This document contains a solution manual for Chapter 2 of a textbook on thermodynamics. It includes:
- An outline of the chapter sections on concept problems, properties and units, force and energy, specific volume, pressure, manometers and barometers, and temperature.
- Solutions to in-text concept questions covering topics like control volumes, mass and energy flows, densities of different phases of water, intensive/extensive properties, and more.
- The solutions provide concise explanations and diagrams to illustrate key concepts in thermodynamics.
- Copyright information is included noting that excerpts may be reproduced for educational purposes with attribution.
This document contains 20 multiple choice problems related to mechanical engineering. The problems cover topics such as fluid mechanics, thermodynamics, heat transfer, and other mechanical engineering principles. They involve calculations related to things like tank volumes, pressure differences, flow rates, heat transfer between substances, and more. The questions provide relevant equations, known values, and ask the reader to determine unknown values or temperatures based on the given information.
Thermodynamics chapter discusses properties of liquids and vapors including:
- Constant pressure process where boiling occurs at a fixed temperature for a given pressure.
- Saturation temperature is the boiling point corresponding to a pressure.
- Quality is the dryness fraction of wet steam.
- Tabulated properties include values for saturated water, steam, compressed water, wet steam and superheated steam.
- Constant pressure, constant volume, isothermal, adiabatic and polytropic processes are examined for non-flow systems involving changes to steam.
This document summarizes information about fans and blowers. It defines fans and blowers, describes common types of fans including axial and centrifugal fans. It discusses fan performance parameters such as pressure, flow rate, power and efficiency. The document also presents relationships called fan laws that describe how these parameters change with speed, size and other variables. Formulas are provided for calculating pressure, power and efficiency. Common applications of fans are also listed.
This document discusses various concepts related to heat, including:
[1] Heat capacity and specific heat capacity, which refer to the amount of heat required to change the temperature of a substance. Specific heat capacity is a material's heat capacity per unit mass.
[2] Latent heat of fusion and vaporization, which is the heat absorbed or released during phase changes between solid, liquid, and gas with no change in temperature.
[3] Methods for determining specific heat capacity experimentally, including the mixture method of transferring heat between substances and the electrical method of applying heat from an electrical source.
Design of machine elements - DESIGN FOR SIMPLE STRESSESAkram Hossain
This document provides solutions to design problems involving the sizing of structural members based on their material properties and applied loads. Problem 1 involves sizing the cross-sectional dimensions of a steel link based on ultimate strength, yield strength, and allowable elongation. Problem 2 is similar but for a malleable iron link. Problem 3 considers a gray iron link. Subsequent problems involve sizing members made of various materials, including steel, cast steel, and bronze, based on factors like ultimate strength, yield strength, and applied tensile, compressive, and shear loads. Check problems 9-13 provide additional practice sizing members and calculating values like number of holes that can be punched or bearing length.
Selection and Design of Condensers
0 INTRODUCTION/PURPOSE
1 SCOPE
2 FIELD OF APPLICATION
3 DEFINITIONS
4 CHOICE OF COOLANT
5 LAYOUT CONSIDERATIONS
5.1 Distillation Column Condensers
5.2 Other Process Condensers
6 CONTROL
6.1 Distillation Columns
6.2 Water Cooled Condensers
6.3 Refrigerant Condensers
7 GENERAL DESIGN CONSIDERATIONS
7.1 Heat Transfer Resistances
7.2 Pressure Drop
7.3 Handling of Inerts
7.4 Vapor Inlet Design
7.5 Drainage of Condensate
8 SUMMARY OF TYPES AVAILABLE
8.1 Direct Contact Condensers
8.2 Shell and Tube Exchangers
8.3 Air Cooled Heat Exchangers
8.4 Spiral Plate Heat Exchangers
8.5 Internal Condensers
8.6 Plate Heat Exchangers
8.7 Plate-Fin Heat Exchangers
8.8 Other Compact Designs
9 BIBLIOGRAPHY
FIGURES
1 DIRECT CONTACT CONDENSER WITH INDIRECT COOLER FOR RECYCLED CONDENSATE
2 SPRAY CONDENSER
3 TRAY TYPE CONDENSER
4 THREE PASS TUBE SIDE CONDENSER WITH INTERPASS LUTING FOR CONDENSATE DRAINAGE
5 CROSS FLOW CONDENSER WITH SINGLE PASS COOLANT
This document discusses transport phenomena and provides examples of momentum, heat, and mass transfer. It defines transport phenomena as irreversible processes involving the transfer of matter, energy, or momentum in fluids not at equilibrium. Examples given are heat conduction, viscosity, diffusion, and electric charge transfer. Analogies are drawn between the transfer of mass, energy, and momentum via diffusion processes. The document then provides the governing equations for heat (Fourier's law), mass (Fick's law), and momentum (Newton's law) transfer and worked examples. It also discusses driving forces, resistances, and the relationships between diffusivity, viscosity, and thermal conductivity.
This document contains multiple problems involving ideal gas processes. The first problem describes a steady flow compressor handling nitrogen with known intake conditions and discharge pressure. It asks to determine the final temperature and work for two process types. The second problem involves air in a cylinder being compressed in a polytropic process with known initial and final pressures and temperatures. It asks to determine the work and heat transfer. The third problem describes a gas turbine expanding helium polytropically and asks to determine the final pressure, power produced, heat loss, and entropy change.
This document contains 20 multiple choice problems related to mechanical engineering. The problems cover topics such as fluid mechanics, thermodynamics, heat transfer, and other mechanical engineering principles. They involve calculations related to things like tank volumes, pressure differences, flow rates, heat transfer between substances, and more. The questions provide relevant equations, known values, and ask the reader to determine unknown values or temperatures based on the given information.
Thermodynamics chapter discusses properties of liquids and vapors including:
- Constant pressure process where boiling occurs at a fixed temperature for a given pressure.
- Saturation temperature is the boiling point corresponding to a pressure.
- Quality is the dryness fraction of wet steam.
- Tabulated properties include values for saturated water, steam, compressed water, wet steam and superheated steam.
- Constant pressure, constant volume, isothermal, adiabatic and polytropic processes are examined for non-flow systems involving changes to steam.
This document summarizes information about fans and blowers. It defines fans and blowers, describes common types of fans including axial and centrifugal fans. It discusses fan performance parameters such as pressure, flow rate, power and efficiency. The document also presents relationships called fan laws that describe how these parameters change with speed, size and other variables. Formulas are provided for calculating pressure, power and efficiency. Common applications of fans are also listed.
This document discusses various concepts related to heat, including:
[1] Heat capacity and specific heat capacity, which refer to the amount of heat required to change the temperature of a substance. Specific heat capacity is a material's heat capacity per unit mass.
[2] Latent heat of fusion and vaporization, which is the heat absorbed or released during phase changes between solid, liquid, and gas with no change in temperature.
[3] Methods for determining specific heat capacity experimentally, including the mixture method of transferring heat between substances and the electrical method of applying heat from an electrical source.
Design of machine elements - DESIGN FOR SIMPLE STRESSESAkram Hossain
This document provides solutions to design problems involving the sizing of structural members based on their material properties and applied loads. Problem 1 involves sizing the cross-sectional dimensions of a steel link based on ultimate strength, yield strength, and allowable elongation. Problem 2 is similar but for a malleable iron link. Problem 3 considers a gray iron link. Subsequent problems involve sizing members made of various materials, including steel, cast steel, and bronze, based on factors like ultimate strength, yield strength, and applied tensile, compressive, and shear loads. Check problems 9-13 provide additional practice sizing members and calculating values like number of holes that can be punched or bearing length.
Selection and Design of Condensers
0 INTRODUCTION/PURPOSE
1 SCOPE
2 FIELD OF APPLICATION
3 DEFINITIONS
4 CHOICE OF COOLANT
5 LAYOUT CONSIDERATIONS
5.1 Distillation Column Condensers
5.2 Other Process Condensers
6 CONTROL
6.1 Distillation Columns
6.2 Water Cooled Condensers
6.3 Refrigerant Condensers
7 GENERAL DESIGN CONSIDERATIONS
7.1 Heat Transfer Resistances
7.2 Pressure Drop
7.3 Handling of Inerts
7.4 Vapor Inlet Design
7.5 Drainage of Condensate
8 SUMMARY OF TYPES AVAILABLE
8.1 Direct Contact Condensers
8.2 Shell and Tube Exchangers
8.3 Air Cooled Heat Exchangers
8.4 Spiral Plate Heat Exchangers
8.5 Internal Condensers
8.6 Plate Heat Exchangers
8.7 Plate-Fin Heat Exchangers
8.8 Other Compact Designs
9 BIBLIOGRAPHY
FIGURES
1 DIRECT CONTACT CONDENSER WITH INDIRECT COOLER FOR RECYCLED CONDENSATE
2 SPRAY CONDENSER
3 TRAY TYPE CONDENSER
4 THREE PASS TUBE SIDE CONDENSER WITH INTERPASS LUTING FOR CONDENSATE DRAINAGE
5 CROSS FLOW CONDENSER WITH SINGLE PASS COOLANT
This document discusses transport phenomena and provides examples of momentum, heat, and mass transfer. It defines transport phenomena as irreversible processes involving the transfer of matter, energy, or momentum in fluids not at equilibrium. Examples given are heat conduction, viscosity, diffusion, and electric charge transfer. Analogies are drawn between the transfer of mass, energy, and momentum via diffusion processes. The document then provides the governing equations for heat (Fourier's law), mass (Fick's law), and momentum (Newton's law) transfer and worked examples. It also discusses driving forces, resistances, and the relationships between diffusivity, viscosity, and thermal conductivity.
This document contains multiple problems involving ideal gas processes. The first problem describes a steady flow compressor handling nitrogen with known intake conditions and discharge pressure. It asks to determine the final temperature and work for two process types. The second problem involves air in a cylinder being compressed in a polytropic process with known initial and final pressures and temperatures. It asks to determine the work and heat transfer. The third problem describes a gas turbine expanding helium polytropically and asks to determine the final pressure, power produced, heat loss, and entropy change.
This document contains a 16 question multiple choice mechanical engineering review problem set. It covers topics including: specific weight calculations, changes in weight due to elevation, pressure and force calculations for scuba diving, determining height using barometer readings, properties of gas mixtures, heat transfer between materials, gas turbine processes, combustion calculations, and thermodynamic processes including changes in temperature, pressure, volume, entropy and heat/work.
Methods of handling Supply air in HVAC Yuri Melliza
The document provides examples of calculations for air conditioning systems that use outside air and recirculated air. It determines parameters such as mass flow rates, cooling and heating capacities, and condensate removal. For the last example, it calculates:
a) The supply air mass flow rate is 14.2 kg/sec
b) The recirculated air mass flow rate is 8.51 kg/sec
c) The outside air mass flow rate is 5.7 kg/sec
d) The condensate removal rate is 0.056 kg/sec
e) The refrigeration capacity of the AC unit is 90.21 tons
Different objects have different heat capacity. Sand has a low heat capacity and gets hot quickly while sea water has a high heat capacity and gets hot slowly. Heat capacity of an object increases when the mass of the object increases. For example, the water in a full kettle takes a longer time to boil compared to the water in a half-fi lled kettle. This shows that water of bigger mass has a higher heat capacity compared to water of smaller mass.
Several daily situations involving heat capacity also discussed.
4.2 Specific Heat Capacity
4.2.1 Explain heat capacity, C.
4.2.2 Define specific heat capacity of a material, c
4.2.3 Experiment to determine:
(i) the specific heat capacity of water
(ii) the specific heat capacity of aluminium
4.2.4 Communicate to explain the applications of specific heat capacity in daily life, material engineering and natural phenomena.
4.2.5 Solve problems involving specific heat capacity
2.2.3 Thermal capacity (heat capacity)
Core
Relate a rise in the temperature of a body to an increase in its internal energy
Show an understanding of what is meant by the thermal capacity of a body
Supplement
• Give a simple molecular account of an increase in internal energy
• Recall and use the equation thermal capacity = mc
• Define specific heat capacity
• Describe an experiment to measure the specific heat capacity of a substance
• Recall and use the equation change in energy = mcΔT
The document summarizes the first law of thermodynamics. It states that energy cannot be created or destroyed, only changed from one form to another. The total amount of energy in the universe remains constant. Energy exists in many forms, including mechanical, heat, light, chemical and electrical. Thermodynamics is the study of energy and how it is transferred or changed from one system to another or between systems and their surroundings.
