Dimensional Analysis

1. Dimensional Analysis Keeping track of units.

2. "Dimensions" are different from "units". • units • M – grams, kg, ton, slug, amu, ... • L – meters, km, feet, miles, ... • T – second, minute, hour, day, ... • dimensions • M – mass • L – length • T – time

3. $100 Million error • Mars Lander

4. Can’t mix apples and oranges. • Only like-dimensioned quantities may be added, subtracted, compared, or equated. • The dimensions are: • An expression may be dimensionally correct but have mixed units: • Convert one unit to another using a conversion factor

5. Are these equations correct? Use dimensions to determine the units of viscosity .

6. Dimensionless units. • What are the dimensions of G, the gravitational constant? • Relative error is dimensionless. • It makes it easy to determine which term is important.

7. Dimensional analysis helps us understand physical dependences. • Look at the period (time) for the oscillation of a mass on a spring. • What does the period depend on? • Mass • Spring constant (M/T2) • Gravity (L/T2) • T  m k g • Dimensions: • The period can’t depend on g because there is no L on left side. • Correct dependence is:

8. Scaling • Examine how different physical magnitudes depend on the size of a system (defined by a length parameter L) • Material properties (e.g., strengths, moduli, densities, coefficients of friction) are held constant. • If you scale the water jug up to a town tank, what problems arise? • If you scale up a chemistry lab reaction to industrial plant size, what problems arise?

9. Dimensionless systems simplify analysis of some problems. • Look at Aerodynamics example • Independent variables • lift force F, • wing area A, • velocity v, • air density  • air viscosity  • Can do experiments varying one variable while holding all others constant. • Takes a long time (& money).

10. Buckingham Pi Theorem • This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables. • If problem has n independent variables • And m "primitive dimensions" (L, M, T) • Then you can simplify to n-m dimensionless variables. • Not all solutions are physically meaningful • Look at Aerodynamics example • Independent variables • F, A, v, ,  • Dimensionless variables = n-m = 5-3 = 2

11. Aerodynamics • The most general power-law expression would be • Dimensionally this look like • To be dimensionless, the powers must all be 0.

12. Aeronautics (continued) • Now add some insight to get physically meaningful solution. • We want to get a variable with the lift force F, so a = 1. • We only have three equations so must eliminate one variable. • Set e = 0. • The exponents become • The solution is

13. Aeronautics (continued) • A second physically meaningful solution has a=0, c=1 • The exponents become • The solution is • These two dimensionless variables are to test aerodynamic shapes in wind tunnels.

14. Diffusion

15. What do these processes have in common? 1) Hydrogen embrittlement of pressure vessels in nuclear power plants 2) Flow of electrons through conductors 3) Dispersion of pollutants from smoke stacks 4) Transdermal drug delivery 5) Influenza epidemics 6) Chemical reactions 7) Absorption of oxygen into the bloodstream

16. They all depend on Diffusion (conduction) What is diffusion? The transport of material--atoms or molecules--by random motion What is conduction? The transport of heat or electrons by random motion.

17. Place a drop of ink into a glass of water. What happens? Brownian motion causes the ink particles to move erratically in all directions. A concentration of ink particles will disperse.

18. Why does random motion cause spreading of a concentration of particles? Because there are more ways for the particles to drift apart than there are for the particles to drift closer together. We can also explain the spreading of a concentration by entropy. The second law of thermodynamics says that systems tend towards maximum entropy – or maximum disorder. Area of high concentration and low/zero concentration is an ordered state and the mixed state is the disordered state!

19. Other examples? Why do metal cooking spoons have plastic handles?

20. Other examples? What happens if someone across the room sprays perfume? Perfume diffusion simulation

21. After adding milk and sugar, why do we stir our coffee? Diffusion is slow! Diffusion Progression Rates: Gas: 10 cm/min Liquids: 0.05 cm/min Solids: 0.00001 cm/min Agitation (or stirring) can move fluids much larger distances in the same amount of time, which can accelerate the diffusion process.

22. Ji is called the flux. It has units of D is called the diffusion coefficient. It has units of In each of these examples, molecules (or heat) are moving down a gradient! (From an area of high concentration to an area of low concentration) Fick’s Law:

23. Diffusivity Values Greater the diffusivity, greater the flux!

24. C(*) N2 time CO2 (constant T & P) Do our definitions of flux make sense? • If double area of capillary, expect the amount of gas transported to double. • Want flux independent of apparatus – normalize by area. • Flux is proportional to the concentration gradient – steeper the gradient, more material transported. • Flux is inversely proportional to capillary length – increasing the distance to travel will decrease the flux.

