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{{Other uses}}
For material flowing through a volume, the '''residence time''' is a measure of how much time the matter spends in it. Examples include fluids in a [[chemical reactor]], specific elements in a [[geochemical]] reservoir, water in a [[catchment]], bacteria in a culture vessel and drugs in human body. A molecule or small parcel of fluid has a single residence time, but more complex systems have a '''residence time distribution (RTD)'''.

There are at least three time constants that are used to represent a residence time distribution. The ''turn-over time'' or ''flushing time'' is the ratio of the material in the volume to the rate at which it passes through; the ''mean age'' is the mean length of time the material in the reservoir has spent there; and the ''mean transit time'' is the mean length of time the material spends in the reservoir.

Applications of residence times or residence time distributions can be found in a wide variety of disciplines including [[environmental science]], [[engineering]], [[chemistry]], and [[hydrology]].

== History ==
The concept of residence time originated in models of chemical reactors. The first such model was an ''axial dispersion model'' by [[Irving Langmuir]] in 1908. This received little attention for 45 years; other models were developed such as the [[plug flow reactor model]] and the [[continuous stirred-tank reactor]], and the concept of a ''washout function'' (representing the response to a sudden change in the input) was introduced. Then, in 1953, [[Peter Danckwerts]] resurrected the axial dispersion model and formulated the modern concept of residence time.<ref name=Nauman>{{cite journal|last1=Nauman|first1=E. Bruce|title=Residence Time Theory|journal=Industrial & Engineering Chemistry Research|date=May 2008|volume=47|issue=10|pages=3752–3766|doi=10.1021/ie071635a}}</ref>

== Distributions ==
[[File:Control volume.svg|thumb|[[Control volume]] with incoming flow rate ''f''<sub>in</sub>, outgoing flow rate ''f''<sub>out</sub> and amount stored ''m'']]

The time that a particle of fluid has been in a [[control volume]] (e.g. a reservoir) is known as its ''age''. In general, each particle has a different age. The frequency of occurrence of the age <math>\tau</math> in the set of all the particles that are located inside the control volume at time <math>t</math> is quantified by means of the (internal) '''age distribution''' <math>I</math>.<ref name=Bolin/>

At the moment a particle leaves the control volume, its age is the total time that the particle has spent inside the control volume, which is known as its ''residence time''. The frequency of occurrence of the age <math>\tau</math> in the set of all the particles that are leaving the control volume at time <math>t</math> is quantified by means of the '''residence time distribution''' or '''exit age distribution''' <math>E</math>.<ref name=Bolin/>

Both distributions are assumed to be positive and to have unitary integral along the age:<ref name=Bolin/>
:<math>\int_0^\infty E(\tau,t)d\tau = \int_0^\infty I(\tau,t)d\tau = 1</math>

In the case of [[Fluid_dynamics#Steady_vs_unsteady_flow|steady flow]], the distributions are assumed to be independent of time, that is <math>\partial_tE=\partial_tI=0 \; \forall t</math>, which may allow to redefine the distributions as simple functions of the age only.

If the flow is steady (but a generalization to non-steady flow is possible<ref name=Schwartz/>) and is [[Continuity_equation#Fluid_dynamics|conservative]], then the exit age distribution and the internal age distribution can be related one to the other:<ref name=Bolin/>
:<math>\left.\begin{aligned}
 \frac{\partial I}{\partial t}=\frac{dm}{dt}=0 & \\
 f_\text{in}=f_\text{out}=f &
\end{aligned}\ \right\} \implies fE=-m\frac{\partial I}{\partial \tau}</math>

Distributions other than <math>E</math> and <math>I</math> can be usually traced back to them. For example, the fraction of particles leaving the control volume at time <math>t</math> with an age greater or equal than <math>\tau</math> is quantified by means of the '''washout function''' <math>W</math>, that is the complementary to one of the cumulative exit age distribution:
:<math>W(\tau,t)=1-\int_0^\tau E(s,t)ds</math>

