# Residence time

(Redirected from Residence time distribution)

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.

## HistoryEdit

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.[1]

## DistributionsEdit

Basic residence time theory treats a system with an input and an output, both of which have flow only in one direction. The system is homogeneous and the substance that is flowing through is conserved (neither created nor destroyed). A small particle entering the system will eventually leave, and the time spent there is its residence time. In a particularly simple model of flow, plug flow, particles that enter at the same time continue to move at the same rate and leave together. In this case, there is only one residence time. Generally, though, their rates vary and there is a distribution of exit times. One measure of this is the washout function ${\displaystyle W(t)}$ , the fraction of particles leaving the system after having been there for a time ${\displaystyle t}$  or greater. Its complement, ${\displaystyle F(t)=1-W(t)}$ , is the cumulative distribution function. The differential distribution, also known as the residence time distribution or exit age distribution,[2]:260–261 is given by

${\displaystyle E(t)=dF(t)/dt.}$

This has the properties of a probability distribution: it is always nonnegative and

${\displaystyle \int _{0}^{\infty }E(t)dt=1.}$ [1][2]:260–261

One can also define a density function based on the flux (mass per unit time) out of the system. The transit time function is the fraction of particles leaving the system that have been in it for up to a given time. It is the integral of a distribution ${\displaystyle I(t)}$ . If, in a steady state, the mass in the system is ${\displaystyle M_{0}}$  and the outgoing flux is ${\displaystyle F_{0}}$ , the distributions are related by

${\displaystyle F_{0}I(t)=-M_{0}{\frac {dE(t)}{dt}}.}$

As an illustration, for a human population to be in a steady state, the deaths per year of people older than ${\displaystyle t}$  years (the left hand side of the equation) must be balanced by the number of people per year reaching age ${\displaystyle t}$  (the right hand side).[3]

## Time constantsEdit

"Residence time" can be a synonym for more than one constant used to represent the distribution.[4]

### Mean residence timeEdit

Some statistical properties of the residence time distribution are frequently used. The mean residence time, or mean age, is given by the first moment of the residence time distribution:

${\displaystyle \tau =\int _{0}^{\infty }tE(t)dt}$ ,

and the variance is given by

${\displaystyle \sigma _{t}^{2}=\int _{0}^{\infty }\left(t-\tau \right)^{2}E(t)dt}$

or by the dimensionless form ${\displaystyle \sigma ^{2}=\sigma _{t}^{2}/\tau ^{2}}$ .[1]

### Mean transit timeEdit

The mean transit time is the first moment of the transit time distribution:

${\displaystyle t_{\mathrm {t} }=\int _{0}^{\infty }tI(t)dt.}$

### Flushing timeEdit

The turnover time, also known as the flushing time,[5] is simply the ratio of mass to flux:

${\displaystyle t_{0}=M_{0}/F_{0}.}$

When applied to liquids, it is also known as the hydraulic retention time (HRT), hydraulic residence time or hydraulic detention time.[6]

### Relationship between timesEdit

It can be shown that, in a steady state, the mean transit time and flushing time are equal (${\displaystyle t_{t}=t_{0}}$ ).[3]

The relationship between ${\displaystyle t_{0}}$  or ${\displaystyle t_{\mathrm {t} }}$  and ${\displaystyle \tau }$  is determined by the type of distribution:[3]

• ${\displaystyle \tau  : 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 bomb test where radioactive material is introduced high in the stratosphere and filters down to the troposphere.
• ${\displaystyle \tau =t_{0}}$ : the frequency functions ${\displaystyle E(t)}$  and ${\displaystyle I(t)}$  are 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 first order chemical reactions (where the reaction rate is proportional to the amount of reactant).
• ${\displaystyle \tau >t_{0}}$ : 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.[3]

## Simple flow modelsEdit

### Plug flow reactorEdit

In an ideal plug flow reactor the fluid elements leave in the same order they arrived, not mixing with those in front and behind. Therefore, fluid entering at time ${\displaystyle t}$  will exit at time ${\displaystyle t+\tau }$ , where ${\displaystyle \tau }$  is the residence time. The fraction leaving is a step function, going from 0 to 1 at time ${\displaystyle \tau }$ . The distribution function is therefore a Dirac delta function at ${\displaystyle \tau }$ .

${\displaystyle E(t)=\delta (t-\tau )\,}$

The mean is ${\displaystyle \tau }$  and the variance is zero.[1]

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 ${\displaystyle E(t)}$  curve arrives earlier than the expected time ${\displaystyle \tau }$  it indicates that there is 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.

### Continuous stirred-tank reactorEdit

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:

${\displaystyle E(t)={\frac {1}{\tau }}\exp \left({\frac {-t}{\tau }}\right).}$

The mean is ${\displaystyle \tau }$  and the variance is 1.[1] A notable difference from the plug flow reactor is that material introduced into the system will never completely leave it.[5]

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 ${\displaystyle E(t)}$  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.