25. ci,0 ci,l l Steady diffusion across a thin film Now let’s use our diffusion equation to predict the concentration profile of a material diffusing across a thin film! Thin film Well-mixed dilute solution with concentration ci,0 Well-mixed dilute solution with concentration ci,l If we are at steady-state (the concentration profile has no time dependence, or in other words, there is no accumulation of i in the film), we have a linear concentration profile.

26. ci,0 ci,c ci,l z=l z=0 z=zc Concentration-dependent diffusion Consider two neighboring thin films with a separation at ci,c: D1 D2 Which diffusivity is greater? How do you know?

27. Unsteady state diffusion Back to a drop of ink in a glass of water… If consider diffusion in the z-direction only: How does the concentration profile change with time? (add ink drop – all ink located at z = 0) t = 0 t z=0

28. Heat Transfer Occurs by three means: • Conduction: • Occurs between two static objects • Heat flows from the hotter to the cooler object • For example, holding a cup of hot coffee • Convection: • Transport of heat via a fluid medium • Currents caused by hot air rising, fan circulating air • Radiation: • Transport of energy as electromagnetic waves; the receiving body absorbs the waves and is warmed • For example, warmth of a fire

29. Heat moves down a temperature gradient! (From an area of high temperature to an area of low temperature) Fourier’s Law: qz is called the heat flux. It has units of k is called the thermal conductivity. It has units of α is called the thermal diffusivity. It is defined as and has units of

30. Thermal Conductivity Values Greater the thermal conductivity, greater the heat flux!

31. Heat Conduction Consider a two-paneled door: TH Tc z wood metal What will the steady-state temperature profile look like? Why? kmetal > kwood

32. κ1 Here’s a heat-conducting bar with a fixed temperature T at each end: T(t,0)=0; T(t,100)=100. 2k1 = k2 . κ2 z=0 z=100 T(t,0)=0 T(t,100)=100 At steady-state: (Constant flux) Therefore, the ratios of the temperature gradients in each section must equal the inverse ratios of the k’s.

33. Gradient transport summary

34. Chemical Engineering Labs Data Analysis Chromatography Lab

35. Write-Up • First, describe what happened, taking care to differentiate between different colors and different solvents. • For the methanol and 2-propanol experiments, calculate flow rates from the purple ink.

36. Dye Flow Rates 5.0 cm 2.9 cm Flow rate of pink = 2.9/5.0 = 0.59

37. Write-Up Continued • Present a graph showing the flow rate of each component as a function of the proportion of 2-propanol in the solvent (approximately 100% for 2-propanol and 0% for methanol). • Assuming that flow rate changes linearly with 2-propanol concentration, draw a line which represents the expected flow rate for each component in various concentrations. Deduce the expected equations of each of these lines.

39. Write-Up Continued Notice that for the purple pen, one ink component flows faster in 2-propanol while the other flows faster in methanol. If we were attempting to separate these two components, and we were really unlucky by choosing the wrong concentration of methanol, we could obtain no separation. Estimate the methanol concentration in 2-propanol of this most undesirable solvent.

40. Chemical Engineering Labs Data Analysis Heat Transfer Lab

41. Heat Transfer Lab Calculate the total electrical energy input for both experiments (with and without the lid): Calculate the water's energy increase for both experiments:

42. Heat Transfer Lab Calculate the efficiencies with and without the lid: What difference did putting a lid on the pot make? If the efficiency is less than 1.0, where did the remaining energy go? Try to think of all possible “losses”. How could you improve the efficiencies?

43. Chemical Engineering Labs Data Analysis Distillation

45. Distillation Use the temperature data and the specific gravity data to determine the weight percent ethanol in each sample. Distillate #1 T = 24.0 º C s.g. = 0.920

46. Information from the Data Table • At 20ºC 45% ethanol has specific gravity = 0.92472 46% ethanol has specific gravity = 0.92257 At 25ºC 45% ethanol has specific gravity = 0.92085 46% ethanol has specific gravity = 0.91868

47. .

48. Determine the % Ethanol in Distillate #1 • At 24ºC the distance between the 45% and 46% alcohol lines is 1.2 cm. • At 24ºC the distance between the 45% and the data point is 0.9 cm. • So • And the % ethanol = 45.75%

49. Grams of Ethanol in each solution Mass of Distillate #1 = volume of the solution * specific gravity Grams of ethanol in Distillate #1 = grams of solution * % alcohol ÷ 100 Moles of ethanol = grams of ethanol ÷ 46.07 g/mol

50. Grams of Water in each Solution Calculate the total moles of water in each sample moles of H2O= grams of H2O ÷ 18.02grams/mole Perform a mole balance analysis for ethanol and water to check whether all material is accounted for.