Residence time (fluid dynamics)

Residence time (also known as removal time) is the average amount of time that a particle spends in a particular system. This measurement varies directly with the amount of substance that is present in the system.

The residence time is a representation of how long it takes for the concentration to significantly change in the sediment.

Residence time is a widely used term that is mostly seen in science, technological and medical disciplines. Every discipline that uses residence time in some way adapts the definition in order to make it more specific to the application to which it is referring. The base definition for residence time also has a universal mathematical equation that can be added to and adapted for different disciplines. This is as follows:

$\tau = \frac{\mbox{System capacity to hold a substance}}{\mbox{Flow rate of the substance through the system}}$

The generic variable form of this equation is as follows:

$\tau = \frac{V}{q}$

where $\tau$ is used as the variable for residence time, V is the capacity of the system, and q is the flow for the system.

Residence time begins from the moment that a particle of a particular substance enters the system and ends the moment that the same particle of that substance leaves the system. The system in question is arbitrary and can be defined as needed according to the application. If the size of the system is changed, the residence time of the system will be changed as well. The larger the system, the larger the residence time, assuming the inflow and outflow rates are held constant. The smaller the system, the shorter the residence time will be, again assuming steady-state conditions.

Inflow and outflow will also have an effect on the residence time of a system. If the inflow and outflow are increased, the residence time of the system will be shorter. However, if the inflow and the outflow of a system are decreased, the residence time will be longer. This is assuming that the concentration of the substance in the system and the size of the system remain constant, and assuming steady-state ^[1] conditions.

Assumptions

When using the residence time equation, a variety of assumptions are made to reduce the complexity of the system being modeled. These assumptions include, but are not limited to: steady state inflow and outflow, constant volume, constant temperature, and uniform distribution of the substance throughout the volume of the system. It is also assumed that chemical degradation (chemical decomposition) does not occur in the system in question and that particles do not adsorb onto surfaces that would hinder their flow. If chemical degradation were to occur in a system, the substance that originally entered the system may react with other existing compounds in the system, causing the residence time to be significantly shorter since the substance would be chemically consumed and effectively be removed from the system before it was able to naturally flow out of the system.

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.

$\tau_{s}=\tau_{s,0} \; e^\frac{E_{s,a}}{R T}$

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

See also

• Residence time distribution
• Lake retention time
• Water cycle: residence times

Notes

1. Steady-state refers to a system in which the given parameter is held constant over time.
2. .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.