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Abstract. Air sampling is a key element in environmental studies and occupational exposure assessments. Air flow control and measurement is critical to the accuracy of air sampling. Commercially available flow controlling orifices were found to lack the required precision and accuracy. Most critical orifice designs require a pressure drop in excess of 47 kPa to ensure a stable flow. To achieve this pressure drop, a special high-power vacuum pump must be used. There is no existing design standard available for small critical orifices or venturi (throat diameter smaller than 12.5 mm). Many efforts have been made to experimentally develop a critical airflow control device with low pressure drop, but the procedure and the products are both labour intensive and inaccurate. Based on the theory of fluid mechanics and experimental optimization, a series of unique critical airflow venturi have been developed to provide a constant air flow rate at low pressure drops (approximately 9 kPa), with variations of less than 1% of the full scale flow rate. These venturi can be widely used in air quality studies and other field for accurate gas flow rate control.
| A1 | area at cross-section 1 | m2 |
| A2 | area at the exit of convergent-divergent nozzle | m2 |
| At | area at the throat | m2 |
| c | discharge coefficient of flow | |
| Ca | coefficient of velocity at the throat | |
| d1 | inlet diameter at cross-section 1 | mm |
| dt | throat diameter | mm |
| D | outside diameter of critical venturi | mm |
| j | number of variables | |
| k | number of levels | |
| L | total length of critical venturi | mm |
| Le | length of entrance of critical venturi | mm |
| Lt | length of throat section of critical venturi | mm |
| Ld | length of divergent section of critical venturi | mm |
| L1 | level 1 of variable | |
| L2 | level 2 of variable | |
| L3 | level 3 of variable | |
| L9 | orthogonal experimental design | mm |
|
mass flow rate | kg/s |
|
maximum mass flow rate | kg/s |
| P1 | upstream pressure | Pa |
| P2 | downstream pressure | Pa |
| Pt | pressure at the throat | Pa |
| Pc | critical pressure | Pa |
| Pdfc | the first critical downstream pressure | Pa |
| Q | volume flow rate | m3/s |
| R | gas constant | (Pa*m3/kg*K) |
| V1 | variable 1 of the optimal design | |
| V2 | variable 2 of the optimal design | |
| V3 | variable 3 of the optimal design | |
| V4 | variable 4 of the optimal design | |
| τ1 | absolute temperature of gas in upstream | K |
| v1 | upstream velocity of gas | m/s |
| vt | velocity of gas at the throat | m/s |
| Τe | radius of entrance of critical venturi | mm |
| α | cone angle of divergent section of critical venturi | |
| ρ1 | gas density in upstream | kg/m3 |
| ρt | gas density at the throat | kg/m3 |
| γ | specific heat ratio | |
| η | pressure recovery efficiency | % |
Many studies have confirmed that high contaminant concentrations in confinement livestock buildings have adverse effects on the health of workers.1,2,3,4 Strategies and technologies for control of particulate and gaseous contaminants, such as scrubbing, ionization, oil application, filtration, aero-deduster, electrostatic precipitation, and biofilters have been studied.5,6 Air contaminant concentration measurements are critical to the occupational exposure assessment and the development and evaluation of air quality control technologies.7 Research activities that characterize air contaminants, define health effects, and determine the effectiveness of engineering control measures are all dependent on air sampling.
The accurate measurement of the gas flow rate is very important in air filter sampling because the contaminant concentration is determined by the ratio of the sampled contaminant quantity to the sampled air volume. One widely used conventional flowmeter in air sampling is the rotameter. Rotameters are sensitive to pressure changes in upstream and downstream air flows.8 Most flowmeters are calibrated at atmospheric pressure, and many require pressure corrections when used at other pressures. When the flowmeter is used in air sampling, it should be downstream of the filter to exclude the possibility of sample losses in the flowmeter. Therefore, the sampled air is at a pressure below atmospheric due to the pressure drop across the filter. Furthermore, if the filter resistance increases due to the accumulation of dust, the pressure correction is not a constant factor. During the sampling period, the filter tends to be plugged and the flow rate may decrease as filter resistance increases.9 These factors make it difficult to measure the flow rate accurately.