==Averages==
The '''mean age''' of all the particles inside the control volume at time ''t'' is the first [[moment (mathematics)|moment]] of the age distribution:<ref name=Bolin/><ref name=Schwartz/>
:<math> \tau_a(t) = \int_0^\infty \tau I(\tau,t) d\tau</math>

The '''mean transit time''' or '''mean residence time''' of the particles leaving the control volume at time ''t'' is the first moment of the residence time distribution:<ref name=Bolin/><ref name=Schwartz/>
:<math> \tau_t(t) = \int_0^\infty \tau E(\tau,t) d\tau.</math>

If the flow is [[Fluid_dynamics#Steady_vs_unsteady_flow|steady]] and [[Continuity_equation#Fluid_dynamics|conservative]], the mean transit time equals the ratio between the amount of fluid contained in the control volume and the flow rate through it:<ref name="Bolin"/>
:<math>\left.\begin{aligned}
 \frac{\partial I}{\partial t}=\frac{dm}{dt}=0 & \\
 f_\text{in}=f_\text{out}=f &
\end{aligned}\ \right\} \implies \tau_t = \frac{m}{f}.</math>
This ratio is commonly known as the '''turnover time''' or '''flushing time'''.<ref name="Monsen"/> When applied to liquids, it is also known as the '''hydraulic retention time''' (''HRT''), ''hydraulic residence time'' or ''hydraulic detention time''.<ref name=Davis>{{cite book|last1=Davis|first1=Mackenzie L.|last2=Masten|first2=Susan J.|title=Principles of environmental engineering and science|date=2004|publisher=McGraw-Hill Higher Education|location=Boston, Mass.|isbn=9780072921861|pages=150, 267, 480, 500}}</ref> In the field of chemical engineering this is also known as the '''Space time''' (or '''space-time''').<ref>Elements of Chemical Reaction Engineering (4th Edition) by H. Scott Fogler, Prentice Hall PTR, 2005. {{ISBN|0-13-047394-4}}</ref> The space time is frequently used in calculation involving chemical reactors and is often a more useful quantity to work with than the average residence time. The average residence time and the space are equal if all of the following conditions are met:<ref name="CEKRD">Chemical Engineering Kinetics and Reactor Design by Charles G. Hill, Jr. John Wiley & Sons Inc, 1977. {{ISBN|978-0471396093}}</ref>
# Temperature and pressure are constant
# The reaction proceeds with constant density
# The volumetric flow rate is evaluated at reactor inlet conditions.
The reciprocal of space time is ''space velocity''.

The mean age and the mean transit time generally have different values:<ref name=Bolin/>
* <math>\tau_a < \tau_t</math>: it takes some time for particles to begin leaving the system. Examples include water in a lake with inlet and outlet on opposite sides; and a [[Nuclear weapons testing|nuclear bomb test]] where [[Nuclear fallout|radioactive material]] is introduced high in the [[stratosphere]] and filters down to the [[troposphere]].
* <math>\tau_a = \tau_t</math>: the frequency functions ''E'' and ''I'' are [[Exponential decay|exponential]], as in the CSTR model above. Such a distribution occurs whenever all particles have a fixed probability per unit time of leaving the system. Examples include [[radioactive decay]] and [[Rate equation#first-order reaction|first order chemical reactions]] (where the reaction rate is proportional to the amount of [[reactant]]).
* <math>\tau_a > \tau_t</math>: most of the particles pass through quickly, but some are held up. This can happen when the main source and sink are very close together or the same. For example, most [[water vapor]] rising from the ocean surface soon returns to the ocean, but water vapor that gets sufficiently far away will probably return much later in the form of rain.<ref name=Bolin/>

==Simple flow models==
Design equations are equations relating the space time to the fractional conversion and other properties of the reactor. Different design equations have been derived for different types of the reactor and depending on the reactor the equation more or less resemble that describing the average residence time. Often design equations are used to minimize the reactor volume or volumetric flow rate required to operate a reactor.<ref name="CEKRD" />

