This document provides an introduction to compressible flow. It defines compressible flow as flow involving significant changes in density. The key concepts covered include: thermodynamic relations for a perfect gas; stagnation properties and how stagnation pressure, temperature, and density are defined; the speed of sound and Mach number; and how flow parameters like temperature and pressure ratios relate to Mach number for isentropic flow of an ideal gas.
This document provides an overview of compressible flow concepts including:
- Thermodynamic relations for perfect gases including equations of state relating pressure, density, temperature, and specific heats.
- Stagnation properties and how stagnation pressure, temperature, and density relate to static properties for isentropic flow.
- The Mach number, defined as the ratio of flow velocity to local speed of sound, and its importance in determining whether flow is compressible.
- How conservation of energy applies to nozzles, with stagnation properties (pressure, temperature) remaining constant for isentropic flow.
Turbulent flows are characterized by chaotic, unpredictable changes in velocity. The document discusses turbulence, including defining turbulence, the transition from laminar to turbulent flow, Reynolds averaging to decompose variables into mean and fluctuating components, and the effects of turbulence on the Navier-Stokes equations. It also examines Reynolds stresses, time-averaged conservation equations for turbulent flow, and modeling approaches like Reynolds averaging to account for turbulent fluctuations and closure problems in the equations.
External forced convection involves fluid flow over solid surfaces. Key topics covered include drag and heat transfer mechanisms of friction and pressure drag, flow over flat plates where correlations are developed for friction coefficient and heat transfer coefficient as a function of Reynolds number, and flow over cylinders, spheres, and tube banks where empirical correlations describe the variations in drag and heat transfer with Reynolds number and surface characteristics.
This document appears to be the title page and table of contents for the second edition of the book "Heat Exchanger Design Handbook" by Kuppan Thulukkanam. The title page provides publication details about the book and its author. The table of contents gives an overview of the book's chapter structure and topics covered, including an introduction to heat exchangers, their classification and selection, thermohydraulic fundamentals, design considerations for different heat exchanger types, and applications.
This document discusses compressible flow and its applications in chemical engineering. It begins by defining compressible flow as fluid flow with significant changes in density, usually when the Mach number is greater than 0.3. It then discusses how to distinguish compressible fluids using the Mach number and provides some historical examples. The effects of compressibility, such as choked flow, shock waves, and changes in density with pressure changes are described. Finally, some applications of compressible flow in chemical engineering are mentioned, such as high-speed gas flow in pipes and nozzles and compressible gas flow in chemical processing industries.
Heat exchangers are devices that transfer heat from one medium to another. The purpose of the heat transfer typically is to lower or raise temperatures in a device.
Introduction to convection
The dimensionless number and its physical significance
Similarity parameters from the differential equation
Dimensional analysis approach and its application
Numerical on Dimensional analysis approach
Review of Navier-Stokes equation
The document discusses various analogies that can be drawn between the transport processes of momentum, heat, and mass. It explains that the basic transport mechanisms are the same and the governing equations are identical in form. Various analogies are presented, including the Reynolds analogy and modifications by Prandtl and von Korman that account for viscous sublayers and buffer layers in turbulent transport.
This document provides an overview of compressible flow concepts including:
- Thermodynamic relations for perfect gases including equations of state relating pressure, density, temperature, and specific heats.
- Stagnation properties and how stagnation pressure, temperature, and density relate to static properties for isentropic flow.
- The Mach number, defined as the ratio of flow velocity to local speed of sound, and its importance in determining whether flow is compressible.
- How conservation of energy applies to nozzles, with stagnation properties (pressure, temperature) remaining constant for isentropic flow.
Turbulent flows are characterized by chaotic, unpredictable changes in velocity. The document discusses turbulence, including defining turbulence, the transition from laminar to turbulent flow, Reynolds averaging to decompose variables into mean and fluctuating components, and the effects of turbulence on the Navier-Stokes equations. It also examines Reynolds stresses, time-averaged conservation equations for turbulent flow, and modeling approaches like Reynolds averaging to account for turbulent fluctuations and closure problems in the equations.
External forced convection involves fluid flow over solid surfaces. Key topics covered include drag and heat transfer mechanisms of friction and pressure drag, flow over flat plates where correlations are developed for friction coefficient and heat transfer coefficient as a function of Reynolds number, and flow over cylinders, spheres, and tube banks where empirical correlations describe the variations in drag and heat transfer with Reynolds number and surface characteristics.
This document appears to be the title page and table of contents for the second edition of the book "Heat Exchanger Design Handbook" by Kuppan Thulukkanam. The title page provides publication details about the book and its author. The table of contents gives an overview of the book's chapter structure and topics covered, including an introduction to heat exchangers, their classification and selection, thermohydraulic fundamentals, design considerations for different heat exchanger types, and applications.
This document discusses compressible flow and its applications in chemical engineering. It begins by defining compressible flow as fluid flow with significant changes in density, usually when the Mach number is greater than 0.3. It then discusses how to distinguish compressible fluids using the Mach number and provides some historical examples. The effects of compressibility, such as choked flow, shock waves, and changes in density with pressure changes are described. Finally, some applications of compressible flow in chemical engineering are mentioned, such as high-speed gas flow in pipes and nozzles and compressible gas flow in chemical processing industries.
Heat exchangers are devices that transfer heat from one medium to another. The purpose of the heat transfer typically is to lower or raise temperatures in a device.
Introduction to convection
The dimensionless number and its physical significance
Similarity parameters from the differential equation
Dimensional analysis approach and its application
Numerical on Dimensional analysis approach
Review of Navier-Stokes equation
The document discusses various analogies that can be drawn between the transport processes of momentum, heat, and mass. It explains that the basic transport mechanisms are the same and the governing equations are identical in form. Various analogies are presented, including the Reynolds analogy and modifications by Prandtl and von Korman that account for viscous sublayers and buffer layers in turbulent transport.
Throttling refers to the process where a fluid flows through a partially open valve, causing a significant pressure drop but constant enthalpy. During throttling, velocity may change slightly due to compressibility but enthalpy remains constant. Throttling is commonly used to control steam turbine speed, determine steam dryness, and in refrigeration plants and gas liquefaction. The Joule-Thomson experiment demonstrated that during throttling, enthalpy remains constant through a porous plug, validating the throttling process model.
1. Fluids deform continuously under any applied shear stress and never stop deforming, unlike solids which will stop deforming after reaching a fixed strain.
2. Gases have much lower intermolecular forces than liquids or solids, requiring a large amount of energy release during phase changes between gas and liquid or solid.
3. The no-slip condition means that fluid velocity is zero at any solid boundary, due to viscosity, resulting in boundary layers and surface drag.
The document discusses different types of flow measurement techniques. It covers topics like the Pitot tube, which measures static and stagnation pressure to determine flow velocity. It also discusses common obstruction flowmeters like the orifice meter, Venturi meter, and nozzle meter that use a constriction to measure flow rate based on pressure differences. Examples are provided to demonstrate calculations of flow rate and pressure drop using equations that depend on parameters like diameter ratio, discharge coefficient, and fluid properties.
This document discusses mass transfer and diffusion. It begins by introducing mass transfer and explaining that it occurs due to concentration gradients, with the rate of transfer proportional to the gradient according to Fick's law. It then draws analogies between heat and mass transfer, discussing conduction, convection, and other concepts. The document provides various equations to calculate mass transfer rates and explains concepts like boundary conditions, diffusion coefficients, and steady diffusion through walls. It is intended to help understand the physical mechanisms and analyze problems involving mass transfer and diffusion.
This document discusses two-phase flow models and compares different pressure drop correlation methods. It begins with an introduction to two-phase flow and important variables like liquid holdup, gas void fraction, and slip velocity. It then describes the different flow patterns or regimes that can occur, including dispersed bubble, stratified smooth, wavy, slug, annular, and spray flows. The document outlines factors that affect flow patterns and discusses how patterns vary between horizontal, upward inclined, and downward inclined pipes. It concludes that selecting the most suitable correlation is key to accurately sizing pipelines for different applications.
This chapter discusses heat conduction through plane walls, cylinders, spheres, and multilayer geometries under steady conditions. It introduces the concept of thermal resistance networks to model conduction and convection resistances. Contact resistance is analyzed, and applications like insulation and fins are discussed. Fins enhance heat transfer by increasing surface area, and the fin equation models temperature variation along a fin.
