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On transients of airflow in mine workings
Mining Science and Technology, 6 (1988) 247 254 Elsevier Science Publishers B.V., Amsterdam
247
Printed in The Netherlands
ON TRANSIENTS OF AIRFLOW IN MINE WORKINGS W. Trutwin
*
Department of Mining Engineering, University of Nottingham, Nottingham (U.K.) (Received August 3, 1987; accepted September 14, 1987)
ABSTRACT
The paper examines the transients in airflow, which are caused by steplike changes in initial and/or boundary conditions. Results of the numerical simulation of air-
flows, based alternatively on the assumptions of incompressibility and compressibility, are presented and compared.
1. I N T R O D U C T I O N
network are considered, the difference between the real, i.e., compressible flow, and a simulation based upon the incompressibility assumption can be interpreted as a quantitative difference. This is more or less important depending on whether the verifications are being based on measurement. It is expected, that substantial qualitative differences between the real, compressible flow and a simulation based on incompressibility will occur if transient states are considered. This can be explained by the fact that the dynamics of incompressible fluid are described by non-linear ordinary differential equations, whereas the compressible fluid, i.e., airflow, is expressed by a non-linear partial differential equation, usually of a hyperbolic type. The implication of the last remark is significant. The main difference between a compressible flow and an incompressible one are the pressure waves and flow waves propagating in an airway due to disturbances, in initial a n d / o r boundary conditions. In these
A consideration of the non-steady flow in mine workings requires an assessment of the limitation of the mathematical model, which is based on the assumption that the medium is incompressible. In many opinions the assumption of incompressibility of the flow is justified, if the conditions prevailing in the mine ventilation network are normal. Under normal conditions it is understood that pressure losses and differences, and changes in air velocities due to transient states, do not exceed the range of 4 - 1 0 4 Pa and _+2 × 101 m/s. Considerations concerning transients caused by explosions, outbursts, etc., are thus excluded. If steady state conditions in the ventilation
* Address for correspondence: Strata Mechanics Research Institut, Polish Academy of Sciences, 27 ul Reymonta, Cracow, Poland. 0167-9031/88/$03.50
© 1988 Elsevier Science Publishers B.V.
248
circumstances it is rather important to realize the extent of the differences between the airflow and a simulation based on incompressibility, particularly, when transients in the network are caused by steplike changes in flow conditions. The basic aim of this paper is to visualize the difference between a compressible flow and an incompressible one. The comparison of both types of flow is based upon numerical simulation, however, it should be mentioned, that the mathematical models used are still liable to experimental verification.
2. MATHEMATICAL MODEL OF COMPRESSIBLE FLOW It is assumed that the flow in an airway is similar to the flow of a compressible fluid in the conduit. Thus, the flow is regarded as one-dimensional, with the corresponding implications concerning the parameters describing the flow. The flow will be considered in a segment of an airway marked by two stations, the inlet and outlet. Other simplifying assumptions are made concerning the cross-sectional area, elevation of the airway to such that only the significant parameters for further analysis will be taken into account.
or. if eqn. (1) is taken into account ap A ~ + ~s (Oq) =O
(3)
where 0 = O(s, t) denotes the density of air. Further consideration will be made with reference to a standard thermodynamic state, the normal state, determined by an air density p0, pressure p0 and temperature T °. Between the current parameters of the flow and the parameters expressed for normal state exist the following relation:
oq = o°q °
(4)
where, as p0= constant and the parameter qO= qO(s ' t) is the normal volume rate of flow, which can also be interpreted as a measure of mass rate of flow. The continuity equation can now be written as below: Op 0° 0q~° - 0 0t+A 0s
(5)
Assuming that disturbances in the flow cause changes in pressure and density an adiabatic transformation of thermodynamic state must be considered. The velocity of sound for the above conditions is determined by the formula: C2 =
Kp
P
2.1. Continuity equation or
The velocity in an one-dimensional flow is defined as follows:
(1)
, ( s . t ) = q(s.
where q(s, t) stands for the volume rate of flow, A(s) the cross-sectional area, s the coordinate measured along the axis of the airway and t is time. The continuity equation is in case of a one-dimensional flow as below
+
= 0
(2)
c2 = xRT
(6)
in which x = Cp/Cv; %, Cv specific heat; R the gas constant, T = T(s, t) the temperature, p = p ( s , t) pressure. Taking into account eqns. (6) and (5), the continuity equation takes the form: 0p Oq ° L 1- ~ +a~- s =0
(7)
providing that the changes in pressure and density of the airflow are small, which complies with normal conditions in the network.