===Plug flow reactor===
In an ideal [[Plug flow reactor model|plug flow reactor]] (PFR) the fluid particles leave in the same order they arrived, not mixing with those in front and behind. Therefore, the particles entering at time ''t'' will exit at time ''t'' + ''T'', all spending a time ''T'' inside the reactor. The residence time distribution will be then a [[Dirac delta function]] delayed of ''T'':
:<math>E(\tau) = \delta(\tau-T)\,</math>
The mean is ''T'' and the variance is zero.<ref name=Nauman/>

The RTD of a real reactor deviates from that of an ideal reactor, depending on the hydrodynamics within the vessel. A non-zero variance indicates that there is some dispersion along the path of the fluid, which may be attributed to turbulence, a non-uniform velocity profile, or diffusion. If the mean of the distribution is earlier than the expected time ''T'' it indicates that there is [[Stagnation point|stagnant fluid]] within the vessel. If the RTD curve shows more than one main peak it may indicate channeling, parallel paths to the exit, or strong internal circulation.

In PFRs, reactants enter the reactor at one end and react as they move down the reactor. Consequently, the reaction rate is dependent on the concentrations which vary along the reactor requiring the inverse of the reaction rate to be integrated over the fractional conversion.

:<math> \tau = C_{AO} \int \frac{1}{(-r_{A})}\,df_{A}</math>

===Batch reactor===
Batch reactors are reactors in which the reactants are put in the reactor at time 0 and react until the reaction is stopped. Consequently, the space time is the same as the average residence time in a batch reactor.

:<math> \tau = N_{AO} \int \frac{1}{(-r_{A})V_{R}}\,df_{A} </math>

===Continuous stirred-tank reactor===
In an ideal [[continuous stirred-tank reactor]] (CSTR), the flow at the inlet is completely and instantly mixed into the bulk of the reactor. The reactor and the outlet fluid have identical, homogeneous compositions at all times. The residence time distribution is exponential:
:<math>E(\tau) = \frac{1}{T} \exp\left(\frac{-\tau}{T}\right).</math>
The mean is ''T'' and the variance is 1.<ref name=Nauman/> A notable difference from the plug flow reactor is that material introduced into the system will never completely leave it.<ref name=Monsen/>

In reality, it is impossible to obtain such rapid mixing, especially on industrial scales where reactor vessels may range between 1 and thousands of cubic meters, and hence the RTD of a real reactor will deviate from the ideal exponential decay. For example, there will be some finite delay before ''E'' reaches its maximum value and the length of the delay will reflect the rate of mass transfer within the reactor. Just as was noted for a plug-flow reactor, an early mean will indicate some stagnant fluid within the vessel, while the presence of multiple peaks could indicate channeling, parallel paths to the exit, or strong internal circulation. Short-circuiting fluid within the reactor would appear in an RTD curve as a small pulse of concentrated tracer that reaches the outlet shortly after injection.
R
eactants continuously enter and leave a tank where they are mixed. Consequently, the reaction proceeds at a rate dependent on the outlet concentration:
:<math> \tau = \frac{C_{A in}- C_{A out}}{(-r_{A})}\ </math>

===Laminar flow reactor===
In a [[laminar flow reactor]], the fluid flows through a long tube or parallel plate reactor and the flow is in layers parallel to the walls of the tube. The velocity of the flow is a parabolic function of radius. In the absence of [[molecular diffusion]], the RTD is<ref name="Colli and Bisang, 2015" >{{Cite journal  |title=Study of the influence of boundary conditions, non ideal stimulus and dynamics of sensors on the evaluation of residence time distributions  |first1=A. N.  |last1=Colli  |first2=J. M.  |last2=Bisang  |journal=Electrochimica Acta  |volume=176  |pages=463-471  |date=September 2015 |doi=10.1016/j.electacta.2015.07.019
}}</ref> 
:<math>E(\tau)=\begin{cases}
 0 & \tau \leq T/2\\
 \dfrac{T^2}{2 \tau^3} & \tau > T/2.
\end{cases}</math>
The variance is infinite. In a real reactor, diffusion will eventually mix the layers so that the tail of the RTD becomes exponential and the variance finite; but laminar flow reactors can have variance greater than 1, the maximum for CTSD reactors.<ref name=Nauman/>

===Recycle reactors===
Recycle reactors are PFRs with a recycle loop. Consequently, they behave like a hybrid between PFRs and CSTRs.