The document discusses internal forced convection in circular pipes. It covers topics like laminar and turbulent flow, hydrodynamic and thermal entry lengths, constant surface temperature and constant surface heat flux conditions, and the fully developed region. It provides equations for average velocity, Reynolds number, Nusselt number, and logarithmic mean temperature difference. Analytic relations are given for velocity profile, pressure drop, and heat transfer coefficients in fully developed laminar flow.
This document contains information about transport phenomena including:
1. An example of calculating the time it takes for a brass kettle to empty based on the mass flow rate out of the kettle.
2. An introduction to Mujeeb UR Rahman, a chemical engineering student, and examples of where he can be found online.
3. An example problem involving calculating the chlorine concentration in a swimming pool based on inputs and outputs of water.
The document summarizes boiling and condensation heat transfer. It defines boiling and evaporation, and classifies different types of boiling including pool boiling and flow boiling. It describes the boiling curve and different boiling regimes like nucleate boiling and film boiling. It provides equations to calculate heat flux, critical heat flux, and minimum heat flux during boiling. It also discusses ways to enhance boiling heat transfer through surface modifications. Finally, it briefly discusses flow boiling in tubes and different flow regimes.
This document discusses boiling and condensation processes. It defines boiling as a liquid to vapor phase change and condensation as a vapor to liquid phase change. The document describes different types of boiling including nucleate, critical heat flux, transition, and film boiling. It also discusses pool boiling and flow boiling. For condensation, it covers film condensation and dropwise condensation. The key applications of boiling and condensation are in heat exchangers and refrigeration systems.
The document defines and provides the significance of 20 dimensionless numbers used in fluid mechanics and heat transfer analyses. It states the variables and equations used to calculate each number, such as the Reynolds number being the ratio of inertia to viscous forces, the Froude number comparing inertia to gravity forces, and the Nusselt number relating convective to conductive heat transfer. The dimensionless numbers described are used to characterize different types of flows and analyze phenomena involving forces, heat and mass transfer, phase changes, lubrication, and more.
The document discusses turbulent flow of fluids in pipes and annuli. It defines laminar and turbulent flow, and introduces the Reynolds number used to characterize flow regimes. It then provides methods to determine friction factors and pressure losses for Newtonian fluids in pipes and annuli under both laminar and turbulent flow conditions. Non-Newtonian fluid models including Bingham plastic and power law models are also covered.
This document provides an overview of turbulent fluid flow, including:
1) It defines laminar and turbulent flow and explains that turbulent flow occurs above a Reynolds number of 2000.
2) It describes methods for characterizing turbulence, including magnitude, intensity, and mixing length theory.
3) It discusses the universal law of the wall and how velocity is distributed in smooth and rough pipes. Friction factors depend on Reynolds number and relative roughness.
4) Experimental results from Nikuradse are presented showing relationships between friction factor and Reynolds number/relative roughness that can be used to model pressure losses in pipes.
Obtain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow.
Have a visual understanding of different flow regions in internal flow, and calculate hydrodynamic and thermal entry lengths.
Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference.
Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar flow.
Determine the friction factor and Nusselt number in fully developed turbulent flow using empirical relations, and calculate the heat transfer rate.
This document discusses various topics related to heat transfer including the three main modes of heat transfer: conduction, convection, and radiation. It provides the governing equations for each, including Fourier's law of conduction, Newton's law of cooling for convection, and Stefan-Boltzmann law for radiation. Specific topics covered include steady state and transient conduction, extended surfaces, heat exchangers, and free and forced convection. Equations are also derived for one-dimensional conduction under steady state conditions with and without heat generation.
This document discusses turbulent fluid flow. It defines turbulence as an irregular flow with random variations in time and space that can be expressed statistically. Turbulence occurs above a critical Reynolds number when the kinetic energy of the flow is enough to sustain random fluctuations against viscous damping. Characteristics of turbulent flow include fluctuating velocities and pressures, and more uniform velocity distributions compared to laminar flow. Turbulence can be generated by solid walls or shear between layers, and can be categorized as homogeneous, isotropic, or anisotropic. Transition from laminar to turbulent flow is also discussed.
This document provides an overview of fluid properties including:
1. Intensive and extensive properties such as temperature, pressure, density, mass, volume, and momentum.
2. The continuum model which treats fluids as continuous substances rather than discrete particles.
3. Equations of state relating pressure, temperature, and density including the ideal gas law.
4. Specific properties including specific volume, specific gravity, and specific weight.
5. Concepts of density, vapor pressure, cavitation, energy, specific heats, coefficients of compressibility and volume expansion, viscosity, and surface tension.
Throttling refers to the process where a fluid flows through a partially open valve, causing a significant pressure drop but constant enthalpy. During throttling, velocity may change slightly due to compressibility but enthalpy remains constant. Throttling is commonly used to control steam turbine speed, determine steam dryness, and in refrigeration plants and gas liquefaction. The Joule-Thomson experiment demonstrated that during throttling, enthalpy remains constant through a porous plug, validating the throttling process model.
1. Fluids deform continuously under any applied shear stress and never stop deforming, unlike solids which will stop deforming after reaching a fixed strain.
2. Gases have much lower intermolecular forces than liquids or solids, requiring a large amount of energy release during phase changes between gas and liquid or solid.
3. The no-slip condition means that fluid velocity is zero at any solid boundary, due to viscosity, resulting in boundary layers and surface drag.
The document discusses different types of flow measurement techniques. It covers topics like the Pitot tube, which measures static and stagnation pressure to determine flow velocity. It also discusses common obstruction flowmeters like the orifice meter, Venturi meter, and nozzle meter that use a constriction to measure flow rate based on pressure differences. Examples are provided to demonstrate calculations of flow rate and pressure drop using equations that depend on parameters like diameter ratio, discharge coefficient, and fluid properties.
This document discusses mass transfer and diffusion. It begins by introducing mass transfer and explaining that it occurs due to concentration gradients, with the rate of transfer proportional to the gradient according to Fick's law. It then draws analogies between heat and mass transfer, discussing conduction, convection, and other concepts. The document provides various equations to calculate mass transfer rates and explains concepts like boundary conditions, diffusion coefficients, and steady diffusion through walls. It is intended to help understand the physical mechanisms and analyze problems involving mass transfer and diffusion.
This document discusses two-phase flow models and compares different pressure drop correlation methods. It begins with an introduction to two-phase flow and important variables like liquid holdup, gas void fraction, and slip velocity. It then describes the different flow patterns or regimes that can occur, including dispersed bubble, stratified smooth, wavy, slug, annular, and spray flows. The document outlines factors that affect flow patterns and discusses how patterns vary between horizontal, upward inclined, and downward inclined pipes. It concludes that selecting the most suitable correlation is key to accurately sizing pipelines for different applications.
This chapter discusses heat conduction through plane walls, cylinders, spheres, and multilayer geometries under steady conditions. It introduces the concept of thermal resistance networks to model conduction and convection resistances. Contact resistance is analyzed, and applications like insulation and fins are discussed. Fins enhance heat transfer by increasing surface area, and the fin equation models temperature variation along a fin.
The document discusses internal forced convection in circular pipes. It covers topics like laminar and turbulent flow, hydrodynamic and thermal entry lengths, constant surface temperature and constant surface heat flux conditions, and the fully developed region. It provides equations for average velocity, Reynolds number, Nusselt number, and logarithmic mean temperature difference. Analytic relations are given for velocity profile, pressure drop, and heat transfer coefficients in fully developed laminar flow.
This document contains information about transport phenomena including:
1. An example of calculating the time it takes for a brass kettle to empty based on the mass flow rate out of the kettle.
2. An introduction to Mujeeb UR Rahman, a chemical engineering student, and examples of where he can be found online.
3. An example problem involving calculating the chlorine concentration in a swimming pool based on inputs and outputs of water.
The document summarizes boiling and condensation heat transfer. It defines boiling and evaporation, and classifies different types of boiling including pool boiling and flow boiling. It describes the boiling curve and different boiling regimes like nucleate boiling and film boiling. It provides equations to calculate heat flux, critical heat flux, and minimum heat flux during boiling. It also discusses ways to enhance boiling heat transfer through surface modifications. Finally, it briefly discusses flow boiling in tubes and different flow regimes.
This document discusses boiling and condensation processes. It defines boiling as a liquid to vapor phase change and condensation as a vapor to liquid phase change. The document describes different types of boiling including nucleate, critical heat flux, transition, and film boiling. It also discusses pool boiling and flow boiling. For condensation, it covers film condensation and dropwise condensation. The key applications of boiling and condensation are in heat exchangers and refrigeration systems.
