Two-dimensional duct flow computations for radial distributions of velocity, temperature, and electrical conductivity are reported. Calculations were carried out for the flow conditions representative of a hydrogen-oxygen combustion driven MHD duct. Results are presented for: profiles of developing flow in a smooth duct, and for profiles of fully developed pipe flow with a specified streamwise shear stress distribution. The predicted temperature and electrical conductivity profiles for the developing flows compare well with available experimental data

Greywall, M. S.

Pian, C. C. P.

English

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https://ntrs.nasa.gov/search.jsp?R=19780020429 2020-03-22T02:48:01+00:00Z
NASA Technical Memorandum 78968
N78-28372(NASA-TM-78968) VELOCITY, TEMPERATURE, AND
ELECTRICAL CONDUCTIVITY PROFILES IN
HYDROGEN-OXYGEN MHD DUCT FLOVS (NASA) 11 p
HC A02/zF A01	 CSCL 20D
I
Unclas
G 3/3 4 27117
•	 VELOCITY, TEMPERATURE, AND ELECTRICAL CONDUCTIVITY
PROFILES IN HYDROGEN-OXYGEN MHD DUCT FLOWS
Mahesh S. Greywall
Wichita State University
Wichita, Kansas 67108
and
Carlson C. P. Pian
Lewis Research Center
Cleveland, Ohio 44135
It
4	 TECHMCAL PAPER to be presented at the 	
f
Winter Annual Meeting
sponsored by American Society of Mechanical Engineers
San Francisco, California, December 10-15, 1978
Ik
VELOCITY, TEMPERATURE, AND ELECTRICAL CONDUCTI^'TY
PROFILES IN HYDROGEN-OXYGEN MHD DUCT FLOWS*
Mahesh S. Greywail
Department of Mechanical Engineering
Wichita State University
Wichita, Kansas 67108
Carlson C. P. Pian
NASA Lewis Research Center
Cleveland, Ohio 44135
ABSTRACT
This paper presents results of two-dimensional duct flow computations for
radial distributions of velocity, temperature, and electrical conductivity.
Calculations were carried out for the flow conditions representative of NASA
Lewis hydrogen-oxygen combustion driven MHD duct. Results are presented for
two sets of computations: (1) profiles of developing flow in a smooth duct,
and (2) profiles of fully developed pipe flow with a specified streamwise shear
stress distribution. The predicted temperature and electrical conductivity
profiles for the developing flows compared well with available experimental data.
I. INTRODUCTION
Magnetohydrodynamic (MHD) power generation utilizes the interaction of an
electrically conducting gas flowing through a duct with an externally applied
magnetic field to convert a portion of the energy of the gas directly into
electricity. An accurate estimation of the fluid's electrical conductivity (a)
during its passage through the MHD duct is crucial to one's ability to predict
the generator performance. Smith (ref. 1) performed a series of tests to
measure the electrical properties of a cesium seeded H2-0 2 combustion gas. The
results of the bulk electrical conductivity measurements were found to be in
good agreement with the one-dimensional calculations of reference 1. To gain
insight into the radial distribution of the electrical conductivity and flow
properties, Wang and Smith (ref. 2) made spectroscopic measurements of transverse
profiles of the absolute integrated intensity from the optically thin lines of
cesium atoms. The radial profiles of the cesium emission coefficients were then
obtained by Abel inversion. These emission coefficients were used to infer
the radial temperature distributions for cases where the density of the cesium
atoms can be assumed uniform across the duct. Alternatively, the radial o and
seed density distributions can also be inferred from th e emission coefficients
by using temperature profiles estimated from theoretical considerations (ref. 2).
^o partially funded under Department of Energy Interagency Agreement No.
EF-77-A-01-2647 and also under NASA Grant NSG-3186.
To add to the theoretical background for the understanding of these flows,
a two-dimensional calculation of these flows was undertaken. This paper
presents the results of preliminary computations for the radial distributions
of velocity, temperature, and electrical conductivity for the duct geometry and
flow conditions typical of the experiments of references 1 and 2: stoichiometric
H2-02 combustion gas seeded with cesium hydroxide flowing through a constant
area duct (2.53 cm radius) with centerline velocity and temperature of the
order of 2500 m/s and 3000 K respectively, heat sink wall temperature of approx-
imately 600K, and burner pressure of 1100 kPa (10.9 atm).
Two different sets of computations were carried out. In the first set of
calculations the initial profiles at the beginning of the duct are varied to
simulate different amounts of boundary layer blockage until the resulting
temperature profile gave the best fit to the experimental data. Also in these
computations, the axial pressure variation was calculated as part of the con-
fined flow soluti.,ms. In the second set of computations, the temperature and
velocity profiles were calculated assuming the limiting case that the flow is
fully developed. Also in the latter set, the shear stress at the wall was ad-
justed to yield the experimentally observed axial pressure variation.
