The vertical line through point I meets the isotherm line t,=90 Cat point 2 and we can obtain h,=123.5 kJ/(kg dry air ), d 2=d=0.012 5 kg/(kg dry air Heater Fig 7-14 Schematic of Drying Device Fig 7-15 Drying Process on h-d chart Drawing a constant enthalpy line through point 2, it intersects with the isotherm line 1=40 Cat (2)The amount of moisture l kg dry air absorbing in the drying chamber Ad=d -d2=d-d=325-12.5=20 g/kg dry air) (3) The amount of dry air needed to absorb l kg water is 50 kg The amount of the heat absorbed in the heater is Q=m(h2-h)=50×(123.5-565)=3350kJ
123 The vertical line through point 1 meets the isotherm line 2 t = 90 ℃at point 2 and we can obtain 2 h =123.5 kJ/(kg dry air）， 2 1 d d = = 0.012 5 kg/(kg dry air ) . Drawing a constant enthalpy line through point 2, it intersects with the isotherm line 3 t = 40 ℃at point 3, 3 d = 0.0325 kg/(kg dry air) . ⑵ The amount of moisture 1 kg dry air absorbing in the drying chamber 3 2 3 1 = − = − = − = d d d d d 32.5 12.5 20 g/(kg dry air) ⑶ The amount of dry air needed to absorb 1 kg water is 1 1 50 kg 0.02 ma d = = = The amount of the heat absorbed in the heater is 2 1 ( ) 50 (123.5 56.5) 3 350 kJ Q m h h = − = − = a Fig 7-14 Schematic of Drying Device Fig 7-15 Drying Process on h d − chart
Chapter 8 Gas and vapor Flow and compressed The adiabatic flow process of gas or vapor in a nozzle and a diffuser is not only widely used in steam turbines, gas turbines and other power equipment, but also applied to ejectors, impeller-type compressors, burners and other thermal devices in ventilation, air conditioning and gas engineering This chapter focuses on how the flow properties vary along the flow direction with the flow area of the channel when gas flows through a nozzle or other devices ideally without friction, and the energy transfer and transformation of gases during this flow process is also emphasized. In addition irreversible flow process with friction and throttling process are also analyzed in brief. Compressor is a machine which can be used to compress various kinds of gases. Gas compressed by a compressor is called compressed gas, its pressure being increased Compressed gas is widely used ngineering, such as all kinds of start-up mechanical, pneumatic conveying in granular materials, metallurgical furnace blast, high pressure oxygen chamber, refrigeration engineering and chemical industry, and so on According to the operational principle of compressors and their structures, compressors can be categorized into piston, impeller (centrifugal, axial flow, rotary positive displacement)and the ejector types of compressor, and so on In terms of the highest pressure of the compressed gases they produces, compressors can be divided into fan(<1 15 kPa), blower(115-350 kPa)and compressor(>350 kPa). The task of thermodynamic analysis on compressors is to calculate the compressor power consumption during the process of compressing a certain amount of gas from its initial state to a predetermined final pressure; and it is to find ways of saving energy. This chapter mainly focuses on piston type compressors. The means to save energy are suggested 8.1 Sonic Speed and Mach number Sonic speed(speed of sound) is an important property in studying compressible fluids flow. It is the speed at which an infinitesimally small pressure wave caused by weak disturbance travels through a continuous medium. In gas medium, the pressure wave propagation process can be approximatel regarded as an isentropic process. It has been investigated in physics that the sonic speed a in compressible fluid is -( where, the subscript s stands for isentropic process For isentropic process of ideal gas, there is Thus (8-2) where K, R and Represents the specific heat ratio, the gas constant and the thermodynamic emperature of an ideal gas, respectively Eq. (8-1)and(8-2) indicate that sonic speed is a property depending on the variety of the medium as well as its state. Thus, the concept of local sonic speed is introduced to distinguish the sonic speed in a medium at different states. The so-called local sonic speed is the sonic speed at a certain state(p,v,T). For example, the sonic speed in the air at t=20℃isa=√14×287×293=343ms while at an altitude of 10,000 m and the temperature drops to [=-50C, the sonic speed in the air is