The document defines and provides the significance of 20 dimensionless numbers used in fluid mechanics and heat transfer analyses. It states the variables and equations used to calculate each number, such as the Reynolds number being the ratio of inertia to viscous forces, the Froude number comparing inertia to gravity forces, and the Nusselt number relating convective to conductive heat transfer. The dimensionless numbers described are used to characterize different types of flows and analyze phenomena involving forces, heat and mass transfer, phase changes, lubrication, and more.
The document discusses turbulent flow of fluids in pipes and annuli. It defines laminar and turbulent flow, and introduces the Reynolds number used to characterize flow regimes. It then provides methods to determine friction factors and pressure losses for Newtonian fluids in pipes and annuli under both laminar and turbulent flow conditions. Non-Newtonian fluid models including Bingham plastic and power law models are also covered.
This document provides an overview of turbulent fluid flow, including:
1) It defines laminar and turbulent flow and explains that turbulent flow occurs above a Reynolds number of 2000.
2) It describes methods for characterizing turbulence, including magnitude, intensity, and mixing length theory.
3) It discusses the universal law of the wall and how velocity is distributed in smooth and rough pipes. Friction factors depend on Reynolds number and relative roughness.
4) Experimental results from Nikuradse are presented showing relationships between friction factor and Reynolds number/relative roughness that can be used to model pressure losses in pipes.
Obtain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow.
Have a visual understanding of different flow regions in internal flow, and calculate hydrodynamic and thermal entry lengths.
Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference.
Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar flow.
Determine the friction factor and Nusselt number in fully developed turbulent flow using empirical relations, and calculate the heat transfer rate.
This document discusses various topics related to heat transfer including the three main modes of heat transfer: conduction, convection, and radiation. It provides the governing equations for each, including Fourier's law of conduction, Newton's law of cooling for convection, and Stefan-Boltzmann law for radiation. Specific topics covered include steady state and transient conduction, extended surfaces, heat exchangers, and free and forced convection. Equations are also derived for one-dimensional conduction under steady state conditions with and without heat generation.
This document discusses turbulent fluid flow. It defines turbulence as an irregular flow with random variations in time and space that can be expressed statistically. Turbulence occurs above a critical Reynolds number when the kinetic energy of the flow is enough to sustain random fluctuations against viscous damping. Characteristics of turbulent flow include fluctuating velocities and pressures, and more uniform velocity distributions compared to laminar flow. Turbulence can be generated by solid walls or shear between layers, and can be categorized as homogeneous, isotropic, or anisotropic. Transition from laminar to turbulent flow is also discussed.
This document provides an overview of fluid properties including:
1. Intensive and extensive properties such as temperature, pressure, density, mass, volume, and momentum.
2. The continuum model which treats fluids as continuous substances rather than discrete particles.
3. Equations of state relating pressure, temperature, and density including the ideal gas law.
4. Specific properties including specific volume, specific gravity, and specific weight.
5. Concepts of density, vapor pressure, cavitation, energy, specific heats, coefficients of compressibility and volume expansion, viscosity, and surface tension.
1. Compressible flows involve significant changes in density and occur in devices where gases flow at very high speeds, requiring both fluid dynamics and thermodynamics.
2. Stagnation properties represent the total energy of a fluid when kinetic and potential energies are included. For high-speed flows, kinetic energy is significant and combined with enthalpy as stagnation enthalpy.
3. In steady flow through a duct like a nozzle, the stagnation enthalpy at the inlet and outlet are equal, so any increase in velocity causes an equivalent decrease in static enthalpy. Bringing the fluid to a stop converts kinetic energy to increased temperature and pressure, defining the stagnation state.
This document discusses thermodynamic properties and relations. Some key points:
- Thermodynamic properties can be measured directly or related through equations based on properties being point functions specified by two intensive properties.
- Maxwell relations relate partial derivatives of properties like pressure, specific volume, temperature and entropy for simple compressible substances.
- The Clapeyron equation relates enthalpy change during a phase change to the slope of the saturation curve on a pressure-temperature diagram.
- Relations are developed to express changes in internal energy, enthalpy and entropy in terms of pressure, specific volume, temperature and specific heats for real gases.
This document discusses thermodynamic properties and relations. Some key points:
- Thermodynamic properties that cannot be directly measured must be related to measurable properties.
- Properties are continuous point functions that have exact differentials and can be written as functions of two independent variables like z(x,y).
- The Maxwell relations relate the partial derivatives of properties like pressure, specific volume, temperature and entropy.
- The Clapeyron equation relates the enthalpy change of phase change to the slope of the saturation curve on a pressure-temperature diagram.
- Specific heats, internal energy, enthalpy and entropy changes can be expressed in terms of pressure, specific volume, temperature and specific he
This document contains 51 short answer questions related to aerodynamics and compressible flow. The questions cover topics like gas dynamics, compressible versus incompressible flow, compressibility, types of compressibility, properties of perfect gases, adiabatic and isentropic processes, Mach number, flow regimes, continuity, momentum, and energy equations. Many questions also focus specifically on nozzle flow, including definitions of different types of nozzles, choking, expansion, under-expanded versus over-expanded nozzles, and nozzle efficiency.
1. The document discusses improving the efficiency of combined cycle power plants through various methods like regeneration, intercooling, and reheating. These processes divide the gas turbine cycle into multiple stages to better approximate ideal cycles.
2. Part-load performance is also important, as efficiency decreases at lower power outputs due to lower air flow rates. Ambient conditions also impact maximum power output and efficiency.
3. Flow rate equations are presented for single and multi-stage turbines, showing how pressure ratio, efficiency, and ambient conditions impact relative flow rates and available power.
This document discusses the volumetric properties of pure fluids. It covers pressure-temperature diagrams and pressure-volume diagrams, including phase change curves, triple points, and critical points. Equations of state are presented that relate pressure, volume, and temperature for fluids in equilibrium. The ideal gas law and virial equations of state are also discussed. Specific heat capacities, isothermal compressibility, and volume expansivity are defined. Several examples problems calculate work, heat, internal energy and enthalpy changes for ideal gas processes including compression, expansion, heating and cooling.
This document summarizes key concepts in thermodynamics including:
- Pressure-temperature and pressure-volume diagrams and important points like the triple point and critical point.
- Equations of state relating pressure, volume, and temperature for pure fluids.
- Ideal gas behavior and equations like the ideal gas law.
- Thermodynamic processes like isothermal, adiabatic, and their calculations.
- Use of concepts and equations to calculate work, heat, internal energy and enthalpy change for processes involving ideal gases.
The document discusses several key concepts related to fluid properties including:
1) Vapor pressure, which is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases, increases with temperature.
2) Surface tension, which is the work required to increase a liquid's surface area.
3) Capillarity, in which liquids rise or fall in narrow tubes due to interactions between surface tension and adhesion to the walls.
4) Fluid pressure, which acts equally in all directions and depends on factors like depth, density, and compressibility. Pressure can be calculated using various equations provided.
- Physics concepts like pressure, volume, temperature and flow are important in anaesthesia to safely manage gases and fluids. Boyle's law and other gas laws allow calculation of gas amounts in cylinders.
- Bernoulli's principle and the Venturi effect explain changes in pressure and velocity of fluids. Heat transfer principles are relevant to patient temperature regulation. Humidity levels impact moisture exchange filters.
- Reynolds number determines laminar versus turbulent flow, important for breathing circuits. The Coanda effect impacts airway distribution. Simple mechanics equations enable calculating work of breathing and syringe pressures.
1) This document discusses heat, work, and the first law of thermodynamics. It defines heat and work as the two ways energy can transfer across the boundary of a closed system, with heat transferring due to a temperature difference and work occurring from a force acting through a distance.
2) The first law of thermodynamics states that the change in a system's internal energy is equal to the net heat transferred to the system plus the net work done by the system. This is illustrated with examples of processes involving only heat transfer, where the energy change equals the net heat.
3) Different types of thermodynamic processes are examined, including isobaric, isochoric, isothermal, and poly
The document provides an overview of topics related to compressible fluid flow, including:
- Continuity, impulse-momentum, and energy equations for compressible fluids under isothermal and adiabatic conditions.
- Basic thermodynamic relationships like the ideal gas law, processes like isothermal and adiabatic, and concepts like internal energy and entropy.
- Propagation of elastic waves in fluids due to compression, and how the velocity of sound depends on factors like pressure, temperature, and fluid properties.
