非同型热机的非线性响应关系及其最大功率对应的效率

Nonlinear constitutive relation and efficiency at maximum power for nonhomotypic heat engine

现有的典型热机模型均表现出同型性:对于循环热机而言,热机的工质与高温热库和低温热库之间的热交换规律具有相同的函数形式;对于自治热机来说,正向速率流和逆向速率流遵从相同的函数形式满足这类条件的热机被称为同型热机.本文提出非同型热机的概念,着重研究非同型循环热机,即热机的工质与高温热库和低温热库之间的热交换规律具有不同的函数形式,并导出这类热机的非线性响应关系:J_m=LA[1+ΛλA-δ(1+λ~2)/2A]+O(A^3),其中J_m和A分别是热机的机械流和亲合力,L和Λ与一阶和二阶昂萨格系数相关,λ和δ分别代表热机的非对称度和非同型度.本文导出非同型循环热机最大功率对应的效率为ηmaxP=ηC/2+η~2/8+2λ(1-Λβξ)+δ(1+λ~2)βξ/16η_C^2+O(η_C^3),其中ηC代表卡诺效率,β是加权逆温度,ξ是流过热机的热流的特征能量.由此可见,最大功率时热机取普适效率(ηC/2+η_C^2/8)的充要条件为2λ(1-Λβξ)+δ(1+λ~2)βξ=0.这表明非同型循环热机不满足现有文献中热机最大功率对应效率存在普适性的充分条件.

Typical heat engine exhibits a kind of homotypy., heat exchanges between the cyclic heat engine and the two reservoirs abide by the same function type. The forward and backward flows for an autonomous heat engine also conform to the same function type. These heat engines are named homotypic heat engines. Nonhomotypic cyclic heat engine, with heat exchange between the engine and the two reservoirs not conforming to the same function type, is considered in the present paper. Nonlinear constitutive relation for nonhomotypic cyclic heat engine is derived.Jm=LA[1＋ΛλA-δ（1＋λ^2）/2A]＋O（A^3）,where Jr. and A are flux related to mechanical movement and affinity of the heat engine, respectively. L and ∧ are related to the first and the second order Onsager coefficients, respectively. λ and δ represent asymmetric parameter and nonhomotypic degree of the heat engine, respectively. A model of nonhomotypic endoreversible heat engine is constructed to confirm the above nonlinear constitutive relation. Efficiency at maximum power for nonhomotypic cyclic heat engine is：ηmaxP=ηC/2＋η^2/8＋2λ（1-Λβξ）＋δ（1＋λ^2）βξ/16ηC^2＋O（ηC^3）,where ηc represents Carnot efficiency. Factor β is weighted reciprocal of temperature of the heat engine. Factor ξ is energy scale of the heat flowing through the heat engine in each cycle. From the above equation, it is found that sufficient condition to achieve universal efficiency （ηC/2＋ηC^2/8） at maximum power is： 2λ（1-Λβξ）＋δ（1＋λ^2）βξ=0.For nonhomotypic cyclic heat engine, this condition challenges symmetric coupling and energy matching condition found in the literature.