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Tracer transport in parallel systems provides a non-

diffusive mechanism contributing to dispersion [4-8].

This stream tube effect has been referred to as a convec-

tive-dispersive process [9,10], causing tracer arrival-time

variance to increase as a quadratic function of travel dis-

tance for convective flow. Since within a single stream

tube all hydraulic properties remain constant, the distri-

bution of travel time or flow velocities between stream

tubes will be the utmost important factors determining

the dispersive process at the outflow mixing surface. In

the previous studies of dispersion, the effect of normal

distributions [11,12], normal or lognormal arrival times

[10,13,14] between the stream tubes on solute transport

were investigated. However, to date there has been no

systematic study on both a pure convective process

within a stream tube and the effect of normal or log-

normal distributions of arrival times between stream

tubes on solute transport. In addition to the travel time

distributions, there would be some other factors such as

sampling time and pulse duration which influence the

dispersive process for pure convective flow.

The objective of this study is (1) to generate the break-

through curves (BTC) of nonreactive solute using a con-

ceptual stream tube model (STM) by assuming pure

convective process and (2) to investigate how the pulse

conditions and various travel time distributions affect the

transport concept of stream tube model using time mo-

ment analysis.

2. Material and Methods

2.1. Modeling of Conceptual Stream Tube

The conceptual stream tube model can be constructed in

two different ways. The first is a diameter-based type to

assemble a finite number of tubes with equal lengths but

different cross-sectional areas or diameters so that dif-

ferent average flow velocities are achieved by allowing

equal flow rate to each tube. The other is a length-based

type to construct the tubes with equal diameter but dif-

ferent lengths in order for the stream tubes to have nor-

mal or lognormal distributions of travel time when an

equal amount of discharge is imposed to each tube. In

this study, the latter case is adopted.

Following assumptions should be made for the con-

ceptual stream tube model:

1) The stream tube model is constructed with a finite

number of bundles of one-dimensional stream tubes with

equal diameter but different lengths.

2) Flow velocity within a single tube is steady and in-

variant.

3) Tracer solution is injected uniformly at the inlet

boundary of each tube.

4) Tracer transport in each tube obeys the convective

flow.

5) Molecular diffusion during pulse input is neglected.

6) Dispersive mixing occurs due to velocity differences

between tubes at the outflow mixing surface by instanta-

neous and complete mixing.

7) Effluents are collected at one common mixing sur-

face regardless the differences in tube lengths.

8) Velocity differences can be realized by normal or

lognormal distributions of travel length or time.

Consider a set of stream tubes which has a normal dis-

tribution of tube length as shown in Figure 1 where each

bundle consisting of the same tube length has a different

tube number. Let the length of ith tube bundle be Li, the

number of ith tube bundle Ni. Then the total number of

max

1

i

i

i

NTT N

. Let the diameter and cross- tubes is

sectional area of each tube be d and A, and injection flow

rate at the inlet boundary Q, injection concentration Co,

pulse duration To. Then the flow velocity and flow rate in

each tube is

vQNTTA

,

qQNTT

LvT and pulse

length 0p

t

. If a STM consists of five different

lengths (30, 40, 50, 60, 70 cm) and sampling time inter-

val Lvt

, then the sampling length interval

and number of pulse length p. Let the

number of sampling J and then sampling time will be

NPL LL

12

J

TJ t

L. For instance, the number of times of

which can have the effluent concentration for ith

bundle ii

TLLL, and the number of times of

required for breakthrough of ith tube ii

TT .

Then the number of tubes with tracer effluent at sampling

time J would be the following:

T NPL

where

iii

iI

NTENIi TJTT

(1)

Therefore the effluent concentration at J will be

00

J

J

qtNTE NTE

CC C

Qt NTT

L

(2)

In order to investigate the effect of travel time distri-

bution on the convective-dispersive process, we gener-

ated three different input probability density functions of

travel times or tube lengths (60, 80, 100, 120, 140 cm) as

was shown in Figure 2 and used them for simulation of

BTCs at the mixing surface based on pure convective

flow within stream tubes. Sensitivity analysis was also

performed to see the effect of sampling time (

= 10,

20 cm) and pulse length (NPL = 1 to 6) on the BTCs.

2.2. Time Moment Analysis

In order to investigate the effect of sampling time, pulse

duration and input distribution of travel length on the

dispersive process along stream tubes, time moment

analysis was performed. The simulated concentrations in

time were first converted to probability density function

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