Critical orifices10 have been widely used in flow rate control for air sampling because they are simpler, reliable and inexpensive. When the pressure drop across the critical orifice is more than 47% of the upstream pressure, the speed of sound is achieved in the throat and the velocity will not change with a further reduction in downstream pressure. Under these conditions, the flow rate is kept constant if upstream conditions are constant. However, commercially available orifices were found to lack the required precision and accuracy because they differed from the nominal flowrate by up to 15%.11 Another disadvantage of most critical orifice designs is that a pressure drop in excess of 47 kPa is required to ensure a stable flow.12 To achieve this pressure drop, a special high power vacuum pump must be used. Some commercial flow limiting orifices even require a vacuum as high as 72 kPa.
Electronic flow controller is another widely used device to measure and control the gas flow with high accuracy. However, compared with critical orifice, this device is much more expensive.
There is an increasing need for small critical airflow control devices with low pressure drop for air quality studies. Many efforts have been made to develop critical flow devices.11,12,13,14 For example, an experimental procedure was used to obtain a constant air flow by finishing disposal glass serological pipette tips.11 The procedure and the products are both labour intensive and inaccurate. There is no existing design standard available for small orifices. The existing critical orifice design standard is only available for opening diameters larger than 12.5 mm.15 In air quality studies, sampling rates often require 20 l/min or lower, which represents a throat opening diameter of 1.6 mm or smaller at critical pressure drop. It is highly desirable to develop an accurate and reliable critical airflow control device to maintain a constant airflow rate at low pressure differentials for air sampling.
The objectives of this project are to: (1) analyze the theory of critical air flow; (2) design the critical air flow venturi; and (3) evaluate and optimize the critical air flow venturi.
Figure 1 shows an example of gas flow through an orifice or a convergent nozzle16. Assume that the flow is isentropic, i.e., the flow is adiabatic and frictionless. Adiabatic condition means that no heat is transferred between the system and its surroundings. The assumption of adiabatic flow can be considered true if the flow occurs very quickly and with a small amount of friction. Isentropic analysis can be applied to high velocity gas flows over short distances where friction and heat transfer are relatively small. This flow can be described with an energy conservation equation17:
(1)
where v1 is the upstream velocity of gas (cross- section 1), ρ1 is the upstream gas density, P1 is the upstream pressure, vt is the velocity of gas at the throat (minimum area), Pt is the pressure at the throat and γ is the specific heat ratio.
Since the flow is assumed to be isentropic, then:
(2)
and:
(3)
where ρt is the gas density at the throat.
The mass conservation equation shows:
(4)
where A1 is the area at cross-section 1, and At is the area at the throat (minimum area) with assuming that the velocities are uniform across the flow area.
Combining Eqns (3) and (4), results in:
(5)
Substituting Eqn(5) into Eqn (1), vt can be expressed as:
(6)
where Ca is the coefficient of velocity at the throat:
(7)
The relationship between the coefficient of velocity at the throat Ca and the pressure ratio Pt/P1 for the different value of diameter ratio dt/d1 is shown in Fig 2, where dt is throat diameter and d1 is the inlet diameter at cross-section 1. These results show that Ca is near unity for all pressure ratios Pt/P1 up to 1.0 provided dt/d1< 0.3 (the error is less than 0.5%). Therefore, the velocity coefficient Ca can be considered to be unity and Eqn (6) can be simplified to:
(8)
The mass flow rate can be obtained by combining Eqns (3), (4) and (8):
(9)
Assume that the downstream pressure is P2. When P2 is greater than the critical pressure Pc, Pt is equal to P2. Then:
(10)
The variation of with the pressure ratio P2/P1 in Eqn (10) is illustrated by the curved line in
Fig 3. When the pressure ratio P2/P1 decreases, the mass flow rate increases. When the
pressure ratio reaches a critical level (0.53), the flow rate reaches a maximum value .
The critical pressure Pc can be determined by differentiating with respect to P2 in Eqn (10)
and setting the result equal to zero. This operation gives:
(11)
At this critical pressure ratio in Eqn (11), Pt = Pc,. Substituting Eqn (11) into Eqn (8), the gas velocity vt at the throat is:
(12)
It equals the speed of sound at the throat condition according to the definition of sound speed. When the downstream pressure is reduced below Pc, downstream pressure cannot transmitted back into the throat of the orifice because the gas in the throat is moving with the same velocity of pressure propagation, i.e., the sound speed. When the downstream pressure P2 is less than Pc, the pressure in the throat will not be affected by the downstream pressure and Pt is always equal to Pc. At this critical condition (P2≤Pc), Pt equals Pc and the mass flow rate reaches the maximum. Substituting Eqn(11) into Eqn(10), the maximum mass flow rate can be obtained with the following equation:
(13)
where T1 is the absolute temperature of the air upstream and R is the gas constant.
