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The Energy Equation 2 CHAPTER The Energy Equation 2 CHAPTER In this chapter the energy equation is derived in its various forms and the stagnation state and stagnation values are defined. Other important relations based on the adiabatic energy equation are also derived. Figure 2.1 shows the flow of a fluid through the control volume. Various thermofluid dynamic parameters at entry and exit sections are shown with subscripts 1 and 2 respectively. For the sake of generality heat (Q) supplied or rejected and work (W) output or input are also shown. Control volume c c 2 1 p 1 W v s 1 p H 2 1 v Z 2 Z 1 H 2 2 Q Datum,Z = 0 FIGURE 2.1 Steady flow through a control volume (open system) The energy Equation (1.8) Q = W + (E2 – E1) in Chapter 1, for a system was derived from the first law of thermodynamics. The energy terms (E) in Equation (1.8) may include gravitational potential energy, kinetic energy, internal energy, strain energy, magnetic energy, etc. As an example we may write 1 E = U + mgZ + mc2 (2.1) 2 ignoring other forms of energy. 36 THE ENERGY EQUATION 37 The differential form of Equation (2.1) is F 1 I dE = dU + mgdZ + md G c2 J (2.2) H 2 K Integrating Equation (2.2) between two given states 2 2 2 1 2 E – E = dE = z dU + mg dZ + m d(c2) 2 1 z z z 1 1 1 2 1 1 E – E = (U – U ) + mg (Z – Z ) + m (c 2 – c 2) (2.3) 2 1 2 1 2 1 2 2 1 Equation (2.3) in (1.8) yields a more general form of the energy equation 1 Q = W + (U – U ) + mg (Z – Z ) + m (c 2 – c 2) (2.4(a)) 2 1 2 1 2 2 1 In terms of specific values 1 q = w + (u – u ) + g(Z – Z ) + (c 2 – c 2) (2.4(b)) 2 1 2 1 2 2 1 2.1 ENERGY EQUATION FOR A NON-FLOW PROCESS As explained before in Section 1.42 (Chapter 1), processes like the expansion and compression of gases in a cylinder with a piston are non-flow processes in closed systems. The potential and kinetic energy terms for such processes are negligible compared to other quantities in the energy equation; and the work term W includes only shaft work. Therefore, Equation (2.4 (a)) reduces to Q = Ws + (U2 – U1) (2.5) dQ = dWs + dU Assumption of perfect gas gives dQ = p dV + mcv dT (2.6) Equation (2.6) on integration yields 2 Q = z p dV + mc (T – T ) (2.7) 1 v 2 1 2 The value of the integral z p dV depends on the type of process, e.g., p = constant, 1 pV = constant, etc. 2.2 ENERGY EQUATION FOR A FLOW PROCESS As stated before in Section 1.41 (Chapter 1) expansion of steam and gas in turbines and compression of air and gases in turbocompressors are examples of flow processes in open systems. In such processes the work term (W) includes the flow work also. W = Ws + (p2V2 – p1V1) (2.8) 38 FUNDAMENTALS OF COMPRESSIBLE FLOW Equations (2.4 (a)) and (2.8) give 1 Q = W + (p V – p V ) + (U – U ) + mg (Z – Z ) + m (c 2 – c 2) s 2 2 1 1 2 1 2 1 2 2 1 1 Q = W + (U + p V ) – (U – p V ) + mg (Z – Z ) + m (c 2 – c 2) s 2 2 2 1 1 1 2 1 2 2 1 putting U + pV = H 1 Q = W + (H – H ) + mg (Z – Z ) + m (c 2 – c 2) s 2 1 2 1 2 2 1 1 1 H + mgZ + mc 2 + Q = H + mgZ + mc 2 + W (2.9 (a)) 1 1 2 1 2 2 2 2 s In terms of specific values 1 1 h + gZ + c 2 + q = h + gZ + c 2 + w (2.9 (b)) 1 1 2 1 2 2 2 2 s Equations (2.9(a)) and (2.9 (b)) are the Steady Flow Energy Equations (SFEE). Generally in flow problems of gases and vapours the magnitude of g(Z2 – Z1) is negligible compared to other quantities. Therefore, 1 1 h + c 2 + q = h + c 2 + w (2.10) 1 2 1 2 2 2 s 2.3 THE ADIABATIC ENERGY EQUATION In some engineering problems the heat transfer (q) during the process is negligibly small and can be ignored. Expansion of gases and vapours in turbines are examples of such processes. For such processes Equations (2.9 (b)) and (2.10) reduce to 1 1 h + gZ + c 2 = h + gZ + c 2 + w (2.11) 1 1 2 1 2 2 2 2 s 1 1 h + c 2 = h + c 2 + w (2.12) 1 2 1 2 2 2 s 2.3.1 Work in Flow Process If the difference in the kinetic energy