Flow friction (f) and Poiseuille number (Po) for single phase flow in

microchannels

The friction factor (f), as mentioned in section 2.1.6 above, is a gage of surface roughness that influences the pressure drop (Δp) and heat transfer. The Poiseuille number (Po) described in section 2.1.7 in Equation 2.10 is a measure in fluid flow to represent the fully developed laminar flow friction data for traditional pipe flow. For fully developed

laminar flow in conventional pipe, the Po is usually independent of Re and has a constant value of Po = fRe = 64 for circular pipe using Darcy friction factor.

The friction coefficient or Fanning friction factor (ff) in heat transfer literature and

the Darcy friction factor (fd) in fluid mechanics are commonly used with their

relationship as fd = 4ff. The conventional conduit flow correlations for f and Po in the

laminar range and f in the post laminar regimes are well established, which are usually offered as a function of Re depending on flow situation and duct cross section. They can be readily theoretically and experimentally derived [61, 63, 68].

It is generally said that the f in microchannel is higher than the conventional pipes. Among the reviewed and summarized key works [24-27, 33, 37-39, 41, 79-192], the f or Po in microchannel was observed both higher, lower, similar, and anarchic (i.e. simultaneously lower, higher, similar or inconclusive) as compared to the classical values. For some of the works this information could not be extracted. The statistics is portrayed in Figure 3.1. The lower values of f and/or Po were observed particularly for gas flow where f decreased with the Knudsen number (Kn) and increased with the Mach numbers (Ma) for Ma > 0.3, which according to some authors might have been caused by the gas compressibility or rarefaction effects [88, 97, 100, 105, 107, 116-117, 132, 136].

In conventional pipe flow, it is well known that the f is influenced by the relative roughness of the pipe walls in turbulent flow regimes. For microchannel flow however, some works reported that the f depends on the wall relative roughness even in laminar flow [81, 110, 123, 133-134, 137-139, 161, 163, 168, 171]. While several authors reported that the Po in laminar microchannel flow is similar to the classical value, a large group however observed the Po to depend on the Re in laminar flow and showed different values than the values for conventional smooth pipe [87, 96, 103, 113, 134, 139, 151, 171, 174].

Friction factor (f) in laminar flow in microchannel

A few f-Re correlations proposed for fully developed laminar flow in straight smooth non-circular narrow channels, mostly in rectangular cross-section, can be found such as one by Hartnett and Kostic (1989) as cited by Kim et al. [166]. The correlation is given by Equation (3.1) below.

2 3 4 5 1 1.3553 1.9467 1.7012 Po Re 24 0.9564 0.2537 AR AR AR f AR AR ⎛ _{− + −} ⎞ = = ⎜_⎜ ⎟_⎟ + − ⎝ ⎠, for Re < 2100, (3.1)

where Re is based on hydraulic diameter and AR is the height-to-width known aspect ratio of the flow channel.

The proposed correlation for fully developed laminar flow in circular microchannel is rare. A correlation for fully developed laminar flow in straight smooth circular micro-tube available in the open literature was proposed by Yu et al. [110] as given in Equation (3.2) below. The authors experimentally investigated the characteristics of nitrogen gas and water flows through silica made micro-tubes in the Reynolds number range of 220 ≤ Re ≤ 19500. They observed lower Darcy f in micro-tubes, which are lower than the established value of 64 for conventional circular pipe. The lower values were from 0.77 to 0.81 times of the conventional value. Although the reported critical Re was 2000, the transition however observed from Re = 1700 to 6000.

50.13

, for R e 2000 (Fully developed laminar flow in micro-tube) Re

f = < . (3.2)

Pressure drop (Δp) in laminar flow in microchannel

The pressure drop (Δp) in fully developed laminar flow in traditional pipe is a direct function of the Darcy friction factor (fd), which is given by Equation (2.7) in

section 2.1.6 above. The Poiseuille law in fully developed laminar flow in traditional pipe relates the pressure drop (Δp) to the volume flow rate (∀ ) as presented by Equation (2.13) in section 2.2.1 above. For developing laminar flow in traditional pipe, in addition to the effect of entrance region, another effect often contributes to the pressure drop is the active apparent friction factor attributed to the flow developing effect, which could be more prominent in narrow flow passage.

