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.
Poland, as for Europe, is a country with an average forest cover of approximately 30%. In these forests, more than 37M m3 of wood, mostly coniferous (over 80%), is harvested per year. In 2012, 4.2M m3 of sawn timber was produced (sawn timber without factory lumber). At the same time, in Poland there are over 8000 sawmills, whereas only about 700 of them saw over 90% of the harvested timber. So much fragmentation is a major cause of low sawmills innovation, particularly of those small ones. However, in recent years, a trend of development in this sector is noticeable, and it is through rationalisation of material and energy economy. One of the methods to increase the technical and economic effectiveness of enterprises involved in woodworking is to build in the combined heat and power system (CHP) plant with the ORC system into the existing infrastructure, which will be matched to the needs of the company. This article presents an analysis of the profitability of the investment based on the example of a medium-sized company sawing approximately 50,000 m3 of timber per year, and the economic analysis was performed for prices and costs valid in Poland. The analysis made for the 1650 kW(el) organic Rankine cycle (ORC) system, has resulted in a profitability index PI = 1.3, on the assumptions that the ORC system operates for 6000 h y(-1), will be purchased at the price of 4500 € kW(el)(-1) and at the price of electricity sales of 130 € MWh(-1).
In the paper, comparison of an analytical and a numerical model of silica gel/water adsorber/desorber is presented. Adsorber/desorber as a part of the two-bed single-stage adsorption heat pump is discussed. The adsorption heat pump under consideration consists of an evaporator, two adsorber/desorber columns and a condenser. Only heat and mass transfer was taken into account during operation of assumed adsorption heat pump. Adsorption equilibrium was described by the Dubinin-Astachov model. The presented mathematical models, both analytical and numerical, were created to describe the temperature, heat and concentration changes in the adsorber/desorber and consequently to describe the performance of the adsorption heat pump. Heat and mass transfer equations of the model were solved analytically and calculation parameters were entered in the common spreadsheet. The numerical model was established comprising of the set of heat and mass balance partial differential equations, together with the initial and boundary conditions and was solved by the numerical method of lines. The developed analytical model is very basic and can be used only for the initial estimates of mean cyclic temperature of the cooling/heating water in the adsorber/desorber bed. Validation of the numerical model shows reasonable agreement between the numerical and experimental averaged bed temperatures and is an indication of the proper mathematical modeling and the accuracy of the numerical analysis.
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