1. The document derives a general differential equation for fluid flow problems in rectangular Cartesian coordinates using a shell momentum balance.
2. It considers flow between parallel plates where the velocity depends only on the x-coordinate and derives expressions for the shear stress distribution, velocity profile, maximum velocity, average velocity, and mass flow rate for a Newtonian fluid.
3. The shear stress is found to be linearly proportional to x, the velocity profile parabolic, the maximum velocity occurs at the center, and the average velocity is 2/3 of the maximum velocity.
This document discusses the process design of shell and tube heat exchangers. It begins by classifying heat exchangers and describing different types of shell and tube heat exchangers such as fixed tube sheet, removable tube bundle, floating head, and U-tube designs. The document then discusses various thermal design considerations for shell and tube heat exchangers, including selections for the shell, tube materials and dimensions, tube layout and count, baffles, and fouling factors. It provides process design procedures and an example problem for designing shell and tube heat exchangers.
This document discusses heat exchangers and one-dimensional steady conduction. It provides equations and examples for conduction through plane slabs, composite walls, cylindrical layers, and spherical layers. It also discusses the purpose of insulation, critical insulation thickness, and factors that affect thermal conductivity. Dimensionless numbers for convective heat transfer like Reynolds number, Prandtl number, and Nusselt number are defined. Empirical relationships are provided for forced and natural convection.
This document discusses fluid flow and provides information on several topics:
1) It describes laminar and turbulent flow, and introduces the Reynolds number which determines the transition between these two flow regimes.
2) It discusses mass balances and the continuity equation which states that the rate of mass input equals the rate of mass output in steady state flow.
3) It derives the overall energy balance equation based on the first law of thermodynamics and describes how to apply this to steady state flow systems.
4) It introduces the mechanical energy balance equation which is useful for analyzing flowing liquids and accounts for kinetic energy, potential energy, and frictional losses.
chapter 4 first law of thermodynamics thermodynamics 1abfisho
This document discusses the first law of thermodynamics for closed systems. It begins by defining the first law as the law of conservation of energy, where energy cannot be created or destroyed, only transformed between states. The energy balance is then analyzed for closed systems, where the internal energy, kinetic energy, potential energy, work and heat transfer across boundaries are considered. Several examples are provided to demonstrate applying the first law to calculate changes in internal energy, work, heat transfer and other properties for closed thermodynamic systems undergoing various processes like constant volume, constant pressure and others.
Here are the key steps to derive the expression for heat of reaction at constant pressure:
1) For a chemical reaction occurring at constant pressure, the enthalpy change (ΔH) is equal to the heat absorbed or released by the system (qP).
2) Enthalpy change (ΔH) is defined as the change in internal energy (ΔU) plus the product of pressure (P) and change in volume (ΔV).
ΔH = ΔU + PΔV
3) For a reaction at constant pressure, the volume change (ΔV) is small and pressure remains constant.
4) From the first law of thermodynamics, the change in internal energy (Δ
This document summarizes an experiment conducted to demonstrate Boyle's law. The experiment was conducted on October 29, 2013 by a student to measure the relationship between the pressure and volume of a gas. Boyle's law states that the pressure and volume of a gas are inversely related when temperature is held constant. The equipment used in the experiment included tanks and valves to control the pressure and volume of the gas sample. The student recorded measurements as the pressure and volume were varied and analyzed the results to validate Boyle's law.
The document discusses heating and cooling curves, which are graphs of temperature over time that can be used to determine phase changes and melting/boiling points of substances. Heating curves show flat regions where temperature remains constant as a substance changes phase from solid to liquid to gas. Cooling curves are the opposite and show phase changes from gas to liquid to solid. The document provides examples of constructing heating and cooling curves by heating/cooling water and stearic acid and measuring temperature changes.
This document discusses the thermal design of a simple boiler. It presents the calculation procedures for boiler design, focusing on heat transfer modes, heat and mass balances, and a worked example. The key points are:
- Heat transfer in boilers occurs via conduction, convection, and radiation. Conduction is not considered in simple calculations.
- Heat and mass balance equations relate the heat input from fuel to the heat output via steam as well as accounting for air and flue gas flows.
- A worked example calculates furnace conditions like flue gas temperature for a methane-fueled boiler, assuming radiation is the only heat transfer mode in the furnace. Tube bank calculations then determine the exit gas
This document summarizes a study on the performance characteristics of counter flow wet cooling towers. The study uses a detailed model to investigate cooling tower performance. It is shown that evaporation contributes the majority (62.5-90%) of the total heat transfer rate from the bottom to the top of the tower. The model accounts for variations in air and water temperatures and properties along the height of the tower. Heat and mass transfer mechanisms are explained using psychrometric charts.
This document provides an overview of Chapter 3 from a thermodynamics textbook. It discusses volumetric properties of pure fluids. The chapter describes phase change processes and diagrams for pure substances. It also discusses property tables that list internal energy, enthalpy, and entropy values for substances, as these relationships are too complex to express through simple equations. Examples are provided on using property tables to determine pressure, temperature, volume changes, and energy changes during phase change processes like vaporization of water.
This document contains a chapter from the textbook "Introduction to Engineering Thermodynamics" by Sonntag and Borgnakke. It provides the solution manual for chapter 4, which was authored by Claus Borgnakke. The chapter covers various thermodynamic concepts related to work, including poltropic processes, boundary work for multi-step processes, and rates of work. It includes over 70 example problems and their step-by-step solutions.
This document discusses key concepts in thermodynamics including heat, temperature, thermodynamics, the first law of thermodynamics, and reversible and irreversible processes. Thermodynamics is the branch of physics concerned with heat, temperature, and their relation to energy and work. Heat is a form of energy that flows from higher to lower temperature, and temperature determines the direction of heat flow. The first law of thermodynamics states that the total energy in an isolated system remains constant, as energy cannot be created or destroyed. Reversible processes can be retraced in the opposite direction, while irreversible processes cannot.
This document provides information about laboratory safety procedures and proper use of personal protective equipment (PPE) in a laboratory setting. It outlines dos and don'ts for safe laboratory practices, including always wearing appropriate eye protection and closed-toe shoes. Basic PPE like safety goggles, respirators, and gloves are described. Common laboratory equipment like scales, moisture meters, calipers and their proper and safe use are explained. Safety precautions for equipment including ovens, infrared thermometers, and multimeters are provided. The document emphasizes ensuring safety in the laboratory and following dos and don'ts.
Gases exist as individual molecules that are in constant random motion. The kinetic molecular theory describes gases as composed of molecules that are separated by large distances and move rapidly in random directions, frequently colliding with one another. The theory states that the average kinetic energy of gas molecules is proportional to the absolute temperature of the gas. Higher temperatures cause molecules to move faster on average with more molecules possessing higher speeds.
The document summarizes Chapter 7 of a textbook on thermodynamics. It includes in-text concept questions, concept problems, sections on heat engines/refrigerators, the second law and processes, Carnot cycles and absolute temperature, finite temperature heat transfer, and ideal gas Carnot cycles. It also includes review problems at the end. The chapter examines concepts related to heat engines, refrigerators, the second law of thermodynamics, and Carnot cycles.
The document discusses the second law of thermodynamics. It introduces Kelvin-Planck's statement that it is impossible to construct an engine that produces work without transferring heat from a hot reservoir and into a cold reservoir. Clausius' statement is also presented, that it is impossible to spontaneously transfer heat from a cold body to a hot body without an external work input. The two statements are shown to be equivalent. Reversible and irreversible processes are defined. The Carnot cycle is described as an ideal reversible thermodynamic cycle consisting of reversible isothermal and adiabatic processes, with its efficiency dependent only on the temperatures of the hot and cold reservoirs.
This document contains a 16 question multiple choice mechanical engineering review problem set. It covers topics including: specific weight calculations, changes in weight due to elevation, pressure and force calculations for scuba diving, determining height using barometer readings, properties of gas mixtures, heat transfer between materials, gas turbine processes, combustion calculations, and thermodynamic processes including changes in temperature, pressure, volume, entropy and heat/work.
Methods of handling Supply air in HVAC Yuri Melliza
The document provides examples of calculations for air conditioning systems that use outside air and recirculated air. It determines parameters such as mass flow rates, cooling and heating capacities, and condensate removal. For the last example, it calculates:
a) The supply air mass flow rate is 14.2 kg/sec
b) The recirculated air mass flow rate is 8.51 kg/sec
c) The outside air mass flow rate is 5.7 kg/sec
d) The condensate removal rate is 0.056 kg/sec
e) The refrigeration capacity of the AC unit is 90.21 tons
Different objects have different heat capacity. Sand has a low heat capacity and gets hot quickly while sea water has a high heat capacity and gets hot slowly. Heat capacity of an object increases when the mass of the object increases. For example, the water in a full kettle takes a longer time to boil compared to the water in a half-fi lled kettle. This shows that water of bigger mass has a higher heat capacity compared to water of smaller mass.
Several daily situations involving heat capacity also discussed.
4.2 Specific Heat Capacity
4.2.1 Explain heat capacity, C.
4.2.2 Define specific heat capacity of a material, c
4.2.3 Experiment to determine:
(i) the specific heat capacity of water
(ii) the specific heat capacity of aluminium
4.2.4 Communicate to explain the applications of specific heat capacity in daily life, material engineering and natural phenomena.
4.2.5 Solve problems involving specific heat capacity
2.2.3 Thermal capacity (heat capacity)
Core
Relate a rise in the temperature of a body to an increase in its internal energy
Show an understanding of what is meant by the thermal capacity of a body
Supplement
• Give a simple molecular account of an increase in internal energy
• Recall and use the equation thermal capacity = mc
• Define specific heat capacity
• Describe an experiment to measure the specific heat capacity of a substance
• Recall and use the equation change in energy = mcΔT
The document summarizes the first law of thermodynamics. It states that energy cannot be created or destroyed, only changed from one form to another. The total amount of energy in the universe remains constant. Energy exists in many forms, including mechanical, heat, light, chemical and electrical. Thermodynamics is the study of energy and how it is transferred or changed from one system to another or between systems and their surroundings.
1. The document derives a general differential equation for fluid flow problems in rectangular Cartesian coordinates using a shell momentum balance.
2. It considers flow between parallel plates where the velocity depends only on the x-coordinate and derives expressions for the shear stress distribution, velocity profile, maximum velocity, average velocity, and mass flow rate for a Newtonian fluid.
3. The shear stress is found to be linearly proportional to x, the velocity profile parabolic, the maximum velocity occurs at the center, and the average velocity is 2/3 of the maximum velocity.
This document discusses the process design of shell and tube heat exchangers. It begins by classifying heat exchangers and describing different types of shell and tube heat exchangers such as fixed tube sheet, removable tube bundle, floating head, and U-tube designs. The document then discusses various thermal design considerations for shell and tube heat exchangers, including selections for the shell, tube materials and dimensions, tube layout and count, baffles, and fouling factors. It provides process design procedures and an example problem for designing shell and tube heat exchangers.
This document discusses heat exchangers and one-dimensional steady conduction. It provides equations and examples for conduction through plane slabs, composite walls, cylindrical layers, and spherical layers. It also discusses the purpose of insulation, critical insulation thickness, and factors that affect thermal conductivity. Dimensionless numbers for convective heat transfer like Reynolds number, Prandtl number, and Nusselt number are defined. Empirical relationships are provided for forced and natural convection.
This document discusses fluid flow and provides information on several topics:
1) It describes laminar and turbulent flow, and introduces the Reynolds number which determines the transition between these two flow regimes.
2) It discusses mass balances and the continuity equation which states that the rate of mass input equals the rate of mass output in steady state flow.
3) It derives the overall energy balance equation based on the first law of thermodynamics and describes how to apply this to steady state flow systems.
4) It introduces the mechanical energy balance equation which is useful for analyzing flowing liquids and accounts for kinetic energy, potential energy, and frictional losses.
chapter 4 first law of thermodynamics thermodynamics 1abfisho
This document discusses the first law of thermodynamics for closed systems. It begins by defining the first law as the law of conservation of energy, where energy cannot be created or destroyed, only transformed between states. The energy balance is then analyzed for closed systems, where the internal energy, kinetic energy, potential energy, work and heat transfer across boundaries are considered. Several examples are provided to demonstrate applying the first law to calculate changes in internal energy, work, heat transfer and other properties for closed thermodynamic systems undergoing various processes like constant volume, constant pressure and others.
