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8 Calculation of the steady state
8 Calculation of the steady state
8.1 Abridged calculation of the steady state The calculation procedure introduced in the present book may be employed for studying water hammer in various pipe-line systems, and also for calculating the steady state of flow. If other methods are used, calculation of the steady state of flow can be a tedious operation, especially for more complex hydraulic systems. The steady state may be calculated either as an independent problem or it may be used to determine the initial state of flow before the calculation of water hammer. In calculating the steady state, one submits any state of flow in the pipe-line system investigated as the initial state and the subsequent course of the unsteady flow in the system is calculated repetitively until such time as the flow becomes steady. The boundary conditions now are identical to those for the required steady state. The procedure described in Chapter 7 can be applied in practice for this purpose, but it is complicated by two problems. In some systems, the time for the flow to become steady is too long; moreover, it is rather difficult to check with sufficient accuracy, whether the steady state has been reached. For these reasons, the method of calculation was modified to allow us to use an abridged calculation of the steady state in conjunction with the calculation of water hammer. The basic procedure remains unchanged, but, instead of the hydraulic system submitted, another system which results in the same steady state is considered in the calculation. The modifications of the system are carried out automatically during the calculation. They are described in Sections 8.2 to 8.4. They lead to a reduction on the time needed for the steady state to be established, to a simplification of its verification and to a simplification of the submission. The course of the unsteady flow in the system substituted does not correspond to that in the original one, but this has no adverse effect, considering the purpose of the calculation.
154
Mod@cutiun of the pipe-line sections
8.2 Modification of the pipe-line sections For the steady state (dQ/dt converts to
=
0, dp/&
=
0), the system of equations (7.2), (7.3)
-dQ= o dx
It follows from the solution of Eq. (8.2) that
Q = Q, where Q, is the constant discharge at the steady state. It follows from Eq. (8. l), after substituting (8.3) for the difference in pressure between the upstream and downstream ends of the pipe-line sections that Ap, =
81~1
-IQ,I Q, n2DS
Consequently, neither the discharge through a pipe-line section nor the difference in pressure for the steady state depends on the wave velocity or on the length I of the section, provided the coefficient of friction 1is also modified so that the value of the product I1 remains unchanged. In the abridged calculation of the steady state, one uses the same wave velocity a, > 0 and the same modified length of the sections 1, = a, At
for all sections. The velocity a, may be chosen. The modified coefficient of friction I , is determined in accordance with equation (7.9), so that the value 1I is maintained. To now ascertain the steady state of flow in the sections, it suffices to check for all sections, whether the discharge at their upstream and downstream ends is the same and whether the difference in pressure at their ends corresponds to the equation (8.4). During the course of the calculation, the program checks after each calculation step, whether the absolute value of the difference of the discharge at the end points of all sections is less than the admissible difference dQ,. The difference in pressure at the upstream and downstream ends of the pipe-line are found from the calculation of water hammer. Then, the difference in pressure at the upstream and downstream ends of the pipe-line is calculated with the aid of equation (8.4). These two differencesare subtracted and the result
Calculation of the steady state
is taken as an absolute value. This absolute value is then compared with the value d p . In addition, after each step, the program ascertains whether the pressure devices, the butterfly valve, the pump and the turbine, provided they form a part the system investigated, are in a steady state. This checking in not required for other damping and pressure devices in the light of their modification (refer to Sects. 8.3 and 8.4). After all the conditions introduced above have been satisfied, the state of the hydraulic system is considered steady.