249
In eqn. (7),
pOc2 a-
KA
(8)
is a constant parameter described as capacitance.
2.2. Equation of motion Assuming that between stations marking the inlet and outlet of the airway that there is no fan, the equation of motion is as follows
[1,21: Dv dz 3p OD t + gP d-ss + 07s + i = 0
(9)
where D / D t = 3/3t + v3/3s, g stands for gravitational acceleration and z = z(s) is the level above datum. Pressure losses due to friction are caused by continuous (airway) as well as local resistances (regulators, bends, etc.). Thus the gradient of pressure losses in eqn. (9) can be expressed by the formula
1 i=X~p
Ivlv 2
i = r Iq°lq °
OF
INCOMPRESSIBLE
Usually the airflow is considered to be incompatible, if changes of pressure due to the flow result in negligible changes of air density. As 0 = constant thus from eqns. (1) and (9) the equation of motion is as follows:
pO 3@
3p
-d T + G
+i=0
or after integration within the interval sl, s2: B--
B(10)
where X = M R e ) is the resistance factor, D t = D,(s) the hydraulic diameter; r = KP/A3; K = K(s) the resistance factor used in mine ventilation and P = P(s) the perimeter of the airway. For the sake of simplicity, the assumption is made that d z / d s = O and A = constant within the airway. Equation (9) may be rewritten in such a way that in certain terms the air velocity is replaced by the rate of flow with the density being eliminated. Further, eqn. (9) may be expressed in the following form: o
c2 3 7 +
3. EQUATION FLOW
+ P2 - P l + w = 0
(12)
where
or by a more conventional formula
po
This equation together with eqn. (7) describes the air flow, which is regarded as a flow of a compressible fluid in mine workings. The unknown variable in this set of equations are the function q 0 = qO(s ' t) and p = p ( s , t).
+ i=0
(11)
p°L A
(13)
denotes the acoustic mass of the airway and
f -ids or w Sz
W
=
=
R lq°lq °
(14)
s1
stands for pressure loss due to frictions; further R = K P L / A 3 is the resistance; L = s 2 s 1 the length of the airway.
4. METHODS OF SOLUTION The basic problem in connection with eqns. (7) and (11) which describe the compressible flow and eqn. (11) concerning the incompressible flow of air in mine workings, is to find the solution for given specified initial and boundary conditions. As the equations are either partial or ordinary differential equations with non-linear terms, there must
250
almost always be employed an approximate numerical method based on corresponding finite difference equations.
t
d~sdt
4.1. Compressible flow The set of partial differential equations, eqns. (7) and (11), concerning the flow in the airway, are of the hyperbolic type. These equations describe the propagation of changes (waves, jumps, discontinuities, etc.) of pressure and rate of flow with a finite velocity up and downstream in the airway. The finite velocity determines lines in the time (t) and space (s) of the flow, which are called the characteristics (Figs. 1 and 2). Therefore the preferred method to solve the dynamic problems concerning compressible flow in conduits is the method of characteristics [3,4]. The linear combination of eqns. (7) and (11) gives the following equation: L = L 1+
(15)
•'L 2 = 0
In this equation 1/k', of the unknown multiplier k', is identified as a pair of velocities + c, - c , which are equal to the velocity of the acoustic waves, travelling up and down stream in the flow. Thus the velocities of propagation of the changes with respect to a fixed point is as follows: ds - v _+ c dt
(16)
ds ÷
- dt
//
i
.
.
If v << c, as in the case considered ds +c (17) dt The pair of equations above determine the gradients of the characteristics. Equations (15) and (17) lead to a set of the so called characteristic forms of simultaneous equations for compressible flow. O0 dq0 1 dp A- d ~ +-c-d-~ + i = 0 ds + -c dt p0 dq0 A dt ds-
ds-
,'Z .
Fig. 2. Discontinuities in the flow due to boundary conditions travelling along characteristics.
.
.
//
.
~
-s
Fig. 1. Discontinuities in pressure from initial conditions travelling along characteristics up and down stream.
