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本文从求解一组含有 B 个未知变量的 B 个方程的角度考察了通风网路问题,其中 B 是网路的分支数。根据基尔霍夫电流定律和电压定律,一般通风网路问题的解必须满足 B 个网路方程。本文选取风量及调节装置的压力作为变量,并将它们分成 B 个因变量(未知的)和 B 个自变量,建立起一组含有 B 个未知变量的 B 个方程。根据这些变量是因变量还是自变量,可将网路中的分支分成四种类型:固定分支、非固定分支、p 固定分支和 q 固定分支。通风网路问题可能包括所有这四种类型的分支。以这种方式定义的通风网路问题,在某些条件下没有唯一解。要有唯一解,则方程组的雅可比矩阵必须非奇异,因此:·以风量作为自变量的分支数必须小于或等于对应一棵生成树的连枝数;·以风量作为自变量的分支所构成的子网路必须不含有任何割集;·以调节装置的压力作为因变量的分支所构成的子网路必须不含有任何割集。不论变量的值如何,上述三个条件均需满足。本文除了给出上述问题的公式外,还给出了应用牛顿法[或称牛顿-拉夫逊(Newton-Raphson)法]求解方程组的步骤,并举例说明。
In this paper, the problem of ventilation network is investigated from the perspective of solving a set of B equations with B unknown variables, where B is the number of branches of the network. According to Kirchhoff’s law of current and voltage law, the solution of the general ventilation network problem must satisfy B network equations. In this paper, we choose the air volume and the pressure of the regulating device as variables, and divide them into B dependent variables (unknown) and B independent variables to establish a set of B equations with B unknown variables. Depending on whether these variables are dependent or dependent, the branches in the network can be divided into four types: fixed, unfixed, p fixed, and q fixed. Ventilation network problems may include all four of these types of branches. The ventilation network problem defined in this way does not have a unique solution under certain conditions. To have a unique solution, the Jacobian matrix of the system of equations must be non-singular so that: • The number of branches with wind volume as independent variable must be less than or equal to the number of branches corresponding to a spanning tree; • The branch with wind volume as an argument The resulting subnetworks must not contain any cut sets. The subnetworks formed by the branches of the regulator pressure as dependent variables must not contain any cut sets. Regardless of the value of the variable, the above three conditions must be met. In addition to the formulas given above, this paper also gives examples of solving equations using the Newton-Raphson method.