A Study of Travel Time for Different Open Channels

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  • Published: 21 March 2020
  • Volume 101 , pages 399–407, ( 2020 )

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An increase in developmental activities such as afforestation, paved surfaces and construction of buildings and other structures leads to an increase in surface runoff and peak discharge from the watershed. These all cause a decrease in detention storage and surface depression and thus diminish the concentration time that flow will take and distributes flow to the adjoining stream quickly rather than that would have taken before development or urbanization. Despite the importance of time of concentration, planners and engineers are often puzzled by different profiles of the channels and their equation available in the literature without knowing the accuracy of each formula. In this paper, kinematic wave theory integrated with the Manning’s equation has been applied for the comparative assessment of the performance of the various cross-sectional channels. The result of different channel profiles toward travel time of flow has been matched for nine channel profiles. Of the nine channel profiles, it was found that the deep rectangular cross-sectional channel possesses the highest time of travel. Therefore, the use of deep rectangular channel yields lesser watershed runoff. The parabolic channel with more depth yields the lesser time of travel; therefore, the use of parabolic channel profile yields larger watershed runoff.

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Introduction

Water is one of the most vital elements for human survival as well as it is needed for agricultural, municipal, industrial and power sectors. To fulfill the demand for water from various end users, it needs to be conveyed from one location to another location. Leakage and losses through the evaporation are the utmost serious forms of water loss in an irrigation canal network. This necessity has impelled the creativity of human to dig a channel, for transporting water as according to their place of wishes and requirements. A channel includes irrigation channels, roadside channels and drainage ditches that are either man-made with regular geometric cross sections or unlined or lined with artificial or natural material to protect against erosion. These channels and small section of parabolic cannels can be built by earthmovers and other types of earth-moving equipment [ 1 ]. The most generally utilized channel areas are rectangular, circular, triangular and trapezoidal. In a hilly watershed which involves a system of overland planes and open channels, the travel time of flow in channels upsets the surface flow attributes of the watershed. Travel time is characterized as the “normal time required for water to go from the highest point of the hillslope by means of the subsurface hillslope to the watershed outlet”. In channels having extended travel time and detention storage, the watershed disturbed would have an extended time of concentration. These types of the watershed have small rainfall intensity duration frequency curves; therefore, the runoff generated from watershed will be lesser. Conversely, in channels having less travel time and lesser detention storage, the runoff generated from watershed will be enormous. Thus, the profile of the channels must have a consequence on the travel time and the detention storage of basin as suggested by Wong [ 2 ]. Henceforth, the channel profile can be utilized as a way to deal with the runoff from a watershed. This is, be that as it may, not honed which might be because of no distributed outcome on the impact of channel profile on the season of travel. This absence of result might be because of the troubles in playing out an exploratory or a numerical review. Then again, with an insignificant cost and in a moderately brief time, the impact of channel profile on the season of travel can be contemplated hypothetically.

Time of concentration (Tc) of the stream is an essential parameter in numerous hydrologic configuration forms. Time of concentration study started with Henderson and Wooding [ 3 ] equation for a solitary plane. However, this theory could not be effortlessly connected in practice because of the fact that it included parameters which were not identified with the physical qualities of an overland. Tc is an essential parameter in seepage framework plan of a watershed. In the method for the kinematic wave hypothesis (KWH), it is conceivable to get a scientific travel time equation for certain exaggerated channel conditions. In case of fluid mass flows through the channels, kinematic wave hypothesis plays a key role to understand the elementary features of the associated wave phenomena. In these hypotheses, mass and momentum equations are combined to yield a kinematic wave (KW) equation. Depending upon the kinematic wave, it is governed by a simple partial differential equation (PDE) with a single unknown function (e.g., the flow or wave height, h) in terms of the two independent variables, namely the time and the space with some coefficients which contains information about the flow characteristics. Wong [ 4 ] built up a condition, which can be connected to different planes having distinctive lengths, slopes, roughnesses, stream regimes, soil sorts and infiltration degrees with various net precipitation Intensities. Further, in light of the KWH, a summed up travel time for an alternate profiled channels was created that is pertinent to the general channels subject to a uniform sidelong inflow and a steady upstream inflow.

Chow [ 5 ] gave different properties of ideal segments and communicated the relations between the segment factors of the most powerfully effective segments for various channel profiles. His review reflected the conveyance of a given flow rate with the least flow area. The limitation of this model was only the condition of uniform discharge. The after effects of his review are still being used for comparing different channel types. The relations obtained for the ideal channel section variables were adjusted by considering distinctive parameters for overland streams. Kinematic wave conditions can be connected to most overland stream circumstances, and it determines its quality by having the possibility of getting physically based procedures without the requirement for any experimentation and they combined the kinematic wave (kw) Tc equation with Manning’s condition and inferred the Tc equation [ 2 and 5 ]. A correlation of conditions is proposed by Woolhiser, and Liggett and USDA [ 6 and 7 ] demonstrated that the rearrangements should be possible by changing precipitation intensity in technique USDA [ 7 ]. USDA [ 7 ] proposed an adjusted kw Tc equation, which was additionally identified with a steady precipitation. Chen and Wong [ 8 ] inferred the condition by coupling the kw Tc with the Darcy–Weisbach equation. Swamee with another co-author [ 9 ] proposed unequivocal conditions for channel section factors considering rectangular, trapezoidal, circular and triangular waterway geometries. The authors have considered the minimization of waterway cost as the target of the work.

Wong [ 10 ] evaluated the significance of overland Tc on the design flow rate and analyzed the execution nine procedures published somewhere around 1946 and 1993, which are planned for overland flow just that is subjected to uniform precipitation. The evaluation contrasts the assessments from the methods and experimental values that are determined under similar conditions for two surfaces, in particular, cement and grass. The evaluation demonstrates that equation that doesn’t represent the rainfall intensity is substantial for a restricted scope of rainfall intensity. The equation that records for the rainfall intensity, for the most part, shows better concurrence with the experimental information. At last, the appraisal gives two rankings of the equations for the two different surfaces in agreement to their precision when contrasted with the experimental information. Chahar [ 11 ] exhibited ideal outline conditions for a parabolic channel/trench. The framework conditions for the lowest earthwork cost and least lined cost segment were in the unequivocal frame and result in ideal measurements of a waterway in single stride calculations. The condition was obtained in the wake of applying the Fibonacci look strategy on a nonlinear unconstrained improvement issue. Easa [ 12 ] demonstrated channel cross area with illustrative sides and even base to more practical (give lesser development cost per unit length) than the trapezoidal cross segment. Munier [ 13 ] processed precisely the reaction time of an open channel which is of extraordinary significance for administration operations on waterway systems, for example, encouraging forward control issues. Chow [ 14 ] enumerates the values of “ n ” for different channels.

The above literature reveals that a significant amount of work was carried out to determine the optimal dimensions of a channel for given site conditions. Several optimization techniques were also used to either minimize the construction cost of a channel or to enhance their conveyance capability. In fact, the conveyance capacity of a channel affects the time of concentration and detention storage characteristics. Therefore, the profiles of channel or tributaries of a given watershed area play a vital role in deciding watershed response to the overland flow, flood management practices and irrigation pattern of the proposed reach. Kinematic wave theory was employed to understand the conveyance characteristics of the overland flow, wherein a modified form of Manning’s equation is used. In the present study, nine standard different channel profiles of open channel [deep rectangular (D); wide rectangular (W); square (S); triangular (T); parabolic (P); vertical curb (V); circular (C); trapezoidal with equal side slopes (E); and trapezoidal with one side vertical (O)] for given discharge, uniform lateral discharge and an endless upstream discharge are equated in terms of travel time.

