Pressure Drop Model and Experimental Variability

Plate and Frame Filter Press

Erin DurkeeYen Nguyen

Dalton Russell

CHE 4002-401 Chemical Engineering Laboratory I: Project III

Oklahoma State UniversityCoach Clint Aichele

Coach Mike ResetaritsCoach Russ Rhinehart

Flow behaves like a transitional flow.  Pressure drop model for the combined set of both laminar and 

turbulent portions

Where: 

a = 0.0484 

b = 10.01 

c = 0.0009 

d = 5.47  

p = 3.41 

CONCLUSION

∆ 𝑃𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑=(𝑎+𝑏𝑁 )∗𝑄+(𝑐+

𝑑𝑁 𝑝 )∗𝑄𝑝

Apparatus Overall Diagram

EXPERIMENTAL EQUIPMENT

      

Suspension Tank

Hand Wheel with Spindle

Overall System

Pumpoutlet Thermometer Pumpoutlet Pressure Gauge

EXPERIMENTAL EQUIPMENT

      

Bypass Stop Valve Pipe Diaphragm Valve Filterinlet Pressure Gauge

Filteroutlet Flow MeterFilter Plate Arrangement

EHS & LP Environmental – water is the only component used

Health – no health hazards

Safety – wear standard lab clothing and safety glasses       – slipping hazard due to water

Loss Prevention – use minimal amount of resources necessary

      

THEORY Pressure drops through two different zones in the filter press:• the in-pipe zone :  

• laminar• turbulent • transitional 

• porous medium of the filter paper: • laminar • turbulent• transitional ∆ 𝑃=∆ 𝑃𝑝𝑖𝑝𝑒+∆ 𝑃𝑝𝑜𝑟𝑜𝑢𝑠

(A sketch to illustrate the two different zones.)

 

THEORY Darcy-Weisbach equation:

Where: • f: Darcy friction factor• L and D: length and inside diameter of the pipeline• ρ: density of water• Q: flow rate of water through the unit

THEORY: In-pipe zone: 

Porous medium zone:

Where:• c, d are turbulent flow unknown constants• N: the number of filter papers used, or the number of 

split flows during the filtering process 

 

Theory : Laminar Flow Data from the Moody diagram (Munson):

Plug back into the Darcy-Weisbach equation:

Poiseuille’s law:

THEORY: In-pipe zone:

Porous medium zone:

Where: • a, b are laminar flow unknown constants

THEORY Laminar only:

Turbulent only:

Combined Flow:

∆ 𝑃 𝑙𝑎𝑚𝑖𝑛𝑎𝑟=(𝑎+ 𝑏𝑁 )∗𝑄

∆ 𝑃𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑=(𝑎+𝑏𝑁 )∗𝑄+(𝑐+

𝑑𝑁 𝑝 )∗𝑄𝑝

∆ 𝑃 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡=(𝑐+𝑑𝑁2 )∗𝑄2

DATA PROCESSING Flow rate: 

Where:• V1, V2 : initial and final readings of volume in gallons

• t1, t2 : initial and final readings of time in minutes

• 6.309*10-5 : unit conversion for the flow rate (Q) from gal/min to m3/s

𝑄=( 6.309∗10−5 )∗𝑉 2−𝑉 1

𝑡2−𝑡 1

DATA PROCESSING Data model: 

Where:• : specific weight of water • (z3-z2) : change in elevation

• KL=7: minor loss coefficient for the water meter equipment 

• Re ≤ 2100:   ;  Re ≥ 4000:  

𝑃2=𝛾 (𝑧 3− 𝑧2+[ 𝑓 𝐿𝐷 +𝐾 𝐿 ] 8𝑄2

𝑔𝜋 2𝐷4 )Sketch to illustrate the points (2) and (3)

EXPERIMENTAL PLANDAY 1 – PRELIMINARY TRIALS

N plates

6 psig

Record Flow Rate in 1 min

Calculate P2

Repeat with new filter papers 

*N = 8, 12, and 16

Number of Plates:

Inlet Pressure (P1): 9 psig 12 psig

EXPERIMENTAL PLANDAY 2 – VARIABILITY AND MODEL TESTING

14 plates

Proper Orientation

Record Flow Rate in 1 min

Calculate P2

Number of Plates:

Inlet Pressure Range (P1): 3 – 12 psig

Random Orientation

EXPECTATION The value of the outlet pressure P2 should be less than 0 psig. 

