1 MECE2320U-THERMODYNAMICS HOMEWORK # 5 Instructor: Dr. Ibrahim Dincer Assignment Date: Thursday, 22 October 2015 Assignment Type: Individual Due Date: Thursday, 29 October 2015 (3.00 pm latest, leave in dropbox 8) 1) As shown in figure, the inlet and outlet conditions of a steam turbine are given. The heat loss from turbine is 35 kJ per kg of steam. a) Show all the state points on T-v diagram b) Write mass and energy balance equations c) Calculate the turbine work 2) As shown in figure, refrigerant R134a enters to a compressor. Write both mass and energy balance equations. Calculate the compressor work and the mass flow rate of refrigerant. 3) As shown in figure, the heat exchanger uses the heat of hot exhaust gases to produce steam. Where, 15% of heat is lost to the surroundings. Exhaust gases enters the heat exchanger at 500°C. Water enters at 15°C as saturated liquid and exit at saturated vapor at 2 MPa. Mass flow rate of water is 0.025 kg/s, and for exhaust gases, it is 0.42 kg/s. The specific heat for exhaust gases is 1.045 kJ/kg K, which can be treated as ideal gas. 1 Turbine 2 ? 1 = 1 ??/? ?1 = 1 ??? ?1 = 300 ℃ ?1 = 40 ?/? ? ??? =? ????? = 35 ??/?? ?2 = 150 ??? ?2 = 0.9 ?2 = 180 ?/? 1 Compressor 2 ???? ???? = 1.3 ?3/??? ?1 = 100 ??? ?1 = −20 ℃ ? ?? =? ? ???? = 3 ?? ?2 = 800 ??? ?2 = 60 ℃ 2 a) Write mass and energy balance equations. b) Calculate the rate of heat transfer to the water. c) Calculate the exhaust gases exit temperature. 4) As shown in figure, two refrigerant R134a streams mix in a mixing chamber. If the mass flow rate of cold stream is twice that of the hot stream. a) Write mass and energy balance equations. b) Calculate the temperature of the mixture at the exit of the mixing chamber c) Calculate the quality at the exit of the mixing chamber 5) As shown in figure, an air conditioning system requires airflow at the main supply duct at a rate of 140 m3/min. The velocity inside circular duct is not to exceed 9 m/s. Assume that the fan converts 85% of electrical energy it consumes into kinetic energy of air. a) Write mass and energy balance equations. b) Calculate the size of electric motor require to drive the fan c) Calculate the diameter of the main duct ?2 = 1 ??? ?2 = 90 ℃ ?1 = 1 ??? ?1 = 30 ℃ ?3 =? ?3 =? 140 ?3/??? 9 ?/? Air Fan

1 MECE2320U-THERMODYNAMICS HOMEWORK # 5 Instructor: Dr. Ibrahim Dincer Assignment Date: Thursday, 22 October 2015 Assignment Type: Individual Due Date: Thursday, 29 October 2015 (3.00 pm latest, leave in dropbox 8) 1) As shown in figure, the inlet and outlet conditions of a steam turbine are given. The heat loss from turbine is 35 kJ per kg of steam. a) Show all the state points on T-v diagram b) Write mass and energy balance equations c) Calculate the turbine work 2) As shown in figure, refrigerant R134a enters to a compressor. Write both mass and energy balance equations. Calculate the compressor work and the mass flow rate of refrigerant. 3) As shown in figure, the heat exchanger uses the heat of hot exhaust gases to produce steam. Where, 15% of heat is lost to the surroundings. Exhaust gases enters the heat exchanger at 500°C. Water enters at 15°C as saturated liquid and exit at saturated vapor at 2 MPa. Mass flow rate of water is 0.025 kg/s, and for exhaust gases, it is 0.42 kg/s. The specific heat for exhaust gases is 1.045 kJ/kg K, which can be treated as ideal gas. 1 Turbine 2 ? 1 = 1 ??/? ?1 = 1 ??? ?1 = 300 ℃ ?1 = 40 ?/? ? ??? =? ????? = 35 ??/?? ?2 = 150 ??? ?2 = 0.9 ?2 = 180 ?/? 1 Compressor 2 ???? ???? = 1.3 ?3/??? ?1 = 100 ??? ?1 = −20 ℃ ? ?? =? ? ???? = 3 ?? ?2 = 800 ??? ?2 = 60 ℃ 2 a) Write mass and energy balance equations. b) Calculate the rate of heat transfer to the water. c) Calculate the exhaust gases exit temperature. 4) As shown in figure, two refrigerant R134a streams mix in a mixing chamber. If the mass flow rate of cold stream is twice that of the hot stream. a) Write mass and energy balance equations. b) Calculate the temperature of the mixture at the exit of the mixing chamber c) Calculate the quality at the exit of the mixing chamber 5) As shown in figure, an air conditioning system requires airflow at the main supply duct at a rate of 140 m3/min. The velocity inside circular duct is not to exceed 9 m/s. Assume that the fan converts 85% of electrical energy it consumes into kinetic energy of air. a) Write mass and energy balance equations. b) Calculate the size of electric motor require to drive the fan c) Calculate the diameter of the main duct ?2 = 1 ??? ?2 = 90 ℃ ?1 = 1 ??? ?1 = 30 ℃ ?3 =? ?3 =? 140 ?3/??? 9 ?/? Air Fan

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Tornado Eddy Investigation Abstract The objective of this lab was to write a bunch of jibberish to provide students with a formatting template. Chemical engineering, bioengineering, and environmental engineering are “process engineering” disciplines. Good abstracts contains real content, such as 560 mL/min, 35 deg, and 67 percent yield. Ideal degreed graduates are technically strong, bring broad system perspectives to problem solving, and have the professional “soft skills” to make immediate contributions in the workplace. The senior lab sequence is the “capstone” opportunity to realize this ideal by integrating technical skills and developing professional soft skills to ensure workforce preparedness. The best conclusions are objective and numerical, such as operating conditions of 45 L/min at 32 deg C with expected costs of $4.55/lb. Background Insect exchange processes are often used in bug filtration, as they are effective at removing either positive or negative insects from water. An insect exchange column is a packed or fluidized bed filled with resin beads. Water flows through the column and most of the insects from the water enter the beads, but some of them pass in between the beads, which makes the exchange of insects non-ideal. Insectac 249 resin is a cation exchange resin, as it is being used to attract cationic Ca2+ from the toxic waste stream. This means the resin is negatively charged, and needs to be regenerated with a solution that produces positively charged insects, in this case, salt water which contains Na+ insects. The resin contains acidic styrene backbones which capture the cationic insects in a reversible process. A curve of Ca2+ concentration concentration vs. time was obtained after a standard curve was made to