Methodology

The construction cost of different channels in any of the hydro/irrigation work is a main cost item, and maximum low cost is accomplished through the economical channel design by considering suitable section shape and dimension. In light of the kwh, Wong [ 15 ] demonstrated that, for discharge in a channel with an inconsequential backwater impact, the celerity ( c ) of the wave going downward to the channel is assumed to be represented by

where t  = duration, x  = lateral distance of channel toward flow direction and α and β are factors associated to flow rate Q and flow area A with relation.

The validity of Eq. ( 1 ) is true only in case of α and β are constants.

Considering a waterway or channel with a steady upstream discharge Q u and a lateral discharge ( q ) that is consistently appropriated along the channel, then flow rate in the channel ( Q ) follows Eq. ( 3 )

On substituting Eq. ( 3 ) in Eq. ( 4 ) and integrating from 0 to t t for time ( t ) and 0 to L c for distance ( x ), the following equation is obtained

If the upstream flow is zero, then Q u  = 0, and thus,

From the law of continuity, the downstream discharge of a channel segment during equilibrium ( Q d ) is related to constant upstream discharge ( Q u ) and uniform lateral discharge ( q ); therefore, Eqs. ( 3 and 4 ) take the following form:

Equation ( 7 ) can be written as

where \( \lambda \) is discharge ratio that relates upstream discharge to lateral discharge as follows:

So as to think about the travel time for the nine different channel profiles (D, W, S, T, P, V, C, E, O) on a combined basis, S-type channel profile has been selected as a reference, because of the fewer parameters involved with it. While comparing, the different channels (D, W, S, T, P, V, C, E, O), assume same lateral discharge and the upstream discharge together with other channel characteristics excluding the profile of channel and size. The ratio of the travel time equations for the (D, W, S, T, P, V, C, E, O) channel profiles by the travel time of the square channel is summarized in Table  1 , and their notation along with figure is described in “ Appendix I ”.

Result and Discussion

The results drawn from the present work are presented in Figs.  1 , 2 and 3 , which show the comparison and variation of time of travel ratio for different cross section geometries of channels with flow depth and slope. Figure  1 a, b represents the variation of travel time ratio with ‘ µ ’ for D- and W-type channel, which shows as the flow depth increases, the travel time ratio increases for D-type channel while it decreases for W-type channel. Figure  1 c, d represents the variation of travel time ratio with ‘z’ for T and V types of channel, which shows that as the channel side slope decreases the time travel ratio first decreases to a certain value and then increases with steep side slope to mild side slope, while Fig.  1 e shows that time travel ratio for the parabolic channel with and without upstream inflow continuously decreasing with an increase in depth of flow. It means that it produces large watershed runoff. Figure  1 f, g represents the variation of time of travel ratio with ‘z’ for T-type channel. E type of channel having travel time ratio decreases continuously with an increase in side slope, while this ratio goes decreasing up to certain minimum value and then starts increasing for the O-type channel. Figures  2 and 3 show the comparative behavior which is discussed in a later section.

figure 1

a Variation of D channel, b W channel, c T channel, d V channel, e P channel, f E channel, g O channel with respect to travel time

figure 2

Variation of O, T and V types of channel with respect to travel time

figure 3

Variation of W, D, S and P channels with respect to travel time

Comparative Assessments of Travel Time for T, E, O and V Channels

The travel time in the T and O types of channel is identical for a channel with “ z ” and “1/ z ”. Figure  2 displays the travel time ratio curves “ t tt / t ts , t tv / t ts and t ttz2 / t ts ”, for z  = 0.1–5. A lesser value of “ z ” characterizes a channel by sharp side slope (SSS), whereas the large value of “ z ” characterizes a channel by the mild side slope (MSS). On the evaluation of Fig.  2 , for T type of channel having more travel time, t tt / t ts varies from 0.970 to 1.232, for V type of channel, t tv /t ts varies from 1.034 to 1.456, and for O channel t ttz2 /t ts ranges from 1.167 to 1.750. Therefore, one can say that the travel time for V- and O-type channels is nearby. Further, figures also indicate that all the curves have a decreasing trend with increasing the value of “ z ”, reach the lowest and then escalate with increasing value of “ z ”.

Figure  2 additionally demonstrates that for channels with SSS (low value of z), the travel time for the V channel is longer and bigger than those for the T channel, which is compared to more prominent stream resistance in the V channel. On the other hand, for channels with MSS (more value of z), the figures demonstrate that the travel time for the T channel is longer and bigger than those for the V channel, which is compared to more prominent stream resistance in the T channel.

Comparative Assessments of Travel Time for Rectangular and Parabolic Channels

Figure  3 shows that ratio of the travel time of D, W and P channel with respect to S channel t tw / t ts , t td / t ts , t ts / t ts and t tp / t ts , for the value of \( \lambda \)  = 0 and 1 and µ  = 0.01–1000. A low value of “ µ ” describes a rectangular or parabolic channel with little flow depth, and large value of “ µ ” describes a rectangular or parabolic channel with huge flow depth. The estimations of the value of λ for 0 and 1 express channels without and with upstream discharge. For the W channel, the ratio of travel time t tw / t ts diminishes with expanding value of µ and λ , which relates to diminishing flow resistance with expanding flow depth and upstream discharge, when contrasted with that of the square channel. On the other hand, for the D channel, the ratio t td / t ts increases with expanding µ and λ , which is related to increasing flow resistance with increasing flow depth and upstream discharge, when contrasted with that of the square channel. Further, as the flow resistance for the W channel is contributed from the base of the channel (i.e., like one side of the D channel), and for the D channel, it is from two sides of the channel, the estimation of t td / t ts for µ  = 100 is around twice that for t tw / t ts for µ  = 0.01. For the P channel, t tp / t ts diminishes with expanding µ and λ , which is related to diminishing flow resistance with expanding flow depth and upstream discharge, when contrasted with that of the square channel. For channels with little flow depth, the flow conditions in the W channel and P channel are comparative. Without a doubt, the figures demonstrate that for little value of “ µ ”, the ratios t tw / t ts and t tp / t ts are near to each other.

Table  2 demonstrates that the alteration in the travel time proportions between λ  = 1 and 100 is little. The impact of the upstream discharge on the travel time ratio is low. The value corresponds to λ  = 100 demonstrating that the travel time among different channel profiles could be as much as sixfolds; accordingly, it proves that travel time has a significant role. Among the nine channel profiles [deep rectangular (D); wide rectangular (W); square (S); triangular (T); parabolic (P); vertical curb (V); circular (C); trapezoidal with equal side slopes (E); and trapezoidal with one side vertical (O)], the one that delivers the longest travel time is the D ( µ  = 100). Henceforth, the utilization of this channel produces littler watershed runoff. The channel that creates the most limited time of travel is the P channel with extensive stream depth ( µ  = 100). Henceforth, the utilization of this channel creates a bigger watershed runoff.

