Lecture 4 - Ideal Gas PDF

Thermofluids I: Thermodynamic “ENGN 2610” Lecture 4 Ideal Gases  Learning outcomes By the end of the lecture, you should gain the following outcomes: Textbook Understand the ide...

Thermofluids I: Thermodynamic “ENGN 2610” Lecture 4 Ideal Gases  Learning outcomes By the end of the lecture, you should gain the following outcomes: Textbook Understand the ideal gas behavior under different conditions. Chapter 3-6 to 3-7 Define the gas constant. Understand the deviation of real gases from the ideal gas behavior. 2  The Ideal-Gas equation of state What is an ideal gas? A model substance of infinitely small particles (point masses) that completely fill a defined space and do not interact. Examples: Air – Nitrogen – Oxygen – CO2 – Argon …. 3  The Ideal-Gas equation of state What is the equation of state? Any equation that relates the pressure, temperature, and specific volume of a substance is called an equation of state. The ideal-gas equation of state predicts the P-v-T behavior of a gas quite accurately within some properly selected region. It has been experimentally observed that the ideal-gas relation given closely approximates the P-v-T behavior of real gases at low densities. At low pressures and high temperatures, the density of a gas decreases, and the gas behaves as an ideal gas under these conditions. 4  The Ideal-Gas 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. Boyle-Mariotte’s Law 𝑃 ∝ 1/𝑣 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. Gay-Lussac’s Law 𝑣∝𝑇 5  The Ideal-Gas equation of state The ideal gas equation of state 𝑃𝑣 = 𝑅𝑇 Where the constant of proportionality R is called the gas constant. 6  Gas constant R The gas constant R is different for each gas and is determined from: 𝑘𝐽 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 [ ] 𝑅= 𝑘𝑚𝑜𝑙. 𝐾 = 𝑅𝑢 [ 𝑘𝐽 ] 𝑘𝑔 𝑀 𝑘𝑔. 𝐾 𝑀𝑜𝑙𝑎𝑟 𝑚𝑎𝑠𝑠 [ ] 𝑘𝑚𝑜𝑙 The constant Ru is the same for all substances, and its value is: 𝑅𝑢 = 8314 𝐽Τ𝑘𝑔. 𝐾 7  Gas constant R 𝑘𝐽 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 [ ] 𝑅= 𝑘𝑚𝑜𝑙. 𝐾 = 𝑅𝑢 [ 𝑘𝐽 ] 𝑘𝑔 𝑀 𝑘𝑔. 𝐾 𝑀𝑜𝑙𝑎𝑟 𝑚𝑎𝑠𝑠 [ ] 𝑘𝑚𝑜𝑙 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. Example: When we say the molar mass of nitrogen is 28, it simply means the mass of 1 kmol of nitrogen is 28 kg Molar mass of O2 is 32. What is the molar mass of air? 8 Gas constant R 𝑅𝑢 𝑘𝐽 𝑅= [ ] 𝑀 𝑘𝑔. 𝐾 9  The Ideal-Gas equation of state The ideal gas equation of state 𝑃𝑣 = 𝑅𝑇 𝑃𝑉 = 𝑚𝑅𝑇 (m is the mass of the substance) OR 𝑃𝑉 = 𝑁𝑅𝑢 𝑇 (N is the mole number) 𝑚 = 𝑀𝑁 [𝑘𝑔] 10  The Ideal-Gas equation of state The ideal gas equation of state for closed systems 𝑃1 𝑉1 𝑃2 𝑉2 𝑚 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝑇1 𝑇2 The relation above can be applied for any process from state 1 to state 2 under the following conditions: The working fluid is ideal gas. The mass is constant in the system. 11  Problems Problem 1 Determine the mass of the air in a room whose dimensions are 4 m x 5 m x 6 m at 100 kPa and 25°C. 12  Is water vapor an ideal gas? At pressures below 10 kPa, water vapor can be treated as an ideal gas, regardless of its temperature. At higher pressures, however, the ideal gas assumption yields unacceptable errors The numbers in the graph represents the percentage of error when water vapor is treated as ideal gas. 13  Compressibility Factor [Z] What is the compressibility factor? It is a correction factor that measures the deviation from the ideal gas behavior. 𝑃𝑉 = 𝑍𝑚𝑅𝑇 ▪ For ideal gases: Z = 1. ▪ For real gases: Z can be greater or less than unity. ▪ The farther away Z is from unity, the more the gas deviates from ideal-gas behavior. 14  Compressibility Factor [Z] Determination of Z Gases behave differently at a given temperature and pressure, but they behave very much the same at temperatures and pressures normalized with respect to their critical temperatures and pressures, called reduced temperature (TR) and reduced pressure (PR). 𝑇 𝑃 𝑇𝑅 = 𝑃𝑅 = 𝑇𝑐𝑟 𝑃𝑐𝑟 Where: Tcr is the critical temperature, Pcr is the critical pressure. If problem is given in terms of P-v and T-v, define the pseudo reduced volume: 𝑣 𝑣𝑅 = 𝑅(𝑇𝑐𝑟 /𝑃𝑐𝑟 ) 15  Chart for compressibility factor It is uploaded to D2L. 16  Compressibility Factor [Z] At very low pressures (PR 2), ideal-gas behavior can be assumed with good accuracy regardless of pressure (except when PR >> 1). The deviation of a gas from ideal-gas behavior is greatest in the vicinity of the critical point. 17  Problems Problem 2 Determine the specific volume of refrigerant-134a at 1 MPa and 50°C, using (a) the ideal-gas equation of state and (b) the generalized compressibility chart, (c) the tables. Compare the values obtained to the actual value of 0.021796 m3/kg and determine the error involved in each case. 18  Summary At the end of this lecture, you should be able to answer the following questions: What is the ideal gas equation of state. What is the relation of P-v-t under the ideal gas behavior. Solve problems of determination of properties for ideal gases. What is the compressibility factor Z. Solve problems of how to compare the properties for ideal and real gases using the compressibility factor chart. 19