-
(18) 1 dp c dt + i = 0 c
dt This set of equations determines the rate of flow q0 and pressure p on the characteristics. Approximate solutions of eqns. (18), for given initial and boundary conditions, can be found by using finite difference methods. A description of the various methods are omitted since they are well documented. Attention must be drawn to the initial and boundary conditions. Initial conditions can be set independently for pressure and rate of flow. However, boundary conditions for pressure and rate of flow cannot be set indepen-
251 dently. For example, if one end of the airway has been sealed, the boundary conditions must satisfy eqn. (9) and therefore the assumption v or q 0 = 0 implies that Op/3s = 0 becomes the boundary condition. For the case when the end of the airway is connected to a chamber, the boundary conditions must fulfill the following equation p 0/) 2 Op J 3~- + 3~s = 0
(19)
This has been derived from eqn. (9) under the assumption that pressure losses and inertia forces can be neglected.
4.2. Incompressible flow Approximative solutions of eqn. (16) can easily be obtained by numerical methods based on finite difference methods. Initial conditions have to be set for the rate of flow and the boundary conditions set to specify the end pressure conditions Pl and P2Analytical solutions of eqn. (16) can be found for very simple cases of flow. For example, if Pl = P2 for t > 0 and pressure loss is given by eqn. (14), from eqn. (16) the following equations describing a transient flow due to stoppage of fan are derived: dq °
(q0)2
~ dt for q 0 > 0
(dq q ° )° 2 - ~ d t
(20a)
for q0 < 0
1
qO - -qO(o )
qO
+~/t
forq°(0),q°>0
1 -
In order to show the nature of a compressible flow due to steplike changes in initial and boundary conditions, and to be able to make a comparison with incompressible flow, a numerical simulation has been carried out. The simulation concerns a flow in an airway, whose ends are connected to chambers in which the pressure remains constant during the transients considered. The transient in air flow through the airway is caused by the instantaneous opening of a door at the inlet of the airway. Initial and boundary conditions for the flow considered are as follows:
Compressible flow p(s,O) = 0 (above barometric pressure) q°(s,O) = 0 p(O,t) = 1 kPa (above barometric pressure) p ( L, t) = 0
(above barometric pressure) q°(O,t), q°( L,t)(these boundary conditions fulfill eqn. 19)
Incompressible flow
q°(0) = 0
qO(O)
(21) ~t
(above barometric pressure) P2(t) = 0 (above barometric pressure). Data concerning the cases under consideration are enclosed in Table 1.
(20)
where ~/= R/B; B = p°L/A stands for the acoustic mass or inductance of the airway. Integration of eqn. (19) gives
1
5. SIMULATION OF A SELECTED FLOW
pl(t) = 1 kPa -
and
1
This solution is very convenient for verification because if y = 1/q °, y(0) = l / q ° ( 0 ) the solution becomes a straight line.
for q°(O), qO __<0
where q°(O) stands for the initial conditions.
TABLE 1 Data concerning simulation Length of airway (m) Cross-sectional area (m2) Perimeter (m) Resistance factor (kg/m3) Barometric pressure (kPa) Air density (kg/m3) Sound velocity (m/s) Total resistance of airway (kg/m7) Adiabatic constant
1000 4 8 10.10 3 103 1.2 346 1.25 1.4
252
r=50
(~)
~0
# E
\ c~ 8
J
\ L=1088
(m)
8 -30
Fig. 3. Transient of volume rate of flow in an airway due to initial and boundary condition described in the tex~
T=2Ia
E Er
0 (D _/ Is_
8 -28
Fig. 4. Enlargement of transient in Fig. 3.
30 <
% cr
121
L=I @@8
(m)
o
8 -30
Fig. 5. Transients of volume rate of flow in fixed points of airway.
(~;)
253
iii II"
-
L ,,
i /"
i ll
i ~,
i~"
l~J"
/
Fig. 6. Transients of pressure in fixed points of airway.
T=~
~3
3~ E
8
Fig. 7. Comparison of transients due to compressible (shaded line) and incompressible flow (solid line).
SO
25
I'
~
'
2S
S@
C~)
TINE
Fig. 8. Difference between incompressible and compressible flow (rate of flow) expressed in relative measure at s = 500 m.