- Additional topics covered include stagnation properties, flow through converging-diverging passages, shock waves, and external aerodynamic flows.
Here are the key steps to solve this problem:
1. Given: TH = 817°C = 817 + 273 = 1090 K
TL = 25°C = 25 + 273 = 298 K
QR = 25 kW
2. Use the Carnot efficiency equation:
η = (TH - TL)/TH = (1090 - 298)/1090 = 0.726
3. Set up an equation for the heat input using the efficiency and heat rejected:
QA = QR/(1-η) = 25000/(1-0.726) = 87500 kW
Therefore, the heat input (QA) required is 87500 kW.
This document discusses various physics concepts and clinical measurements related to anaesthesia. It covers gas laws, critical temperatures, solubility, diffusion, osmosis, electricity, fluid dynamics, and blood pressure monitoring. Key points include definitions of Boyle's law, Charles' law, and Dalton's law of partial pressures. It also discusses non-invasive and invasive blood pressure monitoring techniques.
1) The document discusses heat, work, and the first law of thermodynamics. It defines heat and work as the two types of energy transfer across boundaries of closed systems.
2) The first law of thermodynamics, also called the law of conservation of energy, states that the total energy of a system remains constant, with increases in internal energy equal to net heat and work transfers.
3) Specific examples are provided to illustrate the first law for closed systems undergoing various processes like heating, cooling, and adiabatic changes with and without work. Formulas are derived for calculating internal energy changes based on the first law.
Energy transfer by heat occurs between systems with a temperature difference, even if no work is done. Heat transfer is driven by decreasing temperature, with a higher rate of transfer for larger temperature differences. Heat transfer between a system and its surroundings is represented by Q, with a positive Q indicating heat transfer to the system. For transient processes, the rate of energy transfer to or from a system can be determined by integrating heat (Q) and work (W) terms over time.
Heat and thermodynamics - Preliminary / Dr. Mathivanan VelumaniMathivanan Velumani
The document discusses key concepts in thermodynamics including:
1. Thermodynamic states are characterized by macroscopic properties like temperature, pressure, and volume that determine a system's internal state and interaction with external bodies.
2. Thermal equilibrium exists when temperature is uniform throughout a system, as stated by the zeroth law of thermodynamics.
3. Internal energy (U) is the energy associated with the random, disordered motion of molecules within a system.
1. A cylinder separated by an adiabatic piston initially contains nitrogen on one side and helium on the other at 20°C and 95 kPa.
2. Heat is added to the nitrogen side from a 500°C reservoir until the helium pressure reaches 120 kPa.
3. The final temperature of helium is calculated to be 321.7K, the final volume of nitrogen is 0.2838 m3, and the heat transferred to the nitrogen is 46.6287 kJ. The entropy generation is determined to be 0.057 kJ/K.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
2. Contents of the Chapter
Introduction
Thermodynamic Relations
Stagnation Properties
Speed of Sound and Mach number
One Dimensional Isentropic flow: Effect of area
changes of flow parameters
Converging Nozzle: Effect of back pressure on flow
parameters
Converging-Diverging Nozzles
Normal shock waves; Fanno and Rayleigh lines
Oblique shock waves
2
3. Introduction
Flows that involve significant changes in density are called
compressible flows.
Therefore, ρ(x, y, z) must now be treated as a field variable
rather than simply a constant.
Typically, significant density variations start to appear when the
flow Mach number exceeds 0.3 or so. The effects become
especially large when the Mach number approaches and
exceeds unity.
In this chapter we will consider flows that involve significant
changes in density. Such flows are called compressible flows,
and they are frequently encountered in devices that involve the
flow of gases at very high speeds such as flows in gas turbine
engine components . Many aircraft fly fast enough to involve
compressible flow.
3
4. Gas has large compressibility but when its velocity is low
compared with the sonic velocity the change in density is
small and it is then treated as an incompressible fluid.
When a fluid moves at speeds comparable to its speed of
sound, density changes become significant and the flow is
termed compressible.
Such flows are difficult to obtain in liquids, since high
pressures of order 1000 atm are needed to generate sonic
velocities. In gases, however, a pressure ratio of only 2:1
will likely cause sonic flow. Thus compressible gas flow is
quite common, and this subject is often called gas
dynamics.
4
Introduction
5. Thermodynamic Relations
Perfect gas
A perfect gas is one whose individual molecules interact
only via direct collisions, with no other intermolecular
forces present.
For such a perfect gas, p, ρ, and the temperature T are
related by the following equation of state
p = ρRT
where R is the specific gas constant. For air, R =287J/kg-K◦
.
It is convenient at this point to define the specific volume as
the limiting volume per unit mass,
which is merely the reciprocal of the density.
5
6. The equation of state can now be written as
pυ = RT
which is the more familiar thermodynamic form.
Here R is the gas constant, and
where Ro is the universal gas constant (Ro = 8314J/(kg K))
and M is the molecular weight. For example, for air
assuming M = 28.96, the gas constant is
6
Thermodynamic Relations
7. Then, assuming internal energy and enthalpy per unit mass
e and h respectively,
the specific enthalpy, denoted by h, and related to the other
variables by
For a calorically perfect gas, which is an excellent model
for air at moderate temperatures both e and h are directly
proportional to the temperature.
7
Thermodynamic Relations
8. Therefore we have
where cv and cp are specific heats at constant volume and
constant pressure, respectively.
and comparing to the equation of state, we see that
Defining the ratio of specific heats, γ ≡ cp/cv, we can with a
bit of algebra write
8
Thermodynamic Relations
9. so that cv and cp can be replaced with the equivalent
variables γ and R. For air, it is handy to remember that
First Law of Thermodynamics
Consider a thermodynamic system consisting of a small
Lagrangian control volume (CV) moving with the flow.
Over the short time interval dt, the CV undergoes a process
where it receives work δw and heat δq from its
surroundings, both per unit mass. This process results in
changes in the state of the CV, described by the increments
de, dh, dp . . .
9
Thermodynamic Relations
10. Thermodynamic Relations
The first law of thermodynamics
for the process is
This states that whatever energy
is added to the system, whether
by heat or by work, it must
appear as an increase in the
internal energy of the system.
10
11. Isentropic relations
Aerodynamic flows are effectively inviscid outside of
boundary layers. This implies they have negligible heat
conduction and friction forces, and hence are isentropic.
Therefore, along the pathline followed by the CV in the
figure above, the isentropic version of the first law applies
11
Thermodynamic Relations
12. This relation can be integrated after a few substitutions.
First we note that
and with the perfect gas relation
the isentropic first law becomes
12
Thermodynamic Relations
→
13. The final form can now be integrated from any state 1 to
any state 2 along the pathline.
From the equation of state we also have
Which gives the alternative isentropic relation
13
Thermodynamic Relations
14. Stagnation Properties
Consider a fluid flowing into a diffuser at a velocity ,
temperature T, pressure P, and enthalpy h, etc. Here the
ordinary properties T, P, h, etc. are called the static
properties; that is, they are measured relative to the flow at
the flow velocity.
The diffuser is sufficiently long and the exit area is
sufficiently large that the fluid is brought to rest (zero
velocity) at the diffuser exit while no work or heat
transfer is done. The resulting state is called the
stagnation state.
14
V
15. We apply the first law per unit mass for one entrance, one
exit, and neglect the potential energies. Let the inlet state
be unsubscripted and the exit or stagnation state have the
subscript o.
Since the exit velocity, work, and heat transfer are zero,
The term ho is called the stagnation enthalpy (some
authors call this the total enthalpy).
It is the enthalpy the fluid attains when brought to rest
adiabatically while no work is done.
15
q h
V
w h
V
net net o
o
+ + = + +
2 2
2 2
Stagnation Properties
h h
V
o = +
2
2
16. Stagnation Properties
If, in addition, the process is
also reversible, the process is
isentropic, and the inlet and
exit entropies are equal.
The stagnation enthalpy and
entropy define the stagnation
state and the isentropic
stagnation pressure, Po.
The actual stagnation pressure
for irreversible flows will be
somewhat less than the
isentropic stagnation pressure
as shown in the fig.
16
s s
o =
17. 19
Ideal Gas Result
Rewrite the equation defining the stagnation enthalpy as
h h
V
o − =
2
2
For ideal gases with constant specific heats, the enthalpy
difference becomes
C T T
V
P o
( )
− =
2
2
where To is defined as the stagnation temperature.
