Fundamentals of TRANSPORT MECHANISMs CHE 149. Transport Phenomena Lecture 4. Fundamentals of TRANSPORT MECHANISMs MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB

Objectives of this chapter: 4.0 Fundamentals of Transfer Mechanisms Objectives of this chapter: To evaluate velocities, gradients and diffusivities in simple functions for design purposes; To describe overall conditions at some position along a flow duct using integrated equations; and To define transfer coefficients . MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 2

Mechanism ratio analysis CHE 149. Transport Phenomena Lecture 4.1. Mechanism ratio analysis MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB

Mechanism Ratio Analysis 4.1 Mechanism Ratio Analysis Mechanism Ratio Analysis The mechanism ratio analysis simply requires an awareness of the various mechanisms involved with the process. Valuable information may be obtained from the quotient of differential equations derived from transport expressions. e.g. Navier – Stokes equation of viscous flow MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4

Mechanism Ratio Analysis 4.1 Mechanism Ratio Analysis Mechanism Ratio Analysis Summarizing the mechanisms found in viscous flow ITEM MECHANISM VARIABLE A INERTIA FORCES u2/L B PRESSURE FORCES P/ρL C VISCOUS FORCES νu/L2 MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 5

Mechanism Ratio Analysis 4.1 Mechanism Ratio Analysis Mechanism Ratio Analysis If each item is divided by the inertia term, MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 6

The Stress Membrane Model 4.1 Mechanism Ratio Analysis The Stress Membrane Model The stress membrane model explains the velocity distribution at different radial positions in a flow duct. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 7

The Stress Membrane Model 4.1 Mechanism Ratio Analysis The Stress Membrane Model Laminar Flow -low mean fluid velocity -the fluid moves in the flow direction with no component of velocity in any other direction -stress results in maintaining the flow linear and linear flow produces stress The tube consists of infinite number of concentric tubular stress membrane that tends to confine the flow within the annuli formed between the membranes. The strength of the membranes (or stress in the fluid) increases linearly from zero at the center to a maximum at the wall. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 8

The Stress Membrane Model 4.1 Mechanism Ratio Analysis The Stress Membrane Model 2. Turbulent Flow -high mean fluid velocity -a portion of a fluid may be subject to a force that tends to give a component of velocity other than y-direction The stress membranes would try to resist the force of the fluid. If the kinetic energy of the fluid is high enough to penetrate the membrane, a true eddy will be formed. Since the center of the tube has zero stress, it is the portion where eddy formation would likely occur. Eddy penetration will continue until a region of high stress is encountered, which would overcome the kinetic energy of the eddy. The region of high stress is in the vicinity of the wall since the stress membrane is supported by the wall. The support by the wall extends into the fluid to form the laminar sublayer in which no eddy activity occurs. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 9

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation The stress membrane model can be used to establish a relationship between the Reynolds umber and the stress phenomena occurring in the fluid. For ease of analysis, it is desirable to use mean velocities and gross system dimensions. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 10

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation Consider fluid flow through a circular duct: In laminar flow, average velocity and point velocity is related by the equation; Differentiating the equation with r to get dv/dr MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 11

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation From Newton’s Law of Momentum transport; At the duct wall τy= τy1 and r=r1 Thus, the stress at the wall can now be computed easily using measurable values v, D andμ MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 12

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation Since, the wall stress is also related to pressure drop per unit length of the duct, by force balance on a fluid element we can get; Rate of transfer of momentum to the wall= Rate of momentum transfer from the fluid Mathematically, MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 13

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation For a circular duct A1=πDL and S1=πD2/4. Thus, Rearranging, MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 14

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation In a system in which turbulent flow is also present, the turbulent mechanism must be included in the analysis. From the stress membrane model, the gross mathematical description of the turbulent mechanism is where α is 0.5 for laminar 0.9-1.0 for turbulent 1.0 for plug flow conditions. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 15

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation Dividing the equation for laminar flow by the turbulent flow equation . Each term is a dimensionless ratio of two mechanisms MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 16