:<math> \tau = C_{AO}(R+1) \int \frac{1}{(-r_{A})}\,df_{A} </math>

In all of these equations :<math> -r_{A} </math> is the consumption rate of A, a reactant. This is equal to the rate expression A is involved in. The rate expression is often related to the fractional conversion both through the consumption of A and through any k changes through temperature changes that are dependent on conversion.<ref name="CEKRD" />

===Variable volume reactions===
In some reactions the reactants and the products have significantly different densities. Consequently, as the reaction proceeds the volume of the reaction changes. This variable volume adds terms to the design equations. Taking this volume change into consideration the volume of the reaction becomes:

:<math> V_{R} = V_{R initial}(1-\delta_{A}f_{A}) </math>

Plugging this into the design equations results in the following equations:

====Batch====

:<math> \tau = N_{AO} \int \frac{1}{(-r_{A})V_{R}(1-\delta_{A}f_{A})}\,df_{A} </math>

====Plug flow reactors====

:<math> \tau = C_{AO} \int \frac{1}{(-r_{A})(1-\delta_{A}f_{A})}\,df_{A}</math>

====Continuous stirred-tank reactors====

:<math> \tau = \frac{C_{A in}- C_{A out}}{(-r_{AF})(1-\delta_{A}f_{A})}\ </math>

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.<ref name="CEKRD" />

==Determining the RTD experimentally==
Residence time distributions are measured by introducing a non-reactive [[Dye tracer|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.<ref>{{cite book|last1=Fogler|first1=H. Scott|title=Elements of chemical reaction engineering|date=2006|publisher=Prentice Hall|location=Upper Saddle River, NJ|isbn=978-0130473943|edition=4th}}</ref>
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 [[deconvolution|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]].<ref name="Colli and Bisang, 2011" >{{Cite journal |title=Evaluation of the hydrodynamic behaviour of turbulence promoters in parallel plate electrochemical reactors by means of the dispersion model |first1=A. N.  |last1=Colli  |first2=J. M.  |last2=Bisang |journal=Electrochimica Acta |volume=56  |issue=21 |pages=7312-7318 |date=August 2011 |doi=10.1016/j.electacta.2011.06.047
}}</ref><ref name="Colli and Bisang, 2015" /> 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, <math>M</math>, is introduced into a vessel of volume <math>V</math> and an expected residence 
time of <math>\tau</math>, the resulting curve of <math>C(t)</math> can be transformed into a dimensionless residence time distribution curve by the following relation:

: <math>E(t) = \frac{C(t)}{\int_0^\infty C(t)\ dt}</math>

===Step experiments===
The concentration of tracer in a step experiment at the reactor inlet changes abruptly from 0 to <math>C_0</math>. The concentration of tracer at the outlet is measured and normalized to the concentration <math>C_0</math> to obtain the non-dimensional curve <math>F(t)</math> which goes from 0 to 1:
: <math>F(t) = \frac {C(t)}{C_{0}}</math>.

The step- and pulse-responses of a reactor are related by the following:
: <math>F(t) = \int_{0}^t E(t')\, dt' \qquad E(t) = \frac {dF(t)}{dt}</math>

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===
[[Image:Typical CSTR RTD curve.svg|thumb|right|An RTD curve for a reasonably well-mixed reactor]]
In [[chemical reactor]]s, the goal is to make components react with a high [[yield (chemistry)|yield]]. In a homogeneous, [[first-order reaction]], the probability that an atom or molecule will react depends only on its residence time:
:<math>P_\mathrm{R} = \exp\left(-k t\right)</math>
for a [[rate constant]] <math>k</math>. Given a RTD, the average probability is equal to the ratio of the concentration <math>a</math> of the component before and after:<ref name=Nauman/>
:<math>\overline{P_\mathrm{R}} = a_\mathrm{out}/a_\mathrm{in} = \int_0^\infty \exp\left(-k t\right)E(t)dt.</math>

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
:<math>a_\mathrm{out} = \int_0^\infty a_\mathrm{batch}(t)E(t)dt.</math>
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]].<ref name=Nauman/>

