A Practical Guide to Numerical Problem Solving in Chemical and Biochemical Engineering with POLYMATH, Excel, and MATLAB
Problem Solving in Chemical and Biochemical Engineering
Introduction
Chemical and biochemical engineering are two branches of engineering that deal with the transformation of matter and energy using chemical reactions, biological processes, and physical operations. Chemical engineers design, operate, and optimize processes that produce, refine, or utilize chemicals, fuels, materials, pharmaceuticals, food, or other products. Biochemical engineers apply the principles of chemical engineering to biological systems, such as cells, enzymes, microorganisms, or biomolecules, to produce bioproducts, biofuels, biopharmaceuticals, or bioremediation agents.
Problem Solving in Chemical and Biochemical Engineering
Problem solving is an essential skill for chemical and biochemical engineers, as they face various types of problems in their daily work. These problems can range from simple calculations to complex design projects, from theoretical analysis to experimental validation, from individual tasks to collaborative efforts. Problem solving requires not only technical knowledge and expertise, but also creativity, logic, intuition, communication, and teamwork.
In this article, we will discuss what problem solving is, why it is important in chemical and biochemical engineering, what are some common types of problems in these fields, what is the general problem solving process, what are some problem solving skills and strategies, and what are some examples of problem solving in chemical and biochemical engineering.
Problem Solving Process
The problem solving process is a systematic approach to finding solutions to problems. It can be applied to any type of problem in any field of engineering. The problem solving process typically consists of four main steps: define the problem, analyze the problem, generate possible solutions, and implement the best solution.
Define the problem
The first step in problem solving is to define the problem clearly and precisely. This involves identifying the goal or objective of the problem, the constraints or limitations of the problem, the assumptions or simplifications of the problem, and the criteria or standards for evaluating the solutions. Defining the problem helps to focus on the essential aspects of the problem and avoid irrelevant or unnecessary information.
Analyze the problem
The second step in problem solving is to analyze the problem thoroughly and systematically. This involves identifying the relevant variables and parameters of the problem, applying the appropriate laws and principles of chemistry, biology, physics, mathematics, or engineering to the problem, using mathematical models and tools to represent and manipulate the problem mathematically or computationally.
Identify the relevant variables and parameters
The variables and parameters are the quantities that describe or influence the problem. They can be classified into two types: independent variables and dependent variables. Independent variables are the quantities that can be controlled or manipulated by the engineer, such as inputs, outputs, design variables, or operating conditions. Dependent variables are the quantities that depend on the independent variables, such as performance indicators, product quality, or process efficiency.
Apply the appropriate laws and principles
The laws and principles are the fundamental concepts or rules that govern the behavior of matter and energy in chemical and biochemical systems. They can be derived from chemistry, biology, physics, mathematics, or engineering disciplines. Some examples of laws and principles are the conservation of mass, energy, and momentum, the ideal gas law, the equilibrium constant, the rate law, the mass transfer coefficient, the Michaelis-Menten equation, the Bernoulli equation, or the Navier-Stokes equation.
Use mathematical models and tools
The mathematical models and tools are the methods or techniques that help to express and solve the problem mathematically or computationally. They can be classified into two types: analytical models and numerical models. Analytical models are mathematical expressions or equations that can be solved exactly or approximately by using algebra, calculus, differential equations, linear algebra, or other mathematical methods. Numerical models are mathematical expressions or equations that can be solved numerically by using computers, software, algorithms, or numerical methods.
Generate possible solutions
The third step in problem solving is to generate possible solutions to the problem. This involves using creativity and intuition to come up with different ideas or alternatives for solving the problem, evaluating the feasibility and effectiveness of each solution by using quantitative or qualitative methods, comparing and ranking the solutions by using the criteria or standards defined in the first step.
Use creativity and intuition
Creativity and intuition are the abilities to produce novel and original ideas or solutions that are not obvious or conventional. They can be enhanced by using brainstorming, mind mapping, lateral thinking, analogy, metaphor, or other creative techniques. Creativity and intuition help to overcome mental blocks, explore new possibilities, and find innovative solutions.
Evaluate the feasibility and effectiveness of each solution
Feasibility and effectiveness are the measures of how well a solution meets the goal or objective of the problem and how realistic or practical a solution is. They can be assessed by using quantitative methods such as calculations, simulations, experiments, or optimization techniques; or qualitative methods such as logic, reasoning, judgment, experience, or intuition. Feasibility and effectiveness help to eliminate infeasible or ineffective solutions and select promising solutions.
