Lecture 1: Introduction to analytical methods CHEE 3321 Reading: none 1 Learning objectives (course) 1. Develop mathematical skills for analyzing problems in chemical engineering 2. Apply skills to solve mathematical models of chemical engineering systems 3. Develop a systematic method for modeling chemical engineering systems 2 Learning objectives (lecture) 1. Explain why mathematics is important for chemical engineers 2. Present a general method for generating engineering models 3 Chemical engineers and math Driving question: why do chemical engineers need mathematics? • Describe relationships between quantities in chemical engineering processes (mass, heat, momentum balances) - Algebraic relationships • Examples: - Ordinary differential equations (ODEs): linear, nonlinear • Examples: - Partial differential equations (PDEs): linear, nonlinear • Examples: • Solve and analyze mathematical model to predict system behavior under different conditions 4 Method for generating models 1 1. Describe the problem a. Identify relevant physical and chemical phenomena b. Identify knowns and unknowns: dependent (state) variables, independent variables, physical properties, speci ed operating conditions c. Draw a clear schematic 2. Construct governing equations (microscopic and/or macroscopic) Balance equations: a. Mass balance b. Momentum balance 5 fi c. Energy balance Method for generating models 2 3. Construct governing equations (microscopic and/or macroscopic) • Algebraic balances: • Differential balances: • Steady-state balances: • Identify boundary and/or initial conditions 4. Apply constitutive relations (if any) for uxes, stresses, etc. 5. Apply simplifying assumptions to make the model solvable How do we know how to make assumptions? • Problem statement • Experience in solving similar problems • Intuition (to be developed!) fl 6 Method for generating models 3 6. Solve the model 7. Analyze physical implications of the solution 8. Revise assumptions • Better match experimental data • Assess relevance of assumptions Additional notes: • A complete description is rarely possible • Simple models provide useful insight • Making use of models requires mathematical tools to solve them • Focus of this course: developing these mathematical tools - Detailed physical concepts are not the focus (developed in other classes) - Use simple physics to learn how to analyze model predictions 7 Simple schematic model Balance equation: Schematic drawing: Simple examples • Mass balance: • Momentum balance: • Energy balance: 8 Representative mathematical models First-order reactions in a batch reactor Transient continuous stirred-tank reactor (CSTR) Steady-state conduction Transient conduction Conduction in heat exchange n Conduction with heat generation Reaction and diffusion in a porous catalyst Multiple reactions in a batch reactor 9 fi • • • • • • • • First order liquid phase batch reactor 1 Consider a rst-order liquid phase batch reactor (constant volume) Rate of disappearance of A: Reaction: k: rst-order rate coef cient [1/s] CA: concentration of species A [mol/L] units: Mass balance over time (assume: constant volume reactor): fi fi fi 10 First order liquid phase batch reactor 2 Rewrite as a differential equation: t!0 : Note: if the volume V is constant, this is equivalent to: Initial condition: ( Solve: separate the variables ( fi 11 First order liquid phase batch reactor 3 Apply initial condition: Rearrange solution: Physical dependence: 12 Steady-state conduction in a slab 1 Physical examples: heat loss through insulation; heat transfer through a thin metal or polymer lm Assume that heat ux only depends on the variable z Energy shell balance: q: heat ux [J / m2 s] z!0: / z, Apply constitutive equation: Fourier’s law (Empirical relationship between temperature gradient and heat ux) k: thermal conductivity [J/ m K s] fi fl fl fl 13 Steady state conduction in a slab 2 Combine equations and assume constant k: Boundary conditions: 1. 2. 14 Conduction in a solid sphere 1 Physical examples: rapid quench of a metal ingot; cooling of droplets Energy shell balance: accumulation = in - out + production k: thermal conductivity [J/ m K s] ρ: density [kg/m3] Cp: speci c heat per unit mass [J/ kg K] Boundary conditions: 1. 2. 3. fi 15 Conduction in a solid sphere 2 De ne the thermal diffusivity: Rewrite equation: (linear partial differential equation) fi 16 Conduction in a heat n 1 Physical examples: heat exchanges; ns (“osteoderms”) in stegosaurs (see Farlow et al., Science 192, 1123-1125 (1976)!) Assume that the n is thin, wide, and long: Boundary condition: Newton’s law of cooling (describes heat loss from a solid surface into a liquid) h: heat transfer coef cient [J/m2 s K] fi fi fi fi 17 Conduction in a heat n 2 Energy balance: accumulation = in - out + production Apply Fourier’s law (along n) and Newton’s law of cooling (surface losses): / x, x!0: Boundary conditions: 1. 2. fi fi 18