analyses of the proposed units in their networks show violations of the recommended velocities range. Thus, this work discusses the thermal equipment cost optimization with constraints in the mean shellside and tubeside flow velocities and the pumping cost influence in the network’s final cost.

COSTS

Before defining the *TAC* optimization procedure for the thermal equipment design, it’s worth to note the following about the costs calculated in the present paper.

Installed thermal equipment cost (*Cequip*)

Calculated according to the relationships found in Douglas (1988), as a function of the heat transfer area. When, due to existence of strong temperature crossing (indicated through the correction factor *F* < 0.75), the shell division is necessary, it’s made generating shells with equal areas. In this case, *Cequip* will be considered for the group of shells, calculated by the summation of the costs of each shell.

Utility cost (*Cutil*)

It is considered that the pumping cost is included in the respective utility cost, that is, the pumping costs are considered only in the process streams.

Pumping cost (*Cpump*)

Includes the electric energy cost and the cost of the pumps used to impel the streams. Centrifugal pumps are located upstream relative to each heat exchanger, supplying the mechanical energy individually for each equipment, except when it is necessary shell division. In this case, the pump is located upstream of the first shell of the series, allowing treating the set as only one heat exchanger with multiple passages.

It is considered the use of centrifugal pumps with efficiency (*NI*) of 80% and price FOB, including motor, given by Hall et al. (1988). The installation factor (*IFA*) is considered equal to 1.5, as recommended by Perry (1984).

The annualization of *Cequip* and *Cpump* is accomplished through the application of the Capital Charge Factor (*CCF*) - considered 0.333 year^{-1}. This value implies in a capital rate of return of 15% in 11 years.

The thermal equipment *TAC* is defined as the summation of *Cequip*, *Cutil* and *Cpump*. Finally, the total network cost (*TNC*) is calculated through the summation of all *TAC*.

TOTAL ANNUAL COST OPTIMIZATION

The optimization of a thermal equipment cost, without phase change in the streams, can be formulated starting from the thermofluiddynamical relationships proposed by Jegede and Polley (1992). Together with the traditional equation for the heat transfer area (*A*) as a function of the thermal resistances through the heat transfer process, these equations relate shellside and tubeside pressure drops (D*p _{s}* and D

*p*) to the heat transfer coefficients (

_{t}*h*and

_{s}*h*) and to the mean flow velocities (

_{t}*v*and

_{s}*v*), in the respective streams.

_{t}Employing these relationships, together with the cost equations, it is possible to write an objective function - the thermal equipment *TAC* - describing a trade-off between the thermal equipment and the pumping costs (Equation 1). In this expression, the mean flow velocities are the variables and the physical properties of the streams and their temperatures, the tube inside and outside diameters (*D _{i}* and

*D*), the tube pitch (

_{o}*Pt*) and its bundle layout are specified and assumed constant. It’s not necessary to specify the tube length (

*l*). Once specified the combination pair and its thermal load (

*Q*), the term correspondent to the utility cost is constant and then it can be removed from the objective function.

Subjected to:

(3)

In Equations 2 and 3, *M&S* is the Marshall & Swift cost index, *Fo* is an operational factor, r is the fluid density and *m* is its mass flowrate. The correction factor *Fc* is defined in Douglas (1988), in order to consider cost differences relative to the head type, design pressure and construction materials.

Equations 4 to 8 represent the thermofluiddynamical relationships, where *Kp _{s}*,

*Kp*,

_{t}*K*and

_{t}*K*are constants calculated as functions of the tube diameters (

_{s}*D*and

_{i}*D*), tube pitch (

_{o}*Pt*) and tube pass (

*npt*), with constant stream physical properties (Oliveira, 1995). Equation 6 is the traditional one for the global heat transfer coefficient (

*U*), where

*Rf*and

_{o}*Rf*are the fouling resistances,

_{i}*Q*is the heat load,

*F*is the correction factor of D

*Tln*, that is the logarithmic mean of the temperature differences in the extremities of the equivalent countercurrent flow configuration.

A constraint imposed to the mean flow velocities is that they should satisfy the recommendations proposed by Sinnott (1993), which are presented in Table 1. Values lower than those recommended could originate deposition problems in the tubes and greater values can cause vibration in the tube bundle and erosion.

