a t 1 atm. T h e damage left by the mild explosion a t 23 atm.

, A simple explanation is that the NO contained more than

shown a t center, was much less than that from the brisant det- enough NZimpurity to saturate the liquid and solid phases of
onation a t 1 atm., shown a t right. NO.
T h e apparatus, nevertheless, was damaged by the mild det- T h e explosion behavior of all three experiments with NO
onation a t 23 atm. The control experiment with liquid N2- can now be explained by gas-bubble initiation theory as fol-
0% was made a t the same pressure, 23 atm., to assess how much lows: When N O was boiling a t 1 atm., NO vapor bubbles
damage was unique to N O . T h e inside appearance of the were there to form adiabatic hot spots for the fast chemical
two pipes after shocking Nz-02 and NO, both a t 23 atm., is initiation reactions required by the brisant mode of propa-
shown in Figure 4. Both pipes in this picture have been sawed gation. When the pressure was 3 atm. over nonevaporating
down their mid-planes through the instrument ports. The liquid-solid NO, bubbles of N2 impurity, rather than KO vapor,
inside of the pipe that contained N O was severely oxidized made enough hot spots to propagate the brisant mode,
and bulged, while the control shows no evidence of oxidation When the pressure on liquid NO a t its normal boiling point was
or explosion in the liquid. elevated to 23 atm., however, both the vapor pressure of N O
The bulges caused by the low order explosion in KO a t 23 and the Henry’s law pressure of N2 in KO ivere exceeded, and
atm. (Figure 4, right) are interesting because they occur at, no hot-spot-forming bubbles were in the liquid NO, The
or a short distance past, the instrument ports in the pipe. T h e chemical rate was too slow for the brisant mode and a lower
sensing devices seem to have momentarily increased the reac- order explosion took its place.
tion rate, causing strong pressure surges a t these locations. The
pressure surges, roughly estimated from the strength properties literature Cited
of 304 stainless steel, were about 2000 atm. A velocity of Bowden, F. P., Yoffe, A. D., “Initiation and Growth of Explosion
roughly 2000 meters per second was indicated in the second in Liquids and Solids,” Cambridge University Press, Cambridge,
half of the low order propagation in NO a t 23 atm. Very England, 1952.
Johnston, H. L., Giauque, W. F., J . Am. Chem. Soc. 51, 3194
inefficient conversion of NO to N2 and 0 2 was shown by the (1929).
--
\ - - I -

copious quantities of NO2 fumes that remained after this Lezberg, E. A., Zlatarich, S. A., National Aeronautics and Space
explosion. Administration, NASA Tech. Note D-2878(1965).
Miller, R. O., “Detonation and Two-Phase Flow,” S. S. Penner
I n each experiment with KO, the pressure in the pipe, after and F. A. Williams, eds., p. 65, Academic Press, New York,
condensation of NO, was significantly higher than the vapor 1962.
pressure of N O ; considerable gassing a t 1 atm. occurred in for review November 29, 1967
RECEIVED
each experiment even while Nz was still in the coolant jacket. ACCEPTEDApril 18, 1968

COMMUNICATION

PERF0 RATED- PI PE DI STRI BUTORS

A method for calculating the pressure drop in perforated-pipe distributors by separating the pipe into dis-
crete sections and using the summation method is presented. For short distributors or distributors with a
small number of holes, this method gives an applicable equation which is more reliable than prior equations.

THE
pressure drop in pipe distributors is calculated by Since equal distribution is assumed, the velocity in section 2 is:
separating the pipe into discrete sections and using the Vz = (1 - I/n) VI (2)
summation technique rather than the integral approach to For the case of n holes, the general form of the velocity in sec-
arrive a t the solution. For long distributors with a large tion i can be written as:
number of holes, the final solution presented here does not
contradict prior correlations such as those presented by Lapple
(1951) and Acrivos et al. (1959); however, for short distribu-
vi= (1 - - i, 1) VI (3)