Equation (13) shows that when the pressure ratio P2/P1 is less than the critical pressure ratio
Pc/P1, the mass flow rate is only dependent on the upstream conditions and will not be affected
by the downstream pressure P2. Therefore, the mass flow rate will always equal the maximum
value and remain constant regardless of the pressure changes downstream as long as the
pressure ratio between downstream and upstream is less than the critical pressure ratio.
For a convergent-divergent nozzle, the maximum flow rate can be reached when the pressure at the throat equals the critical pressure. However, the downstream pressure can be greater than the critical pressure because the divergent section can recover some pressure (Fig 4). As the pressure ratio decreases from unity, the flow increases as the back pressure decreases. When the back pressure is reduced to a certain value, the flow rate reaches a limiting flow because the throat is at the choked condition, i.e., the velocity equals to the speed of sound. This back pressure at which the maximum flow rate is reached is usually called the first critical downstream pressure Pdfc.18 The flow rate will remain constant if the back pressure is less than the first critical pressure. That is, the first critical pressure is the highest downstream pressure at which the flow reaches a stable flow. It is obvious that the flow rate remains constant in a convergent-divergent venturi nozzle provided the back pressure is lower than the first critical pressure, because the throat is choked and the velocity in the throat remains sonic. In this case, the maximum flow rate will be the same as Eqn (13).
According to the theoretic analysis and Eqn (13), when a air flow venturi works at a state of critical condition, the volumetric flow rate Q can be determined with the following equation:
(14)
All equations are developed based on the assumptions of isentropic condition and perfect gas. However, there is a friction at real situation, the discharge coefficient c is introduced in the volumetric flow rate.
(15)
From Eqn (15) using At = πdt2/4 and P1/ρ1=RT1, the throat diameter dt can be determined with the equation:
(16)
For example, when the gas to be measured is air and the assumed experimental conditions are T1 = 293K, P1 =1.013x105 Pa, ρ1 =1.19 kg/m3, γ=1.41 the required throat diameter dt can be calculated by the Eqn (17) from the volumetric flow rate Q and the discharge coefficient c:
(17)
The discharge coefficient c depends on the configuration of the venturi. The discharge coefficient c is a function of venturi parameters and it can only be determined experimentally. The process of the throat diameter design is iterative.
According to the theoretic analysis, the pressure ratio P2/P1 for critical flow in the convergent- divergent nozzle can be greater than the theoretic critical pressure ratio in the convergent nozzle (0.53). It is desirable to keep a constant flow for a pressure drop as low as possible. Therefore, the design of the critical venturi is based on the convergent-divergent venturi nozzle.
A preliminary design is shown in Fig 5. Many factors influence the performance of the critical venturi, such as entrance shape, the length of the throat section, the length of the divergent section, and the convergent angle and the divergent angle. It is necessary to optimize the design of the critical venturi to improve the performance.
The objective of optimizing the design is to develop a critical venturi which maintains a constant air flowrate with a low critical pressure drop. The critical pressure drop is defined as the pressure difference between the upstream pressure P1 and the first critical downstream pressure Pdfc at a stable airflow. At the atmosphere condition, the commercial critical orifices require a critical pressure drop of 47 to 72 kPa to maintain a constant airflow rate. To achieve this pressure drop, a special high power vacuum pump must be used. The lower the critical pressure drop required for critical flow, the less expensive and more efficient is the air supply equipment. Therefore, the pressure recovery efficiency η can be used to represent the performance of the critical venturi:
(18)
The authors conducted a number of preliminary experiments to determine the entrance shape and other important factors. According to the preliminary study, the critical venturi with a curved entrance has better performance than the critical venturi with a cone entrance. The critical pressure drop is lower and the discharge coefficient is higher for the critical venturi with a curved entrance. Based on the preliminary study, a modified design of critical venturi with curved entrance is shown in Fig 6.