8.3 Modification of the damping devices Some damping devices extend the time needed to attain the steady state. Therefore we consider, in the abridged calculation, modified devices which lead to the same steady state of flow. They are modified so as to correspond to the steady state during each step of the calculation. Of these devices, the junction without a damping device, the constant pressure and the reservoir remain unchanged. However, the air chamber, the surge tank, the air inlet valve and cavitation are replaced by a junction without damping device. At the steady state, the discharge from the junction into these damping devices is zero (Q, = 0), as for the case of a junction without a damping device (refer to Sect. 5.1). The volume of the liquid in the air chamber and in the surge tank is not calculated, it is derived from the resulting pressure in the junction. It is only determined as late as at the beginning of the calculation of water hammer. For the air inlet valve, it is assumed that it is full of liquid at the steady state. Similarly, it is assumed that there is no cavitation. The submitted parameters of these damping devices are irrelevant for the abridged calculation: they are not used in it. The damping device denoted “overflow” is replaced by a junction without damping device only in case the pressure in the junction is lower than the pressure p,, submitted for the junction with the overflow in the basic state. Otherwise, the “overflow” damping device is considered to have the same function as in the calculation of water hammer (refer to Sect. 5.6). The parameter S submitted is irrelevant in the abridged calculation of the steady state. An integrated damping device is replaced by a junction without damping device, if it is arranged as an air chamber or an air inlet valve. If it is arranged as a chamber with overflow, it is replaced by a junction without damping device only in case the pressure in the junction is lower than at the steady state with Q, = 0 and with the liquid surface at the level of the overflow edge. Otherwise, it is considered as described in Sect. 5.9. The calculation does not include the effect of the inertia of the liquid in the chamber of the damping device; it has no effect in the steady state. The pressure in the junction is then determined by 156
Modijcation of the pressure devices
the relation PJ = Pb
+ CJ IQJI
QJ
+ Qgh,,
(8.6) is defined by
which follows from relations (5.24) and (5.25). The quantity CJ relation (5.21). In the abridged calculation of the steady state, only the submitted value N , is of importance for an integrated damping device; it defines the type of arrangement of the device. For chamber with overflow, when N , = 1, further values which affect the steady state are also significant. The damping device denoted “pressure” is treated as described in Section 5.10. In the junction, however, the pressure determined by function (5.28) for t = 0 is taken to persist indefinitely. Similarly, the damping device “discharge” is treated as described in Section 5.1 1. The discharge from the junction into the damping device takes the permanent value resulting from function (5.29) for t = 0. Of the parameters submitted for the damping devices pressure and discharge only those values are important in the abridged calculation, which determine the pressure or the discharge at time t = 0.
8.4 Modification of the pressure devices The function of some pressure devices depends on time. In the abridged calculation of the steady state, the steady state corresponding to the conditions valid for t = 0 is sought. Hence, .the presssure devices were modified to correspond to these conditions all the time. Some effects which depend on derivatives of time, are not considered, since the resulting steady state does not depend on them. The effect of inertia, though, is considered for the butterfly valve, pumps and turbines, because it favourably influences the attainment of the steady state. For these devices it is necessary to check whether a steady state has been attained. The remaining pressure devices correspond permanently to the steady state. Pressure devices such as the attachment without a pressure device (refer to Sect. 6.1), the closed pipe-line (refer to Sect. 6.2), local loss (refer to Sect. 6.3) and non-return flap valve (refer to Sect. 6.5) are included in the calculation in a similar manner as in the calculation of water hammer. The control valve is considered as described in Section 6.4, but the quantity r~ takes permanently the value resulting from relations (6.9) or (6.10) and (6.11 ) for t = 0. The effect of the butterfly valve is concidered in like manner as in the calculation of water hammer (refer to Sect. 6.6). The oil pump, however, is either permanently switched on or off, as may be seen from the submission for t = 0. In each step of the calculation, the program checks, whether the butterfly valve is in the steady state, i.e. whether 157
