AXIAL PRESSURE CORRELATION FOR ACCELERATING PARTICULATE PIPE FLOW G. J . T R E Z E K A N D
D.
M . F R A N C E
Thermal Systems Dioision, Department of Mechanical Engineering, L’niniversity of CaliJornia, Berkeley, Calt).
+
A semiempirical relation for the axial pressure distribution of the form P* = [In (ax* b ) ] ” ” has been obtained for an accelerating particulate choked flow. Values of the constants a, b, and n were determined from experimental measurements over a range o f particle sizes (1 10, 200, and 390 microns in diameter), duct diameters (0.25, 0.625-, and 0.75-inch i.d.)., and mass loading ratios (0.3 to 7). The values of the constants a, b, ctnd n are essentially independent of particle size and vary slightly with duct diameter. Typically, values of a between - 1 .O and - 1.3, b between 2.1 and 2.4, and n between 1.7 and 2.5 were obtained for the considered range of conditions. Using numerical methods, Runge-Kutta technique, calculations for the axial distribution of the gas dynamk parameters (u,, up, T,, T,, and f) were made employing the semiempirical pressure correlation. The results indicate that for nearly constant L I D ratio ducts, a variation in the duct diameter has virtually no effect on the axial distribution of calculated gas dynamic parameters (up, up, T,, and T,) within the system, while a variation in particle size for a given flow rate and loading ratio has a marked effect on the gas and particle velocities (u, and u p ) . Values of the average fricin the range of 0.001 8 and 0.0035 for the stainless steel and copper ducts, respectively, were tion factor, i, obtained for the range of flow conditions.
n extension of classical one-dimensional gas dynamics to two-phase accelerating particulate flows, involving solid particles which are suspended by turbulence in a gas stream, involves an accounting of the solid-phase momentum and energy. The fluid mechanical behavior of the particulate system is governed by a set of five coupled, simultaneous, ordinary differential equations Lrhich physically represent the folloiring : combined continuity of phases and equation of state, over-all momentum. over-all energy, drag on the solid phase owing to particle velocity lag, and heat transfer between phases. An analytical formulation of this nature was first introduced by So0 (1961, 1967). Unfortunately even when mass flow rates of solid and gas are known, the five governing equations involve six unknown parameters. These are the particle and fluid velocities, U p and fig, the temperatures of particle and fluid phases, T , and T,, the static pressure along the channel, I’, and the friction factor, f. This situation is inherent in the one-dimensional treatment of internal flow problems. I n certain specific instances, such as the flow through a converging nozzle as a first approximation, the wall friction can be neglected, and the resulting set of equations can be solved either in closed form by assuming a linear axial pressure distribution, or by numerical methods which for the most part have employed a Runge-Kutta technique. However, Lvhen it is necessary to include the effect of wall friction, an alterna.te, semiempirical method of solution is used, which involves the experimental determination of one of the unknown parameters, usually the axial static pressure distribution. I t poses the least problem experimentally. Once the pressure is known, the remaining five parameters may be determined using the numerical RungeKutta technique. Applications of this nature have been used successfully for the study of nozzles (Hultberg and Soo, 1965). The semiempirical approach must be employed in the analysis of accelerating particulate pipe flows (Trezek and Soo, 1966) since, as in the case of classical gas dynamic Fanno flow, the wall friction contributes to the acceleration of the flow.
A
YPICAL
OPENIN
AT’ RRE SSURE T A P I/CONNECTION
SELECTOR VALVES PRESSURE TRANSDUCERS
a CEC
C.E.C. CARRIER AMPLlFl ER
/
C.EC. HIGH SPEED RECORDER
Shielded mble __r_
GAGE
Figure 1.
