2023 Asia and Pacific Mathematical Contest in Modeling
2023 年亚太地区建模数学竞赛

The yield of greenhouse crops is affected by various climate factors, including temperature, humidity, and wind speed [1]. Among them, suitable temperature and wind speed are crucial for plant growth [2]. In order to regulate climate factors such as temperature and wind speed inside the glass greenhouse, ventilation systems with greenhouse fans are commonly used in the design of greenhouse, as shown in Figure 1. The position of the greenhouse fan and the speed of the warm air outlet affect the distribution and uniformity of the velocity field and the temperature field in the greenhouse. Therefore, how to optimize the greenhouse fan to obtain suitable wind speed and temperature, and improve their uniformity, is an important issue that needs to be solved in current glass greenhouse design.
温室作物的产量受各种气候因素的影响，包括温度、湿度和风速 [1]。其中，适宜的温度和风速对植物生长至关重要 [2]。为了调节玻璃温室内的温度和风速等气候因素，温室设计中通常使用带有温室风扇的通风系统，如图 1 所示。温室风扇的位置和暖风出口的速度会影响温室中速度场和温度场的分布和均匀性。因此，如何优化温室风机以获得合适的风速和温度，并提高其均匀性，是当前玻璃温室设计中需要解决的重要问题。

Figure 1. Glass greenhouse
图 1.玻璃温室
The glass greenhouse is sealed and placed indoors, without considering external factors such as greenhouse doors, drafts, solar radiation, and other environmental factors. The current design of the glass greenhouse has dimensions of $10\mathrm{m}\times 3\mathrm{m}\times 2\mathrm{m}$$10\mathrm{m}\times 3\mathrm{m}\times 2\mathrm{m}$10mxx3mxx2m10 \mathrm{~m} \times 3 \mathrm{~m} \times 2 \mathrm{~m} (length $\times$$\times$xx\times width $\times$$\times$xx\times height) and the size of the greenhouse fan is $0.5\mathrm{m}\times 0.5\mathrm{m}$$0.5\mathrm{m}\times 0.5\mathrm{m}$0.5mxx0.5m0.5 \mathrm{~m} \times 0.5 \mathrm{~m}, positioned on the left side of the greenhouse. The center of the greenhouse fan is located 1.3 m above the ground, as illustrated in Figure 2. The boundary conditions on the side of the greenhouse fan are set as velocity inlet conditions, blowing warm air at ${40}^{\circ }\mathrm{C}$${40}^{\circ }\mathrm{C}$40^(@)C40^{\circ} \mathrm{C} in the horizontal direction with an average velocity of $2\mathrm{m}/\mathrm{s}$$2\mathrm{m}/\mathrm{s}$2m//s2 \mathrm{~m} / \mathrm{s}. The outer glass and bottom soil of the greenhouse are set as wall conditions, primarily exchanging
玻璃温室是密封的，放置在室内，没有考虑温室门、气流、太阳辐射和其他环境因素等外部因素。目前玻璃温室设计的尺寸为 $10\mathrm{m}\times 3\mathrm{m}\times 2\mathrm{m}$$10\mathrm{m}\times 3\mathrm{m}\times 2\mathrm{m}$10mxx3mxx2m10 \mathrm{~m} \times 3 \mathrm{~m} \times 2 \mathrm{~m} （长 $\times$$\times$xx\times 宽 $\times$$\times$xx\times 、高），温室风扇的尺寸为 $0.5\mathrm{m}\times 0.5\mathrm{m}$$0.5\mathrm{m}\times 0.5\mathrm{m}$0.5mxx0.5m0.5 \mathrm{~m} \times 0.5 \mathrm{~m} ，位于温室的左侧。温室风扇的中心位于离地面 1.3 m 处，如图 2 所示。温室风扇侧面的边界条件设置为速度入口条件，以平均速度 向水平方向吹暖空气 ${40}^{\circ }\mathrm{C}$${40}^{\circ }\mathrm{C}$40^(@)C40^{\circ} \mathrm{C} $2\mathrm{m}/\mathrm{s}$$2\mathrm{m}/\mathrm{s}$2m//s2 \mathrm{~m} / \mathrm{s} 。温室的外玻璃和底土被设置为墙体条件，主要交换
energy with the entire greenhouse through convective heat transfer and conduction [3]. The initial temperature is set at ${20}^{\circ }\mathrm{C}$${20}^{\circ }\mathrm{C}$20^(@)C20^{\circ} \mathrm{C}. When crops are planted inside the greenhouse, it is necessary to consider the canopy resistance of the crops. The crop model can be simplified as a porous medium with dimensions of $8\mathrm{m}\times 2\mathrm{m}\times 0.5\mathrm{m}$$8\mathrm{m}\times 2\mathrm{m}\times 0.5\mathrm{m}$8mxx2mxx0.5m8 \mathrm{~m} \times 2 \mathrm{~m} \times 0.5 \mathrm{~m} (length $\times$$\times$xx\times width $\times$$\times$xx\times height) [4], placed in the center of the greenhouse. The suitable wind speed for crop growth inside the greenhouse is 0.3 $1\mathrm{m}/\mathrm{s}$$1\mathrm{m}/\mathrm{s}$1m//s1 \mathrm{~m} / \mathrm{s}, and the suitable temperature is $23-{26}^{\circ }\mathrm{C}$$23-{26}^{\circ }\mathrm{C}$23-26^(@)C23-26^{\circ} \mathrm{C}.
能量传递给整个温室 [3]。初始温度设置为 ${20}^{\circ }\mathrm{C}$${20}^{\circ }\mathrm{C}$20^(@)C20^{\circ} \mathrm{C} 。当农作物在温室内种植时，有必要考虑农作物的抗冠性。作物模型可以简化为尺寸为 $8\mathrm{m}\times 2\mathrm{m}\times 0.5\mathrm{m}$$8\mathrm{m}\times 2\mathrm{m}\times 0.5\mathrm{m}$8mxx2mxx0.5m8 \mathrm{~m} \times 2 \mathrm{~m} \times 0.5 \mathrm{~m} （长 $\times$$\times$xx\times 、宽 $\times$$\times$xx\times 、高）[4] 的多孔介质，放置在温室的中心。温室内作物生长的适宜风速为 0.3 $1\mathrm{m}/\mathrm{s}$$1\mathrm{m}/\mathrm{s}$1m//s1 \mathrm{~m} / \mathrm{s} ，适宜温度为 $23-{26}^{\circ }\mathrm{C}$$23-{26}^{\circ }\mathrm{C}$23-26^(@)C23-26^{\circ} \mathrm{C} 。

