Flows Newton number

L = Static mixer length, in., or length of pipe in ft IAt = Thickness of heat transfer wali, ft M = Mass flow rate, lb/hr m = Function of fluid properties, such as u, k and cp N, = Shaft speed of rotation, revolutions per second Ne = Newton number, depends on design NF = Force number, consistent units, dimensionless = F/(PN2D4)... 

The power input in stirred tanks can be calculated using the equation P = Ne pnM if the Newton number Ne, which at present still has to be determined by empirical means, is known. For stirred vessels with full reinforcement (bafQes, coils, see e.g. [20]), the only bioreactors of interest, this is a constant in the turbulent flow range Re = nd /v> 5000-10000, and in the non-aerated condition depends only on geometry (see e.g. [20]). In the aerated condition the Newton number is also influenced by the Froude number Fr = n d/g and the gas throughput number Q = q/nd (see e.g. [21-23]). 

Using Eqs. (11) and (2) the stirrer performance and the resultant Newton number Ne for the laminar flow range can be derived with the formula P = 2ti... 

In the range Re > 50 (vessel with baffles) or Re > 5 x 10" (unbaffled vessel), because the Newton number Ne = P/(pn d ) remains constant. In this case, viscosity is irrelevant we are dealing with a turbulent flow region. 

Weber [572] has recently drawn attention to this problem by reminding us of the fact that fully developed turbulence cannot be realized in unbaffied laboratory vessels. Only at Re > 10 does the friction factor Cf in stirrer flow (counterpart of the Newton number) become constant. This can only be attained by stirring water in tanks with D > 1 m ... 

The so-called Newton (or power) number Ne depends on the Reynolds number and the special geometry of the system (vessel, stirrer, baffles, flow resistances). In Fig. 3.7-2 the Newton number is plotted against the Reynolds number for a multiblade stirrer in a vessel without and with baffles. 

The Newton numbers for nearly all other stirrers in the completely tuibulent flow range are between 1/3 and 5 and depend on their flow resistances. 

Computation of the thermal energy dissipated by a stirrer requires knowledge of the Newton number, also called the power number, which depends on the stirrer type and the flow regime characterized by the Reynolds number. The contribution of the stirrer power to the heat balance is examined in more detail in Section 4.3. 

Fig. 23.18 Newton number Ne) for the liquid flow inside the nozzie as a function of the iiquid Reynolds number (Rei)...

For laminar flow a similar differential force balance, balancing the drag force on a blade section and the torque can be formulated. For creeping flow inertial forces are not important and viscous forces completely dominates the flow. The laminar steady drag force on the fluid can be written as F a p,UD, where p, is the fluid viscosity. Substituting U with kND the power becomes P a pN D or = K, where K is the Newton number for laminar flow. K can be rewritten as ... 

The dynamic viscosity, or coefficient of viscosity, 77 of a Newtonian fluid is defined as the force per unit area necessary to maintain a unit velocity gradient at right angles to the direction of flow between two parallel planes a unit distance apart. The SI unit is pascal-second or newton-second per meter squared [N s m ]. The c.g.s. unit of viscosity is the poise [P] 1 cP = 1 mN s m . The dynamic viscosity decreases with the temperature approximately according to the equation log rj = A + BIT. Values of A and B for a large number of liquids are given by Barrer, Trans. Faraday Soc. 39 48 (1943). 

For a Hquid under shear the rate of deformation or shear rate is a function of the shearing stress. The original exposition of this relationship is Newton s law, which states that the ratio of the stress to the shear rate is a constant, ie, the viscosity. Under Newton s law, viscosity is independent of shear rate. This is tme for ideal or Newtonian Hquids, but the viscosities of many Hquids, particularly a number of those of interest to industry, are not independent of shear rate. These non-Newtonian Hquids may be classified according to their viscosity behavior as a function of shear rate. Many exhibit shear thinning, whereas others give shear thickening. Some Hquids at rest appear to behave like soHds until the shear stress exceeds a certain value, called the yield stress, after which they flow readily. 

For Reynolds numbers > 1000, the flow is fully turbulent. Inertial forces prevail and becomes constant and equal to 0.44, the Newton region. The region in between Re = 0.2 and 1000 is known as the transition region andC is either described in a graph or by one or more empirical equations. 

This expression for the terminal velocity (i.e., the constant velocity that the particle ultimately attains), is called Stokes law. When the Reynolds number is high, say usually greater than of the order of 800, the flow becomes turbulent flow and eddies form. It was Newton... 

In a supersonic gas flow, the convective heat transfer coefficient is not only a function of the Reynolds and Prandtl numbers, but also depends on the droplet surface temperature and the Mach number (compressibility of gas). 154 156 However, the effects of the surface temperature and the Mach number may be substantially eliminated if all properties are evaluated at a film temperature defined in Ref. 623. Thus, the convective heat transfer coefficient may still be estimated using the experimental correlation proposed by Ranz and Marshall 505 with appropriate modifications to account for various effects such as turbulence,[587] droplet oscillation and distortion,[5851 and droplet vaporization and mass transfer. 555 It has been demonstrated 1561 that using the modified Newton s law of cooling and evaluating the heat transfer coefficient at the film temperature allow numerical calculations of droplet cooling and solidification histories in both subsonic and supersonic gas flows in the spray. 

Viscosity is then a measure of the resistance of a material to flow. In fact, the inverse of viscosity is given the name fluidicity. A material s resistance to flow increases with its viscosity. Viscosity has been reported using a number of different names. The centimeter-gram-second (CGS) unit of viscosity is called the poise, which is a dyne seconds per square centimeter. Another widely used unit is the pascal (or Pa), which is Newton seconds per square centimeter. In fact, 1 Pa= 10 poise. 

At the phenomenological level, there are enough further relations between the 14 variables to reduce the number to 5 and make the problem determinate. These further relations are the thermodynamic ones and Stokes and Newton s laws of viscosity and heat flow. These lead from the transport equations to the Navier-Stokes equations. It is noted that these are irreversible. 

Global Newton Naphtali and Sandholm (42) Holland (8) High number of trays, Tew components All type mixtures including nonideal Requires good starting values Chemical and reactive systems Two of condenser duty, re boiler duly, reflux, and boilup plus all side product flows, one purity allowed... 

Using the cgs system with the dyne as the unit of force and erg as the unit of work is impractical for many applications (too large or too small numbers to work with), so the mks system of units is preferred. Then the force is in newtons (1 N = 105 dyn), and the unit of work is the joule (1 J = 107 erg). A rate of doing work of 1 J/s is 1 W. A watt is also a measure of electrical power, being 1 A flowing with a potential of 1 V. 

The idealised flow of non-reactive liquids in smooth channels has been considered by founding physicists such as Newton and Poiseuille and is well understood. Slow flow of a liquid is laminar or streamlined while rapid flow is turbulent with the transition occurring when the Reynolds number - the dimensionless parameter (Udphf) in which U is the velocity of the liquid, p and i] are the density and... 

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