determine how many drops from the low cost barium test kit from Aquarium Pharmaceuticals (API)1 bottle #2 would correspond to a certain concentration in solution. A standard curve works by preparing solutions with known concentrations and testing these concentrations using the kit to create a curve of number of drops from bottle #2 (obtained result) vs. concentration of Ca2+ in solution (desired response). The standard curve can then be used for every test on the prototype and in the field, to quickly and accurately obtain a concentration from the test kit. The barium concentration vs. time curve can be used to calculate the exchange capacity of the resin and, in later tests, the regeneration efficiency. The curves must be used to get the total amount of barium removed from the water, m. Seen in Equation 2, the volumetric flow rate of water, , is multiplied by the integral from tinitial to tfinal of the total concentration of Ca2+ absorbed by the resin as a function of time, C. (2) 1 http://aquariumpharm.com/Products/Product.aspx?ProductID=72 , date accessed: 11/26/10 CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 9 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A graphical trapezoid method was used to evaluate the integral and get the final solution in equivalents of Ca2+ per L, it must be noted that there are 2 equivalents per mole of barium, as the charge of the barium insect is +2. An initial exchange capacity was calculated for the virgin resin, and an adjusted exchange capacity was calculated once the resin was regenerated. The regenerated resin capacity was found by multiplying the virgin resin capacity by the regeneration efficiency, expressed in Equation 3. (3) See Appendix A for the calculation of the exchange capacities and the regeneration efficiency. Materials and Methods Rosalie and Peter Johnson of Corvallis established the Linus Pauling Chair in Chemical Engineering to honor Oregon State University’s most famous graduate. Peter Johnson, former President and owner of Tekmax, Inc., a company which revolutionized battery manufacturing equipment, is a 1955 graduate of the College of Engineering.2 The Chair, also known as the Linus Pauling Distinguished Engineer or Linus Pauling Engineer (LPE), was originally designed to focus on the traditional “capstone” senior lab sequence in the former Department of Chemical Engineering. The focus is now extended to all the process engineering disciplines. The LPE is charged with establishing strong ties with industry, ensuring current and relevant laboratory experiences, and helping upperclass students develop skills in communication, teamwork, project management, and leadership. Include details about lab procedures not sufficiently detailed in the SOP, problems you had, etc. The bulk solution prepared to create the standard curve was used in the second day of testing to obtain the exchange capacity of the insectac 249 resin. The solution was pumped through a bathroom scale into the prototype insect exchange column. 45 mL of resin was rinsed and added to the column. The bed was fluidized as the solution was pumped through the resin, but for the creation of the Ca2+ concentration vs. time curve, the solution was pumped down through the column, as illustrated in the process flow diagram seen in Figure 1. Figure 1. Process sketch of the insect exchange column used for the project. Ref: http://www.generon.co.uk/acatalog/Chromatography.html 2 Harding, P. Viscosity Measurement SOP, Spring, 2010. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 10 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A bathroom scale calibration curve was created to ensure that the 150 mL/min, used to calculate the breakthrough time, would be delivered to the resin. The bathroom scale used was a Dwyer brand with flowrates between 0 and 300 cc/min of water. Originally, values between 120 and 180 mL/min were chosen for the calibration, with three runs for each flowrate, however the bathroom scale values were so far away from the measure values the range was extended to 100 to 200 mL/min. The regeneration experiment was performed using a method similar to that used in the water softening experiment, however instead of using a 640 ppm Ca2+ solution to fill the resin, a 6000 ppm Na+ solution was used to eject the Ca2+ from the resin. Twelve samples times were chosen and adjusted as the experiment progressed, with more than half of the samples taken at times less than 10 minutes, and the last sample taken at 45 minutes. The bulk exit solution was also tested to determine the regeneration efficiency. Results and Discussion The senior lab sequence has its roots in the former Department of Chemical Engineering. CHE 414 and 415 were taught in Winter and Spring and included 6 hours of lab time per week. The School has endeavored to incorporate the courses into the BIOE and ENVE curriculum, and this will be complete in 2008-2009. Recent development of the senior lab course sequence is shown chronologically in Fig. 1. In 2006-2007, CHE 414 and 415 were moved to Fall and Winter to enable CHE 416, an elective independent senior project course. Also that year, BIOE students took BIOE 414 in the Fall and BIOE 415 was developed and taught. No BIOE students enrolled in the optional CHE. In 2007-2008, the program transitioned in a new Linus Pauling Engineer and ENVE 414 was offered. Also, approximately 30 percent of BIOE students enrolled in the optional CHE 416. Accommodating the academic calendars of the three disciplines required a reduction in weekly student lab time from 6 to 3 hours. The expected relationship between coughing rate, y, and length of canine, x, is Bx z y Fe− (1) where F is a pre-exponential constant, B is vitamin B concentration and z is the height of an average trapeze artist. 3 The 2008-2009 brings the challenge of the dramatic enrollment increase shown in Fig. 1 and the first offering of ENVE 415. The result, shown on the right in Fig. 1, is the delivery of the senior lab sequence uniformly across the process engineering disciplines. CBEE 416 is expected to drawn approximately of the students that take the 415 courses. In 2007-2008, 414 and 415 were required for CHEs, 414 and 415 for BIOEs, and only 414 for ENVEs. CHE 416 is ostensibly an elective for all disciplines. In 2008-2009, 414 and 415 is required for all disciplines and CHE 416 will be an elective. The content of 414 is essentially 3 Fundamentals of Momentum, Heat, and Mass Transfer, Welty, J.R. et al., 4th edition, John Wiley & Sons, Inc. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 11 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE identical for all three disciplines, 415 has discipline-specific labs, and 416 consists of senior projects with potentially cross-discipline teams of 2 to 4 students. Tremendous labor and struggling with the lab equipment resulted in the data shown in y = –‐0.29x + 1.71 y = –‐0.25x + 2.03 y = –‐0.135x + 2.20 –‐1.5 –‐1.0 –‐0.5 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 ln y (units) x (units) ln y_1 ln y_2 ln y_3 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Case 1 Case 2 Case 3 Slope (units) (a) (b) Figure 1. (a) Data for y and x plotted for various values of z and (b) a comparison of slopes for the 3 cases investigate. The log plot slope yields the vitamin B concentration. The slopes were shown to be significantly at the 90% confidence level, but the instructor ran out of time and did not include error bars. The slope changed as predicted by the Snirtenhoffer equation. Improvements to the lab might include advice on how to legally change my name to something less embarrassing. My whole life I have been forced to repeat and spell it. I really feel that this has affected my psychologically. This was perhaps the worst lab I have ever done in my academic career, primarily due to the fact that there was no lab time. I simply typed in this entire report and filled it with jibberish. Some might think nobody will notice, but I know that …… Harding reads every word. Acknowledgments The author acknowledges his elementary teacher for providing truly foundational instruction in addition and subtraction. Jenny Burninbalm was instrumental with guidance on use of the RT-345 dog scratching device. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 12

Tornado Eddy Investigation Abstract The objective of this lab was to write a bunch of jibberish to provide students with a formatting template. Chemical engineering, bioengineering, and environmental engineering are “process engineering” disciplines. Good abstracts contains real content, such as 560 mL/min, 35 deg, and 67 percent yield. Ideal degreed graduates are technically strong, bring broad system perspectives to problem solving, and have the professional “soft skills” to make immediate contributions in the workplace. The senior lab sequence is the “capstone” opportunity to realize this ideal by integrating technical skills and developing professional soft skills to ensure workforce preparedness. The best conclusions are objective and numerical, such as operating conditions of 45 L/min at 32 deg C with expected costs of $4.55/lb. Background Insect exchange processes are often used in bug filtration, as they are effective at removing either positive or negative insects from water. An insect exchange column is a packed or fluidized bed filled with resin beads. Water flows through the column and most of the insects from the water enter the beads, but some of them pass in between the beads, which makes the exchange of insects non-ideal. Insectac 249 resin is a cation exchange resin, as it is being used to attract cationic Ca2+ from the toxic waste stream. This means the resin is negatively charged, and needs to be regenerated with a solution that produces positively charged insects, in this case, salt water which contains Na+ insects. The resin contains acidic styrene backbones which capture the cationic insects in a reversible process. A curve of Ca2+ concentration concentration vs. time was obtained after a standard curve was made to determine how many drops from the low cost barium test kit from Aquarium Pharmaceuticals (API)1 bottle #2 would correspond to a certain concentration in solution. A standard curve works by preparing solutions with known concentrations and testing these concentrations using the kit to create a curve of number of drops from bottle #2 (obtained result) vs. concentration of Ca2+ in solution (desired response). The standard curve can then be used for every test on the prototype and in the field, to quickly and accurately obtain a concentration from the test kit. The barium concentration vs. time curve can be used to calculate the exchange capacity of the resin and, in later tests, the regeneration efficiency. The curves must be used to get the total amount of barium removed from the water, m. Seen in Equation 2, the volumetric flow rate of water, , is multiplied by the integral from tinitial to tfinal of the total concentration of Ca2+ absorbed by the resin as a function of time, C. (2) 1 http://aquariumpharm.com/Products/Product.aspx?ProductID=72 , date accessed: 11/26/10 CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 9 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A graphical trapezoid method was used to evaluate the integral and get the final solution in equivalents of Ca2+ per L, it must be noted that there are 2 equivalents per mole of barium, as the charge of the barium insect is +2. An initial exchange capacity was calculated for the virgin resin, and an adjusted exchange capacity was calculated once the resin was regenerated. The regenerated resin capacity was found by multiplying the virgin resin capacity by the regeneration efficiency, expressed in Equation 3. (3) See Appendix A for the calculation of the exchange capacities and the regeneration efficiency. Materials and Methods Rosalie and Peter Johnson of Corvallis established the Linus Pauling Chair in Chemical Engineering to honor Oregon State University’s most famous graduate. Peter Johnson, former President and owner of Tekmax, Inc., a company which revolutionized battery manufacturing equipment, is a 1955 graduate of the College of Engineering.2 The Chair, also known as the Linus Pauling Distinguished Engineer or Linus Pauling Engineer (LPE), was originally designed to focus on the traditional “capstone” senior lab sequence in the former Department of Chemical Engineering. The focus is now extended to all the process engineering disciplines. The LPE is charged with establishing strong ties with industry, ensuring current and relevant laboratory experiences, and helping upperclass students develop skills in communication, teamwork, project management, and leadership. Include details about lab procedures not sufficiently detailed in the SOP, problems you had, etc. The bulk solution prepared to create the standard curve was used in the second day of testing to obtain the exchange capacity of the insectac 249 resin. The solution was pumped through a bathroom scale into the prototype insect exchange column. 