Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

1. Advanced Transport Phenomena Module 8 Lecture 34 Problem-solving techniques, aids, philosophy Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. Problem-solving techniques, aids, philosophy

3. EXAMPLE • Design of a heat-exchanger for extracting power from combustion products in a fossil-fuel-fired power plant • Requires blend of fundamental principles of “transport phenomena” (covered earlier) and experimental data

4. PC FURNANCE + HEAT EXCHANGER Pulverized coal combustion stationary power plant (schematic, after Flagan and Friedlander (1978))

5. PC FURNANCE + HEAT EXCHANGER • Design of a heat-exchanger tube exposed to products of pulverized coal combustion requires estimation of • Rate at which energy is transferred from hot combustion gases to tube, • Rate at which each tube wall will accumulate ash (present as particulate matter in PC combusion products)

6. PC FURNANCE + HEAT EXCHANGER Configuration and nomenclature: heat- exchanger tube in a cross-flow of combustion product gas.

7. PC FURNANCE + HEAT EXCHANGER • Tube of circular cross-section, diameter 5 cm • Mainstream gas properties: • T∞ = 1200K • p = 1 atm • U∞ = 10 m/s • Ash particles of two types: • Submicron: dp = 10-1mm atwp = 2 X 10-4 • Supermicron: dp = 20 mm at wp = 1 X 10-2

8. PROBLEM • Estimate rate of energy gain (kW) per meter of tube length • Temperature at outer surface of tube Tw = 800K • Estimate rate at which submicron and supermicron ash accumulate per unit length of cylinder • “fouling” leading to heat-transfer barrier • Requires periodic steam-jet cleaning, or plant shutdown for tube replacement

9. SIMPLIFYING ASSUMPTIONS • Gas flowing around cylinder can be treated as continuum • Tube may be treated as isolated cylinder in cross-flow • Forced-convection dominates natural convection • Viscous dissipation can be neglected in determining gas temperature profile, hence heat-transfer rate from gas to solid

10. SIMPLIFYING ASSUMPTIONS • Mainstream turbulence level not high enough to alter time-averaged energy or mass-transfer rates • For submicron particle transport, analogy between mass and energy-transfer holds • Validity to be confirmed during analysis

11. ENERGY-TRANSFER RATE FROM COMBUSTION PRODUCTS TO TUBE • Average heat-transfer coefficient for this flow configuration (circular cylinder of Lref = dw, transverse to flow): • Important dimensionless parameters: • Experimental data relating these available

12. ENERGY-TRANSFER RATE FROM COMBUSTION PRODUCTS TO TUBE Correlation of heat loss/gain by circular cylinder in a steady cross-flow of air (after McAdams (1954))

13. ENERGY-TRANSFER RATE FROM COMBUSTION PRODUCTS TO TUBE • Since air used for PC combustion contains large amount of nonreactive N2 (MW = 28): • Scale-model experimental data for air should apply to combustion products • Heat-transfer flux calculated via: • Rate then calculated for L = 1 m

14. ENERGY-TRANSFER RATE FROM COMBUSTION PRODUCTS TO TUBE n, m, r for air calculated at 1 atm, and “film” (mean) temperature: Gas density well approximated by perfect gas law (as 0.34 kg/m3): Viscosity of air at 20 atm known, and pressure-independent for perfect gas =>

15. ENERGY-TRANSFER RATE FROM COMBUSTION PRODUCTS TO TUBE • Momentum diffusivity may then be estimated as: • Reynolds’ number may be evaluated as: • Hence, for air flowing over a cylinder:

16. ENERGY-TRANSFER RATE FROM COMBUSTION PRODUCTS TO TUBE • Thermal conductivity, k, also p-independent for perfect gas; thus: • Heat transfer rate for L = 100 cm: or:

17. ENERGY-TRANSFER RATE FROM COMBUSTION PRODUCTS TO TUBE Thus: • Each meter of heat-exchanger tube extracts energy from combustion gases at the rate of 3.1 kW

18. ASH ACCUMULATION ON HE TUBES: FOULING RATE PREDICTIONS • Assume: Laws of particle transport analogous to those governing energy transport • True only if particles can be treated like “heavy gas” • Brownian diffusion analogous to Fourier heat-flux • Needs to be validated a posteriori

19. ASH ACCUMULATION ON HE TUBES: FOULING RATE PREDICTIONS • Functional relationship for particle mass transfer: • For circular cylinder in cross-flow at 1000K: • dp = 10-1mm: Sc ≈ 3 X 104 • dp = 20 mm: Sc≈ 108 at 1500K => at 1000K:

20. ASH ACCUMULATION ON HE TUBES: FOULING RATE PREDICTIONS • Num (Re, Sc) not available in this large Sc-range • Can be estimated from Nuh(Re, Pr) for Pr = Prair ≈ 0.7 by analogy • Nuhgoes as Pr1/3 for Pr ≥ 0.7 at sufficiently high Re • Hence, for submicron ash: • For supermicron ash:

21. ASH ACCUMULATION ON HE TUBES: FOULING RATE PREDICTIONS • Assume each incident particle sticks to wall, hence: wp,w << wp,∞ • Estimation of Dp (1000 K, 1 atm):

22. ASH ACCUMULATION ON HE TUBES: FOULING RATE PREDICTIONS • Particle deposition rate may then be calculated from: • For L = 100 cm, dp = 0.1 mm: = 1.1mg/s (0.035 kg/year)

23. ASH ACCUMULATION ON HE TUBES: FOULING RATE PREDICTIONS • For L = 100 cm, dp= 20mm: • = 0.35 mg/s (0.011 kg/year) • Fouling rates can be converted to deposit thickness if we assume: • Deposit density is known, and • Deposit is uniformly distributed around perimeter.