terms in Equation (2.12) is negligible, it reduces to 2 ws = h1 – h2 = – z dh (2.13) 1 For a perfect gas Equation (2.13) gives 2 ws = – cp z dT 1 ws = cp (T1 – T2) (2.14) For a reversible process Equation (2.13) can be written as 2 dp 2 w = – z = – z v dp (2.15) s 1 ρ 1 THE ENERGY EQUATION 39 2.3.2 Adiabatic Energy Transformation Equations (2.11) and (2.12) are adiabatic energy Equations valid for processes involving both energy transfer (on account of shaft work) and energy transformation. Some adiabatic processes involve only energy transformation, e.g., expansion of gases in nozzles and their compression in diffusers. In these processes shaft work is absent and Equations (2.11) and (2.12) are modified to 1 1 h + gZ + c 2 = h + gZ + c 2 (2.16) 1 1 2 1 2 2 2 2 1 1 h + c 2 = h + c 2 (2.17) 1 2 1 2 2 2 (when change in elevation is ignored). 2.3.3 Stagnation Enthalpy Figure 2.2 shows the deceleration of a gas stream at velocity c, pressure p and temperature T to almost zero velocity (c ~− 0) in an infinitely large settling chamber. p 0 T Flow c, p, T 0 c » 0 h 0 Large settling chamber FIGURE 2.2 Deceleration of a gas to stagnation state Stagnation enthalpy of a gas or a vapour is its enthalpy when it is adiabatically decelerated to zero velocity at zero elevation. Putting h1 = h, Z1 = Z and c1 = c for the initial state and h2 = h0, Z2 = 0, c2 = 0 for the final state in Equation (2.16) the value of the stagnation enthalpy (h0) is obtained. 1 h = h + gZ + c2 (2.18) 0 2 But as mentioned before the magnitude of gZ compared to other quantities is generally negligible. 1 Therefore, h = h + c2 (2.19) 0 2 For an adiabatic energy transformation process stagnation enthalpy remains constant. Therefore, by differentiation Equation (2.19) gives dh + c dc = 0 (2.20) 2.3.4 Stagnation Temperature Stagnation temperature is the temperature of the gas when it is adiabatically decelerated to zero velocity at zero elevation. 40 FUNDAMENTALS OF COMPRESSIBLE FLOW For a perfect gas, this is defined through stagnation enthalpy. Equation (2.19) for a perfect gas gives 1 c T = c T + c2 p 0 p 2 c2 T0 = T + (2.21) 2c p 2 The quantity c /2cp is known as the velocity temperature (Tc) corresponding to the velocity c. T0 = T + Tc (2.22) c2 Tc = (2.23) 2c p From Equation (2.21) T c2 0 = 1 + T 2cTp T c2 0 = 1 + T 21γγRT /(− ) c2 But γRT = a2 and = M2 a2 T 1 c2 Therefore, 0 = 1 + γ− T 2 a2 T γ−1 0 = 1 + M2 (2.24) T 2 2.4 STAGNATION VELOCITY OF SOUND For a given value of the stagnation temperature Equation (1.78) gives the velocity of sound for a perfect gas a0 = γRT0 (2.25) γ − 1 By substituting R = c γ p a0 = ()γ−1 cTp 0 a0 = ()γ−1 h0 (2.26) THE ENERGY EQUATION 41 2.5 STAGNATION PRESSURE Stagnation pressure is the pressure of a fluid which is attained when it is decelerated to zero velocity at zero elevation in a reversible adiabatic (isentropic) process. For given values of static pressure and temperature its value can be derived from the stagnation temperature. γγ/(−1 ) γγ/(− 1 ) p0 F T I F γ − 1 I = G 0 J = G1 + M 2 J (2.27) p H T K H 2 K 2.6 STAGNATION DENSITY For given values of stagnation pressure and temperature of an ideal gas the stagnation density is given by p0 ρ0 = (2.28) RT0 From isentropic relations 1/γ ρ F p I 0 = G 0 J (2.29) ρ H p K 1/()γ − 1 1/()γ − 1 ρ F T0 I F γ − 1 2 I 0 = HG KJ = HG1 + M KJ (2.30) ρ T 2 2.7 STAGNATION STATE Figure 2.3 depicts the deceleration of a gas in both the isentropic and adiabatic processes. It is observed that during the adiabatic deceleration the gas experiences decrease in stagnation pressure and increase in entropy. T p p¢ T 0 0 0 0 0¢ Isentropic Adiabatic p T s FIGURE 2.3 Deceleration of a gas to stagnation conditions The state of a fluid attained by isentropically decelerating it to zero velocity at zero elevation is referred to as the stagnation state.
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