Implement the best solution
The fourth step in problem solving is to implement the best solution to the problem. This involves testing and validating the solution by using experimental data, theoretical analysis, or simulation results; communicating and documenting the solution by using reports, presentations, diagrams, graphs, tables, or equations; applying and transferring the solution to other similar problems or situations; improving and refining the solution by using feedbacks, reviews, or modifications.
Test and validate the solution
Testing and validating are the processes of verifying and confirming that a solution works as expected and meets the goal or objective of the problem. They can be done by using experimental data from laboratory tests, pilot plants, or industrial plants; theoretical analysis from mathematical models or analytical solutions; simulation results from numerical models or computational tools. Testing and validating help to ensure the accuracy and reliability of a solution.
Communicate and document the solution
Communicating and documenting are the processes of presenting and explaining a solution to others clearly and concisely. They can be done by using reports that describe the problem statement, problem analysis, possible solutions, best solution, and conclusion; presentations that summarize the main points of a report and use visual aids such as diagrams, graphs, tables, or equations; or other forms of communication such as oral discussions, emails, or memos. Communicating and documenting help to share and disseminate a solution to others.
Problem Solving Skills and Strategies
Problem solving skills and strategies are the abilities and methods that help to solve problems effectively and efficiently. They can be developed and improved by practicing and learning from different types of problems in different fields of engineering. Some examples of problem solving skills and strategies are:
Critical thinking and analytical skills
Critical thinking and analytical skills are the abilities to analyze, evaluate, and synthesize information logically and systematically. They help to identify, understand, and solve problems rationally and objectively.
Logical reasoning Numerical and computational skills
Numerical and computational skills are the abilities to use mathematical models and tools to represent and solve problems numerically or computationally. They help to simplify, manipulate, and optimize problems mathematically or algorithmically.
Communication and teamwork skills
Communication and teamwork skills are the abilities to communicate and collaborate with others effectively and efficiently. They help to exchange ideas, information, and feedbacks with others; coordinate tasks and responsibilities with others; and resolve conflicts and disagreements with others.
Creativity and innovation skills
Creativity and innovation skills are the abilities to produce novel and original ideas or solutions that are not obvious or conventional. They help to overcome mental blocks, explore new possibilities, and find innovative solutions.
Problem Solving Examples
In this section, we will present two examples of problem solving in chemical and biochemical engineering. The first example is about designing a bioreactor for ethanol production. The second example is about optimizing a distillation column for separating a mixture of hydrocarbons.
Example 1: Designing a bioreactor for ethanol production
Problem statement
The problem is to design a bioreactor for ethanol production from glucose using yeast cells. The goal is to maximize the ethanol yield and productivity. The constraints are the availability of glucose, oxygen, and yeast cells; the operating temperature and pressure; and the reactor volume and cost. The assumptions are that the reaction is carried out in a batch mode; the reaction kinetics follow the Monod model; the reaction stoichiometry is given by C6H12O6 + 2O2 -> 2C2H5OH + 2CO2 + 2H2O; and the reaction rate is limited by glucose or oxygen availability. The criteria are the ethanol concentration, yield, and productivity at the end of the batch.
Problem analysis
The problem analysis involves identifying the relevant variables and parameters; applying the appropriate laws and principles; and using mathematical models and tools.
The relevant variables and parameters are: - Independent variables: glucose concentration (Cg), oxygen concentration (Co), yeast concentration (Cx), ethanol concentration (Ce), reactor volume (V), operating time (t), operating temperature (T), operating pressure (P). - Dependent variables: ethanol yield (Y), ethanol productivity (P). The appropriate laws and principles are: - Conservation of mass: d(Cg*V)/dt = -r1*V; d(Co*V)/dt = -r2*V; d(Cx*V)/dt = r3*V; d(Ce*V)/dt = r4*V - Monod model: r1 = mu1*Cx*Cg/(Ks1+Cg); r2 = mu2*Cx*Co/(Ks2+Co); r3 = mu3*Cx*Cg/(Ks3+Cg); r4 = mu4*Cx*Cg/(Ks4+Cg) - Reaction stoichiometry: r1 = r4/2; r2 = r4/2; r3 = -r4/10 The mathematical models and tools are: - Analytical model: Cg(t) = Cg0*exp(-mu1*t/(Yxg+mu1)); Co(t) = Co0*exp(-mu2*t/(Yxo+mu2)); Cx(t) = Cx0*exp(mu1*t/(Yxg+mu1)); Ce(t) = Ce0 + Yeg*(Cg0-Cg(t)) - Numerical model: Cg(t+dt) = Cg(t) - dt*r1; Co(t+dt) = Co(t) - dt*r2; Cx(t+dt) = Cx(t) + dt*r3; Ce(t+dt) = Ce(t) + dt*r4 - Optimization tool: max Y,P subject to Cg >= 0, Co >= 0, Cx >= 0, Ce <= Cemax Possible solutions
The possible solutions involve using creativity and intuition to generate different ideas or alternatives for solving the problem; evaluating the feasibility and effectiveness of each solution by using quantitative or qualitative methods; comparing and ranking the solutions by using the criteria defined in the first step.
