Frictional Flow

The temperature of the fluid in compressible flow through a conduit of constant cross section may be kept constant by a transfer of heat through the conduit wall. Long, small, uninsulated pipes in contact with air transmit suflicient heat to keep the flow nearly isothermal. Also, for small Mach numbers, the pressure pattern for isothermal flow is nearly the same as that for adiabatic flow for the same entrance conditions, and the simpler equations for isothermal flow may be used. The maximum velocity attainable in isothermal flow is 

The dimensions of a are those of velocity. Comparing Eq. (6.47) with Eq. (6.17) for the acoustic velocity shows that 

for air where = -JlA 1.2 the acoustic velocity is approximately 20 percent larger than a. Corresponding to the true Mach number, an isothermal Mach number may be defined  

The parameter Ni plays the same part in isothermal flow as the Mach number does in adiabatic flow. An isothermal process cannot pass through the limiting 

The basic equation for isothermal flow is simple. It is obtained by introducing the mass velocity into the mechanical-energy balance [Eq. (6.8)] and integrating directly, 

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The treatment of vent flow calculations in most typical relief wstem configurations involves two classes of computation models flow in low-fric tion geometries such as nozzles and frictional flow in pipes and fittings. 

B. X., Frictional flow characteristics of water flowing through rectangular microchannels, Exp. Heat Transfer 7 (1994) 249-264. 

Assuming the absence of any frictional flow effects, the outlet flow, velocity v, is related to the instantaneous depth of liquid within the tank, by the relationship... 

Figure 4-4 Liquid escaping through a hole in a process unit. The energy of the liquid resulting from its pressure in the vessel is converted to kinetic energy, with some frictional flow losses in the hole.

Figure 4-6 Liquid flowing through a pipe. The frictional flow losses between the fluid and the pipe wall result in a pressure drop across the pipe length. Kinetic energy changes are frequently negligible.

Figure 1.12. Angle of friction—flow through slot...

The values of >Pref and UTre are, respectively, (v2/ypg)1/3 and (ypgv)1/3, where yp = (pp—p)/p [13]. The take-off curve allows to estimate by knowing the adimensionalized friction flow velocity the granulometric slice liable to take-off. In fact, the diameter noted T>pcriti is the diameter from which the particles take-off and the diameter T>pcrit2... 

Head of liquid equivalent to the frictional flow resistance in the downcomer and in the passage of the fluid from the downcomer onto the plate. This head is usually relatively small, but it can be estimated as three, kinetic heads for the liquid, based on the linear liquid velocity at the minimum cross-sectional area for downooming liquid flow, or... 

Tverherg, J.C., Effect of Surface Roughness on Fluid Friction, Flow Control, 8,11,1995. 

Peng, X.F., Peterson, G.P. and Wang, B.X., Frictional Flow Characteristics of Water Flowing Through Micro-Channels, Experimental Heat Transfer, 1994, 7, 249-264. 

The velocity held is determined by the characteristic length L0, and velocity w0 e.g. the entry velocity in a tube or the undisturbed velocity of a fluid flowing around a body, along with the density g and viscosity rj of the fluid. While density already plays a role in frictionless flow, the viscosity is the fluid property which is characteristic in friction flow and in the development of the boundary layer. The two material properties, thermal conductivity A and specific heat capacity c, of the fluid are important for the determination of the temperature held in conjunction with the characteristic temperature difference Ai 0. The specihc heat capacity links the enthalpy of the fluid to its temperature. 

Thomas Edward Stanton (1865-1931) was a student of Reynolds in Manchester. In 1899 he became a Professor for Engineering at the University of Bristol. Stanton researched momentum and heat transport in friction flow. He also worked in aerodynamics and aeroplane construction. 

In filtering centrifuges (Fig. 26), the solids settle out as in the decanter. However, the liquid flows in the same direction and out through the perforated bowl. As the liquid flows in the same direction as the solids, it increases the rate of deposition. In addition, as the liquid flows through the cake, the frictional effect results in sharply increased stresses on the cake. At low speeds, the stress because of the frictional flow may be larger than the centrifugal body forces. 

CORRECTION OF BERNOULLI EQUATION FOR FLUID FRICTION. Friction manifests itself by the disappearance of mechanical energy. In frictional flow the quantity... 

Adiabatic frictional flow through a conduit of constant cross section. This process is irreversible, and the entropy of the gas increases, but as shown by Eq. (6.22), since Q 0, the stagnation temperature is constant throughout the conduit. This process is shown in Fig. 6.1b. 

In adiabatic frictional flow, the temperature of the gas changes. The viscosity also varies, and the Reynolds number and friction factor are not actually constant. In gas flow, however, the effect of temperature on viscosity is small, and the effect of Reynolds number on the friction factor / is still less. Also, unless the Mach number is nearly unity, the temperature change is small. It is satisfactory to use an average value for /as a constant in calculations. If necessary,/ can be evaluated at the two ends of the conduit and an arithmetic average used as a constant. 

EQUATIONS FOR ADIABATIC FRICTIONAL FLOW. Equation (6.8) is multiplied by p/p, giving... 

In adiabatic frictional flow the ratio of the inlet and outlet pressures is found by direct integration of Eq. (6.35) between the limits p , p, and iVMa,o> to give... 

All data given for the safety valve refer to the inlet and outlet diameter as well as the smallest cross sectional area. The friction flow factor a was fixed in accordance with the recommendations given for Leung s model. 

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