Q3. Mathematically prove that the ratio of T₁to T₃ is the reverse of the compression ratio.

Q1. Sketch the temperature of the ice cube versus time, as it, solely exposed to air at a room temperature, slowly turns from a solid state to a complete liquid state.

Q2. The project is based on a problem that an insulated container is partially filled with a liquid in gas at an isothermal condition where the fluid temperature is higher than the ice temperature. Then, several ice cubes are dropped into the container (see the image below). Regarding this transient process of heat transfer, you are asked to develop a formula based on the first law of thermodynamics to predict the needed ice mass or needed ice initial temperature to achieve a targeted final temperature at its thermal equilibrium. Assuming that you know the liquid mass, liquid temperature, gas mass, gas temperature and thermodynamics properties of liquid, gas and ice. To simplify the analysis, let us assume that the mass of gas remains unchanged after the addition of ice to the container and the process of heat transfer in the container is isobaric. To derive a practical cquation for the analysis described herein, how do you make assumptions and simplify the first law of the thermodynamics based on the equation shown in Problem 1-Q2? List the notations for each parameter and variable in the equation proposed by you.

Q1. Identify the processes by numbers where the heat addition, heat rejection, compression and expansion are involved in one cycle.

Q2. Using the first law of thermodynamics to derive equations for each term in Q1 in terms of the gas temperature.

Q4. In thermodynamics, we learned that the entropy balance relation can be written as:  where any process (A to B) of the entropy change of ideal gases can be approximated with constant specific heats:Using the equation above to mathematically derive therk, where k is the ratio of cp to cy and r is the compression ratio.

Q5. Derive the cycle thermal efficiency in terms of the gas temperature.

Q6. Derive the cycle thermal efficiency in terms of r and k.

Q1. A boiler consumes 500 kJ/s to produce heat at 1300 K in an environment at 300 K. Dctermine which value below is the maximum work that can be yielded by the heat in the boiler: (a) 192 kW, (b) 115 kW, (c) 385 kW, (d) 500 kW, (e) 650 kW.

Q2. Indoor heat of 800 W is leaving via a wall to outside. The inner wall surface is 20℃ and outer wall surface is 5℃ . The ambient temperature is 0℃ outside. Determine which value below is the rate of exergy destruction within the wall:(a) 0 W (b) 17,500 W (c) 765 W (d) 32,800 W (e) 40 W

 1. Empiricists are in general rather suspicious with respect to any kind of abstract entities like properties, classes, relations, numbers, propositions, etc. They usually feel much more in sympathy with nominalists than with realists (in the medieval sense). As far as possible they try to avoid any reference to abstract entities and to restrict themselves to what is sometimes called a nominalistic language, i.e., one not containing such references. However, within certain scientific contexts it seems hardly possible to avoid them. In the case of mathematics, some empiricists try to find a way out by treating the whole of mathematics as a mere calculus, a formal system for which no interpretation is given or can be given. Accordingly, the mathematician is said to speak not about numbers, functions, and infinite classes, but merely about meaningless symbols and formulas manipulated according to given formal rules. ... Recently the problem of abstract entities bas arisen again in connection with semantics, the theory of meaning and truth. Some semanticists say that certain expressions designate certain entities, and among these designated entities they include not only concrete material things but also abstract entities, e.g. properties as designated by predicates and propositions as designated by sentences. Others object strongly to this procedure as violating the basic principles of empiricism and leading back to a metaphysical ontology of the Platonic kind. (Rudolf Carap, "Empiricism, Semantics, and Ontology."") (哲學小辭典:empiricism經驗論；nominalism名相論；realism實在論；semantics 語意學.)(60%)