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation Each term is a dimensionless ratio of two mechanisms: The two terms on the left represent the ratio which defines the friction factor (f) f = total momentum transfer momentum transfer by turbulence MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 17

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation Each term is a dimensionless ratio of two mechanisms: The right term represents the ratio momentum transfer by molecular transport momentum transfer by turbulence MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 18

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation Since Reynolds number friction factor for laminar flow is MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 19

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation The Reynolds number can also be written in a very useful form by recognizing that the mass flow of fluid through a duct is given by where w=mass flow, v=mean velocity, ρ=fluid density and S=cross-sectional area MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 20

Reynolds Number-Friction Correlation 4.1 Mechanism Ratio Analysis Reynolds Number-Friction Correlation At steady state for flow through a duct of constant cross-section; Using this symbol, the Nre is frequently written as; MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 21

Friction Factor Flow Equations 4.1 Mechanism Ratio Analysis Friction Factor Flow Equations Moody diagram (Appendix C-3, FOUST) is a logarithmic plot of friction factor as a function of Reynolds number over a range of Nre (100 -10,000,000) for flow in smooth tubes. The system used was operating in well-developed flow. - Well-developed flow means that the velocity pattern in the tube is the same at all points along the test length. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 22

Friction Factor Flow Equations 4.1 Mechanism Ratio Analysis Friction Factor Flow Equations Three Regions: Laminar region -ends at NRe=2100 -no incipient eddies have sufficient energy to break through the stress membrane MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 23

Friction Factor Flow Equations 4.1 Mechanism Ratio Analysis Friction Factor Flow Equations Three Regions: 2. Transition region -2100<NRe<10,000 -relatively unstable and laminar flow, turbulent flow and combination of both occurs -fluid behavior in this region is a function of fluid properties, system geometry, system kinematics and system history MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 24

Friction Factor Flow Equations 4.1 Mechanism Ratio Analysis Friction Factor Flow Equations Three Regions: 3. Turbulent region -above NRe=10,000 -eddy activity is violent enough to break the stress membrane -momentum is transferred by eddy activity as well as molecular transport MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 25

Friction Factor Flow Equations 4.1 Mechanism Ratio Analysis Friction Factor Flow Equations The friction factor that appears in the Moody diagram may be calculated using different equations (mostly empirical). For Laminar flow region: For turbulent flow region with Nre between 5,000 and 200,000 MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 26

Friction Factor Flow Equations 4.1 Mechanism Ratio Analysis Friction Factor Flow Equations For Nre between 3,000 and 3,000,000: Equation form that allows the velocity term to appear in the left term only: MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 27

Friction Factor Flow Equations 4.1 Mechanism Ratio Analysis Friction Factor Flow Equations Shapes other than Cylindrical - For duct shapes other than cylindrical, the diameter must be replaced by an appropriately chosen variable or group of variable that describes the system with a single linear dimension that is equivalent in behavior to D. -The equation for Deq may be used as a replacement for D assuming that the mean stress approximates the actual wall stress. Example2.Calculate the velocity of an oil flowing through a 3 in tube. The pressure drop through the tube is 548lb/ft2 per 100ft of tube. Oil properties are μ=5cP and ρ=60lb/ft3. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 28

Friction Factor Flow Equations 4.1 Mechanism Ratio Analysis Friction Factor Flow Equations Example 1. Calculate the pressure drop through 100ft of smooth tubing for an oil flowing at a mean velocity of 8ft/s. The tubing diameter in 3 in, μ=5cP and ρ=60lb/ft3. Example2.Calculate the velocity of an oil flowing through a 3 in tube. The pressure drop through the tube is 548lb/ft2 per 100ft of tube. Oil properties are μ=5cP and ρ=60lb/ft3. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 29

Friction Factor Flow Equations 4.1 Mechanism Ratio Analysis Friction Factor Flow Equations Example 3. Calculate the equivalent diameter for a rectangular duct 3ft high and 5ft wide. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 30