124 Chapter 8 Gas and Vapor Flow and Compressed The adiabatic flow process of gas or vapor in a nozzle and a diffuser is not only widely used in steam turbines, gas turbines and other power equipment, but also applied to ejectors, impeller-type compressors, burners and other thermal devices in ventilation, air conditioning and gas engineering. This chapter focuses on how the flow properties vary along the flow direction with the flow area of the channel when gas flows through a nozzle or other devices ideally without friction, and the energy transfer and transformation of gases during this flow process is also emphasized. In addition, irreversible flow process with friction and throttling process are also analyzed in brief. Compressor is a machine which can be used to compress various kinds of gases. Gas compressed by a compressor is called compressed gas, its pressure being increased. Compressed gas is widely used in engineering, such as all kinds of start-up mechanical, pneumatic conveying in granular materials; metallurgical furnace blast, high pressure oxygen chamber, refrigeration engineering and chemical industry, and so on. According to the operational principle of compressors and their structures, compressors can be categorized into piston, impeller (centrifugal, axial flow, rotary positive displacement) and the ejector types of compressor, and so on. In terms of the highest pressure of the compressed gases they produces, compressors can be divided into: fan (<115 kPa), blower (115 ~ 350 kPa) and compressor (＞350 kPa). The task of thermodynamic analysis on compressors is to calculate the compressor power consumption during the process of compressing a certain amount of gas from its initial state to a predetermined final pressure; and it is to find ways of saving energy. This chapter mainly focuses on piston type compressors. The means to save energy are suggested. 8.1 Sonic Speed and Mach number Sonic speed (speed of sound) is an important property in studying compressible fluids flow. It is the speed at which an infinitesimally small pressure wave caused by weak disturbance travels through a continuous medium. In gas medium, the pressure wave propagation process can be approximately regarded as an isentropic process. It has been investigated in physics that the sonic speed a in a compressible fluid is 2 s s p p a v v = = − （8-1） where, the subscript s stands for isentropic process. For isentropic process of ideal gas, there is, s p p v v = − Thus a pv RT = = （8-2） where , and R T represents the specific heat ratio, the gas constant and the thermodynamic temperature of an ideal gas, respectively. Eq. (8-1) and (8-2) indicate that sonic speed is a property depending on the variety of the medium as well as its state. Thus, the concept of local sonic speed is introduced to distinguish the sonic speed in a medium at different states. The so-called local sonic speed is the sonic speed at a certain state ( , , ) p v T . For example, the sonic speed in the air at t = 20 ℃ is a = = 1.4 287 343 m/s × × 293 ; while at an altitude of 10,000 m and the temperature drops to t =−50 ℃, the sonic speed in the air is
only 299 m/s Another important parameter in the analysis of compressible flow is the Mach number M. It is the ratio of the actual velocity c of a fluid to the speed of sound in the same fluid at the same state Fluid flow regimes are often described in terms of the flow Mach number If c<a, then M <l, the flow velocity is less than the local sonic speed. The flow is subsonic, If c=a, then M=1, the flow velocity is equal to the local sonic speed. The flow is sonic flow, If c>a, then M>l, the flow velocity is greater than the local sonic speed. The fle Ifc? a, then M? 1,(usually, when M>5, the flow is called hypersonic flow. 8.2 One-dimensional Isentropic Steady Flow 8.2.1 Basic Equations of One-dimensional Isentropic Steady Flow As mentioned before, steady-flow process is a process during which a fluid flows through a control volume steadily. That is, the fluid properties can change from point to point within the control volume, but at any point, they remain constant during the entire process. In practical projects, many devices can be treated as steady flow devices. For a steady flow through a duct, usually the properties change only in the flow direction Thus the flow can be approximated as a one-dimensional flow. Actual flows are irreversible and the fluid may exchange heat with its surroundings. However, in general, ducts are well insulated and the fluid flows very fast. The heat transferred to the surroundings is negligible. For simplicity, this duct flow can be considered as a reversible, adiabatic