Residence time

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For matter that flows through volume, dwell time is a measure of how much time is spent in it. Examples include fluids in chemical reactors, specific elements in geochemical reservoirs, catching water, bacteria in the culture vessels and drugs in the human body. A small molecule or packet of liquids has a single residence time, but a more complex system has residence time distribution (RTD) .

There are at least three time constants used to represent the residence time distribution. The turnaround time or flushing time is the ratio of material in volume to the level at which it passes; average age is the average length of time the material in the reservoir has been spent there; and average transit time is the average length of time the material spends in the reservoir.

Application of residence time or residence distribution can be found in various disciplines including environmental science, engineering, chemistry, and hydrology.

Video Residence time

Histori

The concept of residence time comes from the model of a chemical reactor. The first model was the axial dispersion model by Irving Langmuir in 1908. It received little attention for 45 years; other models were developed such as the model of the plug flow reactor and the continuously stirred reactor tank, and the concept of washout function (representing response to sudden changes in input) was introduced. Then, in 1953, Peter Danckwerts evoked the axial deployment model and formulated the concept of modern residence time.

Maps Residence time

Distribution

The basic dwelling theory treats systems with inputs and outputs, both of which flow only in one direction. The system is homogeneous and the substance that flows through is preserved (not created or destroyed). The small particles entering the system will eventually go away, and the time spent there is the time of their stay. In a simple flow model, the plug flow, the particles entering at the same time continue to move at the same rate and go together. In this case, there is only one stay. Generally, their rates vary and there is out time distribution. One measure of this is the function washout ${\ displaystyle W (t)}$ , the fraction of the particles leaving the system after being there for a time ${\ displaystyle t}$ or higher. More, ${\ displaystyle F (t) = 1-W (t)}$ , is the cumulative distribution function . Distribution differential , also known as residence distribution or outgoing age distribution , is given by

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

Ini memiliki sifat dari distribusi probabilitas: itu selalu non-negatif dan

${\ displaystyle \ int _ {0} ^ {\ infty} E (t) dt = 1.}  ÂÂ$

One can also define the density function based on flux (mass per unit time) out of the system. The transit time function is a particle fraction leaving the system which has been in it up to a certain time. This is an integral part of the distribution of ${\ displaystyle I (t)}$ . If, in stable state, the mass in the system is ${\ displaystyle M_ {0}}$ and flux is ${\ displaystyle F_ {0}}$ , the distribution is related to

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Sebagai ilustrasi, untuk populasi manusia berada dalam kondisi stabil, kematian per tahun orang yang lebih tua dari ${\ displaystyle t}  ÂÂ$ tahun (sisi kiri persamaan) harus diimbangi dengan jumlah orang per tahun yang mencapai usia ${\ displaystyle t}  ÂÂ$ (sisi kanan).

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Konstanta waktu

"Time of residence" can be a synonym for more than one constant used to represent the distribution.

Means residence time

Beberapa properti statistik dari distribusi waktu tinggal sering digunakan. Waktu tinggal rata-rata, atau usia rata-rata , diberikan pada saat pertama distribusi waktu tempat tinggal:

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

dan variansnya diberikan oleh

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

atau dengan bentuk tanpa dimensi ${\ displaystyle \ sigma ^ {2} = \ sigma _ {t} ^ {2}/\ tau ^ {2}}  ÂÂ$ .

Rata-rata waktu transit

Waktu transit rata-rata adalah momen pertama dari distribusi waktu transit:

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

Waktu Pembilasan

The waktu perputaran , juga dikenal sebagai flushing time , hanyalah rasio massa untuk fluks:

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

When applied to the liquid, it is also known as the hydraulic retention time HRT , hydraulic residence time or hydraulic detention time >.

Relationship between time

Dapat ditunjukkan bahwa, dalam keadaan stabil, waktu transit rata-rata dan waktu pembilasan sama ( ${\ displaystyle t_ {t} = t_ {0}}  ÂÂ$ ).