===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.<ref name=Noel>{{Cite journal|last=Noel|first=M.|date=1999|title=Some physical properties of water transport in waste material|url=https://www.imwa.info/docs/imwa_1999/IMWA1999_Noel_449.pdf|journal=Mine, Water & Environment|volume=1999 IMWA Congress|pages=|via=}}</ref> [[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 stress|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 [[Acid mine drainage|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.<ref name=Noel /> [[Aquifer#Aquifers versus aquitards|Aquitards]] are clay zones that can have such a degree of impermeability that they partially or completely retard water flow.<ref name=Davis /><ref name=Faybishenko>{{cite book|last1=Faybishenko|first1=Boris|last2=Witherspoon|first2=Paul A.|last3=Gale|first3=John|title=Dynamics of fluids and transport in fractured rock|date=2005|publisher=American Geophysical Union|location=Washington|isbn=9780875904276|pages=165–167|url-access=registration|url=https://archive.org/details/dynamicsoffluids0000unse}}</ref> 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.<ref name=Faybishenko />

===Water treatment===
{{See also|Activated sludge|Waste stabilization pond}}
[[Primary treatment]] for wastewater or drinking water includes settling in a [[Sedimentation (water treatment)|sedimentation]] chamber to remove as much of the solid matter as possible before applying additional treatments.<ref name=Davis /> The amount removed is controlled by the hydraulic residence time (HRT).<ref name=Davis /> When water flows through a volume at a slower rate, less energy is available to keep solid particles entrained in the stream and there is more time for them to settle to the bottom. Typical HRTs for sedimentation basins are around two hours,<ref name=Davis /> although some groups recommend longer times to remove [[wiktionary:micropollutant|micropollutants]] such as pharmaceuticals and hormones.<ref>{{cite journal |last1=Ejhed |first1=H. |last2=Fång |first2=J. |last3=Hansen |first3=K. |last4=Graae|first4=L.|last5=Rahmberg|first5=M.|last6=Magnér|first6=J.|last7=Dorgeloh|first7=E.|last8=Plaza|first8=G.|title=The effect of hydraulic retention time in onsite wastewater treatment and removal of pharmaceuticals, hormones and phenolic utility substances|journal=Science of the Total Environment|date=March 2018|volume=618|pages=250–261|doi=10.1016/j.scitotenv.2017.11.011}}</ref>

[[Water purification|Disinfection]] is the last step in the [[Sewage treatment#Tertiary treatment|tertiary treatment]] of wastewater or drinking water. The types of pathogens that occur in untreated water include those that are easily killed like [[Pathogenic bacteria|bacteria]] and [[Pathogen#Viral|viruses]], and those that are more robust such as [[Protozoan infection|protozoa]] and [[Microbial cyst|cysts]].<ref name=Davis /> The disinfection chamber must have a long enough HRT to kill or deactivate all of them.

===Surface science===
{{See also|Surface science}}
Atoms and molecules of gas or liquid can be trapped on a solid surface in a process called [[adsorption]]. This is an [[exothermic process]] involving a release of [[heat]], and heating the surface increases the probability that an atom will escape within a given time. At a given temperature <math>T</math>, the residence time of an adsorbed atom is given by  
:<math>\tau=\tau_0 \exp\left(\frac{E_\mathrm{a}}{R T}\right),</math>
where <math>R</math> is the [[gas constant]], <math>E_\mathrm{a}</math> is an [[activation energy]], and <math>\tau_0</math> is a prefactor that is correlated with the vibration times of the surface atoms (generally of the order of <math>10^{-12}</math> seconds).<ref name=Somorjai>{{cite book|last1=Somorjai|first1=Gabor A.|last2=Li|first2=Yimin|title=Introduction to surface chemistry and catalysis|date=2010|publisher=Wiley|location=Hoboken, N.J.|isbn=9780470508237|edition=2nd}}</ref>{{rp|27}}<ref name=Hucknall>{{cite book|last1=Hucknall|first1=D.J.|last2=Morris|first2=A.|title=Vacuum technology calculations in chemistry|date=2003|publisher=RSC|location=Cambridge|isbn=9781847552273}}</ref>{{rp|196}}

