Mass time constant

The force on one nucleus due to sPetching or compressing the bond is equal to the force constant of the bond k times the distance between the nuclei x2 — xi). It is equal and opposite to the force acting on the other nucleus, and it is also equal to the mass times the acceleration x by Newton s second law (see section on the hamionic oscillator in Chapter 4). The equations of motion are... 

The time constant R /D, and hence the diffusivity, may thus be found dkecdy from the uptake curve. However, it is important to confirm by experiment that the basic assumptions of the model are fulfilled, since intmsions of thermal effects or extraparticle resistance to mass transfer may easily occur, leading to erroneously low apparent diffusivity values. 

First-Order Lag (Time Constant Element) Next consider the system to be the tank itself. A dynamic mass balance on the tank gives ... 

Time constants. Where there is a capacity and a throughput, the measurement device will exhibit a time constant. For example, any temperature measurement device has a thermal capacity (mass times heat capacity) and a heat flow term (heat transfer coefficient and area). Both the temperature measurement device and its associated thermowell will exhibit behavior typical of time constants. 

G = air flow rate, mass/time L = waste flow rate, volume/time X = concentration of pollutant in waste, massA ohime H = Henry s constant for the pollutant in water, volume/mass... 

In the case of a temperature probe, the capacity is a heat capacity C == me, where m is the mass and c the material heat capacity, and the resistance is a thermal resistance R = l/(hA), where h is the heat transfer coefficient and A is the sensor surface area. Thus the time constant of a temperature probe is T = mc/ hA). Note that the time constant depends not only on the probe, but also on the environment in which the probe is located. According to the same principle, the time constant, for example, of the flow cell of a gas analyzer is r = Vwhere V is the volume of the cell and the sample flow rate. 

In the SFM the reactor is divided into three zones two feed zones fj and (2 and the bulk b (Figure 8.1). The feed zones exchange mass with each other and with the bulk as depicted with the flow rates mi 2, i,3 and 2,3 respectively, according to the time constants characteristic for micromixing and mesomix-ing. As imperfect mixing leads to gradients of the concentrations in the reactor, different supersaturation levels in different compartments govern the precipitation rates, especially the rapid nucleation process. 

This heat balance contains two terms not seen before mgCR represents the mass times specific heat of the agitator and vessel walls and q represents the energy input by the agitator. Although the model is nominally for constant physical properties, Vermeulen and Fortuin found a better fit to the experimental data when they used a slightly different specific heat for the inlet stream (Cp), . 

Mass transfer in a gas-liquid or a liquid-liquid reactor is mainly determined by the size of the fluid particles and the interfacial area. The diffusivity in gas phase is high, and usually no concentration gradients are observed in a bubble, whereas large concentration gradients are observed in drops. An internal circulation enhances the mass transfer in a drop, but it is still the molecular diffusion in the drop that limits the mass transfer. An estimation, from the time constant, of the time it wiU take to empty a 5-mm drop is given by Td = d /4D = (10 ) /4 x 10 = 6000s. The diffusion timescale varies with the square of the diameter of the drop, so... 

Here Q(t) denotes the heat input per unit volume accumulated up to time t, Cp is the specific heat per unit mass at constant pressure, Cv the specific heat per unit mass at constant volume, c is the sound velocity, oCp the coefficient of isobaric thermal expansion, and pg the equilibrium density. (4) The heat input Q(t) is the laser energy released by the absorbing molecule per unit volume. If the excitation is in the visible spectral range, the evolution of Q(t) follows the rhythm of the different chemically driven relaxation processes through which energy is... 

For a thermometer to react rapidly to changes in the surrounding temperature, the magnitude of the time constant should be small. This involves a high surface area to liquid mass ratio, a high heat transfer coefficient and a low specific heat capacity for the bulb liquid. With a large time constant, the instrument will respond slowly and may result in a dynamic measurement error. 