In all the calculat i ons, the thermodynamic and transport properties of the
gas mixture were calculated assuming local thermodynamic equilibrium following
Gordon and McBride (ref. 3) and Svehla and McBride (ref. 4). The electrical
conductivity of the gas was calculated as in Smith and Nichols (ref. 5).
II.	 DEVELOPING FLOW SOLUTIONS
The partially developed entrance flow in a MHD duct was analyzed in the
first set of calculations. The streamwise pressure variations are calculated as
part of the confined flow solutions. The numerical scheme of Patankar and
Spalding (ref. 6) was used. Modifications were added to handle the variable
fluid properties and to calculate the streamwise pressure variation using a
predictor-corrector scheme. For greater details, the reader is referred to the
work in reference 7.
The experimental duct of references 1 and 2 was connected to an expansion
nozzle. The flow consequently entered the duct with the boundary layer al-
ready partially developed. The exact amount of this development is not known.
Hence the following procedure was followed in the specification of the initial
dependent variable profiles at the duct entrance. A two-layer profile is used
for the velocity distribution (see Figure 1). In the centerline region,
a uniform velocity, ut, has been assumed. A 1/7-power-velocity distribution
law is used near the wall. The total enthalpy distribution is assumed to be
similar to that of the velocity. The value of R1 at the channel entrance is
varied to account for the unknown degree of initial boundary layer development.
The values of the centerline velocity and total enthalpy must still be deter-
mined for a given value of R1. This is accomplished by matching the gross flow
features (mass and enthalpy fluxes, implicit in the assumed initial profiles)
to those flow conditions at the combustor/nozzle exit. The rocket option of
reference 3 is used to calculate the total enthalpy flux out of the combustor/
nozzle for the given hydrogen-oxygen ratio, seed fraction (SF), combustor pres-
sure (Pc), isentropic expansion ratio, and combustor efficiency. An initial
guess is made for the values of the centerline velocity and total enthalpy in
the initial profiles. From these, mass and total enthalpy fluxes can be calcu-
lated and compared to those values at the combustor/nozzle exit. The guessed
values are then adjusted and the process is repeated. This iterative procedure
Is continued until the assumed centerline values yields mass and enthalpy
fluxes that agree with the known upstream nozzle exit conditions.
Figure 2 shows the calculated temperature profiles at .09 meters down-
stream of the duct inlet for three values of seed fraction. The axial location
corresponded to the measurement station in reference 2. In these calculations
the chamber pressure equaled 1:0' kPa, wall temperature e^ »1? 4 600K, and the
2
amount of combustor heat loss was taken to be 6% of the total thermal input.
It can be seen that there is a slight drop in the predicted centerline tempera-
tures when the values of SF are increased. This trend is as expected. The
alkali-metal seed is injected into the combustion chamber in the form of CsOH
dissolved in water. As the seed fraction is increased, this diluent will de-
crease the flame temperature and consequently the gas temperature.
For comparison, the measured temperature profiles at the corresponding
seed fractions are also plotted in Figure 2. There ere some run-to-run devia-
tions in the experimental data and the results shown are from typical runs.
The agrecoent in the profile shapes between the developing flow results and the
measured data is due to an adjustment of the amount of boundary layer develop-
nent in the unknown initial profiles, as discussed earlier. There is a general
tendency to overpredict the gas temperature in the near wall region. There are
several possible explanations for these near wall discrepancies. One of the
host probable causes may have originated from the optical viewports which were
cut in the test section walls. These ports may have provoked localized three-
dimensional flow disturbances such as recirculations, flow detachments and/or
sakes. The relatively cold nitrogen gas used to purge the quartz windows may
have also worked its way to the duct wall region to lower the gas temperature
there.
The computed and measured electrical conductivity distributions for three
values of SF's are shown in Figure 3. The conductivity deteriorated badly in
the outer potion of the channel due to the rather thick thermal boundary
layers associated with the heat sink walls.
III.	 FULLY DEVELOPED FLOW CALCULATIONS
The second set of calculations were carried out using r newly developed
numerical technique to compute two-dimensional duct flows. Tio! details are
given in reference 8. Basically the technique is as follows: The total flow
in the duct is partitioned into a sequence of finite streams. The difference
equations to calculate the flow properties are then obtained by applying the
conservation principles directly to the individual streams.
The results presented in this section are for fully developed flows.