Here are the key steps to derive the expression for heat of reaction at constant pressure:
1) For a chemical reaction occurring at constant pressure, the enthalpy change (ΔH) is equal to the heat absorbed or released by the system (qP).
2) Enthalpy change (ΔH) is defined as the change in internal energy (ΔU) plus the product of pressure (P) and change in volume (ΔV).
ΔH = ΔU + PΔV
3) For a reaction at constant pressure, the volume change (ΔV) is small and pressure remains constant.
4) From the first law of thermodynamics, the change in internal energy (Δ
This document summarizes an experiment conducted to demonstrate Boyle's law. The experiment was conducted on October 29, 2013 by a student to measure the relationship between the pressure and volume of a gas. Boyle's law states that the pressure and volume of a gas are inversely related when temperature is held constant. The equipment used in the experiment included tanks and valves to control the pressure and volume of the gas sample. The student recorded measurements as the pressure and volume were varied and analyzed the results to validate Boyle's law.
The document discusses heating and cooling curves, which are graphs of temperature over time that can be used to determine phase changes and melting/boiling points of substances. Heating curves show flat regions where temperature remains constant as a substance changes phase from solid to liquid to gas. Cooling curves are the opposite and show phase changes from gas to liquid to solid. The document provides examples of constructing heating and cooling curves by heating/cooling water and stearic acid and measuring temperature changes.
This document discusses the thermal design of a simple boiler. It presents the calculation procedures for boiler design, focusing on heat transfer modes, heat and mass balances, and a worked example. The key points are:
- Heat transfer in boilers occurs via conduction, convection, and radiation. Conduction is not considered in simple calculations.
- Heat and mass balance equations relate the heat input from fuel to the heat output via steam as well as accounting for air and flue gas flows.
- A worked example calculates furnace conditions like flue gas temperature for a methane-fueled boiler, assuming radiation is the only heat transfer mode in the furnace. Tube bank calculations then determine the exit gas
This document summarizes a study on the performance characteristics of counter flow wet cooling towers. The study uses a detailed model to investigate cooling tower performance. It is shown that evaporation contributes the majority (62.5-90%) of the total heat transfer rate from the bottom to the top of the tower. The model accounts for variations in air and water temperatures and properties along the height of the tower. Heat and mass transfer mechanisms are explained using psychrometric charts.
This document provides an overview of Chapter 3 from a thermodynamics textbook. It discusses volumetric properties of pure fluids. The chapter describes phase change processes and diagrams for pure substances. It also discusses property tables that list internal energy, enthalpy, and entropy values for substances, as these relationships are too complex to express through simple equations. Examples are provided on using property tables to determine pressure, temperature, volume changes, and energy changes during phase change processes like vaporization of water.
This document contains a chapter from the textbook "Introduction to Engineering Thermodynamics" by Sonntag and Borgnakke. It provides the solution manual for chapter 4, which was authored by Claus Borgnakke. The chapter covers various thermodynamic concepts related to work, including poltropic processes, boundary work for multi-step processes, and rates of work. It includes over 70 example problems and their step-by-step solutions.
This document discusses key concepts in thermodynamics including heat, temperature, thermodynamics, the first law of thermodynamics, and reversible and irreversible processes. Thermodynamics is the branch of physics concerned with heat, temperature, and their relation to energy and work. Heat is a form of energy that flows from higher to lower temperature, and temperature determines the direction of heat flow. The first law of thermodynamics states that the total energy in an isolated system remains constant, as energy cannot be created or destroyed. Reversible processes can be retraced in the opposite direction, while irreversible processes cannot.
This document provides information about laboratory safety procedures and proper use of personal protective equipment (PPE) in a laboratory setting. It outlines dos and don'ts for safe laboratory practices, including always wearing appropriate eye protection and closed-toe shoes. Basic PPE like safety goggles, respirators, and gloves are described. Common laboratory equipment like scales, moisture meters, calipers and their proper and safe use are explained. Safety precautions for equipment including ovens, infrared thermometers, and multimeters are provided. The document emphasizes ensuring safety in the laboratory and following dos and don'ts.
Gases exist as individual molecules that are in constant random motion. The kinetic molecular theory describes gases as composed of molecules that are separated by large distances and move rapidly in random directions, frequently colliding with one another. The theory states that the average kinetic energy of gas molecules is proportional to the absolute temperature of the gas. Higher temperatures cause molecules to move faster on average with more molecules possessing higher speeds.
The document summarizes Chapter 7 of a textbook on thermodynamics. It includes in-text concept questions, concept problems, sections on heat engines/refrigerators, the second law and processes, Carnot cycles and absolute temperature, finite temperature heat transfer, and ideal gas Carnot cycles. It also includes review problems at the end. The chapter examines concepts related to heat engines, refrigerators, the second law of thermodynamics, and Carnot cycles.
The document discusses the second law of thermodynamics. It introduces Kelvin-Planck's statement that it is impossible to construct an engine that produces work without transferring heat from a hot reservoir and into a cold reservoir. Clausius' statement is also presented, that it is impossible to spontaneously transfer heat from a cold body to a hot body without an external work input. The two statements are shown to be equivalent. Reversible and irreversible processes are defined. The Carnot cycle is described as an ideal reversible thermodynamic cycle consisting of reversible isothermal and adiabatic processes, with its efficiency dependent only on the temperatures of the hot and cold reservoirs.
The document summarizes key concepts from the second law of thermodynamics:
1) The first law has limitations in predicting whether an energy conversion is possible. The second law addresses this through statements by Kelvin-Planck and Clausius.
2) Kelvin-Planck's statement says a device cannot produce work without transferring heat from a hot reservoir and into a cold one. Clausius' statement says heat cannot spontaneously flow from cold to hot.
3) A reversible process can be reversed without leaving traces. Irreversible processes like friction cause a loss of useful energy.
4) The Carnot cycle uses reversible, isothermal and adiabatic processes between two reservoirs to produce work
Presentation by Bishal about thermodynamicsAASHISH463514
This document discusses the first law of thermodynamics and provides an overview of petrol engines. It defines thermodynamic systems and explains concepts like work done during volume change. The first law states that the heat supplied equals the internal energy change plus work done. The document also summarizes the four stroke process of a petrol engine and provides merits like efficiency and demerits like cost. It concludes by discussing understanding gained about petrol engine operation.
This document contains the solutions to homework problems assigned in a thermodynamics course. It provides instructions for submitting homework solutions, including showing all work. It then lists 7 problems and provides the numerical solutions. The problems cover various thermodynamic concepts like the ideal gas law, polytropic processes, work calculations, property tables and definitions.
The document discusses a case where a new 40-MVA transformer was found to have an elevated power factor during testing. It was returned to the factory, where inspection found treeing on the phase barrier board between the H2 and H3 windings. Power factor and induced voltage tests during teardown identified the source of contamination and explained the earlier high test results. The transformer's winding design and location of the treeing helped determine the cause of the failure.
Applied Thermodynamics Pamphlet 2nd edition, 2018 by musadotomusadoto
This pamphlet is a 2nd edition of the series in A COMPREHESIVE SIMPLIFIED TO THE APPLIED THERMODYNAMICS.It consists of three topics with solved supplementary problems which based on tutorials, lecture notes ,summaries and the university past papers. Please don‘t forget to deeply study by yourself to improve your understanding for winning your open book exams.
Engineering Thermodynamics: Properties of Pure SubstancesMAYURDESAI42
Engineering thermodynamics is a branch of thermodynamics that deals with the practical application of thermodynamic principles and concepts. One of the fundamental topics in engineering thermodynamics is the properties of pure substances.
A pure substance is a material that has a fixed and constant chemical composition, regardless of its physical state. This means that a pure substance cannot be separated into two or more different substances by physical means. Examples of pure substances include water, oxygen, and carbon dioxide.
The properties of a pure substance are critical in thermodynamics because they are used to calculate important thermodynamic parameters, such as enthalpy, entropy, and internal energy. These parameters are used to understand the behavior of systems and predict their response to changes in temperature, pressure, and other conditions.
One of the key properties of a pure substance is its temperature-pressure phase diagram, which provides information about the physical state of the substance under different conditions. For example, water can exist as a solid (ice), a liquid (water), or a gas (steam) depending on the temperature and pressure conditions. This information is critical for understanding the behavior of a substance in different thermodynamic systems, such as power plants, refrigeration and air-conditioning systems, and chemical processes.
Another important property of a pure substance is its enthalpy of vaporization, which is the amount of energy required to convert a unit mass of the substance from a liquid to a gas at a constant temperature. This property is critical in many applications, such as the design of steam power plants, which use the energy stored in steam to generate electricity.
The specific heat capacity of a pure substance is another critical property. It represents the amount of energy required to raise the temperature of a unit mass of the substance by a unit temperature. This property is used to calculate the heat transfer in thermodynamic systems, such as refrigeration and air-conditioning systems.
Another important property of a pure substance is its thermal conductivity, which represents its ability to transfer heat. This property is critical in the design of heat exchangers, where heat is transferred from one fluid to another.
In conclusion, the properties of pure substances play a critical role in engineering thermodynamics. They provide valuable information about the behavior of a substance under different conditions and are used to calculate important thermodynamic parameters, such as enthalpy, entropy, and internal energy. This information is critical for the design and operation of a wide range of thermodynamic systems, such as power plants, refrigeration and air-conditioning systems, and chemical processes.
This document discusses copyright restrictions on sharing and distributing educational materials from a physics textbook. It includes three conceptual questions (conceptests) about thermodynamics topics like free expansion of gases, work done in a thermodynamic cycle, and identifying reversible and irreversible heat engines.
This document discusses copyright restrictions on sharing and distributing teaching materials from a physics textbook. It includes three conceptual questions (conceptests) about thermodynamics topics: free expansion of gases, work done in a thermodynamic cycle, and identifying reversible and irreversible heat engines.
The document discusses reversible and irreversible processes. It defines a reversible process as one that can reverse itself without leaving any trace on the system or surroundings. Reversible processes are idealizations that do not actually occur in nature but can approximate real processes. The Carnot cycle, composed of reversible processes, represents the most efficient heat engine possible between two temperature limits. The Carnot cycle consists of four steps: two isothermal expansion/compression processes and two adiabatic expansion/compression processes. The efficiency of a Carnot engine depends only on the temperatures of the hot and cold reservoirs. Reversible processes set the theoretical maximum efficiency and minimum work consumption that real irreversible engines can approach but not meet.
This document is a seminar report submitted by Mr. Ganesh Vasant Nirgude to the University of Pune on supercritical technology in power plants. It includes a certificate from the college confirming Mr. Nirgude presented the seminar. The report contains an introduction on the basic Rankine cycle and efficiency factors. It also discusses supercritical Rankine cycles, design aspects of supercritical boilers, material selection challenges, and advances in supercritical technology for the future in India.
This document discusses reversible and irreversible processes in thermodynamics. It begins by defining reversible processes as processes that can reverse themselves without leaving a trace on the system or surroundings. Reversible processes are idealized and can never be achieved exactly in reality. The Carnot cycle is then introduced as an ideal reversible thermodynamic cycle composed of four processes: two reversible isothermal processes and two reversible adiabatic processes. The Carnot cycle establishes the maximum possible efficiency between two temperature reservoirs. Real engines are always less efficient than the Carnot reversible engine due to irreversible losses. The document provides examples comparing real and reversible processes.
Chapter two fluid power system lecture notenurcam1
This document provides an overview of fluid mechanics concepts relevant to hydraulic systems. It defines key fluid properties like density, pressure, viscosity, and compressibility. It describes how hydraulic fluids are used to transmit power and lubricate moving parts. The basic principles of energy transfer, Pascal's law, continuity equation, Bernoulli's equation, and frictional losses in pipelines are summarized. Examples of hydraulic applications like jacks and pressure boosters are provided to illustrate concepts. Formulas for determining required piston size, pump flow rate, and hydraulic power delivered to loads are also outlined.
1. A reversible process is an idealized process that can be reversed without leaving any trace on the system or surroundings.
2. Reversible processes do not occur in nature, but serve as theoretical limits that actual irreversible processes can approach.
