Work done during compression/expansion

An ideal Diesel engine receives air at 103.4 kPa and 27°C. Heat added to the air is 1016.6 kJ/kg, and the compression ratio of the engine is 13. Determine (a) the work added during the compression process, (b) the cut-off ratio, (c) the work done during the expansion process, (d) the heat removed from the air during the cooling process, (e) the MEP (mean effective pressure), and (f) the thermal efficiency of the cycle. 

The thermal efficiency of the process (QE) should be compared with a thermodynamically ideal Carnot cycle, which can be done by comparing the respective indicator diagrams. These show the variation of temperamre, volume and pressure in the combustion chamber during the operating cycle. In the Carnot cycle one mole of gas is subjected to alternate isothermal and adiabatic compression or expansion at two temperatures. By die first law of thermodynamics the isothermal work done on (compression) or by the gas (expansion) is accompanied by the absorption or evolution of heat (Figure 2.2). 

To find the work done during the isothermal expansion of a gas, that is, the work done when the gas changes its volume, by expansion or compression, at a constant temperature. A contraction may be regarded as a negative expansion. There are three interesting applications. 

The total work done during the cycle is equal to the sum of these four components. It is represented by the area 1-2-3-4, which is equal to area 1-2-5-6 less area 4-3-5-6. If the compression and expansion are taken as isentropic, the work done per cycle is... 

The adiabatic expansion and compression serve only to change the temperature of tire gas widrout heat being absorbed or evolved, i.e. iso-entropic changes. The heat changes are therefore only related to the work which is done during the isothermal stages, which is given by... 

The expansion of an ideal gas in the Joule experiment will be used as a simple example. Consider a quantity of an ideal gas confined in a flask at a given temperature and pressure. This flask is connected through a valve to another flask, which is evacuated. The two flasks are surrounded by an adiabatic envelope and, because the walls of the flasks are rigid, the system is isolated. We now allow the gas to expand irreversibly into the evacuated flask. For an ideal gas the temperature remains the same. Thus, the expansion is isothermal as well as adiabatic. We can return the system to its original state by carrying out an isothermal reversible compression. Here we use a work reservoir to compress the gas and a heat reservoir to remove heat from the gas. As we have seen before, a quantity of heat equal to the work done on the gas must be transferred from the gas to the heat reservoir. In so doing, the value of the entropy function of the heat reservoir is increased. Consequently, the value of the entropy function of the gas increased during the adiabatic irreversible expansion of gas. 

Equation (1.3) is an expression for the work done as a result of a finite compression or expansion process, t This kind of work can be represented as an area on a pressure-vs.-volume (PV) diagram, such as is shown in Fig. 1.3. In this case a gas having an initial volume Vt at pressure Pi is compressed to volume V2 at pressure P2 along the path shown from 1 to 2. This path relates the pressure at any point during the process to the volume. The work required for the process is given by Eq. (1.3) and is represented on Fig. 1.3 by the area under the curve. The SI unit of work is the newton-meter or joule, symbol J. In the English engineering system the unit often used is the foot-pound force (ft lbr). 

Let us also assume that the gas in the cylinder is at point A to start with, i.e., it is at a temperature T, pressure and volume If it is allowed to expand adiahatically and reversibly so that the final temperature is Tj, then the state of gas will be represented by point/), i.e., the gas now has temperature T2, pressure P, and volume (the path in this instance is opposite to the arrow mark). The line AD represents the path the gas has followed during adiabatic expansion from A to D. The AD curve is in fact the plot of PV = Constant passing through point The work done by the gas is represented by the area under the curve AD (shaded region). If the gas is again adiahatically compressed from/) to 4 (towards the arrow mark), an equal amount of work will have to be done on the gas and the gas would return to its original state. 

Tlie work done by gases occurs during the expansion stroke. Mechanical systems used today make compression equal to the expansion stroke. It is possible to avoid this combination by using a special cam profile adjustment along with the well known Miler principle. If, during the first part of the upward movement of the piston you maintain the admission valve open, you are in fact reducing the compression stroke. Mazda is said to be marketing such a solution soon (fig. 12). 

This means that a reversible change is more efficient than any irreversible process between the same two states. During a compression the irreversible work done to the system is larger than Wrev because part of is wasted in overcoming dissipative forces that oppose the compression. Likewise, during an expansion the irreversible work done by the system is less than VV gj, because part of must overcome dissipa-... 

A Figure 19.5 An irreversible process. Initially an ideal gas is confined to the right half of a cylinder. When the partition is removed, the gas spontaneously expands to fill the whole cylinder. No work is done by the system during this expansion. Using the piston to compress the gas back to its original state requires the surroundings to do work on the system. 

Mechanical equilibrium. If a fluid, at a pressure p, expands by an amount dF against an external pressure p—Sp) the work done by the fluid is p—dp)dV. The corresponding recompression of the fluid requires a pressure, p+Sp), which is larger than p, and the work done. on the fluid is —(p+ ) dF. These two quantities ofwork become equal, and the processes thus satisfy the de tion of reversibility, only in the limit where 6p approaches zero. It will be noted that the same condition maximizes the work done by the fluid during expansion and minimizes the work done on the fluid during compression. 

If the VRLA battery is cycled to any depth of discharge, the electrodes will grow when in the sulfated state and when recharged the electrode thickness will likely shrink. During this expansion or contraction, it is important for the separator material to have sufficient resiliency to maintain intimate contact with the electrode surfaces so that the battery can continue to function. To that end, much work has been done to develop compression and resiliency curves for various AGM materials both in a dry and wet state [33]. 

Here heat has been completely converted into work but the system is not in the same state as it was to start with. The system can be brought to its initial state by reversibly compressing the gas to a pressure of P The gas will give out heat to the heat reservoir so that its temperature is maintained at By the time the pressure P, is reached, an equal amount of work, as was performed by the gas during expansion fromP, to P2, will have to be done on it. Consequently, the gas will return an equal amount of heat to the heat source. Hence, after such a system is made to perform in a cycle, the net effect is that no heat is taken up or given out by the system and no work is done by or on the system. This type of experience is compatible with statement (//) of the second law, which says that it is not possible to convert heat into work by a constant temperature cycle. 

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