n a o ∫ 1 ( − r a ) v r ( 1 − δ a f a ) d f a \int )v_(1-\delta _f_)}}\,df_ plug flow reactors τ = c a o ∫ 1 ( − r a ) ( 1 − δ a f a ) d f a \int )(1-\delta _f_)}}\,df_ continuous stirred-tank reactors τ = c a in − c a out − r a f ( 1 − δ a f a ) -c_(1-\delta _f_)}}\ } generally, when reactions take place in the liquid and solid phases the change in volume due to reaction is not significant enough that it needs to be taken into account. reactions in the gas phase often have significant changes in volume and in these cases one should use these modified equations. determining the rtd experimentally residence time distributions are measured by introducing a non-reactive tracer into the system at the inlet. its input concentration is changed according to a known function and the output concentration measured. the tracer should not modify the physical characteristics of the fluid (equal density, equal viscosity) or the hydrodynamic conditions and it should be easily detectable. in general, the change in tracer concentration will either be a pulse or a step. other functions are possible, but they require more calculations to deconvolute the rtd curve. pulse experiments this method required the introduction of a very small volume of concentrated tracer at the inlet of the reactor, such that it approaches the dirac delta function. although an infinitely short injection cannot be produced, it can be made much smaller than the mean residence time of the vessel. if a mass of tracer, m , is introduced into a vessel of volume v and an expected residence time of τ , the resulting curve of c ( t ) can be transformed into a dimensionless residence time distribution curve by the following relation: e ( t ) = c ( t ) ∫ 0 ∞ c ( t ) d t ^c(t)\,dt}}} step experiments the concentration of tracer in a step experiment at the reactor inlet changes abruptly from 0 to c 0 . the concentration of tracer at the outlet is measured and normalized to the concentration c 0 to obtain the non-dimensional curve f ( t ) which goes from 0 to 1: f ( t ) = c ( t ) c 0 . .} the step- and pulse-responses of a reactor are related by the following: f ( t ) = ∫ 0 t e ( t ′ ) d t ′ e ( t ) = d f ( t ) d t ^e(t')\,dt'\qquad e(t)= a step experiment is often easier to perform than a pulse experiment, but it tends to smooth over some of the details that a pulse response could show. it is easy to numerically integrate an experimental pulse response to obtain a very high-quality estimate of the step response, but the reverse is not the case because any noise in the concentration measurement will be amplified by numeric differentiation. applications chemical reactors an rtd curve for a reasonably well-mixed reactor in chemical reactors, the goal is to make components react with a high yield. in a homogeneous, first-order reaction, the probability that an atom or molecule will react depends only on its residence time: p r = exp ⁡ ( − k t ) }=\exp \left(-kt\right)} for a rate constant k . given a rtd, the average probability is equal to the ratio of the concentration a of the component before and after: p r ¯ = a o u t / a i n = ∫ 0 ∞ exp ⁡ ( − k t ) e ( t ) d t . }}}=a_ }/a_ }=\int _^\exp \left(-kt\right)e(t)\,dt.} if the reaction is more complicated, then the output is not uniquely determined by the rtd. it also depends on the degree of micromixing, the mixing between molecules that entered at different times. if there is no mixing, the system is said to be completely segregated, and the output can be given in the form a o u t = ∫ 0 ∞ a b a t c h ( t ) e ( t ) d t . }=\int _^a_ }(t)e(t)\,dt.} for given rtd, there is an upper limit on the amount of mixing that can occur, called the maximum mixedness, and this determines the achievable yield. a continuous stirred-tank reactor can be anywhere in the spectrum between completely segregated and perfect mixing. the rtd of chemical reactors can be obtained by cfd simulations. the very same procedure that is performed in experiments can be followed. a pulse of inert tracer particles (during a very short time) is injected into the reactor. the linear motion of tracer particles is governed by newton's second law of motion and a one-way coupling is stablished between fluid and tracers. in one-way coupling, fluid affects tracer motion by drag force while tracer does not affect fluid. the size and density of tracers are chosen so small that the time constant of tracers becomes very small. in this way, tracer particles exactly follow the same path as the fluid does. groundwater flow hydraulic residence time (hrt) is an important factor in the transport of environmental toxins or other chemicals through groundwater. the amount of time that a pollutant spends traveling through a delineated subsurface space is related to the saturation and the hydraulic conductivity of the soil or rock. porosity is another significant contributing factor to the mobility of water through the ground (e.g. toward the water table). the intersection between pore density and size determines the degree or magnitude of the flow rate through the media. this idea can be illustrated by a comparison of the ways water moves through clay versus gravel. the retention time through a specified vertical distance in clay will be longer than through the same distance in gravel, even though they are both characterized as high porosity materials. this is because the pore sizes are much larger in gravel media than in clay, and so there is less hydrostatic tension working against the subsurface pressure gradient and gravity. groundwater flow is important parameter for consideration in the design of waste rock basins for mining operations. waste rock is heterogeneous material with particles varying from boulders to clay-sized particles, and it contains sulfidic pollutants which must be controlled such that they do not compromise the quality of the water table and also so the runoff does not create environmental problems in the surrounding areas. aquitards are clay zones that can have such a degree of impermeability that they partially or completely retard water flow. these clay lenses can slow or stop seepage into the water table, although if an aquitard is fractured and contaminated then it can become a long-term source of groundwater contamination due to its low permeability and high hrt. water treatment