Schematic diagram of experimental apparatus
Thus, one cannot neglect the effects of wall friction and still expect to obtain realistic values of the five other parameters. However, the evaluation of the gas dynamic parameters for accelerating particulate f l o w lvould be simplified if a functional relationship for the axial pressure distribution were available. Ideally, such a relationship should reflect the influence of the particle size and mass loading ratio. The evolution of a semiempirical axial pressure correlation is discussed. Subsequent observations of the effect of duct diameter, particle size, and loading ratio on the particulate gas dynamic parameters also are reported. Experiment0 I Study
A blow-down facility (Figure 1) was used to study the accelerating particulate, choked flow in a constant cross-sectional area circular duct. The duct, fed by a converging nozzle, discharged through a quick-opening valve into a large vacuum chamber. The condition of a n adiabatic wall produced a flow situation similar to the Fanno flow of classical gas dynamics. Three physical factors which had the greatest inVOL. 7
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247
’‘Ol i
wall temperature varied at most 0.5% from room temperatuxL.. The exit of the converging nozzle feeding the duct was taken as the reference distance, X * = 0.0. Glass particles of 110, 200, and 390 microns in diameter were used in the experiments. The particles were introduced into the nozzle entrance by means of a gravimetric feeder. Maintenance of a nearly constant particle level in the feeder for the duration of a run provided a constant particle feed rate. The mass flow rate of fluid phase was measured by monitoring the vacuum chamber pressure rise. This consisted of obtaining a chart record of the output of an alphatron vacuum gage which sensed the vacuum chamber pressure. During choked-flow operation, the pressure rise in the chamber was linear with time. This enabled the gas mass flow rate to be calculated from the following relation:
6.0
6, Ib.,/min. Figure 2. Loading ratio and mass flow rate vs. particle flow rate mp Ib,
-
min
Mp*
0 $
hT
Duct I.D., Inch
Particle Size, I./
A
% % %
390 390
&
%
110
0
200
Duct Material copper copper stainless steel stainless steel
fluence on the apparatus design were the choked flow condition, which had to be maintained long enough to allow all axial pressure data to be recorded; the requirement that the particulate phase be introduced into the gas stream with relative ease and its quantity accurately monitored; and the response time of the static pressure taps coupled with the pressure transducers and electronic amplifier and recorder, which must be compatible with item one. The blow-down technique, employed in previous experiments (Trezek and Soo, 1966), was used because it afforded a convenient pressure (atmospheric) which remained constant for long time intervals, and because it greatly simplified the injection and consequent collection of the solid phase. I t was possible to evacuate the chamber from atmospheric to 130 microns of pressure in approximately 20 minutes. This pressure differential resulted in the maintenance of the choked-flow condition for 1 to 7 minutes depending on the length and cross-sectional area of the flow duct. The chamber was equipped with windows so that the discharging flow could be observed. Three flow channels having constant cross-sectional areas were employed. The inside diameters were 0.75, 0.625, and 0.250 inch, and the length-to-diameter ( L I D ) ratios were 320, 270, and 288, respectively. The quick-opening valve placed between the duct exit and the vacuum chamber was pneumatically operated and afforded an excellent means of starting and stopping the flow abruptly. Static pressure taps, consisting of 0.050-inch 0.d. hypodermic tubing, were affixed to the duct wall at 12 locations on the various ducts used. Extreme care was exercised in the placing of taps normal to the duct axis and ensuring that the inside surface of the duct was free from obstructions. Thermocouples, distributed along the duct wall (shown in Figure 1 as a single thermocouple) were used to support the assumption of an adiabatic wall. The 248
I&EC FUNDAMENTALS
