Nozzle for low pressure flash tanks for ore slurry

A wear-resistant flash tank pressure let down nozzle for use in passing an ore slurry into an ore slurry flash tank to release steam from the slurry and reduce the pressure of the slurry. The nozzle has an expansion cone flaring toward the discharge end of the nozzle. The cross-sectional area of a choke section of the nozzle and the exit diameter of the expansion cone are selected to establish a relationship between pressure upstream of the nozzle and pressure in the flash tank so that underflashing, overflashing, and shock waves inside the flash tank are minimized.

BACKGROUND OF THE INVENTION 
This invention relates to the release of pressure from oxidized ore slurry 
in an autoclave circuit. In particular, the invention relates to the 
design of a nozzle system through which ore slurry passes into slurry 
flash tanks. 
Autoclave circuits are used to recover gold from refractory sulfidic ores. 
Ore leaving an autoclave is typically passed to a series of flash tanks 
where pressure is let down and steam is flashed off to cool the slurry, 
and reduce it to atmospheric pressure for further processing. Steam from 
each flash tank is recycled and contacted with autoclave feed slurry in a 
complementary splash condenser, operated at substantially the same 
pressure as the flash tank, for preheating the autoclave ore feed slurry. 
In one particular system the pressure from the autoclave slurry discharge 
is let down in two stages. In the first stage, pressure is let down from 
about 420 psig to about 120 psig. In the second stage, pressure is let 
down from about 120 psig to atmospheric. This second pressure drop 
corresponds to a much greater volume expansion than in the first stage. 
Heretofore this second pressure drop from about 120 psig to atmospheric has 
been accomplished by use of a nozzle system comprising a straight tubular 
choke extending from outside the flash tank to inside the flash tank. The 
choke was surrounded by a ceramic lined blast tube extending further into 
to flash tank, as shown in FIG. 4. As the volume of the slurry expands 
rapidly upon passage through the choke, the blast tube was violently 
impacted with steam entrained with ore slurry. Catastrophic failure of the 
blast tube, resulting in ore slurry damaging and even breaching the low 
pressure flash tank, has occurred. The typical life of such nozzle 
systems, and in particular of the blast tubes, has been relatively short, 
for example, six weeks, depending on operating parameters, ore 
characteristics, and many other factors. Rebuilding and/or replacing such 
nozzle assemblies is expensive in terms of capital costs and in terms of 
process downtime. 
SUMMARY OF THE INVENTION 
Among the several objects of the invention, therefore, are the provision of 
an extended life nozzle system for a low pressure slurry flash tank; the 
provision of an improved apparatus for preheating gold ore slurry prior to 
pressure oxidation and for reducing the pressure of pressure oxidized gold 
ore slurry after pressure oxidation; and the provision of an improved 
process for reducing the pressure of pressure oxidized gold slurry. 
Briefly, therefore, the invention is directed to a wear-resistant flash 
tank pressure let down nozzle for use in passing an ore slurry into an ore 
slurry flash tank to release steam from the slurry and reduce the pressure 
of the slurry. The nozzle has an inlet end, a discharge end, and a tubular 
passageway extending therebetween for passage of the slurry from a 
location outside the flash tank in fluid flow communication with the inlet 
end. The slurry passageway has a choke comprising a zone of the passageway 
in which its cross sectional area is smallest, the passageway flaring with 
respect to the axis thereof toward the discharge end to define an 
expansion cone. The cross-sectional area of the choke and the exit 
diameter of the expansion cone being selected to establish a relationship 
between pressure upstream of the nozzle and pressure in the flash tank so 
that shock waves inside the flash tank are weaker than shock waves inside 
a reference flash tank having identical dimensions and configuration and 
operating under identical conditions except having a reference pressure 
let down nozzle consisting of a straight choke. 