• a, b, c, d, and p are not negative • Laminar flow coefficients: if a ≠ 0, then c = 0 • Turbulent flow coefficients: if b ≠ 0, then d = 0

The graph of pressure drop ∆P vs. flow rate Q should be:• increasing linear line if laminar flow only• positive increasing quadratic curve if turbulent only

POLYMATH - NONLINEAR REGRESSION

POLYMATH REPORT

Laminar  Turbulent Combined 

RESULTS The combined 

flow model best described the data:

Where: 

a = 0.0484 c = 

0.0009 

b  =  10.01d 

= 5.47  

p = 3.41 ∆ 𝑃𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑=(𝑎+

𝑏𝑁 )∗𝑄+(𝑐+

𝑑𝑁 𝑝 )∗𝑄𝑝

3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.64.505.005.506.006.507.007.508.008.509.009.50

10.0010.5011.0011.5012.0012.5013.00

Pressure drop vs. Flow rate when N=8

Data Model

Laminar Model

Turbulent Model

Combined Model

Flow Rate Q (gal/min)

P1-P2 (psig)

VARIABILITY OF MEDIUM

Proper Orientation  Random Orientation

4.5 5 5.5 6 6.5 7 7.5 8 8.5 93.003.504.004.505.005.506.006.507.007.508.008.509.009.50

10.0010.5011.00

Pressure drop vs. Flow rate when N=14

Process Model

Data Model

Flow rate Q (gal/min)

P1-P

2 (p

sig)

4.2 4.7 5.2 5.7 6.2 6.7 7.2 7.7 8.2 8.73.003.504.004.505.005.506.006.507.007.508.008.509.009.50

10.0010.5011.00

Pressure drop vs. Flow rate when N=14

Data Model

Process Model

Flow rate Q (gal/min)

P1-P

2 (p

sig)

STATISTICAL TEST

Preliminary Trials

Two-tailed t-test:  Passed 

R-lag-1 test: Did not pass

large negative

Proper Orientation

Two-tailed t-test:  Passed 

R-lag-1 test: Did not pass

large positive

Random Orientation

Two-tailed t-test:  Did not pass 

R-lag-1 test: Did not pass

large positive

RESIDUAL PLOTS

Preliminary Trials Proper Orientation

1 2 3 4 5 6 7 8 9 10

-0.50

-0.30

-0.10

0.10

0.30

0.50

0.70

0.90

Plot of Residuals when N=14

Trials

Res

idua

ls

PROPAGATION OF UNCERTAINTY Data model:   

ε(p1) (psig) ε(z3) (m) ε(z2) (m) ε(L) (m) ε(D) (m) ε(V2) (m3) ε(V1) (m3) ε(t2) (min) ε(t1) (min)

4.00 0.0010 0.0010 0.0010 0.0010 0.0004 0.0004 0.0083 0.0083

1 -1.42E+00 1.42E+00 -5.63E-02 7.65E+01 -1.79E+05 1.79E+05 2.72E+03 -2.72E+03

1 -1.42E+00 1.42E+00 -5.88E-02 8.03E+01 -1.84E+05 1.84E+05 2.85E+03 -2.85E+03

1 -1.42E+00 1.42E+00 -6.13E-02 8.41E+01 -1.88E+05 1.88E+05 2.99E+03 -2.99E+03

1 -1.42E+00 1.42E+00 -8.61E-02 1.23E+02 -2.26E+05 2.26E+05 4.36E+03 -4.36E+03

1 -1.42E+00 1.42E+00 -9.21E-02 1.32E+02 -2.34E+05 2.34E+05 4.70E+03 -4.70E+03

εΔP, 0.95

81.02

83.14

85.27

105.09

109.65

PROPAGATION OF UNCERTAINTY

Result • Average 95% probable errors:  ε∆P , 0.95  = 136.89• Two sigma limit : 2σ = 0.88

DISCUSSION

Model works well in the range of:• N=8 up to 14 plates• Low to medium-high flow rate

 At a very high flow rate (when control valve shows PI1 ≈ 2 bar):• Accurate inlet pressure readings P1 were hard to obtain

• Residuals between data model and processing model increase• Data started showing outliers, model does not fit data well

REFERENCES1. Munson, Bruce R., Ted H. Okiishi, Wade W. Huebsch, and Alric P. 

Rothmayer. Fundamentals of Fluid Mechanics, 7th edition. Jefferson City: John Wiley & Sons, Inc, 2013. 400-410, 416-431. Print.

2. Rhinehart, R. R. (2016). Lessons on Data Analysis and Model and Procedure Validation.    Oklahoma State University. 

3. Venugopal, Vidhya. Standard Operating Procedures: Experiments in Plate and Frame Filter Press. Oklahoma State University. 

 Pressure Drop Model and Experimental

VariabilityPlate and Frame Filter Press

Erin Durkee

Yen Nguyen

Dalton Russell

Conclusion:  

QUESTIONS?

∆ 𝑃𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑=(0.0484  +10.01𝑁 )∗𝑄+(0.0009+

5.47𝑁3.41 )∗𝑄3.41

Units:• ∆Pcombined (psig)• Q (gal/min)