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

Dorota Majorkowska-Mech,Katarzyna Zwarycz-Makles

doi:10.3390/app8122437

Dorota Majorkowska-Mech, Katarzyna Zwarycz-Makles

Open Access

https://doi.org/10.3390/app8122437

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Journal: Applied Sciences	Publication Date: Dec 1, 2018
Citations: 5	License type: CC BY 4.0

Affiliation: West Pomeranian University of Technology

Abstract

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.

Highlights

In the face of the growing need to search for ecologically safe sources of energy, research on increasing the use of available low-temperature heat sources by means of various physical processes has intensified
The spatial discretization was performed with spaced axial nodes and the partial differential equations (PDEs) were reduced to a set of ordinary differential equations (ODEs)
The geometrical specifications and the operating conditions of the numerical model were adjusted to their counterparts in the tested lab-scale chilling module with the adsorption/desorption bed

Summary

Introduction

In the face of the growing need to search for ecologically safe sources of energy, research on increasing the use of available low-temperature heat sources by means of various physical processes has intensified. One of them is the adsorption process used, among others, in sorption heat pumps. Adsorption heat pump applications in cooling refer to useful cooling [4]; chilled water circuits [5,6]; electric vehicle air conditioners [7]; heat rejection systems, such as dry or wet cooling towers or ground coupled heat exchangers [8]; and solar systems as the driving heat source [9,10]. Heating applications of adsorption heat pumps are found in heating systems, waste heat utilization [3,5], low-temperature heat sources (e.g., ground heat exchangers and geothermal water installations [11]) and driving heat sources (e.g., gas furnace) [12]. Adsorptive storage of CO2 can be an interesting example of another application [13]

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