Calculation of the steady state
or whether the flap is in an extreme position and whether it is kept there by the resulting moment. The effect of the condenser is considered to permanently remain that given by relation (6.28), corresponding to full chambers. Of the parameters submitted, only the value [ is important. The pump and the turbine with fixed characteristics are treated as in the calculation of water hammer (refer to Sects. 6.8 and 6.9). The electric motor or the generator is either permanently connected to or disconnected from the network in the manner corresponding to the submission for t = 0. In each step of the calculation, the attainment of the steady state is checked, i.e., whether IM M,I 5 d M (8.8) This checking is unnecessary when the electric motor or the generator is connected to the network, thus ensuring a constant speed. A turbine with variable characteristics is treated in a similar manner as in the calculation of water hammer (refer to Sect. 6.10). The connection of the generator to the network or its disconnection from it and the setting of the guide and action blades is considered as corresponding permanently to the submission for t = 0. The attainment of the steady state of the turbine is checked in agreement with relation (8.8) in each calculation step. For a governor-controlled turbine (refer to Sect. 6.1 I), the effect of the network is considered to correspond permanently to that which follows from the submission for t = 0. The setting of the guide and action blades is determined by the governor. The function of the governor is modified to correspond permanently to the steady state for t = 0. All derivatives with respect to time are zero. For the calculation of the voltage Uo(t),the relation
+
is used instead of equation (6.81). To determine the voltage U , ( t ) , relation (6.82) is used, but the value of the function & ( t ) for t = 0 is considered to remain constant, so that
(8.10) To determine the voltage U,(t), we use the relation Ul u, = cP
158
(8.11)
Submitting the caIcuIation
which follows from equation (6.83) for the steady state. The voltage U , ( t ) is limited to satisfy condition (6.84). The voltage U , ( t )is determined using relations (6.85) and (6.86). The function f,(t) is considered permanently to retain its value for t = 0, so that
u, = u, +fp(O)
(8.12)
The value U , ( t ) is limited in accordance with relation (6.87). The value f,,,(O) is considered permanently instead of the value of the funciton f,,(t). The position x, of the slide valve of the hydraulic amplifier of the guide wheel and the extension y, of the piston rod of the slave cylinder are determined by numerically solving the system of equations formed by equations (6.88) and
+
(8.13) F,, F,, = 0 Equation (8.13) resulted from Eq. (6.90) for the steady state by assuming that F,
=
0
(8.14)
If a solution compatible with conditions (6.89) and (6.96) is not found, the maximum or minimum values of x, and y, are used. The parameter a defining the position of the guide blades is determined from relation (6.97). If the governor also controls the action blades of the turbine, the voltage U , ( t ) is calculated using functions (6.98) or (6.99). The subsequent procedure is similar to that determining the position of the guide blades. Relations (6.100) and (6.101) are substituted for relations (6.88) and (6.97); the subscripts are substituted for the subscripts o! in the other relations used. The steady state of the governor-controlled turbine is checked in each step of the calculation with the aid of relation (8.8). In the abridged calculation of the steady state, submission of the following values is irrelevant: xoa;xop; Cf;C,; C , ; m a ;Cop;F,; ms;C, * F f p and the functions: f b ( uO); aa(Xa); f d y + a ( y a ) ; f d y - a ( Y a ) ; ap(xfi); f d y + )* f d - p(y 1, because none are employed in the calculation. For the functions f r ( t ) ;f p ( t ) ;fmax(t), only their value for t = 0 is relevant.
f;6f;f$);fc();
8.5 Submitting the calculation The form of submission of the abridged calculation of the steady state does not differ from that of the calculation of water hammer. The difference lies only in the meaning of some quantities. The submitted values of the wave velocity a in the sections are irrelevant, since they are not used in the calculation. The initial discharge Qo in the sections may be chosen arbitrarily. It is convenient to estimate, at least approximately, dis159
Calculation of the steady state