Computations were carried out along the duct until the axial variations of the
velocity and temperature radial profiles became negligible. To save computa-
tional time, the initial profiles at the beginning of the duct were mocked up,
guided by the results of numerical experimentation, to produce fully developed
profiles in about two pipe diameters (approximately 10 cm) from the starting
point.
Due to the nature of the experimental duct, some features of the flow are
difficult to account for during numerical simulation. For example, weak oblique
shock waves were detected in the duct during the experiments of reference 1.
Some of these disturbances originated from the rocket-type nozzle which was
designed to achieve low weight rather than perfectly parallel flow at the
nozzle exit. Pressure disturbances can also arise from the optical viewports
and other "roughness" on the duct walls. Not only is this complex flow
phenomena of shock-boundary layer interaction beyond the scope of our present
two-dimensional computational methods, but also the presence of these weak pres-
sure disturbances confuse the issue as to what fraction of the pressure rise
experimentally observed along the supersonic duct is actually due to friction
at the wall. In this set of calculations, the pressure rise along the duct was
taken as a specified parameter. During the computations the shear stress at
the wall (i.e., surface roughness) was iterated upon until the calculated pres-
sure gradient equaled the specified distribution. Figure 4 shows the experi-
mental data of reference 1 for the streamwise pressure variatinn. To bracket
the data, two pressure gradient lines were drawn as shown. Thus the calcula-
tions were carried out for the specified pressure gradients of 280 and 180
kPa/m. The seed fraction equaled 5%.
Results of the computations are shown in Figure 5. This figure shows the
radial distributions of velocity and temperature for the two values of dp/dx
considered. The shear stress at the wall for 280 kPa/m pressure rise was
1.8 kPa and that for 180 kPa/m was 1.3 kPa. From the velocity distribution
curves we note a slight flattening of the profiles with lower pressure gradient.
This is in accord with what will be expected with lower wall stress (lower wall
roughness) that goes with the lower pressure gradient. The temperature profiles
for the two cases were found to be very close. The difference was too small to
be shown in the figure thus both cases are shown as a single curve.
In Figure 6 we have compared + he results for radial temperature distribu-
tion as calculated in section II and section III with the experimental data.
From this figure we note that the flow 9 cros downstream of the beginning of the
channel, where the measurements were made, is still developing.
IV. SUMMARY
Two-dimensional analyses of the NASA Lewis cesium-seeded H 2-02 MHD duct
flow have been performed. The results of two sets of computations are presented:
(1) profiles of the developing flow in a smooth duct, and (2) profiles of fully
developed pipe flow with specified streamwise shear stress distribution. The
predicted temperature and electrical conductivity profiles for the developing
flows compared well with the experimental results, although some slight dis-
crepancies were observed. The possible sources of these discrepancies are noted.
REFERENCES
1. Smith, J. M., "Results and Progress on the NASA Lewis H2-02 MHD
Program," Presented at the 15th Symposium on the Engineering Aspects of Magneto-
hydrodynamics, The University of Pennsylvania, Philadelphia, PA, May 24-16,
1976.
2. Wang, S. Y. and Smith, J. M., "Spectroscopic Determination of Elec-
trical Conductivity in an MHD Duct from Absolute Intensity Measurements,"
Proceedings 16th S m osium on the Engineering Aspects of Ma netoh drod namics,
nergy Research and Development AdminIstration, pp. V.1.1-V.1.5,
	 .
3. Gordon, S. and McBride, B. J., "Computer Program for Calculation of
Complex Chemical Equilibrium Compositions, Rocket Performance, Incident and
Reflected Shocks, and Chapman-Jouquet Detonations," NASA SP-273, 1967.
4. Svehla, R. A. and McBride, B. J., "FORTRAN IV Computer Program for
Calculations of Thermodynamic and Transport Properties of Complex Chemical
Systems," NASA TN D-7056, 1973.
5. Smith, J. M. and Nichols, L. D., "Estimation of Optimal Operating
Conditions for Hydrogen-Oxygen Cesium Seeded Magnetohydrodynamic Power Genera-
tor," NASA TN D-8374, 1977.
6. Patankar, S. V. and Spalding, D. B., Heat and Mass Trans fer in Boundary
Layers, 2nd ed., Intertext Books, London, 1970.
7. Pian, C. C. P. and Wang, S. Y., "Estimation of the Electrical Con-
ductivity in a Cesium Seeded Hydrogen-Oxygen Maonetohydrodynamic Generator Duct,"
Proposed NASA Technical Paper, 1978.
8. Greywall, M. S., "Two Dimensional Duct Flow Code," ME-GL-78-1,
Department of Mechanical Engineering, Wichita State University, Wichita,
Kansas.
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Velocity, temperature, and electrical conductivity profiles in hydrogen-oxygen MHD duct flows

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