3. The Carnot cycle, composed of reversible processes, achieves the maximum possible efficiency between two temperature reservoirs. No heat engine can have a higher efficiency than a Carnot engine operating between the same reservoirs.
The document is the solutions manual for a thermodynamics textbook. It provides solutions to problems from Chapter 3 on properties of pure substances. The document notes that the solutions manual is proprietary property and may only be distributed to authorized professors and instructors for course preparation. Students are not authorized to use the manual. It then presents solutions to 23 sample problems covering topics like phase change processes, property diagrams, property tables, and the use of equations of state to determine thermodynamic properties of various pure substances like water, refrigerants and ammonia.
Hazard operability study - in Industires and plantJayaKarthic1
The Hazard and Operability (HAZOP) analysis technique is used to identify safety hazards and operability problems in a process plant that could compromise productivity or lead to undesirable consequences. A multi-disciplinary team uses a systematic approach and guide words like "no," "more," and "less" to examine how deviations from the design intent could occur. The team conducted a HAZOP study on a 1 MT benzol plant, identifying deviations and their causes and consequences for processes like transferring benzol to a reboiler. Key findings included pump seal leaks, improper equipment earthing, unscreened vents, and needed instrument maintenance. Recommendations focused on replacing seals, improving grounding, adding vent screens, and
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2. Borgnakke and Sonntag
CONTENT
SUBSECTION PROB NO.
Concept Problems 1-18
Properties and Units 19-22
Force and Energy 23-34
Specific Volume 35-40
Pressure 41-56
Manometers and Barometers 57-77
Temperature 78-83
Review problems 84-89
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3. Borgnakke and Sonntag
In-Text Concept Questions
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4. Borgnakke and Sonntag
2.a
Make a control volume around the turbine in the steam power plant in Fig. 1.1 and
list the flows of mass and energy that are there.
Solution:
We see hot high pressure steam flowing in
at state 1 from the steam drum through a
flow control (not shown). The steam leaves
at a lower pressure to the condenser (heat
exchanger) at state 2. A rotating shaft gives
a rate of energy (power) to the electric
generator set.
W
T
1
2
2.b
Take a control volume around your kitchen refrigerator and indicate where the
components shown in Figure 1.6 are located and show all flows of energy transfer.
Solution:
The valve and the
cold line, the
evaporator, is
inside close to the
inside wall and
usually a small
blower distributes
cold air from the
freezer box to the
refrigerator room.
cb
W
.
Q
.
Q leak The black grille in
the back or at the
bottom is the
condenser that
gives heat to the
room air.
The compressor
sits at the bottom.
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5. Borgnakke and Sonntag
2.c
Why do people float high in the water when swimming in the Dead Sea as compared
with swimming in a fresh water lake?
As the dead sea is very salty its density is
higher than fresh water density. The
buoyancy effect gives a force up that equals
the weight of the displaced water. Since
density is higher the displaced volume is
smaller for the same force.
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6. Borgnakke and Sonntag
2.d
Density of liquid water is ρ = 1008 – T/2 [kg/m3] with T in oC. If the temperature
increases, what happens to the density and specific volume?
Solution:
The density is seen to decrease as the temperature increases.
∆ρ = – ∆T/2
Since the specific volume is the inverse of the density v = 1/ρ it will increase.
2.e
A car tire gauge indicates 195 kPa; what is the air pressure inside?
The pressure you read on the gauge is a gauge pressure, ∆P, so the absolute
pressure is found as
` P = Po + ∆P = 101 + 195 = 296 kPa
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7. Borgnakke and Sonntag
2.f
Can I always neglect ∆P in the fluid above location A in figure 2.12? What does that
depend on?
If the fluid density above A is low relative to the manometer fluid then you
neglect the pressure variation above position A, say the fluid is a gas like air and the
manometer fluid is like liquid water. However, if the fluid above A has a density of
the same order of magnitude as the manometer fluid then the pressure variation with
elevation is as large as in the manometer fluid and it must be accounted for.
2.g
A U tube manometer has the left branch connected to a box with a pressure of 110
kPa and the right branch open. Which side has a higher column of fluid?
Solution:
Since the left branch fluid surface feels 110 kPa and the
right branch surface is at 100 kPa you must go further
down to match the 110 kPa. The right branch has a higher
column of fluid.
cb
Po
Box
H
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8. Borgnakke and Sonntag
Concept Problems
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9. Borgnakke and Sonntag
2.1
Make a control volume around the whole power plant in Fig. 1.2 and with the help
of Fig. 1.1 list what flows of mass and energy are in or out and any storage of
energy. Make sure you know what is inside and what is outside your chosen C.V.
Solution:
Smoke
stack
Boiler
building
Coal conveyor system
Dock
Turbine house
Storage
gypsum
Coal
storage
flue
gas
cb
Underground
power cable
Welectrical
Hot water
District heating
m
Coal
m
m
Flue gas
Storage for later
Gypsum, fly ash, slag
transport out:
Cold return m
m
Combustion air
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10. Borgnakke and Sonntag
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2.2
Take a control volume around the rocket engine in Fig. 1.12. Identify the mass flows
and where you have significant kinetic energy and where storage changes.
We have storage in both
tanks are reduced, mass
flows out with modest
velocities.
Energy conversion in the
combustion process.
gas at high pressure
expands towards lower
pressure outside and thus
accelerates to high
velocity with significant
kinetic energy flowing
out.
Control and mixing
Combustion
Oxydizer Fuel
Nozzle
High speed flow out
11. Borgnakke and Sonntag
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2.3
Make a control volume that includes the steam flow around in the main turbine loop
in the nuclear propulsion system in Fig.1.3. Identify mass flows (hot or cold) and
energy transfers that enter or leave the C.V.
Solution:
The electrical power
also leaves the C.V.
to be used for lights,
instruments and to
charge the batteries.
1 1Hot steam from generator
c Electric
power gen. WT
cb
Welectrical
3
2
5 4Condensate
to steam gen.
cold 76
Cooling by seawater
12. Borgnakke and Sonntag
2.4
Separate the list P, F, V, v, ρ, T, a, m, L, t, and V into intensive, extensive, and non-
properties.
Solution:
Intensive properties are independent upon mass: P, v, ρ, T
Extensive properties scales with mass: V, m
Non-properties: F, a, L, t, V
Comment: You could claim that acceleration a and velocity V are physical
properties for the dynamic motion of the mass, but not thermal properties.
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13. Borgnakke and Sonntag
2.5
An electric dip heater is put into a cup of water and heats it from 20oC to 80oC.
Show the energy flow(s) and storage and explain what changes.
Solution:
Electric power is converted in the heater
element (an electric resistor) so it becomes
hot and gives energy by heat transfer to
the water. The water heats up and thus
stores energy and as it is warmer than the
cup material it heats the cup which also
stores some energy. The cup being
warmer than the air gives a smaller
amount of energy (a rate) to the air as a
heat loss.
Welectric
Q loss
C B
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14. Borgnakke and Sonntag
2.6
Water in nature exists in different phases such as solid, liquid and vapor (gas).
Indicate the relative magnitude of density and specific volume for the three phases.
Solution:
Values are indicated in Figure 2.7 as density for common substances. More
accurate values are found in Tables A.3, A.4 and A.5
Water as solid (ice) has density of around 900 kg/m3
Water as liquid has density of around 1000 kg/m3
Water as vapor has density of around 1 kg/m3 (sensitive to P and T)
Ice cube Liquid drop Cloud*
* Steam (water vapor) can not be seen what you see are tiny drops suspended in air from
which we infer that there was some water vapor before it condensed.
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15. Borgnakke and Sonntag
2.7
Is density a unique measure of mass distribution in a volume? Does it vary? If so, on
what kind of scale (distance)?
Solution:
Density is an average of mass per unit volume and we sense if it is not evenly
distributed by holding a mass that is more heavy in one side than the other.
Through the volume of the same substance (say air in a room) density varies only
little from one location to another on scales of meter, cm or mm. If the volume
you look at has different substances (air and the furniture in the room) then it can
change abruptly as you look at a small volume of air next to a volume of
hardwood.
Finally if we look at very small scales on the order of the size of atoms the density
can vary infinitely, since the mass (electrons, neutrons and positrons) occupy very
little volume relative to all the empty space between them.
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16. Borgnakke and Sonntag
2.8
Density of fibers, rock wool insulation, foams and cotton is fairly low. Why is that?
Solution:
All these materials consist of some solid substance and mainly air or other gas.
The volume of fibers (clothes) and rockwool that is solid substance is low relative
to the total volume that includes air. The overall density is
ρ =
m
V =
msolid + mair
Vsolid + Vair
where most of the mass is the solid and most of the volume is air. If you talk
about the density of the solid only, it is high.
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17. Borgnakke and Sonntag
2.9
How much mass is there approximately in 1 L of engine oil? Atmospheric air?
Solution:
A volume of 1 L equals 0.001 m3, see Table A.1. From Table A.4 the density is
885 kg/m3 so we get
m = ρV = 885 kg/m3 × 0.001 m3 = 0.885 kg
For the air we see in Figure 2.7 that density is about 1 kg/m3 so we get
m = ρV = 1 kg/m3 × 0.001 m3 = 0.001 kg
A more accurate value from Table A.5 is ρ = 1.17 kg/m3 at 100 kPa, 25oC.
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18. Borgnakke and Sonntag
2.10
Can you carry 1 m3 of liquid water?
Solution:
The density of liquid water is about 1000 kg/m3 from Figure 2.7, see also Table
A.3. Therefore the mass in one cubic meter is
m = ρV = 1000 kg/m3 × 1 m3 = 1000 kg
and we can not carry that in the standard gravitational field.
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19. Borgnakke and Sonntag
2.11
A heavy cabinet has four adjustable feet on it. What feature of the feet will ensure
that they do not make dents in the floor?
Answer:
The area that is in contact with the floor supports the total mass in the
gravitational field.
F = PA = mg
so for a given mass the smaller the area is the larger the pressure becomes.
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20. Borgnakke and Sonntag
2.12
The pressure at the bottom of a swimming pool is evenly distributed. Suppose we
look at a cast iron plate of 7272 kg lying on the ground with an area of 100 m2. What
is the average pressure below that? Is it just as evenly distributed as the pressure at
the bottom of the pool?
Solution:
The pressure is force per unit area from page 25:
P = F/A = mg/A = 7272 kg × (9.81 m/s2) / 100 m2 = 713.4 Pa
The iron plate being cast can be reasonable plane and flat, but it is stiff and rigid.
However, the ground is usually uneven so the contact between the plate and the
ground is made over an area much smaller than the 100 m2. Thus the local
pressure at the contact locations is much larger than the quoted value above.
The pressure at the bottom of the swimming pool is very even due to the ability of
the fluid (water) to have full contact with the bottom by deforming itself. This is
the main difference between a fluid behavior and a solid behavior.
Iron plate
Ground
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21. Borgnakke and Sonntag
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2.13
Two divers swim at 20 m depth. One of them swims right in under a supertanker; the
other stays away from the tanker. Who feels a greater pressure?
Solution:
Each one feels the local pressure which is the static pressure only a function of
depth.
Pocean= P0 + ∆P = P0 + ρgH
So they feel exactly the same pressure.
22. Borgnakke and Sonntag
2.14
A manometer with water shows a ∆P of Po/10; what is the column height difference?
Solution:
∆P = Po/10 = ρHg
H = Po/(10 ρ g) =
101.3 × 1000 Pa
10 × 997 kg/m3 × 9.80665 m/s2
= 1.036 m
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23. Borgnakke and Sonntag
2.15
A water skier does not sink too far down in the water if the speed is high enough.
What makes that situation different from our static pressure calculations?
The water pressure right under the ski is not a static pressure but a static plus
dynamic pressure that pushes the water away from the ski. The faster you go, the
smaller amount of water is displaced but at a higher velocity.
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24. Borgnakke and Sonntag
2.16
What is the smallest temperature in degrees Celsuis you can have? Kelvin?
Solution:
The lowest temperature is absolute zero which is
at zero degrees Kelvin at which point the
temperature in Celsius is negative
TK = 0 K = −273.15 oC
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25. Borgnakke and Sonntag
2.17
Convert the formula for water density in In-text Concept Question “e” to be for T in
degrees Kelvin.
