VISCOUS FLOW IN CONDUITS  When we consider viscosity in conduit flows, we must be able to quantify the losses in the flow Fluid Mechanics [ physical interpretation: what are we doing today? ]  The magnitude of these losses will vary significantly depending on many factors, including whether the flow is laminar or turbulent  For most practical purposes, the Reynolds number is such that conduit flows that serve us in everyday life are turbulent  our knowledge of how to quantify losses in conduit flows allows us to optimize performance and efficiency in contained flows from water and oil pipelines, to chemical networks, air supplies, and the conduit network in the human body  Who Cares!?  The complexity of turbulent flows typically necessitate the use of extensive experimental data and empirical formulae

VISCOUS FLOW IN CONDUITS  The pressure drop and head loss in a pipe are dependent on the wall shear stress,  w, between the fluid and pipe surface [ introduction to the moody diagram ]  In turbulent flow the shear stress is a function of fluid density, this is not the case in laminar flow, where it is only dependent on the viscosity,   We can consider the pressure drop,  p, in a steady, incompressible turbulent flow, in a horizontal pipe of diameter, D to be expressed as  here pressure drop is a function of fluid velocity, V, pipe diameter, D, pipe length, l, pipe surface roughness height, , fluid viscosity , and fluid density,  - (1)  Though pipe roughness, , is not a factor for laminar flow, it is included for the accommodation of turbulent flow Fluid Mechanics

VISCOUS FLOW IN CONDUITS  The figure represents the flow in the viscous sub-layer for rough and smooth walls [ introduction to the moody diagram ] Fluid Mechanics

VISCOUS FLOW IN CONDUITS  The factors laid out in (1), are in fact a complete list of influencing parameters on pressure drop, that is to say that other factors such as surface tension, and vapor pressure, etc. do not affect the pressure for the conditions we have assumed, i.e., steady, incompressible, horizontal, and round pipe [ introduction to the moody diagram ] - (2)  We discover that this function representation can be simplified by assuming that pressure drop is proportional to pipe length, this we arrive at through our knowledge of many experiments (not by dimensional analysis), so we factor l /D out  Recalling our dimensional analysis, the number of variables in this problem, k=7, and the number of basic dimensions, m=3, we therefore expect to see 4 dimensionless groups i.e., Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ introduction to the moody diagram ] - (4)  Where f is now a function of two dimensionless terms, the Reynolds number, Re, and the relative roughness,  /D  We can rearrange (3) to construct a term  pD/( l  V 2 /2) that we will refer to as the friction factor, f, i.e., we write (3) now as - (3) - (5) Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ introduction to the moody diagram ]  For laminar, fully developed flow the value of f is only dependent on the Reynolds number, i.e., f=64/Re (no dependence on  /D)  For turbulent flow, there is not as yet an analytical solution for the friction factor, f; rather, results for f are summarized from experiments on the Moody Diagram (or an equivalent curve fitting formula) Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ introduction to the moody diagram ]  Here is the friction factor, f as a function of Reynolds number and relative roughness (  /D) for round pipes Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ introduction to the moody diagram ]  Following are some typical values for pipe wall roughnesses Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ introduction to the moody diagram ]  Now, we recall our energy equation for steady incompressible flow - (6)  When we talk about flow with a constant diameter, D, (i.e., constant velocity, V 1 =V 2, and horizontal, z 1 =z 2, then  p=p 1 -p 2 =  h L ), we can re-write (6), (combined with (4)) to get - (4) - (7)  (7) is referred to as the Darcy Weisbach equation, valid for any fully developed, steady, incompressible pipe flow (horizontal or otherwise) Fluid Mechanics

