Velocity steady-state terminal

At steady-state terminal velocity, the acceleration term in the force balance (B9) on a moving submerged body is zero, and the balance may be written so that gravity forces are balanced by the sum of buoyancy and resistance forces. 

Relaxation time The time necessary for a moving particle to adjust from one given steady state velocity to another, e.g., the time for a falling particle to reach its terminal velocity. It is independent of the nature of the force applied to the particle. 

If a single particle is falling freely under gravity in an infinitely dilute suspension, it will accelerate until it reaches a steady-state velocity. This final velocity is known as the terminal settling velocity (t/t) and represents the maximum useful superficial velocity achievable in a fluidised bed. Thus, the contained particles will be elutriated from the column if the superficial velocity is above Ut, the value of which can be predicted using the Stokes equation... 

Church s equations, 176-177 Combustion instability, 52-57 bulk-coupled, 56-57 pressure-coupled, 52-55 velocity-coupled, 55-56 steady-state, 29-51 prediction, 30 pressure plateaus, 34 propellants, 31-50 termination, 57-64 depressurization, 58-62 fluid-injection, 63-64 L, 62-63... 

That the most likely coarse velocity is equal to the most likely terminal velocity can only be true in two circumstances either the system began in the steady state and the most likely instantaneous velocity was constant throughout the interval, or else the system was initially in a dynamically disordered state, and x was large enough that the initial inertial regime was relatively negligible. These equations are evidently untrue for x —> 0, since in this limit the most... 

That the terminal acceleration should most likely vanish is true almost by definition of the steady state the system returns to equilibrium with a constant velocity that is proportional to the initial displacement, and hence the acceleration must be zero. It is stressed that this result only holds in the intermediate regime, for x not too large. Hence and in particular, this constant velocity (linear decrease in displacement with time) is not inconsistent with the exponential return to equilibrium that is conventionally predicted by the Langevin equation, since the present analysis cannot be extrapolated directly beyond the small time regime where the exponential can be approximated by a linear function. 

So the terminal velocity or flux is a maximum when the terminal acceleration is zero, which implies that the terminal velocity is a constant, which implies that <2(x) is a linear function of x. By definition, the steady state is the state of constant flux. This justifies the above focus on the terms that are linear in x in the study of the steady state. The preceding equations show that the steady state... 

At t = 0, V = 0 and the drag force is zero. As the particle accelerates, the drag force increases, which decreases the acceleration. This process continues until the acceleration drops to zero, at which time the particle falls at a constant velocity because of the balance of forces due to drag and gravity. This steady-state velocity is called the terminal velocity of the body and is given by the solution of Eq. (11-8) with the acceleration equal to zero ... 

Aybers and Tapucu (A4, A5) measured trajectories of air bubbles in water. When surface-active agents continue to accumulate during rise, the terminal velocity may never reach steady state (A4, Bl) and may pass through a maximum (W4). Five types of motion were observed, listed in Table 7.1 with Re based on the maximum instantaneous velocity. Secondary motion of fluid par-... 

Flat metal plates and cylinders driven by tangentially incident detonation waves were examined by Hoskin et al (Ref 5) using a 2-D steady state characteristic code. Their computations for plates or cylinders indicate that metal compressibility has little effect on the terminal velocity imparted to the metal by the expl. Thus the Gurney treatment is found to give essentially the same terminal states as their more sophisticated characteristics computation. This... 

At steady state, that is when the terminal velocity is attained, the accelerating force due to gravity must equal the drag force on the particle F, or (nd3/6)(px — p)g = 3npdun where no is the terminal velocity of the particle. 

In an experiment done to separate proteins by electrophoresis at 20°C, the voltage is switched on at time t = 0. At what time does serum albumin, for which D = 5.94xl0 7 cm2/s and MW = 66,000, reach 99% of its terminal (steady-state) average velocity (See Exercises 3.1 and 3.2.)... 

Tubular Reactors. The simplest model of a tubular reactor, the plug-flow reactor at steady state is kinetically identical to a batch reactor. The time variable in the batch reactor is transformed into the distance variable by the velocity. An axial temperature gradient can be imposed on the tubular reactor as indicated by Gilles and Schuchmann (22) to obtain the same effects as a temperature program with time in a batch reactor. Even recycle with a plug flow reactor, treated by Kilkson (35) for stepwise addition without termination and condensation, could be duplicated in a batch reactor with holdback between batches. 

As outlined above, steady-state theories for the liquid-solid mass transfer are largely classified into two categories i.e., those based on Kolmogoroff s theory and those based on the terminal velocity-slip velocity approach. 

II) Isotachophoresis K separation in a discontinuity between two buffer solutions (the leading and the terminating electrolyte). The sample components are introduced in small quantities in the discontinuity, a d.c. current is applied, and the various components start to move. Consecutive zones are formed and a steady state is attained where each zone moves with equal velocity, hence the name. 

Equation (5.71) allows us to calculate the drag coefficient at steady state for a specific terminal velocity. This formulation is particularly useful because it suggests a strategy for calculating Cd that can easily be applied to very different fluid-fluid systems. In fact, given the particular system under investigation, we can identify the terminal velocity and... 

At steady-state vz = v, the particle reaches its terminal velocity given by... 

It shonld be obvious that the terminal velocity of a bowling ball in air is mnch larger than the terminal velocity of a feather in air. However, in both cases, a steady-state force balance on the object that accounts for buoyancy, gravity, and hydrodynamic drag reveals that log(Dtenninai) P log(psoiid - Pair), where psolid Corresponds to either the bowling ball or the feather. 

---