In an ideal stop flow reactor, the fluid elements go in the same order, not mixed with the front and back. Therefore, liquids come in at ${\ displaystyle t}$ will exit at ${\ displaystyle t \ tau}$ , where ${\ displaystyle \ tau}$ is the time to stay. The leave fraction is a step function, going from 0 to 1 at ${\ displaystyle \ tau}$ . The distribution function is the Dirac delta function in ${\ displaystyle \ tau}$ .

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

Mean adalah ${\ displaystyle \ tau}  ÂÂ$ dan variasinya nol.

The RTD of the real reactor deviates from the ideal reactor, depending on the hydrodynamics in the vessel. A non-zero variance indicates that there is some dispersion along the fluid path, which may be due to turbulence, a non-uniform velocity profile, or diffusion. If the mean of ${\ displaystyle E (t)}$ curve arrives earlier than estimated time ${\ displaystyle \ tau}$ This indicates that there is stagnant fluid inside the vessel. If the RTD curve shows more than one major peak may indicate channeling, parallel paths to the exit, or strong internal circulation.

Continuous stirred reactor reactor

In an ideal continuous stirred tank reactor (CSTR), the flow in the inlet entirely is fully and directly mixed into most reactors. The reactor and the outlet liquid have identical and homogeneous compositions at all times. Distribution of residence time is exponential:

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

Mean is ${\ displaystyle \ tau}$ and its variants are 1. The important difference from the plug flow reactor is that the material inserted into the system will never actually leave it.

In fact, it is not possible to obtain such rapid mixing, especially on an industrial scale where reactor vessels can range between 1 and thousands of cubic meters, and therefore the RTD of the real reactor will deviate from ideal exponential decay. For example, there will be some limited delay before ${\ displaystyle E (t)}$ reaches its maximum value and the length of delay will reflect the mass transfer rate in the reactor. As noted for plug-flow reactors, the initial average will show some of the flu stagnant in the vessel, while the presence of several peaks may indicate channeling, parallel paths to the exit, or strong internal circulation. Short-circuit fluid in the reactor will appear in the RTD curve as a small pulse of concentrated tracers reaching the outlet shortly after the injection.

Laminar flow reactor

Dalam reaktor aliran laminar, cairan mengalir melalui tabung panjang dan alirannya berlapis-lapis sejajar dengan dinding tabung. Kecepatan aliran adalah fungsi jari-jari parabola. Dengan tidak adanya difusi molekuler, RTD adalah

${\ displaystyle E (t) = 0, \ quad t & lt; = \ tau/2}  ÂÂ$

dan

${\ displaystyle E (t) = {\ frac {\ tau ^ {2}} {2t ^ {3}}} \ quad t & gt; \ tau/2.}  ÂÂ$

The variance is unlimited. In a real system, diffusion will eventually blend the layers so that the tail of the RTD becomes exponential and the variance limited; but the laminar flow reactor can have a variance greater than 1, the maximum for the CTSD reactor.

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Specify RTD experimentally

The residence time distribution is measured by introducing non-reactive trackers into the inlet system. The input concentration changes according to the known function and the output concentration is measured. Trackers should not modify the physical characteristics of the fluid (equal density, same viscosity) or hydrodynamic conditions and should be easily detected. In general, the change in the concentration of the tracker will be the pulse or step . Other functions are possible, but they require more calculations to declare RTD curves.