In case of heavy rainfall, inappropriate design of channel may lead to cause of flooding in developing nations where channel is either not properly designed or if designed not sustain the flow of water, which hinder the free flow of excess storms when they occur. Therefore, practically this study revealed that the best channel in extreme rainfall too hole maximum flood by increasing the travel time of the flow increases the lag time and thus save the likely causes of major disaster as well as flow variability among the different channels.

Conclusions

On coupling, the Manning’s equation with kinematic wave parameters ( α and β ) has been used to find the best suited channels for smaller watershed runoff which can add to flood management tools. The estimation of wave parameters ( α and β ) for different channels is efficient in decision support tools in mitigating climate induced hazards such as flood and drought. A new approach for the design of channel network on the basis of the time of travel has been envisaged in the present work using the integration of kinematic wave theory with the Manning’s equations. On the basis of its verification and by means of the derived formulas, the effect of channel profile on the time of travel for nine channel profiles has been compared on a unified basis and the following conclusions were drawn:

It also shows that channel profile can cause a sixfold increase in time of travel.

Of the nine channel profiles, the one that produces the longest time of travel is the deep rectangular channel. Hence, the use of this channel produces smaller watershed runoff.

The channel that produces the shortest time of travel is the parabolic channel with large flow depth. Hence, the use of this channel produces relatively larger watershed runoff.

The order of selecting a channel that produces the longest time of travel to least time of travel is as deep rectangular, a trapezoidal channel with equal side slopes, wide rectangular, square, triangular, circular and parabolic.

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Kumar, V. A Study of Travel Time for Different Open Channels. J. Inst. Eng. India Ser. A 101 , 399–407 (2020). https://doi.org/10.1007/s40030-020-00439-3

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Published : 21 March 2020

Issue Date : June 2020

DOI : https://doi.org/10.1007/s40030-020-00439-3

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SMATS

Travel Time Reliability: How to Measure and Why it is Important?

by Shahrzad Jalali | Jul 16, 2020 | Blog - ITS Systems - Vehicle detection , Traffic Management

flow travel time

We have all experienced traffic delays in our trips to home, work, or vacations. Although sometimes delays are expected ahead of time, and we can add extra time to our trip duration, unexpected delays can create serious problems for travelers, shippers, and businesses, making travel time reliability important to motorists. The unexpected delays can be caused by adverse weather conditions, road closures, or incidents.

Annual Average Travel Time is the measure that is used to report the roads’ traffic congestion. However, most of the time, it is different from what riders would experience every day or what they remember due to unexpected delays. So, what other measures should be reported along with average travel time as measures of congestion?

Travel Time Reliability (TTR) Measures

Travel Time Reliability (TTR) measures help in calculating the unexpected delays. The following measures are the main components of TTR:

1. Travel Time Index (TTI):

Travel Time Index (TTI) is the ratio of Average Travel Time in peak hours to Free-Flow Travel Time. In other words, the Travel Time Index represents the average additional time required for a trip during peak times in comparison with that trip duration in no-traffic condition. For calculating Free-Flow Travel Time, divide the road length by maximum speed limit of the road.

flow travel time

For instance, if the Average and Free-Flow Travel Time are 5 and 4 minutes, respectively, TTI would be 1.25. This value means that your trip will take 25% longer then no congestion condition. TTI can be calculated for different temporal grouping schemes such as X-minute intervals, by time-of-the-day, day-of-the-week, month, and for the entire year. Also, for each of these groups, TTI can be calculated for weekdays and weekends separately.

2. Buffer Index (BI):

Buffer Time is the additional time for unexpected delays that commuters should consider along with average travel time to be on-time 95 percent of the time. Buffer Index is calculating as follow:

flow travel time

The buffer index is expressed as a percentage. For example, if BI and average travel time are 20% and 10 minutes, then the buffer time would be 2 minutes. Since it is calculated by 95 th percentile travel time, it represents almost all worst-case delay scenarios and assures travelers to be on-time 95 percent of all trips.

3. Planning Time Index (PTI):

Planning Time Index is the ratio of the 95th percentile to the free-flow travel time and shows the total time which is needed for on-time arrival in 95 percent of all trips.

flow travel time

The difference between Buffer Index and Planning Time Index is that BI represents the extra delay time that should be added to average travel time, while the PTI indicates the total trip time (average travel time + buffer time). A PTI value of 2.0 for a given period suggests that travelers should spend twice as much time traveling as the free-flow travel time to reach their destination on-time 95 percent of the time. The planning time index is useful because it can be directly compared to the travel time index on similar numeric scales.

Different percentile values can be used instead of the 95 th percentile. This value depends on your desired level of reliability. The lower percentile value results in lower reliability.

4. 90th or 95th Percentile Travel Times

This measure is the most straightforward method that represents the travel time of the most congested day. Since this measure reports in minutes, it is easily understandable for drivers. However, the 90th or 95th Percentile measure can’t be used to compare different trips because of their various length. Also, it is hard to aggregate the trips travel time and report as subarea or citywide average.

5. Percentage of Travel under Congestion (PTC)

The percentage of travel under congestion is defined as the percentage of all vehicles’ miles traveled (VMT) under congested conditions in the specified duration. The PTC measure can be aggregated in the similar temporal fashion described above for TTI.

6. Frequency that Congestion Exceeds Some Expected Threshold

This measure shows the percent of days or times that the congestion exceeds some expected threshold. The threshold can be set on travel time or speed data, especially when you capture the traffic data 24/7. This measure is commonly reported on weekdays peak hours.

The following figure shows TTR Indices on a Travel Time Distribution chart from SMATS iNode :

flow travel time

An example of Travel Time Distribution Chart in iNode

Source: U.S. Federal Highway Administration

Do  these equations and methodologies seem  overwhelming? SMATS’ iNode is designed to translate raw traffic data into r eady-to-use performance metrics such as Travel Time Reliability (TTR) and much more .

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TRID the TRIS and ITRD database

A flow travel time relationship for use in transportation planning

A flow travel time relationship based on concepts of the queueing theory is postulated and the level of agreement with measured data is examined. For some transportation planning purposes, a section of road can be sufficiently described purely in terms of the three parameters of the relationship. Its use leads to a more sophisticated approach to the problem of traffic assignment and enables feedback effects to be readily taken into account. It also provides a logical basis for interzonal assignments to more than one road with full capacity restraint. The nature of the relationship also gives rise to suggested more rational definitions of saturation flow rate and practical capacity (A).