### Laminar flow reactorEdit

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[7]

${\displaystyle E(t)=0,\quad t<=\tau /2}$

and

${\displaystyle E(t)={\frac {\tau ^{2}}{2t^{3}}}\quad t>\tau /2.}$

The variance is infinite. In a real system, 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.[1]

## Determining the RTD experimentallyEdit

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.[8] 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 experimentsEdit

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.[9] 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, ${\displaystyle M}$ , is introduced into a vessel of volume ${\displaystyle V}$  and an expected residence time of ${\displaystyle \tau }$ , the resulting curve of ${\displaystyle C(t)}$  can be transformed into a dimensionless residence time distribution curve by the following relation:

${\displaystyle E(t)={\frac {C(t)}{\int _{0}^{\infty }C(t)\ dt}}}$

### Step experimentsEdit

The concentration of tracer in a step experiment at the reactor inlet changes abruptly from 0 to ${\displaystyle C_{0}}$ . The concentration of tracer at the outlet is measured and normalized to the concentration ${\displaystyle C_{0}}$  to obtain the non-dimensional curve ${\displaystyle F(t)}$  which goes from 0 to 1:

${\displaystyle F(t)={\frac {C(t)}{C_{0}}}}$ .

The step- and pulse-responses of a reactor are related by the following:

${\displaystyle F(t)=\int _{0}^{t}E(t')\,dt'\qquad E(t)={\frac {dF(t)}{dt}}}$

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.

## ApplicationsEdit

### Chemical reactorsEdit

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:

${\displaystyle P_{\mathrm {R} }=\exp \left(-kt\right)}$

for a rate constant ${\displaystyle k}$ . Given a RTD, the average probability is equal to the ratio of the concentration ${\displaystyle a}$  of the component before and after:[1]

${\displaystyle {\overline {P_{\mathrm {R} }}}=a_{\mathrm {out} }/a_{\mathrm {in} }=\int _{0}^{\infty }\exp \left(-kt\right)E(t)dt.}$

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

${\displaystyle a_{\mathrm {out} }=\int _{0}^{\infty }a_{\mathrm {batch} }(t)E(t)dt.}$

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.[1]

### Groundwater flowEdit

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.[10] 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.[10] Aquitards are clay zones that can have such a degree of impermeability that they partially or completely retard water flow.[6][11] 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.[11]

### Water treatmentEdit

Primary treatment for wastewater or drinking water includes settling in a sedimentation chamber to remove as much of the solid matter as possible before applying additional treatments.[6] The amount removed is controlled by the hydraulic residence time (HRT).[6] 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,[6] although some groups recommend longer times to remove micropollutants such as pharmaceuticals and hormones.[12]

Disinfection is the last step in the tertiary treatment of wastewater or drinking water. The types of pathogens that occur in untreated water include those that are easily killed like bacteria and viruses, and those that are more robust such as protozoa and cysts.[6] The disinfection chamber must have a long enough HRT to kill or deactivate all of them.

### Surface scienceEdit

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 ${\displaystyle T}$ , the residence time of an adsorbed atom is given by

${\displaystyle \tau =\tau _{0}\exp \left({\frac {E_{\mathrm {a} }}{RT}}\right),}$

where ${\displaystyle R}$  is the gas constant, ${\displaystyle E_{\mathrm {a} }}$  is an activation energy, and ${\displaystyle \tau _{0}}$  is a prefactor that is correlated with the vibration times of the surface atoms (generally of the order of ${\displaystyle 10^{-12}}$  seconds).[13]:27[14]: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 vacuums.[14]:195

### EnvironmentalEdit

In environmental terms, the residence time definition is adapted to fit with ground water, the atmosphere, glaciers, 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 aquifers 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 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.

### PharmacologyEdit

A large class of drugs are enzyme inhibitors that bind to enzymes 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.[15]:88 This residence time is determined by the kinetics of the interaction and is proportional to the half life of the chemical dissociation.[16] 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.[15]: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 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.[17]: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, lozenges and dry powders.[17]:274

### Queuing theoryEdit

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.

### BiochemicalEdit

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 fermentors.[1]

Biofuel cells utilize the metabolic processes of anodophiles (electronegative bacteria) to convert chemical energy from organic matter into electricity.[18][19][20] A biofuel cell mechanism consists of an anode and a cathode that are separated by an internal 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. [19][20] The HRT for this application is the rate at which the feed molecules are passed through the anodic chamber.[20] 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.[19] The hydraulic residence time (HRT) affects the substrate loading rate of the microorganisms that the anodophiles consume, which affects the electrical output.[20][21] 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.[20] Shorter HRTs support the development of non-exoelectrogenous bacteria which can reduce the 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.[20]

## ReferencesEdit

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