The invention is also directed to an ore slurry flash tank apparatus for 
receiving and holding pressure oxidized ore slurry for reducing the 
pressure of pressure oxidized gold ore slurry. The apparatus has a vessel 
having a bottom, a top, and side walls, and a wear-resistant flash tank 
pressure let down nozzle. The nozzle has an inlet end, a discharge end, 
and a tubular passageway extending therebetween for passage of the slurry 
from a location outside the flash tank in fluid flow communication with 
the inlet end. The slurry passageway has a choke comprising a zone of the 
passageway in which its cross sectional area is smallest, the passageway 
flaring with respect to the axis thereof toward the discharge end to 
define an expansion cone. The cross-sectional area of the choke and the 
exit diameter of the expansion cone are selected to establish a 
relationship between pressure upstream of the nozzle and pressure in the 
flash tank so that shock waves inside the flash tank are weaker than shock 
waves inside a reference flash tank having identical dimensions and 
configuration and operating under identical conditions except having a 
reference pressure let down nozzle consisting of a straight choke. 
In another aspect, the invention is directed to an apparatus for preheating 
gold ore slurry prior to pressure oxidation and for reducing the pressure 
of pressure oxidized gold ore slurry after pressure oxidation. There is a 
flash tank for receiving a volume of pressure oxidized gold ore slurry, 
the flash tank comprising a vessel having a bottom, a top, and side walls, 
and a nozzle on the top of the vessel for passing ore slurry into the 
vessel. The nozzle has an inlet end, a discharge end, and a tubular 
passageway extending therebetween for passage of the slurry from a 
location outside the flash tank in fluid flow communication with the inlet 
end. The slurry passageway has a choke comprising a zone of the passageway 
in which its cross sectional area is smallest, the passageway flaring with 
respect to the axis thereof toward the discharge end to define an 
expansion cone. The cross-sectional area of the choke and the exit 
diameter of the expansion cone are selected to establish a relationship 
between pressure upstream of the nozzle and pressure in the flash tank so 
that shock waves inside the flash tank are weaker than shock waves inside 
a reference flash tank having identical dimensions and configuration and 
operating under identical conditions except having a reference pressure 
let down nozzle consisting of a straight choke. There is a steam outlet 
for passing steam out of the flash tank, and a splash condenser for 
contacting ore slurry with steam prior to pressure oxidation of the ore 
slurry in order to preheat the ore slurry, the splash condenser having a 
steam inlet. There is also conduit for transferring steam from the steam 
outlet of the flash tank to the splash condenser. 
The invention is further directed to a process for reducing the pressure of 
pressure oxidized ore slurry from above about 100 psig to about 
atmospheric. Slurry is passed through a nozzle into a flash tank, the 
nozzle disposed on the top of the flash tank and comprising a receiving 
end and a discharge end, and a slurry passageway extending through the 
nozzle from the receiving end to the discharge end for passage of the 
slurry into the flash tank from a location outside the flash tank. The 
slurry passageway flares outwardly at an angle of about 15.degree. toward 
the discharge end to gradually reduce the pressure of the slurry and to 
direct the slurry such that it impacts a volume of slurry in the bottom of 
the flash tank. 
The invention is also directed to an ore slurry flash tank apparatus for 
receiving and holding pressure oxidized ore slurry for reducing the 
pressure of pressure oxidized gold ore slurry. There is a vessel having a 
bottom, a top, and side walls, and a nozzle on the top of the vessel for 
passing ore slurry into the vessel. The nozzle has an inlet end, a 
discharge end, and a tubular passageway extending therebetween for passage 
of the slurry from a location outside the flash tank in fluid flow 
communication with the inlet end. The slurry passageway comprises a choke 
comprising a zone of the passageway in which its cross sectional area is 
smallest, the passageway flaring with respect to the axis thereof toward 
the discharge end to define an expansion cone. The nozzle has a choke 
diameter of between about 31/2 and about 41/2 inches, an expansion cone 
exit diameter of between about 7 and about 71/2 inches, and an expansion 
cone length of between about 53/4 inches and about 61/4 inches, to 
establish a relationship between pressure upstream of the nozzle and 
pressure in the flash tank so that shock waves inside the flash tank are 
weaker than shock waves inside a reference flash tank operating under 
identical conditions except having a reference pressure let down nozzle 
comprising a straight choke. 
