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

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis The Boundary Layer - This section examines the behavior of fluids prior to the full development of turbulence (well-developed flow) which is assumed in the previous calculations - Examples of this area of study are entry sections of long tubes and large diameter short ducts wherein the velocity profile is not the same at different positions in the flow duct. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 2

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis The Boundary Layer Assume an infinite duct so that volume of the plate has no effect upon the fluid velocity in the duct. At the instant a differential element of moving fluid contacts the leading edge of the flat plate, the velocity of that fluid immediately decreases to zero (τy= ∞). Successive layers from the plate will be retarded as y increases and the stress distributes itself over a layer of some thickness forming the boundary layer. τy= ∞ due to instantaneous deceleration that requires an infinite force. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 3

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis The Boundary Layer -the region in which the fluid velocity is less than vfs (free-stream velocity) -it extends from the leading edge (y=0) to end of plate (y=yt) and from the surface of the plate (x=0) to the boundary layer limit (x=δ). Figure 13.2 Boundary –layer buildup on a flat plate (vertical scale magnified) MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis The Boundary Layer Figure Description: -at y=0,δ=0 and δincreases to a maximum at y=yt -the point stress decreases from the leading edge to the end of the plate or as y increases -NEAR the LEADING EDGE:  δ is small so distance between v=0 and v=vfs is small, thus, stress is maximum regions of high fluid stress inhibit formation of eddies so laminar flow is expected -MOVING in the Y-direction  δ increases (large dx) and vfs is constant, thus, stress is relatively decreased. since stress is decreased, the KE of eddies can now overcome the fluid stress so the bulk of the boundary layer can exist in the turbulent regime (LS, BR, and TC) MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 5

The Boundary Layer Calculations 4.1 Mechanism Ratio Analysis The Boundary Layer Calculations Reynolds number based on x-positions The boundary-layer thickness (δ) is difficult to measure. Since, it is a function of y, for convenience Nre may be calculated using the equation: *Both forms of Reynolds number above have specific uses. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 6

The Boundary Layer Calculations 4.1 Mechanism Ratio Analysis The Boundary Layer Calculations In fully developed flow in cylindrical ducts, the diameter and shear stress are constant with the length of the tube. However, for boundary layer calculations the boundary layer thickness and the shear stress varies with y thus requiring greater number of equations to describe the flow. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 7

Laminar Boundary Layer Equations 4.1 Mechanism Ratio Analysis Laminar Boundary Layer Equations The stress at any point y in terms of (Nre)δ at same point y Boundary layer thickness as function of Nre Mean stress on surface between y=0 to y=y as fxn of (Nre)y *On long plates, as y increases (Nre)y increases. At positions y corresponding to (Nre)y=105 to 106 transition occurs & turbulent boundary layer is formed MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 8

Turbulent Boundary Layer Equations 4.1 Mechanism Ratio Analysis Turbulent Boundary Layer Equations The stress at any point y in terms of (Nre)δ at same point y Boundary layer thickness as function of Nre Mean stress on surface between y=0 to y=y as fxn of (Nre)y MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 9

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Nomenclature: δ-boundary layer thickness (Lx) y-position measured from the leading edge of a plate (y=0) in the direction of flow (Ly) vfs-free-stream velocity (Ly/θ) -Reynolds number based upon the position y, dimensionless -Reynolds number based upon the boundary layer thickness (δ ), dimensionless -stress on the plate at point y downstream from the leading edge -mean value of the stress between y=0 and y=y - friction factors for the boundary layer MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 10

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Boundary Layer Equations (3) for both laminar and turbulent flow assumes that flow is the same for y=0 ato y=y. In practice, the laminar flow layer exist from the leading edge to some critical point, yc, where Nre is between 105 to 106. Thus, the two equations must be used over the range for which they apply. For a plate of total length yt and width, z The mean stress for the laminar portion of the boundary layer that lies between y=0 and y=yc can be calculated by MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 11

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Boundary Layer The force on one side of the plate beween y=0 and y=yc is equal to the stress area product. The equation for plate in turbulent flow is written for y=0 and y=yt. Therefore, the force between yc and yt must be computed and added to the force due to laminar flow between y=o to y=yc. The mean stress between y=0 and y=yt is calculated using: MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 12

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Boundary Layer The force on one side of the plate beween y=0 and y=yt if the entire plate were in turbulent flow is The mean stress for the region between y=0 and y= yc for the turbulent flow can be calculated as MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 13

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Boundary Layer The force applied to one side of the plate in turbulent flow beween y=0 and y=yc is Substituting and rearranging, MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 14

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Boundary Layer Example A flat plate 1.5 m long has an edge facing the flow direction. Air is flowing over the plate at 12 m/s. The air properties are ρ=1.2 kg/m2 andμ=1.8x10-5 N-s/m2. a. Calculate the boundary layer thickness 1.5 m from the leading edge. b. Calculate the point stress 1.5 m from the leading edge. c. Calculate the average stress over the plate from the leading edge to a point 1.5m from the leading edge. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 15

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis The Entry Length The entry length of the tube is an example of the build-up of a boundary layer. Consider a pipe suspended in a region of flow of uniform velocity; Figure 13.3. Boundary-layer buildup in the entry length. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 16

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis The Entry Length Description -at the immediate entrance of the pipe, a boundary layer is set-up at the inside of the tube surface -some length of the tube (Le) is required as the boundary layer grows in thickness until all boundary layers meet at the center of the pipe -thus, any flow within the tube is essentially all boundary layer flow MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 17

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis The Entry Length Description In the ENTRY LENGTH: boundary layer and free stream exists within the tube Le ends where boundary layers meet at the center after which fully developed flow occurs If boundary layer is laminar throughout the entire entry length the fully developed flow is laminar, otherwise, the remainder of the tube will operate in turbulent flow Pressure drop through Le is greater than after fully developed flow due to: -shear stress near the leading edge is greater than local wall stress downstream -fluid in the boundary layer moves slowly than fluid in free stream MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 18

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis The Entry Length Calculations: In the laminar regime, the entry length can be predicted from the following equation: The entry length for turbulently flowing fluid can be predicted from the equation: MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 19

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis The Entry Length Calculations: Calculate the entry length for water at 20oF flowing through a ½ in tube at 0.1ft/s. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 20

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Form Drag Flow past immersed objects fluid changes path to pass around a solid body affecting to deceleration *Skin friction/ Skin drag -tangential stress on a smooth surface that is oriented parallel to the flow direction causing momentum transfer -always present between fluid and surface - significant friction losses occur because of acceleration and deceleration of fluid through immersed objects (solid set in the flow path) MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 21

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Form Drag vs Skin Drag MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 22

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Form Drag at the center of the body, vfs=0stagnation point Note: The summation of all forces on the body due to acceleration and deceleration constitutes the form drag, FD. MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 23

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Form Drag MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 24

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Form Drag MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 25

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Form Drag MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 26

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Form Drag *Drag Coefficient: Correlations are usually presented graphically on logarithmic plots, similar to moody diagram, of CD vs Nre Figure 13.5, Table 13.2 FOUST For Nre=0.1  CD=24/Nre MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 27

MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 4.1 Mechanism Ratio Analysis Form Drag Sample Problem A chimney 100ft high and 5 ft in diameter is subjected to a maximum wind of 100 miles/hr. Calculate the force exerted on the chimney by the wind: μ=0.018cP ρ=0.075lb/ft3 MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB MVOLAURIO, DEOSANTIAGO. ChE Department, CEAT - UPLB 28