Trial pulses

This method requires the introduction of very small volumes of tracers concentrated in the reactor inlet, thus approaching the Dirac delta function. Although very short injections can not be produced, it can be made much less than the average shelf life of the vessel. If the mass tracker, ${\ displaystyle M}$ , inserted into volume vessel ${\ displaystyle V}$ and expected residence time of ${\ displaystyle \ tau}$ , result curve ${\ displaystyle C (t)}$ can be converted into a dimensionless residence time distribution curve with the following relationships:

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

Attempt step

Konsentrasi pelacak dalam percobaan langkah pada masukan reaktor berubah secara tiba-tiba dari 0 hingga ${\ displaystyle C_ {0}}  ÂÂ$ . Konsentrasi pelacak di outlet diukur dan dinormalkan ke konsentrasi ${\ displaystyle C_ {0}}  ÂÂ$ untuk mendapatkan kurva non-dimensi ${\ displaystyle F (t)}  ÂÂ$ yang berjalan dari 0 ke 1:

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

Reaksi langkah dan pulsa reaktor terkait dengan hal-hal berikut:

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

A step experiment is often easier to perform than pulse experiments, but this experiment tends to smooth out some of the details that the pulse response can show. It is very easy to numerically integrate an experimental pulse to obtain an approximate response of a very high-quality step, but vice versa not because every noise in the concentration measurement will be amplified by numerical differentiation.

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Apps

Chemical reactors

Dalam reaktor kimia, tujuannya adalah membuat komponen bereaksi dengan hasil tinggi. Dalam reaksi orde pertama yang homogen, probabilitas bahwa atom atau molekul akan bereaksi hanya bergantung pada waktu tinggalnya:

${\ displaystyle P _ {\ mathrm {R}} = \ exp \ kiri (-kt \ right)}  ÂÂ$

untuk konstanta laju ${\ displaystyle k}  ÂÂ$ . Dengan RTD, probabilitas rata-rata sama dengan rasio konsentrasi ${\ displaystyle a}  ÂÂ$ dari komponen sebelum dan sesudah:

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

If the reaction is more complicated, the output is uniquely determined by the RTD. It also depends on the level of micromixing , mixing between incoming molecules at different times. If there is no mixing, the system is said to be completely separate, and the results can be given in the form

${\ displaystyle a _ {\ mathrm {out}} = \ int_ {0} ^ {\ infty} a _ {\ mathrm {batch}} (t ) E (t) dt.} Â Â$

For RTD grants, there is an upper limit on the amount of mixing that can occur, called maximum mixture , and this determines the results that can be achieved. A continuously stirred continuous tank reactor can be anywhere in the spectrum between completely separate and perfect mixing.

Groundwater flow

Hydraulic residence time (HRT) is an important factor in the transport of environmental toxins or other chemicals through ground water. The amount of time spent by the pollutants through the subsurface described is related to the saturation and soil hydraulic conductivity of the soil or rock. Porosity is another factor that contributes significantly to the mobility of water through the soil (eg toward the water table). The junction between density and pore size determines the rate or magnitude of the flow rate through the medium. This idea can be illustrated by comparison of ways of moving water through clay versus gravel. The retention time through a certain vertical distance in the clay will be longer than through the same distance in the gravel, although both are characterized as high porosity materials. This is because the pore size is much larger in gravel media than in clay, so there is little hydrostatic tension that works against underwater surface gradients and gravity.

Ground water flow is an important parameter to consider in the design of waste rock basins for mining operations. Residual rocks are heterogeneous materials with particles that vary from rock to clay-sized particles, and contain sulphate pollutants that must be controlled in such a way that they do not endanger the quality of the water surface and also runoff does not create surrounding environmental problems. area. Aquitard is a clay zone that can have an impermeable level that partially or completely inhibits the flow of water. These clay lenses can slow or stop seepage into the water table, although if the cracking is contaminated then it can be a source of long-term groundwater contamination due to its low permeability and high HRT.

Water treatment

Primary treatments for wastewater or drinking water include settling in the sedimentation chamber to remove as much of the solid as possible before applying additional treatments. The amount removed is controlled by hydraulic residence time (HRT). As water flows through the volume at a slower rate, less energy is available to keep the solid particles in the flow and there is more time for them to settle to the bottom. Typical HRT for sedimentary basins is about

Source of the article : Wikipedia