  • Record URL: http://arrbknowledge.com
  • Davidson, K B
  • Australian Road Research Board (ARRB) Conference, 3rd, 1966, Sydney
  • Publication Date: 1966
  • Pagination: 183-94
  • Issue Number: 1

Subject/Index Terms

  • TRT Terms: Highway capacity ; Level of service ; Traffic assignment ; Traffic delays ; Traffic flow ; Transportation planning ; Travel time ; Types of roads
  • Geographic Terms: Australia ; Brisbane, Queensland
  • ATRI Terms: Delay ; Level of service ; Road type ; Traffic assignment ; Traffic capacity ; Traffic flow ; Transport planning ; Travel time
  • Subject Areas: Highways; Operations and Traffic Management;

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3.6: 3-6 Route Choice

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  • Associate Professor (Engineering) via Wikipedia

Route assignment , route choice , or traffic assignment concerns the selection of routes (alternative called paths) between origins and destinations in transportation networks. It is the fourth step in the conventional transportation forecasting model, following Trip Generation, Destination Choice, and Mode Choice. The zonal interchange analysis of trip distribution provides origin-destination trip tables. Mode choice analysis tells which travelers will use which mode. To determine facility needs and costs and benefits, we need to know the number of travelers on each route and link of the network (a route is simply a chain of links between an origin and destination). We need to undertake traffic (or trip) assignment. Suppose there is a network of highways and transit systems and a proposed addition. We first want to know the present pattern of travel times and flows and then what would happen if the addition were made.

Link Performance Function

The cost that a driver imposes on others is called the marginal cost. However, when making decisions, a driver only faces his own cost (the average cost) and ignores any costs imposed on others (the marginal cost).

  • \[AverageCost=\dfrac{S_T}{Q}\]
  • \[MarginalCost=\dfrac{\delta S_T}{\delta Q}\]

where \(S_T\) is the total cost, and \(Q\) is the flow.

BPR Link Performance Function

Suppose we are considering a highway network. For each link there is a function stating the relationship between resistance and volume of traffic. The Bureau of Public Roads (BPR) developed a link (arc) congestion (or volume-delay, or link performance) function, which we will term S a (Q a )

\[S_a(Q_a)=t_a(1+0.15\dfrac ({Q_a}{c_a})^4)\]

t a = free-flow travel time on link a per unit of time

Q a = flow (or volume) of traffic on link a per unit of time (somewhat more accurately: flow attempting to use link a )

c a = capacity of link a per unit of time

S a (Q a ) is the average travel time for a vehicle on link a

There are other congestion functions. The CATS has long used a function different from that used by the BPR, but there seems to be little difference between results when the CATS and BPR functions are compared.

Can Flow Exceed Capacity?

On a link, the capacity is thought of as “outflow.” Demand is inflow.

If inflow > outflow for a period of time, there is queueing (and delay).

For Example, for a 1 hour period, if 2100 cars arrive and 2000 depart, 100 are still there. The link performance function tries to represent that phenomenon in a simple way.

Wardrop's Principles of Equilibrium

User Equilibrium

Each user acts to minimize his/her own cost, subject to every other user doing the same. Travel times are equal on all used routes and lower than on any unused route.

  • System optimal

Each user acts to minimize the total travel time on the system.

Price of Anarchy

The reason we have congestion is that people are selfish. The cost of that selfishness (when people behave according to their own interest rather than society's) is the price of anarchy .

The ratio of system-wide travel time under User Equilibrium and System Optimal conditions.

For a two-link network with linear link performance functions (latency functions), Price of Anarchy is < 4/3.

Is this too much? Should something be done, or is 33% waste acceptable? [The loss may be larger/smaller in other cases, under different assumptions, etc.]

Conservation of Flow

An important factor in road assignment is the conservation of flow. This means that the number of vehicles entering the intersection (link segment) equals the number of vehicles exiting the intersection for a given period of time (except for sources and sinks).

Similarly, the number of vehicles entering the back of the link equals the number exiting the front (over a long period of time).

Auto assignment

Long-standing techniques.

The above examples are adequate for a problem of two links, however real networks are much more complicated. The problem of estimating how many users are on each route is long standing. Planners started looking hard at it as freeways and expressways (motorways) began to be developed. The freeway offered a superior level of service over the local street system and diverted traffic from the local system. At first, diversion was the technique. Ratios of travel time were used, tempered by considerations of costs, comfort, and level of service.

The Chicago Area Transportation Study (CATS) researchers developed diversion curves for freeways versus local streets. There was much work in California also, for California had early experiences with freeway planning. In addition to work of a diversion sort, the CATS attacked some technical problems that arise when one works with complex networks. One result was the Moore algorithm for finding shortest paths on networks.

The issue the diversion approach didn’t handle was the feedback from the quantity of traffic on links and routes. If a lot of vehicles try to use a facility, the facility becomes congested and travel time increases. Absent some way to consider feedback, early planning studies (actually, most in the period 1960-1975) ignored feedback. They used the Moore algorithm to determine shortest paths and assigned all traffic to shortest paths. That’s called all or nothing assignment because either all of the traffic from i to j moves along a route or it does not.

The all-or-nothing or shortest path assignment is not trivial from a technical-computational view. Each traffic zone is connected to n - 1 zones, so there are numerous paths to be considered. In addition, we are ultimately interested in traffic on links. A link may be a part of several paths, and traffic along paths has to be summed link by link.

An argument can be made favoring the all-or-nothing approach. It goes this way: The planning study is to support investments so that a good level of service is available on all links. Using the travel times associated with the planned level of service, calculations indicate how traffic will flow once improvements are in place. Knowing the quantities of traffic on links, the capacity to be supplied to meet the desired level of service can be calculated.

Heuristic procedures

To take account of the affect of traffic loading on travel times and traffic equilibria, several heuristic calculation procedures were developed. One heuristic proceeds incrementally. The traffic to be assigned is divided into parts (usually 4). Assign the first part of the traffic. Compute new travel times and assign the next part of the traffic. The last step is repeated until all the traffic is assigned. The CATS used a variation on this; it assigned row by row in the O-D table.

The heuristic included in the FHWA collection of computer programs proceeds another way.

  • Step 0: Start by loading all traffic using an all or nothing procedure.
  • Step 1: Compute the resulting travel times and reassign traffic.
  • Step 2: Now, begin to reassign using weights. Compute the weighted travel times in the previous two loadings and use those for the next assignment. The latest iteration gets a weight of 0.25 and the previous gets a weight of 0.75.
  • Step 3. Continue.

These procedures seem to work “pretty well,” but they are not exact.

Frank-Wolfe algorithm

Dafermos (1968) applied the Frank-Wolfe algorithm (1956, Florian 1976), which can be used to deal with the traffic equilibrium problem.

Equilibrium Assignment

To assign traffic to paths and links we have to have rules, and there are the well-known Wardrop equilibrium (1952) conditions. The essence of these is that travelers will strive to find the shortest (least resistance) path from origin to destination, and network equilibrium occurs when no traveler can decrease travel effort by shifting to a new path. These are termed user optimal conditions, for no user will gain from changing travel paths once the system is in equilibrium.

The user optimum equilibrium can be found by solving the following nonlinear programming problem

\[min \displaystyle \sum_{a} \displaystyle\int\limits_{0}^{v_a}S_a(Q_a)\, dx\]

subject to:

\[Q_a=\displaystyle\sum_{i}\displaystyle\sum_{j}\displaystyle\sum_{r}\alpha_{ij}^{ar}Q_{ij}^r\]

\[sum_{r}Q_{ij}^r=Q_{ij}\]

\[Q_a\ge 0, Q_{ij}^r\ge 0\]

where \(Q_{ij}^r\) is the number of vehicles on path r from origin i to destination j . So constraint (2) says that all travel must take place: i = 1 ... n; j = 1 ... n

\(\alpha_{ij}^{ar}\)= 1 if link a is on path r from i to j ; zero otherwise.

