Abstract
Solution multiplicity of natural ventilation in buildings is of much importance for personnel safety and ventilation design. In this paper, a new mathematical model of buoyancy pressure ventilation for two vertically connected open cavities is presented. Compared with the previous published papers studying two vertically connected open cavities with equal heights and hot source E2 < 0 in the upper room, we study two vertically connected open cavities with unequal heights and hot source E2 < 0 or E2 > 0 in the upper room. By solving and analyzing the equilibrium points and characteristic roots of the differential equations, we analyze the stability of two systems with upward flow pattern and downward pattern and obtain the criteria to determine the stability and existence of solutions for two scenarios. According to these criteria, the multiple steady states of buoyancy ventilation in two vertically connected open cavities with unequal heights and variable strength of hot sources can be obtained. These criteria can be used to design buoyancy ventilation or natural exhaust ventilation systems in two vertically connected open cavities. Compared with two stable states of buoyancy ventilation existing in two vertically connected open cavities with equal heights in the previously published papers, we find that more stable states and unstable states of buoyancy ventilation exist in two vertically connected open cavities with unequal heights in our paper. Finally, bifurcation diagrams and the phase portraits for the two scenarios are given.
Highlights
Nonlinear characteristics exist in systems with various research directions, such as chaotic circuit [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], neural network [16,17,18,19,20,21], image encryption [22,23,24,25]
Natural ventilation is increasingly recognized as an important energy saving technology that provides thermal comfort and a healthy indoor environment for buildings [26]. e main objectives of natural ventilation include the provision of adequate outdoor air, effective removal of indoor air, heat, smoke, and/or odour pollution, and comfortable range of air temperature and humidity
The height of the two cavities can be different; for example, the height of the first floor is often greater than that of the second floor in a building. erefore, it is of practical significance and novelty to study the air flow and temperature solutions of buoyancy ventilation in the cavities when the height of the cavity is not equal
Summary
Nonlinear characteristics exist in systems with various research directions, such as chaotic circuit [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], neural network [16,17,18,19,20,21], image encryption [22,23,24,25]. Based on the equations of air flow equilibrium and heat equilibrium, Li established dynamic differential equations about temperature with two-layer open cavities and obtained two stable equilibrium points and two corresponding stable manifolds [46]. Erefore, it is of practical significance and novelty to study the air flow and temperature solutions of buoyancy ventilation in the cavities when the height of the cavity is not equal. We study the two-layer open cavities with unequal heights and hot sources E1 > 0 and E20, the mathematical model is established, and multiple solutions are obtained. Erefore, we can obtain the heat balance equations of the two cavities shown in the following equations: M1C1ddTt1 − q1C1 T1 − T2 + E1,.
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