charges corresponding to those for the steady state sought, since this shortens the calculation. The initial pressures in the junction may be chosen arbitrarily. For the same reason, it is again convenient to use values estimated to approximate the pressures corresponding to the steady state sought. An exception is the junction with the “constant pressure” damping device. In this junction, the pressure has to be given accurately, since this value represents one of the boundary conditions for the calculation. The accuracy of the calculation of the steady state may be determined by submitting the values d,, > 0; d , > 0; d , > 0. Notes concerning the submission of the parameters of the damping and pressure devices are introduced in Sects. 8.3 and 8.4. The abridged calculation of the steady state is realized, when Z = 3, 4, 7 or 8 is submitted for the type of calculation. The values corresponding to the steady state are then calculated, viz., the pressures in all the junctions and the discharges in all the sections. When the system contains the pressure devices butterfly valve, pump or turbine, additional values are determined for each of these devices, that is, the angle of tilt of the flap, the pump or turbine speed and the setting of the slide valve of the hydraulic amplifier of the guide and action blades. All these values are stored in the memory of the computer after the calculation has been completed and they may be used as the initial values for subsequent calculations of water hammer or abridged calculations of the steady state. For 2 = 3 and 4, the submitted values are used as the initial values for the abridged calculation of the steady state. For Z = 7 and 8, the values determined in the preceding abridged calculation of the steady state are used as the initial values. These values are the discharges, the pressure and, as the case may be, other quantities described above. The submitted values of these quantities are irrelevant in this case, since they are not used in the calculation. With 2 = 3 and 7, the calculation of the junctions is carried out without iteration, with Z = 4 and 8, iteration is applied (refer to Sect. 7.11). For the abridged calculation of the steady state, the wave velocity a, has to be submitted; it has the same value for all the sections (refer to Sect. 8.2). Its value need not correspond to the actual wave velocity in the pipe-line of the system investigated. Nevertheless, the wave velocity has to be chosen suitably, since it substantially affects the duration of the calculation. A reduction of the value a, up to a specific optimum value reduces the duration of the calculation. Further reduction, however, prolongs the duration and, with excessively low values of as, the calculation ceases to converge. An example of the relationship between the calculation time (expressed by the number of calculation steps t / A t ) and the value a, for the hydraulic system presented in Sect. 14.2 is shown in Fig. 8.1. 160
Submitting the calculation
The optimum value a, for a given system can only be found by trial-and-error. In many cases, safisfactory results are obtained with the smallest a, possible, though still such that the following condition is satisfied for all sections: (8.15)
Fig. 8.1 Example of the relationship between the duration of the abridged calculation of the steady state, and the wave velocity.
The value At > 0 may be chosen arbitrarily. The duration of the calculation does not depend on it. It is relevant only in some cases, when the hydraulic system includes a pump or a turbine for which a constant speed is not ensured by an attached network, or when the system contains a butterfly valve. An increase of At in such cases leads to a shortening of the duration of the calculation up to a limit. A further increase of At prolongs the duration and, with an excessively large At, the calculation ceases to converge. The optimum value At is to be found by trial-and-error for every system. The abridged calculation of the steady state is terminated when the steady state has been reached with the required accuracy. Another limit on the duration of the calculation is the attainment of the maximum calculation time tmm.When this value is reached, the calculation is terminated, even if the steady state required has not been attained. Consequently, the value t,,, has to be chosen sufficiently large. A first estimate, with a suitable choice of a, and At, may be made using the relation t,,, = 40NsAt
(8.16)
In the abridged calculation of the steady state, it is usually convenient to make 161
Calculation of the steady state
use of the graphical output of certain parameters on the screen; thus, the steadying of the flow can be observed and the quantities a,, At and tmaxmodified, if necessary. _t
At
150
100
50
01
-
OD1 062 0.03 0.06 At k i Fig. 8.2 Example of the relationship between the duration of the calculation of the steady state, the time interval At of the calculation and the wave velocity a,.
