Building a bridge between industry and theory on the example of a new ventilation system

The paper presents the possibilities of simplified determination of the air volumetric flow rate in ventilation ducts. This problem occurred during the tests of local losses in the elements of a new ventilation system based on ducts with a rounded rectangular cross-section. The presented method requires mathematical modelling of the flow velocity distribution in the ducts. The paper presents four models of the velocity distribution. The necessity of using so many models resulted from the wide coverage of the tested sections: max min 46.88 A A  . Motto An error of 10 percent (or more) in friction factors calculated using the relations in this chapter is the “norm” rather than the “exception.” (Yunus A. Çengel, John M. Cimbala: Fluid Mechanics – Fundamentals and Applications: Chapter 8. Flow in pipes, p. 322)


Introduction
An engineering device (e.g. ventilation, heating and air conditioning systems) can be studied either experimenttally (testing and taking measurements) or analytically (by analysis or calculations). The experimental approach has the advantage that we deal with the actual physical system, and the desired quantity is determined by measurement, within the limits of experimental error. However, this approach is expensive, time-consuming, and often impractical. Besides, the system we are studying may not even exist. For example, the entire ventilation systems of a building must usually be sized before the building is actually built on the basis of the specifications given. The analytical approach (including the numerical approach) has the advantage that it is fast and inexpensive, but the results obtained are subject to the accuracy of the assumptions, approximations, and idealizations made in the analysis. In engineering studies, often a good compromise is reached by reducing the choices to just a few by analysis, and then verifying the findings experimentally.

Ventilation system developed
In ventilation systems, as in air conditioning, the basic cross-sections are rectangular and circular cross-sections. Both types have their own advantages and disadvantages. In this work, ducts with a rounded rectangular cross-section are analysed, this cross-section is to combine the advantages of both above-mentioned shapes of ventilation ducts.  WH  0.8m 0.5m  obtained during simulation in the ANSYS-FLUENT code, where the average velocity in the duct was entered at the duct input 10 m upstream from the presented analysed duct cross-section. In Fig. 2 has been defined nondimensional parameters of rounded rectangle cross-section, which are used to generalize derived equations.
Note that Crr Crec vv  , this is not a surprising result because it was expected for the same average velocity avg v . This result obtained on by numerical simulation testifies more uniform filling of the cross-section of the rectangle with rounded corners.

Models of velocity distribution
The fundamental problem of the experimental studies of system elements properties (mainly local losses) was to determine the flow rate in the ventilation duct From flow rate V we can obtain average velocity avg v in examined cross-section. For rounded rectangle is where: rr A -rounded rectangular area (see Fig. 2)

Basic power-law velocity profile in uniform cross-section
In this model, it is assumed that the velocity distribution over the entire cross-sectional area of the duct is determined by the same formula, known as power-law velocity profile. The adjective "uniform" has been added in the title of this subsection, because in subsections 2.3 and 2.4 the cross-section divided into two parts will be analysed. At the basis of the analysis of velocity distribution in a rounded rectangular lies the assumption that along the axis of the duct symmetry the velocity distribution is determined by the power-law velocity profile. The maximum velocity is at the central point C, i.e. at the intersection of the rectangle's symmetry axis.

Modified power-law velocity profile in uniform cross-section
Modification of power-law velocity profile consists in putting a squares into it, The formula (5) obtained as a result of integration of the distribution function over quarter of duct area (Fig. 3) was also presented in [1,2]. In the rounded rectangle, integration problem creates a cross-section area, i.e. the integration surface, however, this area can be divided into three components that have a shape that allows integration. According to Fig. 3, due to symmetry, only a quarter cross-section was integrated. The volume flow rate at approximation using the basic power-law velocity profile was determined as rr1 V , while at the approximation using modified power-law velocity profile (the next subsection) was marked rr

Basic power-law velocity profile in slotrounded square cross-section
The considerations in subsections 2.1 and 2.2 were tested experimentally. Satisfactory confirmation of the mathematical analysis was obtained for smaller cross-sections of ventilation ducts, . This mainly concerned rounded rectangular cross-sections for 0.5  . Therefore, it was necessary to find a new model of flow velocity distribution in the cross-section. During the tests, it was noticed that in the wide coordinate w range around the central point C, the flow local velocity does not change. This suggests the presumption that the velocity distribution in this area is similar to the velocity distribution in the slot. This observation confirms the well-known fact that the Reynolds number calculated on the basis of hydraulic diameter h D for cross-sections different from circular ones, better reflects the similarity of flows if the cross-section is more similar to circular, e.g. square, hexagon, etc. Therefore, in the new model, it has been assumed that the flow in the channel has a twofold character: one-dimensional as in the slot and two-dimensional as in the circular cross-section (Fig. 6). The cross-section marked in Fig. 6 Fig. 8 shows the velocity distribution solely on the basis of measurements along the shorter axis h of cross-section, because in the assumed model, it was assumed that this axis determines the distribution of the velocity of the flowing air. In this case, the value of C v is the effect of matching the measured points along the h-axis with the model's curve using the least-squares method.
Integration of formula (7) over area rsq A (Fig. 6) was also presented in [3]. Flow rate is determined by formula

