Consider the conditions of Problem 1, homework 5 (it is repeated below) and that the surroundings are at a temperature of 4 C. Instead of mixing the two fluids together consider the following proposal. Heat is allowed to flow from the 3 Kg mass of water through a Carnot heat engine and is rejected to the cooler mass which acts as the low temperature thermal reservoir. Determine the maximum amount of work that could be performed by the heat engine, the entropy produced, and the exergy destroyed in the 3 Kg mass during this operation. Also, determine the total entropy produced and exergy destroyed in the total system. In this case the temperature and internal energy of the mass of water is changing unlike the infinite heat reservoir case. Since the temperature of both the high temperature and low temperature heat input is changing, the Carnot efficiency is also changing. This problem should be solved using a numerical technique that you write using either excel or Matlab. In order to account for this change, you should consider using an energy balance for every 0.2 C change in the temperature of the 3Kg water mass and calculate the Carnot efficiency at the average temperature of the high temperature reservoir (the 3 Kg mass). For example, in the first numerical step the temperature of the water mass would change from 99.5 to 99.3 C and the Carnot efficiency for this temperature step would be T = 372.55 K. You should determine the incremental work done during this step using the Carnot efficiency and the heat rejected using an energy balance on the heat engine. The new temperature of the low temperature reservoir is determined using an energy balance on it. The entropy production and exergy destruction at each step is calculated in a similar incremental manner. You should repeat this process until the water temperature reaches the equilibrium temperature you calculated in problem 1, hwk. 5. (a) Define the dead state you would use in this problem and state your reasons. (b) Calculate the work output and entropy production when the 3 Kg mass of water is brought into equilibrium with the final temperature you calculated in homework 5. (c)Compare the exergy destroyed between the mixing process (Homework 5) and the heat engine process (Homework 6) for the 3 Kg mass and for the total system. (d) Repeat (a) – (c) when there is a variable thermal resistance between the high and low heat reservoirs and the thermal efficiency of the heat engine is 80% of the Carnot Efficiency at any condition. The variable thermal resistance produces a constant 0.15 K temperature difference between the high and low thermal reservoirs and the heat engine. (e) Discuss the implications and the effect of practical considerations, (subproblem d)on the entropy production and exergy destruction terms. Problem 1, homework 5 (Repeated Problem Statement) A 3 Kg mass of water is heated to a temperature of 99.5 C at 1 bar of pressure. It is slowly poured into a heavily insulated beaker containing 7.5 Kg of water at a temperature and pressure of 4 C, 1 bar, respectively. The specific heat of the water, an incompressible material is, c = 4.1 KJ/(Kg K). The beakers are an adiabatic system. The two water mass’ reach an equilibrium temperature. There is no kinetic or potential energy change in this problem. Using the first law determine the final temperature of the water in the beaker. Calculate the entropy production in this process. (Hint: note that this is an incompressible material which means that the density and specific volume are constant)

Question

Consider the conditions of Problem 1, homework 5 (it is repeated below) and that the surroundings are at a temperature of 4 C. Instead of mixing the two fluids together consider the following proposal. Heat is allowed to flow from the 3 Kg mass of water through a Carnot heat engine and is rejected to the cooler mass which acts as the low temperature thermal reservoir. Determine the maximum amount of work that could be performed by the heat engine, the entropy produced, and the exergy destroyed in the 3 Kg mass during this operation. Also, determine the total entropy produced and exergy destroyed in the total system.

In this case the temperature and internal energy of the mass of water is changing unlike the infinite heat reservoir case. Since the temperature of both the high temperature and low temperature heat input is changing, the Carnot efficiency is also changing. This problem should be solved using a numerical technique that you write using either excel or Matlab. In order to account for this change, you should consider using an energy balance for every 0.2 C change in the temperature of the 3Kg water mass and calculate the Carnot efficiency at the average temperature of the high temperature reservoir (the 3 Kg mass). For example, in the first numerical step the temperature of the water mass would change from 99.5 to 99.3 C and the Carnot efficiency for this temperature step would be T = 372.55 K. You should determine the incremental work done during this step using the Carnot efficiency and the heat rejected using an energy balance on the heat engine. The new temperature of the low temperature reservoir is determined using an energy balance on it. The entropy production and exergy destruction at each step is calculated in a similar incremental manner. You should repeat this process until the water temperature reaches the equilibrium temperature you calculated in problem 1, hwk. 5.

(a) Define the dead state you would use in this problem and state your reasons.

(b) Calculate the work output and entropy production when the 3 Kg mass of water is brought into equilibrium with the final temperature you calculated in homework 5.

(c)Compare the exergy destroyed between the mixing process (Homework 5) and the heat engine process (Homework 6) for the 3 Kg mass and for the total system.

(d) Repeat (a) – (c) when there is a variable thermal resistance between the high and low heat reservoirs and the thermal efficiency of the heat engine is 80% of the Carnot Efficiency at any condition. The variable thermal resistance produces a constant 0.15 K temperature difference between the high and low thermal reservoirs and the heat engine.

(e) Discuss the implications and the effect of practical considerations, (subproblem d)on the entropy production and exergy destruction terms.

Problem 1, homework 5 (Repeated Problem Statement)

A 3 Kg mass of water is heated to a temperature of 99.5 C at 1 bar of pressure. It is slowly poured into a heavily insulated beaker containing 7.5 Kg of water at a temperature and pressure of 4 C, 1 bar, respectively. The specific heat of the water, an incompressible material is, c = 4.1 KJ/(Kg K). The beakers are an adiabatic system. The two water mass’ reach an equilibrium temperature. There is no kinetic or potential energy change in this problem.

Using the first law determine the final temperature of the water in the beaker.
Calculate the entropy production in this process. (Hint: note that this is an incompressible material which means that the density and specific volume are constant)