In another aspect, the invention is directed to an ore slurry flash tank 
apparatus for receiving and holding pressure oxidized ore slurry for 
reducing the pressure of pressure oxidized gold ore slurry from between 
about 100 psig and about 140 psig to about atmospheric. There is a vessel 
having a bottom, a top, and side walls, and a nozzle on the top of the 
vessel for passing ore slurry into the vessel. The nozzle has a tubular 
passageway extending therebetween for passage of the slurry from a 
location outside the flash tank in fluid flow communication with the inlet 
end, the slurry passageway comprising a choke comprising a zone of the 
passageway in which its cross sectional area is smallest. The passageway 
flares with respect to the axis thereof toward the discharge end to define 
an expansion cone. The nozzle has a choke diameter of between about 3.8 
and about 4.1 inches corresponding to the smallest cross-section in the 
slurry passageway, the straight section having a length of between about 
91/2 and about 101/2 inches, an expansion cone exit diameter of between 
about 7.1 and about 7.4 inches, an expansion cone length of between about 
6 inches and about 6.2 inches, and an expansion cone length of between 
about 14.degree. and about 16.degree., to establish a relationship between 
pressure upstream of the nozzle and pressure in the flash tank so that 
shock waves inside the flash tank are weaker than shock waves inside a 
reference flash tank operating under identical conditions except having a 
reference pressure let down nozzle comprising a straight choke. 
Other objects and features of the invention will be in part apparent and in 
part pointed out hereinafter.

DETAILED DESCRIPTION OF THE INVENTION 
FIG. 1 illustrates a preferred gold recovery process in which the invention 
is used. This process is described generally in Thomas et al. U.S. Pat. 
No. 5,071,477 and Thomas et al. U.S. Pat. No. 5,489,326, the entire 
disclosures of which are expressly incorporated by reference. According to 
such a process, ore is crushed and wet milled, and the ground ore slurry 
screened for trash or tramp material. The ground ore is thickened by 
removal of excess water in a solid-liquid separation operation. The ore 
slurry is then subjected to pressure oxidation in the presence of sulfuric 
acid using oxygen gas at elevated pressure. It is sometimes necessary to 
add sulfuric acid to facilitate oxidation, so the addition of sulfuric 
acid to the thickened ore slurry is indicated as an optional step. 
Pressure oxidation is typically conducted in a horizontal 
multi-compartmented autoclave, the compartments of which are preferably of 
substantially equal volume. Energy from the exothermic pressure oxidation 
is recovered by heat exchange between the oxidized slurry and acidulated 
feed to the autoclave. As indicated in FIG. 1, this heat exchange is 
preferably effected by letting down the pressure of the oxidized slurry in 
flash tanks in which the nozzles of the current invention are used, and 
using the steam which is flashed from the oxidized slurry to heat the 
autoclave feed, preferably by direct contact in splash condensers 
positioned ahead of the autoclave. 
After it is partially cooled by flashing of steam, the oxidized slurry is 
further cooled and then passed directly to a neutralization operation. 
Here lime and/or other base is added to increase the pH to allow for 
subsequent cyanide leaching. Gold is recovered from the neutralized 
oxidized slurry by, for example, carbon-in-leach cyanidation in a 
continuous countercurrent system. 
Referring to FIG. 2, ground ore slurry is directed to a trash screen 1; ore 
slurry passing through the screen is directed to a mechanical thickening 
device 2, typically a vertical tank of large diameter which provides a net 
vertical flow low enough to permit sedimentation of the solid particles. 
Overflow from the thickener is recycled to the grinding circuit. Thickened 
ore slurry underflow from the thickener is directed to another trash 
screen (not shown) and by a transfer pump 3 to a series of stirred 
acidulation tanks 5, 6 and 7, through which the slurry passes 
continuously. Although three stages are shown, in the preferred embodiment 
there are four stages. A fresh sulfuric acid stream (optional) 4 is added 
to the acidulation tanks in order to release carbon dioxide from the 
carbonate contained in the slurry, and thereby reduce the equivalent 
carbon dioxide levels in the ore. To promote removal of CO.sub.2, 
compressed air may be sparged into the acidulation tanks. 
Residue slurry leaving the acidulation tanks is fed by a transfer pump 8 to 
the first of a series of brick lined splash condensers 9 and 10, in which 
the treated feed slurry for the pressure oxidation step is preheated by 
contact with steam flashed from the oxidized slurry leaving the pressure 
oxidation. The successive splash condensers are each, preferably, 
internally baffled to promote contact between steam and liquid, and are 
respectively operated at progressively higher pressure and temperature. A 
2-stage centrifugal pump 12 is interposed to increase the pressure of the 
slurry between condensers. 