where Vo and To are the vacuum chamber volume and temperature, respectively. Static pressure measurements were recorded on a C.E.C. 5-124 recording oscillograph. The selector valve manifold system allowed each of four C.E.C. 4-326, 0- to 25-p.s.i.a. range pressure transducers to be used to measure pressures at more than one tap along the duct. Direct print chart paper made results immediately available. The response time of the hypodermic tubing joining a pressure tap to the transducer manifold was of the order 5 to 7 seconds. The various experimentally investigated combinations of particle sizes, flow ducts and loading ratios are shown in Figure 2. A complete set of axial static pressure measurements were obtained for each flow situation. As the particle loading ( r l l P ) increases, the mass flow of the carrying gas tends to approach a limit. This is easily seen in the experiments performed Tvith the 0.23-inch i.d. duct, \\.here particle flow rates greater than 6 lb., per minute tended to make the validity of the particulate suspension model questionable. Consistency of the measurements is indicated, as the loading ratio (>Mp*) curves pass through zero, and the clean gas mass flow rate (GtP = 0) decreases approximately as the square of the duct diameter. Axial Pressure Correlation
Typical behavior of the axial static pressure for a choked flow condition at a loading ratio of approximately unity is shown in Figure 3 for the three flow ducts. The regularity of the curves suggests that an analytical relation exists which will predict the behavior of the axial static pressure. Various polynomial forms were considered ; holvever, an accurate curve fit for the entire duct length (x* = 0.0 to 1.0) would involve a polynomial on the order of an eighth degree or higher. In the work of Hultberg and So0 (1965), a 22-degree polynomial \vas needed to represent the axial static pressure in a nozzle for accurate numerical calculations of the remaining gas dynamic parameters to proceed. Thus, polynomial forms are not attractive in the sense that: a polynomial must be generated for each new set of pressure data, and exploratory calculations on systems Lvhere experimental data is lacking would not be feasible; and the over-all calculation time becomes excessive with high-degree polynomials. The axial static pressure could be represented by a semiempirical relation of the form
P* = [In
( U P
+
b)]l”t
2.2
h
*a 1.0 I 0.0
o.2 0
t
L
0.4 x x 0.6
0.2
0.8
1.0
Figure 4. Representation of axial pressure correlation functions
.
0
1 0.2
j ' 0.4
Duct I.D., Inch
I
0.6
0.8
1.0
X*
Figure 3.
'Typical pressure data
Duct I.D., Inch
0 L!
J/4 %
A
'/4
Mil*
1.21 0.896 0.960
=
In [--0.97
.Y*
%
2
E?' '/4
A
MI,* n a 1.21 1.7 -1.221 0.896 2.5 -1.03 0.960 2.3 -1.07
b
I
2.337 2.07 2.119
*
200 390 390
Particle Size,
200 390 390
An indication of the ability of this relation in representing the axial pressure is shown in Figure 4 for the case of the data shown in Figure 3. T h e .values of the parameters a, b , and n are shown in Figure 5 \vhich, in effect, represents all the axial pressure data. Figure 5 !jhO\vs that the parameters a and b approach constant values independent of particle size as the duct diameter decreases. For the clean gas case, the value of n increases from 1.7 for the 0.75-inch i.d. duct to 2.5 for the 0.625- and 0.25-inch ducts. T h e value of n also appears to be independent of loading ratio and to be unique for each duct studied. T h e values determined for the constants of a, b , and n are peculiar to the set of floiv ducts studied. Hoivever, the utility of this functional relationship for the pressure distribution lies in the ability to interpohte betiveen curves to arrive at a first approximation for calculations on other flow systems. If, for example, an axial pressure,distribution is required for a system employing a 0.625-.inch i.d. duct at a loading ratio of 2.0 (a loading ratio \vhere no data are available), the relation
P*
0
Particle Size,
+ 2.02]1'2,57
kvould provide a good first approximation. It is also possible to estimate the values of a, b , and n for systems employing other diameter ducts such as 0.50-inch i.d. and operating over the range of loading ratios. 'The accuracy of the approximated pressure distribution can be readily checked and altered if necessary. The mechanism by which this check is made possible is implicit in the choked-flow situation, namely the Mach number of unity at the duct exit. Using the assumed pressure distribution in the !solution of the differential equations, it is only necessary to calculate the duct exit Mach number. For experimentally considered cases, this value was within 1% of unity. If the assumed pressure distribution does not yield a result \vithin this margin of error? the values of parameters
i
0.0
0.5
1.0
1.5
*2.0 MP
2.5
3.0
3.5
Figure 5. Constants a, b, and n for the axial pressure relation P* = [ l n (ax* b)]'/"
+
0
A
Duct I.D., Inch
Particle Size, /r
?4 % %
200 390 390 110
vl
a, b , and n may be adjusted slightly until the desired result is obtained. The utility of the pressure relationship is shown by analytical studies where it is desirable to gain an insight into the behavior of the gas dynamic parameters for a rang? of duct geometries and particle loading ratios before a n actual system is selected.
