Transcript Slides
ME 475/675 Introduction to Combustion Lecture 26 Plug flow reactor, Problem X5:Constant area and pressure equations Announcements • Midterm 2, November 13, 2015 (2 weeks) • HW 8a, Due now, Example 6.2 Equations • HW 8b, Due Monday, Numerical solution • HW 9 Due Monday, 10/26/15, Problem X5 • Friday, 10/30/2015, Holiday • Student, Alumni, and Faculty Research Opportunities at ORNL • hands-on research in a-real world setting with award winning scientists • Visit: http://www.orau.org/ornl[orau.org] • Or contact: ORNL Education Programs at [email protected], or Leslie Fox at (865) 576-3427 Broader Impact Assignment • Two important ABET Student Learning Objectives: • Students will show: • A recognition of a need for, and an ability to engage in, life long learning (graduate school, continuing education, short courses, technical training, self instruction by reading articles or textbook) • A knowledge of contemporary issues • Two choices, Both due November 6, 2015 • Attend and write a two paragraph summary of this seminar: • Used Nuclear Fuel: Storage, Transportation, and Disposal – Technical, Political and Other Issues • John Wagner, Director, Reactor & Nuclear Systems Division, Oak Ridge National Laboratory • Noon, November 2, 2015, DMS 102 • Hosted by UNR American Nuclear Society’s Student Chapter • President: Kodi Summers [email protected] • Read and write a two paragraph summary of this article • Dependence of Fire Time of Concern on Location of a One-assembly Transport Packages Plug-Flow Reactors • Assumptions • Quasi-one dimensional • All quantities are ≠ 𝑓𝑛(𝑟, 𝜃) 𝑑 • Steady state, =0 𝑑𝑡 • No-viscosity 𝜇 = 0 • Axial turbulent and molecular diffusion are small compared to advection (high enough axial velocity) • If velocity is “constant” then pressure is “constant” • Integrate to find 𝑇 𝑥 , 𝑌𝑖 𝑥 , 𝜌 𝑥 • At each location also need to calculate 𝑚 𝜌 𝑥 𝐴(𝑥) 𝜌 𝑥 𝑅𝑢 𝑇(𝑥) 𝑀𝑊𝑚𝑖𝑥 • 𝑣𝑥 𝑥 = • 𝑃 𝑥 = • Like the transient constant-pressure reactor, but varies with location instead of time. Stationary Reaction Zone What do we expect? (flow from left to right) • Reaction take place is a “small” region Problem X5 (homework) dx 𝑚 m x h Yi h + (dh/dx)dx Yi + (dYi/dx)dx 𝑄" (𝑥) • Consider a constant-area A 𝑑𝐴 𝑑𝑥 " = 0 plug flow reactor. It has an axially-varying heat flux 𝑄 𝑥 applied to the wall, mass flow rate 𝑚 small). 𝑘𝑔 𝑠 𝑊 𝑚2 , and operates a constant pressure P (velocity variations are • The following mass-based reaction is taking place within the reactor with a stoichiometric air/fuel ratio of ν: • 1 kg F + ν 𝑘𝑔 𝑂𝑥 → 1 + ν 𝑘𝑔 𝑃𝑟; • Assume • • • • 𝜔𝐹 𝑥 = 𝑑𝐹 𝑑𝑡 = −𝐴𝐹 𝑒 𝐸 𝑅 − 𝑎 𝑢 𝑇 𝑂𝑥 𝑚 𝐹 𝑛 ν= 𝐴 𝐹 𝑆𝑡 𝑣𝑥2 (2 The mass flow kinetic energy per mass is much less than its enthalpy ≪ ℎ) The fuel F, Oxidizer Ox, and products Pr, have the same 𝑀𝑊 and 𝑐𝑝 (and 𝑐𝑝 ≠ 𝑓𝑛(𝑇)) 𝑜 The oxidizer and product heat of formation are zero, and that of the fuel is ℎ𝑓,𝐹 The inlet equivalence ratio