Solution:
ρ = 1008 – TC/2 [kg/m3]
We need to express degrees Celsius in degrees Kelvin
TC = TK – 273.15
and substitute into formula
ρ = 1008 – TC/2 = 1008 – (TK – 273.15)/2 = 1144.6 – TK/2
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26. Borgnakke and Sonntag
2.18
A thermometer that indicates the temperature with a liquid column has a bulb with a
larger volume of liquid, why is that?
The expansion of the liquid volume with temperature is rather small so by having
a larger volume expand with all the volume increase showing in the very small
diameter column of fluid greatly increases the signal that can be read.
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27. Borgnakke and Sonntag
Properties and units
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28. Borgnakke and Sonntag
2.19
An apple “weighs” 60 g and has a volume of 75 cm3 in a refrigerator at 8oC.
What is the apple density? List three intensive and two extensive properties of the
apple.
Solution:
ρ =
m
V =
0.06
0.000 075
kg
m3 = 800
kg
m3
Intensive
ρ = 800
kg
m3 ; v =
1
ρ
= 0.001 25
m3
kg
T = 8°C; P = 101 kPa
Extensive
m = 60 g = 0.06 kg
V = 75 cm3 = 0.075 L = 0.000 075 m3
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29. Borgnakke and Sonntag
2.20
A steel cylinder of mass 2 kg contains 4 L of liquid water at 25oC at 200 kPa.
Find the total mass and volume of the system. List two extensive and three
intensive properties of the water
Solution:
Density of steel in Table A.3: ρ = 7820 kg/m3
Volume of steel: V = m/ρ =
2 kg
7820 kg/m3 = 0.000 256 m3
Density of water in Table A.4: ρ = 997 kg/m3
Mass of water: m = ρV = 997 kg/m3 ×0.004 m3 = 3.988 kg
Total mass: m = msteel + mwater = 2 + 3.988 = 5.988 kg
Total volume: V = Vsteel + Vwater = 0.000 256 + 0.004
= 0.004 256 m3 = 4.26 L
Extensive properties: m, V
Intensive properties: ρ (or v = 1/ρ), T, P
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30. Borgnakke and Sonntag
2.21
A storage tank of stainless steel contains 7 kg of oxygen gas and 5 kg of nitrogen
gas. How many kmoles are in the tank?
Table A.2: MO2 = 31.999 ; MN2 = 28.013
nO2 = mO2 / MO2 =
7
31.999 = 0.21876 kmol
nO2 = mN2 / MN2 =
5
28.013 = 0.17848 kmol
ntot = nO2 + nN2 = 0.21876 + 0.17848 = 0.3972 kmol
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31. Borgnakke and Sonntag
2.22
One kilopond (1 kp) is the weight of 1 kg in the standard gravitational field. How
many Newtons (N) is that?
F = ma = mg
1 kp = 1 kg × 9.807 m/s2 = 9.807 N
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32. Borgnakke and Sonntag
Force and Energy
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33. Borgnakke and Sonntag
2.23
The “standard” acceleration (at sea level and 45° latitude) due to gravity is
9.80665 m/s2. What is the force needed to hold a mass of 2 kg at rest in this
gravitational field? How much mass can a force of 1 N support?
Solution:
ma = 0 = ∑ F = F - mg
F = mg = 2 kg × 9.80665 m/s2 = 19.613 N
F = mg =>
m =
F
g =
1 N
9.80665 m/s2 = 0.102 kg
m
F
g
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34. Borgnakke and Sonntag
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2.24
A steel piston of 2.5 kg is in the standard gravitational field where a force of 25 N
is applied vertically up. Find the acceleration of the piston.
Solution:
Fup = ma = F – mg
a =
F – mg
m =
F
m – g
=
25 N
2.5 kg – 9.807 m/s2
= 0.193 ms-2
g
F
35. Borgnakke and Sonntag
2.25
When you move up from the surface of the earth the gravitation is reduced as g =
9.807 − 3.32 × 10-6 z, with z as the elevation in meters. How many percent is the
weight of an airplane reduced when it cruises at 11 000 m?
Solution:
go= 9.807 ms-2
gH = 9.807 – 3.32 × 10-6 × 11 000 = 9.7705 ms-2
Wo = m go ; WH = m gH
WH/Wo = gH/go =
9.7705
9.807 = 0.9963
Reduction = 1 – 0.9963 = 0.0037 or 0.37%
i.e. we can neglect that for most applications.
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36. Borgnakke and Sonntag
2.26
A model car rolls down an incline with a slope so the gravitational “pull” in the
direction of motion is one third of the standard gravitational force (see Problem
2.23). If the car has a mass of 0.06 kg find the acceleration.
Solution:
ma = ∑ F = mg / 3
a = mg / 3m = g/3
= 9.80665 (m/s2) / 3
= 3.27 m/s2
g
This acceleration does not depend on the mass of the model car.
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37. Borgnakke and Sonntag
2.27
A van drives at 60 km/h and is brought to a full stop with constant deceleration in
5 seconds. If the total car and driver mass is 2075 kg find the necessary force.
Solution:
Acceleration is the time rate of change of velocity.
a =
dV
dt =
60 × 1000
3600 × 5
= 3.333 m/s2
ma = ∑ F ;
Fnet = ma = 2075 kg × 3.333 m/s2 = 6916 N
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38. Borgnakke and Sonntag
2.28
An escalator brings four people of total 300 kg, 25 m up in a building. Explain what
happens with respect to energy transfer and stored energy.
Solution:
The four people (300
kg) have their potential
energy raised, which is
how the energy is
stored. The energy is
supplied as electrical
power to the motor that
pulls the escalator with a
cable.
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39. Borgnakke and Sonntag
2.29
A car of mass 1775 kg travels with a velocity of 100 km/h. Find the kinetic
energy. How high should it be lifted in the standard gravitational field to have a
potential energy that equals the kinetic energy?
Solution:
Standard kinetic energy of the mass is
KIN = ½ m V2 = ½ × 1775 kg ×
100 × 1000
3600
2
m2/s2
= ½ × 1775 × 27.778 Nm = 684 800 J
= 684.8 kJ
Standard potential energy is
POT = mgh
h = ½ m V2 / mg =
684 800 Nm
1775 kg × 9.807 m/s2 = 39.3 m
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40. Borgnakke and Sonntag
2.30
A 1500-kg car moving at 20 km/h is accelerated at a constant rate of 4 m/s2 up to
a speed of 75 km/h. What are the force and total time required?
Solution:
a =
dV
dt =
∆V
∆t
=> ∆t =
∆V
a =
(75 − 20) 1000
3600 × 5
= 3.82 sec
F = ma = 1500 kg × 4 m/s2 = 6000 N
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41. Borgnakke and Sonntag
2.31
On the moon the gravitational acceleration is approximately one-sixth that on the
surface of the earth. A 5-kg mass is “weighed” with a beam balance on the
surface on the moon. What is the expected reading? If this mass is weighed with a
spring scale that reads correctly for standard gravity on earth (see Problem 2.23),
what is the reading?
Solution:
Moon gravitation is: g = gearth/6
mm
m
Beam Balance Reading is 5 kg Spring Balance Reading is in kg units
This is mass comparison Force comparison length ∝ F ∝ g
Reading will be
5
6
kg
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42. Borgnakke and Sonntag
2.32
The escalator cage in Problem 2.28 has a mass of 500 kg in addition to the people.
How much force should the cable pull up with to have an acceleration of 1 m/s2 in
the upwards direction?
Solution:
The total mass moves upwards with an
acceleration plus the gravitations acts
with a force pointing down.
ma = ∑ F = F – mg
F = ma + mg = m(a + g)
= (500 + 300) kg × (1 + 9.81) m/s2
= 8648 N
F
g
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43. Borgnakke and Sonntag
2.33
A bucket of concrete of total mass 200 kg is raised by a crane with an acceleration
of 2 m/s2 relative to the ground at a location where the local gravitational
acceleration is 9.5 m/s2. Find the required force.
Solution:
F = ma = Fup − mg
Fup = ma + mg = 200 ( 2 + 9.5 ) = 2300 N g
Fup
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44. Borgnakke and Sonntag
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2.34
A bottle of 12 kg steel has 1.75 kmole of liquid propane. It accelerates horizontal
with 3 m/s2, what is the needed force?
Solution:
The molecular weight for propane is M = 44.094 from Table A.2. The force
must accelerate both the container mass and the propane mass.
m = msteel + mpropane = 12 + (1.75 × 44.094) = 90.645 kg
ma = ∑ F ⇒
F = ma = 90.645 kg × 3 m/s2 = 271.9 N
45. Borgnakke and Sonntag
Specific Volume
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46. Borgnakke and Sonntag
2.35
A 15-kg steel gas tank holds 300 L of liquid gasoline, having a density of 800
kg/m3. If the system is decelerated with 2g what is the needed force?
Solution:
m = mtank + mgasoline
= 15 kg + 0.3 m3 × 800 kg/m3
= 255 kg
F = ma = 255 kg × 2 × 9.81 m/s2
= 5003 N
cb
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47. Borgnakke and Sonntag
2.36
A power plant that separates carbon-dioxide from the exhaust gases compresses it
to a density of 110 kg/m3 and stores it in an un-minable coal seam with a porous
volume of 100 000 m3. Find the mass they can store.
Solution:
m = ρ V = 110 kg/m3 × 100 000 m3 = 11 × 10 6 kg
Just to put this in perspective a power plant that generates 2000 MW by burning
coal would make about 20 million tons of carbon-dioxide a year. That is 2000
times the above mass so it is nearly impossible to store all the carbon-dioxide
being produced.
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48. Borgnakke and Sonntag
2.37
A 1 m3 container is filled with 400 kg of granite stone, 200 kg dry sand and 0.2
m3 of liquid 25°C water. Use properties from tables A.3 and A.4. Find the
average specific volume and density of the masses when you exclude air mass and
volume.
Solution:
Specific volume and density are ratios of total mass and total volume.
mliq = Vliq/vliq = Vliq ρliq = 0.2 m3 × 997 kg/m3 = 199.4 kg
mTOT = mstone + msand + mliq = 400 + 200 + 199.4 = 799.4 kg
Vstone = mv = m/ρ = 400 kg/ 2750 kg/m3 = 0.1455 m3
Vsand = mv = m/ρ = 200/ 1500 = 0.1333 m3
VTOT = Vstone + Vsand + Vliq
= 0.1455 + 0.1333 + 0.2 = 0.4788 m3
v = VTOT / mTOT = 0.4788/799.4 = 0.000599 m3/kg
ρ = 1/v = mTOT/VTOT = 799.4/0.4788 = 1669.6 kg/m3
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49. Borgnakke and Sonntag
2.38
One kilogram of diatomic oxygen (O2 molecular weight 32) is contained in a 500-
L tank. Find the specific volume on both a mass and mole basis (v and v ).
Solution:
From the definition of the specific volume
v =
V
m =
0.5
1 = 0.5 m3/kg
v =
V
n =
V
m/M = M v = 32 × 0.5 = 16 m3/kmol
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50. Borgnakke and Sonntag
2.39
A tank has two rooms separated by a membrane. Room A has 1 kg air and volume
0.5 m3, room B has 0.75 m3 air with density 0.8 kg/m3. The membrane is broken
and the air comes to a uniform state. Find the final density of the air.
Solution:
Density is mass per unit volume
m = mA + mB = mA + ρBVB = 1 + 0.8 × 0.75 = 1.6 kg
V = VA + VB = 0.5 + 0.75 = 1.25 m3
ρ =
m
V =
1.6
1.25 = 1.28 kg/m3
A B
cb
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51. Borgnakke and Sonntag
2.40
A 5 m3 container is filled with 900 kg of granite (density 2400 kg/m3 ) and the
rest of the volume is air with density 1.15 kg/m3. Find the mass of air and the
overall (average) specific volume.
Solution:
mair = ρ V = ρair ( Vtot −
mgranite
ρ
)
= 1.15 [ 5 -
900
2400 ] = 1.15 × 4.625 = 5.32 kg
v =
V
m =
5
900 + 5.32 = 0.005 52 m3/kg
Comment: Because the air and the granite are not mixed or evenly distributed in
the container the overall specific volume or density does not have much meaning.
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52. Borgnakke and Sonntag
Pressure
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53. Borgnakke and Sonntag
2.41
The hydraulic lift in an auto-repair shop has a cylinder diameter of 0.2 m. To what
pressure should the hydraulic fluid be pumped to lift 40 kg of piston/arms and 700
kg of a car?
