2 Differential Equations Heat transferMass transferConservation of momentum, thermal energy or massODE(4.1)(4.2)PDE(4.3)
3 ODE Definition Example A 3rd order differential equation for r = r(t) Solutionindependent(4.4)dependent(4.5)(4.6)(4.7)
4 Important Issues Existence of a solution Uniqueness of the solution How to determine a solution
5 Linear Equation (1) Rewrite 4.9 for all t Determine (4.8)(4.9)Rewrite 4.9Determinefor all t(4.10)(4.11)where m(t) is called an integrating factor
6 Linear Equation (2) Multiply both sides of equation 4.10 by m(t) Observe that the left-hand side of eqn 4.12 can be written asor(4.12)(4.13)
7 Linear Equation (3) Equation (4.12) can be rephrase as: Integrate both sides of Equation (4.14) with respect to the independent variable:(4.14)(4.15)where c is the constant of integration
8 Example 1Water containing 0.5 kg of salt per liter is poured into a tank at a rate of 2 l/min, and the well-stirred mixture leaves at the same rate. After 10 minutes, the process is stopped and fresh water is poured into the tank at a rate of 2 l/min, with the new mixture leaving at 2 l/min. Determine the amount (kg) of salt in the tank at the end of 20 minutes if there were 100 liters of pure water initially in the tank.2 l/min½ kg salt/lCA2 l/min, CA (l/min)
10 Example 2Consider a tank with a 500 l capacity that initially contains 200 l of water with 100 kg of salt in solution. Water containing 1 kg of salt/l is entering at a rate of 3 l/min, and the mixture is allowed to flow out of the tank at a rate of 2 l/min. Determine the amount (kg) of salt in the tank at any time prior to the instant when the solution begins to overflow. Determine the concentration (kg/l) of salt in the tank when it is at the point of overflowing. Compare this concentration with the theoretical limiting concentration if the tank had infinite capacity.
12 THEOREMIf the functions p and g are continuous on an open interval a < x < b containing the point x = x0, then there exists a unique function y = f(x) that satisfies the differential equationy’ + p(x)y = g(x)for a < x < b , and that also satisfies the initial conditiony(x0) = y0where y0 is an arbitrary prescribed initial value.
13 Higher ODE Reduces to 1st Order In general, it is sufficient to solve first-order ordinary differential equations of the form
14 reduces the above equation to a linear equation. Nonlinear equations can be reduced to linear ones by a substitution. Example:y’ + p(x)y = q(x)ynand if n ¹ 0,1 thenn(x) = y1-n(x)reduces the above equation to a linear equation.(4.16)(4.17)
15 Example 3Suppose that in a certain autocatalytic chemical reaction a compound A reacts to form a compound B. Further, suppose that the initial concentration of A is CA0 and that CB(t) is the concentration of B at time t. Then CA0 – CB (t) is the concentration of A at time t. Determine CB(t) if CB(0) = CB0.
22 And equation 4.16 becomeswhich integrates to:where m1 is an arbitrary constant to be determined with the given initial t = 0, CB = CB0, then
23 Example of Problem Setup Consider the continuous extraction of benzoic acid from a mixture of benzoic acid and toluene, using water as the extracting solvent. Both streams are fed into a tank where they are stirred efficiently and the mixture is then pumped into a second tank where it is allowed to settle into two layers. The upper organic phase and the lower aqueous phase are removed separately, and the problem is to determine what proportion of the acid has passed into the solvent phase.
24 Example (cont…) List of assumptions Combine the two tanks into a single stageExpress stream-flow rates on solute-free basisSteady flowrate for each phaseToluene and water are immiscibleFeed concentration is constantMixing is efficient, the two streams leaving the stage are in equilibrium with each other given by y = mxComposition stream leaving is the same with the composition in the stageThe stage initially contains V1 liter toluene, V2 liter water and no benzoic acid
25 Problem 1Consider an engine that generates heat at a rate of 8,530 Btu/min. Suppose this engine is cooled with air, and the air in the engine housing is circulated rapidly enough so that the air temperature can be assumed uniform and is the same as that of the outlet air. The air is fed to the housing at 6lb-mole/min and 65oF. Also, an average of 0.20 lb-mole of air is contained within the engine housing and its temperature variation can be neglected. If heat is lost from the housing to its surroundings at a rate of Ql(Btu/min) = 33.0(T-65oF) and the engine is started with the inside air temperature equal to 65oF.Derive a differential equation for the variation of the outlet temperature with time.Calculate the steady state air temperature if the engine runs continuously for indefinite period of time, using Cv = 5.00 Btu/lb-mole oF.
26 Problem 2A liquid-phase chemical reaction with stoichiometry A B takes place in a semi-batch reactor. The rate of consumption of A per unit volume of the reactor is given by the first order rate expressionrA (mol/liter.s) = kCAwhere CA (mol/liter) is the reactant concentration. The tank is initially empty. At time t=0, a solution containing A at a concentration CA0(mol/liter) is fed to the tank at a steady rate f(liters/s). Develop differential balances on the total volume of the tank contents, V, and on the moles of A in the tank, nA .
27 Solving ODEs using Numerical Methods Initial Value Problem (IVP)y’’ = -yxy(0) = 2, y’(0) = 1Boundary Value Problem (BVP)y” = -yxy(0) = 2, y’(1) = 1
28 General ProcedureRe-write the dy and dx terms as Δy and Δx and multiply by ΔxLiterally doing this is Euler’s method
45 Example y΄=x+y, y(0)=0 x yo k1=fi k2=f(x+h/2,y+h/2k1) yn=yo+1/6(k1+2k2+2k3+k4)h0.10.110.2220.20.02140.2210.3440.3560.4930.09180.40.0920.4920.6410.6560.8230.22210.60.8221.0041.0231.2270.42550.80.4261.2261.4481.4701.7200.71811.7181.9902.0172.3221.1201.22.3202.6522.6853.0571.6551.43.0553.4613.5013.9552.3531.63.9534.4484.4985.0523.2501.85.0505.6545.7156.3934.389