In [[vacuum technology]], the residence time of gases on the surfaces of a vacuum chamber can determine the pressure due to [[outgassing]]. If the chamber can be heated, the above equation shows that the gases can be "baked out"; but if not, then surfaces with a low residence time are needed to achieve [[ultra-high vacuum]]s.<ref name=Hucknall/>{{rp|195}}

===Environmental===
In environmental terms, the residence time definition is adapted to fit with ground water, the atmosphere, [[glacier]]s, lakes, streams, and oceans. More specifically it is the time during which water remains within an aquifer, lake, river, or other water body before continuing around the [[hydrological cycle]]. The time involved may vary from days for shallow gravel [[aquifer]]s to millions of years for deep aquifers with very low values for [[hydraulic conductivity]]. Residence times of water in rivers are a few days, while in large lakes residence time ranges up to several decades. Residence times of continental ice sheets is hundreds of thousands of years, of small glaciers a few decades.

Ground water residence time applications are useful for determining the amount of time it will take for a pollutant to reach and [[water contamination|contaminate]] a ground water drinking water source and at what concentration it will arrive. This can also work to the opposite effect to determine how long until a ground water source becomes uncontaminated via inflow, outflow, and volume. The residence time of lakes and streams is important as well to determine the concentration of pollutants in a lake and how this may affect the local population and marine life.

Hydrology, the study of water, discusses the water budget in terms of residence time. The amount of time that water spends in each different stage of life (glacier, atmosphere, ocean, lake, stream, river), is used to show the relation of all of the water on the earth and how it relates in its different forms.

===Pharmacology===
A large class of [[drug]]s are [[enzyme inhibitor]]s that bind to [[enzyme]]s in the body and inhibit their activity. In this case it is the drug-target residence time (the length of time the drug stays bound to the target) that is of interest. Drugs with long residence times are desirable because they remain effective for longer and therefore can be used in lower doses.<ref name=Li>{{cite book|editor-last1=Li|editor-first1=Jie Jack|editor-last2=Corey|editor-first2=E. J.|title=Drug discovery practices, processes, and perspectives|date=2013|publisher=John Wiley & Sons|location=Hoboken, N.J.|isbn=9781118354469}}</ref>{{rp|88}} This residence time is determined by the [[pharmacokinetics|kinetics]] of the interaction and is proportional to the [[half life]] of the [[dissociation (chemistry)|chemical dissociation]].<ref>{{cite book|editor-last1=Keserü|editor-first1=György|editor-last2=Swinney|editor-first2=David C.|editor-last3=Mannhold|editor-first3=Raimund|editor-last4=Kubinyi|editor-first4=Hugo|editor-last5=Folkers|editor-first5=Gerd|title=Thermodynamics and Kinetics of Drug Binding|isbn=9783527335824}}</ref> One way to measure the residence time is in a ''preincubation-dilution'' experiment where a target enzyme is incubated with the inhibitor, allowed to approach equilibrium, then rapidly diluted. The amount of product is measured and compared to a control in which no inhibitor is added.<ref name=Li/>{{rp|87–88}}

Residence time can also refer to the amount of time that a drug spends in the part of the body where it needs to be absorbed. The longer the residence time, the more of it can be absorbed. If the drug is delivered in an oral form and destined for the [[Gastrointestinal tract|upper intestines]], it usually moves with food and its residence time is roughly that of the food. This generally allows 3 to 8 hours for absorption.<ref name=Mitra>{{cite book|editor-last1=Mitra|editor-first1=Ashim K.|editor-last2=Kwatra|editor-first2=Deep|editor-last3=Vadlapudi|editor-first3=Aswani Dutt|title=Drug Delivery|date=2014|publisher=Jones & Bartlett Publishers|isbn=9781449674267}}</ref>{{rp|196}} If the drug is delivered through a [[mucous membrane]] in the mouth, the residence time is short because [[saliva]] washes it away. Strategies to increase this residence time include [[bioadhesive]] [[polymers]], gums, [[lozenge]]s and dry powders.<ref name=Mitra/>{{rp|274}}

===Queuing theory===
Beyond fluid dynamics and chemistry, the definition(s) of residence time can be applied to any [[flow network]], where the flows of generic "resources" is modeled (e.g.: people, cars, money, products). Most notably, the over-mentioned definition of residence time is extended to stationary random processes by averaging on time ([[fluid limit]]), obtaining the so-called [[Little's Law]], which is a prominent relation in [[queueing theory]] and [[supply chain management]]. In the context of queueing theory, the residence time is addressed as ''waiting time'', while in the context of supply chain management it is most often addressed as ''[[lead time]]''.

