Problems 麦克斯韦关系

9.9 Rubber bands are entropic springs. 橡胶带是熵弹簧。

Experiments show that the retractive force f of polymeric elastomers as a function of temperature T and expansion L is approximately given by f(T,L) = aT(L−L0) where a and L0 are constants.
(a) Use Maxwell’s relations to determine the entropy and enthalpy, S(L) and H(L), at constant T and p.
(b) If you adiabatically stretch a rubber band by a small amount, its temperature increases, but its volume does not change. Derive an expression for its temperatureT as a function of L, L0, a, and its heat capacity C = (∂U/∂T).
中文。实验表明，聚合弹性体的回缩力 f 与温度 T 和膨胀率 L 的函数关系近似为 f(T,L) = aT(L-L0)，其中 a 和 L0 为常数。
(a) 利用麦克斯韦关系确定恒定 T 和 p 时的熵和焓 S(L) 和 H(L)。
(b) 如果将橡皮筋绝热拉伸一小段，其温度会升高，但其体积不变。推导其温度 T 与 L、L0、a 及其热容 C = (∂U/∂T) 的函数关系式。
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9.10 Metal elasticity is due to energy, not entropy. 金属弹性是由能量而非熵引起的。

Experiments show that the retractive force f of a metal rod as a function of temperature T and extension L relative to undeformed length L0 is given by f(T,L) =EaΔL/L0, where ΔL = L[1−α(T−T0)]−L0 = L−Lα(T−T0)−L0. a is the cross-sectional area of the rod, E (which has the role of a spring constant) is called Young’s modulus,and α ≈ 10−5 is the linear expansion coefficient. Compute H(L) and S(L). Is the main dependence on L due to enthalpy H or entropy S?
中文。实验表明，金属杆的回缩力 f 与温度 T 和相对于未变形长度 L0 的延伸率 L 的函数关系为 f(T,L) =EaΔL/L0，其中 ΔL = L[1-α(T-T0)]-L0 = L-Lα(T-T0)-L0。a 是杆的横截面积，E（具有弹簧常数的作用）称为杨氏模量，α ≈ 10-5 是线膨胀系数。计算 H(L) 和 S(L)。对 L 的主要依赖是由于焓 H 还是熵 S？

Classical Thermodynamics

Problem 8.13

An air-standard Diesel cycle absorbs 1500 J⋅mol−1 of heat (step DA of Fig. 8.10,which simulates combustion). The pressure and temperature at the beginning of thecompression step are 1 bar and 20°C, and the pressure at the end of the compression step is 4 bar.
Assuming air to be an ideal gas for which CP = (7/2)R and CV = (5/2)R, what are the compression ratio and the expansion ratio of the cycle?

中文

空气标准柴油机循环吸收 1500 J-mol-1 的热量（图 8.10 的 DA 步，模拟燃烧）。压缩步骤开始时的压力和温度分别为 1 巴和 20°C，压缩步骤结束时的压力和温度分别为 1 巴和 20°C。

Problem Otto engine

Air is compressed in an Otto cycle beginning at 35 °C and 0.1 MPa. The maximum temperature of the process is 1100 °C and the compression ratio is 7.
Find (a) the pressure and temperature at all points of the cycle,
(b) the heat that must be supplied to the process per unit mass (kg) of air, the work done per unit mass of air,
(c) the efficiency of the cycle.

中文

空气在 35 °C 和 0.1 MPa 开始的奥托循环中被压缩。过程的最高温度为 1100 °C，压缩比为 7。
求 (a) 循环各点的压力和温度、
(b) 每单位质量（千克）空气必须向该过程提供的热量，以及每单位质量空气所做的功、
和 (c) 循环的效率。

Problem 9.9 with H3’ given as 414.5 kJ/kg (show all steps and equations regarding the vapor compression refrigeration systems accompanied by corresponding diagrams - see both diagrams in Fig. 9.1)

A vapor-compression refrigeration system operates on the cycle of Fig. 9.1. The refrigerant is tetrafluoroethane (Table 9.1, Fig. F.2).
For one of the following sets of operating conditions, determine the circulation rate of the refrigerant,
the heat-transferrate in the condenser, the power requirement, the coefficient of performance of the cycle,
and the coefficient of performance of a Carnot refrigeration cycle operating between the same temperature levels.
(a) Evaporation T = 0°C; condensation T = 26°C; η(compressor) = 0.79; refrigeration rate = 600 kJ·s−1

（显示有关蒸汽压缩制冷系统的所有步骤和方程，并附相应的图表–见图 9.1 中的两个图表）

蒸汽压缩制冷系统按图 9.1 中的循环运行。制冷剂为四氟乙烷（表 9.1，图 F.2）。在下列一组运行条件中，请确定制冷剂的循环速率、冷凝器中的传热系数、功率要求、循环的性能系数以及在相同温度水平之间运行的卡诺制冷循环的性能系数。
(a) 蒸发温度 T = 0°C；冷凝温度 T = 26°C；η（压缩机）= 0.79；制冷速率 = 600 kJ-s-1