T T
V
C
o
P
− =
2
2
18. 20
For the isentropic process, the stagnation pressure can be
determined from
or
The ratio of the stagnation density to static density can be
expressed as
19. Example 2
An aircraft is flying at a cruising speed of 250 m/s at an altitude
of 5000 m where the atmospheric pressure is 54.05 kPa and the
ambient air temperature is 255.7 K. The ambient air is first
decelerated in a diffuser before it enters the compressor.
Assuming both the diffuser and the compressor to be isentropic,
determine (a) the stagnation pressure at the compressor inlet and
(b) the required compressor work per unit mass if the stagnation
pressure ratio of the compressor is 8.
21
20. Solution
High-speed air enters the diffuser and the compressor of an
aircraft. The stagnation pressure of the air and the
compressor work input are to be determined.
Assumptions 1 Both the diffuser and the compressor are
isentropic. 2 Air is an ideal gas with constant specific heats
at room temperature.
Properties The constant-pressure specific heat cp and the
specific heat ratio k of air at room temperature are
cp = 1.005 kJ/kg . K and k = 1.4
Analysis
(a) the stagnation temperature T01 at the compressor inlet can
be determined from
22
21. Then
That is, the temperature of air would increase by 31.1°C
and the pressure by 26.72 kPa as air is decelerated from
250 m/s to zero velocity. These increases in the temperature
and pressure of air are due to the conversion of the kinetic
energy into enthalpy.
23
Solution
22. To determine the compressor work, we need to know the
stagnation temperature of air at the compressor exit T02.
Disregarding potential energy changes and heat transfer,
the compressor work per unit mass of air is determined
from
24
Solution
23. 25
Conservation of Energy for Control Volumes Using
Stagnation Properties
The steady-flow conservation of energy for the above figure is
Since
h h
V
o = +
2
2
24. 26
For no heat transfer, one entrance, one exit, this reduces to
If we neglect the change in potential energy, this becomes
For ideal gases with constant specific heats we write this as
Conservation of Energy for a Nozzle
We assume steady-flow, no heat transfer, no work, one entrance,
and one exit and neglect elevation changes; then the conservation
of energy becomes
1 1 2 2
in out
o o
m h m h
=
=
E& E&
& &
25. 27
But
m m
1 2
= thus h h
o o
1 2
=
Thus the stagnation enthalpy remains constant throughout the
nozzle. At any cross section in the nozzle, the stagnation
enthalpy is the same as that at the entrance. For ideal gases this
last result becomes
T T
o o
1 2
=
Thus the stagnation temperature remains constant through out
the nozzle. At any cross section in the nozzle, the stagnation
temperature is the same as that at the entrance.
Assuming an isentropic process for flow through the nozzle,
we can write for the entrance and exit states
So we see that the stagnation pressure is also constant through
out the nozzle for isentropic flow.
26. Speed of Sound and Mach number
An important parameter in the study of compressible flow
is the speed of sound (or the sonic speed), which is the
speed at which an infinitesimally small pressure wave
travels through a medium.
The pressure wave may be caused by a small disturbance,
which creates a slight rise in local pressure.
To obtain a relation for the speed of sound in a medium,
consider a duct that is filled with a fluid at rest,
28
27. A piston fitted in the duct is now moved to the right with a constant
incremental velocity dV, creating a sonic wave.
The wave front moves to the right through the fluid at the speed of
sound c and separates the moving fluid adjacent to the piston from
the fluid still at rest.
The fluid to the left of the wave front experiences an incremental
change in its thermodynamic properties, while the fluid on the right
of the wave front maintains its original thermodynamic properties,
as shown in Fig.
To simplify the analysis, consider a control volume that encloses the
wave front and moves with it, as shown in the fig. below.
To an observer traveling with the wave front, the fluid to the right
will appear to be moving toward the wave front with a speed of c
and the fluid to the left to be moving away from the wave front with
a speed of c - dV.
29
Speed of Sound and Mach number
29. 31
Speed of Sound and Mach number
Cancel terms and neglect ; we have
dV
2
dh CdV
− =
0
30. 32
Now, apply the conservation of mass or continuity equation
to the control volume.
m AV
= ρ
ρ ρ ρ
ρ ρ ρ ρ ρ
AC d A C dV
AC A C dV Cd d dV
= + −
= − + −
( ) ( )
( )
Cancel terms and neglect the higher-order terms like .
We have
d dV
ρ
Cd dV
ρ ρ
− =
0
Also, we consider the property relation dh T ds v dP
dh T ds dP
= +
= +
1
ρ
31. 33
Let's assume the process to be isentropic; then ds = 0 and
dh dP
=
1
ρ
Using the results of the first law
dh dP CdV
= =
1
ρ
From the continuity equation
dV
Cd
=
ρ
ρ
Now we have
32. 34
Thus dP
d
C
ρ
= 2
Since the process is assumed to be isentropic, the above becomes
By using thermodynamic property relations this can be written
as
where k is the ratio of specific heats, k = CP/CV.
33. 35
Example -3
Find the speed of sound in air at an altitude of 5000 m.
At 5000 m, T = 255.7 K.
C
kJ
kg K
K
m
s
kJ
kg
m
s
=
⋅
=
14 0 287 2557
1000
3205
2
2
. ( . )( . )
.
Ideal Gas Result
For ideal gases →
34. 36
Notice that the temperature used for the speed of sound is the
static (normal) temperature.
Example -4
Find the speed of sound in steam where the pressure is 1 MPa
and the temperature is 350o
C.
At P = 1 MPa, T = 350o
C,
1
s
s
P P
C
v
ρ
∂ ∂
= =
∂
∂
35. 37
Here, we approximate the partial derivative by perturbating
the pressure about 1 MPa. Consider using P±0.025 MPa at
the entropy value s = 7.3011 kJ/kg⋅ K, to find the
corresponding specific volumes.
36. 38
What is the speed of sound for steam at 350o
C assuming
ideal-gas behavior?
Assume k = 1.3, then
C
kJ
kg K
K
m
s
kJ
kg
m
s
=
⋅
+
=
13 0 4615 350 273
1000
6114
2
2
. ( . )( )
.
Mach Number
The Mach number M is defined as M
V
C
=
M <1 flow is subsonic
M =1 flow is sonic
M >1 flow is supersonic
37. 39
Example -5
In the air and steam examples above, find the Mach number if the
air velocity is 250 m/s and the steam velocity is 300 m/s.
M
m
s
m
s
M
m
s
m
s
air
steam
= =
= =
250
3205
0 780
300
6055
0 495
.
.
.
.
The flow parameters To/T, Po/P, ρo/ρ, etc. are related to the flow Mach
number. Let's consider ideal gases, then
T T
V
C
T
T
V
C T
o
P
o
P
= +
= +
2
2
2
1
2
38. 40
but C
k
k
R or
C
k
kR
P
P
=
−
=
−
1
1 1
T
T
V
T
k
kR
o
= +
−
1
2
1
2
( )
and
C kRT
2
=
so T
T
k V
C
k
M
o
= +
−
= +
−
1
1
2
1
1
2
2
2
2
( )
( )
The pressure ratio is given by
39. 41
We can show the density ratio to be
For the Mach number equal to 1, the sonic location, the static
properties are denoted with a superscript “*”. This condition,
when M = 1, is called the sonic condition. When M = 1 and k =
1.4, the static-to-stagnation ratios are
40. Example. Mach Number of Air Entering a Diffuser
Air enters a diffuser shown in Fig. with a velocity of 200
m/s. Determine (a) the speed of sound and (b) the Mach
number at the diffuser inlet when the air temperature is
30°C.
42
42. One-Dimensional Isentropic Flow
During fluid flow through many devices such as nozzles,
diffusers, and turbine blade passages, flow quantities vary
primarily in the flow direction only, and the flow can be
approximated as one-dimensional isentropic flow with good
accuracy.
Effect of Area Changes on Flow Parameters
Consider the isentropic steady flow of an ideal gas through
the nozzle shown below.
44
43. 45
Air flows steadily through a varying-cross-sectional-area duct
such as a nozzle at a flow rate of 3 kg/s. The air enters the
duct at a low velocity at a pressure of 1500 kPa and a
temperature of 1200 K and it expands in the duct to a pressure
of 100 kPa. The duct is designed so that the flow process is
isentropic. Determine the pressure, temperature, velocity,
flow area, speed of sound, and Mach number at each point
along the duct axis that corresponds to a pressure drop of 200
kPa.