To account for this effect, a flow developing pressure loss as given in Equation 2.17 in section 2.2.1 often considered when estimating the pressure drop or friction factor in a pipe flow [5, 18]. For K∞in Equation 2.17, Shah and Sekulic [5], from an analytical solution, presented a constant value of K∞ = 1.28 for fully developed circular duct flow. However, they mentioned that this value may not truly represent the flow in microchannel geometry. In the absence of an established correlation for this K∞ for microchannel geometry, Olsson and Sunden [67] recommended that the Re dependent

K∞ relationship for circular conventional pipe flow proposed by Chen [66] can be employed for microchannel flow. Some authors also supported the use of this relationship for narrow channels [18]. Thus, as applicable, this correlation is used in estimating pressure drop or friction factor in current work.

Friction factor (f) in turbulent flow in microchannel

There are a number of established f-Re correlations available for turbulent flow in traditional pipe as described in section 2.2.1 above. However, such any correlation for turbulent flow in microchannel is not readily available. Kim et al. [166] proposed that, for turbulent flow in both circular and non-circular microchannels, the well known Blasius correlation [5, 11, 64, 68] established for conventional smooth pipes, given in section 2.2.1 in Equation 2.20, could be employed in microchannel flow to represent the f data without much error.

Some modifications to the Blasius Equation 2.20 was also proposed by Phillips [193] by defining it through the apparent Fanning friction factor (ff,app) as a function of Re

as expressed by Equation (3.3), which is said to cover both the developed and developing turbulent flow regimes in conventional pipe flow. Kandlikar et al. [18] has suggested that this modified Blasius Equation (3.3) can also be attempted for microchannel flow.

0.3293 0.268 f,app

1.01612

ReB 0.0929 Re xD ; (Turbulent pipe flow)

f A x D ⎛ ⎞ ⎜− − ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ = = + ⎜ ⎟ ⎝ ⎠ . (3.3)

Yu et al. [110], in their investigation of nitrogen gas and water flows through silica made micro-tubes, proposed the f-Re correlation in the Reynolds number range of 220 ≤ Re ≤ 19500. They observed some lower Darcy f in micro-tubes, which are 0.7 to 0.9 times of that of the values estimated from traditional Blasius equation. They observed the transition-to-turbulent regime over the Reynolds number range of 1700 ≤ Re ≤ 6000. Their proposed correlation is cited below by Equation (3.4).

0.25

0.302

, for 6000 R e 20000 (Developed turbulent flow in micro-tube) Re

f = < < . (3.4)

Webb and Zhang [125] studied the turbulent flow and heat transfer natures of liquid R134a in 9-channel multi-port circular and rectangular microchannels in the Reynolds number range of 5000 ≤ Re ≤ 25000. Their friction data compared well with traditional pipe flow correlations such as Blasius correlation [5, 11, 64, 68] given by Equation (2.20); and Petukov correlation [53, 70] given by Equation (2.22). The authors proposed an f-Re correlation for turbulent flow in circular and rectangular microchannels in form, which is defined by Equation (3.5) below.

0.22

0.0605 Re , for 5000 R e 25000 (Turbulent flow in microchannel)

f = − < < . (3.5)

Comments on friction factor (f) for flow in microchannel

From the above review it is seen that the f values for microchannel flow are about 0.5 to 5 times of that of the f values for traditional pipe flow. This trend of data scatter and discrepancies in microchannel flow are also compiled and reported by some authors [37]. Examples of these deviations in published result are displayed by Figure 1.3.

It is also seen that the differences in the reported results spread among a broad range of diameters or hydraulic diameters as surveyed for single phase flows in microchannels in current study. Important fact is that the study or the proposed correlation for developing flow in either straight or in serpentine bend circular microchannel is not available in the open literature.