Dissipation viscous

The viscous dissipation is defined as mechanical energy which is irreversibly converted to thermal energy due to viscous effects in the fluid. The viscous dissipation is often taken into account by the Brinkman number, Br, which is the ratio of dissipation and heat diffusion  

The exponent d of Br is positive when the fluid in the microchannel is heated and negative when it is cooled. 

Morini and Spiga [36, 50] demonstrated analytically the link between the average Nu and Br. A general relationship for circular and noncircular channels is 

Koo and Kleinstreuer [51] showed for water that for channel sizes below 100 pm the viscous dissipation in the energy equation caimot be neglected. 

---

Note that the total pressure drop consists of 0.5 velocity heads of frictional loss contrihiition, and 1 velocity head of velocity change contrihiition. The frictional contrihiition is a permanent loss of mechanical energy hy viscous dissipation. The acceleration contrihiition is reversible if the fluid were subsequently decelerated in a frictionless diffuser, a 4,000 Pa pressure rise would occur. 

The Br is a measure of the extent to which viscous heating is important relative to an impressed temperature difference. This can be of some concern in the scale-up design, v usually increasing, with other properties remaining constant. A comparison of the Br for a pilot scale (0.05-m screw) and an industrial (0.15-m screw) unit yields values of ca. 0.65, and 5.73, respectively, for the Br with A = 0.5 w/m K and rj = 500 Pa.s at 60 rpm. The numbers suggest that viscous dissipation will be important and will be much more pronounced in the case of an industrial unit. 

When a plastic material is subjected to an external force, a part of the work done is elastically stored and the rest is irreversibly (or viscously) dissipated hence a viscoelastic material exists. The relative magnitudes of such elastic and viscous responses depend, among other things, on how fast the body is being deformed. It can be seen via tensile stress-strain curves that the faster the material is deformed, the greater will be the stress developed since less of the work done can be dissipated in the shorter time. 

Celata et al. (2005) evaluated the effect of viscous heating on friction factor for flow of an incompressible fluid in a micro-channel. By integrating the energy equation over the micro-channel length, a criterion that determines conditions when viscous dissipation effect is signiflcant was obtained ... 

The behavior of liquid flow in micro-tubes and channels depends not only on the absolute value of the viscosity but also on its dependence on temperature. The nonlinear character of this dependence is a source of an important phenomenon - hydrodynamic thermal explosion, which is a sharp change of flow parameters at small temperature disturbances due to viscous dissipation. This is accompanied by radical changes of flow characteristics. Bastanjian et al. (1965) showed that under certain conditions the steady-state flow cannot exist, and an oscillatory regime begins. 

Judy J, Maynes D, Webb BW (2002) Characterization of frictional pressure drop for liquid flows through micro-channels. Int J Heat Mass Transfer 45 3477-3489 Kandlikar SG, Joshi S, Tian S (2003) Effect of surface roughness on heat transfer and fluid flow characteristics at low Reynolds numbers in small diameter tubes. Heat Transfer Eng 24 4-16 Koo J, Kleinstreuer C (2004) Viscous dissipation effects in microtubes and microchannels. Int J Heat Mass Transfer 47 3159-3169... 

Hetsroni et al. (2005) evaluated the effect of inlet temperature, channel size and fluid properties on energy dissipation in the flow of a viscous fluid. For fully developed laminar flow in circular micro-channels, they obtained an equation for the adiabatic increase of the fluid temperature due to viscous dissipation ... 

We can estimate the values of the Brinkman number, at which the viscous dissipation becomes important. Assuming that the physical properties of the fluid are constant, the energy equation for fully developed flow in a circular tube at 7(v = const, is ... 

The effect of viscous dissipation on temperature change along the micro-channel axis is illustrated in Fig. 4.11, where the dependences dT),/ dx on d that correspond to water and isopropanol flows are presented. One can see that under the conditions corresponding to the Judy et al. (2002) experiments = 74.1 pm, L = 114 mm, Ljd = 1,543), the rise of bulk temperature due to viscous dissipation is small enough. So, at d > 100 pm the temperature gradient is dT),/ dx < 1 K/m. In this case, the difference between outlet and inlet temperature is about 0.1 K. Under conditions that are typical for micro-channels of electronic devices L/d r j 102) this difference is about 0.01 K. The rise of temperature due to viscous dissipation is small enough even at water flow in micro-channels with d 20 pm. Thus, for micro-channels with d = 20 pm and L/d = 10, we have Tout — Tin 0.8 K. 

Fig. 4.11 Temperature gradients due to viscous dissipation at Re = 300. Flow in rectangular (7 = 0.1), and square (7= 1) micro-channel. Reprinted from Morini (2005) with permission...

Equation (4.12) indicates the effect of viscous dissipation on heat transfer in micro-channels. In the case when the inlet fluid temperature, To, exceeds the wall temperature, viscous dissipation leads to an increase in the Nusselt number. In contrast, when To < Tv, viscous dissipation leads to a decrease in the temperature gradient on the wall. Equation (4.12) corresponds to a relatively small amount of heat released due to viscous dissipation. Taking this into account, we estimate the lower boundary of the Brinkman number at which the effect of viscous dissipation may be observed experimentally. Assuming that (Nu-Nuo)/Nuo > 10 the follow-... 

For the heating regime at small X+, the heat transferred from the wall to the cold fluid and the heat released due to viscous dissipation lead to an increase in... 

Koo J, Kleinstreuer C (2004) Viscous dissipation effects in micro-tubes and micro-channels. Int J Heat Mass Transfer 47 3159-3169... 

Nusselt number corresponds to negligible viscous dissipation Peclet number Prandtl number... 

Q and R are generation terms for heat and chemical species respectively, while the pressure gradient Vp plays an analogous role for momentum generation. The heat generation arises from viscous dissipation and from reaction heating ... 

Figure 3 illustrates some additional capability of the flow code. Here no pressure gradient is Imposed (this is then drag or "Couette flow only), but we also compute the temperatures resulting from Internal viscous dissipation. The shear rate in this case is just 7 — 3u/3y — U/H. The associated stress is.r — 177 = i/CU/H), and the thermal dissipation is then Q - r7 - i/CU/H). Figure 3 also shows the temperature profile which is obtained if the upper boundary exhibits a convective rather than fixed condition. The convective heat transfer coefficient h was set to unity this corresponds to a "Nusselt Number" Nu - (hH/k) - 1. 

The temperature field for the polymer melt can be determined using an energy balance which includes terms for convection, conduction and viscous dissipation. 

Since variations In the pressure Induced by fluid dynamic effects are negligible for MOCVD reactor flows, the Inlet pressure, Pq, Is used. In formulating the energy balance, the contributions from pressure changes, viscous dissipation and Dufour effects may neglected for MOCVD conditions (14.15) so the equation becomes ... 

---