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# Gear and Runge–Kutta Numerical Discretization Methods in Differential Equations of Adsorption in Adsorption Heat Pump

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Faculty of Civil Engineering and Architecture, Department of Heating, Ventilation and Heat Engineering, West Pomeranian University of Technology Szczecin, Piastow 50, 71-311 Szczecin, Poland
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Faculty of Computer Science and Information Technology, West Pomeranian University of Technology Szczecin, Zolnierska 49, 71-210 Szczecin, Poland
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Author to whom correspondence should be addressed.
Appl. Sci. 2018, 8(12), 2437; https://doi.org/10.3390/app8122437
Received: 21 October 2018 / Revised: 23 November 2018 / Accepted: 26 November 2018 / Published: 1 December 2018
(This article belongs to the Special Issue Sciences in Heat Pump and Refrigeration)
The main aim of this paper was to find the correct method of calculating equations of heat and mass transfer for the adsorption process and to calculate it numerically in reasonable time and with proper accuracy. An adsorption heat pump with a silica gel adsorbent and water adsorbate is discussed. We developed a mathematical model of temperature and uptake changes in the adsorber/desorber comprising the set of heat and mass balance partial differential equations (PDEs), together with the initial and boundary conditions and solved it by the numerical method of lines (NMOL). Spatial discretization was performed with equally spaced axial nodes and the PDEs were reduced to a set of ordinary differential equations (ODEs). We focused on the comparison of results obtained when the set of heat and mass balance ODEs for an adsorber was solved using: (1) the Runge–Kutta fixed step size fourth-order method (RKfixed), (2) the Runge–Kutta–Fehlberg 4.5th-order method with a variable step size (RK45), and (3) the Gear Backward Differentiation Formulae numerical (Gear BDF) methods. In our experience, all three types of ODE numerical methods (RKfixed, RK45, and Gear BDF) can be applied in simple models to model an adsorber with attention on their limitations. The Gear BDF method usually requires much fewer steps than the RK45 method for almost the same calculating time. RK methods require many more steps to obtain results, and the calculating time depends on accuracy or defined time step. Moreover, one should pay attention to the number of nodes or possible oscillations. View Full-Text
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MDPI and ACS Style

Zwarycz-Makles, K.; Majorkowska-Mech, D. Gear and Runge–Kutta Numerical Discretization Methods in Differential Equations of Adsorption in Adsorption Heat Pump. Appl. Sci. 2018, 8, 2437. https://doi.org/10.3390/app8122437

AMA Style

Zwarycz-Makles K, Majorkowska-Mech D. Gear and Runge–Kutta Numerical Discretization Methods in Differential Equations of Adsorption in Adsorption Heat Pump. Applied Sciences. 2018; 8(12):2437. https://doi.org/10.3390/app8122437

Chicago/Turabian Style

Zwarycz-Makles, Katarzyna, and Dorota Majorkowska-Mech. 2018. "Gear and Runge–Kutta Numerical Discretization Methods in Differential Equations of Adsorption in Adsorption Heat Pump" Applied Sciences 8, no. 12: 2437. https://doi.org/10.3390/app8122437

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