254
The results of the simulations are shown in Figs. 3 to 7. The flow pattern during the transient caused by the given initial and boundary conditions is pictured on Fig. 3. The waves of flow travel from the inlet to the outlet, where they reflect, and increase, the rate of flow. The increments of the flow decline with the elapse of time until the flow reaches the steady state. Figure 4 gives the enlargement of the flow pattern at the beginning of the transient. Figure 5 shows the rate of flow at fixed points of the airway. Pressure waves associated with the flow have been pictured on Fig. 6. The amplitudes of the pressure waves decrease when pressure distribution becomes linear between the ends of the airway. Figures 7 and 8 present the quantitative and qualitative difference between the compressible and incompressible flow, which occur under the same conditions. The difference between the flows (Fig. 8) can be interpreted as an error produced if a mathematical model based on the incompressibility assumption was used to simulate compressible flow. Figure 9 presents the comparison of the considered flows during a transient caused by momentary stoppage of the fan. The decay of the flow has been presented in a non-linear form according to eqn. (21).
1
V Cm/~)
/
\
4 E lg 15 @
Transitions of flow and pressure in a compressible medium, as in the case of air in mine workings, show jigsaw like changes which look as though they are superimposed on the transients due to an incompressible flow. The choice as to whether to use, for simulation purposes, the model based on compressibility or incompressibility is mainly a question of interest and of the significance of the phenomena associated with transients to the flow problems under consideration. Attention should be drawn to the effects which the propagation of the waves of flow and pressure may produce in connection with the problems of airborne particles and the mixing of gaseous pollutants.
ACKNOWLEDGEMENTS The author wishes to thank SERC, British Coal and the University of Nottingham, and personally Professor T. Atkinson, Head of Department of Mining Engineering, Mr. I. Longson and Dr. I. Lowndes for the assistance and facilities which enabled the work to be carried out.
REFERENCES
I 3
6. CONCLUSIONS
25
5@ C=) TIME
Fig. 9. Difference between transients of an incompressible (solid line) and compressible flow (dashed line) due to a stoppage of the fan.
1 J. Litwiniszyn, Problem of dynamics of flow in conduit networks. Bull. Acad. Polonaise Sci. Lett., 1(3) (1951): 325-339. 2 W. Trutwin, Use of digital computers for study of non-steady states and automatic control problems in mine ventilation networks. Int. J. Rock Mech. Min. Sci., 9 (1972): 289-323. 3 J.A. Fox, Hydraulic Analysis of Unsteady Flow in Pipe Networks. McMillan Press Ltd., London, 1977. 4 H. Daneshyar, One-Dimensional Compressible Flow. Pergamon Press, Oxford, 1976.
247
Printed in The Netherlands
ON TRANSIENTS OF AIRFLOW IN MINE WORKINGS W. Trutwin
*
Department of Mining Engineering, University of Nottingham, Nottingham (U.K.) (Received August 3, 1987; accepted September 14, 1987)
ABSTRACT
The paper examines the transients in airflow, which are caused by steplike changes in initial and/or boundary conditions. Results of the numerical simulation of air-
flows, based alternatively on the assumptions of incompressibility and compressibility, are presented and compared.
1. I N T R O D U C T I O N
network are considered, the difference between the real, i.e., compressible flow, and a simulation based upon the incompressibility assumption can be interpreted as a quantitative difference. This is more or less important depending on whether the verifications are being based on measurement. It is expected, that substantial qualitative differences between the real, compressible flow and a simulation based on incompressibility will occur if transient states are considered. This can be explained by the fact that the dynamics of incompressible fluid are described by non-linear ordinary differential equations, whereas the compressible fluid, i.e., airflow, is expressed by a non-linear partial differential equation, usually of a hyperbolic type. The implication of the last remark is significant. The main difference between a compressible flow and an incompressible one are the pressure waves and flow waves propagating in an airway due to disturbances, in initial a n d / o r boundary conditions. In these
A consideration of the non-steady flow in mine workings requires an assessment of the limitation of the mathematical model, which is based on the assumption that the medium is incompressible. In many opinions the assumption of incompressibility of the flow is justified, if the conditions prevailing in the mine ventilation network are normal. Under normal conditions it is understood that pressure losses and differences, and changes in air velocities due to transient states, do not exceed the range of 4 - 1 0 4 Pa and _+2 × 101 m/s. Considerations concerning transients caused by explosions, outbursts, etc., are thus excluded. If steady state conditions in the ventilation
* Address for correspondence: Strata Mechanics Research Institut, Polish Academy of Sciences, 27 ul Reymonta, Cracow, Poland. 0167-9031/88/$03.50
© 1988 Elsevier Science Publishers B.V.