Modified power-law velocity profile in slotrounded square cross-section
Based on experimental research, it can be assumed that for large cross-sections the model resulting from the modified power-law velocity profile is better than the model defined by its original version. According to the modified power-law velocity profile, the flow velocity distribution along the coordinate is given by the formula     As in subsection 2.3, the velocity distribution was also determined exclusively on the basis of measurements along the h axis. The result is shown in Fig. 10. Integration of formula (10) over area rsq A (Fig. 6) was also presented in [3]. Flow rate is determined by formula

Measurement only in two points of crosssection area
Approximation was also made based on selected measurement points. The author of this paper is aware of the fact that the measurement along the axis of symmetry of the duct with a small step (4 mm as in the presented example) in industrial conditions is practically impossible. Therefore, the value was determined on the basis of the measured value of the air flow velocity at the appropriate point B located at a distance B h from the duct wall and lying on the axis of symmetry of the ventilation duct -see Fig. 11, [4,5].     (Fig. 11). Similar to the model slot -rounded square with using basic power-law velocity for model slot -rounded square with using modified power-law velocity calculation of m was made. The value of m was determined from the equation  was obtained. The theory presented above refers to the ideal measurement conditions, first of all, there must be no vibrations of the tested duct. This effect, however, cannot be avoided in industrial research. In order to limit the impact of disturbances, the velocity at points C and B was measured many times, it was assumed that 300 measurements would be sufficient. The effect of this assumption is shown in Fig. 13. The configuration of the measurement system is shown in Fig. 14.   Fig. 15. The principle of measuring the distribution of air velocity in the ventilation duct

Conclusions
An analyst working on an engineering problem often finds himself in a position to make a choice between a very accurate but complex model, and a simple but notso-accurate model. The right choice depends on the situation at hand. The right choice is usually the simplest model that yields satisfactory results. Also, it is important to consider the actual operating conditions when selecting equipment. There are many significant real-world problems connected with ventilation, heating, and air conditioning systems that can be analysed with a simple model. But it should always be kept in mind that the results obtained from an analysis are at best as accurate as the assumptions made in simplifying the problem. Therefore, the solution obtained should not be applied to situations for which the original assumptions do not hold.
The average air velocity in the ventilation duct plays a significant role. On the one hand, it should not exceed due to the loudness of the system operation (in industrial power supply ducts can be larger), on the other hand it cannot be too small. During the tests, however, a mistake was made. It was tried, at all costs, to maintain a constant Reynolds number of 5 Re 10  for all cross-sections, changing the flow rate of air. This led to a large dispersion of research results -see Fig.15. Due to the above described behaviour of the air flow, the assumption, which allowed to compare the results was changed. The average air velocity in the channel was taken as the reference value -see Fig.16. Only for the four smallest cross-sections, to preserve values 5 Re 10  tests were carried out at slightly higher velocities.
Duct with dimensions 0.6m 0.2m WH    was chosen because it was possible to verify the flow rate calculated on the basis of the formulas, by measuring the flow rate using a volumetric flowmeter "testo". Flowmeter "testo" was mounted on the fan inlet. This solution allows to use this measurement device for many cross-section measurements. Of course, it is assumed that the amount of incoming air is equal to the amount of exhaust air from duct line.
Equations (4), (6), (8), and (11) used to calculate the flow rate look terrible, but modern computing is doing well with them. The flow rates calculated with the aforementioned equations did not differ from the results "testo" flowmeter by more than 7%. Flowmeter "testo" allows to measure the volume flow rate of air in the range 31 0.12 1.2 m s V   . It has own tubular deflector to calm the input flow. Author presents four velocity distribution models that may be used for analytical determination of the average velocity in the investigated ventilation ducts. It should be noted that the volume flow rate (or average velocity) is the basic parameter for when designing the ventilation systems. During laboratory tests of the ventilation system components, knowledge of average velocity in ducts is a prerequisite, especially for determining local losses.
Thanks to the development and checking of equations (4), (6), (8), and (11), it was possible to quickly test the properties of the elements (for example arches, elbows, diffusers, e.t.c.) for the new ventilation duct system developed.