45 mL of resin was rinsed and added to the column. The bed was fluidized as the solution was pumped through the resin, but for the creation of the Ca2+ concentration vs. time curve, the solution was pumped down through the column, as illustrated in the process flow diagram seen in Figure 1. Figure 1. Process sketch of the insect exchange column used for the project. Ref: http://www.generon.co.uk/acatalog/Chromatography.html 2 Harding, P. Viscosity Measurement SOP, Spring, 2010. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 10 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE A bathroom scale calibration curve was created to ensure that the 150 mL/min, used to calculate the breakthrough time, would be delivered to the resin. The bathroom scale used was a Dwyer brand with flowrates between 0 and 300 cc/min of water. Originally, values between 120 and 180 mL/min were chosen for the calibration, with three runs for each flowrate, however the bathroom scale values were so far away from the measure values the range was extended to 100 to 200 mL/min. The regeneration experiment was performed using a method similar to that used in the water softening experiment, however instead of using a 640 ppm Ca2+ solution to fill the resin, a 6000 ppm Na+ solution was used to eject the Ca2+ from the resin. Twelve samples times were chosen and adjusted as the experiment progressed, with more than half of the samples taken at times less than 10 minutes, and the last sample taken at 45 minutes. The bulk exit solution was also tested to determine the regeneration efficiency. Results and Discussion The senior lab sequence has its roots in the former Department of Chemical Engineering. CHE 414 and 415 were taught in Winter and Spring and included 6 hours of lab time per week. The School has endeavored to incorporate the courses into the BIOE and ENVE curriculum, and this will be complete in 2008-2009. Recent development of the senior lab course sequence is shown chronologically in Fig. 1. In 2006-2007, CHE 414 and 415 were moved to Fall and Winter to enable CHE 416, an elective independent senior project course. Also that year, BIOE students took BIOE 414 in the Fall and BIOE 415 was developed and taught. No BIOE students enrolled in the optional CHE. In 2007-2008, the program transitioned in a new Linus Pauling Engineer and ENVE 414 was offered. Also, approximately 30 percent of BIOE students enrolled in the optional CHE 416. Accommodating the academic calendars of the three disciplines required a reduction in weekly student lab time from 6 to 3 hours. The expected relationship between coughing rate, y, and length of canine, x, is Bx z y Fe− (1) where F is a pre-exponential constant, B is vitamin B concentration and z is the height of an average trapeze artist. 3 The 2008-2009 brings the challenge of the dramatic enrollment increase shown in Fig. 1 and the first offering of ENVE 415. The result, shown on the right in Fig. 1, is the delivery of the senior lab sequence uniformly across the process engineering disciplines. CBEE 416 is expected to drawn approximately of the students that take the 415 courses. In 2007-2008, 414 and 415 were required for CHEs, 414 and 415 for BIOEs, and only 414 for ENVEs. CHE 416 is ostensibly an elective for all disciplines. In 2008-2009, 414 and 415 is required for all disciplines and CHE 416 will be an elective. The content of 414 is essentially 3 Fundamentals of Momentum, Heat, and Mass Transfer, Welty, J.R. et al., 4th edition, John Wiley & Sons, Inc. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 11 Josephine Hornsnogger CBEE 414, Lab Section M 1300–‐1550 April 19, 2010 Oregon State University School of CBEE identical for all three disciplines, 415 has discipline-specific labs, and 416 consists of senior projects with potentially cross-discipline teams of 2 to 4 students. Tremendous labor and struggling with the lab equipment resulted in the data shown in y = –‐0.29x + 1.71 y = –‐0.25x + 2.03 y = –‐0.135x + 2.20 –‐1.5 –‐1.0 –‐0.5 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 ln y (units) x (units) ln y_1 ln y_2 ln y_3 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Case 1 Case 2 Case 3 Slope (units) (a) (b) Figure 1. (a) Data for y and x plotted for various values of z and (b) a comparison of slopes for the 3 cases investigate. The log plot slope yields the vitamin B concentration. The slopes were shown to be significantly at the 90% confidence level, but the instructor ran out of time and did not include error bars. The slope changed as predicted by the Snirtenhoffer equation. Improvements to the lab might include advice on how to legally change my name to something less embarrassing. My whole life I have been forced to repeat and spell it. I really feel that this has affected my psychologically. This was perhaps the worst lab I have ever done in my academic career, primarily due to the fact that there was no lab time. I simply typed in this entire report and filled it with jibberish. Some might think nobody will notice, but I know that …… Harding reads every word. Acknowledgments The author acknowledges his elementary teacher for providing truly foundational instruction in addition and subtraction. Jenny Burninbalm was instrumental with guidance on use of the RT-345 dog scratching device. CBEE 102: ENGINEERING PROBLEM SOLVING AND COMPUTATIONS PROJECT DESCRIPTION 12

