B.A. Buffham

Holdup and dispersion: tracer residence times, moments and inventory measurements

Abstract The inventory function is the quantity of tracer remaining in a continuous-flow system at elapsed time t when steady flow of the tracer is replaced by untraced flow at t = 0. The relations between residence-time distributions, moments and changes in inventory when a tracer is flushed from a system are established. It is shown that inventory measurements could be an attractive way of measuring moments. In particular, the mean residence time is given by the intercept on the baseline of the initial tangent to the inventory curve, and the variance by the area between the inventory curve, the initial tangent and the baseline. It is proposed that dispersion be defined in terms of the variance of the residence-time distribution. This would allow experimentalists to record their results independently of models or theories in addition to comparing their results with the predictions of theories. Methods based on inventory measurements are potentially more accurate than the traditional step- and pulse-response methods. Ways in which inventory measurements might be made are suggested. It is timely that the theory should be presented now because tomographic methods that could be used to measure inventory are starting to appear.

Tracer kinetics: some general properties, the mean residence time and applications to phase and chemical equilibria

Abstract Unambiguous unsteady-state material balances may be written for systems of arbitrary complexity in terms of boundary fluxes and system contents. Such balances provide a basis for the analysis of tracer experiments even when the residence-time concept is not applicable. Counterparts of the familiar integral relationships for residence-time distributions are derived for the more general situation. Circumstances exist for which residence-time distributions may be properly defined in addition to the usual “closed-closed” case. When a system possesses a residence-time distribution the mean residence time is the ratio of the steady-state tracer holdup to the rate of steady tracer transmission through the system. This result is a more general form of the well-known principle that the mean residence time for steady, constant-density, single-phase flow through a closed system is the ratio of the volume to the volumetric flow rate. Examples illustrate how the general principle permits direct evaluation of the mean residence time by steady-state analysis in the presence of such features as density variation, absorbing boundaries and multiple feeds with different compositions. Moreover, the result quoted applies equally to multiphase and reacting systems. When equilibrium obtains in the steady state the mean time is directly related to thermodynamic parameters and transients may be used to determine equilibrium properties without reference to process models.

Capillary viscometry by perturbation of flow and composition

Abstract A new technique for making viscosity measurements on gas mixtures is introduced. The composition and flowrate of a mixture flowing through a capillary tube are perturbed by adding a small stream of perturbation gas. This is usually a pure, individual component of the mixture. The pressure at the inlet of the capillary tube rises when the perturbation gas is added and this pressure increase is proportional to the flowrate change. Because there is empty volume between the point where the perturbation gas is added and the capillary tube, it is some time later that the pressure changes again when the composition of the gas flowing through the tube changes. This second pressure change is proportional to the change in viscosity. The ratio of these two steps of pressure is proportional to d ln μ/dXi where μ is the viscosity and Xi is the mole fraction of component i. An apparatus has been developed which is capable of making measurements with suitable precision and some preliminary data for the nitrogen–argon system at 1.2 bar and 24°C are given.

Gas Adsorption Isotherms from Composition and Flow-Rate Transient Times in Chromatographic Columns II. Effect of Pressure Changes

In the previous paper a method was described for measuring binary adsorption isotherms by a chromatographic method. The method involved making simultaneous small perturbations to the concentration and flow-rate of the stream entering a packed column and observing the changes with time of the composition and flow-rate of the gas leaving the column. The perturbation to the inlet flow was made by adding a small extra stream of gas of known composition to the main flow passing through the columns. The present paper describes the effect of the increased mean pressure in the column caused by the small increase in flow. The effect is surprisingly large. A correction term is derived to compensate for the effect of increased pressure. By reanalysing the results for argon-nitrogen reported in Paper I it is shown that instead of two different isotherms being obtained by making the perturbation with the addition of argon or nitrogen, a single isotherm for each component is obtained.

Limiting forms of the residence time distribution for recycle system.

Abstract The widely held supposition that the residence time distribution of a recycle system always approaches that of an ideal mixer in the limit of high recy Necessary and sufficient conditions for a limit to exist and for this limit to be exponential are derived.

The size and compactness of particles of arbitrary shape : application to catalyst effectiveness factors

Abstract The need to describe particle size and shape occurs in many branches of science and technology. Examples include metal grains in metallurgy, cells in pathology and catalyst pellets in chemical engineering. Shape factors describe a particle simply. They also allow one to infer the performance of complicated shapes from the performance of simpler ones and to correlate performance. Most existing shape factors, such as the ratio of the surface area of the sphere with same volume as a particle to the particles surface area, cannot distinguish between infinite laminae and infinite cylinders (idealized plates and rods). Particle size and compactness are defined here in terms of the particles dimensions and surface curvature. The compactness seems in accord with ones intuitive notion of compactness and is different for infinite laminae and cylinders. When a particle interacts with its surroundings, the geometry affects what happens. A good mathematical model may exist, but solving the model equations for complicated shapes can be difficult even though solutions may be found for simple shapes. The rate of diffusion-impeded chemical reaction is used as an example of compactness as a correlator.