Pressure oxidation is carried out in an autoclave 15, where the slurry is 
passed through a plurality of compartments to provide a retention time of 
the order of 50-80 minutes, where it is contacted in the presence of 
sulfuric acid with oxygen gas at a temperature of between about 
185.degree. and about 225.degree. C., an oxygen partial pressure of at 
least about 25 psig and a total pressure of between about 215 and about 
480 psig. The final acidity of the slurry leaving the last compartment of 
the autoclave is between 5 and 25 grams sulfuric acid per liter of 
solution, and the final emf of the slurry is between about 480 and about 
530 mv. 
Noncondensables and steam generated during the pressure oxidation operation 
are optionally vented through a scrubber. Oxidized slurry leaving the 
autoclave is passed to a series of flash tanks 17 and 18, through control 
valves 17a and 18a, respectively, and through nozzle assemblies 41 and 42. 
In the first flash tank the pressure of the slurry is let down from about 
420 psig to about 120 psig. In the second flash tank the pressure of the 
slurry is let down from about 120 psig to about atmospheric. Steam from 
each flash tank is recycled and contacted with autoclave feed slurry in a 
complementary splash condenser, operated at substantially the same 
pressure as the flash tank, for preheating the feed slurry. Thus, in the 
series as illustrated in the drawing, the first flash tank 17 is coupled 
to the last splash condenser 10, and the second flash tank 18 is coupled 
with the first condenser 9. 
The flash tanks are vessels of generally cylindrical shape having a dished 
bottom, a dished top, and parallel side walls. As shown in FIG. 6, the 
preferred flash tank has a slurry inlet 49, a steam outlet 48, a slurry 
outlet 51, a manhole 52 to permit inspection of the tank interior, a drain 
53, and a blank outlet 54. 
In the first flash tank, where the slurry pressure is let down from about 
420 psig to about 120 psig, there is a volume expansion of the steam of 
from about 3 to about 3.5 times its volume at 420 psig. In the second 
flash tank, where the slurry pressure is let down from about 120 psig to 
about atmospheric, there is a volume expansion of the steam of from about 
8.5 to about 9.5 times its volume at 120 psig, or on the order of 30 times 
its volume at 420 psig. With regard to the second flash tank, or low 
pressure flash tank, it is preferred that the total tank volume be between 
about 1.6 and about 1.9 times the volume of slurry it is to hold at any 
given time. In the preferred embodiment where the volume of slurry in the 
tank is generally maintained between about 9000 and about 10,000 gallons 
(U.S.), and the volume of the tank is about 16,500 gallons, this helps 
ensure a slurry depth adequate to receive and dissipate the energy of 
slurry as it enters the vessel and impacts the slurry surface. 
Referring to FIG. 3, hot oxidized slurry from the flash tank 18 is 
transferred to an intermediate agitated storage tank 23. In order to 
condition the slurry for gold recovery operations, the temperature of the 
hot oxidized slurry is reduced to about 25 to 40.degree. C. by passing the 
slurry, by means of pump 24, through a series of shell and tube coolers 
25. The temperature of the slurry is reduced by exchanging heat from the 
slurry to a cooling water stream. Cooling water is obtained from a 
recirculating system in which the water is recycled through a crossflow, 
induced draft cooling tower 26 by pump 27. 
Cooled oxidized slurry which is discharged from the coolers 25 is fed 
continuously through a series of rubber or epoxy lined agitated 
neutralization tanks 28, 29 and 30, where it is neutralized with a slurry 
of lime and/or other base to raise its pH to the neighborhood of 10 to 12. 
Compressed air 34 is optionally sparged into the slurry in the 
neutralization tanks to convert ferrous iron to ferric iron, as the former 
consumes cyanide in the subsequent carbon-in-leach operation. The 
neutralized slurry is then directed to a carbon-in-leach operation by 
transfer pump 31 where the gold in the oxidized slurry is recovered by, 
for example, conventional carbon-in-leach (C-I-L) cyanidation. 
Turning now to FIGS. 4 and 5 there is shown a prior art nozzle assembly 
employed at location 42 of FIG. 2 from the second (low pressure) flash 
tank. The assembly consists of a straight choke 60 surrounded by a ceramic 
lined titanium blast tube 61. An impact zone is shown at 62 in FIG. 5 
where the blast tube is impacted with steam entrained with ore slurry as 
it rapidly expands upon entering the flash tank. 