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Course: PHYS 5426 — Quantum Statistical Physics Assignment #1 Instructor: Gennady Y. Chitov Date Assigned: January 15, 2014 Due Date: January 29, 2014 Problem 1. Prove [a; f(a†)] = @f(a†) @a† (1) [a†; f(a)] = −@f(a) @a (2) for arbitrary function f of operator which admits a series expansion. The Bose creation/ annihilation operators satisfy the standard commutation relations [a; a†] ≡ aa† − a†a = 1 (3) Hint: From Eqs.(1,2) one can figure out the corresponding commutation relations for the powers of creation/annihilation operators and then prove them by the method of mathematical induction. Note that for an arbitrary operator Aˆ: @A^n @A^ = nAˆn−1. Problem 2. In the presence of a constant external force acting on a one-dimensional oscillating particle its Hamiltonian becomes that of the so-called displaced oscillator, and the Schr¨odinger equation ˆH (q) = E (q) of the problem (cf. lecture notes) can be written in terms of dimensionless variables as ( − 1 2 d2 d2 + 1 2 2 − √ 2  ) () = ” () ; (4) where q = √ ~ m! and E = ~!”. a). Write the Schr¨odinger equation (4) in terms of the creation/annihilation operators of the harmonic oscillator ( = 0)  = √1 2 (a + a†) (5) d d = √1 2 (a − a†) (6) 1 Via a linear transformation to the new creation/annihilation operators ˜a†; ˜a preserving the bosonic commutation relations for ˜a†; ˜a map the problem (4) of the displaced oscillator onto that of a simple harmonic oscillator with new operators (˜a†; ˜a). b). Find the spectrum (eigenvalues) ” (E) of the displaced oscillator. c). Write the normalized eigenstates |n⟩ of the displaced Hamiltonian (4) via a† and the vacuum state |Θ◦⟩ of the new operators, i.e. ˜a|Θ◦⟩ = 0 (7) d). As follows from the completeness of the oscillator’s eigenstates, the vacuum state of the displaced oscillator |Θ◦⟩ can be related to the simple oscillator’s vacuum |0⟩ (i.e., a|0⟩ = 0) as |Θ◦⟩ = Ω(a†)|0⟩ (8) Find (up to a normalization factor) the operator function Ω(a†) relating two vacua. Hint: in working out Eqs.(7,8), employ Eqs.(1,2). Problem 3. Prove from the standard commutation relations ([ai; a † j ]∓ = ij , etc) that ⟨0|aiaja † ka † l |0⟩ = jkil ± ikjl (9) the sign depending on the statistics. Also calculate the vacuum expectation value ⟨0|ahaiaja † ka † l a† m |0⟩. Problem 4. In the formalism of second quantization the two-particle interaction term of the Hamiltonian for spinless fermions is given by ˆ V = 1 2 ∫ ∫ dxdy ˆ †(x) ˆ †(y)V(x; y) ˆ (y) ˆ (x) (10) For the short-ranged interaction V(x; y) = V(|x−y|) ≡ V(r) = e2 exp(−r)=r find ˆ V in the momentum representation. The field operators and the creation/annihilation operators in the momentum representation are related in the usual way, i.e., ˆ †(x) = ∫ dp (2)3 a†(p)e−ipx (11) Note that the limit  → 0 recovers the Coulomb (long-ranged) interaction V(r) = e2=r. What is the Fourier transform V(q) of the Coulomb interaction? 2 Problem 5. The matrix elements of a two-particle interaction from the previous problem can be written as ⟨k3k4|V|k1k2⟩ = (2)3(k1 + k2 − k3 − k4)V(q) (12) where q ≡ k3−k1 is the momentum transfer. Show that the diagonal part of the interaction operator ˆ V found on the previous problem in the k-representation, arises from momentum transfers q = 0 and q = k2−k1. Write down the two interaction terms and identify them as direct (q = 0) and exchange (q = k2 − k1) interactions. Draw the corresponding Feynman diagrams. Problem 6. Find the first correction to the temperature dependence of the chemical potential  of the degenerate ideal electron gas, assuming constant particle concentration ⟨N⟩=V . Express the result in terms of T and the zero-temperature chemical potential ◦. For the calculations the following formula (we set kB = 1) can be used: I ≡ ∫ ∞ 0 f(“)d” e(“−)=T + 1 = ∫  0 f(“)d” + 2 6 T2f′() + O(T4) (13) 3

Course: PHYS 5426 — Quantum Statistical Physics Assignment #1 Instructor: Gennady Y. Chitov Date Assigned: January 15, 2014 Due Date: January 29, 2014 Problem 1. Prove [a; f(a†)] = @f(a†) @a† (1) [a†; f(a)] = −@f(a) @a (2) for arbitrary function f of operator which admits a series expansion. The Bose creation/ annihilation operators satisfy the standard commutation relations [a; a†] ≡ aa† − a†a = 1 (3) Hint: From Eqs.(1,2) one can figure out the corresponding commutation relations for the powers of creation/annihilation operators and then prove them by the method of mathematical induction. Note that for an arbitrary operator Aˆ: @A^n @A^ = nAˆn−1. Problem 2. In the presence of a constant external force acting on a one-dimensional oscillating particle its Hamiltonian becomes that of the so-called displaced oscillator, and the Schr¨odinger equation ˆH (q) = E (q) of the problem (cf. lecture notes) can be written in terms of dimensionless variables as ( − 1 2 d2 d2 + 1 2 2 − √ 2  ) () = ” () ; (4) where q = √ ~ m! and E = ~!”. a). Write the Schr¨odinger equation (4) in terms of the creation/annihilation operators of the harmonic oscillator ( = 0)  = √1 2 (a + a†) (5) d d = √1 2 (a − a†) (6) 1 Via a linear transformation to the new creation/annihilation operators ˜a†; ˜a preserving the bosonic commutation relations for ˜a†; ˜a map the problem (4) of the displaced oscillator onto that of a simple harmonic oscillator with new operators (˜a†; ˜a). b). Find the spectrum (eigenvalues) ” (E) of the displaced oscillator. c). Write the normalized eigenstates |n⟩ of the displaced Hamiltonian (4) via a† and the vacuum state |Θ◦⟩ of the new operators, i.e. ˜a|Θ◦⟩ = 0 (7) d). As follows from the completeness of the oscillator’s eigenstates, the vacuum state of the displaced oscillator |Θ◦⟩ can be related to the simple oscillator’s vacuum |0⟩ (i.e., a|0⟩ = 0) as |Θ◦⟩ = Ω(a†)|0⟩ (8) Find (up to a normalization factor) the operator function Ω(a†) relating two vacua. Hint: in working out Eqs.(7,8), employ Eqs.(1,2). Problem 3. Prove from the standard commutation relations ([ai; a † j ]∓ = ij , etc) that ⟨0|aiaja † ka † l |0⟩ = jkil ± ikjl (9) the sign depending on the statistics. Also calculate the vacuum expectation value ⟨0|ahaiaja † ka † l a† m |0⟩. Problem 4. In the formalism of second quantization the two-particle interaction term of the Hamiltonian for spinless fermions is given by ˆ V = 1 2 ∫ ∫ dxdy ˆ †(x) ˆ †(y)V(x; y) ˆ (y) ˆ (x) (10) For the short-ranged interaction V(x; y) = V(|x−y|) ≡ V(r) = e2 exp(−r)=r find ˆ V in the momentum representation. The field operators and the creation/annihilation operators in the momentum representation are related in the usual way, i.e., ˆ †(x) = ∫ dp (2)3 a†(p)e−ipx (11) Note that the limit  → 0 recovers the Coulomb (long-ranged) interaction V(r) = e2=r. What is the Fourier transform V(q) of the Coulomb interaction? 2 Problem 5. The matrix elements of a two-particle interaction from the previous problem can be written as ⟨k3k4|V|k1k2⟩ = (2)3(k1 + k2 − k3 − k4)V(q) (12) where q ≡ k3−k1 is the momentum transfer. Show that the diagonal part of the interaction operator ˆ V found on the previous problem in the k-representation, arises from momentum transfers q = 0 and q = k2−k1. Write down the two interaction terms and identify them as direct (q = 0) and exchange (q = k2 − k1) interactions. Draw the corresponding Feynman diagrams. Problem 6. Find the first correction to the temperature dependence of the chemical potential  of the degenerate ideal electron gas, assuming constant particle concentration ⟨N⟩=V . Express the result in terms of T and the zero-temperature chemical potential ◦. For the calculations the following formula (we set kB = 1) can be used: I ≡ ∫ ∞ 0 f(“)d” e(“−)=T + 1 = ∫  0 f(“)d” + 2 6 T2f′() + O(T4) (13) 3