and temperature are Φ𝑖𝑛 and 𝑇𝑖𝑛 • Use conservation of species and energy to find equations that can be used to find the axial variation of 𝑌𝑖 𝑥 , 𝑇 𝑥 , 𝜌 𝑥 , 𝑣𝑥 𝑥 End 2015 General Plug-Flow Reactor • What’s different from problem X5? • Area A x , 𝑑𝐴 𝑑𝑥 ≠0 • Flow kinetic energy is not small compared to enthalpy • 𝑣𝑥2 2 ≪ℎ • Species can have different, temperature-dependent properties Conservation Laws • Mass • 𝑚 = 𝜌𝑣𝑥 𝐴 • 𝑑 𝑑𝑥 𝜌𝑣𝑥 𝐴 = 0 • Momentum • 𝑑𝑃 𝑑𝑥 + 𝑑𝑣𝑥 𝜌𝑣𝑥 𝑑𝑥 =0 • Energy (including kinetic) • 𝑑 𝑣2 ℎ+ 2𝑥 𝑑𝑥 + 𝑄" 𝒫 𝑚 =0 • Species • 𝑑𝑌𝑖 𝑑𝑥 = 𝜔𝑖 𝑀𝑊𝑖 𝐴 ,𝑖 𝜌𝑣𝑥 = 1,2, … , 𝑀 Manipulate …. • Use • 𝜔𝑖 = 𝑓𝑛 𝑌𝑖 , 𝑇, 𝑃 • 𝑃 = 𝜌𝑅𝑇; 𝑅 = • 𝑣𝑥 = 𝑅𝑢 1 ; 𝑀𝑊𝑚𝑖𝑥 𝑀𝑊𝑚𝑖𝑥 𝑚 𝜌𝐴 𝑌𝑖 𝑀𝑊𝑖 = • Need 𝜌 𝑥 and 𝑇 𝑥 • Assume 𝑄 " 𝑥 , 𝐴 𝑥 and 𝑚 are given • Find … (page 209) • 𝑑𝑌𝑖 𝑑𝑥 • 𝑑𝑇 𝑑𝑥 • 𝑑𝜌 𝑑𝑥 = = = 𝜔𝑖 𝑀𝑊𝑖 𝐴 ,𝑖 𝜌𝑣𝑥 𝑣𝑥2 𝑑𝜌 = 1,2, … , 𝑀 𝑣𝑥2 𝑑𝐴 + 𝜌𝑐𝑃 𝑑𝑥 𝑐𝑃 𝐴 𝑑𝑥 𝑅𝑢 1−𝑐 𝑀𝑊 𝑃 𝑚𝑖𝑥 − 𝜔𝑖 𝑀𝑊𝑖 ℎ𝑖 𝜌𝑣𝑥 𝑐𝑃 1 𝑑𝐴 𝜌2 𝑣𝑥2 𝐴 𝑑𝑥 − 𝑄"𝒫 𝑚𝑐𝑃 𝜌𝑅𝑢 +𝑣 𝑐 𝑀𝑊 𝑥 𝑃 𝑚𝑖𝑥 𝑃 𝑣2 1+𝑐 𝑥𝑇 𝑃 𝑀𝑊𝑚𝑖𝑥 𝑀𝑊𝑖 𝜔𝑖 ℎ𝑖 − 𝑀𝑊 𝑐𝑃 𝑇 𝑖 −𝜌𝑣𝑥2 𝜌𝑅𝑢 𝑄" 𝒫 +𝑣 𝐴𝑐 𝑀𝑊 𝑥 𝑃 𝑚𝑖𝑥 Problem 6.11 (Homework) • Develop a plug-flow-reactor model using the same chemistry and thermodynamics as in Example 6.1. Assume the reactor is adiabatic. Use the model to: A. Determine the mass flow rate such that the reaction is 99 percent complete in a flow length of 10 cm for 𝑇𝑖𝑛 = 1000𝐾, 𝑃𝑖𝑛 = 0.2 𝑎𝑡𝑚, and Φ𝑖𝑛 = 0.2. The circular duct has a diameter of 3 cm. B. Explore the effects of 𝑇𝑖𝑛 , 𝑃𝑖𝑛 , and Φ𝑖𝑛 on the flow length required for 99 percent complete combustion using the flow rate determined in Part A. • Constant volume, constant pressure, well-stirred, plug-flow? Excel Solution Method • Starting Point • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH.475.675.Combustion/Prob.6.11.start.xlsx • Pay attention to • Integration step size • Avoiding raising negative numbers to a non-integer power mdot phi kg/s 0.00125 dx m 1 x m 0.0001 Wf kmole/m3s -0.002826921 -0.002832083 mdot phi kg/s 0.00125 dx m 1 x m 0.0001 Yfuel kg/kg 0 0.058824 0.0001 0.058819 Wox kmole/m3s -0.045230736 -0.045313333 Yfuel kg/kg 0 0.058824 0.0001 0.058819 Yox Ypr kg/kg kg/kg 0.941176 0 0.941102 7.88E-05 Yox Ypr kg/kg kg/kg 0.941176 0 0.941102 7.88E-05 rho T P u kg/m3 K Pa m/s 0.070669 1000 20260 25.02362 0.070658 1000.148 20259.86 25.0275 Wf kmole/m3s -0.002826921 -0.002832083 Wpr d[Yfuel]/dx d[Yox]/dx d[Ypr]/dx d[rho]/dx dT/dx kmol/m3s 0.048057657 -0.046359 -0.74174399 0.788102987 -0.109386259 1479.404 0.048145416 -0.0464437 -0.7430985 0.789542155 -0.109552871 1482.092 rho T P u kg/m3 K Pa m/s 0.070669 1000 20260 25.02362 0.070658 1000.148 20259.86 25.0275 Wf kmole/m3s -0.002826921 -0.002832083 Wox kmole/m3s -0.045230736 -0.045313333 completion 1-Yf(0.1m)/Yf(0) 0.990606979 Wpr d[Yfuel]/dx d[Yox]/dx d[Ypr]/dx d[rho]/dx dT/dx kmol/m3s 0.048057657 -0.046359 -0.74174399 0.788102987 -0.109386259 1479.404 0.048145416 -0.0464437 -0.7430985 0.789542155 -0.109552871 1482.092 completion 1-Yf(0.1m)/Yf(0) 0.990606979