tors and/or distributors with a small number of holes, this T h e equation representing the frictional pressure loss in each
development gives rise to a n extremely applicable equation section can be written using the Fanning friction factor in the
and one more reliable than the prior equations. form
Although the theory is not new, it is presented to clarify the
development. (p‘ -pP*+l)i - 2f;r‘ (4)
To describe appropriately the pressure forces existing in the
where i = section number.
pipe distributor, friction terms, flow inefficiencies, and momen-
tum recovery contributions must be considered. If the dis-
tributor is divided into n equal sections, the length of each 1
i Section
I
i Section
I
i Section
I
I Section I
section is represented by L/n where L is the distributor length.
If there is no maldistribution and if the orifice holes are of the
same size, A,, the volumetric flow rate from each side port will
be A,V,.
For section 1 in the flow distributor, a mass balance can be
written for the fluid stream (see Figure 1).
t t
Mass in = mass out VO VO
pLViAi = +
PLV~AI PLV~A, Figure 1 . Expanded schematic of pipe distributor

VOL. 7 NO. 4 OCTOBER 1 9 6 8 593

 The total pressure drop over the distributor due to friction must be conserved. Hence the velocity decrease must be
can then be found by summing the pressure drops over each accompanied by a pressure increase. For section i,
section.

Since the fluid a t each port does not exactly make a 90'
2fn(Lln) Vn2 change in direction and does not decelerate exactly to zero,
(5) Equation 9 must be modified to take this inefficiency into
Dgc
account. By incorporating into Equation 9 a constant k,
where AP = ( P I - P,). where 1 > k > 0, the momentum recovery and fluid flow
For a given distributor diameter, and fluid density and inefficiencies may be written in the form
viscosity, the friction factors are solely functions of velocity in
the respective sections using smooth tube correlations. There- k
fore, f may be written as a function of V I , V S , . . . V , in Equa-
(pi+lp- p t ) = (V? -
tion 5, Although this seems to complicate calculations, a For the entire distributor,
simple computer program can easily handle the computations.
T o facilitate hand calculations, an approximate average
friction factor may be defined. O n e method is to calculate
an average fluid velocity in the distributor and calculate fay.
Therefore, Equation 5 reduces to:
Although k is assumed to be a constant for the entire distributor,
it is recognized that practical situations exist-e.g., V , not
constant-which lead to varying velocity and pressure profiles
along the pipe length, hence varying values for k. Simplifying
and combining Equations 3 and 10, the total pressure recovery
for n holes is:
Simplifying Equation 6 :

Adding the pressure drop due to friction and that due to

momentum recovery and fluid inefficiencies, the total change in
Combining Equations 3 and 6 and simplifying, the result is: pressure over the perforated pipe distributor is :

- 2faJV1' 5[ - 6 -
n 1)]2/n
@) -
(g)
P inetion Dgo z =1 n n
k
Equation 8 has the unique advantage over other equations - [l - l/n']V12 (12)
reported in the literature for calculating frictional losses in pipe gc
distributors that it is a function of the number of holes, n.
Equation 12 can be simplified for practical applications.
I n other equations previously presented, either an integral
The summation in the first term can be carried out explicitly
approach was used which reduces to a continuous slot dis-
tributor, or the expression for the pressure drop through the
and is given for all values of n by the quantity ( n 1)(2n 1)/ + +
6n2. This quantity obviously has the limit 1/3 for large values
orifices was substituted in the equation for distributor pressure
of n. If the summation term is taken and plotted with respect
drop to obtain a solution.
to the number of holes, an asymptote of 0.33 exists for large
At each side port, two additional phenomena contribute to
values of n (Figure 3). When the number of holes reaches 20,
the pressure drop calculation. There is a fluid momentum
effect a t each port. The fluid flowing out each port ideally
decelerates to zero in the main direction of flow in the dis-
1.0
tributor, makes a right-angle turn, and flows out the side port.
.9
At each side port, a momentum balance can be made over
a control volume as in Figure 2. Because of a loss of mass of .8
fluid changing direction from X to Y,the conservation of
linear momentum dictates that momentum in the X direction

r - - ---- --- -- 1le I-X

Y
'
't' 0'
0
I
5
' "
10 15 20 25 30 35
' ' 1
vo n
Figure 2. Pressure and velocity vectors over a control Figure 3. Summation term plotted as a
volume at a port function of the number of side ports