VOL. 7
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249
.20r
t I
," O.11 0.4
1
.04
a
,y
.02 ,011 0
"
"
"
0.2
0.4
"
L
"
0.8
0.6
1.0
X*
Figure 6. ities
Comparison of particle veloc-
.2
I
'
.4
'
16
'
.8
'
,lo
X"
Figure 7. Comparison of gas-phase velocities
M,* = 0.5 = 1.0
-I-
-Mp*
ob
-- -
M,* = 2x5
-M,*
Duct I D., Inch
Particle Size, M
Y4
200 390 390
%
%
M,* = 0.5
- - -M,*
= 1.0
2.6 Duct I.D., Inch
Particle Size,
Y4
200 3 90 3 90
5/s
%
P
Pressure Correlation Applied to Solution of the Governing Equations
The differential equations governing a two-phase particulate flow, obtained from a one-dimensional model, have been treated in detail (Hultberg and Soo, 1965; Soo, 1961, 1967; Trezek and Soo, 1966). These relations are summarized in the Appendix. This type of initial value problem is solved as follows: to initiate a numerical Runge-Kutta type of solution to the governing equations in the flow channel, the gas dynamic parameters must be knoivn at the duct entrance, which is precisely the nozzle exit. Thus, it is necessary to solve the governing equations for the nozzle starting with the stagnation conditions as initial conditions for nozzle flow to determine the initial conditions for the flow channel, thereby enabling a solution to proceed along the duct. Values of the gas dynamic parameters ( u ~up, , T,, T,, and f) are obtained at each axial increment, and the exit Mach number is also calculated to check the accuracy of the analysis. T h e Mach number is taken as the ratio of the velocity of the gas phase to the sonic speed of the mixture. This relation is strictly valid only at the duct exit where the Mach number is unity. T h e speed of sound of the two-phase mixture, on which the Mach number of a particulate flow is based, takes the following nondimensional form (Soo, 1967)
am2 = RTo
250
T*_ dP* P* dx* 1 dP* P* dx*
I&EC FUNDAMENTALS
1 dT* T* dw*
A solution of the governing equations using the axial pressure correlation provided exit Mach numbers to within 1% of unity for all the flow parameter combinations considered. These results were also compared with the gas dynamic parameters and exit Mach numbers calculated using the pressure data represented by an eighth-degree, least-squares polynomial. Further validity of the axial pressure correlation was demonstrated through good agreement with the polynomial fit. The use of the pressure correlation also reduced the over-all calculation time. The gas dynamic parameters of pertinent interest are the axial particle and fluid velocity profiles. Calculated results of gas and particle velocities for loading ratios of 0.5, 1.0, and 2.6 are shown in Figures 6 and 7 and in Figure 8 for loading ratios of 0.75 and 2.4. The loading ratio must be used as the independent variable if a valid comparison of the effect of particle and duct diameter on the gas dynamic parameters is to be made. The profiles of Figures 6, 7, and 8 for the three duct sizes being considered were obtained by cross plotting, for the indicated loading ratios, the family of velocity profiles from the data shown in Figure 2. Figure 6 indicates a decrease in particle velocity at all axial duct locations for each constant-loading ratio (M,*) as the duct crosssectional area decreases. For a given loading ratio, the difference in the velocities of the 0.24-inch and 0.625-inch i.d. ducts is small compared with the 0.75-inch i.d. duct. This cannot be completely attributed to the decrease in duct size since the particles used in this case were nearly twice as small as in the
0.25- and 0.625-inch ducts (390 microns us. 200 microns). Hoxvever, changing the duct diameter generally has a negligible effect on particle velocity. The decrease in gas-phase velocity is approxiinately proportional to the decrease in the square of the duct cliameter (Figure 7). This behavior is no longer valid in tlie region \\.here compressibility dominates, that is, x*
> 0.8.
