1. RTD and FLOWs at steady and pulsatile flow in systemic circulation
EXPERIMENTAL METHODS 2010 PROJECT RTD
RTD and FLOWs at steady and pulsatile flow in systemic circulation
Motivation
Dispersion of matter in systemic circulation.
Stimulus response method for system identification.
Flow distribution and flowrate evaluated from transit time distributions.
What is the effect of flow pulsation upon flowrate distribution in a branched system
Task prepared within the project FRVS 90/2010

2. EXPERIMENTAL METHODS 2010 PROJECT RTD
Residence times
INLET
OUTLET
Fast particle (short residence time)
Residence time  = time spend by a particle inside a continuous system between inlet and outlet.
RTD Residence time distribution E() = histogram of residence times  of all particles passing through outlet.
Applications of E(): Prediction of chemical reactions yield, transport of farmaceuticals, evaluation of active and stagnant volumes, diagnostics,… There are generally different RTDs for plasma and RTDs of blood particles. Different RTD exists at steady and pulsatile flow.

3. Stimulus response experiment
EXPERIMENTAL METHODS 2010 PROJECT RTD
Stimulus response experiment
INLET
OUTLET
All particles passing through inlet at time interval (tI,tI+dt) are marked “red”
Unit IMPULSE stimulus
E()=impulse response
Experiments: “Red particles” – tracers (dyes, radionuclides, salts,…) are quickly injected to inlet. Detectors monitor concentration of tracer at outlet (impulse response). In our case the tracer is KCl solution and reflective particles injected from syringe, detectors are electric conductivity probes.

4. Response by Convolution
EXPERIMENTAL METHODS 2010 PROJECT RTD
Response by Convolution
Convolution enables to calculate response to an arbitrary stimulus function. It is possible to calculate impulse response E(t) by deconvolution from measured stimulus and response.
cin(t), cout(t) concentration of tracer at inlet and outlet of a continuous system
cin
Convolution of stimulus and RTD function
cout E
Fourier transform of convolution
t [s]
Mean time of response = mean time of stimulus + mean residence time
Variance of response = variance of stimulus + variance of E

5. RTD of pipe at convective regime
EXPERIMENTAL METHODS 2010 PROJECT RTD
RTD of pipe at convective regime
Laminar flow and short residence times: diffusion can be neglected and residence times can be calculated from velocity field. The diffusion free regime is characterized by
where Q-flow rate, L-pipe length, Dm-diffusion coefficient.
Example: C~1000 at aorta, arteries, vena cava.
Residence time distribution E() is the response to infinitely short impulse (delta function)
cin
E()
cout
Analytical solution exists also for response to a pulse of a finite width (violet line). Time of the first appearance is not affected, only mean response time is increased.
 (s)
First appearance time =4s

6. Laboratory model of circulation
EXPERIMENTAL METHODS 2010 PROJECT RTD
Laboratory model of circulation
Simulated part - pressure or conductivity transducers alternatively
Cresto
Platinum wires

7. NODES of simulation model
EXPERIMENTAL METHODS 2010 PROJECT RTD
NODES of simulation model
Y [m] 19
Data file defines the coordinates x,y,z of all 29 nodes 18 16 15 28 17 27 14 22 20 29 7 9 12 25 24 6 21 26 8 5 10 11 13 4 3
Conductivity probes. Q-flowrate, C-response (defined in form of table) 23 2 1
X [m]

8. ELEMENTS of simulation model
EXPERIMENTAL METHODS 2010 PROJECT RTD
ELEMENTS of simulation model
V5(14,16,15,27,d3,d4,d4)
V6(17,18,19,28,d3,d4,d4)
P6(7,17,d3)
V7(20,22,21,29,d3,d4,d4)
P5(9,14,d3)
P7(6,20,d3)
V3(8,9,10,25,d2,d3,d3) 25
V2(5,6,7,24,d2,d3,d3)
P4(8,11,d3)
P3(4,8,d2)
V4(11,12,13,26,d3,d4,d4)
P2(3,5,d2)
Nodal indices
V1(2,3,4,23,d1,d2,d2)
Data files define connectivity of
PIPES (2-node elements)
DIVISION (3-node elements)
P1(1,2,d1)

9. RESPONSEs generated by PIPES
EXPERIMENTAL METHODS 2010 PROJECT RTD
RESPONSEs generated by PIPES
x2,y2,z2 Q2 ,c2(t) d
x1,y1,z1 ,Q1, c1(t)
Convolution realized by FFT, and by inverse transformation.