U is the heat transfer coefficient, M the mass, Cp the heat capacity and A the heat transfer area. A knowledge and understanding of the appropriate time constants is important in interpreting many of the simulation examples. 

Figure 2.27. Mixing, mass transfer and oxygen consumption in a bubble column bioreactor (Oosterhuis, 1984). Tj - reaction time constant, Xmt - ass transfer time constant, tmix -mixing time constant. ro2 - oxygen consumption rate, Vs - superficial gas velocity.

It is assumed that all the tank-type reactors, covered in this and the immediately following sections, are at all times perfectly mixed, such that concentration and temperature conditions are uniform throughout the tanks contents. Fig. 3.10 shows a batch reactor with a cooling jacket. Since there are no flows into the reactor or from the reactor, the total mass balance tells us that the total mass remains constant. 

Feed temperature Latent heat of vaporization Average steam mass flow Proportional gain Set temperature of tank Time constant of thermocouple Time constant of thermowell constant of integral control 1,TFIN=30,NOCI=3 RESET GOTOl... 

Figure 25. Ion mass peaks at different two-photon energies. Broadenings of trimethylamine (TMA ) ion peaks as a function of the ionization energy. A hv2 > 3.875 eV B hv2 = 3.688 eV C hv2 = 3.607 eV. Excitation energy of paraxylene (PX) in the Si state = 3.90 eV. The broadenings in B and C correspond to time constants of 160 20 and 200 20 ns, respectively. Peaks corresponding to TMA H+ are also observable. Taken with permission from Int. J. Mass Spectrom. Ion Proc. 1994, 131, 233-264.

Change the value of the mass transfer capacity coefficient KLa and observe how this affects the time to reach equilibrium. Is there a relationship between KLa and the system time constant r ... 

Compare the residence time for the L phase, Vl/L, with the time constant for mass transfer, l/KLa. Maintain a constant ratio of these two time constants but vary the individual parameters. How is the approach to steady state influenced by these changes ... 

The second problem to be tackled is data reconciliation for applications in which the dominant time constant of the dynamic response of the system is much smaller than the period in which disturbances enter the system. Under this assumption the system displays quasi-steady-state behavior. Thus, we are concerned with a process that is essentially at steady state, except for slow drifts or occasional sudden transitions between steady states. In such cases, the estimates should be consistent, that is, they should satisfy the mass and energy balances. 

One way of circumventing the difficulties encountered for systems with widely different time constants is to split the reservoirs into two categories. The first category will comprise the reservoirs with short residence times which will be explicitly required to satisfy the constraints of mass conservation. Reservoirs with long residence times will make up the second category which we will treat as source and sinks. Equation (7.3.8) will be transformed into... 

We have shown that the vector mesons in the CFL phase have masses of the order of the color superconductive gap, A. On the other hand the solitons have masses proportional to F%/A and hence should play no role for the physics of the CFL phase at large chemical potential. We have noted that the product of the soliton mass and the vector meson mass is independent of the gap. This behavior reflects a form of electromagnetic duality in the sense of Montonen and Olive [29], We have predicted that the nucleon mass times the vector meson mass scales as the square of the pion decay constant at any nonzero chemical potential. In the presence of two or more scales provided by the underlying theory the spectrum of massive states shows very different behaviors which cannot be obtained by assuming a naive dimensional analysis. 

During the last years, so-called microhotplates (pHP) have been developed in order to shrink the overall dimensions and to reduce the thermal mass of metal-oxide gas sensors [7,9,15]. Microhotplates consist of a thermally isolated stage with a heater structure, a temperature sensor and a set of contact electrodes for the sensitive layer. By using such microstructures, high operation temperatures can be reached at comparably low power consumption (< 100 mW). Moreover, small time constants on the order of 10 ms enable applying temperature modulation techniques with the aim to improve sensor selectivity and sensitivity. 

As any high school student, knows, Newton s second law of motion says that force is equal to mass times acceleration for a system with constant mass M. 

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