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EBIO/ECHM 100 – Homework #6 2015 Due Thursday October 8, 2015 1. Flow rates – in most bio- or chemical engineering problems, you will be working with flow rates rather than a singular amount of mass, moles or volume. When working with gases, a molar flow rate (mol/time) or a volumetric flow rate (volume/time) can be used in the ideal gas law. Consider a mixed gas consisting of 60% Hydrogen, 20% Nitrogen and 20% Carbon Monoxide (CO) which has a molar flow rate of 175 lbmol/min. The temperature is 200oF and pressure is 3 atm gauge. a. Calculate the individual molar flow rates of each component (mol/min) b. Calculate the total volumetric flow rate in ft3/min c. Determine the molar density of the mixed gas (mol/ft3) d. Determine the mass flow rate of the gas (g/min) e. Determine the mass fraction of hydrogen in the gas Read Section 6.5 of your textbook and/or watch the mini-lecture on statistics. That information will help you complete problems 2-4 In previous years, students in EBIO/ECHM 100 would test their ability to brew a repeatable batch of coffee. Let’s say you tried your best to brew three identical batches of coffee – you ground your own beans and measured exactly how many grounds you added to the pot. You took three samples from each batch and measured the absorbance. Representative data for the absorbance readings is given below. Batch 1 Batch 2 Batch 3 Measurement 1 0.343 0.374 0.327 Measurement 2 0.342 0.372 0.327 Measurement 3 0.371 0.375 0.328 2. For each batch of coffee (data in a vertical column), calculate the mean and standard deviation of the absorbance measurement of the three measurements taken from each batch (batch 1 has 3 absorbance measurements of .343, 0.342 and 0.371). Show at least one hand calculation on engineering paper, but you can do the rest in Excel if you wish, using the functions for average, =average(), and standard deviation, =stdev(). For example if you put data from batch 1 in column B, cells B2, B3 and B4 you could tell Excel to compute the average of those numbers by entering the equation =average(B2,B3,B4) in a neighboring cell. An alternate to way compute the average would be to type =(B2+B3+B4)/3. While you get the same answer with both methods, the second method become more cumbersome as your number of samples goes up. 3. Now, calculate the mean and standard deviation for the combination of all nine samples. 4. Why is the standard deviation calculated in #3 greater than those calculated in #2? Discuss this question in terms of experimental factors that could lead to scatter in your data (think back to the steps of making coffee and try to come up with at least 3 factors – these can address sample to sample variations or batch to batch variations). Staple the spreadsheet to the rest of your HW if using Excel. Adjust numbers so they report correct significant figures.

EBIO/ECHM 100 – Homework #6 2015 Due Thursday October 8, 2015 1. Flow rates – in most bio- or chemical engineering problems, you will be working with flow rates rather than a singular amount of mass, moles or volume. When working with gases, a molar flow rate (mol/time) or a volumetric flow rate (volume/time) can be used in the ideal gas law. Consider a mixed gas consisting of 60% Hydrogen, 20% Nitrogen and 20% Carbon Monoxide (CO) which has a molar flow rate of 175 lbmol/min. The temperature is 200oF and pressure is 3 atm gauge. a. Calculate the individual molar flow rates of each component (mol/min) b. Calculate the total volumetric flow rate in ft3/min c. Determine the molar density of the mixed gas (mol/ft3) d. Determine the mass flow rate of the gas (g/min) e. Determine the mass fraction of hydrogen in the gas Read Section 6.5 of your textbook and/or watch the mini-lecture on statistics. That information will help you complete problems 2-4 In previous years, students in EBIO/ECHM 100 would test their ability to brew a repeatable batch of coffee. Let’s say you tried your best to brew three identical batches of coffee – you ground your own beans and measured exactly how many grounds you added to the pot. You took three samples from each batch and measured the absorbance. Representative data for the absorbance readings is given below. Batch 1 Batch 2 Batch 3 Measurement 1 0.343 0.374 0.327 Measurement 2 0.342 0.372 0.327 Measurement 3 0.371 0.375 0.328 2. For each batch of coffee (data in a vertical column), calculate the mean and standard deviation of the absorbance measurement of the three measurements taken from each batch (batch 1 has 3 absorbance measurements of .343, 0.342 and 0.371). Show at least one hand calculation on engineering paper, but you can do the rest in Excel if you wish, using the functions for average, =average(), and standard deviation, =stdev(). For example if you put data from batch 1 in column B, cells B2, B3 and B4 you could tell Excel to compute the average of those numbers by entering the equation =average(B2,B3,B4) in a neighboring cell. An alternate to way compute the average would be to type =(B2+B3+B4)/3. While you get the same answer with both methods, the second method become more cumbersome as your number of samples goes up. 3. Now, calculate the mean and standard deviation for the combination of all nine samples. 4. Why is the standard deviation calculated in #3 greater than those calculated in #2? Discuss this question in terms of experimental factors that could lead to scatter in your data (think back to the steps of making coffee and try to come up with at least 3 factors – these can address sample to sample variations or batch to batch variations). Staple the spreadsheet to the rest of your HW if using Excel. Adjust numbers so they report correct significant figures.