248
circumstances it is rather important to realize the extent of the differences between the airflow and a simulation based on incompressibility, particularly, when transients in the network are caused by steplike changes in flow conditions. The basic aim of this paper is to visualize the difference between a compressible flow and an incompressible one. The comparison of both types of flow is based upon numerical simulation, however, it should be mentioned, that the mathematical models used are still liable to experimental verification.
2. MATHEMATICAL MODEL OF COMPRESSIBLE FLOW It is assumed that the flow in an airway is similar to the flow of a compressible fluid in the conduit. Thus, the flow is regarded as one-dimensional, with the corresponding implications concerning the parameters describing the flow. The flow will be considered in a segment of an airway marked by two stations, the inlet and outlet. Other simplifying assumptions are made concerning the cross-sectional area, elevation of the airway to such that only the significant parameters for further analysis will be taken into account.
or. if eqn. (1) is taken into account ap A ~ + ~s (Oq) =O
(3)
where 0 = O(s, t) denotes the density of air. Further consideration will be made with reference to a standard thermodynamic state, the normal state, determined by an air density p0, pressure p0 and temperature T °. Between the current parameters of the flow and the parameters expressed for normal state exist the following relation:
oq = o°q °
(4)
where, as p0= constant and the parameter qO= qO(s ' t) is the normal volume rate of flow, which can also be interpreted as a measure of mass rate of flow. The continuity equation can now be written as below: Op 0° 0q~° - 0 0t+A 0s
(5)
Assuming that disturbances in the flow cause changes in pressure and density an adiabatic transformation of thermodynamic state must be considered. The velocity of sound for the above conditions is determined by the formula: C2 =
Kp
P
2.1. Continuity equation or
The velocity in an one-dimensional flow is defined as follows:
(1)
, ( s . t ) = q(s.
where q(s, t) stands for the volume rate of flow, A(s) the cross-sectional area, s the coordinate measured along the axis of the airway and t is time. The continuity equation is in case of a one-dimensional flow as below
+
= 0
(2)
c2 = xRT
(6)
in which x = Cp/Cv; %, Cv specific heat; R the gas constant, T = T(s, t) the temperature, p = p ( s , t) pressure. Taking into account eqns. (6) and (5), the continuity equation takes the form: 0p Oq ° L 1- ~ +a~- s =0
(7)
providing that the changes in pressure and density of the airflow are small, which complies with normal conditions in the network.
249
In eqn. (7),
pOc2 a-
KA
(8)
is a constant parameter described as capacitance.
2.2. Equation of motion Assuming that between stations marking the inlet and outlet of the airway that there is no fan, the equation of motion is as follows
[1,21: Dv dz 3p OD t + gP d-ss + 07s + i = 0
(9)
where D / D t = 3/3t + v3/3s, g stands for gravitational acceleration and z = z(s) is the level above datum. Pressure losses due to friction are caused by continuous (airway) as well as local resistances (regulators, bends, etc.). Thus the gradient of pressure losses in eqn. (9) can be expressed by the formula
1 i=X~p
Ivlv 2
i = r Iq°lq °
OF
INCOMPRESSIBLE
Usually the airflow is considered to be incompatible, if changes of pressure due to the flow result in negligible changes of air density. As 0 = constant thus from eqns. (1) and (9) the equation of motion is as follows:
pO 3@
3p
-d T + G
+i=0
or after integration within the interval sl, s2: B--
B(10)
where X = M R e ) is the resistance factor, D t = D,(s) the hydraulic diameter; r = KP/A3; K = K(s) the resistance factor used in mine ventilation and P = P(s) the perimeter of the airway. For the sake of simplicity, the assumption is made that d z / d s = O and A = constant within the airway. Equation (9) may be rewritten in such a way that in certain terms the air velocity is replaced by the rate of flow with the density being eliminated. Further, eqn. (9) may be expressed in the following form: o
c2 3 7 +
3. EQUATION FLOW
+ P2 - P l + w = 0
(12)
where
or by a more conventional formula
po
This equation together with eqn. (7) describes the air flow, which is regarded as a flow of a compressible fluid in mine workings. The unknown variable in this set of equations are the function q 0 = qO(s ' t) and p = p ( s , t).
+ i=0
(11)
p°L A
(13)
denotes the acoustic mass of the airway and
f -ids or w Sz
W
=
=
R lq°lq °
(14)
s1
stands for pressure loss due to frictions; further R = K P L / A 3 is the resistance; L = s 2 s 1 the length of the airway.