FIGS. 6 and 7 show the nozzle of the invention, consisting of first and 
second opposite ends 44 and 45, respectively, and a steam/slurry 
passageway 46 extending through the nozzle from the first end 44 to the 
second end 45. The nozzle is constructed from a material having high 
hardness. One preferred material is a sintered alpha phase silicon carbide 
available from Carborundum (Amherst, N.Y.) under the trade designation 
Hexoloy SA. 
Slurry and steam pass into the nozzle at the first end 44 and out of the 
nozzle at the second end 45 thereof to a location inside the flash tank. 
As shown in FIG. 7, the steam passageway flares outwardly from a location 
generally halfway through the passageway, axially inwardly of the second 
end toward the second end. FIG. 6 is a schematic representation--the 
actual nozzle configuration is more accurately portrayed in FIG. 7. In one 
preferred embodiment, the slurry flow rate through the nozzle is between 
about 100 tons per hour and about 500 tons per hour of ore slurry 
comprising between about 30% and about 70% solids by weight. 
Important nozzle dimensions for the prediction and control of flashing 
behavior include the straight section or choke diameter, the expansion 
cone exit diameter, and either the expansion cone length or the expansion 
half-angle. By careful selection of these dimensions, it has been 
discovered that a relationship can be established between pressure 
upstream of the nozzle and pressure downstream of the nozzle, so that the 
development of shock waves just inside the nozzle exit, which resulted in 
excessive noise and vibration with prior designs, and internal wear can be 
minimized. In particular, the nozzle is designed so that the pressure at 
the discharge end is about the same as the pressure in the tank. The shock 
waves, noise and vibration are substantially reduced in comparison to a 
system operating under identical conditions (i.e., a "reference" flash 
tank), with the only difference being use of a straight choke flash tank 
nozzle (i.e., a "reference" nozzle). The shock waves inside the flash tank 
using the nozzle of the invention are weaker than shock waves inside a 
flash tank using a straight choke, but otherwise identical. Also, 
recondensation which occurred as a result of overflashing, which 
recondensation was deleterious to vapor-liquid separation within the flash 
tank, thereby causing excessive liquid and solids entrainment in the 
recycled steam to the preheat towers, can also be minimized. 
The choke diameter fixes the slurry mass rate of flow entering the flash 
tank at a given absolute pressure (or, alternatively, fixes the upstream 
pressure at a given mass rate of flow) according to the equation: 
##EQU1## 
where m is the mass rate of flow, A.sub.t is the choke cross-sectional 
area, and the remainder of the equation is a unique function of absolute 
pressure within the choke. A derivation of this formula is presented below 
in Appendix A. In one preferred embodiment where the pressure is to be let 
down from between about 14.degree. psia and about 100 psia to about 
atmospheric, and where the mass flow rate is from about 1500 to about 3000 
tons/day solids (50% pulp density), by use of the analysis of the 
invention, the choke diameter is from about 2.8 inches to about 4.6 
inches. 
The expansion cone exit diameter largely fixes the absolute exit pressure 
of the flashing slurry upon entering the flash tank. It is important for 
this exit pressure to match closely the pressure within the flash tank, 
for they are generally not equivalent otherwise. In the preferred 
embodiment where the pressure is to be let down from between about 140 
psia and about 100 psia to about atmospheric, and where the mass flow rate 
is from about 1500 to about 3000 tons/day solids (50% pulp density), by 
use of the analysis of the invention, the expansion cone exit diameter is 
from about 7.7 inches to about 11.5 inches. The expansion half-angle is 
between about 22.degree. and about 30.degree. where the choke diameter and 
expansion cone exit diameter are as described in this preferred 
embodiment. As alluded to above, if the exit pressure is too high (i.e., 
the nozzle does not reduce the pressure far enough), underflashing occurs, 
in which case a significant amount of flashing must occur beyond the 
nozzle. This results in a recirculating flow pattern which causes external 
wear to the nozzle casing. If the exit pressure is too low (i.e., the 
nozzle reduces the pressure too far), overflashing occurs, in which case a 
shock wave develops just inside the nozzle exit resulting in excessive 
noise and vibration, and possibly internal wear. Also, the recondensation 
which must occur as a result of the overflashing may be deleterious to 
vapor-liquid separation within the flash tank, thereby causing excessive 
liquid and solids entrainment in the recycled steam to the preheat towers. 
