Net rate

The rate of dissolving of a solid is determined by the rate of diffusion through a boundary layer of solution. Derive the equation for the net rate of dissolving. Take Co to be the saturation concentration and rf to be the effective thickness of the diffusion layer denote diffusion coefficient by . 

The absorption and emission eases ean be eombined into a single net expression for the rate of photon absorption by reeognizing that the latter proeess leads to photon produetion, and thus must be entered with a negative sign. The resultant expression for the net rate of decrease of photons is ... 

To summarize, the line shape function I(co) produces the net rate of photon absorption... 

For all of the eases eonsidered earlier, a C(t) funetion is subjeeted to Fourier transformation to obtain a speetral lineshape funetion I(co), whieh then provides the essential ingredient for eomputing the net rate of photon absorption. In this Fourier transform proeess, the variable co is assumed to be the frequeney of the eleetromagnetie field experienced by the molecules. The above considerations of Doppler shifting then leads one to realize that the correct functional form to use in converting C(t) to I(co) is ... 

It is generally true and especially clear in the case of the barrier represented by the solid line in Fig. 2.7 that movement in either the forward or the backward direction across the barrier is possible. Hence the net rate of the process is written as the difference between the rate forward (subscript f) and the rate backward (subscript b) ... 

This expression gives us the rate constant for the net rate of forward flow according to the activated complex theory. 

In spite of these obstacles, crystallization does occur and the rate at which it develops can be measured. The following derivation will illustrate how the rates of nucleation and growth combine to give the net rate of crystallization. The theory we shall develop assumes a specific picture of the crystallization process. The assumptions of the model and some comments on their applicability follow ... 

The primary control variables at a fixed feed rate, as in the operation pictured in Figure 8, are the cycle time, which is measured by the time required for one complete rotation of the rotary valve (this rotation is the analog of adsorbent circulation rate in an actual moving-bed system), and the Hquid flow rate in Zones 2, 3, and 4. When these control variables are specified, all other net rates to and from the bed and the sequence of rates required at the Hquid... 

Permeant movement is a physical process that has both a thermodynamic and a kinetic component. For polymers without special surface treatments, the thermodynamic contribution is ia the solution step. The permeant partitions between the environment and the polymer according to thermodynamic rules of solution. The kinetic contribution is ia the diffusion. The net rate of movement is dependent on the speed of permeant movement and the availabiHty of new vacancies ia the polymer. 

Example 5 Radiation in a Furnace Chamber A furnace chamber of rectangular paraUelepipedal form is heated hy the combustion of gas inside vertical radiant tubes hningthe sidewalls. The tubes are of 0.127-m (5-in) outside diameter on 0.305-m (12-in) centers. The stock forms a continuous plane on the hearth. Roof and end walls are refractory. Dimensions are shown in Fig. 5-20. The radiant tubes and stock are gray bodies having emissivities of 0.8 and 0.9 respectively. What is the net rate of heat transmission to the stock by radiation when the mean temperature of the tube surface is SIG C (1500 F) and that of the stock is 649 C (1200 F) ... 

Multiple Reactions When a substance participates in several reactions at the same time, its net rate of decomposition is the algebraic sum of its rates in the individual reactions. Identify the rates of the individual steps with subscripts, dC/dt)i, dC/di)9,. Take this case of three reactions,... 

Free radicals are molecular fragments having one or more unpaired electrons, usually short-lived (milhseconds) and highly reaclive. They are detectable spectroscopically and some have been isolated. They occur as initiators and intermediates in such basic phenomena as oxidation, combustion, photolysis, and polvmerization. The rate equation of a process in which they are involved is developed on the postulate that each free radical is at equihbrium or its net rate of formation is zero. Several examples of free radical and catalytic mechanisms will be cited, aU possessing nonintegral power law or hyperbohc rate equations. 

On Figure 6.1.1, the four consecutive reaction steps are indicated on a vertical scale with the forward reaction above the corresponding reverse reaction. The lengths of the horizontal lines give the value of the rate of reaction in mol/m s on a logarithmic scale. In steady-state the net rates of all four steps must be equal. This is given on the left side with 4 mol/m s rate difference, which is 11 mm long. The forward rate of the first step is 4.35 molW s and the reverse of the first reaction is only 0.35 mol/m s, a small fraction of the forward rate. 

V. For reversible reactions the net rate of reaction can be expressed as the difference between forward and reverse reactions ... 

R = net rate of sensible heating of an air column by solar radiation (about 67 cal/mVs)... 

The net rate of an enzymatie reaetion is usually referred to as its veloeity, V, represented by ... 

The eoneentration of the eomplex ean be obtained from the net rate of disappearanee ... 

The free radieals are CjH O and CH3. The net rate of disap-pearanee of CjHjONOj is ... 

The Temkin-Pyzhev equation deseribes the net rate of synthesis over promoted iron eatalyst as ... 

On the basis of the pseudo-steady state assumption, the net rate of disappearanee is zero, therefore... 

Net rate of flow of mass into fluid element... 

Accounting for the mass flowrate across a face of the element, which is given by the density product, area, and velocity component normal to the face (Figure 10-2), the net rate of flow of mass into the element across its boundaries is given by ... 

If heat ean be removed as fast as it is generated by the reaetion, the reaetion ean be kept under eontrol. Under steady state operating eonditions, the heat transfer rate will equal the generation rate (see Figure 6-26). If the heat removal rate Qj. is less than the heat generation rate Qg (e.g., a eondition that may oeeur beeause of a eooling water pump failure), a temperature rise in the reaetor is experieneed. The net rate of heating of the reaetor eontent is the differenee between Equations 12-44 and 12-45. 

The net rate of mass accumulation within a control volume is equal to the rate at which mass enters the control volume by any process minus the rate at which it leaves the control volume by any process. 

Therefore, the sum of the component balances is the total material balance while the net rate of change of any component s mass within the control volume is the sum of the rate of mass input of that component minus the rate of mass output these can occur by any process, including chemical reaction. This last part of the dictum is important because, as we will see in Chapter 6, chemical reactions within a control volume do not create or destroy mass, they merely redistribute it among the components. In a real sense, chemical reactions can be viewed from this vantage as merely relabeling of the mass. 

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