4. METHODS OF SOLUTION The basic problem in connection with eqns. (7) and (11) which describe the compressible flow and eqn. (11) concerning the incompressible flow of air in mine workings, is to find the solution for given specified initial and boundary conditions. As the equations are either partial or ordinary differential equations with non-linear terms, there must
250
almost always be employed an approximate numerical method based on corresponding finite difference equations.
t
d~sdt
4.1. Compressible flow The set of partial differential equations, eqns. (7) and (11), concerning the flow in the airway, are of the hyperbolic type. These equations describe the propagation of changes (waves, jumps, discontinuities, etc.) of pressure and rate of flow with a finite velocity up and downstream in the airway. The finite velocity determines lines in the time (t) and space (s) of the flow, which are called the characteristics (Figs. 1 and 2). Therefore the preferred method to solve the dynamic problems concerning compressible flow in conduits is the method of characteristics [3,4]. The linear combination of eqns. (7) and (11) gives the following equation: L = L 1+
(15)
•'L 2 = 0
In this equation 1/k', of the unknown multiplier k', is identified as a pair of velocities + c, - c , which are equal to the velocity of the acoustic waves, travelling up and down stream in the flow. Thus the velocities of propagation of the changes with respect to a fixed point is as follows: ds - v _+ c dt
(16)
ds ÷
- dt
//
i
.
.
If v << c, as in the case considered ds +c (17) dt The pair of equations above determine the gradients of the characteristics. Equations (15) and (17) lead to a set of the so called characteristic forms of simultaneous equations for compressible flow. O0 dq0 1 dp A- d ~ +-c-d-~ + i = 0 ds + -c dt p0 dq0 A dt ds-
ds-
,'Z .
Fig. 2. Discontinuities in the flow due to boundary conditions travelling along characteristics.
.
.
//
.
~
-s
Fig. 1. Discontinuities in pressure from initial conditions travelling along characteristics up and down stream.
-
(18) 1 dp c dt + i = 0 c
dt This set of equations determines the rate of flow q0 and pressure p on the characteristics. Approximate solutions of eqns. (18), for given initial and boundary conditions, can be found by using finite difference methods. A description of the various methods are omitted since they are well documented. Attention must be drawn to the initial and boundary conditions. Initial conditions can be set independently for pressure and rate of flow. However, boundary conditions for pressure and rate of flow cannot be set indepen-
251 dently. For example, if one end of the airway has been sealed, the boundary conditions must satisfy eqn. (9) and therefore the assumption v or q 0 = 0 implies that Op/3s = 0 becomes the boundary condition. For the case when the end of the airway is connected to a chamber, the boundary conditions must fulfill the following equation p 0/) 2 Op J 3~- + 3~s = 0
(19)
This has been derived from eqn. (9) under the assumption that pressure losses and inertia forces can be neglected.
4.2. Incompressible flow Approximative solutions of eqn. (16) can easily be obtained by numerical methods based on finite difference methods. Initial conditions have to be set for the rate of flow and the boundary conditions set to specify the end pressure conditions Pl and P2Analytical solutions of eqn. (16) can be found for very simple cases of flow. For example, if Pl = P2 for t > 0 and pressure loss is given by eqn. (14), from eqn. (16) the following equations describing a transient flow due to stoppage of fan are derived: dq °
(q0)2
~ dt for q 0 > 0
(dq q ° )° 2 - ~ d t
(20a)
for q0 < 0
1
qO - -qO(o )
qO
+~/t
forq°(0),q°>0
1 -
In order to show the nature of a compressible flow due to steplike changes in initial and boundary conditions, and to be able to make a comparison with incompressible flow, a numerical simulation has been carried out. The simulation concerns a flow in an airway, whose ends are connected to chambers in which the pressure remains constant during the transients considered. The transient in air flow through the airway is caused by the instantaneous opening of a door at the inlet of the airway. Initial and boundary conditions for the flow considered are as follows:
Compressible flow p(s,O) = 0 (above barometric pressure) q°(s,O) = 0 p(O,t) = 1 kPa (above barometric pressure) p ( L, t) = 0
(above barometric pressure) q°(O,t), q°( L,t)(these boundary conditions fulfill eqn. 19)
Incompressible flow
q°(0) = 0
qO(O)
(21) ~t
(above barometric pressure) P2(t) = 0 (above barometric pressure). Data concerning the cases under consideration are enclosed in Table 1.
