Isentropic work, calculation

If a Mollier diagram (enthalpy-pressure-temperature-entropy) is available for the working fluid the isentropic work can be easily calculated. 

Example 7,6 A stream of ethylene gas at 300°C and 45 bar is expanded adiabaticall in a turbine to 2 bar. Calculate the isentropic work produced. Determine the properti of ethylene by (a) equations for an ideal gas, and (b) appropriate general" correlations. 

After calculating the isentropic work, then calculate the shaft work, brake work, or compressor work, i.e, the work that is delivered to the shaft of the compressor. The compressor work. 

When inlet conditions Ti and Pi and discharge pressure P2 are known, the value of Hi is fixed. In Eq. (4-163) both H2 and W, are unknown, and the energy balance alone does not allow their calculation. However, if the fluid expands reversibly and adiabatically, i.e., isentropi-cally, in the turbine, then S2 = Si. This second equation establishes the final state of the fluid and allows calculation 01 H. Equation (4-164) then gives the isentropic work ... 

Comments Notice that the ideal work (-832.68 kJ/kg K) is not equal to the reversible work (-870.387 kJ/kg) calculated in Example 6.12. This is because the two concepts are not the same. Although both ideal and reversible work refer to a reversible path, these are two different paths. The reversible work calculated in Example 6.1 is the work for reversible operation between inlet state (P, T,), and outlet pressure (P2). The ideal work calculated here is the work for reversible operation between inlet state (P, T,), and outlet state (P2, T2). To reach the outlet temperature reversibly, the system cannot follow an isentropic path. Instead, it follows some different path that exchanges heat with a reservoir at To. 

The explosive energy release (available work) calculation used to obtain the results presented in this chapter involves the assumption of an isentropic expansion of the vaporized fuel. This is a very pessimistic assumption, and calculations by Jankus (73) have indicated that, in some cases, the energy release would be overestimated by as much as a factor of 10. The results of the one experiment performed in this field, the KIWI experiment, support this estimate. Based on the data that have been released and discussions with W. R. Stratton of Los Alamos, it is estimated that the assumption of an isentropic expansion overestimates the energy release from the KIWI experiment by a factor of approximately 6. 

Calculation of Actual Work of Compression For simplicity, the work of compression is calciilated by the equation for an ideal gas in a three-stage reciprocating machine with complete intercoohng and with isentropic compression in each stage. The work so calculated is assumed to represent 80 percent of the actual work. The following equation may be found in any number of textbooks on thermodynamics ... 

Refrigerating capacity is the product of mass flow rate of refrigerant m and refrigerating effect R which is (for isobaric evaporation) R = hevaporator outlet evaporator mJef Powei P required foi the coiTipressiou, necessary for the motor selection, is the product of mass flow rate m and work of compression W. The latter is, for the isentropic compression, W = hjisehatge suction- Both of thoso chai acteristics could be calculated for the ideal (without losses) and for the ac tual compressor. ideaUy, the mass flow rate is equal to the product of the compressor displacement per unit time and the gas density p m = p. 

The first term is due to the irreversible expansion from V, to Vj, and the second term to the isentropic expansion from Vj to Vj. Adamczyk does not actually say how p3 should be chosen. A reasonable choice for seems to be the initial-peak shock overpressure, as calculated from Eq. (6.3.22). The equation presented above can be compared to the results of Guirao et al. (1979). They numerically evaluated the work done by the expanding contact surface. When the difference between... 

Wiedermatm (1986b) presents an alternative method for calculating work done by a fluid. The method uses the lambda model to describe isentropic expansion, and permits work to be expressed as a function of initial conditions and only one fluid parameter, lambda. Unfortunately, this parameter is known for very few fluids. 

Calculate the ideal intermediate pressures and the work required per kilogram of gas. Assume compression to be isentropic and the gas to behave as an ideal gas. Indicate on a temperature-entropy diagram the effect of imperfect imercooling on the work done at each stage. 

In reality, most compressor conditions are neither purely isothermal nor purely isentropic but somewhere in between. This can be accounted for in calculating the compression work by using the isentropic equation [Eq. (8-21)], but replacing the specific heat ratio k by a polytropic constant, y, where 1 < y < k. The value of y is a function of the compressor design as well as the properties of the gas. 

Theoretical methods allow making such calculations for ideal and real gases and gas mixtures under isothermal and frictionless adiabatic (isentropic) conditions. In order that results for actual operation can be found it is neecessary to know the efficiency of the equipment. That depends on the construction of the machine, the mode of operation, and the nature of the gas being processed. In the last analysis such information comes from test work and its correlation by manufacturers and other authorities. Some data are cited in this section. 

This equation now gives us a means to calculate the work [from equation (7.1)], but we still require the outlet temperature. Again we resort to the isentropic approximation. 

A stream of ethylene gas at 260X and 4,100 kPa expands isentropically in a turbine to 140 kE Determine the temperature of the expanded gas and the work produced if the properties of ethyleu are calculated by... 

The decrease in U is the measure of reversible nonflow work that can be extracted from a fluid upon isentropic expansion. The state function U, upon being measured or calculated for various states, becomes useful for engineering design of work-production processes that expand the fluid from one state to another that are tabulated or calculated. 

If the working fluid is steam, calculate the specific entropy at stage inlet from the inlet specific enthalpy and the inlet pressure using either steam tables or equation (16.64). Check whether the steam is initially superheated or saturated by comparing the inlet specific entropy, So, with the saturated specific entropy at inlet pressure, Sglpo). If 5o.i < i,(Po), use equation (16.75) to estimate the initial dryness fraction and Zeuner s equation (15.112) to estimate the isentropic index, y< for the first stage. 

To calculate the work done during the isentropic expansion between these two points, we calculate the change in apparent internal energy, AU, between the two points, which is... 

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