T h e effect of particle size on the gas dynamic parameters was found from experiments with 110- and 390-micron diameter particles accelerating in the 0.25-inch duct. T h e results for the particle and fluid velocity are shobvn in Figure 8 for loading ratios M,* = 0.75 and 2.4. As in the experiments dealing Lvith the comparison of duct diameter, the gas flow rate is sensitive to slight (3 to 4%) changes in particle loading; this effect is pronounced as the duct diameter decreases. Values of the average fr.iction coefficient
s
1
J
=
fLfdx = L O
fdx*
0
are shoivn in Figure 9. T h e values obtained for the 0.25-inch i d . duct are approximately a factor of two lower than those previously reported for the 0.75-inch i.d. ducts (Trezek and Soo, 1966). This may, in part, be due to the fact that the relative roughness for thc. 0.25-inch i d . stainless steel duct is less than the larger diarneter copper ducts. .4 comparison is also made with the values obtained from the relation given by Lel'chuk (1943),
-0
0.4
0.6
0.8
1.0
X*
Figure 8. Effect of particle size and loading ratio on particle and gas velocities
_ _ _ M u * = 0.75
--
for an accelerating subson.ic clean gas flow in a tube. T h e friction coefficient information presented serves as an alternate method of solving the differential equations. This method is used in classical gas dynamic analysis but is not as useful in a particulate floiv system as in an axial pressure relationship. T h e governing equations can be solved if either the average friction factor or the axial pressure distribution is kno\vn; both solutions require the kno\vledge of initial parameter values a t the duct inlet. Hoivever, the solution utilizing friction factor data \vas so highly sensitive to this initial data that it \vas necessary to determine experimentally the pressure a t the duct inlet (nozzle exit condition). This, in effect, defeats the purpose of an analytical means for parametrically analyzing the gas dynamic parameters of a particular flow situation.
M,,* = 2.4 Duct I.D., Inch
Porticle Size,
'/4 '/4
390 110
P
0.02r
0.01
-
0.002
-
& d
A
0.001
Duct I.D., Inch
Appendix
Summary of Particulate Gas Dynamic Equations. For a nonabsorbing fluid phase. the five coupled differential equations take the form indicated below. As a computational convenience, the set of equations is nondimensionalized with respect to the stagnation conditions, since the simple relationship for the velocity of sound of a clean gas is not valid for a particulate flow thus invalidating the usual Mach number nondimensionalizing technique. T h e experiments were performed in a manner such that the stagnation conditions were always knoivn. T h e nondimensionalizing parameters as introduced by So0 (1961), take the folloiving form:
0.2
t
Particle Size, fi 0 34 200 % 390 A 54 390 & 54 110 = 0.079 ( R e y ) ~ , , - " * ~(Lel'chuk, 1 9 4 3 )
Duct Material Capper Copper Stainless steel Stainless steel
The following governing set of nondimensional equations results from the equation of state of the gas
P
=
pyRT
(1)
and the continuity equations,
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7 NO. 2
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1968
251
a combined equation of state and continuity results if the volume occupied by the particles is neglected, that is, p g = pg, thus
where
m, - puo - = poug - RT
A
and in dimensionless form : P*u,*A*
=
T*
The over-all momentum equation which accounts for the net in flux of particle and fluid momentum and the pressure gradient is given by
1 dP* 2 cu * du * --+--L RT* dx* Q -
P* dx*
*2cu * du * u,*~ +M c P + -4cLf -=(I
RT*
dx*
DHR T*
(7)
Consequently the above relations represent a set of five coupled, ordinary, nonlinear differential equations in six unknowns, namely, the fluid and particle velocities, u p and u p ; the fluid and particle temperatures, To and T p ; the static pressure, P ; and the friction coefficient, f. Equations Modified for Duct Flow. The flow system being considered consists of a constant cross-sectional area duct fed by a converging nozzle. A solution through the duct is generated from given initial conditions at the duct entrance which correspond to those at the nozzle exit. Equation 13 is rewritten as