process, that is, an isentropic process. The error caused by irreversibilities can be corrected by exploring experimental factor. In this section, we will discuss the one-dimensional isentropic steady flow at first. It satisfies the following four basic equations (1)Conservation equation of mass (Continuity equation) All the properties of a steady flow any where in a duct do not change with time, and the mass flow rate through any flow area remains constant, that is (8-4a) where A is the cross-sectional area of the duct, m; c is the fluid velocity, m/s andv is the specific volume of the fluid. m /kg Differentiating and dividing the resultant equation by the mass flow rate, we obtain da dc dv A c (8-4b) Eq(8-4a)and Eq. 8-4b are referred to as the continuity equations of steady flow through duct They indicate the relationship among the mass flow rate, the velocity, the specific volume of the fluid and the cross-sectional area of the duct. Note that A, c and v are positive quantities. Thus, whether the cross-sectional area increases or decreases depends on the difference between the change rate of the specific volume and the change rate of the flow velocity. In general, the continuity equation can be applied to analyze any steady flow process of any kind of fluid, whether it is reversible or irreversible (2)Conservation equation of energy As we know, steady flow through any duct satisfies the following energy equation
125 only 299 m/s. Another important parameter in the analysis of compressible flow is the Mach number M. It is the ratio of the actual velocity c of a fluid to the speed of sound in the same fluid at the same state, a c M = (8-3) Fluid flow regimes are often described in terms of the flow Mach number. If c a , then M 1, the flow velocity is less than the local sonic speed. The flow is subsonic; If c a = , then M =1, the flow velocity is equal to the local sonic speed. The flow is sonic flow; If c a , then M 1 , the flow velocity is greater than the local sonic speed. The flow is supersonic; If c a ? ,then M ? 1,（usually, when M 5 , the flow is called hypersonic flow. 8.2 One-dimensional Isentropic Steady Flow 8.2.1 Basic Equations of One-dimensional Isentropic Steady Flow As mentioned before, steady-flow process is a process during which a fluid flows through a control volume steadily. That is, the fluid properties can change from point to point within the control volume, but at any point, they remain constant during the entire process. In practical projects, many devices can be treated as steady flow devices. For a steady flow through a duct, usually the properties change only in the flow direction. Thus the flow can be approximated as a one-dimensional flow. Actual flows are irreversible and the fluid may exchange heat with its surroundings. However, in general, ducts are well insulated and the fluid flows very fast. The heat transferred to the surroundings is negligible. For simplicity, this duct flow can be considered as a reversible, adiabatic process, that is, an isentropic process. The error caused by irreversibilities can be corrected by exploring experimental factor. In this section, we will discuss the one-dimensional isentropic steady flow at first. It satisfies the following four basic equations. ⑴ Conservation equation of mass（Continuity equation） All the properties of a steady flow any where in a duct do not change with time, and the mass flow rate through any flow area remains constant, that is, A c m v = (8-4a) where A is the cross-sectional area of the duct, m2 ; c is the fluid velocity, m/s and v is the specific volume of the fluid, m3 /kg. Differentiating and dividing the resultant equation by the mass flow rate, we obtain d d d 0 A c v A c v + − = (8-4b) Eq. (8-4a) and Eq. 8-4b are referred to as the continuity equations of steady flow through duct. They indicate the relationship among the mass flow rate, the velocity, the specific volume of the fluid and the cross-sectional area of the duct. Note that A,c and v are positive quantities. Thus, whether the cross-sectional area increases or decreases depends on the difference between the change rate of the specific volume and the change rate of the flow velocity. In general, the continuity equation can be applied to analyze any steady flow process of any kind of fluid, whether it is reversible or irreversible. (2) Conservation equation of energy As we know, steady flow through any duct satisfies the following energy equation