The expansion cone length fixes the expansion half-angle for any given set 
of choke and exit diameters. It is important that the expansion cone be 
between ten and twenty centimeters (about four and eight inches) long. 
Shorter than 10 cm, and vapor-liquid equilibrium cannot be assumed during 
flashing. Longer than about 20 cm, and friction losses may become 
significant, thereby invalidating the assumption of isentropic flow. 
Either of these two situations limits the predictability of flashing, and 
are therefore to be avoided. Hence, the optimum length of the expansion 
cone is taken to be about 15 cm (6 in). In the case of the low-pressure 
flash tanks at Barrick Goldstrike, given the necessary choke and exit 
diameters, this results in an expansion half-angle of about 15.degree.. 
This angle also has a very slight effect on exit pressure. However, once 
it is fixed, then the exit diameter may be chosen with confidence from the 
mathematical model of slurry flashing. 
The graph of FIG. 11 illustrates predicted absolute pressure versus length 
within the existing low-pressure flash system at Barrick Goldstrike at the 
design solids flow rate. This graph shows how the target downstream 
pressure may be obtained within a certain expansion cone length, given a 
certain expansion half-angle. This graph depicts the optimum design for 
the low-pressure flash nozzle, with 8.4 bar (120 psi) absolute entrance 
pressure, and 0.8 bar (12 psi) absolute exit pressure. 
Kinetic power due to steam expansion developed within the optimum design at 
the pressures shown in FIG. 11 is depicted in FIG. 12. Note that the 
kinetic power developed is only about half of a megawatt. This is 
approximately 8 times less than the original choke/blast-tube apparatus 
(which self-destructed). 
In one preferred embodiment of the invention shown in FIG. 7 the straight 
section or choke diameter is between about 31/2 and about 41/2 inches, 
preferably between about 3.8 and 4.1 inches. The expansion cone exit 
diameter is between about 7 and about 71/2 inches, preferably between 
about 7.1 and 7.4 inches. The expansion cone length is between about 53/4 
inches and about 61/4 inches, preferably between about 6 and 6.2 inches. 
The expansion half-angle is between about 12.degree. and about 18.degree., 
preferably between about 14.degree. and about 16.degree.. The straight 
section has a length of between about 8 inches and about 12 inches, more 
preferably between about 91/2 inches and about 101/2 inches. 
EXAMPLE 1 
A nozzle as described above and shown in FIG. 7 was made from Hexoloy SA 
available from Carborundum. The nozzle had an inner diameter of 4 inches 
in the first upper segment, and upper segment length of 10 inches, a lower 
segment length of 6.1 inches, the lower segment flaring at an angle of 
15.degree. from an inner diameter of 4 inches to an inner diameter of 7.25 
inches. This nozzle was installed in a low pressure flash tank for 
reducing slurry pressure from about 120 psig to about atmospheric 
pressure. The nozzle was installed without a blast tube. After 30 weeks, 
even without a blast tube, no significant wear was visible on the nozzle 
nor on the flash tank vessel walls. 
As various changes could be made in the above embodiments without departing 
from the scope of the invention, it is intended that all matter contained 
in the above description shall be interpreted as illustrative and not in a 
limiting sense. 
Appendix A 
Formulation of the Equations Governing Flashing Flow of Slurries 
Model assumptions: 
1. Homogeneous flow--vapor, liquid, and solid phases are flowing at the 
same velocity at any point within the system. 
2. Vapor-liquid equilibrium--flashing occurs via a known thermodynamic 
path. 
3. Isentropic flow--the slurry loses no energy to friction. 
4. Adiabatic flow--the slurry gains no heat, and does no work. 
5. Solid-fluid thermal equilibrium--all phases at a uniform temperature at 
any point within the system. 
Homogeneous flow theory provides the simplest technique for analyzing 
multiphase flows, and can be fairly accurate so long as the phases are 
intimately mixed, which is the case in flashing slurries. The assumption 
of vapor-liquid equilibrium is more troublesome. It is known that rapid 
acceleration and pressure changes render equilibrium theory inaccurate for 
describing the discharge of flashing steam-water mixtures through 
orifices, making it necessary to consider the rates of bubble nucleation 
and growth in the superheated liquid. However, controlled expansion 
through nozzles as short as 10 cm can be predicted with surprising 
accuracy. 