Four variables were studied and optimized for the modified venturi nozzle: the radius of the entrance τe for the curved entrance, the length of the throat section Lt, the length of the divergent section Ld, and the cone angle of the divergent section γ. It is obvious that the surfaces should be as smooth as possible. The optimum value for each variable was determined using an experimental design19.
Suppose there are j variables and each variable has k levels, a complete experimental design gives a total of treatments of kj. It is extremely difficult to test each variable at a large number of levels. In this research, variables and levels were chosen according to preliminary experimental results; j is 4 and k is 3, so the total of treatments would be 81. This experimental design of fabricating and testing 81 venturi would be labour intensive and expensive. However, an L9 orthogonal array design can substantially improve the efficiency of the experimental design, shown in Table 1. In this experimental design, V1, V2, V3 and V4 represent four variables τe, Lt, Ld and α of the optimal design and each variable has three levels L1, L2, L3. With this experimental design, only nine venturi samples were needed. Nine bronze venturi samples with different levels were fabricated according to an L9 orthogonal experimental design.
All venturi nozzles were calibrated in the test apparatus, as shown in Fig 7. The experimental apparatus consists of a rotameter, a vacuum pump, two manometers, a damper, a pressure control valve, and the critical venturi which is to be calibrated. The rotameter was upstream from the critical venturi to keep the same inlet pressure for all experiments. Otherwise, the flow rate reading would not be accurate because the flow rate of the rotameter is dependent on the inlet pressure.
Nine venturi samples were fabricated and calibrated. The calibration results are shown in Table 2. The results show that venturi No 6 has the lowest critical pressure drop and has the highest pressure recovery efficiency. According to the data analysis, the optimum throat length, radius of entrance, divergent length, and the divergent angle were selected. According to the statistical analysis and the value of maximum difference in pressure recovery efficiency for each variable, the divergent angle appears to be the most influential variable. The importance of other factors was ranked as follows: the radius of the entrance, the throat length, and the divergent length.
As the L9 orthogonal experimental design did not include various combinations of all factors and levels, a confirmation testing was conducted to validate that the optimized design was indeed the best. A venturi with the optimized design was fabricated and calibrated. The experimental results are shown in Fig 8. Pressure difference refers to the difference of upstream pressure P1 and downstream pressure P1. A comparison of a optimized critical venturi with a non-optimized venturi and a typical critical plate orifice is also shown in the same figure. The critical pressure drop at a stable airflow for a critical plate orifice is at 47 kPa; for a non-optimized critical venturi it is at 23 kPa. But for the optimized critical venturi, the critical pressure drop is only at 9 kPa. It is evident that the optimized design significantly improves the performance by reducing the critical pressure drop.
Compared to the nine samples in Table 2, it is also clear that the optimized critical venturi has better performance than any of the others in the optimizing experiment. The critical pressure drop of the optimized critical venturi at a stable airflow is lower (approximately 9 kPa) and the pressure recovery efficiency is higher (90%). This shows that the critical venturi with the optimized design has the best performance and the optimizing experimental design is successful.
The confirmation testing showed that the critical venturi with the optimized design gave the design flow rate at a low critical pressure drop. Twenty-seven critical venturi with the same specification were fabricated and tested to check the precision. The experimental results show that the critical flow rates of all venturi nozzles are 21 ± 0.3 l/min. The critical pressure drops of all venturi nozzles are below 11 kPa. The pressure recovery efficiencies are between 0.89 - 0.92 and the discharge coefficients are from 0.93 to 0.96. The variation in flowrate is ± 1.5% of the mean flowrate. That is, the maximum difference is 3%. This is consistent with the variation of discharge coefficient. The difference in the performance is probably caused by the difference in the discharge coefficients and inaccuracies in fabrication and experiment. For multipoint air sampling, the item of most concern is the consistency of the flow rate for each sampling head. These critical venturi are being used in a development of a multipoint air sampler for study of spatial distribution of aerosol concentration.
Although the concept of the critical airflow has been studied since the mid-nineteenth century, there are still some questions in its application. This project has attempted to develop an inexpensive and reliable critical venturi for use in air sampling. From the theoretical analysis and experimental calibration, the following conclusions can be drawn.
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