10. RESPONSEs generated by WYES
EXPERIMENTAL METHODS 2010 PROJECT RTD
RESPONSEs generated by WYES
Given c1(t)
x2,y2,z2 Q2,c2(t)
x3,y3,z3 Q3,c3(t)
Intermediate response d1
x4,y4,z4
x1,y1,z1, Q1,c1(t)
final response
It is alternatively possible to characterize the whole wye as a perfectly mixed vessel. In this case the both responses c3 and c2 will be identical
PROBLEMS:
1. Derive impulse response of perfectly mixed vessel Emixed(t)
2. How to modify the responses in the case that detectors record not the tracer concentration but the flowrate of tracer (e.g. radiotracer – counter records rate of decays in the whole outlet).

11. Simulation in MATLAB MATLAB M-files available at http:
EXPERIMENTAL METHODS 2010 PROJECT RTD
Simulation in MATLAB
MATLAB M-files available at http:
Simulation proceeds according to M-file RTD.m in steps
Reading input files xyzq,seq,cp,cv (txt) defining nodes, flowrate and elements
Definition of time scale, time steps, and stimulus function at entry node 1.
Evaluation of responses sequentially in all elements (sequence is defined in vector seq). Responses are evaluated by FFT in functions
function cout=convpipe(t,n,dt,cin,d,l,q)
tmean=pi*d^2*l/(4*q);
for i=1:n
if t(i)<tmean/2
e(i)=0;
else
e(i)=tmean^2/(2*t(i)^3);
end
fcin=fft(cin);
fe=fft(e);
fcout=fcin.*fe;
cout=ifft(fcout)*dt;

12. Simulation in MATLAB MATLAB M-files available at http:
EXPERIMENTAL METHODS 2010 PROJECT RTD
Simulation in MATLAB
MATLAB M-files available at http:
Input data files:
xyzq.txt x y z q (nodal coordinates and flowrates)
seq.txt e1 e2 … (sequence of evaluated elements, +pipes, -wyes)
cp.txt i j d (pipe -indices of nodes and diameter)
cv.txt i j k l d1 d2 d3 (wye - indices of nodes and diameters)
TASK:
1. Modify input files according to parameters of your experiment
2. Compare calculated and measured concentration responses
3. Evaluate flowrates in branches from the shortest/mean transit times

13. Experiments: UVP velocities
EXPERIMENTAL METHODS 2010 PROJECT RTD
Experiments: UVP velocities
Ultrasound Doppler effect for measurement velocity profiles
Piezotransducer is transmitter as well as receiver of US pressure waves operating at frequency 4 or 8 MHz.
Short pulse of US waves is transmitted (repetition frequency 244Hz and more) and crystal starts listening received frequency reflected from particles in fluid.
Time delay of sampling (flight time) is directly proportional to the distance between transducer and the reflecting particle moving with the same velocity as liquid.
Received frequency differs from the transmitted frequency by Doppler shift Δf, that is proportional to the component of particle velocity in the direction of transducer axis.
PROBLEMS:
What is spatial resolution of velocity, knowing speed of sound in water (1400 m/s) and sampling frequency 8 MHz ?
Calculate flowrate in a circular pipe from recorded velocity profile (given angle )

14. Experiments: flowrates
EXPERIMENTAL METHODS 2010 PROJECT RTD
Experiments: flowrates
Flowrate can be identified also from the stimulus response experiments
Mean residence time
Shortest residence time (time of the first appearance)
Cross-correlation of stimulus and response functions

15. LABORATORY REPORT EXPERIMENTAL METHODS 2010 PROJECT RTD
Front page: Title, authors, date
Content, list of symbols
Introduction, aims of project, references
Description of experimental setup
Experiments with injection of tracer at a steady state regime. Record responses and pressures.
Evaluate flowrate from recorded pressure drop. Check Re and flow regime.
Evaluate flowrate from recorded responses and transit times.
Evaluate flowrate from velocity profiles recorded by UVP monitor.
Simulation (steady state). Modify input files and M-files according to measured flowrates and concentration profile at inlet. Evaluate theoretical responses at nodes with conductivity detectors.
Adjust flowrate pulsation and repeat tracer experiments.
Evaluate flowrate distribution at pulsatile flow and compare relative flowrates with steady state case.
Conclusion (identify interesting results and problems encountered)