Note that for a design involving consumption of hot utility with phase change (e.g., a heater using saturated vapor), there is only one degree of freedom - the process stream mean flow velocity – because it is assumed that the vapor condenses in the shellside with negligible mean velocity. In this case, Equations 11 and 12 should be used in substitution to Equations 5 and 8 (Oliveira, 1995), observing that Equation 12 should be solved numerically for the wall temperature (*T _{w}*).

In Equations 11 and 12, *Kaq* is a constant calculated as a function of *D _{o}*, the number of tubes vertically aligned (

*nav*) and physical properties of the changing phase fluid.

*T*is the saturated vapor temperature and

_{v}*DTcm*is the temperature of the cold stream, calculated as the logarithmic mean of the entrance and exit absolute temperatures.

The optimization is accomplished by means of routine Constr.m, available in the MATLAB^{®} optimization package (" The MathWorks, Inc "). This routine looks for the minimum of a constrained non-linear multivariable function using sequential quadratic programming. Once the optimal tubeside and shellside mean flow velocities are known, it is possible to design the heat exchanger (Vieira, 1998).

In spite of not presented in Equations 2 to 10, the ratio between tube length (*l*) and shell diameter (*Ds*) is calculated along the procedure and maintained between 4 and 15, in accordance with Bell’s (1981) recommendation. Lower values could generate equipments with a bad flow distribution in the tubes, while greater ones can cause problems of head loss in the two streams and of mechanical support of the tube bundle.

CASE STUDY

The following example consists of designing the heat exchangers of a network obtained for the problem 4SP1 by Oliveira (1995) with heuristical rules. The process streams (*C1, C2, H1, H2*) are assumed pressurized water, and the utilities are saturated vapor *(UH1)* and cooling water *(UC1), *with fixed temperatures.

The problem data, the proposed network configuration and the design accomplished by Oliveira (1995) are presented in Tables 2 to 4. All thermal equipments are designed with zero pumping cost, considering that there are available shellside and tubeside pressure drops (35 kPa each, for each unit), which are totally employed in the designed equipments.

The results presented in Table 4 show that the procedure completely uses the specified pressure drop, but does not satisfy the recommended ranges of mean flow velocities, except for the first exchanger. This example shows that this procedure, with setting pressure drops, could lead to designs in which some basic aspects don’t have the desired attention.

The algorithm proposed in the present paper allows respecting the velocities and *l/Ds* ratio constraints. Moreover, the pumping cost is included in the *TNC* calculation, making possible its analysis among other costs normally considered. In a first calculation, the problem is solved supposing that each equipment has available pressure drops, at zero cost, of 35 kPa on the shell and tube sides. The results are shown on Table 5 and can be compared with those from Table 4. Note that now the available pressure drop is not completely used in all equipments, but the velocities ranges are satisfied.

In Table 6 are presented results considering no mechanical energy available in the streams. From them, it can be made an important observation: in the network under these conditions, the annual pumping cost is about 30% of the equipment costs. This result is in agreement with "one-third rule", mentioned by Steinmeyer (1996) with respect to a single equipment. According to this heuristics, the annual pumping cost would be approximately a third of the annual equipment cost. Nevertheless, in a HEN there are other costs , mainly the utility cost. In this case, where the utility cost is about three times the equipment cost, the pumping cost is about 9% of the *TNC*.

The proposed design algorithm did not present any convergence problems, solving the example in less than 5s when running on a Pentium 166 computer with 16Mb RAM.

CONCLUSIONS

The result found for the ratio pumping cost to network cost (9% - Table 6) indicates that the pumping cost should be an important parameter in the design of heat exchanger networks. The importance of the pumping cost in HEN increases as the utilities consumption decreases.

The use of arbitrated pressure drops can originate poor designs from the fluiddynamical point of view and does not lead to the optimal exchanger design.

It’s worth to note that, although overestimating the shellside pressure drop, Kern’s method can indeed show the potential hazards of not taking into account the pumping cost in the design optimization stage.

The heat exchanger design procedure here proposed can be used simultaneously with the HEN synthesis. It also leads to better preliminary designs, in the point of view of the total costs and technical constraints, when compared to others procedures available in literature.

