Measurement Techniques, Vol. 43, No. 11, 2000
EFFECT OF THE AIR PHASE OF CLOSED WATER HEATING SYSTEMS ON THE READINGS OF INFERENTIAL FLOWMETERS
E. G. Abarinov, A. V. Mikhnevich, and V. E. Finaev
UDC 621.317
It is shown that the presence of air in the heat carrier in closed heating systems leads to changes in volume flow rate. This effect is evaluated quantitatively.
Specialists involved in operating and maintaining instruments which measure heat use with inferential flowmeters (electromagnetic and ultrasonic flowmeters, differential manometers) – in which flow rate is determined by integrating the velocity of the heat carrier in a pipe – have brought attention to the fact that the heat meters which have been installed at the outlets of open and closed heating systems may show a flow rate greater than the rate recorded at the inlet of the system. This difference could be caused only by an increase occurring in the volume discharge of the carrier at the outlet due to a gas phase (air) in the carrier. We will analyze the effect of the gas phase Γ of the heat carrier on the change in the instantaneous value of volume discharge Q at the outlet of a closed heating system. The instantaneous discharge will be examined in relation to pressure p and temperature T at the inlet and outlet of the system. Under actual operating conditions, the water in the system contains a certain amount of a gas phase (air) in the form of microscopic bubbles. The dimensions of the bubbles fluctuate within broad ranges, from 10–6 mm to 1 mm or more. The smallest microscopic air bubbles (10–6–10–3 mm) are quite stable and are difficult to remove by degassing. The amount of air which is usually present in water varies within a broad range – from 10–3 (degassed water in special closed vessels) to 5% or more in the case of the forced circulation of water in the pipes of water supply systems and heating systems. The main way air enters the pipes is by being sucked in through leaks on the intake sections of circulation pumps. The volume content of air in the water in pipe systems with poor-quality seals may be very substantial – 10% or more. The volume content of the gas phase (air) in a liquid is measured with a change in pressure and temperature in accordance with the Clapeyron equation p T Yg (T , p1 ) = 0 1 Yg (T0 , p0 ), (1) p1 T0 where p0 and T0 are absolute pressure (Pa) and absolute temperature (K) in the initial section of the pipe (at the inlet of the system); p1 and T1 are absolute pressure and absolute temperature in the final section of the pipe (at the outlet of the system); Yg(T0, p0), Yg(T1, p1) are the volume contents of the gas phase (undissolved air) in the liquid at the temperatures and pressure T0, p0 and T1, p1, respectively. If we designate Y1(T0, p0) = Yg0 and Y1(T1, p1) = Yg1, then the volume discharge of the working medium at the inlet Q0 and outlet Q1 of the system can be represented in the form: Q0 = Ql + Yg0Ql = Ql(1 + Yg0);
(2)
Translated from Izmeritel’naya Tekhnika, No. 11, pp. 32–33, November, 2000. Original article submitted June 27, 2000. 0543-1972/00/4311-0959$25.00 ©2000 Plenum Publishing Corporation
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Q1 = Ql + Yg1Ql = Ql(1 + Yg1),
(3)
where Ql is the volume discharge of the liquid (water) without air. The volume discharge of the liquid Ql is nearly constant if we ignore the slight change in the volume of the water that accompanies changes in pressure and temperature. It can be seen from (1) that the quantity Yg1 fluctuates appreciably with changes in temperature and pressure, which leads to a change in the volume discharge of the working medium at the outlet of the system. With allowance for (2) and (3), the relative change in the volume discharge δQ of the working medium at the outlet of a closed system can be represented in the form: δQ =
Q1 − Q0 Ql (1 + Yg1 ) − Ql (1 + Yg0 ) Yg1 − Yg0 . = = 1+ Yg0 Q0 Ql (1 + Yg0 )
(4)
With allowance for (1), Eq. (4) will have the form δQ =
Yg (T0 , p0 ) p0 T1 − 1 . 1 + Yg (T0 , p0 ) p1 T0
(5)
It can be seen from (5) that the expression in the brackets will have the main effect on the relative change δQ in volume discharge at the outlet. Thus, we transform its first term by introducing the concept of pressure loss ∆p, ∆p = p0 – p1,
(6)
and, having expressed the absolute temperature through the temperature of the water at the inlet t0 and outlet t1 in degrees Celsius, T0 = 273 + t0;
T1 = 273 + t1.
Then it will have the form p0 T1 1 1 + t1 / 273 = , p1 T0 1 − δp 1 + t0 / 273
(7)
where δp = ∆p /p0 is the relative pressure loss in the system.
1 Using a Maclaurin series in powers of a small parameter ≈ 1 ± x , we can represent Eq. (7) in the form 1± x p0 T1 tt ∆t = (1 + δp)(1 + t1 / 273)(1 − t0 / 273) = (1 + δp)1 − − 1 02 , 273 273 p1 T0
(8)
where ∆t = t0 – t1 is the difference in temperatures between the inlet and the outlet of the system. Multiplying the expressions in the parentheses and ignoring the second-order infinitesimals t0t1 /2732 and δp(∆t/273), we can write (8) in the form p0 T1 ∆t = 1 + δp − . 273 p1 T0
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(9)
TABLE 1. Dependence of the Relative Change in the Volume Discharge of the Heat Carrier at the Outlet of a Closed System on the Relative Pressure Losses δp = ∆p/p0 and the Temperature Difference Changes in volume discharge of heat carrier (in percents) at the below temperature differences ∆t, °C
Pressure losses δp
10
20
30
40
60
80
100
0.1
0.6
0.2
–0.1
–0.4
–1.1
–1.8
–2.4
0.25
1.9
1.6
1.3
0.9
0.3
–0.4
–1.1
0.35
2.8
2.5
2.2
1.8
1.2
0.5
–0.1
0.5
4.2
3.9
3.5
3.2
2.5
1.9
1.2
0.6
5.1
4.8
4.5
4.1
3.5
2.8
2.1
0.75
6.5
6.2
5.8
5.5
4.8
4.2
3.5
Inserting (9) into (5), we obtain a simple formula for determining the relative change in volume discharge at the oulet of a closed heating system δQ =
Yg (T0 , p0 ) ∆t . δp − 1 + Yg (T0 , p0 ) 273
(10)
It can be seen from (1) that the relative change in volume discharge at the outlet is directly proportional to the volume content of air in the heat carrier, and the volume discharge at the outlet may in fact be greater or less that the volume discharge at the inlet. Table 1 shows the results calculated with Eq. (10) for a 10% volume content of air in the heat carrier. The results were calculated with allowance for realistic values of pressure loss ∆p (1.5–3 atm), pressure at the inlet (4–6 atm), and temperature difference (30–40°C). It can be seen from the table that for realistic values of the relative pressure loss (0.25–0.75) and the difference in temperature between the inlet and outlet (30°C), the increase in volume discharge at the outlet can range from 1.5 to 6%.
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