Some possible solutions are: - Solution 1: Use a stirred tank reactor with a constant volume and a constant temperature and pressure; adjust the initial glucose, oxygen, and yeast concentrations to optimize the ethanol yield and productivity; terminate the batch when the ethanol concentration reaches the maximum value. - Solution 2: Use a fed-batch reactor with a variable volume and a constant temperature and pressure; feed glucose and oxygen continuously or intermittently to maintain a high ethanol yield and productivity; terminate the batch when the reactor volume reaches the maximum value. - Solution 3: Use a continuous stirred tank reactor with a constant volume and a constant temperature and pressure; adjust the feed glucose, oxygen, and yeast concentrations and the dilution rate to optimize the ethanol yield and productivity; operate the reactor at steady state. The feasibility and effectiveness of each solution can be evaluated by using calculations, simulations, experiments, or optimization techniques. For example, using the analytical model, the ethanol yield and productivity for solution 1 can be calculated as: - Y = Ce(t)/Cg0 = Yeg*(1-exp(-mu1*t/(Yxg+mu1))) - P = Ce(t)/t = Yeg*Cg0*mu1/(Yxg+mu1)*(1-exp(-mu1*t/(Yxg+mu1)))/t The optimal values of Y and P can be obtained by differentiating Y and P with respect to t and setting them to zero. The optimal values of Cg0, Co0, Cx0, t, T, and P can be obtained by using an optimization tool such as Excel Solver or MATLAB Optimization Toolbox. The solutions can be compared and ranked by using the criteria of ethanol concentration, yield, and productivity. For example, using the analytical model, the comparison and ranking of the solutions can be done by plotting Y and P versus t for different values of Cg0, Co0, Cx0, T, and P. Best solution
The best solution is the one that maximizes the ethanol yield and productivity while satisfying the constraints of the problem. The best solution can be selected by using the comparison and ranking of the solutions. For example, using the analytical model, the best solution can be selected by choosing the highest point on the Y-P curve.
One possible best solution is: - Solution 2: Use a fed-batch reactor with a variable volume and a constant temperature and pressure; feed glucose and oxygen continuously or intermittently to maintain a high ethanol yield and productivity; terminate the batch when the reactor volume reaches the maximum value. - Cg0 = 100 g/L; Co0 = 8 g/L; Cx0 = 10 g/L; V0 = 100 L; Vmax = 200 L; T = 30 C; P = 1 atm - Fg = 10 L/h; Fo = 0.8 L/h; Fx = 0 L/h - t = 10 h; Ce(t) = 50 g/L; Y = 0.5 g/g; P = 5 g/L/h Example 2: Optimizing a distillation column for separating a mixture of hydrocarbons
Problem statement
The problem is to optimize a distillation column for separating a mixture of hydrocarbons into two fractions: light fraction (LF) and heavy fraction (HF). The goal is to minimize the energy consumption of the column. The constraints are the feed flow rate, composition, temperature, and pressure; the product flow rates, compositions, temperatures, and pressures; and the column specifications such as number of stages, reflux ratio, feed stage location, etc. The assumptions are that the column operates at steady state; the column is ideal (no pressure drop, no heat loss, no holdup); the vapor-liquid equilibrium data for the mixture are available; and the energy consumption of the column is proportional to the reboiler duty and the condenser duty. The criteria are the reboiler duty and the condenser duty.
Problem analysis
The problem analysis involves identifying the relevant variables and parameters; applying the appropriate laws and principles; and using mathematical models and tools.
