# Orthogonal Collocation-based Steady-State and Transient Response Simulation of Two-pass Shell and Tube Heat Exchangers

### Abstract

An orthogonal collocation-based approach to the simulation of the steady-state and transient response of a two-pass shell and tube heat exchanger is presented. The analytical solution of the steady-state temperature profiles in the heat exchanger are derived. These are then used for selection of the parameters of the orthogonal solution such that the Euclidean norm of the error between the analytical and orthogonal collocation solution is minimized. The lumped parameter ordinary differential equations (ODEs) obtained from the application of the orthogonal collocation method are used to simulate the dynamic response of the system using the state transition matrix approach. Very good results were obtained consistent with other reported applications of the orthogonal collocation method to other types of heat exchangers in the literature. The orthogonal collocation approach is attractive in the efficient simulation of single or interconnected heat exchanger systems such as heat exchanger networks or heat exchanger-reactor systems.

### References

Cho , Y. S., and Joseph, B. (1983). Reduced-order Steady State and Dynamic Models, Part I. Development of the Model Reduction Procedure, AIChE J, Vol. 29, (2), pp. 261269.

Correa , J., Marchetti, J. (1987). Dynamic Simulation of Shell-and-Tube Heat Exchangers, HeatTransfer Engineering, Vol 8, pp. 50-59.

Director, S. W., and Rohrer, R.(1972), Introduction To System Theory, McGraw-Hill Book Company, New York.

Ebrahimzadeh , E., Wilding, P., Frankman, D., Fazlollahi, F., and Baxter, L. L. (2016a). Theoretical and Experimental Analysis of Dynamic Plate Heat Exchanger: Non-Retrofit Configuration, Applied Thermal Engineering, Vol. 93, pp. 1006-1019.

Ebrahimzadeh , E., Wilding, P., Frankman, D., Fazlollahi, F., and Baxter, L. L. (2016b). Theoretical and Experimental Analysis of Dynamic Plate Heat Exchanger: Retrofit Configuration, Energy, Vol. 96, pp. 545-560.

Finlayson, B. A. (1980), Nonlinear Analysis in Chemical Engineering, McGraw-Hill Book Company, New York.

Finlayson, B. A. (1972), The Method of Weighted Residuals and Variational Principles, Academic Press, New York.

Fratczak, M., Nowak, P. and Czeczot, J. (2014). Simplified modeling of plate heat exchangers.2014 19th International Conference on Methods and Models in Automation and Robotics, MMAR 2014, DOI: 10.1109/MMAR.2014.6957418.

Fratczak, M., Nowak, P., Czeczot, J. and Metzger, M. (2016). Simplified Dynamical Input-Output Modeling of Plate Heat Exchangers - Case Study, Applied Thermal Engineering, Vol. 98, pp.880-893.

Friedly, J. C.(1972). Dynamic Behaviour of Processes, Prentice-Hall, Englewood Cliffs, N. J.,

Gould, L. A. (1969), Chemical Process Control: Theory and Applications, Addison-Wesley, Reading, Massachusetts, (1969).

Cooper, A. R., and Guttierrez, R. (1979). Heat Exchanger Process Dynamics Review, The Chem. Eng. J., Vol. 17, pp. 13-18.

Gvozdenac, J. (March 9th 2012). Analytical Solution of Dynamic Response of Heat Exchanger, in Heat Exchangers - Basics Design Applications Ed.Mitrovic, J., IntechOpen, DOI: 10.5772/35944.

Kaka, S and Liu, H. (2002). Heat Exchangers: Selection, Rating, and Thermal Design, second edition, Boca Raton CRC Press.

Luo, X., Guan, X. and Roetzel, W. (2003). Analysis of Transient Behaviour of Multipass Shell and Tube Heat Exchangers with the Dispersion Model, Int. J. Heat Mass Transfer, Vol. 46, pp.705-715.

Martens, H. R. (1969). A Comparative Study of Digital Integration Methods, Simulation, Vol. 12, pp. 87-94.

Moler, C. B., and Van Loan, C. F. (1978). Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Rev., Vol. 20, pp. 801-836.

Nevriva, P., Ozana,, S. and Vilimec, L. (2009). The Finite Difference Method Applied for the Simulation of the Heat Exchanger Dynamics, Proc. 13th. WSEAS Int. Conf. on SYSTEMS,22-24 July, pp. 109 - 114.

Omidi, M., Farhadi, M. and Jafari, M. (2017). A Comprehensive Review of Double Pipe Heat Exchangers, Applied Thermal Engineering, Vol. 110, pp. 1075-1090.

Roetzel, W. and Xuan, Y. (1992a). Transient Behaviour of Multipass Shell-and-Tube Heat Exchangers, Int. J. Heat Mass Transfer, Vol. 35, pp. 703-710.

Roetzel, W. and Xuan, Y. (1992b). Analysis of Transient Behaviour of Multipass Shell and Tube Heat Exchangers with the Dispersion Model, Int. J. Heat Mass Transfer, Vol. 35, pp.2953-2962.

Roetzel, W. and Xuan, Y. (1999). Dynamic Behaviour of Heat Exchangers, WIT Press, MA, USA.

Roetzel, W., M. Li and X. Luo. (2002). Dynamic Behaviour of Heat Exchangers, in Advanced Computational Methods in Heat Transfer VII, Eds. B. Sunden and C. A. Brebbia, WIT Press, pp.452-460.

Rosenbrock, H. H. (1962)., The Transient Behaviour of Distillation Columns and Heat Exchangers- A Historical and Critical Review, Trans. Inst. Chem. Eng., Vol. 40, pp. 376-383.

Villadsen, J., and M. L. Michelsen (1978). Solution of differential equation models by polynomial approximation, Prentice-Hall Inc., Englewood Cliffs, N. J.

Sharifi, F., Golkar-Narandji, M.R, and Mehravaran, K. (1995). Dynamic Simulation of Plate Heat Exchangers, Int. Comm. in Heat Transfer, Vol. 22, pp. 213-225.

Srivastava, R. K., and Joseph, B., Simulation of Packed-Bed Separation Processes Using Orthogonal Collocation, Compt. & Chem. Eng., Vol. 8, No. 1, pp. 43-50, (1984).

Williams, A. O. F., and Adeniyi, V. O. (1989), Simulation of the Dynamics of a Steam-heated Heat Exchanger Using Orthogonal Collocation, Proc. Nig. Soc. Chem. Engr. 19th. Ann. Conf.,Badagry-Lagos, pp. 108-115.

Williams, A. O. F., and V. O. Adeniyi (1991), Modelling and Simulation of the Dynamics of Steam heated Heat Exchangers Using Orthogonal Collocation, Modelling, Simulation & Control, B, Vol. 37, No. 3, pp. 1-24.

Williams, A. O. F., and V. O. Adeniyi (2001). Development of Some Fortran 77 Programs for Linear System Analysis, NSE Technical Transactions, Vol. 36, No. 1, pp. 68-80.

Xia, L., De Abreu-Garcia, J. A. and Hartley, T. T. (1991). Modeling and Simulation of a Heat Exchanger, Proc. IEEE Int. Con. on Systems Engineering, 9-11 Aug., 1990, Fairborn, OH,US, pp. 453-456

Young , L. C. (2019), Orthogonal Collocation Revisited, Compt. Methods. Appl Mech & Eng, Vol. 345, pp. 1033-1076.

*Journal of Engineering Research*,

*24*(2), 59-76. Retrieved from http://jer.unilag.edu.ng/article/view/578