Viscous and turbulent flow
When a cylinder of air flows through a duct of circular section its core moves more rapidly than its outer annular shells, these being retarded by the viscous shear stresses set up between them and the rough surface of the duct wall. As flow continues, the energy level of the moving airstream diminishes, the gas expanding as its pressure falls with frictional loss. The energy content of the moving airstream is in the kinetic and potential forms corresponding to the velocity and static pressures. If the section of the duct remains constant then so does the mean velocity and, hence, the energy transfer is at the expense of the static pressure of the air. The magnitude of the loss depends on the mean velocity of airflow, V, the duct diameter d and the kinematic viscosity of the air itself, v. It is expressed as a function of the Reynolds number (Re), which is given by
Vd
(**) = — (15.1)
By means of first principles it is possible to derive the Fanning equation and to state the energy loss more explicitly:
Rr W2
T <152>
Where H = the head lost, in m of fluid flowing (air),
/ = a dimensionless coefficient of friction, g = the acceleration due to gravity in m s-2
V and d have the same meaning as in equation (15.1).
If pressure loss through a length of duct, /, is required we can write * 2/V2p/
Ap = (15.3)
Because pressure, p, equals pgH, p being the density of the fluid.
The Fanning equation provides a simple picture but further examination shows that / assumes different values as the Reynolds number changes with alterations in duct size and mean air velocity, as Figure 15.1 shows. The curves shown for small and large ducts are lines of constant relative roughness (0.01 and 0.000 01, respectively), defined by the ratio
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Reynolds number (Re) Fig. 15.1 Dimensionless coefficient of duct friction, /, is related to the Reynolds number but, without exception, turbulent flow always occurs in duct systems. |
Ks/Dh, where ks is the absolute roughness and Dh is the mean hydraulic inside diameter of the duct, in the same units, see equation (15.14).
A further appeal to first principles yields the equation
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(15.4) |
F=2C(Ref
Again the simplicity is misleading; the determination of C and n requires considerable research effort, as Figure 15.1 implies. C and n are not true constants, but/can be expressed approximately by the Poiseuille formula:
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/ = |
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(15.5) |
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(Re)
This applies only to streamline flow where (Re) is less than 2000. For turbulent flow, Re being greater than 3000, equation (15.4) no longer holds good and instead the Colebrook White function is used
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1 a « I h ~ log 10 |
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1.255 (Re)<f |
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+ |
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3.7d |
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Attempts have been made to rearrange the Fanning equation by making use of equation
(15.6) and an experimental constant but the CIBSE uses a more sophisticated equation, due to Colebrook and White (1937 and 1939):
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N. d |
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10 |
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3.7 d (N3Apd5) |
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In |
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Q = -4(N3Apd ) log |
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Where Q = the rate of airflow in m3 s_1
Ap = the rate of pressure drop in Pa per metre of duct run
D = the internal diameter of the duct in metres ks = the absolute roughness of the duct wall in metres iV3 = 7t2/32p = 0.308 42p’1 N4 = 1.2557t(x/4p = 0.985 67HP“1 p = the density of the air in kg m~3
(j, = the absolute viscosity of the air in kg m_1 s-1
Equation (15.7) is not solvable in a straightforward manner but ASHRAE (1997a) gives a simplified equation, for an approximate determination of the friction factor, due to Altshul and Kiselev (1975) and Tsai (1989), is as follows
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(15.8) |
/’ = 0.11 (ks/Dh + 68/(Re))025
Iff, determined from the above, equals or exceeds 0.018, then/is to be taken as the same as/’. If/’ is less than 0.018 then the value of/is given by
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(15.9) |
/= 0.85/’ + 0.0028
The CIBSE (1986a) has published a chart (see Figure 15.2), that relates volumetric airflow rate, duct diameter, mean air velocity and pressure drop rate. It refers to the following conditions:
Clean galvanised sheet steel ductwork, having joints and seams made in accordance with good commercial practice,
Standard air at 20°C dry-bulb, 43% relative humidity and 101.325 kPa barometric pressure
Air density 1.2 kg itT3
Absolute viscosity 1.8 x 10~5 Ns m2 (or kg nr1 s-1)
Absolute roughness 0.15 mm (as for typical galvanised steel)
Good, approximate corrections to the pressure drop rate (Ap) for changes in air density, arising from variations in barometric pressure (pat) and dry-bulb temperature (t), can be applied by using the following equation:
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(15.10)
The CIBSE (1986a) quotes more refined corrections, taking into account additional factors and relating barometric pressure to changes in altitude.
The influence on the pressure drop rate can be considerable when ducts are made from materials other than galvanised sheet steel. Extensive correction factors are published by CIBSE (1986a) but some typical correction factors are given in Table 15.1. Where a range of values is given in Table 15.1 the value of the factor depends on the equivalent duct diameter.
Using equations (15.7) to (15.9), as appropriate, the pressure drop rate can be determined for a duct of any material, given the necessary details of absolute roughness, by means of the following equation:
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Rate of pressure drop (Pa/m) Fig. 15.2 A simplified duct-sizing chart. |
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Table 15.1 Pressure drop correction factors for ducts of various materials Absolute Correction factors for various pressure drop Material roughness, Јs rates (Pa m-1)
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(15,1)
Flexible ducts are usually made in a spiral form and the pressure drop depends on the material of manufacture and the extent to which the spiral is tightened. Pressure drops can
Be very much greater than those in equivalent, spirally-wound, steel ducting. As an example, a 200 mm diameter, flexible, straight duct made from a three-ply laminate of aluminium and polyester wound on to a helical steel wire fully extended, has a pressure drop rate of about 1.3 Pa m_1 when conveying 100 litres s_1 of standard air. A similar, straight, spirally — wound steel duct has a pressure drop rate of about 0.75 Pa m_1. Manufacturers’ published data should be used. Manufacturers also point out that pressure drop rates vary significantly from published data if flexible ducting is not fully extended. It is claimed that the pressure drop rate could double in a flexible duct that is only 75 per cent extended.
Flexible ducting lined with 25 mm of glass fibre (or the equivalent) is available. Significant attenuation of noise is claimed but there is the risk of noise break-out through the wall of the flexible duct into the surrounding space.
Circular section, permeable cloth ducting is also used, principally for air distribution in industrial applications. Being permeable, the air is diffused into the room uniformly over the entire length of the duct and low velocity air distribution achieved. The recommended materials are polypropylene, polyester and nylon. Cotton should not be used because of its hygroscopic nature and the risk of promoting the growth of microorganisms. About 100 Pa is needed inside the ducting to achieve airflow although with special construction this can be reduced to about 20 Pa. The cloth duct provides a measure of terminal filtration and hence periodic laundering is necessary.
Posted in Engineering Fifth Edition