So constraint (1) sums traffic on each link. There is a constraint for each link on the network. Constraint (3) assures no negative traffic.

Transit assignment

There are also methods that have been developed to assign passengers to transit vehicles. In an effort to increase the accuracy of transit assignment estimates, a number of assumptions are generally made. Examples of these include the following:

  • All transit trips are run on a set and predefined schedule that is known or readily available to the users.
  • There is a fixed capacity associated with the transit service (car/trolley/bus capacity).

flow travel time

Solve for the flows on Links a and b in the Simple Network of two parallel links just shown if the link performance function on link a :

\(S_a=5+2*Q_a\)

and the function on link b :

\(S_b=10+Q_b\)

where total flow between the origin and destination is 1000 trips.

Time (Cost) is equal on all used routes so \(S_a=S_b\)

And we have Conservation of flow so, \(Q_a+Q_b=Q_o=Q_d=1000\)

\(5+2*(1000-Q_b)=10+Q_b\)

\(1995=3Q_b\)

\(Q_b=665;Q_a=335\)

An example from Eash, Janson, and Boyce (1979) will illustrate the solution to the nonlinear program problem. There are two links from node 1 to node 2, and there is a resistance function for each link (see Figure 1). Areas under the curves in Figure 2 correspond to the integration from 0 to a in equation 1, they sum to 220,674. Note that the function for link b is plotted in the reverse direction.

\(S_a=15(1+0.15(\dfrac{Q_a}{1000})^4)\)

\(S_b=20(1+0.15(\dfrac{Q_a}{3000})^4)\)

\(Q_a+Q_b=8000\)

Show graphically the equilibrium result.

flow travel time

At equilibrium there are 2,152 vehicles on link a and 5,847 on link b . Travel time is the same on each route: about 63.

Figure 3 illustrates an allocation of vehicles that is not consistent with the equilibrium solution. The curves are unchanged, but with the new allocation of vehicles to routes the shaded area has to be included in the solution, so the Figure 3 solution is larger than the solution in Figure 2 by the area of the shaded area.

Assume the traffic flow from Milwaukee to Chicago, is 15000 vehicles per hour. The flow is divided between two parallel facilities, a freeway and an arterial. Flow on the freeway is denoted \(Q_f\), and flow on the two-lane arterial is denoted \(Q_a\).

The travel time (in minutes) on the freeway (\(C_f\)) is given by:

\(C_f=10+Q_f/1500\)

\(C_a=15+Q_a/1000\)

Apply Wardrop's User Equilibrium Principle, and determine the flow and travel time on both routes.

The travel times are set equal to one another

\(C_f=C_a\)

\(10+Q_f/1500=15+Q_a/1000\)

The total traffic flow is equal to 15000

\(Q_f+Q_a=15000\)

\(Q_a=15000-Q_f\)

\(10+Q_f/1500=15+(15000-Q_f)/1000\)

Solve for \(Q_f\)

\(Q_f=60000/5=12000\)

\(Q_a=15000-Q_f=3000\)

Thought Questions

  • How can we get drivers to consider their marginal cost?
  • Alternatively: How can we get drivers to behave in a “System Optimal” way?

Sample Problems

Given a flow of six (6) units from origin “o” to destination “r”. Flow on each route ab is designated with Qab in the Time Function. Apply Wardrop's Network Equilibrium Principle (Users Equalize Travel Times on all used routes)

A. What is the flow and travel time on each link? (complete the table below) for Network A

Link Attributes

B. What is the system optimal assignment?

C. What is the Price of Anarchy?

What is the flow and travel time on each link? Complete the table below for Network A:

These four links are really 2 links O-P-R and O-Q-R, because by conservation of flow Qop = Qpr and Qoq = Qqr.

By Wardrop's Equilibrium Principle, the travel time (cost) on each used route must be equal. Therefore \(C_{opr}=C_{oqr}\)

OR \(25+6*Q_{opr}=20+7*Q_{oqr}\)

\(5+6*Q_{opr}=7*Q_{oqr}\)

\(Q_{oqr}=5/7+6*Q_{opr}/7\)

By the conservation of flow principle

\(Q_{oqr}+Q_{opr}=6\)

\(Q_{opr}=6-Q_{oqr}\)

By substitution

\Q_{oqr}=5/7+6/7(6-Q_{oqr})=41/7-6*Q_{oqr}/7\)

\(13*Q_{oqr}=41\)

\(Q_{oqr}=41/13=3.15\)

\(Q_{opr}=2.84\)

\(42.01=25+6(2.84)\)

\(42.05=20+7(3.15)\)

Check (within rounding error)

or expanding back to the original table:

User Equilibrium: Total Delay = 42.01 * 6 = 252.06

What is the system optimal assignment?

Conservation of Flow:

\(Q_{opr}+Q_{oqr}=6\)

\(TotalDelay=Q_{opr}(25+6*Q_{oqr})+Q_{oqr}(20+7*Q_{oqr})\)

\(25Q_{opr}+6Q_{opr}^2+(6_Q_{opr})(20+7(6-Q_{opr}))\)

\(25Q_{opr}+6Q_{opr}^2+(6_Q_{opr})(62-7Q_{opr}))\)

\(25Q_{opr}+6Q_{opr}^2+372-62Q_{opr}-42Q_{opr}+7Q_{opr}^2\)

\(13Q_{opr}^2-79Q_{opr}+372\)

Analytic Solution requires minimizing total delay

\(\deltaC/\deltaQ=26Q_{opr}-79=0\)

\(Q_{opr}=79/26-3.04\)

\(Q_{oqr}=6-Q_{opr}=2.96\)

And we can compute the SO travel times on each path

\(C_{opr,SO}=25+6*3.04=43.24\)

\(C_{opr,SO}=20+7*2.96=40.72\)

Note that unlike the UE solution, \(C_{opr,SO}\g C_{oqr,SO}\)

Total Delay = 3.04(25+ 6*3.04) + 2.96(20+7*2.96) = 131.45+120.53= 251.98

Note: one could also use software such as a "Solver" algorithm to find this solution.

What is the Price of Anarchy?

User Equilibrium: Total Delay =252.06 System Optimal: Total Delay = 251.98

Price of Anarchy = 252.06/251.98 = 1.0003 < 4/3

The Marcytown - Rivertown corridor was served by 3 bridges, according to the attached map. The bridge over the River on the route directly connecting Marcytown and Citytown collapsed, leaving two alternatives, via Donkeytown and a direct. Assume the travel time functions Cij in minutes, Qij in vehicles/hour, on the five links routes are as given.

Marcytown - Rivertown Cmr = 5 + Qmr/1000

Marcytown - Citytown (prior to collapse) Cmc = 5 + Qmc/1000

Marcytown - Citytown (after collapse) Cmr = ∞

Citytown - Rivertown Ccr = 1 + Qcr/500

Marcytown - Donkeytown Cmd = 7 + Qmd/500

Donkeytown - Rivertown Cdr = 9 + Qdr/1000

Also assume there are 10000 vehicles per hour that want to make the trip. If travelers behave according to Wardrops user equilibrium principle.

A) Prior to the collapse, how many vehicles used each route?

