Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 3 Lecture 11

Constitutive Laws: Illustrative Problems 2

LIMITATIONS OF LINEAR LOCAL FLUX VS LOCAL DRIVING FORCE CONSTITUTIVE LAWS  Nonlinear fluids  Nonlocal spatial behavior– Action at a distance  Nonlocal temporal behavior– Fluids with memory  Multiphase effects– nonlinear species drag laws 3

NONLINEAR FLUIDS  Non-Newtonian fluids: dynamic viscosity depends on deformation rate (or extra stress)  e.g., high-mass-loaded slurries, clay-in-water, etc.  Dynamics of such fluids studied under “rheology”  Analogous non-linearities exist in heat & mass transfer  e.g., solute diffusion through human blood 4

NONLOCAL SPATIAL BEHAVIOR  When mfp not negligible compared to overall length over which transport process occurs  e.g., transport across very low-density gases in ducts, electronic conduction in micro crystallites, radiation transport through nearly transparent media 5

NONLOCAL TEMPORAL BEHAVIOR  Fluid responding to previous stresses, stress history  e.g., visco-elastic fluids, such as gels, that partially return to their original state when applied stress is interrupted  Analogous behavior possible for energy & mass transfer  e.g., time lags between imposition of spatial gradients & associated fluxes  Effects associated with finite thermal and concentration wave speeds 6

MULTIPHASE EFFECTS  Single-phase mixture vs multiphase system  e.g., gas with vapors of varying molecular weights vs droplet-laden gas  Not a sharp distinction  Relates to v i – v j, v i – v  Linear laws apply only when these differences are small. 7

Figure on next slide displays the experimentally observed composition dependence of the viscosity  of a mixture of ammonia and hydrogen at 306 K, 1 atm. N.B.: The units used for  in Figure are  P (micro-poise) where 1 poise 1 g/cm∙s. The MKS unit of viscosity is 1 kg/m∙s or 1 Pascal- second. PROBLEM 1 8

9

a.Using Chapman-Enskog (CE) Theory and the Lennard-Jones (LJ) potential parameters, examine the agreement for the two endpoints, i.e., the viscosities of pure NH 3 and pure H 2 at 306 K, 1 atm. b.Using the “square-root rule,” what would you have predicted for that composition dependence of the mixture viscosity in this case? What conclusions do you draw from this comparison? 10 PROBLEM 1

c.How would you expect the NH 3 + H 2 curve to shift if the temperature were increased to 1000 K and the pressure were increased ten-fold? (Discuss the basis for your expectation.) d.Estimate the binary diffusion coefficient from Chapman-Enskog theory. e. Calculate the Schmidt number for an equimolar mixture of H 2 (g) and NH 3 (g). 11 PROBLEM 1 (cm 2 /s)

k(NH 3 (g)) and Calculation Given the tabular data: 12 PROBLEM 1

Calculate k(NH 3 (g)) 1 atm. 306 K Thermal conductivity of NH 3 (g) is calculated to give x poise. 13 PROBLEM 1

14 PROBLEM 1

Therefore 15 PROBLEM 1

Calculation of D NH 3 -H 306 K, 1 atm: and Sc for y 1 = PROBLEM 1

In the present case: 17 PROBLEM 1

From tabular values ( Hirshfelder, Curtiss and Bird (1954)); Therefore: For an equimolar mixture of ammonia and hydrogen: 18 PROBLEM 1

Moreover, Assuming perfect gas behavior: The momentum diffusivity of the mixture is therefore: 19 PROBLEM 1

Corresponding to a diffusivity ratio of : 20 PROBLEM 1

Property estimation for the hydrodesulphurization of Naphtha vapors. For the preliminary design of a chemical reactor to carry out the removal of trace sulfur compounds (e.g., C 4 H 4 S) from petroleum naphtha vapors (predominantly heptane, C 7 H 16 ), it is necessary to estimate the Newtonian viscosity and corresponding momentum diffusivity of the vapor mixture at 660 K, 30 atm; these are rather extreme conditions for which direct measurements are not available. 21 PROBLEM 2

a.Using the vapor composition together with selected results from the kinetic theory of ideal vapors and dense vapors, what are your best estimates for  mix and mix ? 22 PROBLEM 2

b. If this vapor mixture is to be passed at the rate of 2 g/s through each 2.54 cm diameter circular tube ( in a parallel array), calculate the corresponding Reynolds’ number, Re ≡ Ud w / mix within each tube (where  U  is the average vapor-mixture velocity). This dimensionless ratio will be seen to be required to estimate the mechanical energy required to pump the vapor through such tubes. 23 PROBLEM 2

c. Estimate the effective (pseudo-binary) Fick diffusion coefficient for thiophene (C 4 H 4 S) migration through this mixture, and the corresponding diffusivity ratio: Sc ≡  mix /D 3-mix. 24 PROBLEM 2

25 PROBLEM 2

PROPERTY ESTIMATION FOR THE HYDRODESULPHURIZATION OF NAPHTHA VAPORS a. Suppose we need viscosity of b. Further, if The first step is the estimation of the mixture viscosity based on its composition and the properties of its constituents under the anticipated 26

operating conditions (660 K, 30 atm). Tentatively we use: where 27 PROBLEM 2

In what follows, we sequentially consider the viscosities of each of the constituents of the vapor mixture. Viscosity of Hydrogen(g) at 660 K, 30 atm: 28 PROBLEM 2

29 PROBLEM 2

Therefore Using table for L-J 12:6 potential ( Hirschfelder, Curtiss, and Bird (1954)) 30 PROBLEM 2

Therefore We now consider a similar calculation for the species C 7 H 16 (g) and check whether “dense vapor” correction are important 31 PROBLEM 2

Viscosity of n-Heptane at 660 K, 30 atm. 32 PROBLEM 2

Therefore via Hirshfelder et al. (1954) table; cf. 1.22(2.34) =1.065) and 33 PROBLEM 2

Dense Vapor Correction? This can be estimated via the principle of “corresponding states” For C 7 H 16: and From Therefore 34 PROBLEM 2

Table for calc. i M i y i 10 4  i PROBLEM 2 Viscosity of Binary 660 K, 30 atm..

Then Re-Number Calculation: 36 PROBLEM 2

For each d w =2.54 cm tube and If the perfect gas EOS is valid for the mixture, where 37 PROBLEM 2

Now: And Therefore 38 PROBLEM 2

Conclusion: Since for Re=8.26 x 10 3 the flow should be turbulent (under anticipated hydrodesulphurization conditions). 39 PROBLEM 2