===Biochemical===
In [[size-exclusion chromatography]], the residence time of a molecule is related to its volume, which is roughly proportional to its molecular weight. Residence times also affect the performance of [[continuous fermentation|continuous fermentors]].<ref name=Nauman/>

[[Microbial fuel cell|Biofuel cells]] utilize the metabolic processes of anodophiles ([[Electronegativity|electronegative]] bacteria) to convert chemical energy from organic matter into electricity.<ref>{{cite journal|last1=Cheng|first1=Ka Yu|last2=Ho|first2=Goen|last3=Cord-Ruwisch|first3=Ralf|title=Anodophilic Biofilm Catalyzes Cathodic Oxygen Reduction|journal=Environmental Science & Technology|date=January 2010|volume=44|issue=1|pages=518–525|doi=10.1021/es9023833}}</ref><ref name=Chouler>{{cite journal|last1=Chouler|first1=Jon|last2=Di Lorenzo|first2=Mirella|title=Water Quality Monitoring in Developing Countries;  Can Microbial Fuel Cells be the Answer?|journal=Biosensors|date=16 July 2015|volume=5|issue=3|pages=450–470|doi=10.3390/bios5030450|url=http://opus.bath.ac.uk/45910/1/biosensors_05_00450.pdf}}</ref><ref name=Santos>{{cite journal|last1=Santos|first1=João B. Costa|last2=de Barros|first2=Vanine V. Silva|last3=Linares|first3=José J.|title=The Hydraulic Retention Time as a Key Parameter for the Performance of a Cyclically Fed Glycerol-Based Microbial Fuel Cell from Biodiesel|journal=Journal of the Electrochemical Society|date=30 November 2016|volume=164|issue=3|pages=H3001–H3006|doi=10.1149/2.0011703jes}}</ref> A biofuel cell mechanism consists of an [[anode]] and a [[cathode]] that are separated by an internal [[Proton-exchange membrane|proton exchange membrane]] (PEM) and connected in an external circuit with an external load. Anodophiles grow on the anode and consume biodegradable organic molecules to produce electrons, protons, and [[carbon dioxide]] gas, and as the electrons travel though the circuit they feed the external load.<ref name=Chouler /><ref name=Santos /> The HRT for this application is the rate at which the feed molecules are passed through the anodic chamber.<ref name=Santos /> This can be quantified by dividing the volume of the anodic chamber by the rate at which the feed solution is passed into the chamber.<ref name=Chouler /> The hydraulic residence time (HRT) affects the substrate loading rate of the microorganisms that the anodophiles consume, which affects the electrical output.<ref name=Santos /><ref>{{Cite web|url=https://pubs.usgs.gov/wri/wrir-03-4238/|title=Water Quality and the Effects of Changes in Phosphorus Loading, Red Cedar Lakes, Barron and Washburn Counties, Wisconsin|last=Robertson|first=D.M|date=2016|website=United States Geological Survey}}</ref> Longer HRTs reduce substrate loading in the anodic chamber which can lead to reduced anodophile population and performance when there is a deficiency of nutrients.<ref name=Santos /> Shorter HRTs support the development of non-[[exoelectrogen]]ous bacteria which can reduce the [[Faraday efficiency|Coulombic efficiency]] electrochemical performance of the fuel cell if the anodophiles must compete for resources or if they do not have ample time to effectively degrade nutrients.<ref name=Santos />

