Advanced Thermodynamics: Lecture 2 Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Equation of state Any equation that relates the pressure, temperature, and specific volume of a substance is called an equation of state. In 1662, Robert Boyle, an Englishman, observed during his experiments with a vacuum chamber that the pressure of gases is inversely proportional to their volume. In 1802, J. Charles and J. Gay-Lussac, Frenchmen, experimentally determined that at low pressures the volume of a gas is proportional to its temperature. P = R T ν where the constant of proportionality R is called the gas constant. Equation is called the ideal-gas equation of state Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
The gas constant R is di ff erent for each gas is determined from R = R u M where R u is the universal gas constant and M is the molar mass of the gas. The molar mass M can simply be defined as the mass of one mole (also called a gram-mole, abbreviated gmol) of a substance in grams, or the mass of one kmol (also called a kilogram-mole, abbreviated kgmol) in kilograms. Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
The gas constant R is di ff erent for each gas is determined from R = R u M where R u is the universal gas constant and M is the molar mass of the gas. The molar mass M can simply be defined as the mass of one mole (also called a gram-mole, abbreviated gmol) of a substance in grams, or the mass of one kmol (also called a kilogram-mole, abbreviated kgmol) in kilograms. The constant Ru is the same for all substances, and its value is R u = 8 . 31447 kJ / kmol · K Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Gases deviate from ideal-gas behavior significantly at states near the saturation region and the critical point. This deviation from ideal-gas behavior at a given temperature and pressure can be corrected by introducing a correction factor called the compressibility factor Z defined as Z = P ν RT The behavior of di ff erent gases is di ff erent at varying pressures and temperatures. Though if the pressure and temperature are normalize with respective critical points, then at these reduced pressure and temperatures, the behavior of gases are quite similar. P T P R = T R = P crit T crit Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Compressibility factor for various gases Image source: Thermodynamics An Engineering Approach, Cengel and Boles, 7th edition Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Other equations of state Van der Waal’s equation of state P + a ⇣ ⌘ ( ν − b ) = RT v 2 Where, a = 27 R 2 T 2 b = RT crit crit and 64 P crit 8 P crit Beattie-Bridgeman EOS P = R u T c ν + B ) − A ⇣ ⌘ 1 − (¯ ν 2 ν T 3 ν 2 ¯ ¯ ¯ Where, 1 − a ✓ 1 − b ◆ ⇣ ⌘ A = A 0 B = B 0 and ¯ ¯ ν ν Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Forms of Energy Energy can exist in various forms Thermal Kinetic Potential Electric Magnetic Chemical Nuclear The sum of all energies is the total energy of the system E . On per unit mass basis it is denoted as e = E m Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
The total energy can be divided into two,i.e macroscopic and microscopic Macroscopic energy – is defined with respect to some outside reference frame. for example, Potential and Kinetic energies. Microscopic energy – are related to the molecular structure and molecular activity of the system. The sum of all microscopic energies is called internal energy and denoted by U . Hence total energy is given by, E = U + KE + PE = U + mV 2 2 + mgz Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Internal Energy The sensible energy contains the translational, rotational and vibrational energy of the molecules. Image source: Thermodynamics An Engineering Approach, Cengel and Boles, 7th edition Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Internal Energy Image source: Thermodynamics An Engineering Approach, Cengel and Boles, 7th edition Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Energy Internal energy associated with the phase of a system is called latent energy. Internal energy associated with atomic and molecular bonds is called chemical energy. The energy associated with the bonds of the nucleus is called nuclear energy. Mechanical energy can be defined as the form of energy that can be converted to mechanical work completely and directly by an ideal mechanical device such as an ideal turbine. Kinetic and potential energies are the familiar forms of mechanical energy. Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Energy transfer by heat Heat is defined as the form of energy that is transferred between two systems (or a system and its surroundings) by virtue of a temperature di ff erence. A process during which there is no heat transfer is called an adiabatic process. The word adiabatic comes from the Greek word adiabatos, which means not to be passed. Heat is transferred by three mechanisms: conduction, convection, and radiation. Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Mechanisms of Energy transfer by heat Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interaction between particles. Convection is the transfer of energy between a solid surface and the adjacent fluid that is in motion, and it involves the combined e ff ects of conduction and fluid motion. Radiation is the transfer of energy due to the emission of electromagnetic waves (or photons). Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Energy transfer by work Work, like heat, is an energy interaction between a system and its surroundings. If the energy crossing the boundary of a closed system is not heat, it must be work. Work is the energy transfer associated with a force acting through a distance. A rising piston, a rotating shaft, and an electric wire crossing the system boundaries are all associated with work interactions. Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Heat and work are directional quantities, and thus the complete description of a heat or work interaction requires the specification of both the magnitude and direction. Convention: Heat transfer to a system and work done by a system are positive; heat transfer from a system and work done on a system are negative. Both are recognized at the boundaries of a system as they cross the boundaries. That is, both heat and work are boundary phenomena. Both are associated with a process, not a state. Unlike properties, heat or work has no meaning at a state. Both are path functions (i.e., their magnitudes depend on the path followed during a process as well as the end states). Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
First Law of Thermodynamics – known as Conservation of Energy The first law of thermodynamics states that energy can be neither created nor destroyed during a process; it can only change forms. A major consequence of the first law is the existence and the definition of the property total energy E. Image source: Thermodynamics An Engineering Approach, Cengel and Boles, 7th edition Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
The net change (increase or decrease) in the total energy of the system during a process is equal to the di ff erence between the total energy entering and the total energy leaving the system during that process. Energy Change = Energy in − Energy out ∆ E system = E final − E initial = E 1 − E 2 Energy is a property, and the value of a property does not change unless the state of the system changes. ∆ E = ∆ U + ∆ PE + ∆ KE ∆ KE = 1 2 m ( V 2 2 − V 2 ∆ U = m ( u 2 − u 1 ) ∆ PE = mg ( z 2 − z 1 ) 1 ) Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Mechanisms of Energy transfer Heat Transfer Work transfer Mass flow E in − E out = ( Q in − Q out )+( W in − W out )+( E mass in − E mass out ) = ∆ E system Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Cooling of a Hot Fluid in a Tank A rigid tank contains a hot fluid that is cooled while being stirred by a paddle wheel. Initially, the internal energy of the fluid is 800 kJ. During the cooling process, the fluid loses 500 kJ of heat, and the paddle wheel does 100 kJ of work on the fluid. Determine the final internal energy of the fluid. Neglect the energy stored in the paddle wheel. Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
Acceleration of Air by a Fan A fan that consumes 20 W of electric power when operating is claimed to discharge air from a ventilated room at a rate of 0.25 kg/s at a discharge velocity of 8 m/s. Determine if this claim is reasonable. Shivasubramanian Gopalakrishnan sgopalak@iitb.ac.in ME 661
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