Solution:
Force acting on the mass by the gravitational field
F↓ = ma = mg = 740 × 9.80665 = 7256.9 N = 7.257 kN
Force balance: F↑ = ( P - P0 ) A = F↓ => P = P0 + F↓ / A
A = π D2 (1 / 4) = 0.031416 m2
P = 101 kPa +
7.257 kN
0.031416 m2 = 332 kPa
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54. Borgnakke and Sonntag
2.42
A valve in a cylinder has a cross sectional area of 11 cm2 with a pressure of 735
kPa inside the cylinder and 99 kPa outside. How large a force is needed to open
the valve?
Fnet = PinA – PoutA
= (735 – 99) kPa × 11 cm2
= 6996 kPa cm2
= 6996 ×
kN
m2 × 10-4 m2
= 700 N
cb
Pcyl
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55. Borgnakke and Sonntag
2.43
A hydraulic lift has a maximum fluid pressure of 500 kPa. What should the
piston-cylinder diameter be so it can lift a mass of 850 kg?
Solution:
With the piston at rest the static force balance is
F↑ = P A = F↓ = mg
A = π r2 = π D2/4
PA = P π D2/4 = mg ⇒ D2 =
4mg
P π
D = 2
mg
Pπ
= 2
850 kg × 9.807 m/s2
500 kPa × π × 1000 (Pa/kPa)
= 0.146 m
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56. Borgnakke and Sonntag
2.44
A laboratory room keeps a vacuum of 0.1 kPa. What net force does that put on the
door of size 2 m by 1 m?
Solution:
The net force on the door is the difference between the forces on the two sides as
the pressure times the area
F = Poutside A – Pinside A = ∆P A = 0.1 kPa × 2 m × 1 m = 200 N
Remember that kPa is kN/m2.
Pabs = Po - ∆P
∆P = 0.1 kPa
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57. Borgnakke and Sonntag
2.45
A vertical hydraulic cylinder has a 125-mm diameter piston with hydraulic fluid
inside the cylinder and an ambient pressure of 1 bar. Assuming standard gravity,
find the piston mass that will create a pressure inside of 1500 kPa.
Solution:
Force balance:
F↑ = PA = F↓ = P0A + mpg;
P0 = 1 bar = 100 kPa
A = (π/4) D2 = (π/4) × 0.1252 = 0.01227 m2
cb
g
Po
mp = (P − P0)
A
g = ( 1500 − 100 ) × 1000 ×
0.01227
9.80665 = 1752 kg
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58. Borgnakke and Sonntag
2.46
A piston/cylinder with cross sectional area of 0.01 m2 has a piston mass of 100 kg
resting on the stops, as shown in Fig. P2.46. With an outside atmospheric pressure
of 100 kPa, what should the water pressure be to lift the piston?
Solution:
The force acting down on the piston comes from gravitation and the
outside atmospheric pressure acting over the top surface.
Force balance: F↑ = F↓ = PA = mpg + P0A
Now solve for P (divide by 1000 to convert to kPa for 2nd
term)
P = P0 +
mpg
A = 100 kPa +
100 × 9.80665
0.01 × 1000
kPa
= 100 kPa + 98.07 kPa = 198 kPa
Water
cb
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59. Borgnakke and Sonntag
2.47
A cannon-ball of 5 kg acts as a piston in a cylinder of 0.15 m diameter. As the
gun-powder is burned a pressure of 7 MPa is created in the gas behind the ball.
What is the acceleration of the ball if the cylinder (cannon) is pointing
horizontally?
Solution:
The cannon ball has 101 kPa on the side facing the atmosphere.
ma = F = P1 × A − P0 × A = (P1 − P0 ) × A
= (7000 – 101) kPa × π ( 0.152
/4 ) m2
= 121.9 kN
a =
F
m =
121.9 kN
5 kg = 24 380 m/s2
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60. Borgnakke and Sonntag
2.48
Repeat the previous problem for a cylinder (cannon) pointing 40 degrees up
relative to the horizontal direction.
Solution:
ma = F = ( P1 - P0 ) A - mg sin 400
ma = (7000 - 101 ) kPa × π × ( 0.152 / 4 ) m2
- 5 × 9.807 × 0.6428 N
= 121.9 kN - 31.52 N = 121.87 kN
a =
F
m =
121.87 kN
5 kg = 24 374 m/s2
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61. Borgnakke and Sonntag
2.49
A large exhaust fan in a laboratory room keeps the pressure inside at 10 cm water
relative vacuum to the hallway. What is the net force on the door measuring 1.9 m
by 1.1 m?
Solution:
The net force on the door is the difference between the forces on the two sides as
the pressure times the area
F = Poutside A – Pinside A = ∆P × A
= 10 cm H2O × 1.9 m × 1.1 m
= 0.10 × 9.80638 kPa × 2.09 m2
= 2049 N
Table A.1: 1 m H2O is 9.80638 kPa and kPa is kN/m2.
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62. Borgnakke and Sonntag
2.50
A tornado rips off a 100 m2 roof with a mass of 1000 kg. What is the minimum
vacuum pressure needed to do that if we neglect the anchoring forces?
Solution:
The net force on the roof is the difference between the forces on the two sides as
the pressure times the area
F = Pinside A – PoutsideA = ∆P A
That force must overcome the gravitation mg, so the balance is
∆P A = mg
∆P = mg/A = (1000 kg × 9.807 m/s2 )/100 m2 = 98 Pa = 0.098 kPa
Remember that kPa is kN/m2.
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63. Borgnakke and Sonntag
2.51
A 2.5 m tall steel cylinder has a cross sectional area of 1.5 m2. At the bottom with
a height of 0.5 m is liquid water on top of which is a 1 m high layer of gasoline.
This is shown in Fig. P2.51. The gasoline surface is exposed to atmospheric air at
101 kPa. What is the highest pressure in the water?
Solution:
The pressure in the fluid goes up with the depth as
P = Ptop + ∆P = Ptop + ρgh
and since we have two fluid layers we get
P = Ptop + [(ρh)gasoline + (ρh)water] g
Air
Water
1 m
0.5 m
Gasoline
The densities from Table A.4 are:
ρgasoline = 750 kg/m3; ρwater = 997 kg/m3
P = 101 + [750 × 1 + 997 × 0.5]
9.807
1000 = 113.2 kPa
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64. Borgnakke and Sonntag
2.52
What is the pressure at the bottom of a 5 m tall column of fluid with atmospheric
pressure 101 kPa on the top surface if the fluid is
a) water at 20°C b) glycerine 25°C or c) gasoline 25°C
Solution:
Table A.4: ρH2O = 997 kg/m3; ρGlyc = 1260 kg/m3; ρgasoline = 750 kg/m3
∆P = ρgh P = Ptop + ∆P
a) ∆P = ρgh = 997× 9.807× 5 = 48 888 Pa
P = 101 + 48.99 = 149.9 kPa
b) ∆P = ρgh = 1260× 9.807× 5 = 61 784 Pa
P = 101 + 61.8 = 162.8 kPa
c) ∆P = ρgh = 750× 9.807× 5 = 36 776 Pa
P = 101 + 36.8 = 137.8 kPa
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65. Borgnakke and Sonntag
2.53
At the beach, atmospheric pressure is 1025 mbar. You dive 15 m down in the
ocean and you later climb a hill up to 250 m elevation. Assume the density of
water is about 1000 kg/m3 and the density of air is 1.18 kg/m3. What pressure do
you feel at each place?
Solution:
∆P = ρgh,
Units from A.1: 1 mbar = 100 Pa (1 bar = 100 kPa).
Pocean= P0 + ∆P = 1025 × 100 + 1000 × 9.81 × 15
= 2.4965 × 105 Pa = 250 kPa
Phill = P0 - ∆P = 1025 × 100 - 1.18 × 9.81 × 250
= 0.99606 × 105 Pa = 99.61 kPa
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66. Borgnakke and Sonntag
2.54
A steel tank of cross sectional area 3 m2 and 16 m tall weighs 10 000 kg and it is
open at the top. We want to float it in the ocean so it sticks 10 m straight down by
pouring concrete into the bottom of it. How much concrete should I put in?
Solution:
The force up on the tank is from the water
pressure at the bottom times its area. The
force down is the gravitation times mass and
the atmospheric pressure.
F↑ = PA = (ρoceangh + P0)A
F↓ = (mtank + mconcrete)g + P0A
The force balance becomes
Air
Ocean
Concrete
10 m
F↑ = F↓ = (ρoceangh + P0)A = (mtank + mconcrete)g + P0A
Solve for the mass of concrete
mconcrete = (ρoceanhA - mtank) = 997 × 10 × 3 – 10 000 = 19 910 kg
Notice: The first term is the mass of the displaced ocean water. The net force
up is the weight (mg) of this mass called bouyancy, P0 cancel.
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67. Borgnakke and Sonntag
2.55
A piston, mp= 5 kg, is fitted in a cylinder, A = 15 cm2, that contains a gas. The
setup is in a centrifuge that creates an acceleration of 25 m/s2 in the direction of
piston motion towards the gas. Assuming standard atmospheric pressure outside
the cylinder, find the gas pressure.
Solution:
Force balance: F↑ = F↓ = P0A + mpg = PA
P = P0 +
mpg
A
= 101.325 +
5 × 25
1000 × 0.0015
kPa kg m/s2
Pa m2
= 184.7 kPa
gasg
Po
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68. Borgnakke and Sonntag
2.56
Liquid water with density ρ is filled on top of a thin piston in a cylinder with
cross-sectional area A and total height H, as shown in Fig. P2.56. Air is let in
under the piston so it pushes up, spilling the water over the edge. Derive the
formula for the air pressure as a function of piston elevation from the bottom, h.
Solution:
Force balance
H
h
P0 Piston: F↑ = F↓
PA = P0A + mH2Og
P = P0 + mH2Og/A
P = P0 + (H − h)ρg h, V air
P
P0
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69. Borgnakke and Sonntag
Manometers and Barometers
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70. Borgnakke and Sonntag
2.57
You dive 5 m down in the ocean. What is the absolute pressure there?
Solution:
The pressure difference for a column is from Eq.2.2 and the density of water is
from Table A.4.
∆P = ρgH
= 997 kg/m3 × 9.81 m/s2 × 5 m
= 48 903 Pa = 48.903 kPa
Pocean= P0 + ∆P
= 101.325 + 48.903
= 150 kPa
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71. Borgnakke and Sonntag
2.58
A barometer to measure absolute pressure shows a mercury column height of 725
mm. The temperature is such that the density of the mercury is 13 550 kg/m3.
Find the ambient pressure.
Solution:
Hg : L = 725 mm = 0.725 m; ρ = 13 550 kg/m3
The external pressure P balances the column of height L so from Fig. 2.10
P = ρ L g = 13 550 kg/m3 × 9.80665 m/s2 × 0.725 m × 10-3 kPa/Pa
= 96.34 kPa
This is a more common type that
does not involve mercury as the
wall mounted unit to the left.
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72. Borgnakke and Sonntag
2.59
The density of atmospheric air is about 1.15 kg/m3, which we assume is constant.
How large an absolute pressure will a pilot see when flying 2000 m above ground
level where the pressure is 101 kPa.
Solution:
Assume g and ρ are constant then the pressure difference to carry a
column of height 1500 m is from Fig.2.10
∆P = ρgh = 1.15 kg/m3 × 9.807 ms-2 × 2000 m
= 22 556 Pa = 22.6 kPa
The pressure on top of the column of air is then
P = P0 – ∆P = 101 – 22.6 = 78.4 kPa
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73. Borgnakke and Sonntag
2.60
A differential pressure gauge mounted on a vessel shows 1.25 MPa and a local
barometer gives atmospheric pressure as 0.96 bar. Find the absolute pressure
inside the vessel.
Solution:
Convert all pressures to units of kPa.
Pgauge = 1.25 MPa = 1250 kPa;
P0 = 0.96 bar = 96 kPa
P = Pgauge + P0 = 1250 + 96 = 1346 kPa
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74. Borgnakke and Sonntag
2.61
A manometer shows a pressure difference of 1 m of liquid mercury. Find ∆P in kPa.