+edc +gd=+ar Regarding to a steady flow in a short duct, as the change in its potential energy is very small and approximates zero, that is, gdz=0. Heat exchange between the fluid and its surroundings is also negligible. In addition, there is no work output or input. Substituting 8q =0, 8w.=0 and gdz=0 into the above equation, it is simplified to be Integrating the equation above, we get, Ah==Ac Eq.(8-5b)indicates that the increase in the kinetic energy of the fluid comes from the decrease in its enthalpy. It can be rewritten as h,+C=h+c=h+c=constant (8-5c) It indicates that the sum of the enthalpy and the kinetic energy of the fluid on any cross section in the flow direction remains constant for an adiabatic steady flow through a duct In terms of the definition of enthalpy, the energy equation( 8-5a) of the steady flow in the duct will reduce to du +d(pv)=-cdc or du+ pdv +vdp=-cdc Since 8q=du+ pdv, in terms of the first law of thermodynamics, and 8q=0 for the adiabatic flow it can be further reduced to (8-6)sets up the relation between the pressure and the velocity of the flow through a duct. It expresses that for an adiabatic flow through short ducts, the increase in the velocity of the flow(dc>0) will cause the pressure to decrease( dp <0). The decrease in the velocity(dc<0)of the flow will bring about the rise in its pressure The device that increases the velocity of a fluid at the expense of pressure is called nozzle; the device that increases the pressure of a fluid by decreasing the fluid velocity is called diffuser. nozzles are found in many engineering applications, including steam and gas turbines, aircraft and spacecr propulsion systems, and even industrial blasting nozzles and torch nozzles (3)Process equation he process equation of an isentropic process is constant (8-7a) For isentropic process of ideal gases, specific heat ratio x approximately remains constant; For real gases, such as isentropic steam flow through nozzle, Eq (8-7a)can also be adopted to approximate the flow. In this case, K is just an empirical constant without the meaning of specific heat ratio Differential form of the process equation( 8-7a)can be written as 中+xh 0 (8-7b) It indicates that the pressure definitely drops as the flow expands
126 2 s 1 δ d d d δ 2 q h c g z w = + + + Regarding to a steady flow in a short duct, as the change in its potential energy is very small and approximates zero, that is, g zd =0. Heat exchange between the fluid and its surroundings is also negligible. In addition, there is no work output or input. Substituting s q w = 0, 0 and g zd =0 into the above equation, it is simplified to be 1 2 d d 2 − = h c （8-5a） Integrating the equation above, we get, 1 2 2 − = h c （8-5b） Eq. (8-5b) indicates that the increase in the kinetic energy of the fluid comes from the decrease in its enthalpy. It can be rewritten as 2 2 2 2 2 1 1 1 1 1 constant 2 2 2 h c h c h c + = + = + = （8-5c） It indicates that the sum of the enthalpy and the kinetic energy of the fluid on any cross section in the flow direction remains constant for an adiabatic steady flow through a duct. In terms of the definition of enthalpy, the energy equation (8-5a) of the steady flow in the duct will reduce to, d d( ) d u pv c c + = − or d d d d u p v v p c c + + = − Since δq u p v = + d d , in terms of the first law of thermodynamics, and δq = 0 for the adiabatic flow, it can be further reduced to c c v p d d = − (8-6) Eq. (8-6) sets up the relation between the pressure and the velocity of the flow through a duct. It expresses that for an adiabatic flow through short ducts, the increase in the velocity of the flow (dc>0) will cause the pressure to decrease（ d 0 p ）. The decrease in the velocity (dc<0) of the flow will bring about the rise in its pressure. The device that increases the velocity of a fluid at the expense of pressure is called nozzle; the device that increases the pressure of a fluid by decreasing the fluid velocity is called diffuser. Nozzles are found in many engineering applications, including steam and gas turbines, aircraft and spacecraft propulsion systems, and even industrial blasting nozzles and torch nozzles. (3) Process equation The process equation of an isentropic process is pv constant = (8-7a) For isentropic process of ideal gases, specific heat ratio p v c c = approximately remains constant; For real gases, such as isentropic steam flow through nozzle, Eq. (8-7a) can also be adopted to approximate the flow. In this case, is just an empirical constant without the meaning of specific heat ratio. Differential form of the process equation (8-7a) can be written as 0 dp dv p v + = （8-7b） It indicates that the pressure definitely drops as the flow expands