The influence of the choice of the values At and a, on the duration of the calculation of the steady state, which is established after the fading of water hammer, for the hydraulic system presented in Sect. 14.12, is shown in Fig. 8.2. In the calculation, we used the submission contained in Table 14.24 and in the D12.WTH file on the WTHM diskette, where the folloving data were changed. We estimated the initial discharge Qo = 5 m s-'; the initial opening avo = 1.425 rad of the butterfly valve and the constants B,, = 6000 kg m-7, B,- = -4000 kgm-', B,, = 1300 kg m-4 and B,- = - 1000 kg m-4 of the pressure and moment characteristics, respectively. The calculation denoted 2 = 4 was used and the graphical output was realized for each calculation step At. The values a, and At were chosen within the intervals 25 Ia, I150 m s-' and 0.001 IAt I0.046 s, respectively. When the abridged calculation of the steady state serves to determine the initial values for the subsequent calculation of water hammer, the complete submission of data for the calculation of water hammer may already be introduced for the abridged calculation. The subsequent submission for the calculation of water hammer will then consist only of two entries, viz., the lines beginning with numbers 5 and 7, containing the parameters defining the type of calculation and its verbal denotation. 162
8.1 Abridged calculation of the steady state The calculation procedure introduced in the present book may be employed for studying water hammer in various pipe-line systems, and also for calculating the steady state of flow. If other methods are used, calculation of the steady state of flow can be a tedious operation, especially for more complex hydraulic systems. The steady state may be calculated either as an independent problem or it may be used to determine the initial state of flow before the calculation of water hammer. In calculating the steady state, one submits any state of flow in the pipe-line system investigated as the initial state and the subsequent course of the unsteady flow in the system is calculated repetitively until such time as the flow becomes steady. The boundary conditions now are identical to those for the required steady state. The procedure described in Chapter 7 can be applied in practice for this purpose, but it is complicated by two problems. In some systems, the time for the flow to become steady is too long; moreover, it is rather difficult to check with sufficient accuracy, whether the steady state has been reached. For these reasons, the method of calculation was modified to allow us to use an abridged calculation of the steady state in conjunction with the calculation of water hammer. The basic procedure remains unchanged, but, instead of the hydraulic system submitted, another system which results in the same steady state is considered in the calculation. The modifications of the system are carried out automatically during the calculation. They are described in Sections 8.2 to 8.4. They lead to a reduction on the time needed for the steady state to be established, to a simplification of its verification and to a simplification of the submission. The course of the unsteady flow in the system substituted does not correspond to that in the original one, but this has no adverse effect, considering the purpose of the calculation.
154
Mod@cutiun of the pipe-line sections
8.2 Modification of the pipe-line sections For the steady state (dQ/dt converts to
=
0, dp/&
=
0), the system of equations (7.2), (7.3)
-dQ= o dx
It follows from the solution of Eq. (8.2) that
Q = Q, where Q, is the constant discharge at the steady state. It follows from Eq. (8. l), after substituting (8.3) for the difference in pressure between the upstream and downstream ends of the pipe-line sections that Ap, =
81~1
-IQ,I Q, n2DS
Consequently, neither the discharge through a pipe-line section nor the difference in pressure for the steady state depends on the wave velocity or on the length I of the section, provided the coefficient of friction 1is also modified so that the value of the product I1 remains unchanged. In the abridged calculation of the steady state, one uses the same wave velocity a, > 0 and the same modified length of the sections 1, = a, At
for all sections. The velocity a, may be chosen. The modified coefficient of friction I , is determined in accordance with equation (7.9), so that the value 1I is maintained. To now ascertain the steady state of flow in the sections, it suffices to check for all sections, whether the discharge at their upstream and downstream ends is the same and whether the difference in pressure at their ends corresponds to the equation (8.4). During the course of the calculation, the program checks after each calculation step, whether the absolute value of the difference of the discharge at the end points of all sections is less than the admissible difference dQ,. The difference in pressure at the upstream and downstream ends of the pipe-line are found from the calculation of water hammer. Then, the difference in pressure at the upstream and downstream ends of the pipe-line is calculated with the aid of equation (8.4). These two differencesare subtracted and the result
Calculation of the steady state
is taken as an absolute value. This absolute value is then compared with the value d p . In addition, after each step, the program ascertains whether the pressure devices, the butterfly valve, the pump and the turbine, provided they form a part the system investigated, are in a steady state. This checking in not required for other damping and pressure devices in the light of their modification (refer to Sects. 8.3 and 8.4). After all the conditions introduced above have been satisfied, the state of the hydraulic system is considered steady.
