Vapor-Compression Refrigeration System Optimization

[youngenginer]

A simple vapor compression system. Based on constitutive relationships between pressure, temperature, enthalpy, etc., the system has four degrees of freedom. The two pressures in the system(p1 and p2), the level of superheat at the compressor outlet (T2) and the masss flow rate of the refrigerant (r-12) uniquely define the remaining thermodynamic states at each transition points (denoted 1,2,3,4) given the assumptions for an ideal vapor compression refrigeration system are valid. An optimized combination of these variables will result in an efficient system operating at the design point.

Attached is the Problem in PDF, with the picture, and other information need for the problem. I understand the processes, but am unable to generate my objective function and my constraint equations; that is all I need help doing.

I know I put this up as a problem, but an in depth explanation of the work would be greatly appreciated.

Thank You

 

[Valkarie]

. The constraints for a simple vapor compression system are as follows: 1. The mass flow rate of the refrigerant (r-12) must be conserved throughout the system, i.e., m1 = m2 = m3 = m4.2. The total enthalpy must remain constant at all points, i.e., h1 = h2 = h3 = h4.3. The pressure drop across the evaporator must be equal to the pressure drop across the condenser, i.e., p1 - p2 = p3 - p4.4. The temperature at the compressor outlet must be higher than that at the inlet, i.e., T2 > T1. 5. The pressure at each point must be within the limits of the pressure-enthalpy diagram for r-12, i.e., p1≤p2≤p3≤p4. These constraints can be expressed mathematically using the following equations: 1. m1 = m2 = m3 = m4 2. h1 = h2 = h3 = h4 3. p1 - p2 = p3 - p4 4. T2 > T1 5. p1≤p2≤p3≤p4 The objective function for this problem is to maximize the coefficient of performance (COP). This can be expressed mathematically as follows: COP = (Q1 - Q2)/W where Q1 and Q2 are the heat gains and losses in the system and W is the work done by the compressor. Therefore, the objective function is: Maximize COP = (Q1 - Q2)/W