Residence time (fluid dynamics)

Residence time, also known as removal time, is the average amount of time spent in a control volume by the particles of a fluid. Since there's more than one way of averaging the time spent by particles inside the volume, there're also more than one definition of residence time.^[1] In case the flow is stationary and respects continuity, the definition most usually adopted is:

{\text{continuity and stationarity}}\implies \tau :={\frac {m}{f}}

where

$\tau$ is the residence time,
$m$ is the quantity of material contained in the system (the mass, in case of fluid dynamics),
$f$ is the flow (the mass flow rate, in case of fluid dynamics).

Residence time plays an important role in environmental engineering and chemistry. In these fields, not only the (mean) residence time is of interest, but also the whole residence time distribution. Nevertheless, the simple definition just introduced can be employed to quantify the residence times of specific compounds in a mixture only under the hypothesis that no chemical reaction takes place^[1] (otherwise continuity wouldn't be satisfied) and that the compounds concentrations are uniform.

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.

Definition

Fluid dynamic phenomena can be modeled with different degrees of detail. The least detailed and most ubiquitously employed model is that for which we look to just three (functional) variables: the incoming flow $f_{\text{in}}$ , the outcoming flow $f_{\text{out}}$ and the quantity of fluid $m$ stored in the system. Unfortunately, in order to quantify the residence time in the non stationary case, we need a higher degree of detail: we need to know the so-called persistence function:^[1]

A^{*}:\mathbb {R} \times \mathbb {R} _{0}^{+}\rightarrow \mathbb {R} _{0}^{+},\quad \lim _{u\to +\infty }uA^{*}(t,u)=0\ \forall t

The persistence function is axiomatically defined so that its value in (t,u) is the portion of flow $f_{\text{in}}(t-u)$ which is present in the system at time t. From this definition it follows that:

f_{\text{in}}(t)=A^{*}(t,0)

m(t)=\int _{0}^{+\infty }A^{*}(t,u)\operatorname {d} \!u

Without any restrictive assumption, the persistence function allows to define the mean age,^[1] which value in t is the average amount of time that the particles present in the system at time t have spent into the system:

\tau _{a}(t):={\frac {1}{m(t)}}\int _{0}^{+\infty }uA^{*}(t,u)\operatorname {d} \!u

By assuming that the quantity of fluid is conserved (namely, $m,f_{\text{in}},f_{\text{out}}$ satisfy the continuity equation ${\dot {m}}(t)=f_{\text{in}}(t)-f_{\text{out}}(t)$ ) we can express the outflow as a function of $A^{*}$ :

{\text{continuity}}\implies f_{\text{out}}(t)=-\int _{0}^{+\infty }\partial _{u}A^{*}(t,u)+\partial _{t}A^{*}(t,u)\operatorname {d} \!u

Proof

{\begin{aligned}f_{\text{out}}(t)&=f_{\text{in}}(t)-{\dot {m}}(t)=\\&=A^{*}(t,0)-\partial _{t}\int _{0}^{+\infty }A^{*}(t,u)\operatorname {d} \!u=\\&=-[A^{*}(t,u)]_{0}^{+\infty }-\int _{0}^{+\infty }\partial _{t}A^{*}(t,u)\operatorname {d} \!u=\\&=-\int _{0}^{+\infty }\partial _{u}A^{*}(t,u)+\partial _{t}A^{*}(t,u)\operatorname {d} \!u\end{aligned}}

The transit time^[1] is the average amount of time that the particles leaving the system at time t have spent into the system. Under the hypothesis of continuity, the transit time can be expressed in terms of the persistence function:

{\text{continuity}}\implies \tau _{t}(t):=-{\frac {1}{f_{\text{out}}(t)}}\int _{0}^{+\infty }u(\partial _{u}A^{*}(t,u)+\partial _{t}A^{*}(t,u))\operatorname {d} \!u

If the continuity is satisfied and the persistence function is stationary (namely, $\partial _{t}A^{*}(t,u)=0\ \forall t,u$ ), then $m={\text{const}}$ and $f:=f_{\text{out}}=f_{\text{in}}={\text{const}}$ . It can be shown that under these assumptions the definition of transit time converges to the common definition of residence time given in the incipit of this article:

{\text{continuity and stationarity}}\implies \tau _{t}={\frac {m}{f}}={\text{const}}

Proof

The transit time definition can be manipulated integrating by parts:

{\begin{aligned}\tau _{t}(t)&=-{\frac {1}{f_{\text{out}}(t)}}\left(\int _{0}^{+\infty }u\partial _{t}A^{*}(t,u)\operatorname {d} \!u+[uA^{*}(t,u)]_{0}^{+\infty }-\int _{0}^{+\infty }A^{*}(t,u)\operatorname {d} \!u\right)=\\&=-{\frac {1}{f_{\text{out}}(t)}}\left(\int _{0}^{+\infty }u\partial _{t}A^{*}(t,u)\operatorname {d} \!u-m(t)\right)\end{aligned}}

Where at the last step the integrand is null if the persistence function is stationary.

Contrarily to transit time, the mean age doesn't generally converge to $m/f$ .

Applications

Depending on the complexity of the system being modeled and the application for which it is being used, the residence time equation can be altered significantly or even used as a factor.