8.3 Modification of the damping devices Some damping devices extend the time needed to attain the steady state. Therefore we consider, in the abridged calculation, modified devices which lead to the same steady state of flow. They are modified so as to correspond to the steady state during each step of the calculation. Of these devices, the junction without a damping device, the constant pressure and the reservoir remain unchanged. However, the air chamber, the surge tank, the air inlet valve and cavitation are replaced by a junction without damping device. At the steady state, the discharge from the junction into these damping devices is zero (Q, = 0), as for the case of a junction without a damping device (refer to Sect. 5.1). The volume of the liquid in the air chamber and in the surge tank is not calculated, it is derived from the resulting pressure in the junction. It is only determined as late as at the beginning of the calculation of water hammer. For the air inlet valve, it is assumed that it is full of liquid at the steady state. Similarly, it is assumed that there is no cavitation. The submitted parameters of these damping devices are irrelevant for the abridged calculation: they are not used in it. The damping device denoted “overflow” is replaced by a junction without damping device only in case the pressure in the junction is lower than the pressure p,, submitted for the junction with the overflow in the basic state. Otherwise, the “overflow” damping device is considered to have the same function as in the calculation of water hammer (refer to Sect. 5.6). The parameter S submitted is irrelevant in the abridged calculation of the steady state. An integrated damping device is replaced by a junction without damping device, if it is arranged as an air chamber or an air inlet valve. If it is arranged as a chamber with overflow, it is replaced by a junction without damping device only in case the pressure in the junction is lower than at the steady state with Q, = 0 and with the liquid surface at the level of the overflow edge. Otherwise, it is considered as described in Sect. 5.9. The calculation does not include the effect of the inertia of the liquid in the chamber of the damping device; it has no effect in the steady state. The pressure in the junction is then determined by 156
Modijcation of the pressure devices
the relation PJ = Pb
+ CJ IQJI
QJ
+ Qgh,,
(8.6) is defined by
which follows from relations (5.24) and (5.25). The quantity CJ relation (5.21). In the abridged calculation of the steady state, only the submitted value N , is of importance for an integrated damping device; it defines the type of arrangement of the device. For chamber with overflow, when N , = 1, further values which affect the steady state are also significant. The damping device denoted “pressure” is treated as described in Section 5.10. In the junction, however, the pressure determined by function (5.28) for t = 0 is taken to persist indefinitely. Similarly, the damping device “discharge” is treated as described in Section 5.1 1. The discharge from the junction into the damping device takes the permanent value resulting from function (5.29) for t = 0. Of the parameters submitted for the damping devices pressure and discharge only those values are important in the abridged calculation, which determine the pressure or the discharge at time t = 0.
8.4 Modification of the pressure devices The function of some pressure devices depends on time. In the abridged calculation of the steady state, the steady state corresponding to the conditions valid for t = 0 is sought. Hence, .the presssure devices were modified to correspond to these conditions all the time. Some effects which depend on derivatives of time, are not considered, since the resulting steady state does not depend on them. The effect of inertia, though, is considered for the butterfly valve, pumps and turbines, because it favourably influences the attainment of the steady state. For these devices it is necessary to check whether a steady state has been attained. The remaining pressure devices correspond permanently to the steady state. Pressure devices such as the attachment without a pressure device (refer to Sect. 6.1), the closed pipe-line (refer to Sect. 6.2), local loss (refer to Sect. 6.3) and non-return flap valve (refer to Sect. 6.5) are included in the calculation in a similar manner as in the calculation of water hammer. The control valve is considered as described in Section 6.4, but the quantity r~ takes permanently the value resulting from relations (6.9) or (6.10) and (6.11 ) for t = 0. The effect of the butterfly valve is concidered in like manner as in the calculation of water hammer (refer to Sect. 6.6). The oil pump, however, is either permanently switched on or off, as may be seen from the submission for t = 0. In each step of the calculation, the program checks, whether the butterfly valve is in the steady state, i.e. whether 157
Calculation of the steady state