Route A (Marcytown-Rivertown) = Ca = 5 + Qa/1000

Route B (Marcytown-Citytown-Rivertown) = Cb = 5 + Qb/1000 + 1 + Qb/500 = 6 + 3Qb/1000

Route C (Marcytown-Donkeytown-Rivertown)= Cc = 7 + Qc/500 + 9 + Qc/1000 = 16 + 3Qc/1000

At equilibrium the travel time on all three used routes will be the same: Ca = Cb = Cc

We also know that Qa + Qb + Qc = 10000

Solving the above set of equations will provide the following results:

Qa = 8467;Qb = 2267;Qc = −867

We know that flow cannot be negative. By looking at the travel time equations we can see a pattern.

Even with a flow of 0 vehicles the travel time on route C(16 minutes) is higher than A or B. This indicates that vehicles will choose route A or B and we can ignore Route C.

Solving the following equations:

Route A (Marcytown-Rivertown) = Ca = 5 + Qa /1000

Route B (Marcytown-Citytown-Rivertown) = Cb = 6 + 3Qb /1000

Qa + Qb = 10000

We can the following values:

Qa = 7750; Qb = 2250; Qc = 0

B) After the collapse, how many vehicles used each route?

We now have only two routes, route A and C since Route B is no longer possible. We could solve the following equations:

Route C (Marcytown- Donkeytown-Rivertown) = Cc = 16 + 3Qc /1000

Qa+ Qc= 10000

But we know from above table that Route C is going to be more expensive in terms of travel time even with zero vehicles using that route. We can therefore assume that Route A is the only option and allocate all the 10,000 vehicles to Route A.

If we actually solve the problem using the above set of equations, you will get the following results:

Qa = 10250; Qc = -250

which again indicates that route C is not an option since flow cannot be negative.

C) After the collapse, public officials want to reduce inefficiencies in the system, how many vehicles would have to be shifted between routes? What is the “price of anarchy” in this case?

TotalDelayUE =(15)(10,000)=150,000

System Optimal

TotalDelaySO =(Qa)(5+Qa/1000)+(Qc)(16+3Qc/1000)

Using Qa + Qc = 10,000

TotalDelaySO =(Qa2)/250−71Qa+460000

Minimize total delay ∂((Qa2)/250 − 71Qa + 460000)/∂Qa = 0

Qa/125−7 → Qa = 8875 Qc = 1125 Ca = 13,875 Cc = 19,375

TotalDelaySO =144938

Price of Anarchy = 150,000/144,938 = 1.035

  • \(C_T\) - total cost
  • \(C_k\) - travel cost on link \(k\)
  • \(Q_k\) - flow (volume) on link \(k\)

Abbreviations

  • VDF - Volume Delay Function
  • LPF - Link Performance Function
  • BPR - Bureau of Public Roads
  • UE - User Equilbrium
  • SO - System Optimal
  • DTA - Dynamic Traffic Assignment
  • DUE - Deterministic User Equilibrium
  • SUE - Stochastic User Equilibrium
  • AC - Average Cost
  • MC - Marginal Cost
  • Route assignment, route choice, auto assignment
  • Volume-delay function, link performance function
  • User equilibrium
  • Conservation of flow
  • Average cost
  • Marginal cost

External Exercises

Use the ADAM software at the STREET website and try Assignment #3 to learn how changes in network characteristics impact route choice.

Additional Questions

1. If trip distribution depends on travel times, and travel times depend on the trip table (resulting from trip distribution) that is assigned to the road network, how do we solve this problem (conceptually)?

2. Do drivers behave in a system optimal or a user optimal way? How can you get them to move from one to the other.

3. Identify a mechanism that can ensure the system optimal outcome is achieved in route assignment, rather than the user equilibrium. Why would we want such an outcome? What are the drawbacks to the mechanism you identified?

4. Assume the flow from Dakotopolis to New Fargo, is 5300 vehicles per hour. The flow is divided between two parallel facilities, a freeway and an arterial. Flow on the freeway is denoted \(Q_f\), and flow on the two-lane arterial is denoted \(Q_r\). The travel time on the freeway \(C_f\) is given by:

\(C_f=5+Q_f/1000\)

The travel time on the arterial (Cr) is given by

\(C_r=7+Q_r/500\)

(a) Apply Wardrop's User Equilibrium Principle, and determine the flow and travel time on both routes from Dakotopolis to New Fargo.

(b) Solve for the System Optimal Solution and determine the flow and travel time on both routes.

5. Given a flow of 10,000 vehicles from origin to destination traveling on three parallel routes. Flow on each route A, B, or C is designated with \(Q_a\), \(Q_b\), \(Q_c\) in the Time Function Respectively. Apply Wardrop's Network Equilibrium Principle (Users Equalize Travel Times on all used routes), and determine the flow on each route.

\(T_A=500+20Q_A\)

\(T_B=1000+10Q_B\)

\(T_C=2000+30Q_C\)

  • How does average cost differ from marginal cost?
  • How do System Optimal and User Equilibrium travel time differ?
  • Why do we want people to behave in an SO way?
  • How can you get people to behave in an SO way?
  • Who was John Glen Wardrop?
  • What are Wardrop’s Two Principles?
  • What does conservation of flow require in route assignment?
  • Can Variable Message Signs be used to encourage System Optimal behavior?
  • What is freeflow travel time?
  • If a problem has more than two routes, where does the extra equation come from?
  • How can you determine if a route is unused?
  • What is the difference between capacity and flow
  • Draw a typical volume-delay function for a deterministic, static user equilibrium assignment.
  • Can Q be negative?
  • What is route assignment?
  • Is it important that the output travel times from route choice be consistent with the input travel times for destination choice and mode choice? Why?

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What Is a Flow State? How This Highly Focused State Can Boost Productivity

Get in the zone and find your flow.

Shelby is a published lifestyle writer, print and digital journalist, and soul-centered storyteller who specializes in home and decor, mental health and wellness, and travel topics. Her work has appeared in Real Simple, Better Homes and Gardens, Martha Stewart Living, and many more publications and websites. Highlights: * Writing covers home and decor, mental health and wellness, and travel topics * Work has appeared in Real Simple, Better Homes and Gardens, Martha Stewart Living, and more * Former associate magazine editor at American Girl magazine

flow travel time

MirageC/Getty Images

We’re all looking for ways to get more done, whether it’s squeezing in an extra Zoom meeting before lunch or writing a research paper 15 minutes faster than you did yesterday. You may have tried a number of productivity hacks —perhaps turning to Feng Shui for your office or trying the Pomodoro Technique —with some good results.

Tethering your worth to your productivity levels can be unhealthy and toxic : Too much productivity without adequate rest, recovery, or leisure time isn’t great for you. But knowing a few tricks to help boost your productivity when you actually need to get stuff done always comes in handy. And in times when high productivity is essential, you’re going to want to know how to get yourself into a state of “flow.”

Here’s how to harness the power of a flow state of mind, a productivity hack that’s about to change the way you work and create.

What Is Flow State?

“Flow state is a mental state of heightened focus and immersion, where we can maintain our focus [on an activity or task] sometimes for hours at a time,” says Jeff Karp, PhD., author of LIT: Life Ignition Tools and distinguished professor, Brigham and Women’s Hospital, Harvard Medical School. “Some refer to [being in flow] as being 'fully in the zone.’” 