8.2.2 The Relations between the flow velocity and the flow area of the duct From the mass conservation equation(8-4b), we know da d Substituting eq. (8-6)and eq. (b)into eq(a) yields dc (8-8) Eq(8-8)indicates the relation between the flow velocity and the flow area of a duct. It governs the shape of a nozzle or a diffuser for subsonic or supersonic isentropic flow As both A and c are positiv quantities, we can get the following conclusion for isentropic steady flows through nozzles M> M<1 P,> Fig 8-1 Converging nozzle Fig 8-2 Diverging nozze 1. 8-3 Converging-diverging nozzle For subsonic flow(M<1), M-1 is negative, and thus dA and dc must have different signs. That is the flow area of the nozzle must decrease as the velocity of the fluid increases. It means subsonic achieved by a converging duct is the sonic speed, which occurs at the exit of the converging nozzle %e nozzles are converging ducts. The shape is illustrated in Fig 8-1. However, the highest velocity can For supersonic flow(M>1), M-1 is positive, and thus da and dc must have the same signs. That the flow area of the nozzle must as the flow velocity increases. It means supersonic nozzles are diverging ducts. The shape is shown in Fig. 8-2 If a stream flows through a nozzle and it is to be accelerated from subsonic to supersonic,a diverging section must be added to a converging nozzle. Such a duct is called converging-diverging nozzle(also known as Laval nozzle). The smallest flow area of this nozzle is called throat, where the velocity of the flow reaches the sonic The shape of this nozzle is shown in Fig. 8-3 For flows through diffusers, the pressure of the fluid increases and the velocity decreases. The variations of the flow velocity with flow area of diffusers are opposite to those of the flow through nozzles. The variations of the flow velocity with flow area are summarized for flow through nozzles and diffusers in Table 8-1 The relationship between the flow velocity and flow area of nozzles and diffusers Table 8-1 from m<i to m>l M> from m>I to M<l
127 Fig. 8-1 Converging nozzle Fig. 8-2 Diverging nozzle Fig.8-3 Converging-diverging nozzle 8.2.2 The Relations between the Flow Velocity and the Flow area of the Duct From the mass conservation equation (8-4b), we know d d d A v c A v c = − (a) From eq. (8-7b), we can obtain for isentropic flow, d d v p v p = − (b) Substituting eq. (8-6) and eq. (b) into eq. (a) yields d d 2 ( 1) A c M A c = − （8-8） Eq. (8-8) indicates the relation between the flow velocity and the flow area of a duct. It governs the shape of a nozzle or a diffuser for subsonic or supersonic isentropic flow. As both A and c are positive quantities, we can get the following conclusion for isentropic steady flows through nozzles: For subsonic flow ( M 1 ), 2 M −1 is negative, and thus dA and dc must have different signs. That is, the flow area of the nozzle must decrease as the velocity of the fluid increases. It means subsonic nozzles are converging ducts. The shape is illustrated in Fig. 8-1. However, the highest velocity can be achieved by a converging duct is the sonic speed, which occurs at the exit of the converging nozzle. For supersonic flow ( M 1 ), 2 M −1 is positive, and thus dA and dc must have the same signs. That is, the flow area of the nozzle must increase as the flow velocity increases. It means supersonic nozzles are diverging ducts. The shape is shown in Fig. 8-2. If a stream flows through a nozzle and it is to be accelerated from subsonic to supersonic, a diverging section must be added to a converging nozzle. Such a duct is called converging–diverging nozzle (also known as Laval nozzle). The smallest flow area of this nozzle is called throat, where the velocity of the flow reaches the sonic speed. The shape of this nozzle is shown in Fig.8-3. For flows through diffusers, the pressure of the fluid increases and the velocity decreases. The variations of the flow velocity with flow area of diffusers are opposite to those of the flow through nozzles. The variations of the flow velocity with flow area are summarized for flow through nozzles and diffusers in Table 8-1. The relationship between the flow velocity and flow area of nozzles and diffusers Table 8-1 M 1 M 1 Convergent-divergent Nozzle from M 1 to M 1 Convergent-divergent diffuser from M 1 to M 1