Materials Chemistry for Engineers 1. In the van der Waals corrections to the Ideal Gas Law: (P + a/V2)(V – b) = nRT (a) What do a and b correct for from the Ideal Gas Law? (b) How would one determine a and b experimentally? Describe a proposed experiment and data analysis method for your experiment. 2. (a) What are the assumptions of the Ideal Gas Law? How did van der Waal modify these assumptions to come up with his equation of state? (b) what is an equation of state, in general? Describe in your own words. 3. Given the following data: Material a b_____ (l2.atm/mole2) (l/mole) N2 1.39 0.03913 NH3 4.17 0.03107 Aniline 26.50 0.1369 Benzene 18.00 0.1154 (a) Plot P vs. T for each gas using the van der Waals equation of state. Assume that you have a 1 liter volume and 1 mole of gas and plot the temperature on the x-axis from room temperature to 1400 K (pressures should range from about 0 atm to about 120 atm, depending on the gas). Plot the Ideal Gas Law with the other data on one plot. Are the interactions between molecules attractive or repulsive at low temperature? How do you know? What is happening with the gases at high temperature? Is one of the gases different from the others at 1400 K? (b) Discuss the nature of the intermolecular interaction that creates the deviation from ideality for each material. Are there induced dipole-induced dipole interactions, iondipole interactions, etc. for each of the different gases? Draw their chemical structures. 4. Ethane (CH3CH3) and fluoromethane (CH3F) have the same number of electrons and are essentially the same size. However, ethane has a boiling point of 184.5 K and fluoromethane has a boiling point of 194.7 K. Explain this 10 degree difference in boiling point in terms of the van der Waals forces present. Bonus, what is the size of each molecule? Show your calculation/sources.

Materials Chemistry for Engineers 1. In the van der Waals corrections to the Ideal Gas Law: (P + a/V2)(V – b) = nRT (a) What do a and b correct for from the Ideal Gas Law? (b) How would one determine a and b experimentally? Describe a proposed experiment and data analysis method for your experiment. 2. (a) What are the assumptions of the Ideal Gas Law? How did van der Waal modify these assumptions to come up with his equation of state? (b) what is an equation of state, in general? Describe in your own words. 3. Given the following data: Material a b_____ (l2.atm/mole2) (l/mole) N2 1.39 0.03913 NH3 4.17 0.03107 Aniline 26.50 0.1369 Benzene 18.00 0.1154 (a) Plot P vs. T for each gas using the van der Waals equation of state. Assume that you have a 1 liter volume and 1 mole of gas and plot the temperature on the x-axis from room temperature to 1400 K (pressures should range from about 0 atm to about 120 atm, depending on the gas). Plot the Ideal Gas Law with the other data on one plot. Are the interactions between molecules attractive or repulsive at low temperature? How do you know? What is happening with the gases at high temperature? Is one of the gases different from the others at 1400 K? (b) Discuss the nature of the intermolecular interaction that creates the deviation from ideality for each material. Are there induced dipole-induced dipole interactions, iondipole interactions, etc. for each of the different gases? Draw their chemical structures. 4. Ethane (CH3CH3) and fluoromethane (CH3F) have the same number of electrons and are essentially the same size. However, ethane has a boiling point of 184.5 K and fluoromethane has a boiling point of 194.7 K. Explain this 10 degree difference in boiling point in terms of the van der Waals forces present. Bonus, what is the size of each molecule? Show your calculation/sources.

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MAE 241 – Homework 1 Page 1 of 3 MAE 241 – Spring 2019 – Homework 1 Administered 1/11/2019 – Due 11PM, Sunday 1/19/2019 to Gradescope Problem 1 – Review of units The Phoenix with a mass of 400 kg was a spacecraft used for exploration of Mars. Determine the weight of the Phoenix, in N, for the following situations: a. On the surface of Mars where the acceleration of gravity is 3.73 m/s2. b. On Earth where the acceleration of gravity is 9.81 m/s2. Problem 2 – review of chemistry A closed system consists of 0.4 kmol of octane (C8H18) occupying a volume of 2 m3. Determine: a. The weight of the system, in N. b. The molar-based specific volume, in m3/kmol. c. The mass-based specific volume, and m3/kg. Let g = 9.81 m/s2. Problem 3 – review of chemistry A closed vessel having a volume of 2.0 liter holds 2.0 x 1022 molecules of carbon dioxide gas. Determine: a. The number of moles, in kmol b. The mass of CO2 present, in kg and. c. The molar specific volume, in m3/kmol d. The specific volume of the CO2, in m3/kg. Hint: use Table A-1 of your textbook for molecular mass. Problem 4 – Quasistatic Equilibrium The figure below shows a gas contained in a vertical piston–cylinder assembly. A vertical shaft whose cross-sectional area is 0.8 cm2 is attached to the top of the piston. Determine the magnitude, F, of the force acting on the shaft, in N, required if the gas pressure is 300 kPa. The masses of the piston and attached shaft are 30 kg and 0.5 kg, respectively. The piston diameter is D = 10 cm. The local atmospheric pressure is 100 kPa. The piston moves smoothly and slowly at constant velocity in the cylinder and g = 9.81 m/s2. MAE 241 – Homework 1 Page 2 of 3 Problem 5– Measurement of Pressure a. For the tank shown in the figure below derive an expression to compute the Length, L, of the column of fluid as a function of density of the fluid in the manometer, 𝜌, and the atmospheric pressure, Patm. b. Determine the value of L when the manometer liquid is water and the tank contains a gas at a pressure of 1.5 bar. A barometer indicates the local atmospheric pressure is 750 mmHg. You also know that the density of water is 997 kg/m3 and the density of mercury is 13.59 g/cm3. Let g = 9.81 m/s2. Problem 6 – Gage pressure measurement A pressure gage mounted at the inlet to an air compressor indicates that the gage pressure is 60.0 kPa. The absolute pressure of the at the exit of the compressor is 5.5 times the absolute pressure at the inlet. The atmospheric pressure is 1.01 bar. What is the absolute pressure of the gas at the inlet and what is the gage pressure of the gas at the exit? Problem 7 – Unit conversions Perform the following unit conversions. Please do not use an on-line unit converter since this problem is given to you as practice in preparation for what you need to be proficient in: a. 170.8 in3 to L b. 1089.438 ft-lbf to kJ c. 140.0 hp to kW d. 1400.0 lb/h to kg/s e. 41.1488 lbf/in2 to kPa f. 3500.0 ft3/min to m3/s g. 105.0 mile/h to km/h h. 1.4 ton (=2000 lbf) to N Problem 8 – Newton’s second law Estimate the magnitude of the force, in lbf, exerted on a 20-lb goose in a collision of duration 1.5 x 10−3 s with an airplane taking off at 150 miles/h. Assume the bird’s velocity is zero before the collision. MAE 241 – Homework 1 Page 3 of 3 Problem 9 – Temperature conversions On January 3, 2019, in Flagstaff, AZ, the lowest temperature was 14oF at 5 AM and the highest 44oF at 4 PM. a. Express these temperatures in oR, K and oC. b. Determine the temperature change in oF, oR, oC and K from morning low to afternoon high. c. What is the relationship between the temperature changes in oF and oR? d. What is the relationship between the temperature changes in oC and K? Problem 10 – Ideal gas law application A closed deformable system consisting of 2 lb of air undergoes a process during which the relation between pressure and volume is defined by the mathematical expression PVn = Constant. The process begins with P1 = 20 lbf/in2, V1 = 15 ft3 and ends with P2 = 100 lbf/in2. The value of n = 1.3. Determine: a. The final volume, V2, in ft3 b. The specific volume at states 1 and 2, in ft3/lb. c. Use the ideal gas law as you learned in your chemistry course to determine the initial temperature, in °F. Hint: if you need the molecular mass of air please use Table A-1 or A-1E.