As an expert in the phenomenon of flow, Karp explains that flow state is often used within the context of one’s workload, but that it also extends into artistic, sport, or gaming endeavors, too. Even parents and caretakers may find incredible use from tapping into flow. 

“[Psychologist] Mihály Csíkszentmihályi first introduced the concept of ‘flow’ in the 1970s,” says Karp, but it’s just starting to truly hit people’s radars now.

Medical experts have studied the concept of flow, and according to Karp and scientific research, neurotransmitters like dopamine, endorphins, serotonin, norepinephrine, and acetylcholine are the brain-related drivers behind this focused state of mind. Together, these chemicals light up the reward centers of the brain and help the person focus. 

“I think being in flow is correlated with forging new pathways in our brains,” Karp says.

You may be familiar with “hyperfocus,” which is often a symptom of Attention-Deficit/Hyperactivity Disorder (ADHD). It’s during periods of hyperfocus when individuals with ADHD become highly interested and engrossed in a project or task and are able to hone in their attention. While they share some characteristics, hyperfocus and flow aren’t exactly the same thing. 

“I believe hyperfocus is core to the flow state, but being hyperfocused doesn’t guarantee that you are in flow,” Karp explains. “Hyperfocus is like a burst of flow.” In other words, hyperfocus is typically a state that doesn’t persist.

“Flow on the other hand lasts for an extended period—when you're fully engaged—allowing for uninterrupted progress. For people with ADHD, reaching the flow state is often dependent on the presence of high interest or motivation for the activity,” he says. As you strive to reach a flow state, this should be something to keep in mind if you’ve been diagnosed with ADHD.

What Are the Characteristics of Flow?

“Flow is characterized by feeling totally 'plugged in,’” Karp explains. “Distractions are easy to deflect. Flow is also often linked to enhanced learning and performance, which are processes associated with neuroplasticity [the brain’s ability to form new habits and connections]. It feels like you’re sailing through your work under perfect environmental conditions.”

Often, during flow, a person is neither distracted by external, environmental goings on, nor internal sensations and cues—a.k.a interoception . They may not realize that they’re getting hungry, they’re chilly, or that they need to go to the bathroom. They’re totally engrossed in their work and won’t notice it’s time for a snack or bathroom break until the flow state spell is broken. There is a sense of losing oneself in one's work—losing track of time and awareness of what's going on around you due to complete concentration and preoccupation.

“Also, flow doesn’t require someone to be focused on one thing,” Karp adds. “I have been in the flow state many times while bouncing around on things like emails, reviewing manuscripts, and grant applications, and preparing presentations.”

Flow can be experienced while doing something artistic, or while exercising, gaming, studying for an exam, or writing. It enables you to work for an hour or two, or more, at a time without interruption or much effort. Flow state often requires you to be doing something you are good at; something you care about; and something that strikes the right balance of being not so challenging that it's frustrating and not so easy that it's boring.

How Do You Achieve Flow State?

Sailing through your work under perfect conditions and impervious to distractions? Sounds amazing, right? But how do you reach this enviable level of productivity? It comes down to an ideal combination of factors. While they’re not all viewed as positive traits, they can create just the right circumstances.

“I think getting into flow requires the right mix of motivation, environment, interest, and cognitive state,” Karp says. This means that a “pressurized situation,” like a looming deadline or meeting on the horizon, can actually help you get into the flow state.

Counterintuitively, Karp adds that procrastination can be paradoxically helpful at getting you into flow to complete a time-sensitive project. “After avoiding a task for a period of time, the act of finally engaging in it can feel particularly energizing,” Karp says. “For some people, procrastination is the key to flow.”

Yes, that’s right—so-called “negative” productivity traits, like procrastination, can actually help you get into an ideal flow state. The challenge, however, is making sure the stress of procrastinating doesn’t negatively impact these pressurized situations. If you’re a procrastinator who enjoys the feeling of tackling a project at the last minute and racing against the clock, and that’s something that helps you perform even better, putting off tasks until the eleventh hour may be how you get into flow. Maybe you’ve inadvertently been working this way because you know your most focused work comes when you’re tight on time.

What Are the Benefits of Flow?

There are many benefits from being able to achieve and utilize a flow state. As Karp says, flow can help us be efficient in our work while minimizing the toll on our cognitive load or stamina.

“We want to be in flow so that we can get our work done more quickly and leave room for other things we want to engage in,” he says. “It can also help us excel in our work (e.g. the more efficient we become and the better work we do, the greater the chances we can more quickly achieve goals, create value for society, and get promotions and bonuses with our jobs).” 

Additionally, Karp points out that flow can help us minimize downtime in our work while leaving time for a balanced life. If that sounds ideal, flow is something you should try to achieve in your daily life. 

Are There Downsides to Flow?

“One downside is that flow is not [always] associated with purpose or meaning,” Karp says. “So, you can be in a flow state and working on something that isn’t a true priority in our life. But on the flip side, however, when you find that you’re doing something that truly matters to you, that you enjoy, that motivates you, and that aligns with your values, you’ll often find that slipping into flow becomes almost effortless.

“Being in flow state can also make you lose your perception of time,” he adds. That can span from being late for things to feeling as if you’ve missed out on being fully present for even years at a time. If you find that you can easily reach flow state, Karp recommends setting reminders, complete with sounds and/or vibrations, so you can snap out of it when you need to.

And, of course, if the only way you find you can achieve a flow state is to procrastinate, but that procrastination sends you into a stressed-out panic, then this might be the productivity path for you! Not everyone needs to utterly lose themselves in their work or tasks in order to get them done, and that's perfectly OK.

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The thing about travel destinations that have become clichéd is that they started out simply as beautiful or interesting places. Ireland’s Cliffs of Moher, one of the country’s most-visited tourist sights, fit this bill.

Edinburgh … and beyond

I had no idea just how beautiful Edinburgh is. And how old.

But first, let’s take a quick trip around the city’s outskirts. Edinburgh (like Rome) is built on seven hills that surround the city. Holyrood Park is situated to the east of the city centre, not far from the historic Royal Mile, and is dominated by the distinctive shape of the 251m-high Arthur’s Seat, an ancient volcanic peak that broods over the city. Some say it was once the site of Camelot, King Arthur’s fabled court.

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flow travel time

IMAGES

  1. Flow travel time map (from calibrated i; Table 5) for time-area diagram

    flow travel time

  2. Procedure for calculating time-of-travel based on real-time flow

    flow travel time

  3. Speed, Travel Time and Delay as a Function of Flow Rate for

    flow travel time

  4. Deep meridional flow travel-time differences. Top: Average of 15 years

    flow travel time

  5. How to Use a Travel Time Graph

    flow travel time

  6. PPT

    flow travel time

COMMENTS

  1. PDF Chapter 3 Time of Concentration and Time of Concentration and Travel

    Travel time ( Tt ) is the ratio of flow length to flow velocity: L Tt = 3600V [eq. 3-1] where: Tt = travel time (hr) L = flow length (ft) V = average velocity (ft/s) 3600 = conversion factor from seconds to hours. Time of concentration ( Tc ) is the sum of Tt values for the various consecutive flow segments: c = T + T + 1 t 2 Ttm [eq. 3-2]

  2. PDF Lesson 11: Rational Method Step 5: Calculating Time of Concentration

    The travel time would be calculated as follows: So, the travel time for the shallow concentrated flow portion of the hydraulic path is 3.57 minutes. Channel Flow - Lc The last flow regime we need to consider is channel flow. To calculate channel flow, we need to know: Length of channel flow in feet

  3. Hydraulic Design Manual: Time of Concentration

    Travel time and t c are functions of length and velocity for a particular watercourse. A long but steep flow path with a high velocity may actually have a shorter travel time than a short but relatively flat flow path. There may be multiple paths to consider in determining the longest travel time.

