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Advanced Transport Phenomena Module 4 - Lecture 16. Momentum Transport: Flow in Porous Media & Packed Beds. Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras. FLOW IN PORUS MEDIA AND PACKED BEDS.
Module 4 - Lecture 16
Momentum Transport: Flow in Porous Media &
Dr. R. Nagarajan
Dept of Chemical Engineering
Effective diameter of each particle in bed
Appropriate Reynolds number for internal flow
where interstitial mass velocity
Empty-duct (superficial) mass velocity
Function of Re
Correlates well with experimental data (next slide)
Experimentally determined dependence of fixed-bed friction factor fbed on
the bed Reynolds’ number(adapted from Ergun (1952))
For a combustion turbine materials-test program it is desired to expose specimens to a sonic but atmospheric pressure jet of combustion-heated air with a post- combustion (stagnation) chamber temperature of 2000 K.
d. What will the exit (jet) velocity be?
e. What will be the flow rate of axial momentum at the nozzle exit?
f. By what factor will the gas density change in going from the nozzle inlet to the nozzle outlet?
Isentropic gas flow from T0 = 2000 K to sonic speed @ 1 atm. Assume
For a perfect gas EOS:
b. Nozzle shape: see figure
d. Exit (Jet) Velocity
c. Mass Flow Rate
But, for a perfect gas:
e. Flow rate of Axial Momentum, at exit:
Consider a dilute acetylene-air mixture at atmospheric pressure and 300 K with a C2H2 mass fraction of 4.5%. If the heat of reaction of acetylene is 11.52 kcal/gm C2H2, estimate:
This fixes since:
And, since Ma 2 =1(Chapman-Jouguet Condition)
conclusion: Detonation Speed=1.98 km/s ((Ma)1 =5.7). Stagnation Pressure:
Consider the steady axisymmetric flow of hot air in a straight circular tube of radius aw and cross sectional area A
Conditions (at exit):
p=1 atm (uniform)
T= 1500 K (uniform)
aw= 5 mm
Suppose it has been observed that the axial-velocity profile is, in this case, well described by the simple equation:
where U=103 cm/s. Using this observation, the conditions above, answer the following questions:
a. If the molecular mean-free-path in air is approximately given by the equation:
Estimate the prevailing mean free path l and the ratio of l to the duct diameter- i.e., relevant Knudsen number for the gas flow:
What conclusions can you now draw concerning the validity of the continuum approach in this case?
b. Calculate the convective mass flow rate (expressed in g/s) through the entire exit section. For this purpose assume the approximate validity of the “perfect “ gas law, viz,:
Here p is the pressure (expressed in atm), M is the molecular weight (g/g-mole)(28.97 for air), R=82.06 (univ gas const), and T is the absolute temperature (expressed in kelvins). Also note that for this axisymmetric flow a convenient area element is the annular ring sketched below
( where is the unit vector in the z- direction).
c. Also calculate the average gas velocity at the exit section and the corresponding Reynolds’ number;
d. Calculate the convective axial momentum flow rate (expressed in g. cm/s2) through the exit section. Is your result equivalent to Why or why not?
e. Calculate the convective kinetic-energy flow rate (expressed in g. cm2/s3). Is your result equivalent to Why or why not?
f. If, in a addition to the axial component of the velocity vz, the air in the duct also has a swirl component how would this influence your previous estimates ( of mass flow rate, momentum flow rate, kinetic energy flow rate)? Briefly discuss.
g. If the local shear stress is given by the following degenerative form of Newton’s law:
at what radial location does maximize? Calculate the maximum value of and express your result in dyne cm-2 and Newton m-2. Calculate the skin-friction coefficient, cf (dimensionless), at the duct exit. At what radius does take on its minimum value? Can- be regarded as the radial diffusion flux of axial momentum? Why or why not? Does the rate at which work is done by
the stress maximize at either of the two locations found above? Why or why not?
h. Characterize this flow in terms of flow descriptors and defend your choices.
f. would not influence
Newtonian (viscous) Internal
d. Justify the use of an incompressible Newtonian fluid CD(Re, shape) curve to solve Part (a) ( involving the gas air) by showing that is small enough under these conditions to neglect .
Momentum Transfer to (Drag on) Immersed Objects
Drag/meter of axial length=? for objects of transverse dimension 5 cm. in U =10 m/s, air @ 1 atm., 1200 K.
a. Cylinder in Cross flow
Frontal area/meter=5 cm x 100 cm = 5 x 102 cm2 /m
Most of this drag is due to the p(q) distribution- that is, “ form” drag.
b. Plate Normal to flow: check literature
c. Plate Aligned with flow:
In this case
Since ReL<106 (approx.) we expect flow in the momentum defect Boundary Layer to be laminar. Then
But total wetted area/meter=(2)(5x102)=103 cm2/m. Therefore
This drag is entirely due to - i.e., it is “friction drag”
b. Estimate the appropriate value of by assuming that the relevant density is about the arithmetic mean between . How much larger is the effective turbulent momentum diffusivity, , than the intrinsic momentum diffusivity of the jet fluid ?
c. Estimate the downstream distance at which the time-averaged velocity (axial momentum per unit mass) along the jet centerline will be reduced to 10% of the initial jet velocity (axial momentum per unit mass) as a result of momentum diffusion. Compare this to the result that would have been obtained had the jet
remained laminar (with kinematic viscosity ).
d. At this location what is the approximate ratio between the entrained (laboratory air) mass flow and the “primary” (combustion-heated air) jet?
Momentum Transfer: Turbulent Round Jet
Momentum Transfer: Turbulent Round Jet
is determined by . Using
Where does drop to Uj/10. For a turbulent jet:
We find: z=88.6 cm =0.89 m
(This would have been 0.46 km if there had been no turbulent enhancement in momentum diffusion.)
so that at
1. Calculate the time-averaged profile at this location.
2. Can you estimate the centerline, time averaged temperature and CO2(g) concentration at this point? (Itemize and discuss the underlying assumptions.)