Solution:
Hg : L = 1 m; ρ = 13 580 kg/m3 from Table A.4 (or read Fig 2.8)
The pressure difference ∆P balances the column of height L so from Eq.2.2
∆P = ρ g L = 13 580 kg/m3 × 9.80665 m/s2 × 1.0 m × 10-3 kPa/Pa
= 133.2 kPa
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75. Borgnakke and Sonntag
2.62
Blue manometer fluid of density 925 kg/m3 shows a column height difference of
3 cm vacuum with one end attached to a pipe and the other open to P0 = 101 kPa.
What is the absolute pressure in the pipe?
Solution:
Since the manometer shows a vacuum we have
PPIPE = P0 - ∆P
∆P = ρgh = 925 × 9.807 × 0.03
= 272.1 Pa = 0.272 kPa
PPIPE = 101 – 0.272 = 100.73 kPa
cb
Po
Pipe
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76. Borgnakke and Sonntag
2.63
What pressure difference does a 10 m column of atmospheric air show?
Solution:
The pressure difference for a column is from Eq.2.2
∆P = ρgH
So we need density of air from Fig.2.7, ρ = 1.2 kg/m3
∆P = 1.2 kg/m3 × 9.81 ms-2 × 10 m = 117.7 Pa = 0.12 kPa
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77. Borgnakke and Sonntag
2.64
The absolute pressure in a tank is 85 kPa and the local ambient absolute pressure
is 97 kPa. If a U-tube with mercury, density 13550 kg/m3, is attached to the tank
to measure the vacuum, what column height difference would it show?
Solution:
∆P = P0 - Ptank = ρg H
H = ( P0 - Ptank ) / ρg = [(97 - 85 ) × 1000 ] / (13550 × 9.80665)
= 0.090 m = 90 mm
H
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78. Borgnakke and Sonntag
2.65
The pressure gauge on an air tank shows 75 kPa when the diver is 10 m down in
the ocean. At what depth will the gauge pressure be zero? What does that mean?
Ocean H20 pressure at 10 m depth is
Pwater = P0 + ρLg = 101.3 +
997 × 10 × 9.80665
1000 = 199 kPa
Air Pressure (absolute) in tank
Ptank = 199 + 75 = 274 kPa
Tank Pressure (gauge) reads zero at H20 local pressure
274 = 101.3 +
997 × 9.80665
1000 L
L = 17.66 m
At this depth you will have to suck the
air in, it can no longer push itself
through a valve.
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79. Borgnakke and Sonntag
2.66
An exploration submarine should be able to go 4000 m down in the ocean. If the
ocean density is 1020 kg/m3 what is the maximum pressure on the submarine
hull?
Solution:
Assume we have atmospheric pressure inside the submarine then the pressure
difference to the outside water is
∆P = ρLg = (1020 kg/m3 × 4000 m × 9.807 m/s2) / 1000
= 40 012 kPa ≈ 40 MPa
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80. Borgnakke and Sonntag
2.67
A submarine maintains 101 kPa inside it and it dives 240 m down in the ocean
having an average density of 1030 kg/m3. What is the pressure difference
between the inside and the outside of the submarine hull?
Solution:
Assume the atmosphere over the ocean is at 101 kPa, then ∆P is from the 240 m
column water.
∆P = ρLg
= (1030 kg/m3 × 240 m × 9.807 m/s2) / 1000 = 2424 kPa
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81. Borgnakke and Sonntag
2.68
Assume we use a pressure gauge to measure the air pressure at street level and at
the roof of a tall building. If the pressure difference can be determined with an
accuracy of 1 mbar (0.001 bar) what uncertainty in the height estimate does that
corresponds to?
Solution:
ρair = 1.169 kg/m3 from Table A.5
∆P = 0.001 bar = 100 Pa
L =
∆P
ρg
=
100
1.169 × 9.807
= 8.72 m
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82. Borgnakke and Sonntag
2.69
A barometer measures 760 mmHg at street level and 735 mmHg on top of a
building. How tall is the building if we assume air density of 1.15 kg/m3?
Solution:
∆P = ρgH
H = ∆P/ρg =
760 – 735
1.15 × 9.807
mmHg
kg/m2s2
133.32 Pa
mmHg = 295 m
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83. Borgnakke and Sonntag
2.70
An absolute pressure gauge attached to a steel cylinder shows 135 kPa. We want
to attach a manometer using liquid water a day that Patm = 101 kPa. How high a
fluid level difference must we plan for?
Solution:
Since the manometer shows a pressure difference we have
∆P = PCYL - Patm = ρ L g
L = ∆P / ρg =
(135 – 101) kPa
997 kg m-3 × 10 × 9.807 m/s2
1000 Pa
kPa
= 3.467 m
H
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84. Borgnakke and Sonntag
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2.71
A U-tube manometer filled with water, density 1000 kg/m3, shows a height
difference of 25 cm. What is the gauge pressure? If the right branch is tilted to
make an angle of 30° with the horizontal, as shown in Fig. P2.71, what should the
length of the column in the tilted tube be relative to the U-tube?
Solution:
Same height in the two sides in the direction of g.
∆P = F/A = mg/A = Vρg/A = hρg
= 0.25 m × 1000 kg/m3 × 9.807m/s2
= 2452.5 Pa
= 2.45 kPa
h = H × sin 30°
⇒ H = h/sin 30° = 2h = 50 cm
30o
H
h
85. Borgnakke and Sonntag
2.72
A pipe flowing light oil has a manometer attached as shown in Fig. P2.72. What
is the absolute pressure in the pipe flow?
Solution:
Table A.3: ρoil = 910 kg/m3; ρwater = 997 kg/m3
PBOT = P0 + ρwater g Htot = P0 + 997 kg/m3 × 9.807 m/s2 × 0.8 m
= Po + 7822 Pa
PPIPE = PBOT – ρwater g H1 – ρoil g H2
= PBOT – 997 × 9.807 × 0.1 – 910 × 9.807 × 0.2
= PBOT – 977.7 Pa – 1784.9 Pa
PPIPE = Po + (7822 – 977.7 – 1784.9) Pa
= Po + 5059.4 Pa = 101.325 + 5.06 = 106.4 kPa
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86. Borgnakke and Sonntag
2.73
The difference in height between the columns of a manometer is 200 mm with a
fluid of density 900 kg/m3. What is the pressure difference? What is the height
difference if the same pressure difference is measured using mercury, density
13600 kg/ m3, as manometer fluid?
Solution:
∆P = ρ1gh1 = 900 kg/m3 × 9.807 m/s2 × 0.2 m = 1765.26 Pa = 1.77 kPa
hHg = ∆P/ (ρhg g) = (ρ1 gh1) / (ρhg g) =
900
13600 × 0.2
= 0.0132 m = 13.2 mm
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87. Borgnakke and Sonntag
2.74
Two cylinders are filled with liquid water, ρ = 1000 kg/m3, and connected by a
line with a closed valve. A has 100 kg and B has 500 kg of water, their cross-
sectional areas are AA = 0.1 m2 and AB = 0.25 m2 and the height h is 1 m. Find
the pressure on each side of the valve. The valve is opened and water flows to an
equilibrium. Find the final pressure at the valve location.
Solution:
VA = vH2OmA = mA/ρ = 0.1 = AAhA => hA = 1 m
VB = vH2OmB = mB/ρ = 0.5 = ABhB => hB = 2 m
PVB = P0 + ρg(hB+H) = 101325 + 1000 × 9.81 × 3 = 130 755 Pa
PVA = P0 + ρghA = 101325 + 1000 × 9.81 × 1 = 111 135 Pa
Equilibrium: same height over valve in both
Vtot = VA + VB = h2AA + (h2 - H)AB ⇒ h2 =
hAAA + (hB+H)AB
AA + AB
= 2.43 m
PV2 = P0 + ρgh2 = 101.325 + (1000 × 9.81 × 2.43)/1000 = 125.2 kPa
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88. Borgnakke and Sonntag
2.75
Two piston/cylinder arrangements, A and B, have their gas chambers connected
by a pipe. Cross-sectional areas are AA = 75 cm2 and AB = 25 cm2 with the piston
mass in A being mA = 25 kg. Outside pressure is 100 kPa and standard
gravitation. Find the mass mB so that none of the pistons have to rest on the
bottom.
Solution:
P
Po
o
cb
Force balance for both pistons: F↑ = F↓
A: mPAg + P0AA = PAA
B: mPBg + P0AB = PAB
Same P in A and B gives no flow between them.
mPAg
AA
+ P0 =
mPBg
AB
+ P0
=> mPB = mPA AA/ AB = 25 × 25/75 = 8.33 kg
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89. Borgnakke and Sonntag
2.76
Two hydraulic piston/cylinders are of same size and setup as in Problem 2.75, but
with negligible piston masses. A single point force of 250 N presses down on
piston A. Find the needed extra force on piston B so that none of the pistons have
to move.
Solution:
AA = 75 cm2 ;
AB = 25 cm2
No motion in connecting pipe: PA = PB
Forces on pistons balance
Po
Po
cb
A B
FB
FA
PA = P0 + FA / AA = PB = P0 + FB / AB
FB = FA ×
AB
AA
= 250 ×
25
75 = 83.33 N
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90. Borgnakke and Sonntag
2.77
A piece of experimental apparatus is located where g = 9.5 m/s2 and the
temperature is 5°C. An air flow inside the apparatus is determined by measuring
the pressure drop across an orifice with a mercury manometer (see Problem 2.79
for density) showing a height difference of 200 mm. What is the pressure drop in
kPa?
Solution:
∆P = ρgh ; ρHg = 13600 kg/m3
∆P = 13 600 kg/m3 × 9.5 m/s2 × 0.2 m = 25840 Pa = 25.84 kPa
g
Air
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91. Borgnakke and Sonntag
Temperature
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92. Borgnakke and Sonntag
2.78
What is a temperature of –5oC in degrees Kelvin?
Solution:
The offset from Celsius to Kelvin is 273.15 K,
so we get
TK = TC + 273.15 = -5 + 273.15
= 268.15 K
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93. Borgnakke and Sonntag
2.79
The density of mercury changes approximately linearly with temperature as
ρHg = 13595 − 2.5 T kg/ m3 T in Celsius
so the same pressure difference will result in a manometer reading that is
influenced by temperature. If a pressure difference of 100 kPa is measured in the
summer at 35°C and in the winter at −15°C, what is the difference in column
height between the two measurements?
Solution:
The manometer reading h relates to the pressure difference as
∆P = ρ L g ⇒ L =
∆P
ρg
The manometer fluid density from the given formula gives
ρsu = 13595 − 2.5 × 35 = 13507.5 kg/m3
ρw = 13595 − 2.5 × (−15) = 13632.5 kg/m3
The two different heights that we will measure become
Lsu =
100 × 103
13507.5 × 9.807
kPa (Pa/kPa)
(kg/m3) m/s2 = 0.7549 m
Lw =
100 × 103
13632.5 × 9.807
kPa (Pa/kPa)
(kg/m3) m/s2 = 0.7480 m
∆L = Lsu - Lw = 0.0069 m = 6.9 mm
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94. Borgnakke and Sonntag
2.80
A mercury thermometer measures temperature by measuring the volume
expansion of a fixed mass of liquid Hg due to a change in the density, see
problem 2.35. Find the relative change (%) in volume for a change in
temperature from 10°C to 20°C.
Solution:
From 10°C to 20°C
At 10°C : ρHg = 13595 – 2.5 × 10 = 13570 kg/m3
At 20°C : ρHg = 13595 – 2.5 × 20 = 13545 kg/m3
The volume from the mass and density is: V = m/ρ
Relative Change =
V20– V10
V10
=
(m/ρ20) - (m/ρ10)
m/ρ10
=
ρ10
ρ20
– 1 =
13570
13545 – 1 = 0.0018 (0.18%)
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95. Borgnakke and Sonntag
2.81
Density of liquid water is ρ = 1008 – T/2 [kg/m3] with T in oC. If the temperature
increases 10oC how much deeper does a 1 m layer of water become?
Solution:
The density change for a change in temperature of 10oC becomes
∆ρ = – ∆T/2 = –5 kg/m3
from an ambient density of
ρ = 1008 – T/2 = 1008 – 25/2 = 995.5 kg/m3
Assume the area is the same and the mass is the same m = ρV = ρAH, then we
have
∆m = 0 = V∆ρ + ρ∆V ⇒ ∆V = - V∆ρ/ρ
and the change in the height is
∆H =
∆V
A =
H∆V
V =
-H∆ρ
ρ
=
-1 × (-5)
995.5 = 0.005 m
barely measurable.