or whether the flap is in an extreme position and whether it is kept there by the resulting moment. The effect of the condenser is considered to permanently remain that given by relation (6.28), corresponding to full chambers. Of the parameters submitted, only the value [ is important. The pump and the turbine with fixed characteristics are treated as in the calculation of water hammer (refer to Sects. 6.8 and 6.9). The electric motor or the generator is either permanently connected to or disconnected from the network in the manner corresponding to the submission for t = 0. In each step of the calculation, the attainment of the steady state is checked, i.e., whether IM M,I 5 d M (8.8) This checking is unnecessary when the electric motor or the generator is connected to the network, thus ensuring a constant speed. A turbine with variable characteristics is treated in a similar manner as in the calculation of water hammer (refer to Sect. 6.10). The connection of the generator to the network or its disconnection from it and the setting of the guide and action blades is considered as corresponding permanently to the submission for t = 0. The attainment of the steady state of the turbine is checked in agreement with relation (8.8) in each calculation step. For a governor-controlled turbine (refer to Sect. 6.1 I), the effect of the network is considered to correspond permanently to that which follows from the submission for t = 0. The setting of the guide and action blades is determined by the governor. The function of the governor is modified to correspond permanently to the steady state for t = 0. All derivatives with respect to time are zero. For the calculation of the voltage Uo(t),the relation
+
is used instead of equation (6.81). To determine the voltage U , ( t ) , relation (6.82) is used, but the value of the function & ( t ) for t = 0 is considered to remain constant, so that
(8.10) To determine the voltage U,(t), we use the relation Ul u, = cP
158
(8.11)
Submitting the caIcuIation
which follows from equation (6.83) for the steady state. The voltage U , ( t ) is limited to satisfy condition (6.84). The voltage U , ( t )is determined using relations (6.85) and (6.86). The function f,(t) is considered permanently to retain its value for t = 0, so that
u, = u, +fp(O)
(8.12)
The value U , ( t ) is limited in accordance with relation (6.87). The value f,,,(O) is considered permanently instead of the value of the funciton f,,(t). The position x, of the slide valve of the hydraulic amplifier of the guide wheel and the extension y, of the piston rod of the slave cylinder are determined by numerically solving the system of equations formed by equations (6.88) and
+
(8.13) F,, F,, = 0 Equation (8.13) resulted from Eq. (6.90) for the steady state by assuming that F,
=
0
(8.14)
If a solution compatible with conditions (6.89) and (6.96) is not found, the maximum or minimum values of x, and y, are used. The parameter a defining the position of the guide blades is determined from relation (6.97). If the governor also controls the action blades of the turbine, the voltage U , ( t ) is calculated using functions (6.98) or (6.99). The subsequent procedure is similar to that determining the position of the guide blades. Relations (6.100) and (6.101) are substituted for relations (6.88) and (6.97); the subscripts are substituted for the subscripts o! in the other relations used. The steady state of the governor-controlled turbine is checked in each step of the calculation with the aid of relation (8.8). In the abridged calculation of the steady state, submission of the following values is irrelevant: xoa;xop; Cf;C,; C , ; m a ;Cop;F,; ms;C, * F f p and the functions: f b ( uO); aa(Xa); f d y + a ( y a ) ; f d y - a ( Y a ) ; ap(xfi); f d y + )* f d - p(y 1, because none are employed in the calculation. For the functions f r ( t ) ;f p ( t ) ;fmax(t), only their value for t = 0 is relevant.
f;6f;f$);fc();
8.5 Submitting the calculation The form of submission of the abridged calculation of the steady state does not differ from that of the calculation of water hammer. The difference lies only in the meaning of some quantities. The submitted values of the wave velocity a in the sections are irrelevant, since they are not used in the calculation. The initial discharge Qo in the sections may be chosen arbitrarily. It is convenient to estimate, at least approximately, dis159
Calculation of the steady state
charges corresponding to those for the steady state sought, since this shortens the calculation. The initial pressures in the junction may be chosen arbitrarily. For the same reason, it is again convenient to use values estimated to approximate the pressures corresponding to the steady state sought. An exception is the junction with the “constant pressure” damping device. In this junction, the pressure has to be given accurately, since this value represents one of the boundary conditions for the calculation. The accuracy of the calculation of the steady state may be determined by submitting the values d,, > 0; d , > 0; d , > 0. Notes concerning the submission of the parameters of the damping and pressure devices are introduced in Sects. 