Analysis of the velocity distribution in different types of ventilation system ducts

The paper presents the results obtained during the preliminary studies of circular and rectangular ducts before testing the properties elements (elbows, tees, etc.)of rectangular with rounded corners ducts. The fundamental problem of the studies was to determine the flow rate in the ventilation duct. Due to the size of the channel it was decided to determine the flow rate based on the integration of flow velocity over the considered cross-section. This method requires knowledge of the velocity distribution in the cross section. Approximation of the measured actual profile by the classic and modified Prandtl power-law velocity profile was analysed. 1 Preliminary remarks The testing of ventilation systems components and their design can be greatly simplified by finding analytical solutions which determine the velocity distribution of air flow in a duct. Control Group of Mechanical Engineering Faculty UTP in Bydgoszcz in cooperation with Nucair Technologies Sp. z o.o., Solec Kujawski, Poland conducts research of new type ventilation ducts of rectangular with rounded corners cross-section. This duct type has significant operational advantages, but the aerodynamic phenomena occurring in these ducts have not been studied yet. 2 Circular cross-section of duct In the state of art literature, the ventilation ducts with a circular cross-section are well examined in the terms of analysis. The ducts with rectangular cross-section are also well described. This paper presents the results of own research related to these two types of channels. Fig. 1. Average velocity avg v for circular cross-section During the study of pressure drop (hydraulic resistance), one of the main problems has been to determine the flow rate of air. These tests were aimed to determine the flow rate based on velocity distribution in a duct. In fluid flow, it is convenient to work with an average velocity avg v which remains constant in incompressible flow when the cross-section of the duct is constant (Fig. 1). A semi-empirical dependence was used due to the lack of accurate analytical solutions. It is known as Prandtl power-law velocity profile, which is expressed as [1, 2] for a duct of circular cross-section


Preliminary remarks
The testing of ventilation systems components and their design can be greatly simplified by finding analytical solutions which determine the velocity distribution of air flow in a duct. Control Group of Mechanical Engineering Faculty UTP in Bydgoszcz in cooperation with Nucair Technologies Sp. z o.o., Solec Kujawski, Poland conducts research of new type ventilation ducts of rectangular with rounded corners cross-section. This duct type has significant operational advantages, but the aerodynamic phenomena occurring in these ducts have not been studied yet.

Circular cross-section of duct
In the state of art literature, the ventilation ducts with a circular cross-section are well examined in the terms of analysis. The ducts with rectangular cross-section are also well described. This paper presents the results of own research related to these two types of channels. During the study of pressure drop (hydraulic resistance), one of the main problems has been to determine the flow rate of air. These tests were aimed to determine the flow rate based on velocity distribution in a duct. In fluid flow, it is convenient to work with an average velocity avg v which remains constant in incompressible flow when the cross-section of the duct is constant (Fig. 1).
A semi-empirical dependence was used due to the lack of accurate analytical solutions. It is known as Prandtl power-law velocity profile, which is expressed as [1,2] for a duct of circular cross-section , r , R are the quantities set out in

Analysis of the Prandtl power-law velocity profile
The first doubts have risen during the numerical analysis of the problem in the ANSYS-FLUENT environment. The curve illustrating the velocity distribution along a diameter in Fig. 2 suggests that the tangent to velocity profile in the axis point of the circular cross-section duct is perpendicular to the axis. This can never be achieved when using the Prandtl approximation model, because after its differentiation with respect to r we obtain 1 1 1 hence substituting into (2) 0 r  , we get

Fig. 2. Numerical solution in ANSYS-FLUENT
Because previously were studied rectangular ducts and rectangular ducts with rounded corners, where well they fits the modified Prandtl power-law velocity profile in the form for the comparative analysis of the ducts shapes was adopted precisely this equation. Earlier, a comparison of velocity distributions of the both equations was done.
The results of this analysis are shown in The figure Fig. 3 shows that the difference of distribution decreases with the increase of n . We can use smaller parameter n by using distribution equation which includes   2 r R . Thanks to this operation, the function of distribution   v r will be the same for all types of ducts.

Analysis of the tests results
The study involved a linear section of the duct composed of 12 same segments with a radius 0 2 m R ,  and length 1 m. The measurements were performed along the horizontal diameter. Fig. 4 presents the results of measurements and the approximation of this course by the two methods mentioned above. The approximation in this case turned out to be better by classic Prandtl equation. This conclusion has been drawn from the analysis of the sum of squares of deviations for all measuring point.

Analytical calculations
Later in the research, the average velocity for the modified Prandtl distribution was attempted to be determined. The following equation (5) was used for this purpose  

Rectangular cross-section
Rectangular cross-section with rounded corners is a combination of parts of the rectangular cross-section and circular cross-section. Therefore, this type of ducts was part of compare analysis.

The axial velocity distributions
Experimental study of velocity distribution is made for a rectangular cross-section with width 0 5 m W ,  and height 0 25 m H ,  . The study involved a linear section of the duct composed of 12 same segments of 1 m length. The profiles were measured on symmetry axes. The traversing step of the thermo-anemometer probe was 4 mm w h     . During the measurements, the same velocity C v of the air flow at the point of intersection of the two axes should be provided. The test results are shown in the figures Fig. 5 and Fig. 6. The figures Fig. 5 and Fig. 6 also present the approximating curves of the measured results by the formulas: Approximating formulas were adopted in the same form as for a circular cross-section. Radius r was replaced by w and h respectively. Consequently, flow velocity at any point in cross section is given by On the basis of the formula (11), the average velocity

Determining the average velocity using log-Chebyshev method
The duct is divided into rectangular areas (Fig. 7) by using the log-Chebycheff method. Velocity is measured at corners of these rectangulars.