Since the inlet velocity is low, the stagnation properties equal
the static properties.
T T K P P kPa
o o
= = = =
1 1
1200 1500
,
44. 46
After the first 200 kPa pressure drop, we have
ρ = =
⋅
=
P
RT
kPa
kJ
kg K
K
kJ
m kPa
kg
m
( )
( . )( . )
.
1300
0 287 11519
3932
3
3
45. 47
A
m
V
kg
s
kg
m
m
s
cm
m
cm
= =
=
( . )( . )
.
ρ
3
39322 310 77
10
24 55
3
4 2
2
2
C kRT
kJ
kg K
K
m
s
kJ
kg
m
s
= =
⋅
=
14 0 287 11519
1000
680 33
2
2
. ( . )( . )
.
M
V
C
m
s
m
s
= = =
310 77
680 33
0 457
.
.
.
Now we tabulate the results for the other 200 kPa increments
in the pressure until we reach 100 kPa.
46. 48
Summary of Results for Nozzle Problem
V
∞
Step P
kPa
T
K m/s
ρ
kg/m3
C
m/s
A
cm2
M
0 1500 1200 0 4.3554 694.38 0
1 1300 1151.9 310.77 3.9322 680.33 24.55 0.457
2 1100 1098.2 452.15 3.4899 664.28 19.01 0.681
3 900 1037.0 572.18 3.0239 645.51 17.34 0.886
4 792.4 1000.0 633.88 2.7611 633.88 17.14 1.000
5 700 965.2 786.83 2.5270 622.75 17.28 1.103
6 500 876.7 805.90 1.9871 593.52 18.73 1.358
7 300 757.7 942.69 1.3796 551.75 23.07 1.709
8 100 553.6 1139.62 0.6294 471.61 41.82 2.416
Note that at P = 797.42 kPa, M = 1.000, and this state is the
critical state.
47. We note from the Nozzle Example
that the flow area decreases with
decreasing pressure down to a
critical-pressure value where the
Mach number is unity, and then it
begins to increase with further
reductions in pressure.
The Mach number is unity at the
location of smallest flow area,
called the throat .
Note that the velocity of the fluid
keeps increasing after passing the
throat although the flow area
increases rapidly in that region.
This increase in velocity past the
throat is due to the rapid decrease
in the fluid density.
49
48. The flow area of the duct considered in this example first
decreases and then increases. Such ducts are called
converging–diverging nozzles. These nozzles are used to
accelerate gases to supersonic speeds and should not be
confused with Venturi nozzles, which are used strictly for
incompressible flow.
50
49. 51
Now let's see why these relations work this way. Consider the
nozzle and control volume shown below.
The first law for the control volume is
dh VdV
+ =
0
The continuity equation for the control volume yields
d dA
A
dV
V
ρ
ρ
+ + =
0
Also, we consider the property relation for an isentropic process
Tds dh
dP
= − =
ρ
0
50. 52
and the Mach Number relation dP
d
C
V
M
ρ
= =
2
2
2
Putting these four relations together yields
dA
A
dP
V
M
= −
ρ
2
2
1
( )
Let’s consider the implications of this equation for both nozzles
and diffusers. A nozzle is a device that increases fluid velocity
while causing its pressure to drop; thus, d > 0, dP < 0.
V
Nozzle Results dA
A
dP
V
M
= −
ρ
2
2
1
( )
Subsonic
Sonic
Supersonic
: ( )
: ( )
: ( )
M dP M dA
M dP M dA
M dP M dA
< − < <
= − = =
> − > >
1 1 0 0
1 1 0 0
1 1 0 0
2
2
2
51. 53
To accelerate subsonic flow, the nozzle flow area must first
decrease in the flow direction. The flow area reaches a
minimum at the point where the Mach number is unity. To
continue to accelerate the flow to supersonic conditions, the
flow area must increase.
The minimum flow area is called the throat of the nozzle.
We are most familiar with the shape of a subsonic nozzle.
That is, the flow area in a subsonic nozzle decreases in the
flow direction.
52. 54
A diffuser is a device that decreases fluid velocity while
causing its pressure to rise; thus, d < 0, dP > 0.
V
Diffuser Results dA
A
dP
V
M
= −
ρ
2
2
1
( )
Subsonic
Sonic
Supersonic
: ( )
: ( )
: ( )
M dP M dA
M dP M dA
M dP M dA
< − > >
= − = =
> − < <
1 1 0 0
1 1 0 0
1 1 0 0
2
2
2
To diffuse supersonic flow, the diffuser flow area must first
decrease in the flow direction. The flow area reaches a minimum
at the point where the Mach number is unity. To continue to
diffuse the flow to subsonic conditions, the flow area must
increase. We are most familiar with the shape of a subsonic
diffuser. That is the flow area in a subsonic diffuser increases in
the flow direction.
54. 56
Equation of Mass Flow Rate through a Nozzle
Let's obtain an expression for the flow rate through a
converging nozzle at any location as a function of the pressure
at that location. The mass flow rate is given by
m AV
= ρ
The velocity of the flow is related to the static and stagnation
enthalpies.
V h h C T T C T
T
T
o P P
o
= − = − = −
2 2 2 1
0 0
( ) ( ) ( )
and
55. 57
Write the mass flow rate as
m AV o
o
=
ρ
ρ
ρ
We note from the ideal-gas relations that
ρo
o
o
P
RT
=
56. 58
What pressure ratios make the mass flow rate zero?
Do these values make sense?
Now let's make a plot of mass flow rate versus the static-to-
stagnation pressure ratio.
0.00 0.20 0.40 0.60 0.80 1.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.16
P*/Po
m
[kg/s]
P/Po
Dia.=
1 cm
To=
1200 K
Po=
1500 kPa
57. 59
This plot shows there is a value of P/Po that makes the mass flow rate
a maximum. To find that mass flow rate, we note
The result is
So the pressure ratio that makes the mass flow rate a maximum is the
same pressure ratio at which the Mach number is unity at the flow cross-
sectional area. This value of the pressure ratio is called the critical
pressure ratio for nozzle flow. For pressure ratios less than the critical
value, the nozzle is said to be choked. When the nozzle is choked, the
mass flow rate is the maximum possible for the flow area, stagnation
pressure, and stagnation temperature. Reducing the pressure ratio below
the critical value will not increase the mass flow rate.
58. 60
Using
The mass flow rate becomes
When the Mach number is unity, M = 1, A = A*
Taking the ratio of the last two results gives the ratio of the area
of the flow A at a given Mach number to the area where the
Mach number is unity, A*.
What is the expression for mass flow rate when the nozzle is choked?
59. 61
Then
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
M
A/A*
From the above plot we note that for each A/A* there are two
values of M: one for subsonic flow at that area ratio and one for
supersonic flow at that area ratio. The area ratio is unity when
the Mach number is equal to one.
60. 62
Consider the converging nozzle
shown below. The flow is supplied
by a reservoir at pressure Pr and
temperature Tr. The reservoir is
large enough that the velocity in the
reservoir is zero.
Let's plot the ratio P/Po along the
length of the nozzle, the mass flow
rate through the nozzle, and the exit
plane pressure Pe as the back
pressure Pb is varied. Let's consider
isentropic flow so that Po is constant
throughout the nozzle.
Effect of Back Pressure on Flow through a Converging Nozzle
61. Now we begin to reduce the back pressure and observe the
resulting effects on the pressure distribution along the length of
the nozzle, as shown in the Fig. above.
If the back pressure Pb is equal to P1, which is equal to Pr, there is
no flow and the pressure distribution is uniform along the
nozzle.
When the back pressure is reduced to P2, the exit plane pressure
Pe also drops to P2. This causes the pressure along the nozzle to
decrease in the flow direction.
When the back pressure is reduced to P3 (= P*, which is the
pressure required to increase the fluid velocity to the speed of
sound at the exit plane or throat), the mass flow reaches a
maximum value and the flow is said to be choked.
63
Effect of Back Pressure on Flow through a Converging Nozzle
62. Further reduction of the back pressure to level P4 or below does
not result in additional changes in the pressure distribution, or
anything else along the nozzle length.
Under steady-flow conditions, the mass flow rate through the
nozzle is constant and can be expressed as
Solving for T from and for P from
and substituting
64
Effect of Back Pressure on Flow through a Converging Nozzle
63. Thus the mass flow rate of a particular fluid through a nozzle is
a function of the stagnation properties of the fluid, the flow area,
and the Mach number.