(20)
where ~/= R/B; B = p°L/A stands for the acoustic mass or inductance of the airway. Integration of eqn. (19) gives
1
5. SIMULATION OF A SELECTED FLOW
pl(t) = 1 kPa -
and
1
This solution is very convenient for verification because if y = 1/q °, y(0) = l / q ° ( 0 ) the solution becomes a straight line.
for q°(O), qO __<0
where q°(O) stands for the initial conditions.
TABLE 1 Data concerning simulation Length of airway (m) Cross-sectional area (m2) Perimeter (m) Resistance factor (kg/m3) Barometric pressure (kPa) Air density (kg/m3) Sound velocity (m/s) Total resistance of airway (kg/m7) Adiabatic constant
1000 4 8 10.10 3 103 1.2 346 1.25 1.4
252
r=50
(~)
~0
# E
\ c~ 8
J
\ L=1088
(m)
8 -30
Fig. 3. Transient of volume rate of flow in an airway due to initial and boundary condition described in the tex~
T=2Ia
E Er
0 (D _/ Is_
8 -28
Fig. 4. Enlargement of transient in Fig. 3.
30 <
% cr
121
L=I @@8
(m)
o
8 -30
Fig. 5. Transients of volume rate of flow in fixed points of airway.
(~;)
253
iii II"
-
L ,,
i /"
i ll
i ~,
i~"
l~J"
/
Fig. 6. Transients of pressure in fixed points of airway.
T=~
~3
3~ E
8
Fig. 7. Comparison of transients due to compressible (shaded line) and incompressible flow (solid line).
SO
25
I'
~
'
2S
S@
C~)
TINE
Fig. 8. Difference between incompressible and compressible flow (rate of flow) expressed in relative measure at s = 500 m.
254
The results of the simulations are shown in Figs. 3 to 7. The flow pattern during the transient caused by the given initial and boundary conditions is pictured on Fig. 3. The waves of flow travel from the inlet to the outlet, where they reflect, and increase, the rate of flow. The increments of the flow decline with the elapse of time until the flow reaches the steady state. Figure 4 gives the enlargement of the flow pattern at the beginning of the transient. Figure 5 shows the rate of flow at fixed points of the airway. Pressure waves associated with the flow have been pictured on Fig. 6. The amplitudes of the pressure waves decrease when pressure distribution becomes linear between the ends of the airway. Figures 7 and 8 present the quantitative and qualitative difference between the compressible and incompressible flow, which occur under the same conditions. The difference between the flows (Fig. 8) can be interpreted as an error produced if a mathematical model based on the incompressibility assumption was used to simulate compressible flow. Figure 9 presents the comparison of the considered flows during a transient caused by momentary stoppage of the fan. The decay of the flow has been presented in a non-linear form according to eqn. (21).
1
V Cm/~)
/
\
4 E lg 15 @
Transitions of flow and pressure in a compressible medium, as in the case of air in mine workings, show jigsaw like changes which look as though they are superimposed on the transients due to an incompressible flow. The choice as to whether to use, for simulation purposes, the model based on compressibility or incompressibility is mainly a question of interest and of the significance of the phenomena associated with transients to the flow problems under consideration. Attention should be drawn to the effects which the propagation of the waves of flow and pressure may produce in connection with the problems of airborne particles and the mixing of gaseous pollutants.
ACKNOWLEDGEMENTS The author wishes to thank SERC, British Coal and the University of Nottingham, and personally Professor T. Atkinson, Head of Department of Mining Engineering, Mr. I. Longson and Dr. I. Lowndes for the assistance and facilities which enabled the work to be carried out.
REFERENCES
I 3
6. CONCLUSIONS
25
5@ C=) TIME
Fig. 9. Difference between transients of an incompressible (solid line) and compressible flow (dashed line) due to a stoppage of the fan.
1 J. Litwiniszyn, Problem of dynamics of flow in conduit networks. Bull. Acad. Polonaise Sci. Lett., 1(3) (1951): 325-339. 2 W. Trutwin, Use of digital computers for study of non-steady states and automatic control problems in mine ventilation networks. Int. J. Rock Mech. Min. Sci., 9 (1972): 289-323. 3 J.A. Fox, Hydraulic Analysis of Unsteady Flow in Pipe Networks. McMillan Press Ltd., London, 1977. 4 H. Daneshyar, One-Dimensional Compressible Flow. Pergamon Press, Oxford, 1976.