NOMENCLATURE

Latin Letters

*A *heat transfer area (m^{2})

*Bs* baffle space (m)

*CCF *capital cost factor

*Cel* electricity cost (US$/kW.h)

*Cequip *installed thermal equipment cost (US$/yr)

*Cp* specific heat (J/kg.^{o}C)

*Cpump* pumping cost (US$/yr)

*Cutil* utility cost (US$/yr)

*D* tube diameter (m)

*Ds *shell diameter (m)

*F *correction factor for the effective Temperature difference in configurations with multiple passages in the tubes

*Fc *correction factor for equipment cost

*Fo* operational factor or plant annual activity (s/yr)

*g* gravity acceleration (m/s^{2})

*h *heat transfer coefficient (W/m^{2}.^{o}C)

*IFA* correction factor for equipment installation

*k* thermal conductivity (W/m.^{o}C)

*K* constant for the relationship of pressure drop

*Kaq* constant for the relationship of heat transfer coefficient in a stream with phase change

*Kp* constant for the relationship of pressure drop

*l* tube length (m)

*m *mass flowrate (kg/s)

*M&S *Marshall and Swift cost index

*N* total number of tubes

*nav *number of tubes vertically aligned

*Nb* baffle number

*NI* pump efficiency

*npt* number of passages in the tubes

*Ns* number of shells in series

*Pt *tube pitch (m)

*Q *heat load (W)

*Rf *fouling resistance (m^{2}.^{o}C/W)

*T *temperature (^{o}C)

*TAC* themal equipment total annual cost (US$/yr)

*TNC* total network cost (US$/yr)

*U* global heat transfer coefficient

*v *mean flow velocity (m/s)

Greek Letters

*l* latent heat of condensation (J/kg)

*Dp* pressure drop (Pa)

*DTcm *referential mean temperature of the stream (^{o}C), here assumed as the logarithmic mean of the entrance and exit absolute temperatures of the stream

*DTln* logarithmic mean of the temperature difference in the extremities of the thermal equipment (^{o}C)

*m* dynamic viscosity (Pa.s)

*r *density (kg/m^{3})

Indexes

*i *inside

*in* entrance

*inf *inferior limit

*o *outside

*out* exit

*s* shellside

*sup* superior limit

*t* tubeside

*v* saturated vapor

*w *tube wall

ACKNOWLEDGEMENTS

The authors acknowledge CNPq for the financial support during the development of the present paper.

REFERENCES

Bell,K.J., Preliminary Design of Shell and Tube Heat Exchangers. In: Kakaç,S., Bergles,A.E. Mayinger,F., Heat Exchangers - Thermal - Hydraulic Fundamentals and Design. Hemisphere Publishing Corporation, Washington (1981).

Douglas,D.Q., Conceptual Design of Chemical Processes. McGraw Hill Books, New York (1988).

Hall,R.S., Vatavuk,W.M., Matley, J. Estimating Process Equipment Costs. Chem.Eng., 95(17), p. 66-75 (1988).

Jegede,F.O., Polley,G.T. Optimum Heat Exchanger Design. Trans.IChemE. Chem.Eng.Res.Des., 70, p. 133-141 (1992).

Kern,D.Q., Process Heat Transfer. McGraw Hill Books, New York (1950).

Linnhoff,B., Hindmarsh,E. The Pinch Design Method for Heat Exchanger Networks, Chem.Eng.Sci., 38, p. 745-763 (1983).

Oliveira,S.G., A Influência do Projeto na Síntese de Redes de Trocadores de Calor., M.Sc. Thesis, EQ/UFRJ, Brazil (1995).

Perry,R.H., Green,D., Chemical Engineers' Handbook., McGraw Hill Books, New York (1997).

Ravagnani,M.A.S.S., Projeto e Otimização de Redes de Trocadores de Calor., D.Sc. Thesis, FEQ/UNICAMP, Brazil (1994).

Sinnott,R.K., Coulson & Richardson's Chemical Engineering., 6 (Design), Pergamon Press, New York (1993).

Steinmeyer,D., Understanding DP and DT in Turbulent Flow Heat Exchangers., Chem. Eng. Prog., p. 49-55 (1996).

Vieira,A.J.M., Síntese e Projeto de Redes de Trocadores de Calor com Considerações Fluidodinâmicas., M.Sc. Thesis, EQ/UFRJ, Brazil (1998).

* To whom correspondence should be addressed.