MAE 241 – Homework 1 Page 1 of 3 MAE 241 – Spring 2019 – Homework 1 Administered 1/11/2019 – Due 11PM, Sunday 1/19/2019 to Gradescope Problem 1 – Review of units The Phoenix with a mass of 400 kg was a spacecraft used for exploration of Mars. Determine the weight of the Phoenix, in N, for the following situations: a. On the surface of Mars where the acceleration of gravity is 3.73 m/s2. b. On Earth where the acceleration of gravity is 9.81 m/s2. Problem 2 – review of chemistry A closed system consists of 0.4 kmol of octane (C8H18) occupying a volume of 2 m3. Determine: a. The weight of the system, in N. b. The molar-based specific volume, in m3/kmol. c. The mass-based specific volume, and m3/kg. Let g = 9.81 m/s2. Problem 3 – review of chemistry A closed vessel having a volume of 2.0 liter holds 2.0 x 1022 molecules of carbon dioxide gas. Determine: a. The number of moles, in kmol b. The mass of CO2 present, in kg and. c. The molar specific volume, in m3/kmol d. The specific volume of the CO2, in m3/kg. Hint: use Table A-1 of your textbook for molecular mass. Problem 4 – Quasistatic Equilibrium The figure below shows a gas contained in a vertical piston–cylinder assembly. A vertical shaft whose cross-sectional area is 0.8 cm2 is attached to the top of the piston. Determine the magnitude, F, of the force acting on the shaft, in N, required if the gas pressure is 300 kPa. The masses of the piston and attached shaft are 30 kg and 0.5 kg, respectively. The piston diameter is D = 10 cm. The local atmospheric pressure is 100 kPa. The piston moves smoothly and slowly at constant velocity in the cylinder and g = 9.81 m/s2. MAE 241 – Homework 1 Page 2 of 3 Problem 5– Measurement of Pressure a. For the tank shown in the figure below derive an expression to compute the Length, L, of the column of fluid as a function of density of the fluid in the manometer, 𝜌, and the atmospheric pressure, Patm. b. Determine the value of L when the manometer liquid is water and the tank contains a gas at a pressure of 1.5 bar. A barometer indicates the local atmospheric pressure is 750 mmHg. You also know that the density of water is 997 kg/m3 and the density of mercury is 13.59 g/cm3. Let g = 9.81 m/s2. Problem 6 – Gage pressure measurement A pressure gage mounted at the inlet to an air compressor indicates that the gage pressure is 60.0 kPa. The absolute pressure of the at the exit of the compressor is 5.5 times the absolute pressure at the inlet. The atmospheric pressure is 1.01 bar. What is the absolute pressure of the gas at the inlet and what is the gage pressure of the gas at the exit? Problem 7 – Unit conversions Perform the following unit conversions. Please do not use an on-line unit converter since this problem is given to you as practice in preparation for what you need to be proficient in: a. 170.8 in3 to L b. 1089.438 ft-lbf to kJ c. 140.0 hp to kW d. 1400.0 lb/h to kg/s e. 41.1488 lbf/in2 to kPa f. 3500.0 ft3/min to m3/s g. 105.0 mile/h to km/h h. 1.4 ton (=2000 lbf) to N Problem 8 – Newton’s second law Estimate the magnitude of the force, in lbf, exerted on a 20-lb goose in a collision of duration 1.5 x 10−3 s with an airplane taking off at 150 miles/h. Assume the bird’s velocity is zero before the collision. MAE 241 – Homework 1 Page 3 of 3 Problem 9 – Temperature conversions On January 3, 2019, in Flagstaff, AZ, the lowest temperature was 14oF at 5 AM and the highest 44oF at 4 PM. a. Express these temperatures in oR, K and oC. b. Determine the temperature change in oF, oR, oC and K from morning low to afternoon high. c. What is the relationship between the temperature changes in oF and oR? d. What is the relationship between the temperature changes in oC and K? Problem 10 – Ideal gas law application A closed deformable system consisting of 2 lb of air undergoes a process during which the relation between pressure and volume is defined by the mathematical expression PVn = Constant. The process begins with P1 = 20 lbf/in2, V1 = 15 ft3 and ends with P2 = 100 lbf/in2. The value of n = 1.3. Determine: a. The final volume, V2, in ft3 b. The specific volume at states 1 and 2, in ft3/lb. c. Use the ideal gas law as you learned in your chemistry course to determine the initial temperature, in °F. Hint: if you need the molecular mass of air please use Table A-1 or A-1E.

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A real gas will behave most like an ideal gas under conditions of __________. A) high temperature and high pressure B) high temperature and low pressure C) low temperature and high pressure D) low temperature and low pressure E) STP

A real gas will behave most like an ideal gas under conditions of __________. A) high temperature and high pressure B) high temperature and low pressure C) low temperature and high pressure D) low temperature and low pressure E) STP

B) high temperature and low pressure