VISCOUS FLOW IN CONDUITS  Of course, the implicit dependence on f requires an iterative solution (not an issue with programmable calculators or computers)  We must remember to exercise caution when utilizing either Colebrook’s expression or the Moody diagram, as results can only be taken with assurances of a 10% accuracy [ introduction to the moody diagram ]  In (7), f is had from the Moody diagram, or may be more conveniently calculated through an expression that is valid for some portion of the diagram - (7)  The Colebrook expression is valid for the entire non-laminar range of the Moody diagram - (8) Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ minor losses ]  Expressions like Darcy Weisbach’s (7), are useful in computing headloss over long sections of straight pipe  Typical pipe networks however contain many bends, tees, joints, and valves  The flow will experience losses though such sections (mostly due to losses associated with changes in flow geometry and direction—momentum losses)  Considering the overall head loss in the system, the losses associated with these sections are usually minor, thus their being termed “minor” losses (relative to the more significant friction losses incurred over long stretches of straight pipe) Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ minor losses ]  Theoretical evaluations of the losses through each valve and fitting in a system are as yet not plausible, therefore the head loss data for such components has been determined by experiment  The most common method for determining the head losses or pressure drops across these elements is to specify a loss coefficient, K L, defined as - (9)  where  then we write - (10) - (11) Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ minor losses: entrance flow conditions ]  What typically happens (the essence of vena contracta):  The flow separates from the corner (basically it can’t make the turn)  The max velocity at (2) will be greater than (3) and as a result the pressure drops, if the flow could put the brakes on and convert that saved velocity to pressure, you would have an ideal pressure distribution—and no losses, this is not what happens Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ minor losses: entrance flow conditions ] Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ minor losses: exit flow conditions ]  Here the entire kinetic energy of the exiting fluid, V 1, is dissipated through viscous effects as the incoming stream mixes with ambient water and eventually comes to rest (V 2 =0), thus exit losses from (1) to (2) are typically equal to one velocity head (V 2 /2g), or K L = Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ minor losses: exit flow conditions ]  The sudden expansion loss mechanism can actually be evaluated analytically Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ minor losses: exit flow conditions ]  Here - (12) - (13) - (14)  and, we know- (15)  which can be rearranged as - (16)  of course in this development Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] GIVEN: REQD: Determine the pressure at (1) if a: no losses considered, b: just major losses considered, c: all losses considered Water flows at 60 o F from the basement to the second floor under the following conditions Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] SOLU: 1. Let us write the energy equation for this flow - (E1)  from which we can rearrange for p 1 as - (E2) a: [NO LOSSES CONSIDERED] Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] SOLU: 2. With no losses h L goes to 0, so - (E3)  or - **(ans a:) **NB- (ans a:) we note here how 8.67 psi of the pressure drop is due to change in elevation and 2.07 psi is due to increase in kinetic energy a: [NO LOSSES CONSIDERED] Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] SOLU: - (E1)  and we can compute h L from (11) (D-W) - (E4) b: [ONLY MAJOR LOSSES CONSIDERED] 1. We of course still apply (E1) Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] SOLU: - (E5)  and we can compute hL from (11) (D-W) - (E4) b: [ONLY MAJOR LOSSES CONSIDERED] 2. From given data we assemble  going to the Moody, we write 3. Applying (E1), we simplify to Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] SOLU: - (E5)  which becomes b: [ONLY MAJOR LOSSES CONSIDERED] - **(ans b:) **NB- (ans b:) of this pressure drop, 10.6 psi is due to pipe friction Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] SOLU: - (E6)  which becomes c: [MAJOR and MINOR LOSSES CONSIDERED] 1. Applying (E1), we simplify to Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] SOLU: - (E7) c: [MAJOR and MINOR LOSSES CONSIDERED] 2. Here, the 21.3 psi is due to elevation change, kinetic energy change, and the major losses we have accounted for from a: through b: Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] SOLU: - (E8) c: [MAJOR and MINOR LOSSES CONSIDERED] 3. Now we pick up the loss coefficients for all the minor losses in the system Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] SOLU: c: [MAJOR and MINOR LOSSES CONSIDERED] 4. Thus, summing from a: and b: - **(ans c:) Fluid Mechanics

VISCOUS FLOW IN CONDUITS [ example 1: determination of pressure drop ] c: [MAJOR and MINOR LOSSES CONSIDERED] Let us examine the behaviour of the pressure through the system, note that not all losses are irreversible (like friction and momentum loss), losses due to elevation and velocity changes are reversible Fluid Mechanics