# Conversion from circular to rectangular section

An airstream having a circular section is the most efficient way of containing the airflow because the section of the jet then has the minimum ratio of perimeter to area, P/A. For this reason, the best way of ducting airflow is in ducts of circular section. If rectangular sections are used the comers of the ducts contain turbulence and represent a loss of energy. This is made worse if the ducts are of very large aspect ratio. The CIBSE (1986a) and HVCA (1998) recommend that the aspect ratio should not exceed 4 but engineering prudence suggests that the maximum should be 3. Similar considerations apply to the use of flat oval duct. Bearing in mind the above restriction on aspect ratio, it is not always good practice to regard a rectangular duct section as the best way of using building space. It may be better to use multiple spirally wound ducts of circular section.

Ducts should be sized initially to give circular sections with diameters read from a duct sizing chart to the best accuracy possible. After this, the sizes may be converted to the standard sizes of circular duct commercially available using CIBSE (1986a) data, or to rectangular or flat oval dimensions, if necessary.

The conversion from circular to rectangular section should be done so that the rectangular duct has the same surface roughness and conveys the same volumetric airflow rate with the same rate of pressure drop, as does the circular duct. An alternative approach, of little value in commercial air conditioning, is to convert so that the rectangular duct has the same surface roughness, mean velocity and pressure drop rate but carries a different volumetric airflow rate. The starting basis is the Fanning formula (equation (15.2)) and the derivation for the useful conversion, giving equal volumetric airflow rate and pressure drop is as follows:

The rectangular equivalent having the same volumetric airflow rate and the same rate of pressure drop:

H=%r <I5-2>

Multiplying by pg to convert to pressure

4*-^ ,15,3)

The mean hydraulic diameter, Dh, is defined by

Ј>h = AIP (15.14)

Where A is the internal cross-sectional area and P is the internal perimeter and, for the case of circular ducts this is (nd2/4)/(nd) which is d/4. Thus d equals 4AIP and a substitution can be made in equation (15.13):

OfIV2 p

Apt = —^anc*’ since V = QIA, we have

For each duct, circular and rectangular, it is stipulated that the volumetric airflow rate, Q, must be the same and, since the pressure drop rates and surface roughnesses are also required to be equal Apt, f p and I are also the same. Hence conversion is achieved by equating VA3IP for the circular and rectangular sections:

N3d6 _ (ab)

4 3nd 2(a + b)

Where a and b are the sides of the rectangular duct. Re-arranging the equation yields the solution required:

If the surface roughnesses of the circular and rectangular ducts are not the same their friction factors are different. Denoting these by/c and/r, respectively, they may be incorporated in equation (15.15) to yield

D = <1516>

A similar approach yields the following equation for flat oval ducts having overall dimensions of a x b, the same surface roughness and conveying the same volumetric airflow rate with the same rate of pressure drop:

Y [71a + 2(b — a)]fr

Equations are seldom used for conversion. Tables are published by CIBSE (1986a) which are commonly used.

In terms of the smoothest and quietest likely airflow, other factors being equal, the preferred duct section shapes are as follows, in order of preference:

(1) spirally-wound circular

(2) circular duct rolled from flat sheet

(3) square

(4) rectangular or flat oval with aspect ratios not exceeding 3.

Rectangular ductwork should never be used with high velocity systems; it is expensive and noisy.

Posted in Engineering Fifth Edition