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Note for Basics of Mechanical Engineering - BME by Yash Yadav

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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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www.vidyarthiplus.com definition of heat is that a body never contains heat. Rather, heat can be identified only as it crosses the boundary. Similar to work, heat is also a form of energy transfer occurring at the boundary of the system and is a path function. 2.12 Sign Convention of Heat • Heat given into a system is positive • Heat coming out of the system is negative Fig. 2.8 Sign convention of work 2.13 Modes of Heat Exchange Conduction, convection and radiation are the three possible modes of heat transfer between systems and between system and its surroundings. Conduction occurs without bulk movement of molecules. Energy transfer in conduction is due to lattice vibration and free electron movement. It is the predominant mode of heat transfer in solids. Convection occurs with bulk movement of molecules and therefore, occurs in gases and liquids. If the bulk movement or flow is due to an external device, it is known as forced convection. In the absence of an external device the flow is due to the difference in density caused by the temperature difference. This mode is known as natural convection. Bodies separated by a distance may exchange heat in the form of electromagnetic waves without the participation of the intervening medium. It is known as radiation. It is generally a surface phenomenon. Sometimes as in the case of gas mixtures containing carbon dioxide and water vapour it is a volume phenomenon. 2.14 Sensible and Latent Heat It is known that a substance can exists in three phases namely solid, liquid and gas. When a substance is heated or cooled temperature of the substance increases or decreases respectively unless there is any phase change. Quantity of heat added or removed to change the temperature by unit degree is known as specific heat. For solids and liquids same quantity of heat is required to cause unit degree rise for both constant pressure heating as well as constant volume heating as Thermodynamics 1 [MIME 3110] 15 www.vidyarthiplus.com

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