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Note for Basics of Mechanical Engineering - BME by Yash Yadav

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www.vidyarthiplus.com Chapter 2 WORK AND HEAT In the previous chapter, the different thermodynamic systems and their characteristics were discussed. To undergo a change of state, the system has to interact with its surroundings. Work and heat transfers across the boundaries cause these changes. In this chapter various forms of work and modes of heat transfers are discussed. 2.1 Work as Defined in Mechanics work is done when the point of application of a force moves in the direction of the force. The product of the force and the distance moved in the direction of the force is equal to the amount of the work done. This simple definition of work confines only to the area of mechanics and can not be extended to the more complex problems in thermodynamics. Hence a new definition should be introduced to cover mechanical as well as the other forms of work. 2.2 The Thermodynamic Definition of Work Positive work is done by a system, during a given process, when sole effect external to the system could be reduced to the lifting of a mass. Consider a gas expanding in a piston cylinder arrangement as given in Figure 2.1. Here no mass is actually lifted against gravity. But if the existing surroundings is fitted with an arrangement as given in the Figure 2.2, there is a possibility of lifting the mass. Hence work is said to be done by the system. While exploring the possibility of lifting a mass the effects that are external to the system alone must be taken into account. For example, a lift with a person and a suitcase is considered as a system. If the person lifts the suitcase, it should not be taken into account, because this event occurs within the system. Thermodynamics 1 [MIME 3110] 10 www.vidyarthiplus.com

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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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