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96. Borgnakke and Sonntag
2.82
Using the freezing and boiling point temperatures for water in both Celsius and
Fahrenheit scales, develop a conversion formula between the scales. Find the
conversion formula between Kelvin and Rankine temperature scales.
Solution:
TFreezing = 0 oC = 32 F; TBoiling = 100 oC = 212 F
∆T = 100 oC = 180 F ⇒ ToC = (TF - 32)/1.8 or TF = 1.8 ToC + 32
For the absolute K & R scales both are zero at absolute zero.
TR = 1.8 × TK
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97. Borgnakke and Sonntag
2.83
The atmosphere becomes colder at higher elevation. As an average the standard
atmospheric absolute temperature can be expressed as Tatm = 288 - 6.5 × 10−3 z,
where z is the elevation in meters. How cold is it outside an airplane cruising at
12 000 m expressed in Kelvin and in Celsius?
Solution:
For an elevation of z = 12 000 m we get
Tatm = 288 - 6.5 × 10−3 z = 210 K
To express that in degrees Celsius we get
TC = T – 273.15 = −63.15oC
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98. Borgnakke and Sonntag
2.84
Repeat problem 2.77 if the flow inside the apparatus is liquid water, ρ ≅ 1000
kg/m3, instead of air. Find the pressure difference between the two holes flush
with the bottom of the channel. You cannot neglect the two unequal water
columns.
Solution: Balance forces in the manometer:
P P1. 2
·
h
h1
2
H
(H - h2) - (H - h1) = ∆hHg = h1 - h2
P1A + ρH2Oh1gA + ρHg(H - h1)gA
= P2A + ρH2Oh2gA + ρHg(H - h2)gA
⇒ P1 - P2 = ρH2O(h2 - h1)g + ρHg(h1 - h2)g
P1 - P2 = ρHg∆hHgg - ρH2O∆hHgg = 13 600 × 0.2 × 9.5 - 1000 × 0.2 × 9.5
= 25 840 - 1900 = 23940 Pa = 23.94 kPa
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99. Borgnakke and Sonntag
2.85
A dam retains a lake 6 m deep. To construct a gate in the dam we need to know
the net horizontal force on a 5 m wide and 6 m tall port section that then replaces
a 5 m section of the dam. Find the net horizontal force from the water on one side
and air on the other side of the port.
Solution:
Pbot = P0 + ∆P
∆P = ρgh = 997× 9.807× 6 = 58 665 Pa = 58.66 kPa
Neglect ∆P in air
Fnet = Fright – Fleft = Pavg A - P0A
Pavg = P0 + 0.5 ∆P Since a linear pressure variation with depth.
Fnet = (P0 + 0.5 ∆P)A - P0A = 0.5 ∆P A = 0.5 × 58.66 × 5 × 6 = 880 kN
F
Fleft
right
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100. Borgnakke and Sonntag
2.86
In the city water tower, water is pumped up to a level 25 m above ground in a
pressurized tank with air at 125 kPa over the water surface. This is illustrated in
Fig. P2.86. Assuming the water density is 1000 kg/m3 and standard gravity, find
the pressure required to pump more water in at ground level.
Solution:
∆P = ρ L g
= 1000 kg/m3 × 25 m × 9.807 m/s2
= 245 175 Pa = 245.2 kPa
Pbottom = Ptop + ∆P
= 125 + 245.2
= 370 kPa
cb
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101. Borgnakke and Sonntag
2.87
The main waterline into a tall building has a pressure of 600 kPa at 5 m elevation
below ground level. How much extra pressure does a pump need to add to ensure
a water line pressure of 200 kPa at the top floor 150 m above ground?
Solution:
The pump exit pressure must balance the top pressure plus the column
∆P. The pump inlet pressure provides part of the absolute pressure.
Pafter pump = Ptop + ∆P
∆P = ρgh = 997 kg/m3 × 9.807 m/s2 × (150 + 5) m
= 1 515 525 Pa = 1516 kPa
Pafter pump = 200 + 1516 = 1716 kPa
∆Ppump = 1716 – 600 = 1116 kPa
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102. Borgnakke and Sonntag
2.88
Two cylinders are connected by a piston as shown in Fig. P2.88. Cylinder A is
used as a hydraulic lift and pumped up to 500 kPa. The piston mass is 25 kg and
there is standard gravity. What is the gas pressure in cylinder B?
Solution:
Force balance for the piston: PBAB + mpg + P0(AA - AB) = PAAA
AA = (π/4)0.12 = 0.00785 m2; AB = (π/4)0.0252 = 0.000 491 m2
PBAB = PAAA - mpg - P0(AA - AB) = 500× 0.00785 - (25 × 9.807/1000)
- 100 (0.00785 - 0.000 491) = 2.944 kN
PB = 2.944/0.000 491 = 5996 kPa = 6.0 MPa
P
B
GAS
A Oil
Po
cb
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103. Borgnakke and Sonntag
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2.89
A 5-kg piston in a cylinder with diameter of 100 mm is loaded with a linear
spring and the outside atmospheric pressure of 100 kPa as shown in Fig. P2.89.
The spring exerts no force on the piston when it is at the bottom of the cylinder
and for the state shown, the pressure is 400 kPa with volume 0.4 L. The valve is
opened to let some air in, causing the piston to rise 2 cm. Find the new pressure.
Solution:
A linear spring has a force linear proportional to displacement. F = k x, so
the equilibrium pressure then varies linearly with volume: P = a + bV, with an
intersect a and a slope b = dP/dV. Look at the balancing pressure at zero volume
(V -> 0) when there is no spring force F = PA = PoA + mpg and the initial state.
These two points determine the straight line shown in the P-V diagram.
Piston area = AP = (π/4) × 0.12 = 0.00785 m2
400
106.2
2
1
0 0.4
P
V
0.557
2P
a = P0 +
mpg
Ap
= 100 kPa +
5 × 9.80665
0.00785 Pa
= 106.2 kPa intersect for zero volume.
V2 = 0.4 + 0.00785 × 20 = 0.557 L
P2 = P1 +
dP
dV ∆V
= 400 +
(400-106.2)
0.4 - 0 (0.557 - 0.4)
= 515.3 kPa
105. 2 Borgnakke and Sonntag
CONTENT CHAPTER 3
SUBSECTION PROB NO.
Concept problems 1-15
Phase diagrams, triple and critical points 16-24
General tables 25-64
Ideal gas 65-84
Compressibility factor 85-97
Equations of state 98-106
Review problems 107-125
Linear interpolation 126-130
Computer tables 131-135
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106. 3 Borgnakke and Sonntag
In-Text Concept Questions
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107. 4 Borgnakke and Sonntag
3.a
If the pressure is smaller than Psat at a given T, what is the phase?
Refer to the phase diagrams in Figures 3.6
and 3.7. For a lower P you are below the
vaporization curve and that is the superheated
vapor region. You have the gas phase.
ln P
T
Vapor
L
S
Critical Point
S
3.b
An external water tap has the valve activated by a long spindle so the closing
mechanism is located well inside the wall. Why is that?
Solution:
By having the spindle inside the wall the coldest location with water when the valve
is closed is kept at a temperature above the freezing point. If the valve spindle was
outside there would be some amount of water that could freeze while it is trapped
inside the pipe section potentially rupturing the pipe.
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108. 5 Borgnakke and Sonntag
3.c
What is the lowest temperature (approximately) at which water can be liquid?
Look at the phase diagram in Fig. 3.7. At the
border between ice I, ice III and the liquid region
is a triple point which is the lowest T where you
can have liquid. From the figure it is estimated to
be about 255 K i.e. at -18oC.
T ≈ 255 K ≈ - 18°C
ln P
T
V
L
S
CR.P.
lowest T liquid
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109. 6 Borgnakke and Sonntag
3.d
Some tools should be cleaned in water at a least 150oC. How high a P is needed?
Solution:
If I need liquid water at 150oC I must have a pressure that is at least the saturation
pressure for this temperature.
Table B.1.1: 150oC, Psat = 475.9 kPa.
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110. 7 Borgnakke and Sonntag
3.e
Water at 200 kPa has a quality of 50%. Is the volume fraction Vg/Vtot < 50% or
> 50%?
This is a two-phase state at a given pressure without looking in the table we know
that vf is much smaller than vg.
From the definition of quality we get the masses from total mass, m, as
mf = (1 – x) m, mg = x m
The volumes are
Vf = mf vf = (1 – x) m vf, Vg = mg vg = x m vg
So when half the mass is liquid and the other half is vapor the liquid volume is
much smaller that the vapor volume.
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111. 8 Borgnakke and Sonntag
3.f
Why are most of the compressed liquid or solid regions not included in the printed
tables?
For the compressed liquid and the solid phases the specific volume and thus
density is nearly constant. These surfaces are very steep nearly constant v and there
is then no reason to fill up a table with the same value of v for different P and T.
3.g
Why is it not typical to find tables for Ar, He, Ne or air like an Appendix B table?
The temperature at which these substances are close to the two-phase
region is very low. For technical applications with temperatures around
atmospheric or higher they are ideal gases. Look in Table A.2 and we can see the
critical temperatures as
Ar : 150.8 K He: 5.19 K Ne: 44.4 K
It requires a special refrigerator in a laboratory to bring a substance down
to these cryogenic temperatures.
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112. 9 Borgnakke and Sonntag
3.h
What is the percent change in volume as liquid water freezes? Mention some effects
in nature and for our households the volume change can have.
The density of water in the different phases can be found in Tables A.3 and A.4
and in Table B.1.
From Table B.1.1 vf = 0.00100 m3/kg
From Table B.1.5 vi = 0.0010908 m3/kg
Percent change: 100
vi – vf
vf
= 100 ×
0.0010908 – 0.001
0.001 = 9.1 % increase
Liquid water that seeps into cracks or other confined spaces and then freezes
will expand and widen the cracks. This is what destroys any porous material exposed
to the weather on buildings, roads and mountains. It can burst water pipes and crack
engine blocks (that is why you put anti-freeze in it).
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113. 10 Borgnakke and Sonntag
3.i
How accurate is it to assume that methane is an ideal gas at room conditions?
From Table A.2: Tc = 190.4 K, Pc = 4.60 MPa
So at room conditions we have much higher T > Tc and P << Pc so this is the
ideal gas region. To confirm look in Table B.7.2
100 kPa, 300 K, v = 1.55215 m3/kg
Find the compressibility factor (R from Table A.5) as
Z = Pv/RT =
100 kPa × 1.55215 m3/kg
0.5183 kJ/kg-K × 300 K
= 0.99823
so Z is 1 with an accuracy of 0.2% better than most measurements can be done.
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114. 11 Borgnakke and Sonntag
3.j
I want to determine a state of some substance, and I know that P = 200 kPa; is it
helpful to write PV = mRT to find the second property?
NO. You need a second property.
Notice that two properties are needed to determine a state. The EOS can give you a
third property if you know two, like (P,T) gives v just as you would get by entering a
table with a set (P,T). This EOS substitutes for a table when it is applicable.
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115. 12 Borgnakke and Sonntag
3.k
A bottle at 298 K should have liquid propane; how high a pressure is needed? (use
Fig. D.1)
To have a liquid the pressure must be higher than or equal to the saturation pressure.
There is no printed propane table so we use the compressibility chart and Table A.2
Propane Table A.2: Tc = 369.8 K, Pc = 4.25 MPa
The reduced temperature is:
Tr =
T
Tc
=
298
369.8 = 0.806,
for which we find in Fig. D.1: Pr sat = 0.25
P = Pr sat Pc = 0.25 × 4.25 MPa = 1.06 MPa
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for
testing or instructional purposes only to students enrolled in courses for which this textbook has been
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116. 13 Borgnakke and Sonntag
3.l
A bottle at 298 K should have liquid propane; how high a pressure is needed? (use
the software)
To have a liquid the pressure must be higher than or equal to the saturation pressure.
There is no printed propane table but the software has propane included
Start CATT3, select cryogenic substances, propane
select calculator, select T, x = (25C, 0) P = 0.9518 MPa
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for
testing or instructional purposes only to students enrolled in courses for which this textbook has been
adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108
of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.