Equation isentropic

Four methods are used to estimate the energy of explosion for a pressurized gas Brode s equation, isentropic expansion, isothermal expansion, and thermodynamic availability. Brode s method21 is perhaps the simplest approach. It determines the energy required to raise the pressure of the gas at constant volume from atmospheric pressure to the final gas pressure in the vessel. The resulting expression is... 

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 ... 

These equations are consistent with the isentropic relations for a perfect gas p/po = (p/po), T/To = p/poY. Equation (6-116) is valid for adiabatic flows with or without friction it does not require isentropic flow However, Eqs. (6-115) and (6-117) do require isentropic flow The exit Mach number Mi may not exceed unity. At Mi = 1, the flow is said to be choked, sonic, or critical. When the flow is choked, the pressure at the exit is greater than the pressure of the surroundings into which the gas flow discharges. The pressure drops from the exit pressure to the pressure of the surroundings in a series of shocks which are highly nonisentropic. Sonic flow conditions are denoted by sonic exit conditions are found by substituting Mi = Mf = 1 into Eqs. (6-115) to (6-118). 

Equation (6-128) does not require fric tionless (isentropic) flow. The sonic mass flux through the throat is given by Eq. (6-122). With A set equal to the nozzle exit area, the exit Mach number, pressure, and temperature may be calculated. Only if the exit pressure equals the ambient discharge pressure is the ultimate expansion velocity reached in the nozzle. Expansion will be incomplete if the exit pressure exceeds the ambient discharge pressure shocks will occur outside the nozzle. If the calculated exit pressure is less than the ambient discharge pressure, the nozzle is overexpanded and compression shocks within the expanding portion will result. 

The available isentropic head is usually calculated by computer, using any of the various equations of state. In the absence of such facihty, a quick and reasonably reliable calculation follows. In fact, this calculation is valuable as a cross-check on other methods because it is likely to be accurate within a few percent. 

By plotting Hugoniot curves in the pressure-particle velocity plane (P-u diagrams), a number of interactions between surfaces, shocks, and rarefactions were solved graphically. Also, the equation for entropy on the Hugoniot was expanded in terms of specific volume to show that the Hugoniot and isentrope for a material is the same in the limit of small strains. Finally, the Riemann function was derived and used to define the Riemann Invarient. 

Because the Griineisen ratio relates the isentropic pressure, P, and bulk modulus, K, to the Hugoniot pressure, P , and Hugoniot bulk modulus, K , it is a key equation of state parameter. 

Ruoff (1967) first showed how the coefficients of the shock-wave equation of state are related to the zero pressure isentropic bulk modulus, and its first and second pressure derivatives, K q and Kq, via... 

Moreover, upon comparing (4.32) with (4.14), it can be seen that (Jeanloz and Grover, 1988) the Birch-Murnaghan equation (4.32) with a2 = 0 describes the isentropic equation of state provided the linear shock-particle velocity relation (4.5) describes the Hugoniot. In combination, these require that... 

Understanding such interaction is important both in predicting the amplitudes of shock waves transmitted across interfaces (in the case where the equations of state of all materials are known), and in determining release isentropes or reflected Hugoniots (when measurement of the equation of state is needed). Consider first a shock wave in material A being transmitted to a... 

For many condensed media, the Mie-Gruneisen equation of state, based on a finite-difference formulation of the Gruneisen parameter (4.18), can be used to describe shock and postshock temperatures. The temperature along the isentrope (Walsh and Christian, 1955) is given by... 

Equation (A.26) holds, for example, in materials which obey the isentropic elastic relationship... 

With incompressibile fluids, the value of the acoustic speed tends toward infinity. For isentropic flow, the equation of state for a perfect gas can be written ... 

For isentropic flow, the energy equation can be written as follows, noting that the addition of internal and flow energies can be written as the enthalpy (h) of the fluid ... 

The polytropic efficiency in a turbine can be related to the isentropic efficiency and obtained by combining the previous two equations... 

Since the compressibility does not change the isentropic temperature rise, it should be factored out of the AT portion of the equation. To achieve this for moderate changes in compressibility, an assumption can be made as follows ... 

It has played a dual role, one in Equation 2.18 on specific heat ratio and the other as an isentropic exponent in Equation 2.53. In the previous calculation of the speed of sound. Equation 2.32, the k assumes the singular specific heat ratio value, such as at compressor suction conditions. When a non-perfect gas is being compressed from point 1 to point 2, as in the head Equation 2.66, k at 2 will not necessarily be the same as k at 1. Fortunately, in many practical conditions, the k doesn t change very... 