8.3 and 8.4. The abridged calculation of the steady state is realized, when Z = 3, 4, 7 or 8 is submitted for the type of calculation. The values corresponding to the steady state are then calculated, viz., the pressures in all the junctions and the discharges in all the sections. When the system contains the pressure devices butterfly valve, pump or turbine, additional values are determined for each of these devices, that is, the angle of tilt of the flap, the pump or turbine speed and the setting of the slide valve of the hydraulic amplifier of the guide and action blades. All these values are stored in the memory of the computer after the calculation has been completed and they may be used as the initial values for subsequent calculations of water hammer or abridged calculations of the steady state. For 2 = 3 and 4, the submitted values are used as the initial values for the abridged calculation of the steady state. For Z = 7 and 8, the values determined in the preceding abridged calculation of the steady state are used as the initial values. These values are the discharges, the pressure and, as the case may be, other quantities described above. The submitted values of these quantities are irrelevant in this case, since they are not used in the calculation. With 2 = 3 and 7, the calculation of the junctions is carried out without iteration, with Z = 4 and 8, iteration is applied (refer to Sect. 7.11). For the abridged calculation of the steady state, the wave velocity a, has to be submitted; it has the same value for all the sections (refer to Sect. 8.2). Its value need not correspond to the actual wave velocity in the pipe-line of the system investigated. Nevertheless, the wave velocity has to be chosen suitably, since it substantially affects the duration of the calculation. A reduction of the value a, up to a specific optimum value reduces the duration of the calculation. Further reduction, however, prolongs the duration and, with excessively low values of as, the calculation ceases to converge. An example of the relationship between the calculation time (expressed by the number of calculation steps t / A t ) and the value a, for the hydraulic system presented in Sect. 14.2 is shown in Fig. 8.1. 160
Submitting the calculation
The optimum value a, for a given system can only be found by trial-and-error. In many cases, safisfactory results are obtained with the smallest a, possible, though still such that the following condition is satisfied for all sections: (8.15)
Fig. 8.1 Example of the relationship between the duration of the abridged calculation of the steady state, and the wave velocity.
The value At > 0 may be chosen arbitrarily. The duration of the calculation does not depend on it. It is relevant only in some cases, when the hydraulic system includes a pump or a turbine for which a constant speed is not ensured by an attached network, or when the system contains a butterfly valve. An increase of At in such cases leads to a shortening of the duration of the calculation up to a limit. A further increase of At prolongs the duration and, with an excessively large At, the calculation ceases to converge. The optimum value At is to be found by trial-and-error for every system. The abridged calculation of the steady state is terminated when the steady state has been reached with the required accuracy. Another limit on the duration of the calculation is the attainment of the maximum calculation time tmm.When this value is reached, the calculation is terminated, even if the steady state required has not been attained. Consequently, the value t,,, has to be chosen sufficiently large. A first estimate, with a suitable choice of a, and At, may be made using the relation t,,, = 40NsAt
(8.16)
In the abridged calculation of the steady state, it is usually convenient to make 161
Calculation of the steady state
use of the graphical output of certain parameters on the screen; thus, the steadying of the flow can be observed and the quantities a,, At and tmaxmodified, if necessary. _t
At
150
100
50
01
-
OD1 062 0.03 0.06 At k i Fig. 8.2 Example of the relationship between the duration of the calculation of the steady state, the time interval At of the calculation and the wave velocity a,.
The influence of the choice of the values At and a, on the duration of the calculation of the steady state, which is established after the fading of water hammer, for the hydraulic system presented in Sect. 14.12, is shown in Fig. 8.2. In the calculation, we used the submission contained in Table 14.24 and in the D12.WTH file on the WTHM diskette, where the folloving data were changed. We estimated the initial discharge Qo = 5 m s-'; the initial opening avo = 1.425 rad of the butterfly valve and the constants B,, = 6000 kg m-7, B,- = -4000 kgm-', B,, = 1300 kg m-4 and B,- = - 1000 kg m-4 of the pressure and moment characteristics, respectively. The calculation denoted 2 = 4 was used and the graphical output was realized for each calculation step At. The values a, and At were chosen within the intervals 25 Ia, I150 m s-' and 0.001 IAt I0.046 s, respectively. When the abridged calculation of the steady state serves to determine the initial values for the subsequent calculation of water hammer, the complete submission of data for the calculation of water hammer may already be introduced for the abridged calculation. The subsequent submission for the calculation of water hammer will then consist only of two entries, viz., the lines beginning with numbers 5 and 7, containing the parameters defining the type of calculation and its verbal denotation. 162