Engineering

Residence time is widely used across all engineering disciplines, including chemical engineering, biological systems engineering, biomedical engineering, civil engineering, environmental engineering and geological engineering. The residence time formula is adapted for each of these disciplines depending on the system, the complexity, and the substance involved.

In environmental engineering, residence time applies to water treatment and wastewater treatment. It refers to the amount of time that water spends in a batch reactor, plug flow reactor, completely mixed flow reactor (CMFR), and/or flocculation tanks. Batch reactors, plug flow reactors, and CMFR’s are used in wastewater treatment plants as a means of treating wastewater. Flocculation tanks are part of drinking water treatment facilities where the chemically treated water needs enough time to form flocs. Flocs are colloidal particles that have combined with a coagulant in order to form large enough particles that will eventually settle out in the next phase of water treatment. before reaching the sedimentation basin. These processes are dependent on an adapted version of residence time. In this situation, the important parameter is how long a concentration of fluid needs to remain in the system to be adequately treated.

C=C_{{o}}e^{{-k\tau }}

where:

C is the concentration
C₀ is the initial Concentration
k is the reaction rate constant
$\tau$ = batch reactor residence time

Here the residence time is being used to determine the changing concentration of a contaminant in a system. This residence time is based on the inflow, outflow, volume, initial concentration of contaminant, the added chemical for treatment, and the rate at which the reactions take place. This is particularly useful for a flash mixer in a water treatment facility to determine if too little or too much of a chemical is initially being introduced into the system.

Aerospace

In aerospace engineering, residence time ( $\tau _{{s}}$ ) refers to the quantity of time required to conduct outgassing of accumulated gasses in vacuum environment. The amount of residence time required to achieve outgassing is directly dependent on the temperature of the environment. The higher the temperature, the less residence time in the vacuum environment is required to outgas the same quantity of material. Many vacuum chambers are wrapped with heaters to increase the temperature and thus "bake out" the outgassing molecules.

\tau _{{s}}=\tau _{{s,0}}\;e^{{\frac {E_{{s,a}}}{RT}}}

where:

$\tau _{{s}}$ is the residence Time [s]
$\tau _{{s,0}}$ is the residence Time at 273.15 kelvin [s]
$E_{{s,a}}$ is the activation energy for desorption of contaminant [J/kmol]
$R$ is the universal gas constant [J/(kmol * T)]
$T$ is the temperature [T]

The equation for $\tau _{{s}}$ is referenced incorrectly in MANY books. Pisacane and others have a negative sign before $E_{{s,a}}$ . This is incorrect as it would cause additional heating to increase the time required for outgassing. The reference residence time, $\tau _{{s,0}}$ , is typically assumed to be 1.7 x 10^-13 (seconds), with experimental values typically between 10^-12 and 10^-14 (seconds). The activation energy, $E_{{s,a}}$ , is material dependent and ranges for 400 to 100,000 (Joules/Kmole).^[2]

Environmental

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.

Pharmaceutical

For the medical field, residence time often refers to the amount of time that a drug spends in the body. This is dependent on an individual’s body size, the rate at which the drug will move through and react within the person’s body, and the amount of the drug administered. The mean residence time (MRT) of a drug deviates from the previous equations as it is based on a statistical derivation. This still runs off a steady-state volume assumption but then uses the area under a distribution curve to find the average drug dose clearance time. The distribution is determined by numerical data derived from either urinary or plasma data collected. Each drug will have a different residence time based on its chemical composition and technique of administration. Some of these drug molecules will remain in the system for a very short time while others may remain for a lifetime. Since individual molecules are hard to trace, groups of molecules are tracked and the distribution of these is plotted to find a mean residence time. The distribution is given by the following equation:

MRT={\frac {1}{N}}\sum _{{i=1}}^{m}t_{i}n_{i}

where:

$m$ is the total number of groups
$t_{i}$ is the average time in the body of the i^th group
$n_{i}$ is the number of molecules in the i^th group
$N$ is the total number of molecules introduced into the system

Notes

1 2 3 4 5 Schwartz, Stephen E. (1979). "Residence times in reservoirs under non-steady-state conditions: application to atmospheric SO2 and aerosol sulfate". Tellus. 31 (6): 530–547. doi:10.1111/j.2153-3490.1979.tb00935.x.
↑ .Pisacane, Vincent L. (2008). The space environment and its effects on space systems. Reston, VA: American Institute of aeronautics and Astronautics. ISBN 978-1-56347-926-7.

References

Davis, M; Masten, S (2004). Principles of environmental engineering and science. New York: McGraw Hill.
Leckner, Bo; Ghirelli, Frederico (2004). "Transport equation for local residence time of a fluid.". Science Direct. 59 (3): 513–523. doi:10.1016/j.ces.2003.10.013.
Montgomery, C, & Reichard, J. (2007). Environmental geology. United States: McGraw Hill.
Rowland, M, & Tozer, T. (1995). Clinical pharmacokinetics. Philadelphia, PA: Lippincott Williams & Wilkins
Wolf, David; Resnik, William (1963). "Residence time distribution in real systems". Industrial & Engineering Chemistry :Fundamentals. 2 (4): 1.

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Residence time (fluid dynamics)

Definition

Applications

Engineering

Aerospace

Environmental

Pharmaceutical

See also

Notes

References