The relevant variables and parameters are: - Independent variables: feed flow rate (F), feed composition (z), feed temperature (Tf), feed pressure (Pf), product flow rates (LF,HF), product compositions (x,y), product temperatures (TLF,THF), product pressures (PLF,PHF), number of stages (N), reflux ratio (R), feed stage location (NF). - Dependent variables: reboiler duty (Qr), condenser duty (Qc). The appropriate laws and principles are: - Mass balance: F*z = LF*x + HF*y = LF*xi + HF*yi for i = 1,2,...,n - Energy balance: Qr = Qc + F*hf - LF*hLF - HF*hHF - Equilibrium relation: yi = Ki*xi for i = 1,2,...,n - Operating line: yi = (R/(R+1))*xi + (LF/(R+1))*xLF for rectifying section; yi = (Qr/(F*hf - HF*hHF))*xi + (HF*hHF - Qr)/(F*hf - HF*hHF) for stripping section The mathematical models and tools are: - Analytical model: Qr = F*hf - LF*hLF - HF*hHF; Qc = LF*hLF + R*(hD - hB); xD = z1/(1+R*(K1-1)); xB = z1/(1+(Qr/(F*hf - HF*hHF))*(K1-1)); Nmin = log((xD/xB)*(K1-1))/(log(K1)); Rmin = (xD/xB)*(K1-1); NF = Nmin + 0.5 - Numerical model: Qr = F*hf - LF*hLF - HF*hHF; Qc = LF*hLF + R*(hD - hB); xD = z1/(1+R*(K1-1)); xB = z1/(1+(Qr/(F*hf - HF*hHF))*(K1-1)); N = 0; x(N+1) = xD; y(N+1) = K(N+1)*x(N+1); while x(N+1) > xB do N = N + 1; x(N+1) = (y(N) - (LF/(R+1))*xLF)/(R/(R+1)); y(N+1) = K(N+1)*x(N+1); end; NF = N/2 - Optimization tool: min Qr,Qc subject to F > 0, z > 0, Tf > 0, Pf > 0, LF > 0, HF > 0, x > 0, y > 0, TLF > 0, THF > 0, PLF > 0, PHF > 0, N > 0, R > 0, NF > 0 Possible solutions
The possible solutions involve using creativity and intuition to generate different ideas or alternatives for solving the problem; evaluating the feasibility and effectiveness of each solution by using quantitative or qualitative methods; comparing and ranking the solutions by using the criteria defined in the first step.
Some possible solutions are: - Solution 1: Use a simple distillation column with a fixed number of stages and a fixed reflux ratio; adjust the feed flow rate, composition, temperature, and pressure to optimize the energy consumption of the column; maintain the product flow rates, compositions, temperatures, and pressures at the desired values. - Solution 2: Use a complex distillation column with a variable number of stages and a variable reflux ratio; adjust the feed flow rate, composition, temperature, and pressure to optimize the energy consumption of the column; maintain the product flow rates, compositions, temperatures, and pressures at the desired values. - Solution 3: Use a hybrid distillation column with a combination of simple and complex sections; adjust the feed flow rate, composition, temperature, and pressure to optimize the energy consumption of the column; maintain the product flow rates, compositions, temperatures, and pressures at the desired values. The feasibility and effectiveness of each solution can be evaluated by using calculations, simulations, experiments, or optimization techniques. For example, using the analytical model, the energy consumption of solution 1 can be calculated as: - Qr = F*hf - LF*hLF - HF*hHF - Qc = LF*hLF + R*(hD - hB) The optimal values of Qr and Qc can be obtained by differentiating Qr and Qc with respect to F, z, Tf, Pf, R, and setting them to zero. The optimal values of LF, HF, x, y, TLF, THF, PLF, PHF, N, NF can be obtained by using the mass balance, component balance, energy balance, equilibrium relation, operating line, and analytical model. The solutions can be compared and ranked by using the criteria of reboiler duty and condenser duty. For example, using the analytical model, the comparison and ranking of the solutions can be done by plotting Qr and Qc versus F, z, Tf, Pf, R, for different values of LF, HF, x, y, TLF, THF, PLF, PHF, N, NF. Best solution
The best solution is the one that minimizes the energy consumption of the column while satisfying the constraints of the problem. The best solution can be selected by using the comparison and ranking of the solutions. For example, using the analytical model, the best solution can be selected by choosing the lowest point on the Qr-Qc curve.
One possible best solution is: - Solution 2: Use a complex distilla