The maximum mass flow rate can be determined by
differentiating the above equation with respect to Ma and
setting the result equal to zero. It yields Ma = 1.
Since the only location in a nozzle where the Mach number can
be unity is the location of minimum flow area (the throat), the
mass flow rate through a nozzle is a maximum when Ma = 1 at
the throat. Denoting this area by A*, we obtain an expression for
the maximum mass flow rate by substituting Ma = 1
65
Effect of Back Pressure on Flow through a Converging Nozzle
64. A relation for the variation of flow area A through the nozzle
relative to throat area A* can be obtained by combining
equations f or and for the same mass flow rate and
stagnation properties of a particular fluid. This yields
Another parameter sometimes used in the analysis of one-
dimensional isentropic flow of ideal gases is Ma*, which is
the ratio of the local velocity to the speed of sound at the
throat:
66
Effect of Back Pressure on Flow through a Converging Nozzle
65. where Ma is the local Mach number, T is the local
temperature, and T* is the critical temperature.
Solving for T and for T* and substituting, we get
Note that the parameter Ma* differs from the Mach number
Ma in that Ma* is the local velocity nondimensionalized with
respect to the sonic velocity at the throat, whereas Ma is the
local velocity nondimensionalized with respect to the local
sonic velocity.
67
Effect of Back Pressure on Flow through a Converging Nozzle
67. A plot of versus Pb /P0 for a
converging nozzle is shown in
Fig. below.
Notice that the mass flow rate
increases with decreasing Pb /P0,
reaches a maximum at Pb = P*,
and remains constant for Pb /P0
values less than this critical
ratio. Also illustrated on this
figure is the effect of back
pressure on the nozzle exit
pressure Pe. We observe that
69
Effect of Back Pressure on Flow through a Converging Nozzle
68. To summarize, for all back pressures lower than the critical
pressure P*, the pressure at the exit plane of the converging
nozzle Pe is equal to P*, the Mach number at the exit plane is
unity, and the mass flow rate is the maximum (or choked) flow
rate.
Because the velocity of the flow is sonic at the throat for the
maximum flow rate, a back pressure lower than the critical
pressure cannot be sensed in the nozzle upstream flow and
does not affect the flow rate.
70
Effect of Back Pressure on Flow through a Converging Nozzle
69. 71
1. Pb = Po, Pb /Po = 1. No flow occurs. Pe = Pb, Me=0.
2. Pb > P* or P*/Po < Pb /Po < 1. Flow begins to increase as
the back pressure is lowered. Pe = Pb, Me < 1.
3. Pb = P* or P*/Po = Pb /Po < 1. Flow increases to the choked
flow limit as the back pressure is lowered to the critical
pressure. Pe = Pb, Me=1.
4. Pb < P* or Pb /Po < P*/Po < 1. Flow is still choked and does
not increase as the back pressure is lowered below the
critical pressure, pressure drop from Pe to Pb occurs outside
the nozzle. Pe = P*, Me=1.
5. Pb = 0. Results are the same as for item 4.
Consider the converging-diverging nozzle shown below.
Effect of Back Pressure on Flow through a Converging Nozzle
70. Example. Effect of Back Pressure on Mass Flow Rate
Air at 1 MPa and 600°C enters a converging nozzle, shown
in Fig., with a velocity of 150 m/s. Determine the mass
flow rate through the nozzle for a nozzle throat area of 50
cm2 when the back pressure is (a) 0.7 MPa and (b) 0.4
MPa.
72
72. 74
The critical-pressure ratio is determined from Table 1 (or Eq.
below)
to be P*/P0 = 0.5283.
That is, Pt = Pb = 0.7 MPa, and Pt /P0 = 0.670. Therefore, the flow
is not choked. From Table 1 at Pt /P0 = 0.670, we read Mat =
0.778 and Tt /T0 = 0.892.
74. Converging–Diverging Nozzles
When we think of nozzles, we ordinarily think of flow
passages whose cross-sectional area decreases in the flow
direction. However, the highest velocity to which a fluid
can be accelerated in a converging nozzle is limited to the
sonic velocity (Ma = 1), which occurs at the exit plane
(throat) of the nozzle.
Accelerating a fluid to supersonic velocities (Ma > 1) can
be accomplished only by attaching a diverging flow
section to the subsonic nozzle at the throat. The resulting
combined flow section is a converging– diverging nozzle,
which is standard equipment in supersonic aircraft and
rocket propulsion.
76
75. Forcing a fluid through a converging–diverging nozzle is no
guarantee that the fluid will be accelerated to a supersonic velocity.
In fact, the fluid may find itself decelerating in the diverging
section instead of accelerating if the back pressure is not in the
right range.
The state of the nozzle flow is determined by the overall pressure
ratio Pb/P0. Therefore, for given inlet conditions, the flow through a
converging–diverging nozzle is governed by the back pressure Pb.
77
Converging–Diverging Nozzles
76. Consider the converging–
diverging nozzle shown in Fig.
A fluid enters the nozzle with a
low velocity at stagnation
pressure P0. When Pb =P0 (case
A), there is no flow through the
nozzle.
This is expected since the flow
in a nozzle is driven by the
pressure difference between the
nozzle inlet and the exit.
Now let us examine what
happens as the back pressure is
lowered.
78
Converging–Diverging
Nozzles
78. 3. When PC > Pb > PE, the fluid that achieved a sonic velocity at the
throat continues accelerating to supersonic velocities in the
diverging section as the pressure decreases. This acceleration
comes to a sudden stop, however, as a normal shock develops
at a section between the throat and the exit plane, which causes
a sudden drop in velocity to subsonic levels and a sudden
increase in pressure.
The fluid then continues to decelerate further in the remaining
part of the converging–diverging nozzle. Flow through the
shock is highly irreversible, and thus it cannot be approximated
as isentropic. The normal shock moves downstream away from
the throat as Pb is decreased, and it approaches the nozzle exit
plane as Pb approaches PE.
80
Converging–Diverging Nozzles
79. When Pb = PE, the normal shock forms at the exit plane of the
nozzle. The flow is supersonic through the entire diverging
section in this case, and it can be approximated as isentropic.
However, the fluid velocity drops to subsonic levels just
before leaving the nozzle as it crosses the normal shock.
4. When PE > Pb > 0, the flow in the diverging section is
supersonic, and the fluid expands to PF at the nozzle exit with
no normal shock forming within the nozzle. Thus, the flow
through the nozzle can be approximated as isentropic.
When Pb = PF, no shocks occur within or outside the nozzle.
When Pb < PF, irreversible mixing and expansion waves occur
downstream of the exit plane of the nozzle. When Pb > PF,
however, the pressure of the fluid increases from PF to Pb
irreversibly in the wake of the nozzle exit, creating what are
81
Converging–Diverging Nozzles
80. 82
Example -7
Air leaves the turbine of a turbojet engine and enters a
convergent nozzle at 400 K, 871 kPa, with a velocity of 180 m/s.
The nozzle has an exit area of 730 cm2
. Determine the mass
flow rate through the nozzle for back pressures of 700 kPa, 528
kPa, and 100 kPa, assuming isentropic flow.
The stagnation temperature and stagnation pressure are
T T
V
C
o
P
= +
2
2
81. 83
For air k = 1.4 and from Table or using the equation below the critical
pressure ratio is P*/Po = 0.528. The critical pressure for this nozzle is
P P
kPa kPa
o
*
.
. ( )
=
= =
0528
0528 1000 528
Therefore, for a back pressure of 528 kPa, M = 1 at the nozzle exit and
the flow is choked. For a back pressure of 700 kPa, the nozzle is not
choked. The flow rate will not increase for back pressures below 528
kPa.
82. 84
For the back pressure of 700 kPa,
P
P
kPa
kPa
P
P
B
o o
= = >
700
1000
0 700
.
*
Thus, PE = PB = 700 kPa. For this pressure ratio Table 1 gives
ME = 0 7324
.
T
T
T T K K
E
o
E o
=
= = =
0 9031
0 9031 0 9031 4161 3758
.
. . ( . ) .
C kRT
kJ
kg K
K
m
s
kJ
kg
m
s
V M C
m
s
m
s
E E
E E E
=
=
⋅
=
= =
=
14 0 287 3758
1000
388 6
0 7324 388 6
284 6
2
2
. ( . )( . )
.
( . )( . )
.
83. 85
ρE
E
E
P
RT
kPa
kJ
kg K
K
kJ
m kPa
kg
m
= =
⋅
=
( )
( . )( . )
.