where the hydraulic diameter is defined as
DH = 4
area circumference
du,* - - - EDIu,* - u p * / (u,* dx * ug*up*
(8)
and the friction coefficient is given in terms of the wall shear stress as
- up*)
(17)
and Equation 1 5 after substituting for T , with Equation 6 assumes a similar form, namely
dTP* _ _-- (BG(P*A*u,* dx *
-
+ D,(T,*'
Tp*)
-
Tp*4))/up*
(9) From the first law of thermodynamics, the over-all energy equation accounting for convection from the wall to the gas and radiation from the wall to the particulate phase takes the form :
B,A*[T,* - ( T *
+ u,**)]dx* + c D M * (Tu*4- Tp*4)dx*
(18) Differentiating the combined continuity and Equation of State 6 yields dT,* T,* dug* T,* dA* = --++-+--dx* ug* dx* A* dx*
T,* dP*
(19)
~
P* dx*
The energy Equation 10 can also be differentiated, and on substitution of Equations 19 and 6 into the resulting differentiated form of the energy equation, the term d T p * / d x * is eliminated. This yields the following:
CUP
(10) where the coefficients B , and DR resulting from dimensionalizing are given by
cpDRMp*
[T,*4 - T
2Mp*up*% d* dx *
*4]
- M,*c, _ _ _d _ Tp* c
cup*
dx*
- P*u,* dA* - A*u,* C } / ( A * P *
dx *
+ 2 up*)
dx *
and
The fourth relation governs the phase interaction. Under the assumptions that particles d o not interact and steady-state drag is applicable in accelerating flow, the steady-state gassolid interaction relation becomes:
where
The term dA*/dx* vanishes for constant cross-sectional area flows. The gas temperature, T,, can then be determined from Equation 6. The remaining unknown (f)is then found from the momentum Equation 7-namely
For the case studied in which the wall temperature remained approximately constant at ambient temperature, both parameters (B, and DE)may be taken as zero. The error introduced by this simplification is insignificant. Nomenclature
A a
Taking into account the amount of heat a particle receives by convection from the gas and by radiation from the wall, the fifth governing relation is given in terms of an energy balance on a particle-namely, 252
l&EC FUNDAMENTALS
a,
b
BG
flow area, sq. ft. coefficient in the axial pressure relation, dimensionless = sonic velocity of mixture, ft./sec. = coefficient in the axial pressure relation, dimensionless = convection heat transfer parameter, between solid particles and gas, dimensionless = =
C
convection heat transfer parameter, between wall and fluid, dimensionless = drag coefficient, dimensionless = specific heat at constant pressure of gas, B.t.u./lb.,-
CP
= specific heat of solid particles, B.t.u./lb.,-OR.
Bw CD
=
O R .
D
DH DR d
ED
f
7k L MP * mb‘ ?izp
n (Shu)DH
diameter of flow passage, ft. hydraulic diameter of flow passage, ft. = radiation parameter, wall to particle, dimensionless = diameter of solid particles, ft. = drag parameter, between solid particles and gas, dimensionless = local friction factor, dimensionless = average friction factor, dimensionless = thermal conductivity of gas, B.t.u./sec.-ft.-OR. = length of flow duct, ft. = loading ratio, mass of solids to mass of air, dimensionless = mass flow rate of gas phase, lb.,/min. = mass flow rate of particulate phase, lb.,/min. = exponent in i.he axial pressure relation, dimensionless = =
=
Nusselt number of flow passage,
(LF), . ,
di-
mensionless = Nusselt number of convection between solid and gas phase,
(y),
dimensionless
= Reynolds number of relative motion between
= = = = =
= = =
solid particles and gas, dimensionless (particle Jug* - U,*/m,d Reynolds number), A!-%* static pressure, lb.f/sq. ft. initial static pressure, lb.f/sq. ft. gas constant of the fluid, ft.-lb.f/lb.,-OR. temperature of gas phase, OR. initial static temperature, O R . temperature of solid particles, OR. wall temperature, OR. characteristic: velocity (2gcTo)’’*, ft./sec.