Gas adsorption isotherms from composition and flow-rate transient times in chromatographic columns I. Basic theory and a binary experimental test

The theory of a new method of determining multicomponent gas adsorption isotherms is outlined. The method involves making a perturbation to the composition of a particular mixture of gas flowing through a chromatographic column and recording the variation with time of the flow-rate, in addition to the variation of the composition, of the gas leaving the column. Proper interpretation of these measurements enables the change in adsorption of all of the components in the column to be determined. As an illustration, an experimental apparatus was built to determine the adsorption of nitrogen-argon mixtures on 5A molecular sieve at 50 °C. The method is potentially fast, possibly taking only a few minutes for each composition. For a particular carrier gas composition, the change in composition was made in two ways: one by adding a small stream of argon, and the other by adding a small stream of nitrogen. Comparison of the results obtained by these two experiments showed a systematic difference between them. This difference was probably due to the pressure in the column changing when the small stream of perturbation gas was added.

Perturbation viscometry of gas mixtures: fitting a model to logarithmic viscosity gradients

Abstract Perturbation viscometry is a recently developed method which measures the logarithmic gradient of the viscosity–composition curve for gas mixtures using a variant of capillary-tube viscometry. A gas mixture flowing through a capillary has its composition perturbed by the addition of a small flow of gas, normally one of the pure components of the gas mixture. Two pressure changes, the first due to the change in flowrate and second due to the change in viscosity are seen at the capillary. The logarithmic viscosity gradient is calculated from the ratio of these two pressure changes. Integration of logarithmic viscosity gradients measured for the full composition range yields the mixture viscosity relative to the viscosity of either component of the gas mixture. This method is attractive because, for measurements of equal precision, integration of the gradients is potentially much more precise than conventional methods that measure the absolute viscosities directly. Integration of the logarithmic viscosity gradients could be accomplished either by application of the trapezium rule or numerical integration of a polynomial fitted to the data. Here a method with better theoretical foundations is presented. The method fits a differential form of the Sutherland equation to the logarithmic viscosity gradients. The fitted parameters of the equation are then used to generate the viscosity ratios by substitution into the normal form of the Sutherland equation. The quality of fit obtained may be used to test of the consistency of the experimental data. Experimental data gathered for the mixtures argon–nitrogen, helium–argon and helium–nitrogen at 98.5°C are analysed using this procedure. The internal consistency of the data is checked by direct comparison of calculated values for the gradients generated from the fitted Sutherland parameters with actual experimental data. The calculated relative viscosities are compared with extant data and theoretical predictions and show good agreement. Also the calculated Sutherland parameters are well within the scatter of the parameters obtained from alternative sources.

Remote sensing of the flux responses of a gas–solid catalytic micro-reactor

Abstract A new method for measuring the rates of heterogeneous catalytic and possibly other gas–solid reactions is introduced in this paper. It is based on measuring the “flux response” of a reactor. The flux response of a continuous-flow gas–solid reactor is defined as the way the net rate at which molecules leave the reactor changes when some input, or other, variable is changed. The flux response depends on the combined effect of adsorption, reaction and desorption of the gases in the reactor. Apparatus for measuring the flux response is described. The apparatus design is based on that of apparatus used in the sorption-effect method for measuring gas adsorption. Modifications to that apparatus to make it suitable for investigating the flux response of a catalytic micro-reactor are reported. Flux-response experiments have been carried out using the catalytic decomposition of methanol over platinized alumina as the demonstration reaction. The flux response measurements were complemented by mass spectrometric analysis of the reactor effluent. The experiments reported here confirm that the flux response can disclose some of the detail of the adsorption–reaction–desorption mechanism of heterogeneous catalysis. Conversions are calculated from the experimental flux responses.

On the residence-time distribution for a system with velocity profiles in its connections with the environment

A point-stimulus point-response function is defined that allows rigorous expressions to be written for the residence time distribution for a system where there are complicating factors such as inlet concentration profiles and inlet and outlet velocity profiles. This function is capable of being measured directly; the residence time distribution and other response functions may be calculated from it. It is not possible to calculate the residence time distribution from the results of experiments in which tracer is generated and/or measured in situ at locations were there are significant velocity profiles unless further information is available.