Figure 2. Schematic diagram of glass greenhouse structure
图 2.玻璃温室结构示意图
Question 1: Please establish a mathematical model for the temperature and wind speed distribution inside a glass greenhouse without crops. Display the distribution of wind speed and temperature at a cross-section of the greenhouse at a height of 0.5 meters.
问题 1： 请为没有农作物的玻璃温室内的温度和风速分布建立一个数学模型。显示温室横截面上 0.5 米高处的风速和温度分布。

Question 2: Please establish a mathematical model for the temperature and wind speed distribution inside a glass greenhouse with planted crops. Present the distribution of wind speed and temperature at two cross-sections within the greenhouse: one at a height of 0.5 meters (crop canopy level) and another at a height of 0.1 meters (inside the crop canopy). Analyze whether the conditions are suitable for crop growth.
问题 2： 请为种植农作物的玻璃温室内的温度和风速分布建立一个数学模型。显示温室内两个横截面的风速和温度分布：一个在 0.5 米的高度（作物冠层水平），另一个在 0.1 米的高度（作物冠层内）。分析条件是否适合作物生长。

Question 3: Please provide the temperature and wind speed distribution inside the glass greenhouse for the following two scenarios and compare them with solution presented in the second question. In Scenario One, increase the velocity of warm air outlet from $2\mathrm{m}/\mathrm{s}$$2\mathrm{m}/\mathrm{s}$2m//s2 \mathrm{~m} / \mathrm{s} to $3\mathrm{m}/\mathrm{s}$$3\mathrm{m}/\mathrm{s}$3m//s3 \mathrm{~m} / \mathrm{s}. In Scenario Two, lower the position of the greenhouse fan by moving it from 1.3 m to 1 m .
问题 3：请提供以下两种情况下玻璃温室内的温度和风速分布，并将它们与第二个问题中提出的解决方案进行比较。在场景 1 中，将暖空气出口的速度从 $2\mathrm{m}/\mathrm{s}$$2\mathrm{m}/\mathrm{s}$2m//s2 \mathrm{~m} / \mathrm{s} 增加到 $3\mathrm{m}/\mathrm{s}$$3\mathrm{m}/\mathrm{s}$3m//s3 \mathrm{~m} / \mathrm{s} 。在场景二中，通过将温室风扇的位置从 1.3 m 移动到 1 m 来降低温室风扇的位置。

Question 4: Can your team further optimize the greenhouse fan design of glass greenhouse from the number of greenhouse fan, location, wind speed, blowing temperature, specifications and different crops and other factors.
问题 4：您的团队能否从温室风扇的数量、位置、风速、吹风温度、规格和不同作物等因素进一步优化玻璃温室温室风扇的设计。

[1] Singh M C, Singh J P, Pandey S K, et al. Factors affecting the performance of greenhouse cucumber cultivation-a review[J]. International Journal of Current Microbiology and Applied Sciences, 2017, 6(10): 2304-2323.
[1] Singh MC、Singh JP、Pandey S K 等人。影响大棚黄瓜栽培性能的因素综述[J].国际当代微生物学与应用科学杂志， 2017， 6（10）： 2304-2323.
[2] Liu Y, Li D, Wan S, et al. A long short-term memory-based model for greenhouse climate prediction[J]. International Journal of Intelligent Systems, 2022, 37(1): 135-151.
[2] Liu Y， Li D， Wan S， et al.一种基于长短期记忆的温室气候预测模型[J].智能系统学报， 2022， 37（1）： 135-151.
[3] Norton T, Sun D W, Grant J, et al. Applications of computational fluid dynamics (CFD) in the modelling and design of ventilation systems in the agricultural industry: A review[J]. Bioresource technology, 2007, 98(12): 2386-2414.
[3] Norton T、Sun D W、Grant J 等人。计算流体力学 （CFD） 在农业通风系统建模和设计中的应用研究进展[J].生物资源技术， 2007， 98（12）： 2386-2414.
[4] Fatnassi H, Boulard T, Poncet C, et al. Optimisation of greenhouse insect screening with computational fluid dynamics[J]. Biosystems Engineering, 2006, 93(3): 301-312.
[4] Fatnassi H、Boulard T、Poncet C 等人。基于计算流体动力学的温室昆虫筛选优化[J].生物系统工程， 2006， 93（3）： 301-312.