UQ
=
velocity of gas, ft./sec.
UP
= velocity of particles, ft./sec.
X
=
space coordinate in the direction of motion, ft.
GREEKLETTERS = dynamic gas viscosity, lb.f-sec./sq. ft. I.r = gas phase density, lb.,/cu. ft. PO = particle density, Ib.,/cu. ft. PP e = emissivity of particle, dimensionless = Stefan-Boltzmann constant, B.t.u./sec.-sq. U = wall shear stress, lb.f/sq. ft. rw
~L-OR.~
SUBSCRIPTS g = gas phase 0 = reference condition = particle phase P = condition at wall W SUPERSCRIPT * = dimensionless variables Literature Cited
Hultberg, 4.J., Soo, S. L., Astronautics Acta 11 (3), 207-16 (1965). Lel’chuk, V. L., “Heat Transfer and Hydraulic Flow Resistance for Streams of Hich Velocitv.” Natl. Advisorv Comm. Aeron.. NACA TM-1054,7943. Orr, Clyde, “Particulate Technology,” p. 147, Macmillan, New York, 1966. Schlichting, H., “Boundary Layer Theory,” 4th ed., McGrawHill, New York, 1960. Soo, S. L., “Fluid Dynamics of Multiphase Systems,” Sect. 7.2. 280-93. Blaisdell Publishinv Co.. LtTaltham. Mass. 1967. Sob, S. L.’, A.I.Ch.E. J . 7 (3),-384-91 (1961).’ Trezek, G. J., Soo, S. L., “Gas Dynamics of Accelerating Particulate Flow in Circular Ducts,” Proc. Heat Transfer Fluid Mech. Inst., 1966. RECEIVED for review May 29, 1967 ACCEPTED January 4, 1968 , I
The continued investigation of the accelerating gas-solid suspension was made possible with funds from a faculty research grant while the authors were at the Gas Dynamic Laboratory, Northwestern University, Evanston, Ill.
TURBULEINT FLOW IN ROUGH PIPES J. M. ROBERTSON, J. D. M A R T l N , ’ A N D T. H . BURKHART2
Department of Theoretical and Apfllied Mechanics, University of Illinois, Urbana, Ill.
61801
Surface roughn’ess effects on frictional and temporal-mean velocity distributions in conduit and other flows past rigid boundaries are considered and modes of correlating transitional effects between smooth and rough flow are indicated. From study of air flow in 8-inch “natural” roughness and a 3-inch sand-roughened pipes, frictional pressure drop, near wall velocity profiles and turbulence profiles for pipe Reynolds number range of 1.3 )( lo4 to 28 X l o 4 are presented. These verify transition functions, clarify an uncertainty in regard to pipe factor formulation, and present new information on turbulence structure.
HE nature of turbulent flow near rough and smooth surfaces T i s a long-standing problem of fluid mechanics. Whereas for laminar-viscous flows past solid boundaries surface roughness has no effect (so long as it does not change the general surface contour appreciably), for turbulent flow the nature of the flow is intimately associated with the surface roughness. Knowledge of the nature of turbulent flow along smooth surfaces is extensive and reasonably complete, as to both the
Present address, Cornsdl Aeronautical Laboratory, Buffalo, N. Y . 2 Present address, McDoiinell-Douglas, St. Louis, Mo.
temporal-mean velocity distribution and the general characteristics of the turbulence. I n many cases the surface over which flows occur are rough and, in spite of extensive studies, there is still much to be learned from the mean-flow standpoint. As to the nature of the turbulence appearing near rough surfaces there is very little information. Characteristics of Mean-Flow Occurrences
For conduit and flat-plate boundary-layer flows, surface roughness effects appear in the frictional and velocity profile formulations. I t is conventional to present these two results VOL. 7
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2
MAY 1960
253