24. DEFENDING THE ASSUMPTIONS: ROLE OF PARTICLE THERMOPHORESIS • Force experienced by a small particle in a non-isothermal gas, results in a thermophoretic contributionrwp(cp)T to particle mass flux, where: (aTD)pthermophoretic diffusivity ̴ (3/4) ng (independent of particle size for dp<< l) • Thermophoresis can increase submicron particle rate by > 200X

25. DEFENDING THE ASSUMPTIONS: ROLE OF PARTICLE THERMOPHORESIS • Phoresis increases convection-diffusion rates by a factor of: where, in this case:

26. DEFENDING THE ASSUMPTIONS: ROLE OF PARTICLE THERMOPHORESIS • Thermophoretic drift speed towards heat-exchanger tube: • Leading to: Or equivalently (Re-insensitive)

27. DEFENDING THE ASSUMPTIONS: ROLE OF PARTICLE THERMOPHORESIS (αTLe)pratio of thermophoretic particle diffusivity to heat diffusivity of combustion products • appr. 0.42 for 10-1 diameter SiO2-particles in 1500K, 1 atm gases • Unlikely to change much between 1500K & 1000K; hence:

28. DEFENDING THE ASSUMPTIONS: ROLE OF PARTICLE THERMOPHORESIS • F (thermophoretic suction) ≈ 256! • For diffusing species with large Scp-values, augmentation factor reduced by a factor of the order of: • Improved estimate of F (thermophoresis) = 207 • Enhanced deposition rate of 0.1 mm sized particles = 7.2 kg/yr

29. DEFENDING THE ASSUMPTIONS: ROLE OF PARTICLE THERMOPHORESIS • Corresponding capture efficiencies (100 hcap): 0.004without thermophoresis, 0.8 with • Thermophoresis dominant => angular distribution of deposition rate of submicron ash fraction follows temperature distribution

30. DEFENDING THE ASSUMPTIONS: SINGLE-PHASE FLOW • Valid for submicron (10-1 ) diameter ash fraction • But not for 20 mm (dominated by inertial effects) • > 100X enhancement in particle capture • Dramatic effects on hcap

31. DEFENDING THE ASSUMPTIONS: SINGLE-PHASE FLOW Correlation of inertial capture of particles by a circular cylinder in cross-flow (Israel and Rosner (1983))

32. DEFENDING THE ASSUMPTIONS: SINGLE-PHASE FLOW Representative heat-exchanger tube fouling-rate conditions; (a) particle size dependence of the capture fraction, : (b) size distribution of mainstream particle mass loading. Overall mass fouling rate will be proportional to the integral of the product of these two functions.

33. DEFENDING THE ASSUMPTIONS: SINGLE-PHASE FLOW • Stokes’ number, Stk, determines validity of single-phase assumption • Stk = tp/tflow • tflow= ½ dw/U∞ (dynamical inverse Damkohler number)

34. DEFENDING THE ASSUMPTIONS: SINGLE-PHASE FLOW • Stk evaluated at T∞= 1200K • For submicron particles (dp = 10-1mm), Dp (1200K, 1 atm) estimated from hard-sphere kinetic theory: • For supermicronparticles (dp = 20 mm), Dp (1200K, 1 atm):

35. DEFENDING THE ASSUMPTIONS: SINGLE-PHASE FLOW • Hence: • Stk acceptably small for submicron particles • But for supermicron particles, above critical value (1/8) for inertial impaction • 20mm particles cannot follow fluid streamlines around cylinder

36. DEFENDING THE ASSUMPTIONS: SINGLE-PHASE FLOW • Impaction rate: • Bracketed quantity  ash mass-flow rate through (upstream) projected area of tube

37. DEFENDING THE ASSUMPTIONS: SINGLE-PHASE FLOW • Inserting appropriate values: • Inertial deposition rate = 0.28 g/s (exceeding diffusional estimate by 8.2 X 105) • If all 20 particles that strike the HE tube stick, fouling rate (per meter of tube) = 9 metric tons/ year!

38. RECOMMENDATIONS ON PROBLEM-SOLVING • Write relevant conservation principles in general form • Sketch the problem • Select convenient control volumes & control surfaces • Don’t probe unnecessarily deeply; if a “black box” approach would suffice, stop with that

39. RECOMMENDATIONS ON PROBLEM-SOLVING • Drop terms that are “identically zero” or “relatively insignificant” compared to terms retained • Maintain cumulative list of all assumptions, expressed in the form of quantitative equalities & inequalities

40. RECOMMENDATIONS ON PROBLEM-SOLVING • Solve resulting equations, checking for reasonableness at each step • Insert numerical values (and units) to obtain the result sought • Document sources of numerical data, conversion factors • Use power-of-ten notation to enable order-of-magnitude estimations • Validate assumptions & simplifications • Relax approximations that cannot be defended • Examine solution for reasonableness & implications