700
0 287 3758
6 4902
3
3
Then
. ( )( . )
( )
.
m A V
kg
m
cm
m
s
m
cm
kg
s
E E E
=
=
=
ρ
6 4902 730 284 6
100
134 8
3
2
2
2
For the back pressure of 528 kPa,
P
P
kPa
kPa
P
P
E
o o
= = =
528
1000
0528
.
*
84. 86
This is the critical pressure ratio and ME = 1 and PE = PB = P* = 528 kPa.
T
T
T
T
T T K K
E
o o
E o
= =
= = =
*
.
. . ( . ) .
08333
08333 08333 4161 346 7
And since ME = 1,
V C kRT
kJ
kg K
K
m
s
kJ
kg
m
s
E E E
= =
=
⋅
=
14 0 287 346 7
1000
3732
2
2
. ( . )( . )
.
ρ ρ
E
P
RT
kPa
kJ
kg K
K
kJ
m kPa
kg
m
= = =
⋅
=
*
*
*
(528 )
( . )( . )
.
0 287 346 7
53064
3
3
85. 87
. ( )( . )
( )
.
m A V
kg
m
cm
m
s
m
cm
kg
s
E E E
=
=
=
ρ
53064 730 3732
100
144 6
3
2
2
2
For a back pressure less than the critical pressure, 528 kPa in this case,
the nozzle is choked and the mass flow rate will be the same as that for
the critical pressure. Therefore, at a back pressure of 100 kPa the mass
flow rate will be 144.6 kg/s.
Example -8
Air enters a converging–diverging nozzle, shown in Fig., at 1.0 MPa and
800 K with a negligible velocity. The flow is steady, one-dimensional,
and isentropic with k = 1.4. For an exit Mach number of Ma = 2 and a
throat area of 20 cm2
, determine (a) the throat conditions, (b) the exit
plane conditions, including the exit area, and (c) the mass flow rate
through the nozzle.
90. 92
Normal Shocks
In some range of back pressure, the fluid that achieved a sonic
velocity at the throat of a converging-diverging nozzle and is
accelerating to supersonic velocities in the diverging section
experiences a normal shock.
The normal shock causes a sudden rise in pressure and
temperature and a sudden drop in velocity to subsonic levels.
Flow through the shock is highly irreversible, and thus it
cannot be approximated as isentropic. The properties of an
ideal gas with constant specific heats before (subscript 1) and
after (subscript 2) a shock are related by
91. 93
We assume steady-flow with no heat and work interactions and
no potential energy changes. We have the following
Conservation of mass
1 1 2 2
2 2 2 2
AV AV
V V
ρ ρ
ρ ρ
=
=
r r
r r
92. 94
Conservation of energy
2 2
1 2
1 2
1 2
1 2
2 2
:
o o
o o
V V
h h
h h
for ideal gases T T
+ = +
=
=
r r
Conservation of momentum
Rearranging and integrating yield
1 2 2 1
( ) ( )
A P P m V V
− = −
r r
&
Increase of entropy
2 1 0
s s
− ≥
Thus, we see that from the conservation of energy, the stagnation
temperature is constant across the shock. However, the stagnation
pressure decreases across the shock because of irreversibilities. The
ordinary (static) temperature rises drastically because of the conversion
of kinetic energy into enthalpy due to a large drop in fluid velocity.
93. 95
We can show that the following relations apply across the shock.
2
2 1
2
1 2
2
1 1
2
2
1 2 2
2
2 1
2 2
1
1 ( 1)/ 2
1 ( 1)/ 2
1 ( 1) / 2
1 ( 1) / 2
2/( 1)
2 /( 1) 1
T M k
T M k
M M k
P
P M M k
M k
M
M k k
+ −
=
+ −
+ −
=
+ −
+ −
=
− −
The entropy change across the shock is obtained by applying the
entropy-change equation for an ideal gas, constant properties, across
the shock:
2 2
2 1
1 1
ln ln
p
T P
s s C R
T P
− = −
94. 96
We can combine the conservation
of mass and energy relations into a
single equation and plot it on an h-s
diagram, using property relations.
The resultant curve is called the
Fanno line, and it is the locus of
states that have the same value of
stagnation enthalpy and mass flux
(mass flow per unit flow area).
Likewise, combining the
conservation of mass and
momentum equations into a single
equation and plotting it on the h-s
diagram yield a curve called the
Rayleigh line.
95. 97
The points of maximum entropy on these lines (points a and b)
correspond to Ma = 1. The state on the upper part of each curve is
subsonic and on the lower part supersonic.
The Fanno and Rayleigh lines intersect at two points (points 1 and
2), which represent the two states at which all three conservation
equations are satisfied. One of these (state 1) corresponds to the state
before the shock, and the other (state 2) corresponds to the state after
the shock.
Note that the flow is supersonic before the shock and subsonic
afterward. Therefore the flow must change from supersonic to
subsonic if a shock is to occur. The larger the Mach number before
the shock, the stronger the shock will be. In the limiting case of Ma =
1, the shock wave simply becomes a sound wave. Notice from Fig.
that entropy increases, s2 >s1. This is expected since the flow through
the shock is adiabatic but irreversible.
98. 100
Example -9
Air flowing with a velocity of 600 m/s, a pressure of 60 kPa, and a
temperature of 260 K undergoes a normal shock. Determine the
velocity and static and stagnation conditions after the shock and the
entropy change across the shock.
The Mach number before the shock is
1 1
1
1 1
2
2
600
1000
1.4(0.287 )(260 )
1.856
V V
M
C kRT
m
s
m
kJ s
K
kJ
kg K
kg
= =
=
⋅
=
r r
99. 101
For M1 = 1.856, Table 1 gives
1 1
1 1
0.1597, 0.5921
o o
P T
P T
= =
For Mx = 1.856, Table 2 gives the following results.
2 2
2
1 1
2 2
2
1 1 1
0.6045, 3.852, 2.4473
1.574, 0.7875, 4.931
o o
o
P
M
P
P P
T
T P P
ρ
ρ
= = =
= = =
100. 102
From the conservation of mass with A2 = A1.
2 2 1 1
1
2
2
1
600
245.2
2.4473
V V
m
V m
s
V
s
ρ ρ
ρ
ρ
=
= = =
r r
r
r
2
2 1
1
2
2 1
1
60 (3.852) 231.1
260 (1.574) 409.2
P
P P kPa kPa
P
T
T T K K
T
= = =
= = =
1
1 2
1
1
260
439.1
0.5921
o o
o
T K
T K T
T
T
= = = =
1
1
1
1
60
375.6
0.1597
o
o
P kPa
P kPa
P
P
= = =
101. 103
The entropy change across the shock is
2 2
2 1
1 1
ln ln
P
T P
s s C R
T P
− = −
( ) ( )
2 1 1.005 ln 1.574 0.287 ln 3.852
0.0688
kJ kJ
s s
kg K kg K
kJ
kg K
− = −
⋅ ⋅
=
⋅
You are encouraged to read about the following topics in the text:
•Oblique shocks
2
2 1
1
375.6 (0.7875) 295.8
o
o o
o
P
P P kPa kPa
P
= = =
102. Example 10. Shock Wave in a Converging–Diverging Nozzle
If the air flowing through the converging–diverging nozzle of
Example 8 experiences a normal shock wave at the nozzle exit plane
(see fig below), determine the following after the shock: (a) the
stagnation pressure, static pressure, static temperature, and static
density; (b) the entropy change across the shock; (c) the exit
velocity; and (d) the mass flow rate through the nozzle. Assume
steady, one-dimensional, and isentropic flow with k = 1.4 from the
nozzle inlet to the shock location.
104
103. 105
Analysis (a) The fluid properties at the exit of the nozzle just before the
shock (denoted by subscript 1) are those evaluated in Example 8 at the
nozzle exit to be
The fluid properties after the shock (denoted by subscript 2) are
related to those before the shock through the functions listed in Table
2. For Ma1= 2.0, we read
106. Oblique Shocks
Not all shock waves are normal shocks (perpendicular to
the flow direction). For example, when the space shuttle
travels at supersonic speeds through the atmosphere, it
produces a complicated shock pattern consisting of inclined
shock waves called oblique shocks.
Reading Assignment
Read about oblique shock waves: characteristic features,
governing equations, calculation o properties;
Textbook to Read
[YUNUS A. CENGEL] Fluid Mechanics. Fundamentals
and Applications
108