基于室内供暖温度调控的数学建模研究

Study on Mathematical Modeling Based on Indoor Heating Temperature Control

冬季北方部分地区采取热水供热的供暖方式,研究如何控制进水流量使得室内温度维持在一个相对稳定的范围具有实际意义。针对空间温度分布问题,本文将三维空间节点化,建立了空间节点控制模型,将空间节点分为热源节点与非热源节点,分别基于热传导方程与非热源节点由雅可比迭代法进行求解,得到该房间各高度所在平面的温度分布图可知,从南北方向来看,每个高度的分布图都呈现靠近南、北墙两端温度相对较高、中间相对较低分布;而从东西方向来看,则呈现中间温度相对较高、两端温度相对较低的分布。另外,高度越高,热量分布相对越均匀。针对当水流量变化的问题,我们对暖气片产热量进行修正,以0.1m^(3)/h为间隔求解不同进水流量的出水温度,然后利用空间节点控制模型求解房间的动态温度分布,得到平均温度随水流量的变化趋势:随着进水口流量每增加温度提高的幅度逐渐变小。对于最小耗水量问题,本文以最小累计耗水量为单一目标,在约束条件下进行求解。并在室内平均温度为21℃、标准差为1.5℃、进水口流量在0.1-1.0m^(3)/h的约束条件利用lingo求解得到最小进水量为13.59m^(3)。

In winter,some parts of northern China adopt the hot water heating mode,and it is of practical significance to study how to control the inflow flow to maintain the indoor temperature in a relatively stable range.In view of spatial temperature distribution,a spatial node control model is established to divide spatial nodes into heat source and non-heat source nodes and solve the heat distribution equation with relatively high temperature distribution at the ends of the south and north wall;from the east-west direction,the distribution with relatively high intermediate temperature and relatively low temperature at each end.In addition,the higher the height,the more evenly the heat distribution is.When the problem of water flow change,we correct the radiator heat production,with 0.1 m3/h as the interval of different inlet water temperature,and then use the space node control model to solve the dynamic temperature distribution of the room,the average temperature with water flow trend:as the inlet flow each increase temperature increase gradually smaller.For the minimum water consumption problem,the minimum cumulative water consumption as a single goal is solved under constraints.The average indoor temperature of 21℃,standard deviation of 1.5℃and water inlet flow of 0.1-1.0m3/h were solved by lingo to obtain the minimum water inflow of 13.59m3.