  4. A Study of Travel Time for Different Open Channels

    Travel time is characterized as the "normal time required for water to go from the highest point of the hillslope by means of the subsurface hillslope to the watershed outlet". In channels having extended travel time and detention storage, the watershed disturbed would have an extended time of concentration.

  5. PDF Time of Concentration

    Equation 2B-3.01 where: Tt = travel time, hours l = flow length, ft V = average velocity, ft/s 3,600 = conversion factor, seconds to hours Surface water flow through the watershed occurs as three different flow types: sheet flow, shallow concentrated flow, and open channel flow.

  6. Travel Time Reliability: How to Measure and Why it is Important?

    For calculating Free-Flow Travel Time, divide the road length by maximum speed limit of the road. For instance, if the Average and Free-Flow Travel Time are 5 and 4 minutes, respectively, TTI would be 1.25. This value means that your trip will take 25% longer then no congestion condition.

  7. PDF Appendix C

    Travel time (Tt) is the time it takes runoff to travel from one location to another in a watershed (subreach) and is a component of time of concentration (Tc), which is the time for runoff to travel from the hydraulically most distant point of the watershed to a point of interest within the watershed.

  8. The Theoretical Basis of A Flow-travel Time Relationship for Use in

    The flow-travel time relationship, first proposed in 1966, has been found to have all the required characteristics of such a relationship; however, beyond a loose association with queueing theory, no adequate theoretical basis had been found. This paper demonstrates that the relationship can be derived from either of two queueing models, both ...

  9. Continuous measurement of flow direction and streamflow based on travel

    Masoud Bahreinimotlagh c Add to Mendeley https://doi.org/10.1016/j.jhydrol.2022.128917 Get rights and content • Monitoring river dynamics using three underwater tomographic systems. • Estimating continuous cross-sectional mean velocity and runoff based on travel-time. •

  10. PDF G. Computation of Travel Time Metrics

    Both the travel time and TTI distributions were developed following the procedure discussed above. Applying Equation (6) for the 95th percentile for Section 2: 95. th. percentile travel time = 95. th. percentile TTI * free flow travel time = 1.837 * 5.840 = 10.728 . which matches the actual 95. th. percentile travel time developed straight from ...

  11. NRCS Overland Flow Travel Time Calculation

    Overland flow travel time given by the formula developed by the Natural Resources Conservation Service (NRCS) on the basis of kinematic wave theory can underpredict for short, steep, flow planes having low flow resistance and high degrees of imperviousness. Underestimation is caused by inaccurate representation of average rainfall intensity for ...

  12. Table 2. Free flow travel time and capacity for each link

    Download Table | Free flow travel time and capacity for each link. from publication: A Two-Stage Algorithm for Origin-Destination Matrices Estimation Considering Dynamic Dispersion Parameter for ...

  13. A flow travel time relationship for use in transportation planning

    A flow travel time relationship based on concepts of the queueing theory is postulated and the level of agreement with measured data is examined. For some transportation planning purposes, a section of road can be sufficiently described purely in terms of the three parameters of the relationship. Its use leads to a more sophisticated approach ...

  14. PDF 3.2.2 Rational Method

    Sheet Flow Travel Time. Sheet flow is the shallow mass of runoff on a planar surface with a uniform depth across the sloping surface. This usually occurs at the headwater of streams over relatively short distances, rarely more than about 130 m (400 ft), and possibly less than

  15. A hazard-based model to derive travel time under congested conditions

    Researchers identified congested conditions when the average travel time exceeds 1.33 or 1.66 times free-flow travel time (FFTT). It is well known that travel time under congested conditions (Tc) is more sensitive to land use, road geometry, and traffic control characteristics than the FFTT.

  16. Time of Concentration (TOC) Estimation

    The sheet flow (or overland flow) happens at the beginning of a flow path where usually the depth of flow is less than 0.1 ft.The sheet flow travel time can be estimated via the equation in Figure 1A where the sheet flow length L should not be exceed 100ft. One way to estimate the sheet flow length L is to apply McCuen-Spiess equation as shown in Figure 1B.

  17. 3.6: 3-6 Route Choice

    The travel time on the arterial (Cr) is given by \(C_r=7+Q_r/500\) (a) Apply Wardrop's User Equilibrium Principle, and determine the flow and travel time on both routes from Dakotopolis to New Fargo. (b) Solve for the System Optimal Solution and determine the flow and travel time on both routes. 5.

  18. PDF CE-093 Rational Method Hydrological Calculations

    n = overland flow resistance factor t = travel time (time of concentration) in min For S.I. units: t = [2.19nL/(S1/2)]0.467 Where: L = length of flow path in m S = surface slope in m/m n = overland flow roughness coefficient t = travel time (time of concentration) in min

  19. Discussion on Influencing Factors of Free-flow Travel Time in Road

    open access Abstract Road traffic impedance is an important part of traffic assignment and has a direct impact on the urban transportation planning, especially on the optimization of urban road network. The BPR function has been the classical traffic impedance since 1960s, which pays more attention on the link level.

  20. Full article: Exploring Correlations between Travel Time Based Measures

    The free-flow travel time is the time taken to traverse a road link under free-flow conditions (for example, the 15 th percentile travel time based on all day observations). It is used to compute planning time index (PTI) (FHWA, Citation 2006 ; Lyman & Bertini, Citation 2008 ; Sisiopiku et al., Citation 2012 ) and travel time index (TTI) (Lyman ...

  21. What Is a Flow State? How This Highly Focused State Can Boost Productivity

    Medical experts have studied the concept of flow, and according to Karp and scientific research, neurotransmitters like dopamine, endorphins, serotonin, norepinephrine, and acetylcholine are the brain-related drivers behind this focused state of mind. Together, these chemicals light up the reward centers of the brain and help the person focus.

  22. Analyzing the effects of congestion on planning time index

    Here, congestion is the dependent variable and defined as the ratio of traffic speed over a one-hour period to the free flow speed multiplied by 100. The independent variable of this study is the planning time index (PTI) which is defined as the ratio of the 95th percentile of travel time to the free-flow travel time.

  23. Flow.travel

    Flow Travel focuses on creating bespoke travel experiences for discerning people. We specialise in authentic, customised trips in Africa and beyond, including India, the . ... We believe in personalised consultation taking the time to understand your dreams and interests. This in-depth understanding forms the basis for a truly tailored itinerary.

  24. PM Opens Key Phase Of Dwarka Expressway, Will Ease Delhi-Gurugram Travel

    The expressway will cut the travel time by at least 20 minutes for 90,000 commuters who take this route every day. ... which aims to improve traffic flow between Delhi and Gurugram on NH-48. The ...