The analysis of Hawthorne and Davis [1] for irreversible a/s cycles is developed using the criteria of component irreversibility, firstly for the simple cycle and subsequently for the recuperative cycle. In the main analyses, the isentropic efficiencies are used for the turbomachinery components. Following certain significant relationships, alternative expressions, involving polytropic efficiency and. tc and jcj, are given, without a detailed derivation, in equations with p added to the number. 

The isentropic temperature rise for maximum specific work (J , ) is obtained by differentiating Eq. (3.11) with respect to x and equating the differential to zero, giving... 

The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

Initial shock-wave overpressure can be determined from a one-dimensional technique. It consists of using conservation equations for discontinuities through the shock and isentropic flow equations through the rarefaction waves, then matching pressure and flow velocity at the contact surface. This procedure is outlined in Liepmatm and Roshko (1967) for the case of a bursting membrane contained in a shock tube. From this analysis, the initial overpressure at the shock front can be calculated with Eq. (6.3.22). This pressure is not only coupled to the pressure in the sphere, but is also related to the speed of sound and the ratio of specific heats. 

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... 

Adiabatic compression (termed adiabatic isentropic or constant entropy) of a gas in a centrifugal machine has the same characteristics as in any other compressor. That is, no heat is transferred to or from the gas during the compression operation. The characteristic equation... 

To analyze compressible flow through chokes it is assumed that the entropy of the fluid remains constant. The equation of isentropic flow is... 

All other coefficient equations are identically zero. The second of Eq. (1-101), combined with the equation of conservation of mass, Eq. (1-77), shows that the motion is isentropic ... 

Assuming isentropic expansion of the combustion gases through the nozzle and Pe = Ptt, the exhaust velocity can be determined from the equation... 

Solution In a reversible adiabatic expansion, 6qrev = T dS = 0. Thus, the process is isentropic, or one of constant entropy. To obtain an equation relating p, V and T, we start with... 

The velocity uw = fkP2v2 is shown to be the velocity of a small pressure wave if the pressure-volume relation is given by Pifi = constant. If the expansion approximates to a reversible adiabatic (isentropic) process k y, the ratio of the specific heats of the gases, as indicated in equation 2.30. 

It has been seen in deriving equations 4.33 to 4.38 that for a small disturbance the velocity of propagation of the pressure wave is equal to the velocity of sound. If the changes are much larger and the process is not isentropic, the wave developed is known as a shock wave, and the velocity may be much greater than the velocity of sound. Material and momentum balances must be maintained and the appropriate equation of state for the fluid must be followed. Furthermore, any change which takes place must be associated with an increase, never a decrease, in entropy. For an ideal gas in a uniform pipe under adiabatic conditions a material balance gives ... 

The gas continues to expand isentropically and the pressure ratio w is related to the flow area by equation 4,47. If the cross-sectional area of the exit to the divergent section is such that >r 1 = (10,000/101.3) = 98.7, the pressure here will be atmospheric and the expansion will be entirely isentropic. The duct area, however, has nearly twice this value, and the flow is over-expanded, atmospheric pressure being reached within the divergent section. In order to satisfy the boundary conditions, a shock wave occurs further along the divergent section across which the pressure increases. The gas then expands isentropically to atmospheric pressure. 

The isentropie expansion of the gas to atmospheric pressure. The gas now expands isentropically from P, to Pa (= 101.3 kN/m2) and the flow area increases from A to the full bore of 0.07 m2. Denoting conditions at the outlet by suffix a, then from equation 4.46 ... 

From equation 8,37. work done in isentropic compression of 1 kg of gas... 

Equation B.28 is useful to estimate the polytropic coefficient n if the inlet and outlet pressures are known, along with an estimate of the isentropic efficiency. Knowing the polytropic coefficient allows the outlet temperature for a real gas compression to be estimated from Equation B.26. 

Generally, the efficiency of steam turbines decreases with decreasing load. The overall turbine efficiency can be represented by two components the isentropic efficiency and the mechanical efficiency. The mechanical efficiency reflects the efficiency with which the energy that is extracted from steam is transformed into useful power and accounts for mechanical frictional losses, heat losses, and so on. The mechanical efficiency is high (typically 0.95 to 0.99)6. However, the mechanical efficiency does not reflect the efficiency with which energy is extracted from steam. This is characterized by the isentropic efficiency introduced in Figure 2.1 and Equation 2.3, defined as ... 

Equation 23.7 is based on the actual change in steam enthalpy across the turbine. Although both Equations 23.6 and 23.7 have the same form, their coefficients have completely different meanings. Comparing Equations 23.6 and 23.7, it becomes apparent that the slope of the linear Willans Line (Equation 23.6) is related to the isentropic enthalpy change and turbine isentropic efficiency9. 

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