Down-going equation

We will call this the down-going equation. Remember this A plot of this equation is shown in Figure 2.2. 

You will very often need to find a rate constant or a half-life from real data. In the old days (before Microsoft), it was difficult to plot these data as a curved line as shown in Figure 2.2 and to fit a curved line to it such that one could read off the rate constant, k. Thus, it was handy to convert these data into a linear form and to fit a straight line. We can do this easily by taking the natural logarithm of the down-going equation and getting... 

For the up-going equation, one cannot do a simple linearization like we could for the down-going equation. In this case, the equation can be made linear if, and only if, Cmax is known or can be assumed. In this case,... 

Dr. Woodward May I ask a question of some of our colleagues here as a point of information You use the phrase "irreversible carbon laydown, yet the equation for carbon plus steam has arrows going in both directions. I wonder if anybody has any comments on whether it is irreversible or, if you inadvertently start to get some carbon down, is it reversible Can you steam it out ... 

Just as equation (4.35) allows one to go up the ladder to obtain n + ) from n), equation (4.36) allows one to go down the ladder to obtain n — 1) from n). This lowering procedure maintains the normalization of each of the eigenvectors. 

Sn-i,u equation (6.44) allows one to go down the ladder and obtain Sn i,i from Snh Taking the positive square root in going from equation (6.43) to... 

Here all components of the vector e,-, except the z-th one which is unity, are equal to zero. Because the molecular reaction (30) is induced by chemical interaction of groups Aj and A- its rate constant apparently equals k ia-a + a ja ). It is possible to write down an infinite set of kinetic equations corresponding to the scheme (30) for the concentrations C(l a f) of molecules of different composition and functionality which are present in the reaction system at the moment t. To solve these equations it is convenient to go over to the equivalent equation ... 

If you find yourself going blank or taking a wrong turn midway through then do not be afraid to tell the examiners that you cannot remember and would they mind moving on. No one will mark you down for this as you have already supplied them with the equation and the viva will move on in a different direction. 

It may help to write the equation down first to remind yourself which functions go where. The simple point of this diagram is that it linearizes the Michaelis-Menten graph and so makes calculation of and Krn much easier as they can be found simply by noting the points where the line crosses the y and x axes, respectively, and then taking the inverse value. 

If going down the concentration gradient—C2 is less than Ci—the first term in equation 5.3 will be negative (remember AG = -i Tln Ae,), and if going up the concentration gradient—Q is less than C2—the first term will be positive. 

The methods most generally used for the calculation of activity coefficients at intermediate pressures are the Wilson (1964) and UNIQUAC (Abrams and Prausnitz, 1975) equations. Wilson s equation was used by Sato et al. (1985) to predict the composition of fhe condensate gas stripped from a packed bed fermenter at 30°C, whilst Beck and Bauer (1989) used the UNIQUAC equation, with temperature-dependent parameters given by Kolbe and Gmehling (1985), for their model of an anaerobic gas-solid fluidized bed fermenter at 36°C. In this case it was necessary to go beyond the temperature range of fhe source data down to 16°C in order to predict the composition of the fluidizing gas leaving the condenser. 

For very high Cq the poison-free Monod equation just can t apply, for even if there is plenty enough food the cells will crowd out each other, and growth will slow down and will eventually stop. So, for very high cell concentration, we must go to product poison kinetics. 

Both the foregoing steps, assessing your internal and external worlds, are concerned with the considerations that you put into the equation in formulating your ideas for action. Often this is done perfunctorily, even unconsciously. I fancy doing that . That would go down well . 1 could never do that . They would never agree . There wouldn t be the money . In the process of originating an idea you should take a serious look at the assumptions you are making about yourself and your external world and not let them go unquestioned. 

To derive a conversion factor, consult a table that presents unit equalities, such as Table i.i.Then multiply the given quantity by the conversion factor, and voila, the units are converted. Always be careful to write down your units.They are your ultimate guide, telling you what numbers go where and whether you are setting up the equation properly. 

The ideal gas law explains why a hot-air balloon can remain aloft. According to the equation, as the temperature of a gas (T) goes up (and the pressure and volume remain constant) the number of particles of that gas (n) must go down. 

Computational fluid dynamics enables us to investigate the time-dependent behavior of what happens inside a reactor with spatial resolution from the micro to the reactor scale. That is to say, CFD in itself allows a multi-scale description of chemical reactors. To this end, for single-phase flow, the space resolution of the CFD model should go down to the scales of the smallest dissipative eddies (Kolmogorov scales) (Pope, 2000), which is inversely proportional to Re-3/4 and of the orders of magnitude of microns to millimeters for typical reactors. On such scales, the Navier-Stokes (NS) equations can be expected to apply directly to predict the hydrodynamics of well-defined system, resolving all the meso-scale structures. That is the merit of the so-called DNS. 

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