594 l&EC PROCESS DESIGN AND DEVELOPMENT

the value of the summation is approximately 0.35; however, Nomenclature
as n decreases below 10, the value sharply increases. Further-
more, when n is sufficiently large, [l - l/n2] in the second A1 distributor cross-sectional area, sq. ft.
=
term in Equation 12 goes to 1. For sufficiently large numbers A, port cross-sectional area, sq. ft.
=
D distributor diameter, ft.
=
of side ports, Equation 12 reduces to f Fanning friction factor
=
g, gravitational constant, 32.2 (ft. lb.) (lb. force sec.’)
=
i = section number
k = momentum recovery correction factor
L = distributor length, ft.
Lapple (1951) has derived the pressure drop through a n = number of side ports or orifices
perforated pipe distributor assuming ideal momentum re- PI = pressure a t distributor inlet, p.s.i.
covery-i.e., k = 1-and uniform flow of fluid along the entire P, = pressure a t closed end of distributor, p.s.i.
AP = total pressure drop over distributor, p.s.i.
length of the pipe distributor-Le., the ports are very close to V = velocity, ft./sec.
one another, simulating a continuous slot distributor. Lapple’s V, = velocity through side ports, ft./sec.
pressure drop equation for a horizontal distributor is given as: VI = inlet velocity to distributor, ft./sec.
X,Y = direction vectors
pL = liquid density, lb./cu. ft.

This is equivalent to Equation 13 for k = 1 and n very large.

literature Cited
For small values of n-Le., less than 10-Equation 12 would
be more appropriate to use than Equation 14. Acrivos, A., Babcock, B. D., Pigford, R. L., Chem. Eng. Sci. 10,
The value of k to be used in the momentum recovery term in 112-24 (1959).
Equation 12 still needs additional investigation. Acrivos et a/. Lapple, C. E., “Fluid and Particle Mechanics,” 1st ed., pp. 14-15,
University of Delaware, Newark, Del., 1951.
(1959) present some values of k for air systems, ranging from Soucek, E., Zelnick, E. W., Trans. A m . SOC.Civil Eng. 110, 1357-401
approximately 0.6 to 0.9 and apparently not correlating with (1 945).
fluid maldistribution. O n the same basis, data based on the
work of Soucek and Zelnick (1945) in long 6-inch square E. J. G R E S K O V I C H
channels with square side ports were compared with the air J. T. O’BARA
system. The value of k for water varied from approxi-
Esso Research and Engineering Co.
mately 0.7 to 0.4 with increasing maldistribution. I t has been Florham Park, N . J . 07932
suggested that k may be better correlated with fluid velocity
or pressure drop through the side ports ; however, additional RECEIVED
for review February 1, 1968
data are needed to substantiate this. ACCEPTED May 20, 1968

CORRESPONDENCE

O P T I M A L ADIABATIC BED REACTOR W I T H COLD SHOT COOLING

SIR: The optimal design of adiabatic bed reactors with The state of the process stream a t each stage can be de-
cold shot cooling has been treated in detail by Lee and scribed by the set of state variables: entrance conversion g,
Aris (1963). I n a recent communication MalengC and Viller- exit conversion g’, exit temperature t’, cumulative relative
maux (1967) have shown that the optimizing algorithm pro- mass flow rate A/XN, cumulative profit per unit of mass flow
posed by Lee and Aris does not lead to the optimal design through that stage P. The decisions to be made a t each stage
conditions; in fact, by a direct search method on the set of are the entrance temperature, t , and the holding time, e.
six decision variables appearing in the expression for the profit The following set of equations results, corresponding to
of a three-bed reactor they could substantially improve the Equations 15, 14,16, and 18 of Lee and Ark, respectively :
profit as obtained by Lee and Aris. However, here it is
shown that even the solution of Malengt and Villermaux does
not yield the true optimum and that neither of the previous
solutions, although giving a profit close to the maximum profit,
leads to the optimal design conditions.
Although the optimizing algorithm used by Lee and Aris
fails, their mathematical formulation of the problem is correct
and it suits perfectly a discrete maximum principle approach.
I n this note we use the notation of Lee and Aris, although this
notation is more appropriate to a dynamic programming
formulation than to the maximum principle formulation used
by us. A stage consists of the catalyst bed and the preceding
bypass mixing chamber or the preceding heater (for the Nth Equations 1 to 5 are implicit forms of the “performance
stage). equations” of Fan and Wang (1964). Since the total mass

VOL. 7 NO. 4 OCTOBER 1 9 6 8 595