Tham khảo tài liệu 'design and optimization of thermal systems episode 1 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Modeling of Thermal Systems 147 a set of simultaneous ordinary differential equations arise. For instance the temperatures T1 T2 T3 . Tn of n components of a given system are governed by a system of equations of the form F1 T1 T2 T3 Tj t dĩ d2 F2 TVT2J3 . TnS d Fn T1 T2 T3 . T T ax where the F s are functions of the temperatures and thus couple the equations. These equations can be solved numerically to yield the temperatures of the various components as functions of time T see Example . Partial differential equations are obtained for distributed models. Thus Equation is the applicable energy equation for three-dimensional steady conduction in a material with constant properties. Similarly one-dimensional transient conduction in a wall which is large in the other two dimensions is governed by the equation dT a dT PC37 i l k F dr dx I dx if the material properties are taken as variable. For constant properties the equation becomes lT l T ỈK a dx2 where a k pC is the thermal diffusivity. Similarly equations for two- and threedimensional cases may be written. For convective transport the energy equation is written for a two-dimensional constant property transient problem with negligible viscous dissipation and pressure work as PCP ar dr ar k d2r j- dr dx dy dr2 dy2 where Cp is the specific heat of the fluid at constant pressure and u and V are the velocity components in the x and y directions respectively. Partial differential 148 Design and Optimization of Thermal Systems equations that govern most practical thermal systems are amenable to a solution by analytical methods in very few cases and numerical methods are generally necessary. Finite-difference and finite-element methods are the most commonly employed techniques for partial differential equations. Ordinary differential equations can often be solved analytically particularly if the equation is linear. The integral formulation is based on an integral statement of the conservation .
