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Note for Basics of Mechanical Engineering - BME by Yash Yadav

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www.vidyarthiplus.com 2.3 Units of Work and Power In the international system (SI), the unit of force is Newton (N) and that of distance is metre (m). Hence the unit of work is Nm which is also given a special name Joule. In most of the applications large quantity of work is involved. Therefore kJ is commonly used. Rate of doing work is known as power. Hence its unit is Nm/S or J/S which is again given a special name Watts(W). 2.4. Sign Convention of Work • Work done by the system on the surroundings is considered as positive work. • Work done on the system by the surroundings is taken as negative work. 2.5. Displacement Work Consider a piston cylinder arrangement as given in the Figure 2.4. If the pressure of the fluid is greater than that of the surroundings, there will be an unbalanced force on the face of the piston. Hence, the piston will move towards right. Thermodynamics 1 [MIME 3110] 11 www.vidyarthiplus.com

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www.vidyarthiplus.com Force acting on the piston = Pressure × Area = pA ∴ Work done = Force × distance = pA × dx = pdV where dV - change in volume. This work is known as displacement work or pdV work corresponding to the elemental displacement dx. To obtain the total work done in a process, this elemental work must be added from the initial state to the final state. Mathematically, . 2.6 Evaluation of Displacement Work 2.6.1. Constant Pressure Process Figure 2.5 shows a piston cylinder arrangement containing a fluid. Let the fluid expands such that the pressure of the fluid remains constant throughout the process. Figure 2.6 shows the process in a p-V diagram. The mathematical expression for displacement work can be obtained as follows: = p(V2 – V1) ...(2.1) This expression shows that the area under a curve in a p-V diagram gives work done in the process. 2.6.2. Constant volume process Consider a gas contained in a rigid vessel being heated. Since there is no change in volume, the displacement work . Thermodynamics 1 [MIME 3110] 12 www.vidyarthiplus.com

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www.vidyarthiplus.com 2.6.3 Hyperbolic process Let the product of pressure and volume remains constant at all the intermediate states of a process. In the p-V diagram it will be a hyperbola as given in Figure 2.7. 2 W2 = 1 ∫ pdV 1 2 ∫ CdV where C=pV = 1 2 = C 1 ∫ V dV 1 = C ln (V2/V1) w2 = p1V1ln(V2/V1) (or) p2V2ln (V2/V1) 1 ...(2.2) For Ideal gases when temperature remains constant, pV will be constant i.e., isothermal process are hyperbolic processes for an ideal gas. 2.6.4 Polytropic Process Any process can be represented by the general form pVn = constant. Based on the valve of n, the process differs as given below;For other values of n, the process is known as polytropic process. Figure 2.8 shows the polytropic process of various possible polytropic index ‘n’ on p-V coordinates. Expression for displacements work for a polytropic process can be obtained as follows : 2 ∫ pdV W2 = 1 1 2 = C ∫V n dV where C = pVn 1 Thermodynamics 1 [MIME 3110] 13 www.vidyarthiplus.com

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www.vidyarthiplus.com 2 ∫ = C V − n dV 1 2  V − n +1  = C    − n + 1 1 2  CV − n +1 − CV1 − n +1  =  2  − n +1  1 =  p 2V2 nV2 − n +1 − p1V1 nV1 − n +1    − n +1   since C = p1V1n = p2Vn2 1 =  p 2V2 − p1V1   − n +1    ...(2.3) 2.7 Work is a Path Function Consider a working substance initially occupying 0.2 m3 at 1 bar as represented by state 1 in the Figure 2.9. Let the system changes its state such that the final volume is 0.05m3 and pressure 2 bar. The change of state may occur along the paths 1A2,1B2 or 1C2. As mentioned earlier, area under the curve representing the process in a p-V diagram gives the work done in the process. Comparing the area under the paths 1A2, 1B2 and 1C2, it is clear that the work done in these paths are different. Hence it can be concluded that the amount of work done is not only a function of the end states of a process, but also the path followed between the states. Therefore work is a path function. 2.8 Additivity of Work Over Processes If a system is taken through two or more number of processes, the total work done is the sum of work done in the individual processes. Let a system executes three processes as shown in Figure 2.10. The total work done, 1 ...(2.4) W4 = 1W2 + 2W3 + 3W4 2.11 Heat Heat is the interaction between systems which occurs by virtue of their temperature difference when they communicate. If a system, at a given temperature is brought in contact with another system (or surroundings) at a lower temperature, it can be observed that heat is transferred from the system at the higher temperature to the system at lower temperature. This heat transfer occurs solely because of the temperature difference between the two systems. Another important aspect of the Thermodynamics 1 [MIME 3110] 14 www.vidyarthiplus.com

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