In flashing flow through nozzles, pressure drops are generally very large, 
and thus friction becomes an insignificant source of entropy. Furthermore, 
in the absence of heat sources or turbines, the adiabatic assumption can 
be safely made. Finally, the rate of heat transfer from finely ground 
solids can be safely assumed high enough to ensure thermal equilibrium 
between the solid and fluid phases. 
Derivation of the model equations: 
The basic equations for steady one-dimensional homogeneous equilibrium flow 
in a duct are: 
##EQU2## 
where m is the slurry mass rate of flow, .rho. and v are the slurry 
density and specific volume, u is velocity, A and P are the duct 
cross-sectional area and perimeter, .tau..sub.w is the average wall shear 
stress, dq/dz and dw/dz are the rate of heat input and work output per 
unit length of duct, respectively, z is the vertical coordinate, and 
.theta. is the angle of inclination of the duct to the vertical. Either 
.rho. or 1/v may be used to express slurry density. For our purposes, v is 
more convenient. 
Equation (2) may be rewritten as an explicit equation for the pressure 
drop: 
##EQU3## 
The three terms on the right side can then be regarded as frictional, 
accelerational, and gravitational components of the pressure drop: 
##EQU4## 
In the absence of significant friction losses, and when gravitational 
effects are negligible: 
##EQU5## 
and Newton's second law of motion can be stated: 
##EQU6## 
Assuming steady flow, and since each phase within the slurry shares the 
same velocity (homogeneous flow): 
##EQU7## 
where G is the rate of slurry mass flux, or the slurry mass velocity. 
Combining equations (1) and (8): 
##EQU8## 
Expanding this differential: 
##EQU9## 
By definition: 
##EQU10## 
where x is the steam quality or mass fraction of the total water occurring 
as steam, S is the pulp density or solids mass fraction within the slurry, 
and the subscripts f, g, fg and s refer to liquid water (fluid), steam 
(gas), the difference between the two, and solids, respectively. For a 
vapor-liquid system, specific volume is a unique function of pressure, 
thus: 
where: 
##EQU11## 
Typically, the steam quality gradient, dx/dz, may be calculated from the 
energy equation by equating heat transfer to latent heat changes. However, 
if significant flashing occurs, quality is a function of both enthalpy, h, 
and pressure, thus: 
##EQU12## 
By definition: 
##EQU13## 
Also, by Euler's rule: 
##EQU14## 
where: 
##EQU15## 
where C.sub.ps is the heat capacity of the solids, and the temperature T 
is a unique function of pressure from the vapor-liquid equilibrium. Thus, 
combining equations (13), (14) and (15), the steam quality gradient is 
expressed: 
##EQU16## 
and combining this with equation (12) gives the specific volume gradient: 
##EQU17## 
In the absence of a significant potential energy gradient and under 
adiabatic conditions, equation (2) may be rearranged to solve for the 
enthalpy gradient: 
##EQU18## 
where h.sub.0 is the "stagnation enthalpy." Combining equations (1), (8), 
and (18) gives the enthalpy gradient as a function of pressure drop: 
##EQU19## 
which, when combined with equation (17), gives the final expression for 
the specific volume gradient: 
##EQU20## 
which, when combined with equation (10), gives the pressure drop as a 
function of system operating variables and steam table data only: 
##EQU21## 
The choking condition is thus: 
##EQU22## 
where M is the Mach number. Hence, the critical slurry mass rate of flow 
is determined by: 
##EQU23## 
where the subscript t denotes conditions in the "throat" or narrowest 
section of the choke. Finally, combining equations (16) and (19) results 
in the final expression for steam quality gradient: 
##EQU24## 
Equations (24) and (27) are the working equations of the model. With a 
given choke or nozzle profile and the steam tables, these two equations 
may be integrated numerically to determine the critical slurry mass rate 
of flow, and the corresponding fluid properties, anywhere in the nozzle. 
Quantities of interest for plotting vs length along the nozzle include 
pressure, steam quality, Mach number, and the rate of kinetic energy, or 
kinetic power: 
##EQU25##