==See also==
{{div col}}
*[[Anaerobic digestion#Residence time]]
*[[Baseflow residence time]]
*[[Estuarine water circulation#Residence time]]
*[[Lake retention time]]
*[[Micromixing]]
*[[RTD studies of plug flow reactor]]
*[[Space time (chemical engineering)]]
*[[Water cycle#Residence times]]
{{div col end}}

== References ==
{{Reflist|colwidth=35em|refs=
<ref name=Bolin>{{cite journal|last1=Bolin|first1=Bert|last2=Rodhe|first2=Henning|title=A note on the concepts of age distribution and transit time in natural reservoirs|journal=Tellus|date=February 1973|volume=25|issue=1|pages=58–62|doi=10.1111/j.2153-3490.1973.tb01594.x}}</ref>
<ref name=Schwartz>{{cite journal| last=Schwartz |first=Stephen E. |year=1979 |title=Residence times in reservoirs under non-steady-state conditions: application to atmospheric SO2 and aerosol sulfate |journal=Tellus |volume=31 |issue=6 |pages=530–547 |doi=10.3402/tellusa.v31i6.10471 }}</ref>
<ref name=Monsen>{{cite journal|last1=Monsen|first1=Nancy E.|last2=Cloern|first2=James E.|last3=Lucas|first3=Lisa V.|last4=Monismith|first4=Stephen G.|title=A comment on the use of flushing time, residence time, and age as transport time scales|journal=Limnology and Oceanography|date=September 2002|volume=47|issue=5|pages=1545–1553|doi=10.4319/lo.2002.47.5.1545}}</ref>
}}

== Further reading ==
{{Refbegin}}
*{{Cite book|first1=M|last1=Davis|last2=Masten|first2=Susan|year=2013|title=Principles of environmental engineering and science|place=New York|publisher=McGraw Hill|isbn=9780077492199}}
*{{Cite journal|first1=Bo|last1=Leckner|first2=Frederico|last2=Ghirelli|title=Transport equation for local residence time of a fluid |journal=Chemical Engineering Science|volume=59|issue=3|year=2004|pages=513–523|doi=10.1016/j.ces.2003.10.013}}
*{{cite book|last1=Lee|first1=Peter I.D.|last2=Amidon|first2=Gordon L.|chapter=2. Time constant approach|pages=15–60|title=Pharmacokinetic analysis : a practical approach|date=1996|publisher=Technomic Pub.|location=Lancaster, Penn.|isbn=9781566764254}}</ref>
*{{cite journal| first1 = R.B. |last1=MacMullin |first2= M. |last2=Weber | title = The theory of short-circuiting in continuous-flow mixing vessels in series and kinetics of chemical reactions in such systems | journal = Transactions of American Institute of Chemical Engineers | year = 1935 | volume = 31 | issue = 2 | pages = 409&ndash;458 }}
*{{cite book|last1=Montgomery|first1=Carla W.|title=Environmental Geology|date=2013|publisher=McGraw-Hill Education|isbn=9781259254598|edition=10th}}
*{{cite book | last1 = Nauman | first1 = E. Bruce | chapter = Residence Time Distributions | title = Handbook of Industrial Mixing: Science and Practice | publisher = Wiley Interscience | year = 2004| pages = 1&ndash;17 | isbn = 0-471-26919-0}}
*{{cite book|last1=Rowland|first1=Malcolm|last2=Tozer|first2=Thomas N.|title=Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications|date=2011|publisher=Lippincott Williams and Wilkins|location=New York, NY|isbn=9780781750097|edition=4th}}
*{{cite journal|last1=Wolf|first1=David|last2=Resnick|first2=William|title=Residence Time Distribution in Real Systems|journal=Industrial & Engineering Chemistry Fundamentals|date=November 1963|volume=2|issue=4|pages=287–293|doi=10.1021/i160008a008}}
{{Refend}}

== External links ==
* [https://www.certara.com/2013/03/28/mean-residence-time-mrt-understanding-how-long-drug-molecules-stay-in-the-body/ Mean residence time (MRT): Understanding how long drug molecules stay in the body]
*[http://www.lenntech.com/wwtp/calculate-hrt.htm Calculate the Hydraulic Retention Time] (Lenntech)

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