Lab: Pressure-Flow Relationships

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Lab 3 of 2LB
Lab 4 of 40B
Pressure-Flow Relationships
Courtesy of Yeuh-feng Chiang

Bernoulli’s equation

P + ½ r v² + r gz = constant

h = 0.00890 poise (dyne-sec/cm²) at 25^oC

The viscosity of a fluid is very sensitive to temperature. For water, the viscosity changes about 2.5% per ^oC. (it gets less viscous at higher temperature)

Poiseuille flow.

The discharge rate of fluid from the end of the tube is

Q = D V/D t = (p r²)v

The discharge rate through the capillary tube is related to the pressure difference at the two ends by

Q = (p r⁴/8h L)(P₁-P₂)	(Poiseuille Law)
P = P₀ + r gz	(P₀= atmospheric pressure)
Q = (p r⁴/8h L)( P₀ + r gz - P₀ ) = (p r⁴r g/8h L)z
Q = D V/D t = -A(dz/dt)	(minus sign is due to decreasing z)
dz/z = -(p r⁴r g /8h LA)dt
z = z₀e^{-(p r⁴r g /8h LA)t}

Part 1. Calibration

Use the MPLI software from the main menu.

Select O – Other Options followed by I – Select Inputs

Use the arrow keys and the space-bar to turn off inputs B and C.

Select Z – Calibrate followed by Z – Calibrate Input.

Let the computer guide you through.

Make sure the water level is just at the top of the capillary tube exit to define the zero of pressure. Tip the column to empty water out the overflow tube, or add water from the faucet as needed. Try to get within about 0.1 cm of the capillary top. Record the water surface height on the scale attached to the water column.
After setting the zero scale value. Fill the water column up to the overflow drain. Again, record the height on the scale, and give the second calibration. The pressure of the full column is r gDz, calculate it!
Here you may assume that gravitational constant, g= 9.80´10²(cm/sec²) , and density of water r = 1.00 (g/cm³).

	Unit	COLD WATER
The initial height, z₁
Pressure at the initial height, P₁		0
The final height, z₂
Dz = \|z₂ - z₁\|
density of water, r
Pressure at the final height, P₂ = rgDz

Calibrate Input

COLD WATER

Calibrated by:

Date calibrated:

Input Label:

Input unit:

Reading #1 in Volts:

Entry #1:

Reading #2 in Volts:

Entry #2:

Calculate slope:

Part 2. Pressure versus time data

Select M – Monitor Input from the main menu.

DO NOT FORGET TO ENABLE STORING DATA IN MEMORY

Set the time length as 2.5 minutes (use 02:30 since is MM:SS).

Select 150 readings (one per second).

Turn the valve full on, establishing flow, then hit Enter.

DO NOT TOUCH ANYTHING! Wait for the time it takes to record data.
Plot the data graph on screen and see you have the right data or not.

SAVE THE DATA to disk as instructed.

Part 3. Data Analysis – Room Temp Flow

Exit MPLI and start Graphical Analysis.

First READ IN THE DATA and plot "ln P versus t".

The data was stored in the MPLI directory, so you need to

Select F – File Options in the main menu
Select C – Change Directory and change the directory to

C:\APPLCTNS\MPLI

Select L – Load Data From Disk

And use the arrow keys to move to the file you want.

Use Graphical Analysis to plot the modified data "ln P versus t".
This should be a straight line.
Print out one and only one graph of "ln P versus t", and indicate the water temperature.
Use this slope to estimate the viscosity.

We should expect

ln P(t) - ln P(0) = (p r⁴r g/8h LA) t

Y = MX + B

So fitting a slope we get

M = (p r⁴r g)/(8h LA)	h = - [p r⁴r g/8(slope)LA]
Using A = p d²/4 for the column	h = - [r⁴r g/2(slope)Ld²]

STOP HERE TO MEASURE the capillary length L and the inner diameter of the column . The diameter of the capillary itself is hard to measure, but you can use the value 3mm (r = 0.15 cm).

Calculate the value of h

Capillary length, L = __________ cm

Inner diameter of the column, d = ___________ cm

Inner radius of the capillary, r = 0.15 cm

	Temperature, T, (^oC)	Slope of "ln P(t) vs time"	h _measured, unit: ( )	h _{CRC data table}, unit: ( )
Ice water
Cold water
Hot water

Question 1: Does water become more or less viscous at high temperature?

Requirement:

(0.5) Print out one Graph "h _measured vs Temperature."

(0.5) Use the above data table, describe the trend of your measured viscosity with temperature.

Question 2: What is the fractional change of viscosity per ^oC?

Requirement: Use the above data table, and this formula

(1.5) Calculate three percentage values of your measured [(Dh/h_i)/DT]

from ice water to cold water

from cold water to hot water

from ice water to hot water

Additional Question: Please write down a simple conclusion of the whole Lab 4.

Requirement:

(0.5) Write it in no more than 4 sentences.

(0.5) Use some of your data tables and answered questions to make some quantitative argument.

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| Lab 3 of 40B|Main Menu Page| Lab 5 of 40B|

COLD WATER
Calibrated by:
Date calibrated:
Input Label:
Input unit:

Reading #1 in Volts:
Entry #1:
Reading #2 in Volts:
Entry #2:
Calculate slope: